CORRELATIONS FROM GALACTIC FOREGROUNDS IN THE FIRST-YEARWILKINSON MICROWAVE ANISOTROPY PROBE DATA

Pavel D. Naselsky,1Igor D. Novikov,

1,2and Lung-Yih Chiang

1

Received 2005 June 23; accepted 2005 December 29

ABSTRACT

We study a specific correlation in the spherical harmonic multipole domain for cosmic microwave background(CMB) analysis. This group of correlations between�l ¼ 2n, where n ¼ 1; 2; : : : , is caused by a symmetric signalin the Galactic coordinate system. A phase filter targeting such correlation therefore helps remove the localized brightpointlike sources in the Galactic plane and the strong diffuse component down to the CMB level. We illustrate thesignificance of these correlations and apply this estimator on some derived CMB maps with foreground residuals. Inaddition, we show that our proposed estimator significantly damps the phase correlations caused by Galactic fore-grounds. This investigation provides understanding of mode correlations caused by Galactic foregrounds, which isuseful for paving the way for foreground-cleaning methods for the CMB.

Subject headinggs: cosmic microwave background — cosmology: observations — methods: data analysis

1. INTRODUCTION

Separation of the cosmic microwave background (CMB) sig-nal from extragalactic and Galactic foregrounds (GF) is one ofthe most challenging problems for all the CMB experiments,including the ongoing NASAWMAP (Wilkinson Microwave An-isotropy Probe) and the upcoming ESA Planckmission. The GFproduces the major (in amplitude) signal in the raw maps, whichis localized at a rather small latitude band b < 30�. To avoid anycontribution of the GF to the derived CMB map, starting fromtheCOBE (Cosmic BackgroundExplorer) toWMAP experiments,a set of masks and disjoint regions of the map are in use for ex-traction of the CMB anisotropy power spectrum (Bennett et al.2003a, 2003b, 2003c; Hinshaw et al. 2003; Tegmark et al. 2003;Eriksen et al. 2004). The question is, what kind of assumptionabout the properties of the foregrounds should we apply for thedata processing and what criteria determine the shape and area ofthe mask and the model of the foregrounds? To answer thesequestions we need to know the statistical properties of the GF todetermine the strategy of the CMB signal extraction from theobservational data sets.

These questions are even more pressing for the CMB polar-ization. Unlike temperature anisotropies, our knowledge aboutthe polarized foregrounds is still considerably poor. In addition,we have yet to obtain a reasonable trulywhole-skyCMB anisot-ropy map for statistical analysis, while obtaining a whole-skypolarization map seems to be a more ambitious task. Modelingthe properties of the foregrounds thus needs to be done for achiev-ing the main goals of the Planckmission: to find the CMB anisot-ropy and polarization signals for thewhole skywith unprecedentedangular resolution and sensitivity.

Apart from modeling the foregrounds, Tegmark et al. (2003,hereafter TOH03) propose the ‘‘blind’’ method for separation ofthe CMB anisotropy from the foreground signal. Their method(see also Tegmark & Efstathiou 1996) is based on minimizingthe variance of the CMB plus foreground signal with multipole-dependent weighting coefficientsw(l) on theWMAPK–Wbands,

using 12 disjoint regions of the sky. This leads to their foreground-cleaned map (FCM), which seems to be clean from most fore-ground contamination, and the Wiener-filtered map (WFM), inwhich the instrumental noise is reduced by Wiener filtration. Italso provides an opportunity to derive themaps for combined fore-grounds (synchrotron, free-free, dust emissions, etc.). Both FCMsand WFMs show certain levels of non-Gaussianity (Chiang et al.2003; Schwarz et al. 2004), which can be related to the residualsof the GF (Naselsky et al. 2004a). Therefore, we believe that itis imperative to develop and refine the ‘‘blind’’ methods forthe Planckmission, not only for better foreground separation in theanisotropy maps, but also to pave the way for separating CMBpolarization from the foregrounds.

The development of ‘‘blind’’ methods for foreground cleaningcan be performed in twoways. One is to clarify the multipole andfrequency dependency of various foreground components, includ-ing possible spinning dust, for a high multipole range and at thePlanck High-Frequency Instrument (HFI) frequency range. Theother requires additional information about the morphology ofthe angular distribution of the foregrounds, including knowledgeabout their statistical properties in order to construct a realistichigh-resolution model of the observable Planck foregrounds.Since the morphology of the CMB and foregrounds is closelyrelated to the phases (Chiang 2001) of al,m coefficients from thespherical harmonic expansion�T (�; �), this problem can be re-formulated in terms of analysis of phases of the CMB and fore-grounds, including their statistical properties (Chiang & Coles2000; Chiang et al. 2002a, 2004; Coles et al. 2004; Naselsky et al.2003a, 2003b, 2004b).

In Naselsky &Novikov (2005), it is reported that a major partof the GF produces a specific correlation in the spherical har-monic multipole domain at�l ¼ 4, i.e., between modes al,m andalþ4;m. The series of �l ¼ 4 correlations from the GF requiresmore investigation. This paper is thus devoted to further analysisof the statistical properties of the phases of theWMAP foregroundsfor such correlation.We concentrate on the question as to what thereason is for the 4n correlation in the WMAP data, and whethersuch correlation can help us to determine the properties of the fore-grounds, in order to separate them from the CMB anisotropies.

In this paper we develop the idea proposed by Naselsky &Novikov (2005) and demonstrate that the pronounced symmetry

1 Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark;naselsky@nbi.dk, novikov@nbi.dk, chiang@nbi.dk.

2 Astro Space Center of Lebedev Physical Institute, Profsoyuznaya 84/32,Moscow, Russia.

617

The Astrophysical Journal, 642:617–624, 2006 May 10

# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.

of the GF (in the Galactic system of coordinates) is the maincause not only for the �l ¼ 4n correlation, but also for all evenmultipoles with some alternation in the sign of the estimator. Theestimator designed in Naselsky & Novikov (2005) and the gen-eralized form illustrated in this paper can help us understand GFmanifestation in the harmonic domain, leading to the develop-ment of a ‘‘blind’’ method for foreground cleaning. In combina-tion with the multifrequency technique proposed in Tegmark &Efstathiou (1996; TOH03), the removal of the 2n correlation ofphases can be easily used as an effective method of determina-tion of the CMB power spectrum without a Galactic mask anddisjoint regions for theWMAP data and especially for the futurePlanck data. It can serve as a complementary method to the in-ternal linear combination method (Bennett et al. 2003c; Eriksenet al. 2004) and to the TOH03 method as well, in order to de-crease the contamination of the GF in the derived maps. Such akind of correlation should be observed by the Planck missionand will help us to understand the properties of the GF in detail,as it can play a role as an additional test for the foreground mod-els for the Planck mission.

This paper is organized as follows. In x 2 we describe the d�l;mestimator, which we call the phase filter, for 4n correlation in thecoefficients al,m and generalize the filter to correlation between�l ¼ 2n. In x 3 we give a detailed account on why such corre-lations appear. In x 4 we show the maps reconstructed from thefilter and discuss the connection between the 2n correlation andtheWMAP foreground symmetry. We examine the power spec-tra of the maps from the filter and their correlations in x 5. Theconclusion is in x 6.

2. THE �l ¼ 4n CORRELATIONAND ITS GENERALIZATION

2.1. The 4n Correlation and the Phase Filter

As is shown in Naselsky & Novikov (2005), the Galactic sig-nal reveals the 4n correlation. We recapitulate the definition ofthe estimator, which we call hereafter the ‘‘phase filter’’ (PF), aspecific combination of the spherical harmonic coefficients al,m ,

d�l;m ¼ al;m �al;m�� ��alþ�;m

�� �� alþ�;m; d�l;m¼0 ¼ 0; ð1Þ

where mj j � l and the coefficients al;m ¼ jal;mjexp (i�l;m) aredefined in the standard way,

�T (�; �) ¼Xlmax

l¼2

Xl

m¼�l

al;m�� ��exp (i�l;m)Yl;m(�; �): ð2Þ

Here�T (�; �) are the whole-sky anisotropies at each frequencyband, � and � are the polar and azimuthal angles of the polarcoordinate system, Yl;m(�; �) are the spherical harmonics, andjal;mj and �l,m are the moduli and phases of the l, m harmonics.The superscript� in d�l;m characterizes the shift of the l-mode ind�l;m. In Naselsky & Novikov (2005) the authors concentrate onthe series of correlation for� ¼ 4n, where n ¼ 1; 2; 3; : : : , basedon the fact that the signal of the Galaxy lies mostly close to the� ¼ �/2 plane. The phase filter in equation (1) is related to thephases of the multipoles of the �T (�; �) signal:

d�l;m ¼ al;m�� �� e i�l;m � ei�lþ�;m

� �¼ al;m 1� ei��

� �¼ 2al;m sin

��

2

� �exp i

��� �

2

� �� �; ð3Þ

where �� � �lþ�;m � �l;m. If al,m is Gaussian with randomphases, then the phase difference �� is random as well. The dl,mare then composed of al,m, which are random Gaussian them-selves, and a subtraction of two random unit vectors; therefore,dl,m are uncorrelated with each other. The amplitude hjdl;mji ¼hjal;mjj1� exp (i��)ji ¼

ffiffiffi2

phjal;mji. Hence, the power fordl ,m ,

hjdl;mj2i, is twice that for al ,m. On the other hand, if the phasesare not random, the map synthesized from the phase filter is sim-ply a map of the original al,m with phases rotated by an angle(��� �)/2 and the amplitudes decreased by a factor 2jsin (��/2)j.For �� ! 0,

d�l;m ’ 2al;me�i�=2 sin

��

2

� �: ð4Þ

Therefore, for some al,m modes whose amplitudes are to be de-creased significantly by the phase filter, we concentrate on thecondition �� ! 0.The PF is a nonlinear filter defined in the space of phases, for

which an analogy can be drawn with linear filtering for noise orpoint-source subtraction, such as the Wiener filter, the matchedfilter (Tegmark & de Oliveira-Costa 1998), and the adaptive top-hat filter (Chiang et al. 2002b). If for some range of multipolesthe power of the instrumental noise is negligible in comparisonwith that of CMB, the PF will reveal the power of the CMB. Ingeneral, the signal reconstructed by the PF is a combination ofthe CMB and instrumental noise.

2.2. Generalization of the PF

We would like to point out that the particular choice of the� ¼ 4n parameter of the filter in Naselsky & Novikov (2005)reflects only partially the properties of Galactic foregrounds andthe CMB plus noise signal in the Galactic system of coordi-nates. The properties of this filter are based on the symmetry ofthe associated Legendre polynomials and correlation propertiesof the foregrounds. First, the corresponding angular correlationlength for the foregrounds�f is significantly larger than that forthe CMBplus instrumental noise�g, i.e.,�f 3�g. Second, weexploit the properties of the Legendre polynomials for generali-zation of the � ¼ 4 filter, taking into account that in the Ga-lactic plane area (� ’ �/2, where � is the polar angle), all evenLegendre polynomials have maximal values, and their corre-sponding phases are close to each other or different by �.We generalize the PF as

d�l;m ¼ al;m � (�1)kal;m�� ��alþ2k;m

�� �� alþ2k;m; d�l;m¼0 ¼ 0; ð5Þ

where k is the order of correlation and the optimal values are 1and 2. One can see that for k ¼ 2n this filter reproduces the formin Naselsky & Novikov (2005), but for k ¼ 2nþ 1 the secondterm of the filter changes signs. Now all the even�l correlationsin phases are included in equation (5). We illustrate in x 3 thereason for such generalization.

3. WHY DO THE CORRELATIONSOF THE EVEN �l APPEAR?

In this section we describe the general properties of the� ¼ 2nperiodicity of the Galactic signal, taking into account the sym-metry of the Legendre polynomialsPl,m andPlþ�;m and the corre-lation of the Galactic foregrounds and the CMB plus noise signalsfor even�. To illustrate the properties of the d�¼2n

l;m , we adopt thefollowing model of the signal without losing generality. We as-sume for simplicity that all the pixels in the map are of equal area.

NASELSKY, NOVIKOV, & CHIANG618 Vol. 642

We use a polar system of coordinates, and the �j and �j mark theposition of the jth pixel in themap. Let us assume that the Galacticsignal is localized in the �-direction and is confined in the beltregion with half-width !. We single out the set of pixels p of thisregion and denote it asS ¼ p(�j; �j)j �/2� ! � �j � �/2þ !

;

thus, the map of the Galactic signal in S is

�T (�; �) ¼ Tj�D(cos �� cos �j)�

D(�� �j); ð6Þ

where �D is the Dirac �-function and the amplitude of the signalat pixel (�j; �j) is Tj. Note that we do not specify the localizationin the �-direction of the signal in S. In addition, we assume thatthe Tj is the sum of Galactic foreground signal T

fj , CMB T CMB

j ,and instrumental noise T n

j , where we denote Tcj � T CMB

j þ T nj .

It is important to note that the statistical properties of the fore-ground T f are different from T c, in terms of both amplitudes andpixel-pixel correlations hTjTki. In particular, we assume T

fj 3

Tcj within S, while T

fj TTc

j outside S. Using such a proposedmodel of the signal in the map, we can obtain the correspondingal,m coefficients of the spherical harmonic expansion,

al;m ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2l þ 1

4�

(l � m)!

(l þ m)!

s Xj

TjPml (cos �j)e

�im�j ; ð7Þ

which can be represented as a sum of foreground Fl,m coeffi-cients and the CMB plus noise coefficients cl,m.

In order to understand the nature of the 2n periodicity of theGalactic foreground, let us rewrite equation (1) in terms of co-efficients al,m from equation (7),

d�l;m ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2l þ 1

4�

(l � m)!

(l þ m)!

s Xj

Tje�im�jGm

l;�¼2n(cos �j); ð8Þ

where

Gml;�¼2n(cos �j) ¼ Pm

l (cos �j)

� (�1)nPmlþ2n(cos �j)M

ml (� ¼ 2n);

Mml (�) ¼P

�;� T�T�Pml (cos ��)P

ml (cos ��)e

im (����� )P�;� T�T�P

mlþ2n(cos ��)P

mlþ2n(cos ��)e

im (����� )

" #1=2

: ð9Þ

We examine the properties of equation (9) at the coordinates��; �� ’ �/2, for whichPm

l (cos ��) ’ Pml (0) and (see Gradshteyn

& Ryzhik 2000)

Pmlþ2n(0) ¼

2mffiffiffi�

p

� nþ 1þ l � mð Þ=2ð Þ� �n� l � mþ 1ð Þ=2ð Þ :

ð10Þ

The properties of the Legendre polynomials Pmlþ2n(0) depend

on the n, l, and m. First, for n ¼ 0 only even l þ m contribute tothe Pm

l (0), while all the odd l þ m formally correspond to

Pml (0) ¼ 0.We concentrate on the even�l correlations� ¼ 2n,

where n ¼ 1; 2; : : : ,

Pmlþ�(0) ¼ (�1)nPm

l (0)

Q j¼nj¼1 jþ l � mþ 1ð Þ=2½ �Q j¼n

j¼1 jþ l � mð Þ=2½ �: ð11Þ

Substituting equation (11) into equation (9), we obtainMml (� ¼

2n) ¼ 0, and the major part of the signal related to the Galac-tic plane at � ¼ �/2 vanishes because of the design of the PF.

Fig. 1.—The D(�; �) maps ( from the top to the bottom) synthesized from thePF forWMAP K, Ka, Q, V, and W maps. Note that the color-bar limits from topto bottom are ½�0:50; 0:50�, ½�0:50; 0:50�, ½�0:50; 0:50�, ½�0:40; 0:40�, and½�0:38; 0:50� mK, respectively.

GALACTIC FOREGROUND CORRELATIONS IN WMAP DATA 619No. 2, 2006

However, in reality the Galactic foregrounds stretch well beyondthe Galactic plane, and having equation (9) we can investigatehow they can be removed by the PF.

For that we discuss the properties of the Legendre polyno-mials for ��; �� 6¼ �/2 in equation (9), but l�� 31. In this casewe use the asymptotic form of the Legendre polynomials,

Pmlþ�(cos �) ’

2ffiffiffiffiffiffiffiffiffiffisin �

p �(l þ�þ mþ 1)

�(l þ�þ 3=2)

; cos l þ�þ 1

2

� ��þ 1

2m�� �

4

� �

’ Pml (cos �) cos (2n�)

Q j¼2nj¼1 ( jþ l þ m)Q j¼2n

j¼1 ( jþ l þ 1=2)

; 1� tan (2n�) tan l þ 1

2

� ��þ 1

2m�� �

4

� �� �:

ð12Þ

Once again, if � ¼ �/2� �, where �T�/2, then from equa-tion (12)weget cos (2n�)’ (�1)n cos (2n� )� (�1)n, tan (2n�) �� tan (2n� )T1, and Gm

l;�¼2n(cos �j) ! 0 in equation (9).

4. THE D(�; �) MAP FROM WMAP DATA

As the PF breeds a new set of al,m, we can also synthesize suchprocessed al,m into a whole-sky map. A nontrivial aspect of thePF is that it significantly decreases the brightest part of the Gal-axy image in the WMAP K–W maps. In the following analysiswe use a particular case n ¼ 2 so that� ¼ 4, although it can bedemonstrated that for other values the results of the analysis donot change significantly as long as� � lnoise, where lnoise is themultipole number in the spectrum for which the instrumentalnoise starts dominating over the GF signal. The optimal values,however, to best filter the GF are n ¼ 1, 2, 3, and 4.We show in Figure 1 how the PF transforms the GF image in

the WMAP K–W maps.3 These maps are synthesized from thed�l;m from the WMAP K–W bands for � ¼ 4 and lmax ¼ 512:

D(�; �) ¼Xlmax

l¼2

Xl

m¼�l

d�l;mYl;m(�; �): ð13Þ

Note that the temperature amplitudes are significantly reducedin each map. It should be emphasized that theD(�; �) map is nota temperature anisotropy map, as the phases are altered.We also apply the PF to the WMAP foreground maps at the

Q, V, and W bands. These foreground maps are the sum of syn-chrotron, free-free, and dust emissions. As these foreground

Fig. 2.—From top to bottom the D(�; �) map for Q-, V-, and W-band fore-grounds, respectively, with lmax ¼ 46 and� ¼ 4. The bottom is theD(�; �) mapfrom the V and W band map difference V�W. The color-bar limits are½�0:40; 0:50�, ½�0:19; 0:50�, ½�0:09; 0:29�, and ½�0:1; 0:1� mK, respectively.

Fig. 3.—Top, difference between V and W bands V�W; middle, same map,but with allm ¼ 0modes inal,m set to zero;bottom,D(�; �) map from the top panel.The color-bar limits are ½�0:4; 0:4�, ½�0:4; 0:4�, and ½�0:36; 0:4�mK, respectively.

3 See http://lambda.gsfc.nasa.gov/product/map.

NASELSKY, NOVIKOV, & CHIANG620 Vol. 642

maps do not contain the CMB signal and instrumental noise,they allow us to estimate the properties of the GF in detail.

In Figure 2 we plot the D(�; �) maps for the Q-, V-, andW-band foregrounds (� ¼ 4) for themultipole range l � 46. Thisrange is determined by the resolution of the WMAP foregroundmaps (l � 50). As one can see from these maps, the GF follows4n multipole correlation, which removes the brightest part ofthe signal from ½�1:31; 8:78� down to ½�0:40; 0:50�mK for theQ band, from ½�0:54; 3:82� down to ½�0:19; 0:50� mK for theV band, from ½�0:33; 2:30� to ½�0:09; 0:29�mK for theW band,and from ½�0:24; 1:72� to ½�0:1; 0:1� mK for the D(�; �) map,the difference between the VandW foregrounds. Note that theselimits are related to the brightest positive and negative spots inthe maps, while diffuse components have significantly smalleramplitudes. To show the high-resolutionD(�; �) map,which char-acterizes the properties of the foregrounds in the Vand W bands,in Figure 3we plot themap of the differenceV�Wbands and thecorresponding D(�; �) map for l � 50. Note that the V�Wmapdoes not contain the CMB signal, but for high l the properties ofthe signal are determined by the instrumental noise.

5. THE POWER SPECTRUM AND CORRELATIONSOF THE D MAP

5.1. What Is Constructed from the d�l;m?

To characterize the power spectrum of the D(�; �) maps weintroduce the definitions

D(l ) ¼ 1

2l þ 1

Xl

m¼�l

d�l;m

��� ���2; ð14Þ

G(l ) ¼ l(l þ 1)

2�D(l ): ð15Þ

Fig. 4.—Power spectrumG(l ) ¼ D(l )l(l þ 1)=(2�) forD(�; �) map (W band,solid line with asterisks) in comparison with the G(l ) power of the D(�; �) mapfor the FCM (thick solid line). The dashed line represents the power spectrumG(l ) for the D(�; �) map for the difference between the Q and W bands Q�W,and the dash-dotted line is for V�W.

Fig. 5.—Power spectrum G(l ) � D(l )l(l þ 1)/(2�) for the D map of theWMAP W band (solid line) in comparison with twice the power of the FCM(dotted line), 2Cll(l þ 1)/(2�). The dashed line represents the G(l ) of the D mapfrom the phase filtering on the difference V�W.

Fig. 6.—The Cs(l ) and Si(l ) trigonometric moments for the cross-correlationof phases between the TOH03 FCM and WFM (top pair). The solid line rep-resents the limit when the phases are identical. The middle pair is the Cs(l ) andSi(l ) trigonometric moments for the FCMwith a phase difference �lþ4;m ��l;m.The bottom pair is Cs(l ) and Si(l ) trigonometric moments for the WFM with aphase difference �lþ4;m � �l;m.

GALACTIC FOREGROUND CORRELATIONS IN WMAP DATA 621No. 2, 2006

As we have pointed out in x 2.1, if the derived d�l;m signal isGaussian, then D(l ) contains all the statistical properties of thesignal and is twice the power spectrum of the underlying field.For non-Gaussian signal, D(l ) characterizes the diagonal ele-ments of the correlation matrix. From Figure 4 it can be clearlyseen that forWMAP foregrounds, especially for theVandWbands,the power spectra of D(�; �) are significantly smaller than thepower of the CMB, which we estimate simply by using the powerof the TOH03 FCM, transformed by the PF as

DFCM(l ) ¼1

2l þ 1

Xl

m¼�l

cl;m �cl;m�� ��clþ�;m

�� �� clþ�;m

����������2

; ð16Þ

assuming that the FCM is fairly clean from the foreground sig-nal. An important point of analysis of the WMAP foregroundsis that for the Vand W bands the PF significantly decreases theamplitude of the GF, practically by 1–2 orders of magnitudebelow the CMB level.

The most intriguing question related to 4n correlation of themap derived from the WMAP Vand W band signals is, what isreconstructed by the d�l;m? The next question is why the powerspectra of d�l;m of the V and W bands shown in Figure 5 arepractically at the same level at the range of multipoles l � 100(where we can neglect the contribution from instrumental noiseat both channels and the differences of the antenna beams). Theequivalence of the powers for these two signals, shown in Fig-ure 5, suggests that these derived maps are related to pure CMBsignal (which we assume to be frequency independent).

In the following analysis we discuss what kinds of combina-tions are presented in the d�l;m between amplitudes and phases ofthe CMB signal and the foregrounds between theWMAP VandW bands. As was mentioned in x 2, the PF is designed as a linearestimator of the phase difference�lþ�;m � �l;m, if the phase dif-ference is small. Let us introduce the model of the signal at eachband a

( j )l;m ¼ cl;m þ F

( j )l;m, where cl,m is the frequency-independent

CMB signal and F( j )l;m is the sum over all kinds of foregrounds for

each band j (synchrotron, free-free, dust emission, etc.).According to the investigation above on the foregroundmod-

els, it is realized that without the ILC (internal linear combina-tion) signal the d�l;m estimation of the foregrounds, especially forthe V and W bands, corresponds to the signal4

d�;( f )l;m ¼ Fl;m �

Fl;m

�� ��Flþ�;m

�� �� Flþ�;m; ð17Þ

the power of which is significantly smaller than that of theCMB,

d�;(CMB)l;m ¼ cl;m �

cl;m�� ��clþ�;m

�� �� clþ�;m: ð18Þ

In terms of moduli and phases of the foregrounds at each fre-quency band,

Fl;m ¼ Fl;m

�� �� exp (i�l;m);

cl;m ¼ cl;m�� �� exp (i�l;m); ð19Þ

where �l,m and �l,m are the phases of foreground and the CMB,respectively. From equation (19) we get

d�;( f )l;m ¼ Fl;m

�� �� ei�l;m � ei�lþ�;m� �

; ð20Þ

and practically speaking, we get �l;m ¼ �lþ�;m. Thus, takingthe 2n correlation into account, we can conclude that it reflectsdirectly strong correlation of the phases of the foregrounds, deter-mined by theGF.Moreover, if any foreground-cleanedCMBmapsderived from different methods were to display the 2n correlationof phases, it would be evident that foreground residuals still de-termine the statistical properties of the derived signal.

5.2. 2n Phase Correlation of the D Map

One of the basic ideas for comparison of phases of two sig-nals is to define the trigonometric moments for the phases �l 0;mand �l,m as

Cs(l; l 0) ¼ 1ffiffil

pXl

m¼1

cos �l 0;m ��l;m

� �;

Si(l; l 0) ¼ 1ffiffil

pXl

m¼1

sin �l 0;m ��l;m

� �; ð21Þ

where l � l 0. We apply these trigonometric moments to investi-gate the phase correlations for the TOH03 FCM and WFM. Forthat, we simply substitute l ¼ l 0 in equation (21) and define �l ,mas the phase of the FCM and�l,m as that of theWFM. The resultof the calculations is presented in Figure 6.From Figure 6 it can be clearly seen that the FCM has strong

�l ¼ 4 correlations starting from l ’ 40, which rapidly increase4 Hereafter we omit the mark of channel j to simplify the formulae.

Fig. 7.—The F(l ) function for the TOH03 FCM (left) and the WFM (right). The dotted lines represent the �2/ffiffil

plimits.

NASELSKY, NOVIKOV, & CHIANG622 Vol. 642

for l > 40, while for the WFM these correlations are signifi-cantly damped.

However, the d�l;m estimator allows us to clarify the propertiesof phase correlations for the low multipole range. The idea is toapply the d�l;m estimator to the FCM and WFM and to comparethe power spectra of the signals obtained before and after that.

According to the definition of the d�l;m estimator, the powerspectrum of the signal is given by equation (16), which now hasthe form

D(l ) ¼ 2

l

Xm

cl;m�� ��2 1� cos (�lþ�;m � �l;m)

�: ð22Þ

Fig. 8.—The Cs(l ) and Si(l ) trigonometric moments for the FCM at �l ¼ 2(top pair). The second pair is for�l ¼ 2 after d�¼4

l;m filtration. The third and fourthpairs are for �l ¼ 1 before and after d�¼4

l;m filtration, respectively.

Fig. 9.—Comparison between the FCMandWFMbefore and after implemen-tation of the PF. From top to bottom: DFCM(�; �) map for the FCM, DWFM(�; �)for the WFM, the difference between DFCM(�; �) and DWFM(�; �), and the differ-ence between FCM and WFM. All the top four maps are plotted with color-barlimits ½�0:5; 0:5� mK. The third pair is redrawn from the second pair, but withnarrower color-bar limits ½�0:1; 0:1� mK. For all the maps lmax ¼ 500.

GALACTIC FOREGROUND CORRELATIONS IN WMAP DATA 623No. 2, 2006

The last term in equation (22) corresponds to the cross-correlation between the l, m and l þ 4, m modes, which shouldvanish for Gaussian random signals after averaging over the re-alization. For a single realization of the random Gaussian pro-cess, this term is nonzero because of the same reason as thewell-known ‘‘cosmic variance,’’ implemented for estimation ofthe errors of the power spectrum estimation (see Naselsky et al.2004a). Thus,

D(l ) ’ 2

l

Xm

cl;m�� ��2; ð23Þ

and the error of D(l ) is of the order of

�D(l )

D(l )’ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l þ 1=2p : ð24Þ

To evaluate qualitatively the range of possible non-Gaussianityof the FCM and WFM, in Figure 7 we plot the function F(l ) ¼2½D(l )� 2C(l )�/½D(l )þ C(l )� for the FCM andWFM, in whichwe mark the limits �2/

ffiffil

p. As one can see, the ranges of multi-

poles with potential non-Gaussianity are l ¼ 3–4, l ¼ 21 24,and l ’ 100 150 for the WFM. Nonrandomness on some of themultipole modes is mentioned in Chiang & Naselsky (2006).

At the end of this section we would like to demonstrate thatapplication of the PF to maps with foreground residuals, such asthe FCM, provides additional cleaning. In Figure 8 we presentthe Cs(l ) and Si(l ) trigonometric moments for the FCMwith theshift of the multipoles l 0 ¼ l þ 2. One can see that the � ¼ 2correlation of phases is strong (practically, they are at the samelevel as the � ¼ 4 correlations). However, after d�l;m filtrationthese correlations are significantly decreased.

The implementation of the PF to the non-Gaussian signal sig-nificantly decreases these correlations. The properties of the PFdescribed can manifest themselves more clearly in terms of im-ages of the CMB signal. In Figure 9 we plot the results of themaps with d�l;m implemented on the FCM and WFM, in orderto demonstrate how the PF works on the non-Gaussian tails ofthe derived CMB maps. In Figure 9 we can clearly see that themorphology of the D(�; �) maps is the same, and the differencebetween DFCM(�; �) and DWFM(�; �) is related to point-sourceresiduals localized outside the Galactic plane (see the thirdpanel). A direct subtraction of the WFM from the FCM revealssignificant contamination of the GF residuals and non-Galacticpoint sources (the third from the bottom and bottom maps). Thesecond from the bottom map corresponds to the difference be-

tween DFCM(�; �) and DWFM(�; �), for which the amplitudes ofthe signal are represented in the color-bar limit �0.1 mK. Onecan see that the GF is removed down to the noise level. In com-bination with the phase analysis we can conclude that the imple-mentation of the PF looks promising as an additional cleaningof the GF residuals and can help investigate the statistical prop-erties of derived CMB signals in more detail.

6. CONCLUSION

In this paper we examine a specific group of correlations be-tween l, which is used as an estimation of the statistical prop-erties of the foregrounds in theWMAPmaps. These correlationsamong phases in particular are closely related to symmetry ofthe GF (in the Galactic coordinate system). An important pointof the analysis is that for the foregrounds the correlations ofphases for the total foregrounds at the V and W bands have aspecific shape when �l;m ’ �lþ�;m, where � ¼ 4n and n ¼1; 2; 3; : : : . These correlations can be clearly seen in theWbandof the WMAP data sets down to lmax ¼ 512 and must be takeninto account for modeling of the foreground properties for theupcoming Planckmission. We apply the PF to the TOH03 FCM,which contains strong residuals from the GF, and show that theseresiduals are removed from the DWFM(�; �) map. Moreover, inthat map the statistics of the phases display statistics closer toGaussian than the original FCM (no correlation of phases be-tween different l, m modes except between l þ�, m and l, m,which is chosen as a basic one, defined by the form of the PF).In this paper we do not describe in detail the properties of

the signal derived by the PF from the WMAP V and W bands.Further developments of the method, including multifrequencycombination of the maps and CMB extraction by the PF, will bein a separate paper. To avoid misunderstanding and confusion,here we stress again that anyD(�; �) maps synthesized from thed�l;m are by no means the CMB signals (since the phases of thethese signals are not the phases of true CMB), and the true CMBcan be obtained after multifrequency analysis, which is the sub-ject of the forthcoming paper.

We thank H. K. Eriksen, F. K. Hansen, A. J. Banday, C.Lawrence, K. M. Gorski, and P. B. Lilje for their commentsand critical remarks. We acknowledge the use of NASA LegacyArchive forMicrowave Background Data Analysis (LAMBDA)and the maps. We also acknowledge the use of the HEALPix(Gorski et al. 1999) and theGLESP package (Doroshkevich et al.2005) to produce al,m from the WMAP data sets.

REFERENCES

Bennett, C. L., et al. 2003a, ApJ, 583, 1———. 2003b, ApJS, 148, 1———. 2003c, ApJS, 148, 97Chiang, L.-Y. 2001, MNRAS, 325, 405Chiang, L.-Y., & Coles, P. 2000, MNRAS, 311, 809Chiang, L.-Y., Coles, P., & Naselsky, P. D. 2002a, MNRAS, 337, 488Chiang, L.-Y., Jørgensen, H. E., Naselsky, I. P., Naselsky, P. D., Novikov, I. D.,& Christensen, P. R. 2002b, MNRAS, 335, 1054

Chiang, L.-Y., & Naselsky, P. D. 2006, Int. J. Mod. Phys. D, 15, in pressChiang, L.-Y., Naselsky, P. D., & Coles, P. 2004, ApJ, 602, L1Chiang, L.-Y., Naselsky, P. D., Verkhodanov, O. V., & Way, M. J. 2003, ApJ,590, L65

Coles, P., Dineen, P., Earl, J., & Wright, D. 2004, MNRAS, 350, 989Doroshkevich, A. G., Naselsky, P. D., Verkhodanov, O. V., Novikov, D. I.,Turchaninov, V. I., Novikov, I. D., Christensen, P. R., & Chiang, L.-Y. 2005,Int. J. Mod. Phys. D, 14, 275

Eriksen, H. K., Banday, A. J., Gorski, K. M., & Lilje, P. B. 2004, ApJ, 612,633

Gorski, K. M., Hivon, E., & Wandelt, B. D. 1999, in Evolution of Large-ScaleStructure: from Recombination to Garching, ed. A. J. Banday, R. S. Sheth, &L. Da Costa (Garching: ESO), 37

Gradshteyn, I. S., & Ryzhik, I. M. 2000, Table of Integrals, Series, and Products(6th ed.; New York: Academic)

Hinshaw, G., et al. 2003, ApJS, 148, 135Naselsky, P.D.,Chiang, L.-Y.,Olsen, P.,&Verkhodanov,O.V. 2004a, ApJ, 615, 45Naselsky, P. D., Doroshkevich, A., & Verkhodanov, O. V. 2003a, ApJ, 599, L53———. 2004b, MNRAS, 349, 695Naselsky, P. D., & Novikov, I. D. 2005, Int. J. Mod. Phys. D, 14, 1769Naselsky, P. D., Verkhodanov, O. V., Chiang, L.-Y., & Novikov, I. D. 2003b,ApJ, submitted (astro-ph /0310235)

Schwarz, D. J., Starkman, G. D., Huterer, D., & Copi, C. J. 2004, Phys. Rev.Lett., 93, 221301

Tegmark, M., & de Oliveira-Costa, A. 1998, ApJ, 500, L83Tegmark, M., de Oliveira-Costa, A., & Hamilton, A. 2003, Phys. Rev. D, 68,123523 (TOH03)

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

NASELSKY, NOVIKOV, & CHIANG624