Multi-resolution internal template cleaning: An application to the Wilkinson Microwave Anisotropy Probe 7-yr polarization data
R. Fernández-Cobos, P. Vielva, R.B. Barreiro, E. Martínez-González
aa r X i v : . [ a s t r o - ph . C O ] N ov Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 26 October 2018 (MN L A TEX style file v2.2)
Multi-resolution internal template cleaning: An applicationto the
Wilkinson Microwave Anisotropy Probe
R. Fern´andez-Cobos , ⋆ , P. Vielva , R.B. Barreiro , E. Mart´ınez-Gonz´alez Instituto de F´ısica de Cantabria, CSIC-Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain. Dpto. de F´ısica Moderna, Universidad de Cantabria, Avda. los Castros s/n, 39005 Santander, Spain.
Accepted Received ; in original form
ABSTRACT
Cosmic microwave background (CMB) radiation data obtained by different experi-ments contain, besides the desired signal, a superposition of microwave sky contribu-tions. We present a fast and robust method, using a wavelet decomposition on thesphere, to recover the CMB signal from microwave maps. An application to
WMAP polarization data is presented, showing its good performance particularly in very pol-luted regions of the sky. The applied wavelet has the advantages of requiring littlecomputational time in its calculations, being adapted to the
HEALPix pixelizationscheme, and offering the possibility of multi-resolution analysis. The decompositionis implemented as part of a fully internal template fitting method, minimizing thevariance of the resulting map at each scale. Using a χ characterization of the noise,we find that the residuals of the cleaned maps are compatible with those expectedfrom the instrumental noise. The maps are also comparable to those obtained fromthe WMAP team, but in our case we do not make use of external data sets. In addi-tion, at low resolution, our cleaned maps present a lower level of noise. The E-modepower spectrum C EEℓ is computed at high and low resolution; and a cross power spec-trum C T Eℓ is also calculated from the foreground reduced maps of temperature givenby
WMAP and our cleaned maps of polarization at high resolution. These spectraare consistent with the power spectra supplied by the
WMAP team. We detect theE-mode acoustic peak at ℓ ∼ CDM model. TheB-mode power spectrum C BBℓ is compatible with zero.
Key words: methods: data analysis - cosmic microwave background
Component separation is a critical aspect in the analy-sis of cosmic microwave background (CMB) data. A goodcharacterization of the data is a prerequisite to the ade-quate estimation of cosmological parameters. This need be-comes crucial when, as happens in B-mode detection ex-periments, foreground amplitudes are well above the sig-nal (e.g., Tucci M. et al. 2005). Two physical galactic pro-cesses are the major contaminants to CMB polarized sig-nal: synchrotron radiation and thermal dust. Both appearat large scales, are highly anisotropic and the spatial vari-ation of their emissivity is smooth. Besides, extragalacticemission also contaminates this cosmological signal: pointsources and clusters are compact objects, roughly isotropi-cally distributed in the sky and every single object has a par- ⋆ e-mail: [email protected] ticular frequency dependence. Most of the component sepa-ration methods take into account only diffuse components,assuming that we are previously masking the brightest pointsources or subtracting them by, typically, fitting approaches(see Herranz & Vielva 2010, for a recent review).Current and future experiments (Rubi˜no-Mart´ın et al.2008; Brown et al. 2009; Sievers et al. 2009; Arnold et al.2010; Kogut et al. 2006; Grainger et al. 2008;Charlassier et al. 2008) are able to measure CMB po-larization anisotropies with such precision that foregroundcontamination have become the major limitation when wetry to analyze the data. This is the principal reason to investeffort and time in developing new techniques for separatingcomponents. The goal of all the proposed methods is toseparate or, at least, to identify CMB anisotropies from theother components. The range of proposals includes internallinear combinations (ILC), Bayesian methods and indepen- c (cid:13) R. Fern´andez-Cobos et al. dent component analysis (see Delabrouille & Cardoso 2007,for a recent review).There is abundant literature that includes ap-plications of the various methods related to somepolarization experiments in vogue as, for instance,
PLANCK (Leach et al. 2008; Efstathiou et al. 2009;Betoule et al. 2009; Baccigalupi et al. 2004) and
WMAP (Gold et al. 2011; Delabrouille J. et al. 2009; Kim et al.2009; Bonaldi et al. 2007; Maino et al. 2007; Eriksen et al.2006, 2008).The method that we present in this paper is situatedin the context of the internal linear combinations and it isa bet for a template cleaning in which coefficients are fittedin the space of a particular wavelet that enables a multi-resolution analysis. A fitting by scales allows, in practice,some effective variation of the coefficients in the sky, which isan advantage over the template cleaning in real space. In thissense, our approach based on wavelet space effectively lies inbetween standard linear combination techniques applied inthe real space and more sophisticated parametric methods(e.g., Eriksen et al. 2006, 2008; Stivoli et al. 2010).Our approach is a fast procedure that especially showsits effectiveness in polluted regions, such as those that ap-pear in polarization experiments.This paper is structured as follows. The methodologyis described in detail in Section 2. We set out an analysis ofthe low-resolution polarization
WMAP data in Section 3. Insection 4, we show the treatment for high-resolution
WMAP data in order to obtain the C EEℓ and C TEℓ spectra. Finally,we present the conclusions and discussion in section 5.
In this work, we present a multi-resolution internal templatecleaning (MITC) method for foreground removal. This isthe initial step of the map cleaning process in the
SEVEM method (Mart´ınez-Gonz´alez et al. 2003; Leach et al. 2008)to the case of polarization.For many purposes, it is a key point to have CMB mapsat several frequencies instead of a single map. For instance,it would serve as a consistency check to verify whether anydetected feature of the data is actually monochromatic ornot (as, for instance, the case for non-gaussianity analysis).Another advantage of the method is that we do notneed a thorough knowledge of foregrounds, because we ob-tain all the information to construct different templatesfrom the data. Furthermore, this procedure preserves theoriginal resolution of the CMB component. But the down-side is that the internal templates are noisy, so we in-crease the total noise level when we remove them fromthe data. This circumstance results, for instance, in an in-crease in the error bars of the power spectrum at high mul-tipoles. An alternative would be to incorporate externaltemplates, created from data from other independent ob-servations or based on theoretical arguments. However, thecurrent knowledge of foreground emissions, in polarization,is not substantiated with suitable ancillary data set, andfor that reason, this option is not considered in this case.This situation may change in the future with the informa-tion expected to be provided by experiments like
PLANCK
Figure 1.
Outline of construction of the detail coefficients atresolution j ( d i ) as the substraction of the approximation coeffi-cients at resolution j -1 ( y i ), from the approximation coefficientsat resolution j ( x i ). (Tauber et al. 2010), C-BASS (King et al. 2011) or
QUI-JOTE (Rubi˜no-Mart´ın et al. 2008).
HEALPix wavelet
Wavelets are a powerful tool in signal analysis andare extensively used in many astrophysics applications.Several examples of implementation of component sep-aration methods which employ very diverse waveletscan be found in the literature (e.g., Ghosh et al. 2011;Delabrouille J. et al. 2009; Gonz´alez-Nuevo et al. 2006;Vielva et al. 2003; Hansen et al. 2006). They are localizedwave functions, that allow for a multi-resolution treatmentof the data. This fact represents an advantage over othercomponent separation methods because it allows us to varythe effective emissivity of foregrounds.We use the so-called
HEALPix wavelet, HW,(Casaponsa B. et al. 2011), a discrete and orthogonalwavelet that provides a multi-scale decomposition on thesphere adapted to the
HEALPix pixelization (G´orski et al.2005). The resolution of a map in the
HEALPix tessellationis given in terms of the N side parameter, defined so that thenumber of pixels needed to cover the sphere is N = 12 N side .The resolution j of a map is a number such that 2 j = N side .A CMB map is decomposed in the wavelet coefficient spacein a series of maps from the resolution of the original mapto the lowest resolution considered. All of these maps, ex-cept the lowest resolution one, are called details. The lastone is called the approximation, and is constructed by de-grading the original map to the appropiate resolution, i.e.,to calculate the approximation coefficient at resolution j -1at a given position i we take the average of the four daugh-ter pixels at resolution j . The way that different detail mapsare built is illustrated in figure 1. At each resolution j , detailcoefficients are calculated as the substraction of the approxi-mation coefficients at resolution j -1 from the approximationcoefficients at resolution j . Both this process and the math-ematical formalism of this wavelet is carefully explained inCasaponsa B. et al. (2011), where the HW is used to putconstraints on the f NL parameter from WMAP data.In that paper, it is said that the reconstruction of a map c (cid:13) , 000–000 ulti-resolution internal template cleaning M ( x i ) can be written as M ( x i ) = n j − X p =0 λ j ,p φ j ,p ( x i ) ++ X m =1 J − X j = j n j − X p =0 γ m,j,p ψ m,j,p ( x i ) , (1)where λ j ,ℓ and γ m,j,ℓ are the approximation and detail co-efficients respectively, φ j,p ( x i ) is the scaling function and ψ m,j,p ( x i ) refers to the wavelet functions. The j index takesvalues from the highest resolution J to the approximationresolution j .The advantage of this wavelet with respect to others, inaddition to its straightforward implementation, lies in thespeed of the involved operations. The computational timefor the wavelet decomposition is of the order of the num-ber of pixels ( ∼ N pix ) whereas, for example, for the conti-nous wavelet transform of the spherical Mexican hat wavelet(Mart´ınez-Gonz´alez et al. 2002) or the needlets (Baldi et al.2009) this time is of the order of ∼ N / pix . The signal b T j ( p ) at resolution j is constructed by subtract-ing a linear combination of different templates t ij from theoriginal signal, T j , as follows b T j ( p ) = T j ( p ) − N t X i =1 β ij t ij ( p ) , (2)where N t is the total number of templates and p is a pixelindex.An internal template is formed as the difference of twomaps of the same resolution, corresponding to different fre-quencies, in units of thermodynamic temperature.The variance of the cleaned map is optimally minimizedat each scale to obtain the coefficients β ij or, equivalently,the quadratic quantity χ j = X p n b T j ( p ) C − [ b T j ( p )] t o , (3)where C − is the inverse of the covariance matrix calculatedas the sum of contributions of the CMB and instrumentalnoise (both, from the map to be cleaned and the templates).From the previous discussion, it is obvious that the ap-proach to produce an optimal recovery of the CMB wouldrequire a certain knowledge of this signal, via its correla-tions. However, a more robust estimator, without a prioriknowledge of the signal to be estimated, may be built byconsidering only the instrumental noise correlations.We have checked, however, that the gain in the CMBrecovery, by including the information related to the instru-mental characteristics is, in practice, very little. Even more,in some situations (as it is the case of the WMAP full resolu-tion data, see section 4) the instrumental noise informationis limited to the autocorrelation. Therefore, in this work wehave decided to perform the internal template fitting withuniform weights for all the pixels at each scale, which impliesto minimize the following quantity: E j = X p " T j ( p ) − N t X i =1 β ij t ij ( p ) . (4)Finally, we recover a single map performing the waveletsynthesis. It can be written as b T ( ~x ) = T ( ~x ) − N t X i =1 N res X j =1 γ ij ( ~x ) t ij ( ~x ) , (5)where N res denotes the number of involved resolutions and γ ij some new coefficients given as linear combinations of β ij coefficients which are the result of the synthesis process. WMAP
DATA
The instrumental noise in
WMAP polarization is knownto be correlated (Jarosik et al. 2011). Although the
WMAP data are typically given at a
HEALPix resolution of N side =512, a more accurate version of the pixel-to-pixel correla-tion is only available at low resolution, namely, N side = 16.Taking into account this difference, we have performed thecleaning of the WMAP data in two cases: for low and highresolution maps. In this section, we analyse the maps at N side = 16.The WMAP data are composed by, at least, a superpo-sition of CMB, synchrotron and thermal dust emissions. The
WMAP team proposed a template fitting in the pixel spaceto clean the foreground emission in the Ka, Q, V and Wmaps, using as templates the K band (for the synchrotron)and a low resolution version of the Finkbeiner et al. (1999)model for the thermal dust, with polarization direction de-rived from starlight measurements (Gold et al. 2011).In our approach, we use only a synchrotron template,constructed as K-Ka. The reason for neglecting the thermaldust template is because a previous analysis in real spaceshows that its coefficients are much smaller than the cor-responding ones for the synchrotron template. We clean Q and U polarization components independently minimizingthe variance of the cleaned maps of the Q1, Q2, V1 and V2differencing assemblies (DAs). The wavelet decompositionis carried out down to resolution j = 3 for the data map,thus, in addition to the approximation, we have a single de-tail map at j = 4. Best fitting coefficients for the consideredDAs are given in table 1. We apply the WMAP polarizationanalysis mask that excludes a 26% of the sky.
Since the CMB polarization signal is clearly subdominantin the
WMAP low resolution data, it is hard to establish acriterion to evaluate the goodness of the cleaning process,and to perform comparisons with different solutions.We have decided to evaluate this goodness by compar-ing the cleaned map with the expected signal for a noisy skyfollowing the
WMAP instrumental noise characteristics. Inthis sense, a good compatibility with the noise propertieswould indicate that foregrounds have been satisfactorily re-duced.We generate a set of 10 simulations of the noise maps c (cid:13) , 000–000 R. Fern´andez-Cobos et al.
Frequency band Q1 Q2 V1 V2Detail ( j = 4) Q Stokes 0.092 0.103 0.036 0.023 U Stokes 0.074 0.117 0.020 0.048Approximation ( j = 3) Q Stokes 0.244 0.259 0.081 0.125 U Stokes 0.241 0.236 0.085 0.112
Table 1.
Template cleaning coefficients for Q and U Stokes parameters and DAs for the low resolution case.
Figure 2.
Upper panels show the χ distributions of our cleaned maps and the bottom ones present the same results for the WMAP procedure. The solid line (red) corresponds to the theoretical curve of a χ with as many degrees of freedom as pixels outside the mask(i.e., the effective number of pixels in Q and U maps: 4518). The dashed line is the distribution calculated from simulations of our cleanedmaps. The vertical line shows the χ value of the data maps in each case. The columns correspond to different frequency bands, fromleft to right: Q, V and W bands. resulting from our MITC method at Q, V and W frequencybands, M r ( p ), with r ∈ { , ..., } , in order to construct a χ distribution, calculating each value as χ r = X p,q M r ( p ) N − Obs ( p, q ) M tr ( q ) , (6)where N Obs is the noise correlation matrix. A number ofsimulations of the order of a million is required to estimatethis matrix so that the distribution converges to the the-orical curve of a χ distribution with as many degrees offreedom as pixels outside the mask in Q and U maps (inthis case, we have 4518 degrees of freedom). This distribu-tion characterizes the expected noise level at each frequencymap. We can associate the χ value of the data with relativelevels of signal. We can say that the cleaned maps containmore than just noise (typically foreground residuals, sincethe CMB is subdominant compared to the noise at thesescales) if the data value is much higher than typical valuesof the distribution. Conversely, we can ensure that our mapsare compatible with the expected noise and that residualsare small if the data value falls within the distribution. The Frequency band Q V W χ Cleaned χ F orered
Table 2.
Different values of χ computed with our seven-year cleaned maps per frequency band ( χ Cleaned ) and with
WMAP seven-year foreground reduced maps per frequency band( χ F orered ). DA Q1 Q2 V1 V2 χ Cleaned
Table 3.
Different values of χ computed with our seven-yearcleaned maps per DA. χ values for each band are listed in table 2 and for eachDA in table 3.Our test is based on the assumption that the CMB con-tribution is negligible. We have tested that the CMB pro-vides a very small contribution (a shift of ∼
10 units of χ ) c (cid:13) , 000–000 ulti-resolution internal template cleaning Figure 3.
The combinations of the raw W-band maps. Theo-retical curve of the χ distribution is represented by a solid redline and the χ values are shown by successive vertical lines, fromleft to right: W2-W4, W1-W2, (W1+W3)-(W2+W4), W2-W3,W1-W4, W3-W4 and W1-W3. by generating 10 simulations with CMB and instrumen-tal noise of the cleaned maps. These simulations have beenused to compute another χ distribution with the matrixthat we have already calculated with only the noise compo-nent. When distributions are compared with each other weobserve this typical deviation. Thus, the CMB contributionto the value of the χ of the data is negligible and, there-fore, any significant deviation from the mean value has tobe assigned to foreground residuals.An indirect comparison can be made between the WMAP procedure and our MITC method through the rela-tive possitions of the χ value of the data with respect to thedistribution. As seen in figure 2, we obtain that the valueof the cleaned maps is fully compatible with instrumentalnoise at Q and V frequency bands. At W band the χ valueof the cleaned data is in the tail of the distribution probablydue to the presence of foreground residuals. The deviationis even larger when the WMAP procedure is used. A sig-nificant improvement is also found at Q band since the χ value is shifted from 2 σ to 0 . σ when our MITC method isused.In addition, although we use a template that is noisierthan the ones used by the WMAP team, the noise levels ofour cleaned maps are lower. We have measured a differenceof about a 10% in terms of the standard deviation of thedata maps (this difference is confirmed by instrumental noisesimulations).In order to check further the apparent excess of signalat W band obtained by the two approaches, we computedanalytically the noise covariance matrix of different com-binations of the raw W-band DAs maps which contain, inprinciple, only a combination of instrumental noise. Withthese covariance matrices, based on the full-sky covariancematrices of each DA, a χ value of the data maps is obtained.It is shown in figure 3 that these maps are still compatiblewith the expected noise. However, it is significant that allvalues are to the left of the distribution and that the mostdeviated ones involve W2, followed by W4. We have also analysed the distribution of the χ values of the single-yearforeground reduced maps supplied by the WMAP team foreach DA at the W band, obtaining values more deviated to-wards the tails for the W2 and W4 DAs. This may suggest anot good enough characterization of the instrumental noisefor these DAs. C EEℓ and C BBℓ
We carry out an estimation of the polarization spectrum us-ing our cleaned maps of the Q1, Q2, V1 and V2 DAs. Apseudo cross-power spectrum ˆ D ABℓ between any two differ-encing assemblies A and B can be calculated asˆ D ABℓ = X ℓ ′ M ABℓℓ ′ | p ℓ ′ | B Aℓ ′ B Bℓ ′ h C ABℓ ′ i + h N ABℓ i , (7)where A, B ∈ { Q , Q , V , V | A = B } ; and, in the case ofan EE power spectra,ˆ C ABℓ = 12 ℓ + 1 ℓ X m = − ℓ e Aℓm e B ∗ ℓm , (8)where e ℓm are the spherical harmonic coefficients of the E-mode. Assuming a circular beam response, we denote thebeam of the A map as B Aℓ and the window function of the HEALPix pixel by p ℓ ; h N ABℓ i is the noise cross-power spec-trum. The bias introduced by this term comes from the inter-nal template fitting procedure. It is small and controled bysimulations. Finally, the coupling kernel matrix M ℓℓ ′ is de-scribed in Hivon et al. (2002) and, for the case of the polar-ization components, in Appendix A of Kogut et al. (2003).This procedure is usually referred to as MASTER estima-tion. An estimator, ˆ C ℓ , can be computed as a linear combi-nation of the six different spectra weighted by the inverse oftheir variances in the following way:ˆ C ℓ = X i σ i ! − X i σ i ˆ C iℓ , (9)where i = AB and σ i = σ A σ B . These variances are givenby the WMAP team in the
LAMBDA web site .The resulting power spectra are shown in figure 4. Fromthe C EEℓ spectrum we can say that most of the values arecompatible with zero, so there is almost no signal exceptperhaps for low multipoles ℓ .
6. As expected, the B-modespectrum C BBℓ signal is compatible with zero. Both spectraare compatible with those that the
WMAP team supplies.Our error bars are larger than those obtained by the
WMAP team, because of the use of an estimator that is not optimal,a pseudo-spectrum, whereas the
WMAP team uses a pixel-base likelihood.
WMAP
DATA
In this section, we analyze
WMAP data maps at N side =512. This approach allows us to study the cleaning at smallerscales where, a priori , the correlation of the noise is less im-portant. So then we only take into account the noise covari-ance matrix of each pixel. In this case, the cleaning method http://lambda.gsfc.nasa.gov/c (cid:13) , 000–000 R. Fern´andez-Cobos et al.
Figure 4.
Polarization power spectrum EE (upper pannel) andBB (bottom pannel) for low resolution analysis. Circles are thespectrum supplied by
WMAP team and asterisks represent ourestimation. The fiducial model is plotted by the solid line. based on the wavelet space is applied using two different in-ternal templates. The first one is constructed as K-Ka andaccounts for the synchrotron radiation. The second one isbuilt as V1-W3 and attempts to characterize the thermaldust. The V and W DAs to be cleaned have been selectedby having a lower noise. As in the low resolution case, thewavelet decomposition is carried out until resolution j = 3for the approximation map, hence we have additionally 6different detail maps in this high resolution case. Again, the WMAP polarization mask is used. Best fitting coefficientsare shown in table 4.From the cleaned Q and U maps we study the powerspectra, C EEℓ and C BBℓ , as in the previous section. The spec-trum error bars are estimated from 10 noise simulations. Ingeneral, the CMB contribution is neglegible compared to thenoise one. Bins are taken as a weighted average of the mul-tipoles involved. These weights are calculated as the inverseof the variance of each C ℓ .The spectra are compatible with those that the WMAP team has obtained. Our error bars are larger at high multi-poles, since, at this scale range, the number of effective cross-spectra is much smaller than the one used by the
WMAP team (where all the W-band DAs are available). Neverthe-less, as seen in figure 2, our cleaned maps seem to presenta lower level of contamination than those supplied by the
WMAP team. We expected a better foreground removalsince the templates used are closer to the foreground signaldistribution accross the sky in our case. However, whetherthis may have an impact on the determination of the cos-mological parameters is not clear and would require an ex-haustive analysis which is outside the scope of this paper.Finally, the two points accounting for the largest scalesof the spectra are taken from figure 4, since the correlationof the noise at these scales is important, and it has been bet-ter modeled in the previous section, where a more accurateversion of this information was available.
Figure 5.
Polarization power spectrum EE (upper pannel) andBB (bottom pannel) for high resolution analysis. The circles arethe spectrum supplied by the
WMAP team and the asterisksrepresent our estimation. The solid line represents the fiducialmodel for C EEℓ and the zero value for C BBℓ . The resulting power spectra are presented in figure 5.Our outcome is compatible with the
WMAP team analysis,where it is possible to distinguish the acoustic peak around ℓ ∼
400 in the E-mode spectrum. As expected, the B-modepower spectrum is compatible with zero.Our independent approach can be seen as a confirma-tion of the previously detection reported by the
WMAP team.A similar procedure is applied to determine the corre-lation between temperature and E-mode polarization data.We carry out the analysis using our cleaned maps of the Q and U Stokes parameters and the foreground reducedtemperature maps that the
WMAP team supplies. In thiscase, the combination of two maps of the same DA is al-lowed and the equation 7 has the same form as long aswe add that, in C ABℓ , A refers to the DA of temperaturemaps and B to the DA of polarization maps, with A, B ∈{ Q , Q , V , W , W , W } . For the temperature maps, weuse the temperature analysis mask that the WMAP teamsupplies and the MASTER estimation is computed as is de-scribed in Kogut et al. (2003). c (cid:13) , 000–000 ulti-resolution internal template cleaning DA Template Stokes Q1 Q2 V2 W1 W2 W4Detail ( j = 9) (K-Ka) Q Stokes -0.0175 -0.0067 0.0024 0.0754 0.0072 -0.0222 U Stokes 0.0525 0.0173 0.0085 -0.0869 0.1171 -0.0562(V1-W3) Q Stokes -0.0008 0.0023 -0.0015 0.0013 -0.0015 0.0054 U Stokes 0.0007 0.0006 0.0022 0.0005 -0.0226 -0.0106Detail ( j = 8) (K-Ka) Q Stokes -0.0238 0.0185 0.0135 -0.0046 0.0076 0.0311 U Stokes -0.0080 -0.0176 -0.0059 -0.0513 0.0138 -0.0196(V1-W3) Q Stokes -0.0044 -0.0003 -0.0008 -0.0048 0.0020 0.0071 U Stokes 0.0012 0.0028 0.0042 0.0013 0.0030 0.0012Detail ( j = 7) (K-Ka) Q Stokes 0.0006 0.0045 -0.0091 -0.0227 0.0288 -0.0224 U Stokes -0.0208 0.0196 0.0038 -0.0263 0.0047 -0.0142(V1-W3) Q Stokes 0.0054 0.0004 -0.0002 -0.0053 -0.0002 0.0069 U Stokes -0.0017 0.0006 0.0032 0.0019 -0.0036 0.0009Detail ( j = 6) (K-Ka) Q Stokes 0.0224 -0.0112 -0.0073 0.0302 -0.0057 0.0407 U Stokes 0.0278 0.0172 0.0104 0.0144 0.01797 0.0076(V1-W3) Q Stokes 0.0079 0.0014 -0.0039 -0.0070 -0.0095 -0.0038 U Stokes 0.0066 -0.0051 0.0016 -0.0012 -0.0096 0.0179Detail ( j = 5) (K-Ka) Q Stokes 0.0351 0.0729 0.0702 -0.0325 0.0253 -0.1171 U Stokes 0.0567 0.0523 0.0670 0.1166 0.0255 -0.0225(V1-W3) Q Stokes -0.0131 0.0012 0.0207 -0.0070 -0.0578 0.0508 U Stokes -0.0080 0.0009 -0.0071 -0.0362 -0.0006 0.0363Detail ( j = 4) (K-Ka) Q Stokes 0.2292 0.0571 0.0548 -0.0835 0.3029 -0.2239 U Stokes 0.0741 0.1172 -0.0844 0.1445 -0.2241 -0.2723(V1-W3) Q Stokes -0.0708 0.0774 -0.1389 0.0386 0.1316 -0.0135 U Stokes -0.0440 0.0359 -0.0397 0.1985 -0.2028 0.2320Approximation ( j = 3) (K-Ka) Q Stokes 0.1732 0.2976 0.1988 0.1413 0.1234 0.2612 U Stokes 0.1397 0.2315 0.1674 0.1607 0.4529 0.5386(V1-W3) Q Stokes -0.2852 0.2030 0.6050 0.5513 0.2601 -1.1963 U Stokes 0.2087 0.1022 -0.0048 0.5052 0.0160 -0.3867
Table 4.
Template cleaning coefficients for Q and U Stokes parameters and different frecuency bands for the high resolution case.
Figure 6.
Polarization power spectrum TE. Circles are the spec-trum supplied by
WMAP team and asterisks represent our esti-mation. The fiducial model is plotted by the solid line.
Since the CMB cosmic variance in temperature con-tributes significantly, we cannot ignore it this time in thecalculation of the error bars. So we generate 10 simula-tions of CMB plus noise, which undergo the same process ofcleaning and combination to get the cross power spectrum,with which we estimate the error bar. These errors are rela-tively larger with respect to the WMAP results, due to theless number of effective cross spectra. In addition, we re-mark that, while we use a pseudo-spectrum, low resolutionanalysis is performed through a pixel-base likelihood by the
WMAP team. The resulting cross power spectrum is presented in fig-ure 6, and it is compatible with the power spectrum thatthe
WMAP team supplies.
We introduce an internal template cleaning method thatuses a wavelet decomposition on the sphere. Among its ad-vantages, it is included the possibility of multi-resolutionanalysis, allowing an effective variation of the spectral indexin the sky. Much lower computational time is needed thanwith other widely-used continous wavelets. In addition, agood treatment of incomplete sky coverage is given becauseof the compact support of the
HEALPix wavelet.The MITC method result is a set of some cleaned mapsat several frequencies that can be used, for instance, to ver-ify whether any detected feature of the data is actuallymonochromatic or not. The exclusive use of internal tem-plates allows us to analyze the maps without making anyprior assumptions about the foregrounds in polarization.However, although the implementation that is shown in thiswork make use of only internal templates, it can be triviallyextended to deal with external templates as well.We perform an analysis of 7-year
WMAP data obtain-ing outcomes that are compatible with the
WMAP teamresults. Furthermore, we have hints of better cleaning of theQ-band map at large scales and we have obtained cleanedmaps, at least, as good as those that the
WMAP team sup-plies for V and W bands. Let us remark that our approachdoes not make use of any additional template: everything isobtained from
WMAP data. We have checked that, although c (cid:13) , 000–000 R. Fern´andez-Cobos et al. we use noisier templates, the instrumental noise levels of thefinal cleaned maps are lower than those of the maps providedby the
WMAP team.High resolution maps are also analysed. In agreementwith the
WMAP team, we find an E-mode detection at ℓ ∼ CDM model. We alsoobtain that the B-mode level is compatible with zero. Theseindependent findings are a confirmation of the result alreadypresented by the
WMAP team.The clean maps produced at this work, both at lowand high resolution, are available at the following website:http://max.ifca.unican.es/cobos/WMAP7yrPOL
ACKNOWLEDGMENTS
Authors acknowledge partial financial support from theSpanish
Ministerio de Ciencia e Innovaci´on
ProjectsAYA2010-21766-C03-01 and Consolider-Ingenio 2010CSD2010-00064. RFC thanks financial support from Span-ish CSIC for a
JAE-predoc fellowship. PV thanks financialsupport from the Ram´on y Cajal program. The authorsacknowledge the computer resources, technical expertiseand assistance provided by the
Spanish SupercomputingNetwork (RES) node at Universidad de Cantabria. Weacknowledge the use of
Legacy Archive for MicrowaveBackground Data Analysis (LAMBDA) and the assistanceprovided by Benjamin Gold by e-mail. The HEALPix pack-age was used throughout the data analysis (G´orski et al.2005).
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