The joint large-scale foreground-CMB posteriors of the 3-year WMAP data
H. K. Eriksen, C. Dickinson, J. B. Jewell, A. J. Banday, K. M. Gorski, C. R. Lawrence
aa r X i v : . [ a s t r o - ph ] S e p Draft version October 29, 2018
Preprint typeset using L A TEX style emulateapj v. 08/22/09
THE JOINT LARGE-SCALE FOREGROUND—CMB POSTERIORS OF THE 3-YEAR WMAP DATA
H. K. Eriksen , C. Dickinson , J. B. Jewell , A. J. Banday , K. M. G´orski , C. R. Lawrence (Dated: Received - / Accepted -) Draft version October 29, 2018
ABSTRACTUsing a Gibbs sampling algorithm for joint CMB estimation and component separation, we computethe large-scale CMB and foreground posteriors of the 3-yr WMAP temperature data. Our parametricdata model includes the cosmological CMB signal and instrumental noise, a single power law fore-ground component with free amplitude and spectral index for each pixel, a thermal dust template witha single free overall amplitude, and free monopoles and dipoles at each frequency. This simple modelyields a surprisingly good fit to the data over the full frequency range from 23 to 94 GHz. We obtaina new estimate of the CMB sky signal and power spectrum, and a new foreground model, including ameasurement of the effective spectral index over the high-latitude sky. A particularly significant resultis the detection of a common spurious offset in all frequency bands of ∼ − µ K, as well as a dipolein the V-band data. Correcting for these is essential when determining the effective spectral index ofthe foregrounds. We find that our new foreground model is in good agreement with template-basedmodel presented by the WMAP team, but not with their MEM reconstruction. We believe the lattermay be at least partially compromised by the residual offsets and dipoles in the data. Fortunately, theCMB power spectrum is not significantly affected by these issues, as our new spectrum is in excellentagreement with that published by the WMAP team. The corresponding cosmological parameters arealso virtually unchanged.
Subject headings: cosmic microwave background — cosmology: observations — methods: numerical INTRODUCTION
A major challenge in CMB research is component sepa-ration, which can be summarized in two questions. First,how can we separate reliably the valuable cosmologicalsignal from confusing foreground emission? Second, howcan we propagate accurately the errors induced by thisprocess through to the final analysis products, such asthe CMB power spectrum and cosmological parameters?During the last few years, a new analysis frameworkcapable of addressing these issues in a statistically con-sistent approach has been developed. This frameworkis Bayesian in nature, and depends critically on theGibbs sampling algorithm as its main computational en-gine. The pioneering ideas were described by Jewell et al.(2004) and Wandelt et al. (2004), and later implementedfor modern CMB data sets for temperature and po-larization by Eriksen et al. (2004a) and Larson et al.(2007), respectively. Applications to the 1-yr and 3-yr WMAP data (Bennett et al. 2003a; Hinshaw et al.2007; Page et al. 2007) were described by O’Dwyer et al.(2004), and Eriksen et al. (2006, 2007a). These papersmainly focused on the cosmological CMB signal, andadopted the foreground corrected data provided by theWMAP team.Recently this algorithm was extended to include inter- email: [email protected] Institute of Theoretical Astrophysics, University of Oslo, P.O.Box 1029 Blindern, N-0315 Oslo, Norway Centre of Mathematics for Applications, University of Oslo,P.O. Box 1053 Blindern, N-0316 Oslo Jet Propulsion Laboratory, California Institute of Technology,Pasadena CA 91109 Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild-Str.1, Postfach 1317, D-85741 Garching bei M¨unchen, Germany Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478Warszawa, Poland nal component separation capabilities by Eriksen et al.(2007b). Using very general parameterizations of theforeground components, this method produces the fulljoint and exact foreground-CMB posterior, and there-fore allows us both to estimate each component sepa-rately through marginalized statistics and to propagatethe foreground uncertainties through to the final CMBproducts. The implementation of this algorithm used inthis paper is called “Commander”, and is a direct de-scendant of the code presented by Eriksen et al. (2004a).In this Letter, we apply the method to the 3-yr WMAPtemperature observations (Hinshaw et al. 2007). Thisdata set, with five frequency bands, allows only verylimited foreground models; however, the analysis pro-vides a powerful demonstration of the capabilities of themethod. For a comprehensive analysis of a controlledsimulation with identical properties to this data set, seeEriksen et al. (2007b). DATA
We consider the 3-yr WMAP temperature data, pro-vided on Lambda in the form of sky maps from ten“differencing assemblies” covering the frequency rangebetween 23 and 94 GHz. Since our current implemen-tation of the Gibbs foreground sampler can only handlesky maps with identical beam response (Eriksen et al.2007b), we downgrade each of these maps to a commonresolution of 3 ◦ FWHM and repixelize at a HEALPix resolution of N side = 64, corresponding to a pixel sizeof 55 ′ . These ten maps are then co-added by frequencyinto five single frequency band maps at 23, 33, 41, 64and 94 GHz (K, Ka, Q, V and W-bands, respectively). http://lambda.gsfc.nasa.gov http://healpix.jpl.nasa.gov The power from the instrumental noise is less than 1%of the CMB signal at ℓ = 50 in the V- and W-bands, andless than 2% at ℓ = 100 (Eriksen et al. 2007b). To regu-larize the noise covariance matrix at high spatial frequen-cies, we added 2 µ K per 3 ◦ pixel of uniform white noise.This noise is insignificant at low multipoles, but domi-nates the signal near the spherical harmonic truncationlimit of ℓ max = 150. We then have five frequency maps ata common resolution of 3 ◦ FWHM, with signal-to-noiseratio of unity at ℓ ∼ ℓ max = 150.We choose to include such high l ’s in the analysis fortwo reasons. First, our main goal is an accurate approx-imation of the CMB likelihood at ℓ ≤
50. In order toensure that the degradation process (i.e., smoothing andnoise addition) does not significantly affect these multi-poles, it is necessary to go well beyond ℓ ∼ χ val-ues in the ecliptic plane (Eriksen et al. 2007b), wherethe instrumental noise is higher because of WMAP’sscanning strategy. However, since these high χ val-ues are caused by unmodelled smoothed, random, whitenoise, they do not indicate a short-coming of the signalmodel, but only a slight under-estimation of the statis-tical errors on small angular scales. This has been con-firmed by otherwise identical analyses at both 4 ◦ and6 ◦ FWHM. We present the 3 ◦ FHWM case here as acompromise between angular resolution and accuracy ofthe noise model. This issue will further suppressed withadditional years of WMAP observations and, eventually,high-sensitivity Planck maps.We impose the base WMAP Kp2 sky cut(Bennett et al. 2003b) on the data, but not thepoint source cuts. The base mask is downgraded fromits native N side = 512 resolution to N side = 64 byexcluding all low-resolution pixels for which any one ofits sub-pixels is excluded by the high-resolution mask. Atotal of 42 081 pixels are included, or 85.6% of the sky.The frequency bandpass of each map is modelled as atop-hat function, and implemented in terms of effectivefrequency as a function of spectral index as described byEriksen et al. (2006). The frequency specifications of theWMAP radiometers are given by Jarosik et al. (2003). MODEL AND METHODS
We adopt the following simple parametric model T ν ( p )for the observed signal (measured in thermodynamictemperatures) at frequency ν and pixel p , T ν ( p ) = s ( p ) + m ν + X i =1 m iν [ˆ e i · ˆ n ( p )] ++ b " t ( p ) a ( ν ) (cid:18) νν dust0 (cid:19) . + f ( p ) a ( ν ) (cid:18) νν (cid:19) β ( p ) . (1)The first term is the cosmic CMB signal, character-ized by a frequency-independent spectrum and a co-variance matrix in spherical harmonic space given bythe power spectrum, h a ∗ ℓm a ℓ ′ m ′ i = C ℓ δ ℓℓ ′ δ mm ′ , where s ( p ) = P ℓ,m a ℓ Y ℓm ( p ). The second and third termsdenote a free monopole and three dipole amplitudes at each frequency. We use the standard Cartesian basisvectors projected on the sky, { , x , y , z } , as basis func-tions for these four modes. The fourth term representsa template-based dust model scaled by a spectral indexof β = 1 .
7, in which t ( p ) is the dust template (FDS) ofFinkbeiner et al. (1999), evaluated at ν dust0 = 94 GHz,and a ( t ) is the conversion factor between antenna andthermodynamic temperatures. The last term is a sin-gle foreground component with a free amplitude f ( p )and spectral index β ( p ) at each pixel. The reference fre-quency for this component is ν = 23 GHz.The free parameters in this model are: 1) spherical har-monic coefficients a ℓm of the CMB amplitude s ; 2) CMBpower spectrum coefficients C ℓ ; 3) monopole and dipoleamplitudes at each band; 4) the amplitude of the dusttemplate; and 5) amplitudes and spectral indices of thepixel foreground component.The WMAP data do not have sufficient power to con-strain this simple completely by themselves, as there is avery strong degeneracy between the foreground compo-nent amplitudes at each pixel and the free monopole anddipole coefficients at each band (Eriksen et al. 2007b).For this reason, we introduce two priors in addition tothe Jeffereys’ ignorance prior discussed by Eriksen et al.(2007b). First, we impose a Gaussian prior on the spec-tral indices of β ∼ − ± .
3: A direct fit of the 408MHz template (Haslam et al. 1981) to the WMAP K-band data for Kp2 sky coverage implies an index of -3(Davies et al. 2006), and Davies et al. (1996) determineda typical range for high latitude spectral indices between408 MHz and 1420 MHz of -2.8 to -3.2. Note that thisprior has a noticeable effect only at high Galactic lati-tudes, where the absolute foreground amplitude is low.At low galactic latitudes, the data dominate the priorby up to a factor of ∼
50, and any potential bias in thenear-plane free-free regions is negligible.Second, we impose an implicit spectral index orthogo-nality prior on the monopole and dipole coefficients, asdescribed by Eriksen et al. (2007b), projecting out thefrequency component of these coefficients that matchesthe free spectral index map, thus effectively determiningthe zero-level of the foreground amplitude map. We alsotried an alternative approach, first estimating the Q, V,and W-band monopole and dipole coefficients separatelygiven a crude estimate of the spectral index map, andthen estimating all other parameters given these coeffi-cients. Results were very similar. Thus, the two priorsadopted in this analysis have a very weak effect on allmain results.Having defined our model and priors, we map out thejoint posterior distribution using the foreground Gibbssampler described by Eriksen et al. (2007b). We referthe interested reader to that paper for full details of thealgorithm, and for a comprehensive analysis of a realisticsimulation corresponding to the same data and modelused in this Letter.Finally, we estimate a new set of cosmological parame-ters within the standard ΛCDM model. For this analysis,we follow the approach of Eriksen et al. (2007a), and re-place the low- ℓ part of the WMAP likelihood with a newBlackwell-Rao Gibbs-based estimator (Chu et al. 2005).No ancillary data sets beyond the 3-yr WMAP temper-ature and polarization data are included in the analysis. Fig. 1.—
Marginal posterior mean maps in Galactic coordinates.Rows from top to bottom show the CMB reconstruction, the fore-ground amplitude, and the foreground spectral index, respectively.
The CosmoMC code (Lewis & Bridle 2002) is used as themain MCMC engine. RESULTS
Figure 1 shows the marginal posterior mean maps forthe CMB sky signal, the foreground amplitude and theforeground spectral index. Table 1 gives the correspond-ing results for the monopole and dipole coefficients foreach frequency band. The FDS dust template ampli-tude relative to 94 GHz and an assumed spectral indexof β = 1 . b = 0 . ± . χ computed for each Gibbssample. A χ value exceeding χ = 15 corresponds torejection of the model in that pixel at 99% statisticalsignificance. Two features are clearly visible in this plot.First, the ecliptic plane, or rather, WMAP’s scanningstrategy, is clearly visible, and this is mainly due to theunmodelled smoothed noise component at high ℓ ’s, asdiscussed in Section 2.Second, there are clearly visible structures near theGalactic plane, and in particular around regions withknown high free-free emission (e.g., the Orion and Ophi-ucus regions). This is likely due to the fact that a singlepower-law is not a good approximation to the sum of TABLE 1Monopole and dipole posterior statistics
Monopole Dipole X Dipole Y Dipole ZBand ( µ K) ( µ K) ( µ K) ( µ K)K-band − . ± . . ± . − . ± . . ± . − . ± . . ± . . ± . − . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . Note . — Means and standard deviations of themarginal monopole and dipole posteriors.
Fig. 2.—
Mean χ map computed over posterior samples inGalactic coordinates. A value of χ = 15 is high at the 99% sig-nificance level. many foregrounds with comparable amplitudes. Thus,we have clear evidence of modelling errors in this so-lution, and we therefore strongly emphasize that thequoted error bars presented in this paper include for-mal statistical errors only, and not systematic, model-dependent uncertainties. These issues are discussed indepth by Eriksen et al. (2007b), who find similar be-haviour for a controlled, simulated data set.Although χ is high, the CMB solution obtained is ev-idently good. First of all, in the ecliptic poles, where theWMAP instrumental noise is suppressed by the scanningstrategy, the χ distribution is essentially perfect. Thisimplies that the signal model as such is adequate at highlatitudes. Second, the CMB map is virtually withoutsignatures of residual foregrounds. (Note that the signalinside the Galactic plane is partially reconstructed usinghigh-latitude information and the assumption of isotropy.The signal on scales smaller than the mask size is washedout because it is not possible to predict these from higherlatitudes.) Finally, the foreground amplitude and spec-tral index maps correlate very well with known templatesof synchrotron and free-free emission. For instance, thespectral index near the Gum nebula and near the Velaregions are close to β = − .
1, as expected for free-freeemission.The single most surprising aspects of the solution arethe monopole and, possibly, the V-band dipole coeffi-cients, listed in Table 1. Most notably, there is a strongdetection of a roughly 13 µ K offset common to all fre-quency bands. Formally speaking, these offset valuesare only optimal within the current model; however, thistype of signal is not degenerate with any other compo-nent in the model. Further, we have attempted to fitseveral other models assuming no offsets at one or more
Fig. 3.—
Difference maps of the total “Commander” W-bandforeground model with the WMAP MEM model (top panel) andwith the WMAP template fit model (bottom panel). bands. These all result in strong, visible residuals inthe CMB map, and considerably higher χ values over-all. Finally, very similar results have been obtained byother researchers through other methods, although, toour knowledge, these results have not yet been publishedin the literature. We therefore believe that the monopoleand dipole coefficients presented here are more optimaleven in an absolute sense than those obtained by WMAPbased on a cosecant fit to a plane parallel galaxy model(Bennett et al. 2003b).Figure 3 shows the difference maps between theCommander W-band foreground model and the MEMand template-based foreground models of Hinshaw et al.(2007). Clearly, our foreground model agrees surpris-ingly well with the simple template fits, but not with theMEM solution. One possible explanation for this is thatalthough the MEM approach of Hinshaw et al. (2007)does attempt to estimate spectral indices for each pixel,it does not include monopole or dipole components in itsmodel. Therefore, the MEM solution is plausibly com-promised by the non-zero offset detected here, at leastin part. In addition, the MEM method could be biasedby the initial subtraction of the (foreground contami-nated) ILC estimate of the CMB anisotropy from thefrequency maps, and the use of the 408 MHz data asa prior. Conversely, Commander could be compromisedto some extent by the use of power law spectral indicesfor the combined low frequency foreground component.Nevertheless, the difference is surprising given that theW-band foreground is expected to be mostly comprisedof thermal dust emission.The marginal maximum posterior CMB power spec-trum is shown in Figure 4, together with the maximum-likelihood/ pseudo- C ℓ hybrid spectrum computed by the See discussion lead by P. Leahy athttp://cosmocoffee.info/viewtopic.php?t=631. l C l l ( l + ) / π ( µ K ) Best-fit Λ CDMWMAPCommander
Fig. 4.—
The CMB temperature power spectra obtained in thispaper (red) and that by the WMAP team (black). The best-fit ΛCDM model spectrum of Spergel et al. (2007) is shown as adashed line.
WMAP team. Perhaps the most notable difference isin the ℓ = 21 multipole, which looks anomalous in theWMAP spectrum, as noted by other authors.The cosmological parameters corresponding to theCommander spectrum for a standard six-parameterΛCDM model are Ω b h = 0 . ± . m = 0 . ± . A s ) = 3 . ± . h = 0 . ± . τ = 0 . ± . b h = 0 . ± . m = 0 . ± . A s ) = 3 . ± . h = 0 . ± .
032 and τ = 0 . ± . CONCLUSIONS
We have presented the first exact Bayesian jointforeground-CMB analysis of the 3-yr WMAP data. Wehave established a new estimate of both the CMB skysignal and the power spectrum, a detailed foregroundmodel consisting of a foreground amplitude and spectralindex map and a dust template amplitude, and also pro-vided new estimates of the residual monopole and dipolecoefficients in the WMAP data.The detection of significant non-zero offsets in theWMAP data is the new result of the greatest immedi-ate importance for the CMB community. These newmonopole and dipole estimates could have a significantimpact on several previously published results, espe-cially those concerning the foreground composition inthe WMAP data. For example, our foreground modelis in excellent agreement with the simple template fitspresented by Hinshaw et al. (2007), but not with theirMEM reconstruction.Taking a longer perspective, the most important as-pect of this analysis is a demonstration of feasibility ofexact and joint foreground-CMB analysis. This will beessential for Planck, whose high sensitivity and angularresolution demand more accurate foreground separationthan WMAP. Considering the flexibility, power, and ac-curacy of the method employed in this paper, togetherwith its unique capabilities for propagating uncertaintiesaccurately all the way from the postulated foregroundmodel to cosmological parameters, we believe that thisshould be the baseline analysis strategy for Planck onlarge angular scales, say ℓ . ∼ hke.We acknowledge use of the HEALPix software (G´orski et al. 2005) and analysis package for derivingthe results in this paper. We acknowledge use of theLegacy Archive for Microwave Background Data Analy-sis (LAMBDA). This work was partially performed at theJet Propulsion Laboratory, California Institute of Tech-nology, under a contract with the National Aeronauticsand Space Administration. HKE acknowledges financialsupport from the Research Council of Norway.hke.We acknowledge use of the HEALPix software (G´orski et al. 2005) and analysis package for derivingthe results in this paper. We acknowledge use of theLegacy Archive for Microwave Background Data Analy-sis (LAMBDA). This work was partially performed at theJet Propulsion Laboratory, California Institute of Tech-nology, under a contract with the National Aeronauticsand Space Administration. HKE acknowledges financialsupport from the Research Council of Norway.