Amplitude-Phase Analysis of Cosmic Microwave Background maps

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

arXiv:astro-ph/0007133v2 27 Jul 2000

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

AMPLITUDE-PHASEANALYSISOFCOSMIC

MICROWAVEBACKGROUNDMAPS

P.Naselsky2,D.Novikov1,3andJosephSilk1.1AstronomyDepartment,UniversityofOxford,NAPL,KebleRoad,

OxfordOX13RH,UK2TheoreticalAstrophysicsCenter,JulianeMariesVej30,DK-2100

Copenhagen,Denmark

Astro-SpaceCenterofP.N.LebedevPhysicalInstitute,Profsouznaya84/32,

Moscow,Russia,4RostovStateUniversity,Zorge5,Rostov-on-Don,Russia

Abstract

WeproposeanovelmethodfortheextractionofunresolvedpointsourcesfromCMB

maps.ThismethodisbasedontheanalysisofthephasedistributionoftheFourier

componentsfortheobservedsignalandunlikemostothermethodsofdenoisingdoesnot

requireanysigni cantassumptionsabouttheexpectedCMBsignal.Theaimofourpaper

istoshowhow,usingouralgorithm,thecontributionfrompointsourcescanbeseparated

fromtheresultingsignalonallscales.Webelievethatthistechniqueispotentiallyavery

powerfultoolforextractingthistypeofnoisefromfuturehighresolutionmaps.

Subjectheadings:cosmicmicrowavebackground,cosmology,statistics,observations.3

1Introduction

ObservationsoftheCosmicMicrowaveBackground(CMB)isfundamentalforourunder-standingtheprimordialinhomogeneityoftheUniverse.AfterthesuccessfulCOBEexperi-ment,attentionhasbeenfocusedontheinvestigationofsmallscaleperturbations,thatcanprovideuniqueinformationaboutthemostimportantcosmologicalparameters.OneofthemajorproblemsinthemodernCMBcosmologyistoseparatenoiseofvariousorigins(suchasdustemission,synchrotronradiationandunresolvedpointsources(seee.g.Bandayetal.1996))fromtheoriginalcosmologicalsignal.ManyauthorshavealreadyappliedvariousmethodssuchasWiener ltering(TegmarkandEfstathiou1996,BouchetandGispert1999),maximumentropytechnique(Hobsonetal.1999),radicalcompression(Bondetal.1998),power ltering(Gorskiet.al.1997,Naselskyet.al.1999)andwavelettechniques(e.g.Sanzetal.1999)toextractnoisefromtheCMBdata.

Allofthesetechniqueshavebeentestedforremovingthenoisefromtherealobservational

data.Itisnecessarytonotethat,fordi erentstrategiesandfordi erentexperiments,di erentschemescouldbechosenasmostappropriate.Thechoiceofthealgorithmalsodependsontheparticulartypeofforegroundemissiontobeextracted.

2

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

Theaimofourpaperistoovercometheproblemofdetectingandextractingtheback-

groundofunresolvedpointsourcesfromtheoriginalmap.Themeasuredsignalinthereal

observationaldataisalwayssmoothedwithsome lteringangleθfbecauseofthe nalantenna

beamresolution.Therefore,unresolvedpointsourcescouldmakeasigni cantcontribution

totheresultingsignalonallscales.Thistypeofnoiseshouldberemovedfromtheoriginal

mapbeforeanysubsequentanalysisismade.

Recently(Cayonetal.1999)haveproposedtheuseofisotropicwaveletsforremoving

noiseintheformofpointsources.Theirtechniqueisbasedonthefact,thatthe eldin

thevicinityofthesourceshouldbeintheformoftheantennapro le.Unfortunatelythe

GaussianCMB eldcanalsoformrealpeakswiththesamepro le,sothatalotof’arti cial

sources’couldbefoundusingthistechnique.Besides,theantennapro leisnotnecessarily

isotropic(indeed,asaruleitisveryanisotropic).Therefore,isotropicwaveletsshouldnot

beconsideredasanabsolutecureagainstsuchatypeofnoise.

Inthispaperweconsideranapproach,whichisbasedonthedistributionofphases.

Theideaofusingphasesofrandom eldswasintroducedbyA.Melottetal(1991);Coles

andChiang(2000a,b)fortheLargeScaleStructureformationintheUniverse.Belowwe

developthephase-amplitudeanalysismethodforinvestigationoftheCMBanisotropyand

foreground.Theoutlineofthepaperisasfollows.Insection2webrie yreviewthebasic

de nitions,considerasimulatedone-dimensionalscanoftheCMB rstwithasinglepoint

source,thenwithabackgroundofsuchasources.Insection3wegeneralizeourresults

intotwo-dimensionalmaps.Finally,wesuggestanalgorithmfordenoising.Insection4we

discusstheresultsandpotentialofthemethodforanalyzinghighresolutionmaps.

2Pointsourcesinone-dimensionalscans.

Inthissectionweconsider1DCMBscanswithabackgroundofpointsources.Thisapproach

couldbeveryusefulfordataanalysisofone-dimensionalexperimentswithhighresolution

(suchasRATAN600).Weextendthisdiscussiontotwo-dimensionalexperiments(suchas

thenewgenerationofinterferometerexperiments)insection3.Theinvestigationofpoint

sourcesisespeciallyeasyinonedimension,canbeeasilygeneralizedintotwo-dimensional

mapsandwillhelpustounderstandtheadvantageoftheproposedtechnique.

De nitions

In1Dthedeviationofthetemperaturefromitsmeanvalue T=T T inascanis

describedbythesimpleFourierseries:

T(θ)= kakcos(kθ)+bksin(kθ)(1)

wherekisanintegernumberandθcanbeexpressedintermsofoftherealangleonthesky

(θsky)asfollows:θ=θtot

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

T(θ)= Ts(θ)+ Tn(θ)(2)

wheresandndenotesignalandnoiserespectively.Therefore,theFouriertransformcompo-

nentsak,bkcanbealsoexpressedasasumofFourierdecompositionofthesetwoterms:

nak=ask+ak,

nbk=bsk+bk.(3)

ThestatisticallyisotropicdistributionoftheCMBtemperatureanisotropyissupposed

tobeintheformofarandomGaussian eldwiththepowerspectrumPCMB(k),which

sdeterminedbytheappropriatecosmologicalmodel.Thecoe cientsask,bkdependonthe (θ θ ,θ)andtheactualreal-spectrumoftheCMB,theantenna lteringfunctionFf

izationoftherandomGaussianprocessonthesky.Ingeneral,theyobeytheformulae:

sss askak′ = bkbk′ =δkk′F(k,kf)PCMB(k).Here,F(k,kf)istheFouriertransformofthe l-

teringfunctionandθfistheantennaresolutionangle.kfisawavenumberwhichcorresponds

tothisresolution:kf=1/θf.Inoursimulationsweusetheusualexpressionforak,bk:

ask=αkF

bsk=βkF11CMB(k),2

CMB(k),2(4)

whereαk,βkareindependentGaussiannumberswithzeromeanandunitdispersion.

Inthispaperweconsiderthenoiseintheformofisolatedunresolvedpointsources.

Thismeansthattheaveragedistancebetweensourcesislargerthantheresolutionscaleθf.

Therefore,theshapeofthe’noise’ eldaroundthepointsourcedeterminedbythe ltering

functionF:

Tn(θ)= Nps γjδ(θ θ)F

21j=1(θ θj,θf)(5)

whereγj,θjaretheamplitudeandthepositionofthej-thpointsource,respectively,andNps

isthetotalnumberofpointsourcesintheconsideredscan.Accordingtoequation[5],the

Fouriercomponentsofthenoisecanbedescribedbythefollowingverysimpleandconvenient

formulae:

ank=Nps j=1γjcos(kθj)F1

(6)

2(k,kf).

Forfurtherinvestigationwehavetointroducethephase: http://www.77cn.com.cning

equations[1,3]onecanwrite:

4

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

k=arctan bk

nask+ak (7)

Iftheresulting eldatthescaleskisdominatedbytheGaussianCMBsignal(Sk/Nk>>

s1),then k≈arctg(bsk/ak).Inthiscasethephasesofthek-thharmonicsarerandom

independentuncorrelatedvalues,uniformlydistributedfrom0to2π.Ontheotherhand,

ifthesignalatthesescalesismuchsmallerthanthenoise,thenthedistributionofphases

isdeterminedbythepositionsandamplitudesofpointsourcesonthescan.InFig.1,we

presentthespectrumofCMBinonedimensionPCMB(k)forthestandardCDMmodel

togetherwiththespectrumofpointsources.BothspectraaresmoothedwiththeGaussian2 lteringfunctionF(k,kf)=exp( k

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

locationθ1andamplitudeγ1(see

Fig.2).

Fig.2Upperpanel:simulatedCMB eldon100oscan(1ocorresponds

to≈0.03oonthesky)(dashedline)andthe eldfromasinglepointsource

(solidline).Lowerpanel:thesameastheupperone,butwithbetter

resolution.The eldinthevicinityofthepointsourcebehaveslikean

ordinaryGaussian uctuation.

Thecontributionfromthissourcetotheresulting eldaccordingtoequation[5]isthen:

Tn=kmax k=1γ1cos(k(θ θ1))F1

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

Itsu cestohaveonlytwophases(forexample kand k+1,k>kd)to ndthelocation

ofthesourceθ1:

θ1= k+1 k(10)

InFig.3weshowthebehaviorofthephases k,1<k<kmaxtogetherwiththephasesof

thesource.Forsmallvaluesofk:k<<kdthephasesaredistributeduniformlyandatlarge

kwecande nitelyseetheregularstructurethatisconsistentwithequation

[9].

Fig.3Thephasesofthepointsource(circles)andphasesoftheresulting

signal:CMB+Pointsource(crosses).

InFig.4wealsoshowthepositionsofmaximaforallharmonics.Locationofthemaxima

forthek-thharmoniccanbefoundbytheformulae:

kθmax= k+2π n

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

Fig.4Positionsofmaximaforeachharmonic.Eachpointrepresentsthe

positionsofmaximaforthek-thharmonic.Forsmallktheyaredistributed

uniformly(accordingtotheGaussiandistributionoftheCMB).Thelarge

dotshowsthelocationofthesource.

Theremainingpartoftheproblemisto ndtheamplitude-γ1.Letusde nedthe eld

Tkd(θ)asapartofthe eld T(θ)thatconsistsonlyofthehighharmonics:

T(θ)=kdkmax akcos(kθ)+bksin(kθ)(12)

k=kd

Usingtheformulae[8],wenowcanwritedowntheobviousrelation:

γ1= T(θ1)/kdkmax F(k,kf)(13)

k=kd

Therefore,accordingto[3,6],wehavefoundthecontributionfromthissourcetoall

harmonicsfromk=1tok=kmax.

Backgroundofpointsources

Inthissubsectionwegeneralizeouralgorithmtothecasewherethereareanunknown

numberofpointsourcesintheconsideredscan.Inasituationlikethis,wehaveto ndnot

onlypositionsandamplitudesofeachsourcebutalsothetotalnumberofthem:Nps.

8

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

Webelievethatmanydi erenttechniquesbasedontheresultsoftheprevioussubsection

couldbeproposedtosolvethisproblem.Wesuggestasimpleiterationscheme.Ashasbeen

alreadynoticedabove,wecanconsiderthe eld Tkd,whichconsistsonlyofhighharmonics.

Therefore,onlypointsourcesmakeacontributiontothis eld:

Tkd(θ)

=Nps =k=kd

j=1γjk=kdkmax kmax nankcos(kθ)+bksin(kθ)=F1

2(k)(15)

Accordingto[14,15],onecanwrite:

kd(θ)= T

j=1Nps kmax kd k=1γjcos(kd(θ θj))cos(k(θ θj))F(k,kl)

21(16)

Ifwecanputkl<<kd<<kmax,thenthesecondtermontherighthandsideofequation

[16]issmalland:

Nps

kd(θ)≈ T

j=1 γjkmax kd k=1cos(k(θ θj))F(k,kl)1

2.(18)

Thecontributionfromthissourcetothe eldanditsinterferencewithothersourcesisnow

removed.Thisallowsusto ndmorepreciselythenexthighestmaximum.Therefore,we kdand ndθ2,γ2andsoon(Fig.5).applythesameproceduretothe eld T1

9

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

initial

fieldafter 6

iterations

after 3iterationsafter 9iterations

kdFig.5Theiterationscheme.Eachpanelrepresentstheresiduals Ti

kdkd fromtheinitial eld To= Tafterthei-thiteration.

kd2kd kd)2 )becomessignif-)= ( T((σiWeperformtheseiterationsuntilthedispersionσiikdicantlysmallerthenσo(Fig.6).Thetotalnumberofiterationsthatisneededtosigni cantly

reducetheinitialdispersiongivesusapproximatelythenumberofpointsourcesNpsandeach

iterationgivesthelocationθiandtheamplitudeγiofthei-thsource.Note,that

kd2)(σi= j2γj,γj<γi(19)

androughlyspeaking,inFig.7wecanseethecumulativedistributionofpointsourcesover

thepowerγ.Finally,sincewehavethepositionθiandamplitudeγi,thecontributiontothe

eldfromallpointsourcesmayberemovedinthesamemanner,aswasdoneforasingle

pointsourceintheprevioussubsection.

10

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

1

0.1

0.01

0.0010123456789

Fig.6kdwitheachiteration.Thedecreaseofthedispersionfor

T

Fig.7The nalresult.Theinitial eldofpointsources(solidline),

restored eldbyourmethod(dashedline),andresiduals(dottedline).

11

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

3Pointsourcesintwodimensions.

Inthissectionwebrie ydescribeourresultsintwodimensions.Withoutlossofgenerality

wemayconsiderasmallregionoftheskyandassumethatthegeometryisapproximately

at.Underthisassumption,thepartofthedetectedsignalwhichisdeterminedbythenoise

associatedwithNpspointsourcescanberepresentedaccordingtotheprevioussectionby

writing:

Tn( x)=

Nps j=1γj kF1 k x)+bnsin( ank x)=kcos(k k

dimensionthisdependenceislinear). d)δr)2,(( kmax kwhereδristhedistancefromthepeak(inone

12

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

CMB

noiseCMB + noise

CMB + noise

after filtration

Fig.8Simulatedskymapsof10o×10o.

13

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

after 10 iterations

after 70 iterations

after 30 iterations

after 100 iterations

Fig.9Noisemaps(i.e.removedsources)afterdi erentnumbersofiterations.Thesizeandshadingofeachsourceis

proportionaltoitsamplitude.

14

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

Thisa ectstheneighboringpeaksandcanchangetheiramplitudes.Therefore,this d)δrij)2<<γiapproximationworksif(( kmax k

We propose a novel method for the extraction of unresolved point sources from CMB maps. This method is based on the analysis of the phase distribution of the Fourier components for the observed signal and unlike most other methods of denoising does not req

Bouchet,F.R.1999,MNRAS,306,232.Gorski,K.M.,Proceedingsofthe31-stRecontresdeMarionAstrophysicsMeeting,p.77,1997,astro-ph/9701191.

Guiderdony,B.1999,astro-ph/9903112

Melott,A.,S.ShandarinandR.Scherrer,ApJ.377,79,1991.

Novikov,D.I.,Naselsky,P.D.,Jorgensen,H.E.,Christensen,P.R.,Novikov,I.D.,Norgaarrd-Nielsen,H.U.,astro-ph/0001432

Sanz,J.L.,Barreiro,R.B.,Cayon,L.,Martinez-Gonzalez,E.,Ruiz,G.A.,Diaz,F.J.,Ar-gueso,F.,Silk,J.,andL.To olatti,1999,astro-ph/9909497

Tegmark,M.&Efstathiou,G.1996,MNRAS,281,1297.

16

本文来源：https://www.bwwdw.com/article/p4vi.html