Planck 2015 results X. Diffuse component separation: Foreground maps

A&A 594, A10 (2016) DOI: 10.1051/0004-6361/201525967 c ESO 2016 Astronomy & Astrophysics Planck 2015 results Special feature Planck 2015 results X. Diffuse component separation: Foreground maps Planck Collaboration: R. Adam82, P. A. R. Ade97, N. Aghanim65, M. I. R. Alves107,11,65, M. Arnaud80, M. Ashdown76,6, J. Aumont65, C. Baccigalupi95, A. J. Banday107,11, R. B. Barreiro71, J. G. Bartlett1,73, N. Bartolo33,72, E. Battaner109,110, K. Benabed66,106, A. Benoît63, A. Benoit-Lévy27,66,106, J.-P. Bernard107,11, M. Bersanelli36,53, P. Bielewicz90,11,95, J. J. Bock73,13, A. Bonaldi74, L. Bonavera71, J. R. Bond10, J. Borrill16,101, F. R. Bouchet66,99, F. Boulanger65, M. Bucher1, C. Burigana52,34,54, R. C. Butler52, E. Calabrese103, J.-F. Cardoso81,1,66, A. Catalano82,79, A. Challinor68,76,14, A. Chamballu80,18,65, R.-R. Chary62, H. C. Chiang30,7, P. R. Christensen91,39, D. L. Clements61, S. Colombi66,106, L. P. L. Colombo26,73, C. Combet82, F. Couchot78, A. Coulais79, B. P. Crill73,13, A. Curto71,6,76, F. Cuttaia52, L. Danese95, R. D. Davies74, R. J. Davis74, P. de Bernardis35, A. de Rosa52, G. de Zotti49,95, J. Delabrouille1, F.-X. Désert59, C. Dickinson74, J. M. Diego71, H. Dole65,64, S. Donzelli53, O. Doré73,13, M. Douspis65, A. Ducout66,61, X. Dupac42, G. Efstathiou68, F. Elsner27,66,106, T. A. Enßlin86, H. K. Eriksen69, E. Falgarone79, J. Fergusson14, F. Finelli52,54, O. Forni107,11, M. Frailis51, A. A. Fraisse30, E. Franceschi52, A. Frejsel91, S. Galeotta51, S. Galli75, K. Ganga1, T. Ghosh65, M. Giard107,11, Y. Giraud-Héraud1, E. Gjerløw69, J. González-Nuevo22,71, K. M. Górski73,112, S. Gratton76,68, A. Gregorio37,51,58, A. Gruppuso52, J. E. Gudmundsson104,93,30, F. K. Hansen69, D. Hanson88,73,10, D. L. Harrison68,76, G. Helou13, S. Henrot-Versillé78, C. Hernández-Monteagudo15,86, D. Herranz71, S. R. Hildebrandt73,13, E. Hivon66,106, M. Hobson6, W. A. Holmes73, A. Hornstrup19, W. Hovest86, K. M. Huffenberger28, G. Hurier65, A. H. Jaffe61, T. R. Jaffe107,11, W. C. Jones30, M. Juvela29, E. Keihänen29, R. Keskitalo16, T. S. Kisner84, R. Kneissl41,8, J. Knoche86, M. Kunz20,65,3, H. Kurki-Suonio29,47, G. Lagache5,65, A. Lähteenmäki2,47, J.-M. Lamarre79, A. Lasenby6,76, M. Lattanzi34,55, C. R. Lawrence73, M. Le Jeune1, J. P. Leahy74, R. Leonardi9, J. Lesgourgues67,105, F. Levrier79, M. Liguori33,72, P. B. Lilje69, M. Linden-Vørnle19, M. López-Caniego42,71, P. M. Lubin31, J. F. Macías-Pérez82, G. Maggio51, D. Maino36,53, N. Mandolesi52,34, A. Mangilli65,78, M. Maris51, D. J. Marshall80, P. G. Martin10, E. Martínez-González71, S. Masi35, S. Matarrese33,72,44, P. McGehee62, P. R. Meinhold31, A. Melchiorri35,56, L. Mendes42, A. Mennella36,53, M. Migliaccio68,76, S. Mitra60,73, M.-A. Miville-Deschênes65,10, A. Moneti66, L. Montier107,11, G. Morgante52, D. Mortlock61, A. Moss98, D. Munshi97, J. A. Murphy89, P. Naselsky92,40, F. Nati30, P. Natoli34,4,55, C. B. Netterfield23, H. U. Nørgaard-Nielsen19, F. Noviello74, D. Novikov85, I. Novikov91,85, E. Orlando111, C. A. Oxborrow19, F. Paci95, L. Pagano35,56, F. Pajot65, R. Paladini62, D. Paoletti52,54, B. Partridge46, F. Pasian51, G. Patanchon1, T. J. Pearson13,62, O. Perdereau78, L. Perotto82, F. Perrotta95, V. Pettorino45, F. Piacentini35, M. Piat1, E. Pierpaoli26, D. Pietrobon73, S. Plaszczynski78, E. Pointecouteau107,11, G. Polenta4,50, G. W. Pratt80, G. Prézeau13,73, S. Prunet66,106, J.-L. Puget65, J. P. Rachen24,86, W. T. Reach108, R. Rebolo70,17,21, M. Reinecke86, M. Remazeilles74,65,1, C. Renault82, A. Renzi38,57, I. Ristorcelli107,11, G. Rocha73,13, C. Rosset1, M. Rossetti36,53, G. Roudier1,79,73, J. A. Rubiño-Martín70,21, B. Rusholme62, M. Sandri52, D. Santos82, M. Savelainen29,47, G. Savini94, D. Scott25, M. D. Seiffert73,13, E. P. S. Shellard14, L. D. Spencer97, V. Stolyarov6,102,77, R. Stompor1, A. W. Strong87, R. Sudiwala97, R. Sunyaev86,100, D. Sutton68,76, A.-S. Suur-Uski29,47, J.-F. Sygnet66, J. A. Tauber43, L. Terenzi96,52, L. Toffolatti22,71,52, M. Tomasi36,53, M. Tristram78, M. Tucci20, J. Tuovinen12, G. Umana48, L. Valenziano52, J. Valiviita29,47, F. Van Tent83, P. Vielva71, F. Villa52, L. A. Wade73, B. D. Wandelt66,106,32, I. K. Wehus73,69,?, A. Wilkinson74, D. Yvon18, A. Zacchei51, and A. Zonca31 (Affiliations can be found after the references) Received 26 February 2015 /Accepted 22 February 2016 ABSTRACT Planck has mapped the microwave sky in temperature over nine frequency bands between 30 and 857 GHz and in polarization over seven frequency bands between 30 and 353 GHz in polarization. In this paper we consider the problem of diffuse astrophysical component separation, and process these maps within a Bayesian framework to derive an internally consistent set of full-sky astrophysical component maps. Component separation dedicated to cosmic microwave background (CMB) reconstruction is described in a companion paper. For the temperature analysis, we combine the Planck observations with the 9-yr Wilkinson Microwave Anisotropy Probe (WMAP) sky maps and the Haslam et al. 408 MHz map, to derive a joint model of CMB, synchrotron, free-free, spinning dust, CO, line emission in the 94 and 100 GHz channels, and thermal dust emission. Full-sky maps are provided for each component, with an angular resolution varying between 7. 05 and 1◦. Global parameters (monopoles, dipoles, relative calibration, and bandpass errors) are fitted jointly with the sky model, and best-fit values are tabulated. For polarization, the model includes CMB, synchrotron, and thermal dust emission. These models provide excellent fits to the observed data, with rms temperature residuals smaller than 4µK over 93% of the sky for all Planck frequencies up to 353 GHz, and fractional errors smaller than 1% in the remaining 7% of the sky. The main limitations of the temperature model at the lower frequencies are internal degeneracies among the spinning dust, free-free, and synchrotron components; additional observations from external low-frequency experiments will be essential to break these degeneracies. The main limitations of the temperature model at the higher frequencies are uncertainties in the 545 and 857 GHz calibration and zero-points. For polarization, the main outstanding issues are instrumental systematics in the 100–353 GHz bands on large angular scales in the form of temperature-to-polarization leakage, uncertainties in the analogue-to-digital conversion, and corrections for the very long time constant of the bolometer detectors, all of which are expected to improve in the near future. Key words. ISM: general – cosmology: observations – polarization – cosmic background radiation – diffuse radiation – Galaxy: general ?Corresponding author: I. K. Wehus; [email protected] Article published by EDP Sciences A10, page 1 of 63 A&A 594, A10 (2016) 1. Introduction This paper, one of a set associated with the 2015 release of data from the Planck1mission (Planck Collaboration I 2016), presents a coherent astrophysical model of the microwave sky in both temperature and polarization, as derived from the most recent Planck observations. For temperature, the analysis also incorporates the 9-yr WMAP observations (Bennett et al. 2013) and a 408 MHz survey (Haslam et al. 1982), allowing the separation of synchrotron, free-free, and spinning dust emission. In March 2013, the Planck Consortium released its first temperature measurements of the microwave sky, summarized in terms of nine frequency maps between 30 and 857 GHz (Planck Collaboration I 2014). The richness of these data has enabled great progress in our understanding of the astrophysical composition of the microwave sky. The current Planck data release presents additionally high-sensitivity, full-sky maps of the polarized microwave sky, offering a fresh view on both cosmological and astrophysical phenomena. With increased data volume and quality comes both greater scientific potential and more stringent requirements on model complexity and sophistication. The current Planck data release is more ambitious than the 2013 release in terms of component separation efforts, accounting for more astrophysical effects and components. In this round, three related papers summarize the Planck 2015 component separation products and approaches. First, cosmic microwave background (CMB) reconstruction and extraction are discussed in Planck Collaboration IX (2016). Second, this paper presents the diffuse astrophysical foreground products derived from the 2015 Planck observations, both in temperature and polarization. Third, Planck Collaboration XXV (2016) discusses the scientific interpretation of the new low-frequency Planck foreground products. The main goal of the current paper is to establish a single, internally coherent and global parametric model of the microwave sky, simultaneously accounting for all significant diffuse astrophysical components and relevant instrumental effects using the Bayesian Commander analysis framework (Eriksen et al. 2004, 2006,2008). As such, our discussion does not focus on any single emission component, but rather emphasize the global picture. In the 2013 data release, the same framework was applied to the Planck temperature measurements for frequencies between 30 and 353 GHz, considering only angular scales larger than 400full-width half-maximum (FWHM). This resulted in lowresolution CMB, CO, and thermal dust emission maps, as well as a single low-frequency foreground component combining contributions from synchrotron, free-free, and spinning dust emission (Planck Collaboration XII 2014). Here we extend that analysis in multiple directions. First, instead of 15.5 months of temperature data, the new analysis includes the full Planck mission data, 50 months of Low Frequency Instrument (LFI) and 29 months of High Frequency Instrument (HFI) data, in both temperature and polarization. Second, we now also include the 9-yr WMAP observations between 23 and 94 GHz and a 408 MHz survey map, providing enough frequency constraints to decompose the lowfrequency foregrounds into separate synchrotron, free-free, and spinning dust components. Third, we now include the Planck 1Planck (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states and led by Principal Investigators from France and Italy, telescope reflectors provided through a collaboration between ESA and a scientific consortium led and funded by Denmark, and additional contributions from NASA (USA). 545 and 857 GHz frequency bands, allowing us to constrain the thermal dust temperature and emissivity index with greater precision, thereby reducing degeneracies between CMB, CO, and free-free emission. At the same time, we find that the calibration and bandpass measurements of these two channels represent two of the most important sources of systematic uncertainty in the analysis. Fourth, the present analysis implements a multiresolution strategy to provide component maps at high angular resolution. Specifically, the CMB is recovered with angular resolution 50FWHM (Planck Collaboration IX 2016), thermal dust emission and CO J=2→1 lines are recovered at 7. 05 FWHM, and synchrotron, free-free, and spinning dust are recovered at 1◦FWHM. The resulting parameter fits define the Planck 2015 baseline astrophysical model in temperature and polarization. We emphasize, however, that these models are not unique, but instead represent minimal physically well-motivated models that are able to reproduce the current data. As in the 2013 data release, the CMB solutions derived, using this Bayesian approach, form the basis of the Planck 2015 CMB temperature likelihood on large angular scales. This is described in detail in Planck Collaboration XI (2016), which also presents a detailed characterization of the low-multipole CMB angular power spectrum. The low-frequency astrophysical model presented here is used as input for the temperature-topolarization bandpass mismatch corrections for the LFI polarization maps (Planck Collaboration II 2016). The paper is organized as follows. Section 2gives an overview of the computational framework implemented in the Commander code. Section 3describes the data selection and processing. Section 4, gives an overview of the relevant astrophysical components and systematic effects. Sections 5and 6give the main temperature and polarization products. We summarize in Sect. 7. 2. Algorithms 2.1. Data, posterior distribution and priors Most of the results derived in this paper are established within a standard Bayesian analysis framework, as implemented in the Commander code, in which an explicit parametric model, s(θ), is fitted to a set of observations, d, either by maximizing or mapping out the corresponding posterior distribution, P(θ|d)=P(d|θ)P(θ) P(d)∝ L(θ)P(θ).(1) Here θdenotes some general set of free parameters in the model, L(θ)=P(d|θ) is the likelihood, and P(θ) denotes a set of priors on θ. The evidence, P(d), is a constant with respect to the parameter set, and is neglected in the following. The data are defined by a set of pixelized frequency-channel sky maps, d={dν}, comprising the three Stokes parameters I,Q and U. In this paper, however, we analyse temperature and polarization separately; therefore the data vector comprises either Ior {Q,U}. We start by assuming that the data at a given frequency ν may be described as a linear sum of signal sνand noise nν, dν=sν+nν,(2) where nνis assumed to be Gaussian-distributed with a known covariance matrix Nν. For the signal, we adopt the following parametric expression: sν(θ)=sν(ai, βi, gν,mν,∆ν) (3) =gν Ncomp X i=1 Fi ν(βi,∆ν)ai+Tνmν,(4) A10, page 2 of 63 Planck Collaboration: Planck 2015 results. X. where aiis an amplitude map for component iat a given reference frequency, βiis a general set of spectral parameters for the same component, gνis a multiplicative calibration factor for frequency ν,∆νis a linear shift in the bandpass central frequency, and mνis a set of template correction amplitudes, such as monopole, dipole, or zodiacal light corrections for temperature, or calibration leakage templates for polarization. The corresponding spatially fixed templates are organized column-wise in a template matrix Tν. The mixing matrix, Fi ν(βi,∆ν), accounts for the effect of spectral changes as a function of frequency for component i, parametrized by βi, as well as bandpass integration effects and unit conversions. For numerical stability, all internal calculations are performed in units of brightness temperature, and ais therefore naturally defined in the same units at some specified reference frequency. The posterior distribution takes the usual form, P(θ|dν)=P(dν|ai, βi, gν,mν,∆ν,C`)P(ai, βi, gν,mν,∆ν,C`) (5) =L(ai, βi, gν,mν,∆ν)P(ai)P(βi)P(acmb|C`), where we have included the CMB power spectrum, C`, and also implicitly adopted uniform priors on gν,∆ν,mν, and C`. Because the noise is assumed to be Gaussian and independent between frequency channels, the likelihood reads L(ai, βi, gν,mν,∆ν)∝exp −1 2X ν [dν−sν(θ)]TN−1[dν−sν(θ)] (6) Likewise, we further assume the CMB signal to be Gaussian distributed with a covariance matrix, S(C`), given by the power spectrum, and the corresponding CMB prior factor therefore reads P(acmb|C`)=e−1 2aT cmbS−1(C`)acmb √|S(C`)|·(7) The only undefined factors in the posterior are the amplitude and spectral parameter priors, P(ai) and P(βi). These represent the most difficult problem to handle from a conceptual point of view, since the prior is to some extent a matter of personal preference. However, we adopt the following general practices in this paper. First, for low-resolution analyses that include fitting of template amplitudes (e.g., monopoles and dipoles), we always impose a strict positivity prior, i.e., ai>0, on all signal amplitudes except the CMB. Without such a prior, there are large degeneracies between the zero-points of the amplitude maps and the individual template amplitudes. Second, for the high angular resolution analysis, we fix the template amplitudes at the low-resolution values and disable the positivity prior, in order to avoid noise bias. Third, to further break degeneracies, we adopt fiducial values for the monopole, dipole, and calibration factors for a few selected channels, effectively imposing a set of external priors from CMB dipole measurements and H icrosscorrelation (Planck Collaboration VIII 2014) to anchor the full solution. Fourth, for the spectral parameters, we adopt Gaussian priors with means and variances informed by the high signal-tonoise values observed in the Galactic plane, which for all practical purposes are independent of the adopted priors. Intuitively, we demand that a map of the spectral parameter in question should not be much different in the data-dominated and the priordominated regions of the sky. Fourth, one of the components in the temperature model is free-free emission, which has two free parameters, namely the effective emission measure, EM, and the electron temperature, Te. The latter of these is very poorly constrained with the current data set except in the central Galactic plane, and we therefore adopt a smoothness prior on this paper to increase the effective signal-to-noise ratio, demanding that is must be smooth on 2◦FWHM scales. This in turn has a large computational cost by making the overall foreground parameter estimation process non-local, and Teis therefore only varied in fast maximum-likelihood searches, not in expensive sampling analyses. Its effect on other parameters is, however, minimal, precisely because of its low signal-to-noise ratio. Finally, in addition to these informative priors, we adopt a Jeffreys prior for the spectral parameters in order to suppress prior volume effects (Jeffreys 1946;Eriksen et al. 2008;Dunkley et al. 2009). 2.2. Gibbs sampling and posterior maximization As described above, the posterior distribution contains many millions of free (both non-Gaussian and strongly correlated) parameters for Planck – ranging from 11 million in the following low-resolution analysis to 200 million in the corresponding highresolution analysis – and mapping out this distribution poses a significant computational problem. Indeed, no direct sampling algorithm exists for the full distribution, and the only computationally efficient solution currently known is that of Gibbs sampling, a well-known textbook algorithm in modern statistical analysis (e.g., Gelman et al. 2003). The underlying idea of this method is that samples from a complicated multivariate distribution may be drawn by iteratively sampling over the corresponding conditional distributions, which usually have much simpler, and often analytic, sampling algorithms. This framework was originally introduced to the CMB analysis field by Jewell et al. (2004) and Wandelt et al. (2004), and subsequently developed into a fully functional computer code called Commander by Eriksen et al. (2004,2008). For the problem in question in this paper, this algorithm may be schematically translated into an explicit set of sampling steps through the following Gibbs chain: ai←P(ai|βi, gν,mν,∆ν,C`) (8) βi←P(βi|ai, gν,mν,∆ν,C`) (9) gν←P(gν|ai, βi,mν,∆ν,C`) (10) mν←P(mν|ai, βi, gν,∆ν,C`) (11) ∆ν←P(∆ν|ai, βi, gν,mν,C`) (12) C`←P(C`|ai, βi, gν,mν,∆ν).(13) Here “←” denotes drawing a sample from the distribution on the right-hand side. After some burn-in period, the theory of Gibbs sampling guarantees that the joint set of parameters is indeed drawn from the correct joint distribution. For a full description of the various steps in the algorithm, see Eriksen et al. (2008). While no fully functional alternatives to Gibbs sampling have been established for this full joint distribution to date, Gibbs sampling alone by no means solves all computational problems. In particular, this algorithm is notorious for its slow convergence for nearly degenerate parameters, since it by construction only moves through parameter space parallel to coordinate axes. For this reason, we implement an additional posterior maximization phase, in which we search directly for the posterior maximum point rather than attempt to sample from the full distribution. The resulting solution may then serve either as a final product in its own right, by virtue of being a maximum-posterior estimate, or as the starting position for a regular Gibbs sampling analysis. The crucial point, though, is that special-purpose nonlinear A10, page 3 of 63 A&A 594, A10 (2016) search algorithms can be used in this phase, moving in arbitrary directions through parameter space, and individual optimization combinations may be introduced to jointly probe directions with particularly strong degeneracies. Perhaps the single most important example in this respect is the parameter combination between component amplitudes, detector calibrations, and bandpass uncertainties, {ai, gν,∆ν}, all three of which essentially correspond to scaling parameters. However, since both gνand aiare conditionally linear parameters, and only ∆νis truly nonlinear, it is possible to solve analytically for gνor ai,conditioning on any given fixed value of ∆ν. Consequently, one can set up a nonlinear Powell-type search (Press et al. 2002) for ∆ν, in which the optimal values of either gνor aiare quickly computed at each iteration in the search. A second example is the electron temperature discussed above, for which non-local optimization is feasible, whereas a full-blown sampling algorithm is too expensive to converge robustly. In this situation, fixing the parameter at its maximum-posterior value is vastly preferable compared to adding an unconverged degree of freedom in the full sampler. Even with this optimization phase, however, there is always an inherent danger of the algorithm being trapped in a local posterior maximum. Indeed, with a distribution involving millions of highly correlated parameters, it is exceedingly difficult to prove that the derived solution is the true global posterior maximum. As a partial solution to this problem, we initialize the search using different starting positions, and carefully monitor the convergence properties of the chains. 3. Data selection and processing The primary data used in this paper are the 2015 Planck temperature and polarization sky maps (Planck Collaboration VI 2016;Planck Collaboration VIII 2016). For the temperature analysis we additionally include the 9-yr WMAP observations2(Bennett et al. 2013) and a full-sky 408 MHz survey map (Haslam et al. 1982), with the goal of individually resolving synchrotron, free-free, and spinning dust emission. For WMAP, we adopt the beam-symmetrized frequency maps for the foreground-dominated Kand Ka-bands, to mitigate beam artifacts around compact sources, but we use the standard maps for the CMB-dominated Q-, V-, and W-bands, because of their more accurate noise description. At the lowest frequency, we adopt the destriped version of the 408 MHz survey map recently published by Remazeilles et al. (2015). In order to maximize our leverage with respect to bandpass measurement uncertainties and line emission mechanisms, we employ individual detector and detector set (“ds”) maps for all Planck frequencies between 70 and 857 GHz, and differencing assembly (“DA”) maps for WMAP. However, the polarization analysis employs frequency maps in order to maximize signalto-noise ratio and to minimize correlated noise from destriper mapmaking uncertainties. Intensity-to-polarization leakage from bandpass mismatch between detectors is suppressed through the use of precomputed leakage templates (Planck Collaboration II 2016;Planck Collaboration VIII 2016). For LFI, these templates are based on a preliminary version of the foreground products presented in this paper. The full set of clean channels used in this analysis is summarized in Table 1in terms of centre frequencies, resolution, and noise levels. In total, 32 individual detector and detector set maps3and frequency maps are included in the 2http://lambda.gsfc.nasa.gov 3For uniformity, we refer to the 70 GHz horn pair maps as “detector set” maps in this paper, with the {ds1, ds2, ds3} maps corresponding to horns {18+23, 19+22, 20+21}, respectively. 30 100 300 Frequency [GHz] 0.1 1 Mean brightness temperature [µKRJ] 30 44 70 K Ka Q V W HFI measurement LFI band WMAP band Best-fit power law Fig. 1. Zodiacal light extrapolation from HFI to LFI and WMAP frequency channels in terms of full-sky mean brightness temperature. The dotted line shows the power-law fit to the HFI observations between 100 and 353 GHz, s(ν)=0.70 µKRJ (ν/100 GHz)1.31, and the vertical grey lines indicate the central frequencies of the LFI and WMAP frequency bands. temperature analysis, and seven frequency maps in the polarization analysis. For the Planck HFI channels, a model of zodiacal light emission is subtracted from the time-ordered data prior to mapmaking (Planck Collaboration VIII 2016). In addition, in this paper we apply a small correction to the low-frequency LFI and WMAP channels by scaling the effective HFI 100 GHz zodiacal light correction map (i.e., uncorrected minus corrected map) to each frequency according to a power law fitted to frequencies between 100 and 353 GHz (Planck Collaboration XIV 2014), as illustrated in Fig. 1; the actual template amplitudes relative to the 100 GHz correction map (in thermodynamic units) are listed in Table 2. Although the magnitude of this correction is small, with a maximum amplitude of 2 µK in the 70 GHz map, applying no correction at all below 100 GHz results in a visually noticeable bias in the derived CO J=1→0 map at high Galactic latitudes, in the characteristic form of the zodiacal light. Extending the zodiacal light model to low frequencies efficiently eliminates this structure. In our 2013 release, colour corrections and unit conversions for all Planck channels were based on individual bandpass profiles as measured on the ground before launch (Planck Collaboration V 2014;Planck Collaboration IX 2014). However, as discussed in detail in Sects. 2,4.3, and 5, during the component separation process we find that systematic uncertainties in the nominal bandpasses induce significant residuals between data and model, and it is necessary to fit for these bandpass uncertainties in order to obtain statistically acceptable fits. For WMAP, we adopt the nominal bandpasses4for the first DA within each frequency band, and fit for the remaining DA bandpasses within each frequency. For the 408 MHz survey, we adopt a delta function response at the nominal frequency. The unit conversion factors between thermodynamic and brightness 4As described by Bennett et al. (2013), the WMAP bandpasses evolved during the 9 yr of WMAP observations, resulting in slightly lower effective full-mission frequencies as compared to the nominal bandpasses. We correct for these small shifts in the present analysis by shifting the bandpasses accordingly. A10, page 4 of 63 Planck Collaboration: Planck 2015 results. X. χ2 101103105 217-ds1 -5 5 µKcmb 217-ds2 -5 5 µKcmb 353-ds1 -20 20 µKcmb 353-2 -20 20 µKcmb 353-7 -20 20 µKcmb 353-8 -20 20 µKcmb 545-1 -0.03 0.03 MJy sr−1 857-1 -0.05 0.05 MJy sr−1 857-3 -0.05 0.05 MJy sr−1 857-4 -0.05 0.05 MJy sr−1 Fig. 2. χ2(top) and residual maps, dν−sν(bottom), for a Commander analysis that includes all Planck channel maps. These residual maps correspond to channels that are rejected from the baseline analysis due to instrumental systematics; the labels in the top left corner in each panel indicates frequency channels. No regularization noise has been added to the high-frequency channels in this case. The sharp ring-like features at high Galactic latitudes correspond to far sidelobe residuals; the broad features extending between the north and south ecliptic poles correspond to destriping errors; and the Galactic plane features correspond to calibration, bandpass, and modelling residuals. All maps (and all other full-sky plots later in the paper) are shown in Galactic coordinates adopting a Mollweide projection with the Galactic centre (l=0◦) located in the centre of each panel. A10, page 5 of 63 A&A 594, A10 (2016) Table 1. Overview of data sets. Frequency Detector Noise rms Min smooth Instrument [GHz] label Resolution σT(1◦) or σP(400) scale Units Reference Temperature Planck LFI . . . . . . . . . . 30 (all) 32. 04 2.8 400µKPlanck Collaboration VI (2016) 44 (all) 27. 01 3.0 400µK 70 ds1 13. 06 3.8 400µK ds2 13. 03 4.0 400µK ds3 13. 00 4.1 400µK Planck HFI . . . . . . . . . 100 ds1 9. 07 0.9 400µKPlanck Collaboration VIII (2016) ds2 9. 07 0.8 400µK 143 ds1 7. 02 0.7 7. 05µK ds2 7. 02 0.7 7. 05µK 5 7. 02 0.9 7. 05µK 6 7. 02 1.1 7. 05µK 7 7. 02 1.0 7. 05µK 217 1 5. 00 1.8 7. 05µK 2 5. 00 1.9 7. 05µK 3 5. 00 1.7 7. 05µK 4 5. 00 1.8 7. 05µK 353 ds2 4. 09 4.5 7. 05µK 1 4. 09 3.5 7. 05µK 545 2 4. 07 0.01 7. 05 MJy sr−1 4 4. 07 0.01 7. 05 MJy sr−1 857 2 4. 04 0.01 7. 05 MJy sr−1 WMAP . . . . . . . . . . . . 23 K 5305.9 600µKBennett et al. (2013) 33 Ka 4004.3 600µK 41 Q1 3105.3 600µK Q2 3105.1 600µK 61 V1 2106.4 600µK V2 2105.5 600µK 95 W1 1308.84 600µK W2 13010.1 600µK W3 13010.6 600µK W4 13010.1 600µK Haslam et al. . . . . . . . . 0.408 5601.1 600KHaslam et al. (1982), Remazeilles et al. (2015) Polarization Planck LFI . . . . . . . . . . 30 32. 04 7.5 400µKPlanck Collaboration VI (2016) 44 27. 01 7.5 400µK 70 13. 03 4.8 400µK Planck HFI . . . . . . . . . 100 9. 05 1.3 100µKPlanck Collaboration VIII (2016) 143 7. 02 1.1 100µK 217 5. 00 1.6 100µK 353 4. 09 6.9 100µK Notes. The top section lists all detector and detector set (“ds”) maps included in the temperature analysis, and the bottom section lists all frequency maps included in the polarization analysis. temperatures and flux density per area are tabulated for both nominal and fitted bandpass profiles in Table 3. Some Planck detector maps are affected more strongly by systematic errors than others (Planck Collaboration VIII 2016), and in the following temperature analysis we exclude the worst channels in order not to compromise the overall solution. Out of a total of 31 potential Planck detector and detector set maps, 10 are removed from further analysis, while the remaining 21 are listed in Tables 1and 3. The 10 removed maps are shown in Fig. 2in the form of a (dν−sν) residual map, where sνis a signal model based on the global and spectral parameters derived in Sect. 5, but with amplitudes re-fitted to all 42 channels. The top panel shows the corresponding χ2map, defined as χ2(p)=X ν dν(p)−sν(p) σν(p)!2 ·(14) A10, page 6 of 63 Planck Collaboration: Planck 2015 results. X. PM99.6 PM61 Fig. 3. Processing masks (PM) used in the joint temperature analysis, including 99.6% and 61% of the sky, respectively. The former is used for calibration and bandpass estimation, and the latter for monopole and dipole estimation. Including any one of these ten maps in the full joint analysis increases the χ2of the total fit far beyond what is allowed by random statistical fluctuations. Several different classes of systematic effects may be seen in these plots. Starting with the 217-ds1 map, we see a large-scale red-blue pattern, aligned with the Planck scanning strategy and crossing through the Ecliptic poles (Planck Collaboration I 2014). The same spatial pattern is seen in the 217-ds2, 353-ds1, 353-2, 353-7, 353-8, 857-3, and 857-4 maps as well, with varying amplitudes and signs. These are due to low-amplitude destriping errors induced in the mapmaking process (Planck Collaboration VIII 2016), and only become visually apparent after component separation removes the dominant Galactic signal. These residuals may therefore, at least in principle, be suppressed by iterating between mapmaking and component separation, essentially performing a joint mapmaking/component separation χ2fit for the destriping offsets. A second example of residual systematics is seen in the 857-1 and 857-4 maps, and to a lesser extent in the 545-1 map, in the form of sharp features at high Galactic latitudes. These correspond to far sidelobe (FSL) contamination at the 0.05 MJy sr−1level. Third, and somewhat more subtly, one can see the effect of the very-long time constants (VLTC) discussed in Planck Collaboration VIII (2016), particularly south of the Galactic plane in 353-2, 353-7, and 545-1. In each of these maps, the Galactic plane appears to have been smeared along the scanning direction. As in the case of destriping errors, combining mapmaking and component separation should prove very powerful in suppressing both FSL and VLTC errors in future analyses. While several channels are omitted from the analysis, it is important to note that the Planck HFI temperature observations are strongly signal-dominated at all angular scales above 70FWHM. Our data cuts therefore have very little effect on the full error budget, which is dominated by modelling and systematic errors for most parameters. The main cost of these cuts Table 2. Zodiacal light template coefficients for WMAP and LFI frequencies relative to the 100 GHz HFI zodical light template in thermodynamic temperature units. Band Amplitude Planck 30 . . . . . . . . . 0.15 44 . . . . . . . . . 0.28 70 . . . . . . . . . 0.56 WMAP K. . . . . . . . . 0.11 Ka . . . . . . . . 0.19 Q. . . . . . . . . 0.25 V. . . . . . . . . 0.45 W. . . . . . . . . 0.89 comes in the form of reduced internal redundancy. In particular, having access to only one clean 857 GHz channel limits our ability to determine its bandpass and calibration coefficients. This situation will improve in the next Planck data release, when the remaining three 857 GHz channels are better cleaned at the time domain level. The current analysis is carried out in a number of stages according to angular resolution, in which higher-resolution stages implement a simpler foreground model than lower-resolution stages. For temperature, each full-resolution map is downgraded from its native resolution to 1◦(all channels), 400(all Planck channels), and 7. 05 (HFI channels above 100 GHz) by deconvolving the intrinsic instrumental beam profile and convolving with a Gaussian beam of the appropriate size before repixelizing at HEALPix5resolutions Nside =256, 256, and 2048, respectively (Górski et al. 2005). For polarization, the corresponding smoothing scales are 400and 100FWHM, pixelized at Nside =256 and 1024, respectively. The instrumental noise is assumed to be spatially uncorrelated and Gaussian for all channels, with a spatially-varying rms given by the scanning strategy of the experiment and the instantaneous sensitivity of each detector. The low-resolution noise rms map is found by convolving the high-resolution rms with the appropriate smoothing kernel, properly accounting for its matrix-like nature, and retaining only the diagonal element of the resulting covariance matrix. For the 545 and 857 GHz channels, we additionally add 0.01 MJy sr−1of uniform white noise, to prevent known residual far sidelobe and destriping contamination from propagating to lower frequencies through the thermal dust temperature and spectral index and contaminating both the CMB and CO solutions. Similarly, for the 408 MHz channel we add regularization noise, equal to 1% of the amplitude of the map, to account for low-level residuals not captured by the white noise model described above. Finally, two different processing masks are employed in the temperature analysis (PM99.6 and PM61), removing 0.4% and 39% of the sky, respectively, as shown in Fig. 3. PM99.6 is generated by thresholding the (smoothed) χ2map of a preliminary analysis6at 104, removing only the very brightest outliers in the data set; this mask is used for bandpass estimation and gain calibration of the 545-4 and 857-2 channels. PM61 is generated as the product of the χ2map resulting from an analysis without bandpass corrections thresholded at the 5σlevel, and the 9-yr WMAP point source mask. This mask is used for calibration estimation of the CMB frequencies, and for monopole and dipole estimation. 5http://healpix.sourceforge.net/ 6The preliminary analysis was similar to the one presented in this paper, but without application of any processing mask for calibration purposes. A10, page 7 of 63 A&A 594, A10 (2016) Table 3. Unit conversion coefficients between thermodynamic and brightness temperature and between thermodynamic temperature and flux density per unit area for each channel, with and without the bandpass corrections described in Sect. 5. Uc[KCMB/KRJ]Uc[MJy sr−1/KCMB] Frequency Detector Instrument [GHz] label Nominal Fitted Change [%] Nominal Fitted Change [%] Planck LFI . . . . . 30 . . . 1.0212 1.0217 0.1 23.510 24.255 3.2 44 . . . 1.0515 1.0517 0.0 55.735 56.094 0.6 70 ds1 1.1378 1.1360 −0.2 132.07 129.77 −1.7 ds2 1.1361 1.1405 0.4 129.75 135.37 4.3 ds3 1.1329 1.1348 0.2 126.05 128.57 2.0 Planck HFI . . . . . 100 ds1 1.3090 1.3058 −0.2 244.59 241.58 −1.2 ds2 1.3084 1.3057 −0.2 243.77 241.22 −1.1 143 ds1 1.6663 1.6735 0.4 365.18 368.68 1.0 ds2 1.6754 1.6727 −0.2 369.29 368.00 −0.4 5 1.6961 1.6910 −0.3 380.12 377.68 −0.6 6 1.6829 1.6858 0.2 373.34 374.74 0.4 7 1.6987 1.6945 −0.2 381.25 379.26 −0.5 217 1 3.2225 3.2203 −0.1 486.02 485.87 −0.0 2 3.2378 3.2336 −0.1 486.39 486.10 −0.1 3 3.2296 3.2329 0.1 486.89 485.85 −0.2 4 3.2183 3.2161 −0.1 486.01 487.11 0.2 353 ds2 14.217 14.261 0.3 287.89 287.62 −0.1 1 14.113 14.106 −0.1 288.41 288.45 0.0 545 2 164.54 168.94 2.7 58.880 57.953 −1.6 4 167.60 174.13 3.9 58.056 56.732 −2.3 857 2 9,738.8 10,605.8.9 2.3449 2.1974 −6.3 WMAP . . . . . . . . 23 K 1.0134 ··· ··· 14.985 ··· ··· 33 Ka 1.0284 ··· ··· 31.556 ··· ··· 41 Q1 1.0437 ··· ··· 47.880 ··· ··· Q2 1.0433 1.0439 0.1 47.179 48.226 2.2 61 V1 1.0974 ··· ··· 97.343 ··· ··· V2 1.1006 1.1001 −0.1 101.87 101.20 −0.7 94 W1 1.2473 ··· ··· 209.61 ··· ··· W2 1.2500 1.2458 −0.3 213.65 209.38 −2.0 W3 1.2440 1.2460 0.1 207.88 209.92 1.0 W4 1.2488 1.2453 −0.3 211.78 208.18 −1.7 Haslam . . . . . . . . 0.408 . . . 1.0000 ··· ··· 0.0051 ··· ··· 4. Model survey Before presenting the results of our analysis, it is useful to review the various model components that are relevant for this work, with the goal of building intuition concerning important degeneracies and residuals that may be observed in various goodness-of-fit statistics. We first review each of the astrophysical sky signal components for both temperature and polarization, as summarized in Table 4and Fig. 4, and then each of the most important instrumental effects. 4.1. Sky components in intensity CMB – The CMB is given by a perfect blackbody with only a single spectral parameter, namely the CMB temperature. We adopt a mean value of Tcmb =2.7255 ±0.0006 K (Fixsen 2009), and notice that the uncertainty in this value is sufficiently small to justify its use as a delta function prior. In the current paper, we neglect both Rayleigh scattering and higher-order relativistic effects (Planck Collaboration XXVII 2014;Lewis 2013); these could be accounted for in future work. The resulting CMB brightness temperature is plotted as a dashed line in each panel of Fig. 4, with an amplitude of 70 µK, corresponding to the CMB rms at 1◦FWHM resolution, providing a useful consistent visual reference for other components. Synchrotron – Diffuse synchrotron emission is generated by relativistic cosmic-ray electrons spiraling in the Galactic magnetic field. This radiation may be highly polarized, with a maximum polarization fraction of 0.75. Both theoretical models and observations suggest that the synchrotron spectrum is well approximated by a power law with an index of βs≈ −3 at frequencies above 20 GHz, but with significant flattening at low frequencies. In this paper we adopt a fixed spectral template for the synchrotron spectrum. We use a spectrum extracted from the GALPROP z10LMPD_SUNfE synchrotron model from Orlando & Strong (2013), as described in Planck Collaboration XXV (2016). For brevity we refer to this as the GALPROP model A10, page 8 of 63 Planck Collaboration: Planck 2015 results. X. 0,01 0,1 1 10 100 1000 Frequency (GHz) 10-3 1001031061091012 Brightness temperature (µK) ACMB = 70 µK As = 30 K, α = 0.1 As = 30 K, α = 1 As = 30 K, α = 10 Synchrotron 0,0001 0,001 0,01 0,1 1 10 100 1000 Frequency (GHz) 10-3 100103106109 Brightness temperature (µK) ACMB = 70 µK EM = 0.01 cm-3 pc, Te = 7000 K EM = 1 cm-3 pc, Te = 500 K EM = 1 cm-3 pc, Te = 7000 K EM = 1 cm-3 pc, Te = 20000 K EM = 100 cm-3 pc, Te = 7000 K Free-free 0,1 1 10 100 1000 Frequency (GHz) 10-2 10-1 100101102103104 Brightness temperature (µK) ACMB = 70 µK Asd = 100 µK, νp = 11 GHz Asd = 100 µK, νp = 21 GHz Asd = 100 µK, νp = 31 GHz Spinning dust 100 200 300 400 500 Frequency (GHz) 0 20 40 60 80 Brightness temperature (µK) ACMB = 70 µK Ad = 100 µK, βd = 1.5, Td = 21 K ACO = 50 K km/s ACO = 50 K km/s, h217/100 = 0.50 ACO = 50 K km/s, h353/100 = 0.20 CO 10 100 1000 Frequency (GHz) 10-1 100101102 Brightness temperature (µK) ACMB = 70 µK Ad = 100 µK, βd = 1.0, Td = 21 K Ad = 100 µK, βd = 1.5, Td = 16 K Ad = 100 µK, βd = 1.5, Td = 21 K Ad = 100 µK, βd = 1.5, Td = 26 K Ad = 100 µK, βd = 2.0, Td = 21 K Thermal dust 0,1 1 10 100 1000 Frequency (GHz) -80 -40 0 40 80 Brightness temperature (µK) ACMB = 70 µK ySZ = 10-7 ySZ = 10-6 ySZ = 10-5 Thermal SZ Fig. 4. Spectral energy densities (SEDs) for the main astrophysical components included in the present analysis, in brightness temperature. From left to right and top to bottom, panels show: (1) synchrotron emission; (2) free-free emission; (3) spinning dust emission; (4) CO line emission; (5) thermal dust emission; and (6) the thermal Sunyaev-Zeldovich effect. For each case, several parameter combinations are shown to illustrate their effect on the final observable spectrum. Vertical grey bands indicate the centre frequencies of the observations listed in Table 1, but for clarity true bandwidths are suppressed. In each panel, the black dashed line shows the CMB brightness temperature corresponding to a thermodynamic temperature of 70 µK, the CMB rms at 1◦FWHM angular scale. In the fourth (CO) panel, the dotted line indicates the SED of thermal dust emission with typical parameter values. from now on. A description of the GALPROP7,8code can be found in Moskalenko & Strong (1998), Strong et al. (2007), Orlando & Strong (2013) and references therein. We allow this spectrum to be rigidly shifted in log ν–log Sspace with a single global frequency shift parameter, α, for the full sky; see Table 4 for the explicit definition. The main effect of such translation, 7http://galprop.stanford.edu/ 8https://sourceforge.net/projects/galprop however, is to modify the synchrotron amplitude at 408 MHz, leaving βsat frequencies above 20 GHz essentially constant and equal to −3.11 for any realistic shift parameter; see Fig. 4. Thus, with the current data the synchrotron amplitude is determined almost exclusively by the 408 MHz survey, while the frequency spectrum is determined by the GALPROP model, with the only free spectral parameter being the relative normalization between the 408 MHz and higher frequency channels. Several A10, page 9 of 63 A&A 594, A10 (2016) choices, physically well-motivated models are preferred over phenomenological models. Three examples illustrate our approach. First, for synchrotron emission we have found that a broken power-law model, i.e., a power-law model with fixed but different spectral indices above and below some break frequency (say, 3 GHz) provides an equally good χ2fit as the GALPROP model do. A straight power-law spectrum, on the other hand, does not fit the data, because the 408 MHz map appears dimmer than expected from a straight extrapolation from 23 GHz to 408 MHz, assuming the spectral index of β≈ −3.0 to −3.2 required to fit higher frequencies. The reasons for preferring the GALPROP model are therefore not data-driven, but rather that it requires less tuning (e.g., no choice of break frequency, no free spectral indices, very weak dependency on νs p), and that it is based on a physically well-motivated calculation (Strong et al. 2011;Orlando & Strong 2013). The cost, however, is less flexibility for tracing real spatial variations in the synchrotron spectral index, and thereby possibly greater cross-talk between synchrotron, free-free, and spinning dust. Nevertheless, in the absence of sufficiently strong data to disentangle these variations, we consider this a lesser evil than introducing a very strong statistical degeneracy between synchrotron, free-free, and spinning dust. A second important example is the spinning dust component, which is currently implemented in terms of two independent SpDust components, one with a free peak frequency, νp, per pixel, and one with a spatially constant peak frequency, for a total of three free effective spinning dust parameters per pixel. While the introduction of the first component is unavoidable when combining the Planck and WMAP observations, the necessity of the second is more subtle. Without it we invariably find highly significant residuals (many tens to a few hundreds of microkelvin in the Galactic plane) in the 60–70 GHz frequency range, with dust-like morphology. This strongly suggests a model problem with either spinning or thermal dust (or both), but so far the only acceptable solution we have found is effectively to “widen” the SpDust spectrum slightly, which is most easily implemented by adding a second independent component. Introducing, say, a flatter thermal dust index at around 100 GHz tends to exacerbate rather than ameliorate the problem. We emphasize, however, that the current two-component spinning dust model is a purely phenomenological model introduced in the absence of physically well-motivated alternatives. We fully anticipate that additional low-frequency data or further theoretical developments will improve this model significantly in the future. Third, we currently adopt a one-component greybody model with free emissivity index and temperature for thermal dust emission. Experimenting with various two-component alternatives, we find only one absolute requirement on the thermal dust model for frequencies up to 857 GHz, namely that at least three free thermal dust parameters per pixel are required to achieve an acceptable fit to the high-frequency observations. Whether those are {Ad, βd,Td},{A1 d,T1 d,A2 d}, or even {A1 d, β1 d,A2 d}(!) is not clear from the current data set10. On the other hand, it is clear that additional parameters are not required to model the current data set. For now, we prefer the one-component model simply because it has fewer global parameters than a corresponding two-parameter model, i.e., it requires no spatially fixed βdand Tdfor a second component, and therefore requires less tuning. This is not to be taken as a statement on the relative physical merits of the two models, however. Additional high-frequency observations are required to distinguish between them. 10 Superscripts 1 and 2 refer to two independent greybody components. 5.2. Data preview Before presenting the actual baseline results, it is useful to visually consider a few specific data combinations in order to gain some intuition regarding the power of these data for constraining specific parameters. In Fig. 6, we show four different sky maps, each of which highlights an important and distinct feature in the data. A similar discussion based on an internal linear combination (ILC) method is presented in Sect. 3 of Planck Collaboration XXV (2016). Starting with the top panel, we plot the effective power-law index of the combined Planck, WMAP, and 408 MHz data, when fitting only a single power-law model at low-frequencies, as opposed to fitting individual synchrotron, free-free, and spinning dust components. All other components are defined by the usual baseline model. On the low-frequency side, this approach is thus identical to the 2013 model (Planck Collaboration XII 2014), although the data volume and frequency range are significantly increased. Despite these differences, the 2013 and 2015 lowfrequency spectral index maps agree reasonably well, with a mean and standard deviation difference of 0.16 ±0.14. The main features in this spectral index map can be broadly characterized into two types of spatial signatures. First, we see many distinct regions (e.g., the Gum nebula, Orion, Zeta Oph, and the Cygnus complex) with a shallow spectral index of β≈ −2 and morphology as expected from various free-free tracers (e.g., Alves et al. 2015). Second, there are extended dark regions surrounding the Galactic plane, with very steep spectral indices of β.−3.6, and morphology similar to that of thermal dust emission. In addition, it is possible to see some weak hints of the North Galactic Spur, a strong synchrotron emission region, but since we adopted a synchrotron-like prior of β∼N(−3,0.2) for the low-frequency power-law index in this analysis, it is difficult to distinguish this region from the prior-dominated background. The main point, however, is that even with such minimal modelling, there is strong evidence for at least three distinct physical components at low frequencies, namely: synchrotron, with β≈ −3; free-free, with β≈ −2; and a dust-like component, with β.−3.6. Among all the spectra shown in Fig. 4, only a spinning dust component with νp.30 GHz can reasonably account for the latter. In the second panel we show the ratio between the 857 and 545 GHz frequency maps, masking all pixels for which the 545 GHz amplitude is smaller than ten times its monopole. The spatial variations seen in this map cannot be explained either in terms of calibration or bandpass errors (because it is a ratio map, and either of those errors primarily affects the scale, not the morphology) or in terms of absolute offsets (because of the high mask threshold). They are robust features of the Galactic thermal dust emission, and must be explained by any Galactic model that include these observations. In fact, within our baseline model, these features can only be explained in terms of a spatially varying dust temperature. To be specific, the thermal dust appears hotter (i.e., has larger 857-to-545 ratio; see Fig. 4) near the Galactic centre than in the outer Galaxy (quadrants 2 and 3). Adopting a spatially constant dust temperature is no longer possible, and the only reason that the corresponding 2013 analysis could produce meaningful results with such an assumption was that it considered frequencies only up to 353 GHz, and also focused primarily on high Galactic latitudes (Planck Collaboration XII 2014). The third panel shows the straight difference between the 100-ds1 and 100-ds2 maps. Because of the very short lever arm between these two frequencies, essentially all continuous A10, page 16 of 63 Planck Collaboration: Planck 2015 results. X. βlf −3.6 −2.0 857/545 2.5 3.5 100ds1−2 −10 10 µKcmb W3−W2 −100 100 µKcmb Fig. 6. Top left: effective low-frequency foreground spectral index as measured from the combination of Planck, WMAP, and 408 MHz, with no attempt to disentangle synchrotron, free-free, and spinning dust emission into separate components. However, higher-frequency components (CMB, CO, thermal dust, etc.) are fitted component-by-component, as in the baseline model. Note the very steep spectral indices, βlf .−3.6, near the Galactic plane, with dust-like morphology. These can only be reasonably explained in terms of spinning dust. Top right: ratio between the 857 and 545 GHz frequency maps, smoothed to 1◦FWHM, highlighting the spatially varying temperature of thermal dust. The mask is defined by any region for which the 545 GHz amplitude is smaller than ten times the 545 GHz monopole. Bottom left: difference between the 100-ds1 and 100-ds2 detector maps, smoothed to 1◦, demonstrating the presence of CO J=1→0 emission in these channels. Bottom right: difference between the WMAP W3 and W2 differencing assembly maps, smoothed to 1◦FWHM. The excess signal near the Galactic centre is due to line emission in the 94 GHz channels. The peak amplitude of the difference map is 740 µK. emission mechanisms cancel in this difference, leaving only the sharp line emission from CO J=1→0, as well as instrumental noise. This demonstrates the power of employing detector maps in the following analysis as opposed to frequency maps; using a fine-grained data set vastly increases our ability to extract line emission. Whereas CO line emission was studied extensively in the Planck 2013 release, an additional new 94–100 GHz line feature is included in the current release (see Sect. 4.1). The primary contribution to this component is visualized in the bottom panel of Fig. 6, in terms of the WMAP W3−W2 difference map. The small but bright features near the Galactic centre (the maximum amplitude is 740 µK) are line emission within the W-band, as discussed in Sect. 4.1. This line is also covered by the two Planck 100 GHz channels. To prevent this additional emission from contaminating the main CO estimates, and to achieve an acceptable fit for the WMAP W-channels, we fit for this additional line emission component at 94/100 GHz. 5.3. Baseline model We are now ready to present the Planck 2015 baseline diffuse astrophysical component model in temperature, as derived from the Planck, WMAP, and 408 MHz observations. An overview of the data products delivered through the Planck Legacy Archive (PLA)11 is given in Table 5, including file names and FITS extension numbering, as well as summary statistics in the form of posterior maximum, mean, and rms values. We start our review by considering data that are smoothed to a common resolution of 1◦FWHM, representing the most complete data set and model possible, given the current data set. Processing these observations through the analysis pipeline outlined in Sect. 2, and adopting the baseline model described in Sect. 4.1, we obtain the set of maximum-posterior astrophysical parameter maps shown in the top panels of Figs. 7–19. The bottom panels show the corresponding rms maps found by computing the standard deviation over the ensemble of Gibbs samples. Please note that these rms maps account only for statistical errors from instrumental noise, and not for systematic uncertainties due to modelling errors. They are therefore only meaningful for pixels for which the goodness-of-fit is acceptable. Most importantly, they do not represent the true errors in the Galactic plane, where modelling errors dominate statistical errors. Instrumental parameters, as well as monopole and dipole coefficients, are listed in Table 6, and Fig. 20 provides a visual comparison of the calibration factors for each Planck and WMAP 11 http://pla.esac.esa.int/pla A10, page 17 of 63 A&A 594, A10 (2016) Table 5. Summary of full-sky foreground products available from the PLA. Posterior outside LM93 FITS νref File extension Parameter [GHz/band] Pmax Mean RMS Unit Temperature at 1◦FWHM, Nside =256 AME . . . . . . . . . . . . . . . . 0 Asd1 22.8 93 ±118 92 ±118 11 ±3µKRJ νsd1 ··· 19 ±1 19 ±1 2.0±0.8 GHz 1Asd2 41 14 ±21 18 ±22 4.1±2.8µKRJ CMB . . . . . . . . . . . . . . . . 0 Acmb ··· 3±67 3 ±67 1.5±0.8µKcmb CO . . . . . . . . . . . . . . . . . 0 ACO10 100-ds1 0.3±1.3 0.4±1.3 0.06 ±0.05 KRJ km s−1 1ACO21 217-1 0.22 ±0.57 0.29 ±0.57 0.04 ±0.01 KRJ km s−1 2ACO32 353-ds2 0.16 ±0.21 0.26 ±0.26 0.05 ±0.01 KRJ km s−1 dust . . . . . . . . . . . . . . . 0 Ad545 163 ±228 163 ±228 0.66 ±0.11 µKRJ Td··· 21 ±2 21 ±2 1.1±0.7 K βd··· 1.53 ±0.05 1.51 ±0.06 0.05 ±0.03 ··· freefree . . . . . . . . . . . 0 EM ··· 15 ±35 13 ±35 2.3±2.4 cm−6pc Te··· 7000 ±11 7000 ±11 ··· K Synchrotron . . . . . . . . 0 As0.408 20 ±15 20 ±15 1.1±0.2 KRJa SZ . . . . . . . . . . . . . . . . . 0 Asz ··· 1.4±1.4b2.0±1.3b0.8±0.2b10−6ysz xlinec. . . . . . . . . . . . . 0 A94/100 100-ds1 0.09 ±0.06 0.9±0.8 0.7±0.6µKcmb Temperature at 7. 05 FWHM, Nside =2048 CO21d. . . . . . . . . . . . . . 0 ACO21 217-1 0.2±0.8··· ··· KRJ km s−1 ThermalDustd. . . . . . . . 0 Ad545 0.2±0.8··· ··· µKRJ βd··· 1.54 ±0.07 ··· ··· ··· Polarization at 400FWHM, Nside =256 SynchrotronPold. . . . . 0 Pe s30 12 ±9··· ··· µKRJ Polarization at 100FWHM, Nside =1024 DustPold. . . . . . . . . . . 0 Pe d353 8 ±10 ··· ··· µKRJ Notes. Each entry in the first column corresponds to one multi-column and (optionally) multi-extension FITS file, named COM_CompMap_{label}-commander_{nside}_R2.00.fits. The various columns in each extension list the posterior maximum, mean, and rms maps, in that order, when available. The values reported in Cols. 5 to 7 in this table are the mean and standard deviations of these posterior statistic maps. (a)The data file unit is µKRJ but for convenience we list numbers in KRJ in this table. (b)Evaluated only over the Coma and Virgo regions. (c)This is the 94/100 GHz line emission component. (d)Only the full-mission maps are summarized in this table, but the data files also include corresponding maps for half-mission, half-year, and half-ring data splits. (e)Data files contains Stokes Qand Uparameters, not the polarization amplitude, P=pQ2+U2, listed here. detector map12. Global (i.e., spatially uniform) astrophysical parameters are listed in Table 7. The full marginal uncertainties of these parameters are dominated by modelling, not statistical errors. For this reason, the tabulated uncertainties are computed through realistic end-to-end simulations, as implemented in the Planck 2015 FFP8 simulations (Planck Collaboration XII 2016). These simulations were analysed blindly as part of the CMB validation efforts (Planck Collaboration IX 2016), using the exact same machinery as used in this paper. Further, they are based on a significantly different foreground model than the baseline model adopted here, and they therefore at least partially account for modelling errors, as well as known systematic and mapmaking uncertainties. The only differences in the fitted model compared to the present baseline are that it includes only one 12 The re-calibration factors listed in Table 6for 545 and 857 GHz refer to multiplicative factors in native map units, MJy sr−1. When making comparison with similar calibration factors derived from corresponding maps defined in units of µK, one additionally has to account for differences in unit conversion due to bandpass shifts, as listed in Table 3. spinning dust component and no 94/100 GHz line emission or SZ components. Based on these simulations, we estimate the uncertainties on the calibration and bandpass measurements directly from the simulations, comparing output parameters against known inputs. The monopole and dipole uncertainties are, however, first based on the statistical errors derived from the Gibbs chains, and then rounded up (where necessary) to be no smaller than the corresponding FFP8 simulation uncertainties13. Thus they correspond to the maximum of the instrumental and the modelling errors. Furthermore, we reemphasize that the reported monopole and dipole values are conditional with respect to the nominal Planck, H iand synchrotron priors, as defined by Planck Collaboration VI (2016), Planck Collaboration VIII (2016), and Wehus et al. (2016). The uncertainties in the CO line ratios listed in Table 7, which are model-dominated, are also taken from the FFP8 simulations. The 94/100 GHz line ratios, however, are noise 13 Based on the FFP8 simulations we never report monopole (dipole) uncertainties smaller than 1 µK (0.1µK). A10, page 18 of 63 Planck Collaboration: Planck 2015 results. X. ACMB -250 0 250 µKCMB σCMB 0 3 µKCMB Fig. 7. Maximum posterior (top) and posterior rms (bottom) CMB intensity maps derived from the joint baseline analysis of Planck, WMAP, and 408 MHz observations. The two circular regions close to the North Galactic Pole in the rms map correspond to the Coma and Virgo clusters, for which the thermal SZ efect is fitted together with the primary diffuse components. Note also that the rms map includes statistical errors alone, not modelling errors, and they are therefore only meaningful in regions where the corresponding χ2is acceptable; see Fig. 22. Both panels employ linear colour scales. A10, page 19 of 63 A&A 594, A10 (2016) As 10 30 100 300 KRJ @ 408 MHz σs 0 3 KRJ @ 408 MHz Fig. 8. Maximum posterior (top) and posterior rms (bottom) synchrotron intensity maps derived from the joint baseline analysis of Planck, WMAP, and 408 MHz observations. The top panel employs a non-linear high dynamic range colour scale, while the bottom panel employs a regular linear colour scale. A10, page 20 of 63 Planck Collaboration: Planck 2015 results. X. Aff 0 10 100 1000 cm−6pc σff 0 10 cm−6pc Fig. 9. Maximum posterior (top) and posterior rms (bottom) free-free emission measure maps derived from the joint baseline analysis of Planck, WMAP, and 408 MHz observations. The top panel employs a non-linear high dynamic range colour scale, while the bottom panel employs a regular linear colour scale. A10, page 21 of 63 A&A 594, A10 (2016) Asd 0.01 0.1 1 10 mKRJ @ 30 GHz σsd1 0 20 µKRJ @ 22.8 GHz Fig. 10. Maximum posterior (top) and posterior rms (bottom) spinning dust intensity maps derived from the joint baseline analysis of Planck, WMAP, and 408 MHz observations. The top panel shows the sum of the two spinning dust components in the baseline model, evaluated at 30 GHz, whereas the bottom shows the standard deviation of only the primary spinning dust component, evaluated at 22.8 GHz. Note that the top panel employs a non-linear high dynamic range colour scale, while the bottom panel employs a regular linear colour scale. A10, page 22 of 63 Planck Collaboration: Planck 2015 results. X. Ad 0.01 0.1 1 10 mKRJ @ 545 GHz σd 0 1 µKRJ @ 545 GHz Fig. 11. Maximum posterior (top) and posterior rms (bottom) thermal dust intensity maps derived from the joint baseline analysis of Planck, WMAP, and 408 MHz observations. The top panel employs a non-linear high dynamic range colour scale, while the bottom panel employs a regular linear colour scale. A10, page 23 of 63 A&A 594, A10 (2016) A94/100 0 10 100 µKcmb @ 100-ds1 σ94/100 0 10 µKcmb @ 100-ds1 Fig. 12. Maximum posterior (top) and posterior rms (bottom) 94/100 GHz line emission maps derived from the joint baseline analysis of Planck, WMAP, and 408 MHz observations. The top panel employs a non-linear high dynamic range colour scale, while the bottom panel employs a regular linear colour scale. A10, page 24 of 63 Planck Collaboration: Planck 2015 results. X. ACO10 0 10 100 KRJ km s−1 σCO10 0 0.5 KRJ km s−1 Fig. 13. Maximum posterior (top) and posterior rms (bottom) CO J=1→0 line emission maps derived from the joint baseline analysis of Planck, WMAP, and 408 MHz observations. The top panel employs a non-linear high dynamic range colour scale, while the bottom panel employs a regular linear colour scale. A10, page 25 of 63 A&A 594, A10 (2016) 0.408 −1 1 K K −10 10 µK 30 −10 10 µK Ka −10 10 µK Q1 −10 10 µK Q2 −10 10 µK 44 −10 10 µK V1 −10 10 µK V2 −10 10 µK 70-ds1 −10 10 µK 70-ds2 −10 10 µK 70-ds3 −10 10 µK W1 −20 20 µK W2 −20 20 µK W3 −20 20 µK W4 −20 20 µK 100-ds1 −2 2 µK 100-ds2 −2 2 µK 143-ds1 −2 2 µK 143-ds2 −2 2 µK 143-5 −2 2 µK 143-6 −2 2 µK 143-7 −2 2 µK 217-1 −5 5 µK 217-2 −5 5 µK 217-3 −5 5 µK 217-4 −5 5 µK 353-ds2 −10 10 µK 353-1 −10 10 µK 545-2 −0.05 0.05 MJy/sr 545-4 −0.05 0.05 MJy/sr 857-2 −0.01 0.01 MJy/sr Fig. 21. Residual maps, dν−sν, for each detector data set included in the baseline joint Planck, WMAP, and 408 MHz temperature analysis. All panels employ linear colour scales. The label in the top left corner of each panel indicates frequency channel. A10, page 32 of 63 Planck Collaboration: Planck 2015 results. X. Table 6. Monopoles, dipoles, calibration factors and bandpass corrections derived within the baseline temperature model. Frequency Detector Monopole Xdipole Ydipole Zdipole Calibration Bandpass shift Survey [GHz] label [ µK] [ µK] [ µK] [ µK] [%] [GHz] Planck LFI . . . . . 30 . . . −17 ±1 0a0a0a−0.3±0.1f0.3±0.1 44 . . . 11 ±1 0.5±0.2−0.3±0.1 0.5±0.1 0.3±0.1f0.1±0.1 70 ds1 16 ±1 0.5±0.1−1.1±0.1 1.1±0.1 0.0±0.1f−0.4±1.0 ds2 16 ±1 0.5±0.1−1.0±0.1 1.1±0.1 0.1±0.1f1.1±1.0 ds3 16 ±1−0.1±0.1−0.9±0.1 0.8±0.1−0.1±0.1f0.5±1.0 Planck HFI . . . . . 100 ds1 9a0a0a0a0.11 ±0.02 −0.5±0.7 ds2 8 ±1 0.0±0.1−0.1±0.1 0.3±0.2 0.08 ±0.02 −0.4±0.6 143 ds1 21a0a0a0a0a0.7±0.2 ds2 21 ±1 0.0±0.1 0.0±0.1−0.1±0.1−0.04 ±0.02 −0.2±0.2 5 21 ±1−0.5±0.1 0.0±0.1−0.1±0.1 0.09 ±0.02 −0.5±0.2 6 21 ±1−0.4±0.1 0.0±0.1−0.1±0.1 0.12 ±0.02 0.3±0.2 7 20 ±1−0.2±0.1 0.0±0.1−0.0±0.1 0.01 ±0.02 −0.4±0.2 217 1 68 ±1−0.8±0.1−2.6±0.1 2.9±0.1 0a−0.1±0.1 2 68 ±1−0.7±0.1−2.8±0.1 3.1±0.1 0.01 ±0.03 −0.1±0.1 3 67 ±1−1.0±0.1−2.6±0.1 3.0±0.1 0.00 ±0.03 0.1±0.1 4 68 ±1−0.4±0.1−2.7±0.1 3.0±0.1 0.04 ±0.03 −0.1±0.1 353 ds2 447 ±5−3±1−6±1 6 ±1 0.3±0.1 0.3±0.1 1 449 ±6−4±1−16 ±1 15 ±1 0.8±0.1−0.0±0.1 545 2 0.37a,c 0a0a0a−2.8e2.0e 4 0.36 ±0.01c0a0a0a−3.2e2.8e 857 2 0.62 ±0.01c0a0a0a1.7e5.8e WMAP . . . . . . . . 23 K −8±1−4.5±2.0 1.6±0.5−3.7±0.4 0a0a 33 Ka 3b−0.7±0.6−4.7±0.2 3.8±0.1 0.1±0.1 0a 41 Q1 2 ±1 0.5±0.3−4.6±0.1 3.5±0.1−0.1±0.1 0a Q2 2 ±1 0.4±0.3−4.8±0.1 3.7±0.1 0.2±0.1 0.3±0.1 61 V1 1 ±1 0.2±0.1−5.5±0.1 4.2±0.1 0.1±0.1 0a V2 1 ±1 0.0±0.1−5.5±0.1 4.2±0.1 0.3±0.1−0.1±0.1 94 W1 −5±2 0.3±0.1−5.3±0.2 4.1±0.2 0.3±0.1 0a W2 −5±2 0.1±0.1−5.0±0.1 3.9±0.2 0.4±0.1−0.7±0.3 W3 −6±2 0.2±0.1−6.0±0.2 4.3±0.3−0.1±0.1 0.3±0.3 W4 −5±2−0.0±0.1−6.1±0.1 5.2±0.2 0.1±0.1−0.6±0.3 Haslam . . . . . . . . 0.408 . . . 8.9b,d3.2b,d0.7b,d−0.8b,d0a0a Notes. (a)Fixed at reference value. (b)Fixed at values derived by Wehus et al. (2016). (c)Unit is MJy/sr. (d)Unit is K. (e)For a detailed discussion of bandpass and calibration uncertainties at 545 and 857 GHz, see Sect. 4.3.(f)Adjusted for the well-understood LFI “beam normalization factor” (see Planck Collaboration II 2016). applicable, because informative priors (most notably the positivity prior) eliminate large parts of the parameter space. A parameter that is prior-dominated therefore does not reduce the number of degrees of freedom by unity, but only a fraction of unity. Second, smoothing the data to a common resolution of 1◦ FWHM introduces noise correlations between pixels, and these are not accounted for in the noise description. The overall χ2distribution will therefore be broader than a corresponding distribution with no correlated noise. To estimate the effective number of degrees of freedom, we therefore fit a scaled χ2distribution to the empirical χ2distribution, including only the very cleanest parts of the sky, where actual foreground contamination is minimal. We adopt the conservative PM61 mask for this. The resulting best-fit χ2distribution reads χ2 20.3(x)∝x 1.3720.3/2−1 e−x 2,(18) and we accordingly estimate that the empirical number of degrees of freedom is 20, and the correlated noise scaling factor is 1.37. The former of these suggests that our model effectively contains 32−20 =12 free parameters, not 14 as obtained by naively counting free parameters per pixel. In other words, the combined effect of all priors is to remove 2 of 14 degrees of freedom, indicating that the model is indeed highly data driven. he latter number suggests that the white noise approximation underestimates the true noise level by 37%. The 99% confidence χ2 range from the analytic fit is 11 < χ2<59. The middle panel of Fig. 22 shows the mask obtained by thresholding the χ2map smoothed with a 1◦Gaussian kernel at a value of 50.14 This mask is called the 93% likelihood mask (LM93), and constitutes the primary confidence mask for the Planck 2015 model. Also, together with the CMB solution in Fig. 7, this mask defines the main inputs to the Planck 2015 low-`CMB temperature likelihood (Planck Collaboration XI 2016). We note that this mask removes many bright extragalactic point sources at high Galactic latitudes; the algorithm adopted in this paper treats point sources and diffuse emission identically through pixel-by-pixel fits, and any subsequent analysis of the resulting component maps should take these sources into account either through explicit masking, as done here, or by post-processing fits. The second column of Table 8lists the rms of each residual map outside the LM93 mask. The third column lists the ratio between these rms values and the corresponding instrumental noise rms, as listed in Table 1. The fourth column shows the monopole and dipole corrected median fractional 14 Because of the additional smoothing, χ2 smooth >50 corresponds roughly to a 5σoutlier, not 2.5σas it does in the unsmoothed χ2map. A10, page 33 of 63 A&A 594, A10 (2016) Table 7. Posterior mean and rms values for spatially constant parameters in the temperature model. Detectors Quantity or Band Value Synchrotron freq scale factor, α. . . . . . . . . . . . 0.26b Spinning dust secondary peak freq, νsd2 p. . . . . . 33.35 GHzb CO J=1→0 line ratio . . . . . . . . . . . . . . . 100-ds1 1a 100-ds2 1.02 ±0.01 CO J=2→1 line ratio . . . . . . . . . . . . . . . 217-1 1a 217-2 1.07 ±0.01 217-3 1.15 ±0.01 217-4 1.18 ±0.01 CO J=3→2 line ratio . . . . . . . . . . . . . . . 353-ds2 1a 353-1 1.3±0.1 94/100 GHz line ratio . . . . . . . . . . . . . . . . 100-ds1 1a 100-ds2 1.4±0.3 W1 4.6±3.2 W2 4.2±2.9 W3 5.3±3.7 W4 4.3±3.0 Notes. The uncertainties of the synchrotron frequency scaling factor and the secondary spinning dust peak frequency are strongly dominated by modelling errors that are difficult to quantify properly, and are omitted here; see main text for further discussion. The quoted line ratios refer to amplitudes relative to the respective reference band. (a)Reference channel. (b)Only the maximum-likelihood value is provided; see main text. residual, (dν−sν)/(dν−Tνmν), this time evaluated inside the LM93 mask. As seen in the third column, the rms values of the residual map are close to the instrumental noise for most channels, indicating an excellent goodness-of-fit, not only in terms the integrated χ2, but also channel-by-channel. In accordance with the above discussion, we see that a number of channels have residuals that are lower than the instrumental noise, with the 408 MHz and 857-2 channels being the two most striking examples. This is expected, because the χ2only has 20 degrees of freedom, whereas there are 32 individual frequency channels; the normalized mean residuals must therefore sum to less than one per channel. However, a ratio much smaller than unity typically indicates that the corresponding channel has a much higher effective signal-to-noise ratio with respect to some signal parameter than all other channels combined. In our case, the 408 MHz and 857-2 channels strongly dominate the synchrotron and thermal dust amplitudes, respectively. At first sight, one might suspect these values to indicate the presence of worrisome parameter degeneracies, which typically also can result in residuals lower than the instrumental noise. However, from the parameter maps shown in Figs. 7–19, it is visually obvious that the synchrotron and thermal dust emission amplitude maps are not degenerate with any components. Rather than degeneracies, these low residual values indicate that the current data set is non-redundant with respect to these two amplitude maps; the 408 MHz map determines almost exclusively the synchrotron amplitude, and the 857-2 map determines almost exclusively the thermal dust amplitude. The main problem with these low residuals is therefore not degeneracies, but rather lack of robustness with respect to systematics; any systematic error that may be present in the 408 MHz and 857-2 channels can and will propagate into the respective foreground amplitude maps. In order to improve on this situation in the future, recovering the currently systematics contaminated 857-1, 857-3, and 857-4 channels is χ2 12 50 LM93 0 20 40 60 χ2 0 5 10 15 Number of pixels (103) PM61 mask χ2 20.3(x/1.36) Fig. 22. Top:χ2per pixel for joint baseline Planck, WMAP, and 408 MHz intensity analysis. Middle: confidence mask derived by smoothing the χ2map to 1◦FWHM, and thresholding at a value of χ2 max =50. Its primary application is for the low-`2015 Planck temperature likelihood, and it is accordingly denoted LM93 (93% likelihood mask); see Planck Collaboration XI (2016) for further details. Bottom: histogram of χ2values outside the conservative PM61 mask. The grey dashed line shows the best-fit χ2distribution with a variable degree of freedom and scaling, used to account for prior and noise modelling effects; see Sect. 5.3 for further discussion. imperative on the high-frequency side, and incorporating additional low-frequency observations (between, say, 1 and 20 GHz) is critical on the low-frequency side. A similar effect is seen for a number of other channels, if not equally strongly. For instance, we see that the WMAP K-band and Planck 30 GHz channels have rms ratios of 0.38 and 0.55, respectively, and these dominate the two spinning dust amplitudes, Asd1 and Asd2. The 100-ds2 channel has a ratio of 0.76, A10, page 34 of 63 Planck Collaboration: Planck 2015 results. X. Table 8. Goodness-of-fit statistics for the temperature analysis evaluated at 1◦FWHM. Rms outside LM93 Frac. res. inside Map [µK] σres ν/σinst νLM93 [%] Planck 30 . . . . . . . . . 1.56 0.55 0.08 44 . . . . . . . . . 2.51 0.83 0.42 70-ds1 . . . . . . 3.67 0.96 0.78 70-ds2 . . . . . . 3.88 0.97 0.82 70-ds3 . . . . . . 3.98 0.97 0.80 100-ds1 . . . . . 1.00 1.11 0.18 100-ds2 . . . . . 0.61 0.76 0.08 143-ds1 . . . . . 0.72 1.02 0.08 143-ds2 . . . . . 0.68 0.97 0.08 143-5 . . . . . . . 1.03 1.14 0.15 143-6 . . . . . . . 1.17 1.06 0.12 143-7 . . . . . . . 1.04 1.04 0.10 217-1 . . . . . . . 1.70 0.94 0.06 217-2 . . . . . . . 1.90 1.00 0.09 217-3 . . . . . . . 1.68 0.98 0.08 217-4 . . . . . . . 1.64 0.91 0.05 353-ds2 . . . . . 4.28 0.95 0.03 353-1 . . . . . . . 2.11 0.60 0.02 545-2 . . . . . . . 8.89a0.88 0.14 545-4 . . . . . . . 9.04a0.90 0.13 857-2 . . . . . . . 1.39a0.13 0.00 WMAP K . . . . . . . . . 2.26 0.38 0.03 Ka . . . . . . . . 4.52 1.05 0.56 Q1 . . . . . . . . 5.22 0.98 0.64 Q2 . . . . . . . . 5.09 0.99 0.67 V1 . . . . . . . . 6.29 0.98 1.70 V2 . . . . . . . . 5.62 1.02 1.37 W1 . . . . . . . . 7.90 0.89 1.44 W2 . . . . . . . . 9.74 0.96 1.74 W3 . . . . . . . . 9.57 0.90 1.67 W4 . . . . . . . . 10.15 1.00 1.77 Haslam 0.408 . . . . . . . 0.12b0.10 0.03 Notes. The second column shows the rms residual outside the 93% Commander likelihood mask for each channel, while the third column shows the same, but normalized with respect to the instrumental noise rms listed in Table 1. The fourth column lists the median fractional residual in the complementary 7% of the sky, covering the Galactic plane region. (a)Unit is kJy sr−1.(b)Unit is K. and defines the CO J=1→0 amplitude together with 100-ds1. Finally, the 353-1 channel has a ratio of 0.60, and this channel has the greatest pull on the dust emissivity index, βd. The single most important conclusion from Table 8, however, is that the baseline Planck temperature sky model is an excellent fit to the observed data, in agreement with the visual impression of Fig. 21. The residuals are largely consistent with instrumental noise over 93% of the sky, and the median fractional residual in the complementary 7% of the sky is; smaller than 0.2% for all HFI channels; smaller than 1% for all LFI channels; and smaller than 2% for all WMAP channels. 5.4. High-resolution component maps We now consider the high-resolution intensity maps derived using the same pipeline as above, but with a reduced data set and astrophysical model. Specifically, we only include channels from 143 GHz and above, all smoothed to a common resolution of 7. 05 FWHM. The signal model includes CMB, thermal dust, and CO J=2→1 and 3 →2 emission lines coadded into one map, similar to the Type-3 map in the 2013 data release15. We fix all global instrumental parameters on the values listed in Table 6, and the thermal dust temperature, Td, to the low-resolution solution, upgraded in harmonic space (to avoid pixelization effects) to a HEALPix resolution of Nside =2048. The only free spectral parameter per pixel is now the thermal dust emissivity index, βd. The resulting amplitude maps are shown in the top panels of Figs. 23–25, while the bottom panels show the so-called halfring half-difference maps, i.e., half-difference between two maps derived from independent half-ring maps (Planck Collaboration VI 2016;Planck Collaboration VIII 2016); these provide a direct estimate of the instrumental noise present in the full-mission maps. Figure 26 shows the same for the high-resolution dust spectral index. We note that while the Galactic centre appears negative in the low-resolution CMB solution shown in Fig. 7, it appears positive in the corresponding high-resolution CMB shown in Fig. 23. This qualitative difference demonstrates the importance of modelling errors in the Galactic plane. At high resolution, our model includes only CMB, CO and thermal dust, but no dedicated lowfrequency component. Any residual free-free contributions to frequencies at or above 143 GHz is therefore necessarily interpreted as a combination of CMB and CO, both of which have redder spectral shapes than thermal dust. In the low-resolution solution, on the other hand, the main problem lies in the interplay between CO and thermal dust modelling errors and highfrequency calibration uncertainties. 5.5. Comparison with independent data products To further validate the baseline model presented in Sect. 5.3, we now compare the derived products with similar maps produced either by independent observations or through different analysis techniques, focusing in particular on spinning and thermal dust and CO emission. Synchrotron and free-free (and spinning dust) are addressed separately in a dedicated companion paper, and we refer the interested reader to Planck Collaboration XXV (2016) for full details. A short summary of that analysis includes the following main points. 1. The synchrotron estimates derived by the WMAP team (Bennett et al. 2013) using either Markov chain Monte Carlo (MCMC) or maximum entropy methods (MEM) typically have 50–70% higher amplitudes at high Galactic latitudes compared to that derived in this paper, and this increases to factors of several when including the Galactic plane. This is compensated by about twice as much spinning dust in our model compared to the WMAP models. 2. The free-free model derived in the current analysis correlates well with Hαobservations. For instance, the scaling factor between the two maps in the Gum Nebula is (8.2±1.3) µK R−1. For comparison, Dickinson et al. (2003) found values ranging between 8.2 and 13.1µK R−1, depending on the exact position within the Gum Nebula. 3. The free-free map also shows good morphological agreement with respect to radio recombination line (RRL) observations (Alves et al. 2015), although the derived amplitude is notably higher in our map. The six brightest objects have a mean relative amplitude ratio of 1.14 ±0.04, whereas the ten next have an amplitude ratio of 1.36 ±0.08. Considering that RRLs are in principle a very clean probe of free-free 15 Although our high-resolution CO map formally is a weighted avarage of J=2→1 and J=3→2 line emission, the former vastly dominates, and we therefore refer to the map as CO J=2→1. A10, page 35 of 63 A&A 594, A10 (2016) 120◦60◦0◦300◦240◦ −75◦ −60◦ −45◦ −30◦ −15◦ 0◦ 15◦ 30◦ 45◦ 60◦ 75◦ Full ACMB −300 0 300 µKCMB 120◦60◦0◦300◦240◦ −75◦ −60◦ −45◦ −30◦ −15◦ 0◦ 15◦ 30◦ 45◦ 60◦ 75◦ ∆HR ACMB −30 0 30 µKCMB Fig. 23. High-resolution maximum-posterior (top) and half-ring half-difference (bottom) CMB amplitude maps. The latter provides an estimate of the instrumental noise in the primary map in the top panel. Both panels employ linear colour scales. A10, page 36 of 63 Planck Collaboration: Planck 2015 results. X. 120◦60◦0◦300◦240◦ −75◦ −60◦ −45◦ −30◦ −15◦ 0◦ 15◦ 30◦ 45◦ 60◦ 75◦ Full ACO21 0 10 100 KRJ km s−1 120◦60◦0◦300◦240◦ −75◦ −60◦ −45◦ −30◦ −15◦ 0◦ 15◦ 30◦ 45◦ 60◦ 75◦ ∆HR ACO21 −1 0 1 KRJ km s−1 Fig. 24. High-resolution maximum-posterior (top) and half-ring half-difference (bottom) CO J=2→1 amplitude maps. The latter provides an estimate of the instrumental noise in the primary map in the top panel. Note that the top panel employs a non-linear high dynamic range colour scale, while the bottom panel employs a regular linear colour scale. A10, page 37 of 63 A&A 594, A10 (2016) 120◦60◦0◦300◦240◦ −75◦ −60◦ −45◦ −30◦ −15◦ 0◦ 15◦ 30◦ 45◦ 60◦ 75◦ Full Ad 0.01 0.1 1 10 mKRJ @ 545 GHz 120◦60◦0◦300◦240◦ −75◦ −60◦ −45◦ −30◦ −15◦ 0◦ 15◦ 30◦ 45◦ 60◦ 75◦ ∆HR Ad −2 0 2 mKRJ @ 545 GHz Fig. 25. High-resolution maximum-posterior (top) and half-ring half-difference rms (bottom) thermal dust amplitude maps. The latter provides an estimate of the instrumental noise in the primary map in the top panel. Note that the top panel employs a non-linear high dynamic range colour scale, while the bottom panel employs a regular linear colour scale. A10, page 38 of 63 Planck Collaboration: Planck 2015 results. X. 120◦60◦0◦300◦240◦ −75◦ −60◦ −45◦ −30◦ −15◦ 0◦ 15◦ 30◦ 45◦ 60◦ 75◦ Full βd 1.3 1.4 1.5 1.6 1.7 1.8 120◦60◦0◦300◦240◦ −75◦ −60◦ −45◦ −30◦ −15◦ 0◦ 15◦ 30◦ 45◦ 60◦ 75◦ ∆HR βd -0.1 -0.05 0 0.05 0.1 Fig. 26. High-resolution maximum-posterior (top) and half-ring half-difference rms (bottom) thermal dust spectral index maps. The latter provides an estimate of the instrumental noise in the primary map in the top panel. Both panels employ linear colour scales. A10, page 39 of 63 A&A 594, A10 (2016) 30 100 300 Frequency [GHz] −0.4 −0.2 0.0 0.2 0.4 Normalized difference, (sData −sComm)/sData LFI HFI WMAP Fig. 27. Fractional difference between the template amplitudes derived when fitting the GASS H isurvey data at high Galactic latitudes to: (1) the raw Planck and WMAP temperature maps; and to (2) the sum of the spinning and thermal dust models derived by Commander in this paper. emission, the most likely explanation for this discrepancy is leakage between synchrotron, spinning dust, CO, and freefree in the current solution. On the other hand, the derived RRL amplitudes also depends sensitively on the assumed electron temperature, and raising /Te by ≈20% would resolve much of the difference. In addition, it is worth noting that other component separation techniques, including FastMEM, CCA, and both the 9-yr WMAP MCMC and MEM analyses, all derive free-free amplitudes consistent with the Commander result (Planck Collaboration Int. XXIII 2015). 5.5.1. Dust template amplitude consistency by H Icross-correlation We next perform an internal consistency test of the Commander dust model in the range from 23 to 353 GHz, as defined by the sum of the SpDust2 components described in Sect. 4.1 and thermal dust, by cross-correlating our dust model against GASS H iobservations (McClure-Griffiths et al. 2009;Kalberla et al. 2010) covering 18% of the high Galactic latitude sky near the South Galactic pole, following the procedure of Planck Collaboration Int. XVII (2014). In particular, we compare the resulting template amplitudes against the corresponding amplitudes derived directly from cross-correlation with the raw Planck and WMAP sky maps. Figure 27 shows the fractional difference between the resulting template amplitudes for each frequency band. Overall, the agreement is satisfactory with typically 20% differences in the 20–70 GHz range, in which spinning dust provides a larger contribution to the frequency spectrum than thermal dust. At higher frequencies, where thermal dust emission starts to dominate, the agreement improves further to around 5%, and to within 1 σin terms of statistical uncertainties. 5.5.2. Dust SED consistency by H Iand internal Planck cross-correlations Next, we consider the robustness and consistency of the thermal dust SED model, as parametrized in terms of the two greybody parameters, βdand Td. Specifically, we compare the 100 300 1000 3000 Frequency [GHz] −0.4 −0.2 0.0 0.2 0.4 Normalized difference, (sHI −sComm)/sComm βd= 1.54 Td= 22.8 K LFI (fitted) HFI (fitted) WMAP (fitted) DIRBE (not fitted) Fig. 28. Fractional difference of the mean thermal dust SEDs as derived by cross-correlation with the GASS H isurvey data at high Galactic latitudes, updated with the latest Planck 2015 temperature sky maps, (Planck Collaboration Int. XVII 2014) and by Commander in this paper. The dotted horizontal lines indicate fractional differences of ±10 %. For comparison purposes, we also show the extrapolation to the 100, 140, and 240 µm DIRBE frequencies. These observations are not included in the fits performed in this paper; see Sect. 5.5.2 for further discussion. new SED estimates derived in this paper with corresponding estimates derived by H icross-correlation at high latitudes in Planck Collaboration Int. XVII (2014), and by internal Planck cross-correlations at intermediate Galactic latitudes in Planck Collaboration Int. XXII (2015), both of which have been updated with the latest Planck 2015 sky maps. Figure 28 compares the mean thermal dust SED derived from H i–CMB cross-correlation and the Commander estimates at high Galactic latitudes in terms of the fractional difference, (sHi−sComm)/sComm. The two sets of best-fit thermal dust spectral parameters are (βd,TK)Comm =(1.54,22.8 K) and (βd,TK)Hi=(1.54,21.4 K), respectively, and the two models agree point-by-point to 5–10% between 100 and 857 GHz. At 70 GHz the difference is 50%, and this is due to different spinning dust modelling; as already shown in Fig. 27, the sum of spinning and thermal dust agree to 20% in this range between the two methods. Note also that 70 GHz is very close to the foreground minimum, and these numerically large relative differences therefore correspond to small absolute differences. This test provides a robust estimate of residual systematic errors in the Commander thermal dust model from potential zerolevel and dipole uncertainties in the high-frequency HFI channels arising from zodiacal light emission and CIB residuals, as discussed in Sect. 3. Because the H ianalysis is insensitive to such errors, we take the 1–2 K difference between the two as an estimate of the systematic uncertainty on the Commander thermal dust temperature at high Galactic latitudes. At frequencies above 857 GHz we also plot the extrapolation of the new Commander model into the COBE-DIRBE wavelengths of 240, 140, and 100 µm (Hauser et al. 1998). Here we clearly see that the current model breaks down beyond the Planck frequencies, with a fractional difference of 40% between the Commander model and the DIRBE 100 µm observations. Including the DIRBE channels in the fit would of course reduce these fractional residuals dramatically, but only at a very significant cost of increasing the residuals at lower frequencies A10, page 40 of 63 Planck Collaboration: Planck 2015 results. X. 1.3 1.4 1.5 1.6 1.7 1.8 Thermal dust index, βd 0 20 40 60 80 100 Number of regions Cross-corr; PIP XXII Commander Fig. 29. Comparison of the thermal dust spectral index, βd, estimated by internal Planck map cross-correlations over HEALPix Nside =8 pixels in Planck Collaboration Int. XXII (2015) and those presented in this paper. The best-fit Gaussian distributions to the two histograms have mean and standard deviations of βComm d=1.53 ±0.03 and βcross-corr d=1.51 ±0.06, respectively. between 70 and 353 GHz. The simple one-component greybody thermal dust model adopted in this paper is not able to simultaneously fit this entire frequency range, because of both intrinsic complexity of the dust particle population and because of residual systematics and calibration uncertainties in the DIRBE and high-frequency Planck data. Integration of these channels requires substantial additional work, and is beyond the scope of the current paper. For a first analysis of similar type, see Planck Collaboration Int. XXIX (2016). Next, we turn to intermediate Galactic latitudes, for which the signal-to-noise ratio is higher, but also the astrophysical composition is richer. In this case we therefore compare our results with the outputs from the internal Planck template crosscorrelation analysis of Planck Collaboration Int. XXII (2015). In short, this analysis estimates the SED parameters by crosscorrelating the Planck 353 GHz channel with lower frequencies over circular patches of 10◦radius. Figure 29 compares the histogram of βdderived using this method with the corresponding Commander estimates over the same sky region. The agreement is very good, and with averages and dispersions of βComm d=1.53 ±0.03 and βcross-corr d=1.51 ±0.06, respectively. When interpreting the widths of these distributions, it is useful to return to the thermal dust spectral index maps shown in Figs. 17 and 26. These maps are quite uniform, and, indeed, at the current level of leakage between thermal dust, CO, compact objects and residual offsets, there is little convincing evidence for true spatial variation in βdin the results presented here. If this conjecture is true, the widths of the histograms shown in Fig. 29 are primarily expressions of analysis uncertainties in the form of instrumental noise, parameter degeneracies and systematic errors, rather than true spatial variation. 5.5.3. CO line emission Finally, we compare the CO J=1→0, 2 →1 and 3 →2 maps derived in Sects. 5.3 and 5.4 with independently derived maps and products. As described in Planck Collaboration XIII (2014), Planck has implemented a multi-algorithm approach to CO extraction, configuring the MILCA (Hurier et al. 2013) and Commander algorithms for dedicated CO reconstruction. In 2013, this resulted in three different types of CO maps. In short, the Type-1 CO maps are built from individual bolometer maps within single frequencies, and as such are only weakly dependent on foreground extrapolations, but this insensitivity comes at a high cost in terms of instrumental noise. The Type-2 maps are built per CO line from a small sub-set of frequencies, carefully selected to be optimal for CO extraction. Since more than one frequency is involved, a more elaborate foreground model is required, such as explicit modelling of CMB, dust and freefree, although several simplifications are imposed, such as the assumption of constant dust temperature and spectral indices. Finally, the Type-3 map corresponds to a maximum signal-tonoise CO extraction in which a complete foreground model is fitted with Commander, as described in this paper, but with only a single CO amplitude per pixel and otherwise only spatially fixed line ratios accounting for scaling between frequencies. In the present release, the MILCA-based Type-1 and Type2 maps have been updated with the latest data, while the Commander-Ruler-based Type-3 map from 2013 has been superseded by the high-resolution Commander-only J=2→1 map presented in Sect. 5.4. In addition to these high-resolution maps, we of course also provide the low-resolution line maps discussed in Sect. 5.3. Table 9summarizes the CO-related data products provided in the current release, including angular resolution, instrumental noise, and analysis assumptions. We start by comparing the maps derived with MILCA and Commander, both with each other and with the CO J=1→0 survey presented by Dame et al. (2001). The full-sky J=1→0 and 2 →1 maps are shown in Figs. 30 and 31, while zoom-ins of the Orion region are shown in Fig. 32. All maps are smoothed to a common resolution of 150in these plots, except the Dame et al. survey, which has an intrinsic resolution of about 200. For reference, the 2013 Type-3 map is shown in the bottom panel of Fig. 30. The first three panels of Fig. 33 show T–Tcorrelation plots between each of the three CO line maps and the Dame et al. survey, all smoothed to a common resolution of 1◦FWHM; the fourth panel shows similar correlations between the highand low-resolution Planck products. Please note that all axes are linear, and this figure therefore highlights the very brightest CO objects on the sky. The main points to take away from these scatter plots, and the maps in Figs. 31 and 32, are the following. 1. The Type-2 and low-resolution Commander J=1→0 maps agree very well internally, and also correlate strongly with the Dame et al. survey. However, they both show an overall multiplicative scaling factor of about 1.4 relative to Dame et al. This level of amplitude difference is similar to what was observed in the 2013 release (Planck Collaboration XIII 2014), and is due to a combination of bandpass uncertainties in the Planck observations and the overall 10% calibration uncertainty in the Dame et al. survey. The Type1J=1→0 map shows bigger differences with respect to the Dame et al. survey, both in the scatter plot and the Orion zoom-in. Possible explanations include the presence of a second significant line emission mechanism, such as 13CO J=1→0 (at 110 GHz), or, possibly, thermal dust leakage. 2. In the CO J=2→1 case, the Type-2 map shows some evidence of contamination, both in the form of significant curvature in the scatter plot (top right panel of Fig. 33), and as notable diffuse emission along the Galactic plane in the Orion region and full-sky map. The agreement between the A10, page 41 of 63 A&A 594, A10 (2016) −10 −5 0 5 10 µKCMB −10 −5 0 5 10 µKCMB −20 −10 0 10 20 µKRJ @ 30 GHz −20 −10 0 10 20 µKRJ @ 30 GHz −20 −10 0 10 20 µKRJ @ 353 GHz −20 −10 0 10 20 µKRJ @ 353 GHz Fig. 37. 20◦×20◦polarization zooms centred on the south ecliptic pole with Galactic coordinates (l,b)=(276◦,−30◦) of CMB (top row), synchrotron (middle row), and thermal dust emission (bottom row). Left and right columns show Stokes Qand Uparameters, respectively. The object in the lower left quadrant is the Large Magellanic Cloud (LMC). A10, page 48 of 63 Planck Collaboration: Planck 2015 results. X. AP s 0 20 40 60 80 100 µKRJ @ 30 GHz AP d 3 10 30 100 µKRJ @ 353 GHz Fig. 38. Planck polarization amplitude maps, P=pQ2+U2. The top panel shows synchrotron emission at 30 GHz, smoothed to an angular resolution of 400, and the bottom panel shows thermal dust emission at 353 GHz, smoothed to an angular resolution of 100. A10, page 49 of 63 A&A 594, A10 (2016) Fig. 39. Planck polarization angle maps for synchrotron emission, smoothed to 400FHWM (top) and thermal dust emission, smoothed to 100 FWHM (bottom). Light blue and red colours indicate polarization angles aligned with meridians (ψ=0◦) and parallels (ψ=90◦), respectively, while yellow and purple indicate polarization angles rotated by −45 and +45◦with respect to the local meridian in the HEALPix polarization angle convention. Colours are saturated at 10 µKRJ. In the same region, we also clearly see the Large Magellanic Cloud (LMC) in both synchrotron and thermal dust emission (see Planck Collaboration XXV 2016 for a detailed analysis of the LMC), but only very weakly in the CMB maps. Indeed, the primary signature of the LMC is a slight increase in variance rather than a systematic bias. This is quite reassuring in terms of CMB reconstruction fidelity, since the LMC represents one of the richest astrophysical objects on the sky. In this respect it is worth recalling that we fix all calibration and spectral parameters (thermal dust index and temperature and synchrotron spectrum) in the polarization analysis to those derived in the temperature analysis. Several attempts have been made at re-estimating these parameter independently from the polarization observations, but we find that the resulting parameters invariably become significantly biased by the same large-scale systematics that are responsible for the remaining large-scale residuals in the CMB map (Planck Collaboration II 2016;Planck Collaboration VIII 2016;Planck Collaboration IX 2016). However, other analyses that explicitly exploit spatial correlations (e.g., template fitting) to suppress such systematics have been able to produce robust results, and yield only small differences between the temperature and polarization spectral A10, page 50 of 63 Planck Collaboration: Planck 2015 results. X. indices. For instance, Planck Collaboration Int. XXII (2015) reports full-sky estimates of βd=1.51 ±0.01 for intensity and βd=1.59 ±0.02 for polarization, corresponding to a difference of only 3.6σeven after averaging over most of the sky. Thus, assuming identical temperature and polarization spectral indicies is a very good approximation at the precision level of the current data, considering the additional stability with respect to instrumental systematic errors it provides. 6.1.1. Goodness-of-fit We now consider the statistical goodness-of-fit of this simple baseline model, following the same procedure as for the temperature analysis. First, Fig. 40 shows the residual maps dν−sν, for each of the seven Planck frequency maps, all smoothed to a common resolution of 400FWHM angular scale. Please note that the colour range is linear between ±5µK, and the same in all panels. As expected, we see that 143 GHz is the most sensitive frequency channel, followed by the 100 and 217 GHz channels. In addition to instrumental noise, these channels also exhibit large-scale patterns tracing the Planck scanning strategy at the .0.5µK level. Although small in an absolute sense, it is important to recall that the expected peak-to-peak amplitude of a cosmological signal from reionization corresponding to an optical depth of, say, τ≈0.07 is also about 0.5µK (Planck Collaboration XI 2016). As a result, we do not consider the CMB polarization map presented here suitable for cosmological parameter estimation on large angular scales. Instead, the Planck 2015 low-`polarization likelihood relies only on the 30, 70, and 353 GHz data, for which instrumental systematics are subdominant (Planck Collaboration VI 2016;Planck Collaboration VIII 2016;Planck Collaboration XI 2016). Next, the top panel of Fig. 41 shows the corresponding χ2map, co-adding over both frequencies and Stokes parameters. Compared to the temperature case, it is here much easier to determine the appropriate number of degrees of freedom, since no spectral parameters are fitted to the data, and no positivity priors are imposed on the amplitudes. Specifically, there are in total 14 data points (two Stokes parameters in seven frequencies) and 6 free parameters (two Stokes parameters in three components), resulting in a net 8 degrees of freedom. The nominal 95% confidence region for this number of degrees of freedom is χ2=(2,17). As usual, the Galactic plane is the most significant feature in the χ2map. Furthermore, when comparing this χ2map (and the individual frequency residual maps) with the various component amplitude maps derived in the temperature analysis, we find strong correlations between the χ2map and the CO emission maps. This is expected from the mapmaking analyses presented in Planck Collaboration XI (2016), and, as already noted, a general recommendation regarding these maps is to reject any pixels with significant CO intensity contribution, because of temperature-to-polarization leakage. The Commander polarization mask (CPM) is accordingly defined as the product of the (smoothed and thresholded) χ2map shown in the top panel of Fig. 41, and the low-resolution Commander CO J=1→0 intensity map thresholded at 0.5 KRJ km s−1. The resulting mask is shown in the middle panel of Fig. 41, and excludes 17% of the sky. The bottom panel of Fig. 41 shows a histogram of the χ2 values outside the CPM83 mask, with the best-fit χ2distribution with variable number of degrees of freedom and width, fully analogous to the temperature case in Sect. 5.3. In this case, the best-fit distribution has 7.9 degrees of freedom, in excellent 30 Q 30 U 44 Q 44 U 70 Q 70 U 100 Q 100 U 143 Q 143 U 217 Q 217 U 353Q −5 0 5 µK 353U −5 0 5 µK Fig. 40. Polarization residual maps, dν−sν. Each row corresponds to one frequency map, with 30 GHz in the top row and 353 GHz in the bottom row; left and right columns show the Stokes Qand Uparameters, respectively. All panels employ the same linear colour scale. agreement with the theoretical expectation of 8, while the width rescaling factor that accounts for correlated noise and smoothing is 1.15, indicating that the white noise approximation underestimates the noise by 15% due to correlations. Table 10 lists the rms of the residual maps for each frequency, analogous to Table 8for temperature, averaged over the two Stokes Qand Uparameters. The third column in this table shows the ratio between these rms values and instrumental noise; again, we observe good agreement with expectations. As for the temperature case, the values for the 30 and 353 GHz channels are significantly lower than unity, because these two frequencies dominate the synchrotron and thermal dust amplitude parameters, respectively. A10, page 51 of 63 A&A 594, A10 (2016) χ2 0 26 CPM83 0 10 20 30 χ2 0 5 10 15 Number of pixels (103) CPM83 mask χ2 7.9(x/1.15) Fig. 41. Top:χ2per pixel for the polarization analysis of Planck observations between 30 and 353 GHz, summed over Stokes Qand Uparameters. Middle:Commander polarization mask (CPM), defined as the product of the CO J=1→0 emission map thresholded at 0.5 KRJ km s−1, and the smoothed χ2map thresholded at a value of 26. This mask retains a total of 83% of the sky. Bottom: histogram of χ2values outside the conservative CPM83 mask. The grey dashed line shows the best-fit χ2distribution with a variable degree of freedom and scaling, used to account for noise modelling effects. Next, we assess the impact of temperature-to-polarization leakage from the CMB temperature monopole and dipole and from Galactic temperature emission by computing the synchrotron and thermal dust amplitude maps when adopting two different HFI leakage models. The first is simply the default template set adopted for the Planck 2015 release, corresponding to the results already discussed, while the second is the experimental template set described in Planck Collaboration VIII (2016) and Planck Collaboration XI (2016). From the resulting Table 10. Goodness-of-fit statistics for polarization analysis. Rms outside CPM83 Frequency [GHz] σres ν[µK] σres ν/σinst ν 30 . . . . . . . . . . . 2.12 0.28 44 . . . . . . . . . . . 7.59 1.01 70 . . . . . . . . . . . 4.85 1.01 100 . . . . . . . . . . . 1.18 0.90 143 . . . . . . . . . . . 0.90 0.81 217 . . . . . . . . . . . 1.52 0.95 353 . . . . . . . . . . . 2.93 0.42 amplitude maps, we perform the following steps: first compute the polarization amplitude, P; smooth this to 3◦FWHM for synchrotron and 1◦FWHM for thermal dust; and finally compute the difference, P2−P1, and fractional difference, (P2−P1)/P1, maps. These are shown in Fig. 42. Here we see that the absolute polarization amplitude difference between the two leakage models is around 1 µKRJ for both synchrotron and thermal dust at high Galactic latitudes, increasing to a few tens of µKRJ in the Galactic plane. Accordingly, the fractional residuals are .10% at high Galactic latitudes, and they increase to about 30% in the central Galactic plane. Finally, we comment on the polarization fractions that may be derived from these maps. First of all, we emphasize that the delivered products are maps of the Stokes Qand Uparameters, not of polarization amplitude and polarization angle and fractions. This choice is primarily driven by the fact that the Stokes parameters are linear, and therefore have much simpler noise properties than the corresponding nonlinear parameters. Second, when computing polarization fractions, P/I, it is of utmost importance to recognize and account for the considerable uncertainty in this quantity with respect to the zero-level of the corresponding temperature map. To make this point concrete, we show in the top panel of Fig. 43 the naive polarization fraction derived directly from the delivered Commander thermal dust intensity and polarization maps. This map saturates the colour scale over extended regions in the southern Galactic hemisphere, nominally suggesting a polarization fraction well above 20%. However, as discussed both in Sect. 3and Planck Collaboration VIII (2016), there is an offset in the zero-level of the zodiacal light emission of 34 µK in the current 353 GHz temperature data. Correcting for this offset results in the polarization fraction shown in the lower panel of Fig. 43, which shows significantly smaller values. Further, the raw statistical uncertainty of the 353 GHz Galactic emission zero level from H icross-correlation alone is 23 µK (Planck Collaboration VIII 2016). The conclusion is therefore that any analysis that relies directly on the polarization fraction, as opposed to the much more stable polarization amplitude, needs to account for the significant uncertainties in the Galactic emission zero-level at 353 GHz. 6.2. Synchrotron and thermal dust angular power spectra One of the most important goals of modern CMB cosmology is to detect primordial B-mode polarization on large angular scales, a direct observable signature of inflationary gravitational waves. The main obstacles in this search are the polarized synchrotron and thermal emission signals discussed above. In order to quantify the magnitude of this problem, we compute in this section their angular power spectra, and compare them to the expected A10, page 52 of 63 Planck Collaboration: Planck 2015 results. X. Ps 2−Ps 1 −1 0 1 µKRJ @ 353 GHz Pd 2−Pd 1 −1 0 1 µKRJ @ 30 GHz Ps 2−Ps 1 Ps 1 −0.1 0 0.1 Pd 2−Pd 1 Pd 1 −0.1 0 0.1 Fig. 42. Difference maps (top) and fractional difference maps (bottom) between the synchrotron (left) and thermal dust (right) polarization solutions derived with two different HFI temperature-to-polarization leakage templates. The synchrotron polarization amplitude maps are smoothed to 3◦FHWM before computing absolute and fractional differences, and the thermal dust polarization amplitude maps are smoothed to 1◦. Maps labelled by a subscript “1” correspond to the default leakage templates used in the Planck 2015 release, and maps labelled by a subscript “2” correspond to the experimental leakage templates; see Planck Collaboration VIII (2016) for further discussion. primordial CMB spectrum. A more comprehensive analysis of the same type, but based only on the 353 GHz frequency channel, was recently published in Planck Collaboration Int. XXX (2016). We employ the same cross-spectrum power spectrum estimator as used in Planck Collaboration Int. XXX (2016), but introduce two specific changes. First, we adopt the so-called common mask from the CMB component separation analysis presented in Planck Collaboration IX (2016), rather than the CO mask employed in the original paper, and second, we consider three different mask apodization scales (1, 2, and 5◦FWHM) as opposed to only 5◦FWHM as in Planck Collaboration Int. XXX (2016). The EE and BB spectra resulting from the evaluation using 1◦FWHM apodization are shown in the left and right panels of Fig. 44, respectively, both plotted in terms of D`=C``(`+1)/2π in thermodynamic units. Red data points show the angular power spectra for thermal dust emission at 353 GHz, and green points show synchrotron emission at 30 GHz. Each spectrum is binned with ∆`=25, and the plotted uncertainties are the standard deviation of the single-`spectrum values within each bin. Black solid lines indicate the best-fit ΛCDM spectrum (Planck Collaboration XI 2016), and (in the BB panel only) the dashed black line shows the spectrum for a tensor-to-scalar ratio of r=0.05. Dotted coloured lines indicate the best-fit power-law fit, D`=q(`/80)α, to each foreground spectrum, where the pivot scale of `0=80 is chosen to match that used in Planck Collaboration Int. XXX (2016). The corresponding best-fit parameters are tabulated in Table 11 for all three apodization scales, and including multipoles in the range `=(10,150) for synchrotron and `= (10,300) for thermal dust emission. No synchrotron results are shown for the 5◦FWHM apodization scale. In this case, the effective sky fraction is too small to allow a robust estimate of the synchrotron spectrum. For thermal dust emission these parameters may be compared to the results presented in Planck Collaboration Int. XXX (2016), although a few caveats are in order. In particular: (1) the masks used in the two analyses are different, and the mask adopted in this paper effectively removes more sky around bright point sources after apodization; (2) the map considered in the present analysis is the Commander thermal dust map, whereas the original analysis considered the raw 353 GHz map; (3) the multipole ranges adopted for the parameter fits are slightly different; and (4) we make the fit to the single-`power spectrum, not the binned spectrum. Nevertheless, we see that the results derived here are in good agreement with those found in Planck Collaboration Int. XXX (2016). In particular, when considering the same apodization scale of 5◦FWHM, we recover an identical BB/EE ratio of A10, page 53 of 63 A&A 594, A10 (2016) 0µK 34 µK 0 0.2 Fig. 43. Thermal dust polarization fraction for Galactic emission zerolevel corrections of 0 µK (top) and 34 µK (bottom). A value of 34 µK corresponds to our current best estimate of the residual zodiacal light offset in the 353 GHz channel (Planck Collaboration VIII 2016). The statistical uncertainty on the Galactic emission zero-level from H i cross-correlation is 0.0067 MJy sr−1or 23 µKCMB. 0.53 ±0.01, and the EE power-law index agree to 0.5σ. For BB, the spectral index difference is slightly larger, but still within 2σ. The power spectrum amplitudes, on the other hand, are different because of the different effective sky fractions of the two corresponding apodized masks. Comparing the different apodization scales, we note both that the BB/EE ratio increases slightly, and that the power-law indices steepens slightly, as the mask smoothing scale increases. This is due to thermal dust emission being a highly non-isotropic and non-Gaussian field, as discussed in Planck Collaboration Int. XXX (2016). It is not surprising that its statistical properties may vary between the Galactic plane and the high Galactic latitudes. In addition, there is an algorithmic uncertainty in the form of so-called E-to-Bleakage, due to ambiguous polarization modes near the mask edge. This leakage is far stronger for foregrounds than for CMB, simply because the foreground field by construction is at its maximum near the mask boundary. As a result, it is important to specify the properties of the analysis mask when summarizing the power spectrum of a foreground field, as demonstrated in Table 11. Overall, however, the mask dependence on the angular power spectrum is modest, and D`provides a useful summary for foreground fields as well as for the CMB field. Indeed, one of the interesting results reported by Planck Collaboration Int. XXX (2016) was the asymmetry between the Band E-mode thermal dust power spectra, with a power ratio of BB/EE ≈0.5. This has strong implications for the underlying astrophysics, and indicates the presence of significant filamentary structures on intermediate angular scales. In this paper, we find that the same holds also for synchrotron emission, with an even stronger Table 11. Best-fit power-law parameters to the angular power spectra of synchtrotron (at 30 GHz) and thermal dust emission (at 353 GHz) as a function of mask apodization. Synchrotron Thermal dust q[µK2 CMB]αq[µK2 CMB]α Common mask; apod =1◦FWHM; feff sky =0.73 EE . . . . . . 3.7±0.2−0.44 ±0.07 354 ±6−0.53 ±0.02 BB . . . . . . 1.3±0.2−0.31 ±0.13 208 ±4−0.59 ±0.02 BB/EE . . . 0.36 ±0.06 0.59 ±0.01 Common mask; apod =2◦FWHM; feff sky =0.68 EE . . . . . . 3.2±0.2−0.49 ±0.08 285 ±5−0.53 ±0.02 BB . . . . . . 1.1±0.2−0.02 ±0.17 161 ±3−0.62 ±0.02 BB/EE . . . 0.34 ±0.07 0.56 ±0.01 Common mask; apod =5◦FWHM; feff sky =0.55 EE . . . . . . 188 ±3−0.44 ±0.02 BB . . . . . . 99 ±2−0.51 ±0.03 BB/EE . . . 0.53 ±0.01 CO mask; apod =5◦FWHM; feff sky =0.73; Planck Int. XXX (2014) EE . . . . . . 328 ±3−0.43 ±0.02 BB . . . . . . ··· −0.46 ±0.02 BB/EE . . . 0.53 ±0.01 Notes. The parameters are defined by the model D`=q(`/80)α, and the fits include multipoles between `=10 and 150 for synchrotron, and between `=10 and 300 for thermal dust emission. All uncertainties are statistical, and do not account for systematic or modelling uncertainties. The last case is reproduced from Table 1 in Planck Collaboration Int. XXX (2016). asymmetry of BB/EE ≈0.35. Thus, polarized synchrotron emission appears to be more strongly aligned along filamentary structures than thermal dust. We also find similar power-law indices for synchrotron emission as for thermal dust, with α≈ −0.4. However, the uncertainties are relatively larger, because of the lower signal-to-noise ratio of the 30 GHz channel compared to the 353 GHz channel. These power-law fits can be used to model the total foreground level as a function of both multipole moment and frequency. This is illustrated in Fig. 45 for the 1◦FWHM apodization case in terms of iso-contour plots of the following amplitude ratio, f=sDs `(ν)+Dd `(ν) DCMB ` (21) =v u tqs` 80 αsss(ν) ss(30 GHz) +qd` 80 αdsd(ν) sd(353 GHz) DCMB ` ,(22) where subscripts “s” and “d” refer to synchrotron and thermal dust. The frequency spectra, ss(ν) and sd(ν), are the synchrotron (GALPROP) and thermal dust (one-component greybody) spectra defined in Table 4converted to thermodynamic units, with parameters defined by the average parameters listed in Table 5. This function is thus simply a model of the foreground-to-CMB amplitude ratio as a function of multipole and frequency. Considering first the EE case shown in the left panel of Fig. 45, we note several interesting features. First, the horizontal ripples seen at `&100 correspond to the CMB acoustic oscillations. Next, we see that the foregrounds-to-CMB ratio is smaller than unity for all multipoles above `&40 for frequencies around 70 GHz, and smaller than 10% for `&200. Also, recall that the A10, page 54 of 63 Planck Collaboration: Planck 2015 results. X. 30 100 300 Multipole moment, ` 10−310−210−1100101102103 Power spectrum, D`[µK2] EE Best-fit ΛCDM Thermal dust @ 353 GHz Synchrotron @ 30 GHz 30 100 300 Multipole moment, ` 10−310−210−1100101102103 Power spectrum, D`[µK2] BB r= 0.00 r= 0.05 Thermal dust @ 353 GHz Synchrotron @ 30 GHz Fig. 44. Angular EE (left panel) and BB (right panel) power spectra for polarized synchrotron (at 30 GHz) and thermal dust emission (at 353 GHz), evaluated with 1◦FWHM apodization and including a total effective sky fraction of 73% of the sky. The dashed lines show the best-fit powerlaw models to each case, and the solid black lines shows the best-fit ΛCDM power spectrum as fitted to temperature observations only (Planck Collaboration XI 2016;Planck Collaboration XIII 2016). The dashed black line in the BB panel shows the spectrum for a model with a tensor-toscalar ratio of r=0.05. 30 100 300 Frequency [GHz] 2 10 100 1000 Multipole moment, ` EE 0.1 0.3 1 3 10 10 10 30 30 100 100 30 100 300 Frequency [GHz] 2 10 100 1000 Multipole moment, ` BB 1 3 10 30 30 100 100 r= 0.00 r= 0.05 Fig. 45. Amplitude ratio between total polarized foregrounds and CMB as a function of both multipole moment and frequency, defined by f(`, ν)=[Cfg `(ν)/CCMB `]1/2, as defined Eq. (22) with parameters derived from 73% of the sky. The left and right panels show the EE and BB spectra, and the black and red contours in the latter corresponds to tensor-to-scalar ratios of r=0.0 and 0.05, respectively. corresponding power spectrum ratio goes as the square of these ratios, and we thus find that polarized foregrounds have a small effect on the EE spectrum at multipoles above a few hundred, in agreement with the results pesented in Planck Collaboration XI (2016). However, we also see that the same is by no means true at low multipoles; the foregrounds-to-CMB ratio is larger than 3 throughout the reionization peak for `=2–10. The right panel of Fig. 45 shows the corresponding ratio for BB, but in this case two different contour sets are plotted; one for the standard ΛCDM with a vanishing tensor-to-scalar ratio (black contours), and one with a tensor-to-scalar ratio of r=0.05 (red contours). The peak around `≈1000 corresponds to the signature of weak gravitational lensing, converting E-modes into B-modes, whereas the “plateau” at low multipoles in the red contours corresponds to additional primordial fluctuations from inflationary gravitational waves. First of all, we see that foregrounds are sub-dominant to the lensing signal at multipoles above `&200 for frequencies around 70 GHz in this model, although they never fall below the 10% level. Second, for a vanishing tensor-to-scalar ratio the foreground-to-CMB around the recombination peak of `≈100 is about 3, and at the reionization peak, below `.10, it is about 100. Increasing the tensorto-scalar ratio to r=0.05 decreases these numbers to about 2 and 20, respectively. Before concluding this section, several caveats regarding the above observations are in order. First of all, it is important to remember that the angular power spectra reported here are computed over a large sky fraction including 72% of the sky. For a dedicated B-mode experiment, it obviously makes sense to consider more conservative masks. Second, it is also important to bear in mind that the angular spectra presented here covers only a limited multipole range, and the extrapolation to small angular scales is therefore associated with considerable uncertainty. Clearly, extrapolating actual observations that are made between A10, page 55 of 63 A&A 594, A10 (2016) AQ s AU s 0.39×KQ 0.39×KU ∆Q -20 -10 0 10 20 µKRJ @ 30 GHz ∆U -20 -10 0 10 20 µKRJ @ 30 GHz Fig. 46. Comparison of the Planck polarized synchrotron map (top) and the 9-yr WMAP K-band map, scaled to 30 GHz assuming a spectral index of βs=−3.2 (middle); the bottom row shows the difference between the two maps. All maps are smoothed to a common resolution of 2◦FWHM. `≈10–100 to `≈1000 for synchrotron emission implies strong assumptions regarding the foreground composition of both diffuse foregrounds and compact objects. 6.3. Comparison with independent data products We now turn to consistency tests based on external (or at least independently derived) data products. Of course, given the pioneering nature of the Planck polarization observations, the number of available external cross-checks is significantly sparser compared to the temperature case. On the low-frequency side, the WMAP K-band data represents an excellent comparison for the synchrotron map, while no products of comparable data quality exists on the high-frequency side at the moment. This lack of polarized dust measurements has of course been a major limitation for the entire CMB field for a long time, and the WMAP solution to this problem was to construct a polarized dust template by combining the FDS thermal dust intensity map (Finkbeiner et al. 1999) with polarization directions from starlight polarization observations (see Page et al. 2007 for full details). We compare our new polarized thermal dust and synchrotron maps with the WMAP maps/templates in Figs. 46 and 47, and show corresponding T–Tscatter plots in Figs. 48 and 49. Starting with the synchrotron case, we see first of all in Fig. 48 that the overall pixel-to-pixel scatter between the Planck synchrotron map and the WMAP K-band map is substantial. Indeed, based on this full-sky scatter plot, any synchrotron spectral index between βs=−3.4 and −3.0 appears consistent with the observations. Adopting a mean value of βs= −3.2, and assuming an effective K-band frequency of 22.6 GHz, this translates into a total scaling factor of 0.39 between Kband and 30 GHz16. This scaling factor has been applied to the K-band map shown in Fig. 46, and it also allows us to form a 16 The Planck 2015 foreground product maps are defined at sharp frequencies, and not as bandpass-averaged channel maps. The relevant comparison for synchrotron emission is therefore indeed 30 GHz, and not the effective frequency of the Planck 30 GHz band, which is 28.4 GHz. A10, page 56 of 63 Planck Collaboration: Planck 2015 results. X. AQ d AU d AQ WMAP AU WMAP ∆Q -20 -10 0 10 20 µKRJ @ 353 GHz ∆U -20 -10 0 10 20 µKRJ @ 353 GHz Fig. 47. Comparison of the Planck polarized thermal dust map at 353 GHz (top) and the WMAP polarized dust template, scaled to 353 GHz assuming a scaling factor of 480 µK (middle). The bottom row shows the difference between the two maps. All maps are pixelized at a HEALPix resolution of Nside =16. meaningful residual map, as seen in the bottom row of the same figure. The relative residuals between the Planck and WMAP synchrotron maps are clearly substantial, with amplitudes reaching 5 µK at high Galactic latitudes, and with a morphology clearly associated with the scanning strategy of either Planck or WMAP, both of which have symmetries defined by the Ecliptic reference frame. Furthermore, the residuals are very large-scale in nature, and obviously dominated by the two lowest multipoles, `=2 and 3. It is therefore natural to consider what effects may cause such large-scale features. Starting with Planck, a large suite of null-tests and simulations, specifically targeting large-scale systematics in the LFI observations, is presented in Planck Collaboration II (2016). One noteworthy conclusion from that work is a significant null-test failure in the 44 GHz polarization frequency map for `=2–4, and for two 70 GHz surveys. In addition, the HFI channels between 100 and 217 GHz are also affected by low level residual systematics (Planck Collaboration VIII 2016). As a result, these observations are currently excluded from the Planck 2015 low-`likelihood, which instead only relies on the 30, 70 and 353 GHz channels (Planck Collaboration XI 2016). These large-scale 44 GHz modes are likely to contribute significantly to the residuals seen in Fig. 46. For WMAP, on the other hand, the statistical uncertainties in the WMAP EE `=2 and BB `=3 modes are very large (Bennett et al. 2013), due to the combination of the differential detectors of WMAP, and an opening angle of 141◦ between the A and B side reflectors. Although these uncertainties are appropriately described by the low-resolution WMAP covariance matrices, it is algorithmically non-trivial to account for this effect properly in component separation at higher resolution, and they are also likely to contribute to the residuals in Fig. 46. These differences are also discussed in Planck Collaboration XXV (2016), with consistent conclusions. However, that analysis proceeds with co-adding the Planck and WMAP data sets A10, page 57 of 63