Planck 2013 results. XV. CMB power spectra and likelihood
^{1} APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France
^{2} Aalto University Metsähovi Radio Observatory, Metsähovintie 114, 02540 Kylmälä, Finland
^{3} African Institute for Mathematical Sciences, 68 Melrose Road, 7945 Muizenberg, Cape Town, South Africa
^{4} Agenzia Spaziale Italiana Science Data Center, via del Politecnico snc, 00133 Roma, Italy
^{5} Agenzia Spaziale Italiana, Viale Liegi 26, 00198 Roma, Italy
^{6} Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, UK
^{7} Astrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZuluNatal, Westville Campus, Private Bag X54001, 4000 Durban, South Africa
^{8} Atacama Large Millimeter/submillimeter Array, ALMA Santiago Central Offices, Alonso de Cordova 3107, Vitacura, Casilla 763 0355 Santiago, Chile
^{9} CITA, University of Toronto, 60 St. George St., Toronto, ON M5S 3H8, Canada
^{10} CNRS, IRAP, 9 Av. colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France
^{11} California Institute of Technology, Pasadena, California, USA
^{12} Centre for Theoretical Cosmology, DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
^{13} Centro de Estudios de Física del Cosmos de Aragón (CEFCA), Plaza San Juan, 1, planta 2, 44001 Teruel, Spain
^{14} Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, California, USA
^{15} Consejo Superior de Investigaciones Científicas (CSIC), 28006 Madrid, Spain
^{16} DSM/Irfu/SPP, CEASaclay, 91191 GifsurYvette Cedex, France
^{17} DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
^{18} Département de Physique Théorique, Université de Genève, 24 Quai E. Ansermet, 1211 Genève 4, Switzerland
^{19} Departamento de Física Fundamental, Facultad de Ciencias, Universidad de Salamanca, 37008 Salamanca, Spain
^{20} Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo s/n, Oviedo, Spain
^{21} Department of Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, Ontario, Canada
^{22} Department of Astrophysics/IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
^{23} Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California, USA
^{24} Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada
^{25} Department of Physics and Astronomy, Dana and David Dornsife College of Letter, Arts and Sciences, University of Southern California, Los Angeles CA 90089, USA
^{26} Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
^{27} Department of Physics, Florida State University, Keen Physics Building, 77 Chieftan Way, Tallahassee, Florida, USA
^{28} Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, 00014 Helsinki, Finland
^{29} Department of Physics, Princeton University, Princeton, New Jersey, USA
^{30} Department of Physics, University of California, Berkeley, California, USA
^{31} Department of Physics, University of California, One Shields Avenue, Davis, California, USA
^{32} Department of Physics, University of California, Santa Barbara, California, USA
^{33} Department of Physics, University of Illinois at UrbanaChampaign, 1110 West Green Street, Urbana, Illinois, USA
^{34} Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi di Padova, via Marzolo 8, 35131 Padova, Italy
^{35} Dipartimento di Fisica e Astronomia, Università degli Studi di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
^{36} Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, via Saragat 1, 44122 Ferrara, Italy
^{37} Dipartimento di Fisica, Università La Sapienza, P.le A. Moro 2, 000185 Roma, Italy
^{38} Dipartimento di Fisica, Università degli Studi di Milano, via Celoria, 16, 20133 Milano, Italy
^{39} Dipartimento di Fisica, Università degli Studi di Trieste, via A. Valerio 2, 34127 Trieste, Italy
^{40} Dipartimento di Fisica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, 00133 Roma, Italy
^{41} Dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, 000133 Roma, Italy
^{42} Discovery Center, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
^{43} Dpto. Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain
^{44} European Southern Observatory, ESO Vitacura, Alonso de Cordova 3107, Vitacura, Casilla 19001 Santiago, Chile
^{45} European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo, s/n, Urbanización Villafranca del Castillo, Villanueva de la Cañada, Madrid, Spain
^{46} European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
^{47} Finnish Centre for Astronomy with ESO (FINCA), University of Turku, Väisäläntie 20, 21500, Piikkiö, Finland
^{48} Haverford College Astronomy Department, 370 Lancaster Avenue, Haverford, Pennsylvania, USA
^{49} Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, 00014 Helsinki, Finland
^{50} INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy
^{51} INAF – Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monte Porzio Catone, Italy
^{52} INAF – Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, 34131 Trieste, Italy
^{53} INAF Istituto di Radioastronomia, via P. Gobetti 101, 40129 Bologna, Italy
^{54} INAF/IASF Bologna, via Gobetti 101, 40129 Bologna, Italy
^{55} INAF/IASF Milano, via E. Bassini 15, 23011 Milano, Italy
^{56} INFN, Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
^{57} INFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185 Roma, Italy
^{58} IPAG: Institut de Planétologie et d’Astrophysique de Grenoble, Université Joseph Fourier, Grenoble 1/CNRSINSU, UMR 5274, Grenoble, 38041, France
^{59} ISDC Data Centre for Astrophysics, University of Geneva, ch. d’Ecogia 16, Versoix, Switzerland
^{60} IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India
^{61} Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, UK
^{62} Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA
^{63} Institut Néel, CNRS, Université Joseph Fourier Grenoble I, 25 rue des Martyrs, 38042 Grenoble, France
^{64} Institut Universitaire de France, 103 bd SaintMichel, 75005 Paris, France
^{65} Institut d’Astrophysique Spatiale, CNRS (UMR 8617) Université ParisSud 11, Bâtiment 121, 91405 Orsay, France
^{66} Institut d’Astrophysique de Paris, CNRS (UMR 7095), 98bis Bd Arago, 75014 Paris, France
^{67} Institute for Space Sciences, 077125 BucharestMagurale, Romania
^{68} Institute of Astronomy and Astrophysics, Academia Sinica, 10617 Taipei, Taiwan
^{69} Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
^{70} Institute of Mathematics and Physics, Centre for Cosmology, Particle Physics and Phenomenology, Louvain University, 1348 LouvainlaNeuve, Belgium
^{71} Institute of Theoretical Astrophysics, University of Oslo, Blindern, 0315 Oslo, Norway
^{72} Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, 38205 La Laguna, Tenerife, Spain
^{73} Instituto de Física de Cantabria (CSICUniversidad de Cantabria), Avda. de los Castros s/n, 39005 Santander, Spain
^{74} Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, USA
^{75} Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
^{76} Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
^{77} LAL, Université ParisSud, CNRS/IN2P3, 91898 Orsay, France
^{78} LERMA, CNRS, Observatoire de Paris, 61 Avenue de l’Observatoire, 75014 Paris, France
^{79} Laboratoire AIM, IRFU/Service d’Astrophysique – CEA/DSM – CNRS – Université Paris Diderot, Bât. 709, CEASaclay, 91191 GifsurYvette Cedex, France
^{80} Laboratoire Traitement et Communication de l’Information, CNRS (UMR 5141) and Télécom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France
^{81} Laboratoire de Physique Subatomique et de Cosmologie, Université Joseph Fourier Grenoble I, CNRS/IN2P3, Institut National Polytechnique de Grenoble, 53 rue des Martyrs, 38026 Grenoble Cedex, France
^{82} Laboratoire de Physique Théorique, Université ParisSud 11 & CNRS, Bâtiment 210, 91405 Orsay, France
^{83} Lawrence Berkeley National Laboratory, Berkeley, California, USA
^{84} MaxPlanckInstitut für Astrophysik, KarlSchwarzschildStr. 1, 85741 Garching, Germany
^{85} McGill Physics, Ernest Rutherford Physics Building, McGill University, 3600 rue University, Montréal, QC, H3A 2T8, Canada
^{86} MilliLab, VTT Technical Research Centre of Finland, Tietotie 3, 02044 Espoo, Finland
^{87} National University of Ireland, Department of Experimental Physics, Maynooth, Co. Kildare, Ireland
^{88} Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
^{89} Observational Cosmology, Mail Stop 36717, California Institute of Technology, Pasadena CA 91125, USA
^{90} Optical Science Laboratory, University College London, Gower Street, London, UK
^{91} SBITPLPPC, EPFL, 1015 Lausanne, Switzerland
^{92} SISSA, Astrophysics Sector, via Bonomea 265, 34136 Trieste, Italy
^{93} School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK
^{94} School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK
^{95} Space Sciences Laboratory, University of California, Berkeley, California, USA
^{96} Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, 369167 KarachaiCherkessian Republic, Russia
^{97} Stanford University, Dept of Physics, Varian Physics Bldg, 382 via Pueblo Mall, Stanford, California, USA
^{98} SubDepartment of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
^{99} Theory Division, PHTH, CERN, 1211, Geneva 23, Switzerland
^{100} UPMC Univ. Paris 06, UMR7095, 98bis Boulevard Arago, 75014 Paris, France
^{101} Université de Toulouse, UPSOMP, IRAP, 31028 Toulouse Cedex 4, France
^{102} University of Granada, Departamento de Física Teórica y del Cosmos, Facultad de Ciencias, 18071 Granada, Spain
^{103} Warsaw University Observatory, Aleje Ujazdowskie 4, 00478 Warszawa, Poland
Received: 26 March 2013
Accepted: 5 May 2014
This paper presents the Planck 2013 likelihood, a complete statistical description of the twopoint correlation function of the CMB temperature fluctuations that accounts for all known relevant uncertainties, both instrumental and astrophysical in nature. We use this likelihood to derive our best estimate of the CMB angular power spectrum from Planck over three decades in multipole moment, ℓ, covering 2 ≤ ℓ ≤ 2500. The main source of uncertainty at ℓ ≲ 1500 is cosmic variance. Uncertainties in smallscale foreground modelling and instrumental noise dominate the error budget at higher ℓs. For ℓ < 50, our likelihood exploits all Planck frequency channels from 30 to 353 GHz, separating the cosmological CMB signal from diffuse Galactic foregrounds through a physically motivated Bayesian component separation technique. At ℓ ≥ 50, we employ a correlated Gaussian likelihood approximation based on a finegrained set of angular crossspectra derived from multiple detector combinations between the 100, 143, and 217 GHz frequency channels, marginalising over power spectrum foreground templates. We validate our likelihood through an extensive suite of consistency tests, and assess the impact of residual foreground and instrumental uncertainties on the final cosmological parameters. We find good internal agreement among the highℓ crossspectra with residuals below a few μK^{2} at ℓ ≲ 1000, in agreement with estimated calibration uncertainties. We compare our results with foregroundcleaned CMB maps derived from all Planck frequencies, as well as with crossspectra derived from the 70 GHz Planck map, and find broad agreement in terms of spectrum residuals and cosmological parameters. We further show that the bestfit ΛCDM cosmology is in excellent agreement with preliminary PlanckEE and TE polarisation spectra. We find that the standard ΛCDM cosmology is well constrained by Planck from the measurements at ℓ ≲ 1500. One specific example is the spectral index of scalar perturbations, for which we report a 5.4σ deviation from scale invariance, n_{s} = 1. Increasing the multipole range beyond ℓ ≃ 1500 does not increase our accuracy for the ΛCDM parameters, but instead allows us to study extensions beyond the standard model. We find no indication of significant departures from the ΛCDM framework. Finally, we report a tension between the Planck bestfit ΛCDM model and the lowℓ spectrum in the form of a power deficit of 5–10% at ℓ ≲ 40, with a statistical significance of 2.5–3σ. Without a theoretically motivated model for this power deficit, we do not elaborate further on its cosmological implications, but note that this is our most puzzling finding in an otherwise remarkably consistent data set.
Key words: cosmic background radiation / cosmological parameters / cosmology: observations / methods: data analysis
© ESO, 2014
1. Introduction
This paper, one of a set associated with the 2013 release of data from the Planck^{1} mission (Planck Collaboration I 2014–Planck Collaboration XXXI 2014), describes the CMB power spectra and the corresponding likelihood that we derive from the Planck data.
The power spectrum of the cosmic microwave background (CMB) is a unique signature of the underlying cosmological model (e.g., Spergel et al. 2003; Hinshaw et al. 2013). It has been measured over the whole sky by COBE and WMAP, and over smaller regions by groundbased and suborbital experiments (e.g., Tristram et al. 2005; Jones et al. 2006; Reichardt et al. 2009; Fowler et al. 2010; Das et al. 2011; Keisler et al. 2011; Story et al. 2013; Das et al. 2014). By mapping the whole sky to scales of a few arc minutes, Planck now measures the power spectrum over an unprecedented range of scales from a single experiment. To estimate cosmological parameters from the power spectrum requires a likelihood function that propagates uncertainties.
In this paper we describe the power spectra obtained from the Planck temperature data, as well as the associated likelihood function. Since the probability distribution of the power spectrum is nonGaussian at large scales, we follow a hybrid approach to construct this likelihood (Efstathiou 2004, 2006), using a Gibbs sampling based approach at low multipoles, ℓ, and a pseudoC_{ℓ} technique at high multipoles (Hivon et al. 2002) as well as an approximation to the quadratic maximum likelihood estimator (Rocha et al. 2011, 2010) at high multipoles.
The highℓ part of the Planck likelihood (ℓ ≥ 50) is based on power spectra estimated from each Planck detector in the frequency range 100 to 217 GHz, allowing careful assessment of each detector’s response to the sky emission. We implement three independent likelihood methods. The first, used in the distributed likelihood code, estimates the power spectrum at every multipole, together with the associated covariance matrix. The second takes a simplified form, binning the spectra, and is used to explore the stability of the results with respect to different instrumental and astrophysical systematic effects, the third one takes a CMB map estimated from component separation techniques (hence using all Planck frequency maps) and estimates the band powers and associated covariance matrices. The methods give consistent results.
Unresolved extragalactic foregrounds make a significant contribution to the power spectra at high multipoles. We develop a model for these foregrounds, designed to allow the Planck likelihood to be combined with high resolution data from the Atacama Cosmology Telescope (ACT) and the South Pole Telescope (SPT). We combine frequencies and model unresolved foregrounds in a physical way, as in e.g., Shirokoff et al. (2011); Dunkley et al. (2011); Reichardt et al. (2012), performing component separation at small scales at the power spectrum level. On large scales, ℓ < 50, Galactic contamination is more significant. We use the Planck temperature maps in the range 30 ≤ ν ≤ 353 GHz to separate Galactic foregrounds in the maps, and then estimate the full probability distribution of the CMB power spectrum.
This paper is structured as follows. In Sect. 2 we describe the pseudoC_{ℓ} likelihoods, and in Sect. 3 set up the foreground model. The power spectra and derived cosmological parameters are presented in Sects. 4 and 5, and an assessment of their accuracy and robustness is made in Sects. 6 and 7. In Sect. 8 we describe the lowℓ likelihood, and conclude by presenting the complete Planck likelihood in Sect. 9.
Detectors used to make the maps for this analysis.
2. Highℓ likelihoods
The Planck maps consist of the order 5 × 10^{7} pixels for each detector (Górski et al. 2005), so a likelihood described directly at the pixel level would be too time consuming. A significant compression of data can be achieved with minimal information loss using pseudoC_{ℓ} power spectra, even in the case of incomplete sky coverage. Here we describe the form of the likelihood function of the compressed data, given a sky signal and instrumental model.
Following Hamimeche & Lewis (2008), we assume a Gaussian form of the likelihood (Bond et al. 2000) based on pseudospectra that have been corrected to account for partial sky masking (Hivon et al. 2002) (for an overview on highℓ Likelihoods see e.g. Rocha et al. 2010). We use a “finegrained” data description, computing spectra of maps from individual detectors or detector sets. Table 1 describes the 13 maps used in the analysis, spanning 100 to 217 GHz, which are compared with the general purpose “ds0” maps used in input to the component separation methods. These maps are based on the Planck nominal mission. In the case of the polarised detector sets (PSB type in Table 1), the mapmaking algorithm uses the four combined detectors to produce maps of the T, Q, U Stokes parameters, but only the temperature maps produced from each PSB detector set are used in the likelihood analyses. We compute the spectra at these multiple frequencies to simultaneously constrain the CMB and foreground contributions. We choose these frequencies as a tradeoff between adding further information, and adding further complexity to the foreground model, which would be needed to include the adjacent 70 and 353 GHz channels (see Sect. 3 for further discussion). In our baseline analysis the spectra are computed at each multipole, together with an estimate of the full covariance matrix with offdiagonal errors between different spectra and multipoles. As in the WMAP analysis, we use only crossspectra between detectors, alleviating the need to accurately model the mean noise contribution.
In this section, we begin with a reminder of the pseudospectrum approach, and describe our baseline likelihood distribution, hereafter referred to as the CamSpec likelihood. We then show how a compression of spectra within a given frequency can be achieved with negligible loss of information. We describe the signal and instrument model, including detector noise properties, calibration, and beam uncertainties.
Next, we describe an alternative, simpler, form of the likelihood, hereafter referred to as Plik, based on binned power spectra with an inverseWishart distribution. This does not require the precomputation of large covariance matrices, so changing the sky or instrument modelling is straightforward. This simpler form of the likelihood is used to assess the robustness of our likelihood methodology with respect to technical choices and astrophysical foreground modelling.
In Sect. 7.5 we also compare these likelihoods to the Likelihood estimated from band power spectra computed directly from CMB maps estimated by multifrequency component separation (Planck Collaboration XII 2014). The band powers and covariance matrices are estimated with the XFaster approach (Rocha et al. 2011, 2010).
2.1. The CamSpec likelihood
The temperature maps produced from individual detector sets are first weighted with apodised Galactic and point source masks that are common within a given frequency band (see Sect. 3) for details. On top of these masks crafted to reject astrophysical contamination, each detector set has a small number of unobserved pixels that are added to the corresponding detector set masks.
Let us define as the spherical harmonic coefficients of the weighted temperature map of detector set i. The pseudospectrum at multipole ℓ, for the detector pair (i,j), is then given by (1)where the dagger, †, denotes the Hermitian transpose. This is related to the “deconvolved” spectrum, Ĉ^{Tij}, by a coupling matrix, (2)For an isotropic signal on the sky, the ensemble average of these deconvolved spectra are equal to the spectra of the theoretical models (including CMB and isotropic unresolved foregrounds) that we wish to test. For completeness, the coupling matrices are given explicitly in Appendix A.1.
In the first method, CamSpec, we form the deconvolved spectra Ĉ_{ℓ} without any prior smoothing of the pseudospectra . Even for the largest sky masks used in our analysis (see Sect. 3), the coupling matrices are nonsingular. The deconvolution requires the evaluation of ~ coupling matrices for a data set with N_{map} sky maps, which takes a moderate, but not excessive, amount of computer time.
A more challenging computational task is to compute the covariances of the pseudospectra, i.e., Here we need to compute coupling matrices, and the problem rapidly becomes computationally intractable even for relatively low values of N_{map}. For the moment we assume that these covariance matrices are available and describe their computation in Appendix A.4. We use the notation to denote a column vector for which the index p of a single element X_{p} denotes the map combination (i,j) and multipole ℓ. We denote the covariance matrix of this vector as (3)As explained later, the deconvolved detector set crossspectra given by Eq. (2) can be efficiently combined within a given frequency pair after a small effective recalibration, taking their respective isotropised effective beam transfer function and noise levels into account (see Appendix A.3 for the detailed procedure). Here, like in all Planck papers, the effective beam means the beam, which accounts for the combined effect of the instrument and of the complete processing chain. Covariance estimates of these combined spectra can be deduced from those of the detector set crossspectra. The covariance matrix is computed for a fixed fiducial model, and we approximate the likelihood as a Gaussian, described in Appendix A.5. The likelihood thus takes the form p = e^{− S} with For the current analysis we include the following (deconvolved) spectrum combinations, (4)coupled to a parametric model of the CMB and foreground power spectra. The multipole ranges we select, as well as the masks, depend on frequency, as described in Sect. 5. The tailored selection of multipole ranges and masks we adopt here is designed to keep the Galactic contamination as low as possible in both real and harmonic space, while keeping a reasonably low cosmic variance. This procedure allows us to keep the Galactic emission model simple (see Sect. 3.2). We do not include the 100 × 143 and 100 × 217 spectra since these spectra carry little additional information about the primary CMB anisotropies, but would require us to solve for additional unresolved foreground parameters. This tradeoff of information versus complexity was also considered for the use of the 70 GHz and 353 GHz data, which we choose not to include except for crosschecks.
The fiducial covariance matrix is composed of the blocks shown in Fig. 1. The offdiagonal blocks in this matrix accurately account for the correlations between the power spectra at different frequencies.
Fig. 1
Covariance matrix blocks used in the likelihood, accounting for the correlations between crossspectra estimated from the 100, 143, and 217 GHz channels. 

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This description would be sufficient for perfectly known calibrations and beam transfer functions of each detector sets’ crossspectra. Planck Collaboration VII (2014) describes in detail these uncertainties, and shows that for each detector set pair, (i,j), the effective beam transfer function can be expressed as (5)described further in Appendix A.6, with n_{modes} beam error eigenmodes , and their covariance matrix. These modes are then combined into generalised beam eigenmodes corresponding to the spectra . The associated covariance matrix is used to construct a Gaussian posterior distribution of the eigenmodes, which allows marginalisation over the beam uncertainties.
Finally, in the construction of the covariance matrix, one needs to accurately specify the contribution of the instrumental noise. Even if there is no bias on the spectra due to instrumental noise, having removed autospectra, the latter dominates the covariance matrix on small scales. Fortunately, the Planck scanning strategy at the ring level allows us to make estimates of the noise pseudospectra from halfring difference maps (see Planck Collaboration XII 2014). These halfring difference maps, together with the knowledge of the noise variance per pixel for each detector set, can be used to derive the noise contribution to the covariance matrix with good accuracy (see Appendix A.8 for details).
2.2. The Plik likelihood
We now describe the alternative form of the likelihood, inspired by Cardoso et al. (2008), used for crosschecks and robustness tests. We start from the fullsky exact likelihood for a Gaussian signal, which for N_{map} detector maps is given by where θ is a vector containing the parameters of the signal model, and Ĉ_{ℓ} are the empirical angular spectra. denotes the Kullback divergence between two nvariate zeromean Gaussian distributions with covariance matrices A and B, and is given by As already noted, a sky cut introduces offdiagonal couplings between different multipoles. In this method we bin the power spectra in such a way that these offdiagonal terms of the covariance are negligible. This is adequate to model sources with slowly varying spectra, such as foregrounds, and the CMB anisotropies for standard cosmologies. In this case, the likelihood takes the form (6)where the angular spectra Ĉ_{ℓ} for each crossfrequency spectrum have been averaged into Q spectral bins using spectral windows w_{q}(ℓ) (q = 1,...,Q), with Here w_{q}(ℓ) denotes the window function for the qth bin, and the same symbol, C, is used to denote binned or unbinned spectra. The effective number of modes in the qth bin is We adopt a spectral binning defined by The Plik bin width is Δℓ = 9 from ℓ = 100 to ℓ = 1503, then Δℓ = 17 to ℓ = 2013, and finally Δℓ = 33 to ℓ_{max} = 2508. This ensures that correlations between any two bins are smaller than 10%.
While this binned likelihood approximation does not fully capture all couplings between different multipoles, it has a notable advantage in computational speed, and it agrees well with the primary likelihood. It is therefore very well suited for performing an extensive suite of robustness tests, as many more parameters can be considered in a short time. Further, instrumental effects can be investigated quickly to assess the agreement between pairs of detectors within a frequency channel, such as individual detector calibrations and beam errors.
A specific example is the impact of (effective) beam uncertainty parameters on the likelihood. This can be investigated by reexpressing the model covariance matrices as (7)where C_{ℓ}(θ) is the model covariance including both signal and noise, and B_{ℓ}(γ) is a diagonal matrix encoding the beam and calibration errors with elements given by^{2}(8)Here, are the eigenmodes of the (auto)spectra, similar to Eq. (5). Note that Eq. (7) does not contain the mean beam transfer function, since it is already included in the empirical spectra. Thus, using Eq. (7) Plik approximates the crossspectrum beam errors as the harmonic mean of the corresponding autospectrum beam errors, under the assumption that B_{ℓ} is diagonal between detectors. This approximate factorisation is intrinsically linked to the assumed Kullback shape of the Plik likelihood, and is later demonstrated to work well for both simulations and data.
The Plik likelihood method also provides a direct estimate of the detector noise power spectra as it can include the empirical autospectra, and we find that these noise estimates are in good agreement with the noise spectra used to construct the CamSpec likelihood covariance matrix. The method can also produce a binned CMB power spectrum independent of the underlying cosmological model, providing a direct quality assessment of the foreground model parametrisation. In practice, we proceed in two steps. First, we jointly estimate the noise together with all other parameters using both auto and crossspectra. Then we fix the noise estimates, and use the fiducial Gaussian approximation to explore the remaining free parameters excluding the autospectra, optionally including only specific data combinations.
3. Foreground emission model and sky masks
3.1. Sky masks
The Galactic emission varies strongly in both complexity and strength across the sky. It is therefore necessary to find a balance between maximizing the sky coverage to reduce statistical uncertainties, and establishing a simple yet efficient foreground model. In this paper, we threshold an ILCsubtracted 353 GHz temperature map to define a basic set of diffuse Galactic masks, which form a sequence of increasing sky fraction, to minimise the contribution from diffuse dust emission. We refer to them using the percentage of the sky retained: G22, G35, G45, G56, G65. We also use a point source mask, labelled PS96, which is based on the union of the point sources detected from the channels in the range 100 to 353 GHz. The point source flux cut is not critical, since the amplitudes of the Poisson contributions of unresolved sources are allowed to vary over a wide range in the likelihood analysis. Thus, we do not impose tight priors from source counts and other CMB experiments on the Poisson amplitudes. All theses masks are shown in Fig. B.1.
For Planck, we need to estimate the covariance matrices to percent level precision. For temperature spectra, and in the absence of point source holes, this precision can be achieved with sharp, nonapodised Galactic masks (Efstathiou 2004). However, the inclusion of point source holes introduces nonnegligible lowℓ power leakage, which in turn can generate errors of several percent in the covariance matrices. In order to avoid such power leakage, we also derive a series of apodised masks. For the Galactic masks, we proceed as follows. First, we smooth each mask with a fivedegree Gaussian beam, and zero any pixels below a threshold of 0.15 (chosen to approximately keep the zeroed area the same as that of the original, unapodised mask). We then subtract 0.15 from the remaining pixels, and rescale the resulting mask by 1 / (1 − 0.15). This procedure is iterated a further two times. In order to retain sufficient sky area for the most conservative sky mask, a slightly less aggressive version of mask G22 was used to seed the apodisation process for that case. Each point source is apodised to 30′ FWHM, resulting in the PSA82 point source mask. The resulting set of apodised masks are shown in Fig. B.2.
For all cosmology analyses, we use three of the union of one of the apodised galactic masks and of the apodised point sources mask – CL31, CL39, and CL49, which are shown in Fig. 2. Table 2 summarizes the various masks and the corresponding sky fraction they retain.
Fig. 2
Set of masks (CL31, CL39, CL49) used for the likelihood analyses, in Mollweide projection. 

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3.2. Galactic emission
The contamination from diffuse Galactic emission at low to intermediate multipoles can be reduced to low levels compared to CMB anisotropies by a suitable choice of masking. However, even with conservative masking, the remaining Galactic emission at high multipoles is nonnegligible compared to other unresolved components, such as the cosmic infrared background (CIB) anisotropies at 143 and 217 GHz. A clear way of demonstrating this is by differencing the power spectra computed with different masks, thereby highlighting the differences between the isotropic and nonisotropic unresolved components.
Series of masks used in this paper.
Let us define , a notation we shall use throughout this paper. Figure 3 shows (up to ℓ ≤ 1400) the 217 GHz power spectrum difference for the union masks mask0 ≡ G22∪PSA82 and mask1 ≡ G35∪PSA82, minus the corresponding difference for the 143 GHz frequency channel. Any isotropic contribution to the power spectrum (CMB, unresolved extragalactic sources etc.) cancels in such a maskdifferenced power spectrum, leaving a nonisotropic signal of Galactic origin. Nevertheless, at low multipoles, a mask differenced power spectrum is still severely affected by cosmic variance of the primordial CMB; but this can be eliminated by forming a “doubledifference” power spectrum between frequencies. This is what is plotted at ℓ ≤ 1400 in Fig. 3, rescaling to account for the small dust contribution at 143 GHz. Above ℓ = 1400, the noise in the 143 GHz power spectra becomes significant, and so we plot directly the 217 maskdifferenced power spectrum. The figure gives an accurate estimate of the dust contribution within mask1 to the 217 GHz spectra over the entire multipole range plotted. In the same figure, these difference spectra are compared to the unbinned maskdifferenced 857 GHz power spectrum, scaled to 217 GHz adopting a multiplicative factor^{3} of (9.93 × 10^{5})^{2}; the dotted line shows a smooth fit to the unbinned spectrum. The agreement between this prediction and the actual dust emission at 217 GHz is excellent, and this demonstrates conclusively the existence of a smallscale dust emission component with an amplitude of ~5 − 15 μK^{2} at 217 GHz if mask1 is used.
Fig. 3
Differences between power spectra evaluated from the union masks G22∪PSA82 and G35∪PSA82 (≡ mask0 and mask1 respectively in the legend), showing the presence of Galactic dust. For ℓ ≤ 1400 the spectra show the 217−143 “doubledifferenced” power spectrum, rescaled to correct for dust emission at 143 GHz. For ℓ > 1400 the 217 mask differenced power spectrum is plotted. The blue line shows the 857 GHz maskdifferenced power spectrum scaled to 217 GHz as described in the text. The green dotted line shows the bestfit of Eq. (9) scaled to 217 GHz. This is the model that we use to subtract dust emission at high multipoles from the 143 × 143, 143 × 217 and 217 × 217 spectra with suitably adjusted scaling factors. 

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For cosmological parameter analysis this smallscale dust component must be taken into account, and several approaches may be considered:

1.
Fit to a template shape, e.g., as shown by the dotted line inFig. 3.

2.
Reduce the amplitude by further masking of the sky.

3.
Attempt a component separation by using higher frequencies.
The main disadvantage of the third approach is a potential signaltonoise penalty, depending on which frequencies are used, as well as confusion with other unresolved foregrounds. This is particularly problematic with regards to the CIB, which has a spectrum very similar to that of Galactic dust. In the following we therefore adopt the two former solutions.
It is important to understand the nature of the small scale dust emission, and, as far as possible, to disentangle this emission from the CIB contribution at the HFI cosmological frequencies. We use the 857 GHz power spectrum for this purpose, noting that the dust emission at 857 GHz is so intense that this particular map provides an effectively noisefree dust emission map. In Fig. 4 we again show the 857 GHz mask power spectrum difference, but this time plotted on a loglog scale. The solid line shows the corresponding bestfit model defined by (9)with A = 5.729 × 10^{8}μK^{2}, α = 0.169, ℓ_{c} = 90.5, and γ = 0.427. At high multipoles this fit asymptotically approaches C_{ℓ} ∝ ℓ^{2.6}, which is compatible with previous knowledge about diffuse Galactic emission, i.e., a powerlaw behaviour with an index close to − 3 extending to high multipoles (see e.g., MivilleDeschênes et al. 2007). Note that throughout the paper, we use CMB thermodynamic units.
Fig. 4
857 GHz maskdifferenced power spectrum (points), interpreted as Galactic dust emission. The solid line shows the bestfit model defined by Eq. (9). 

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The upper panel in Fig. 5 shows the 857 GHz spectra for the four Galactic masks (G22, G35, G45 through to G56) with the pointsource mask applied. They are compared to the 857 GHz CIB power spectrum from Planck Collaboration XVIII (2011), which for ℓ > 500 can be described approximately as ℓC_{ℓ} ≈ 7 ×10^{6}μK^{2}. The best fit models to Eq. (9) are also shown (with α, ℓ_{c} and γ fixed to the values derived above, but with the amplitude allowed to vary) fitted to ℓ ≤ 500, where we expect diffuse dust emission to be dominant. The lower panel of Fig. 5 shows the same power spectra after subtracting the bestfit dust model. After subtracting the Galactic dust component, the recovered power spectra are consistent with the CIB measured in Planck Collaboration XVIII (2011) for all masks. The excess at high multipoles may be due to a combination of aliasing of large scale power through the point source masks at ℓ ≳ 3000, Galactic point sources, and uncertainties in the 857 GHz beams.
Fig. 5
Top: 857 GHz power spectra for the four different masks defined in the text. The dotted lines show the bestfit model defined by Eq. (9) fit to ℓ ≤ 500, capturing the Galactic dust. An estimate of the CIB power spectrum is shown in orange points (Planck Collaboration XVIII 2011). Bottom: power spectra after subtracting the Galactic dust model. The horizontal purple line shows a constant amplitude of 7 × 10^{6}μK^{2}, which roughly describes the Planck CIB results. 

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The model explains by construction the “doubledifference” plot shown in Fig. 3. Specifically, this emission is consistent with a cirruslike power spectrum, C_{ℓ} ∝ ℓ^{2.6}, extrapolated to high multipoles. Furthermore, the results of Fig. 5 demonstrate that over a wide area of sky, we can understand the 857 GHz power spectrum in terms of a “universal” cirrus spectrum together with an isotropic CIB component. These results provide strong evidence that an extragalactic CIB component dominates over the diffuse Galactic emission at multipoles ℓ > 700 over the full range of HFI frequencies outside the G22 region (i.e. over the “cleanest” ~20% of sky). Although this is demonstrated for 857 GHz in Fig. 5, it applies over the entire HFI frequency range since Galactic dust and CIB have similar spectra. We therefore use the dust template of of Eq. (9) with suitably adjusted scaling factors to remove the highℓ contribution from dust from the empirical 143 × 143, 143 × 217 and 217 × 217 spectra, which are used in CamSpec.
We take a different approach for the Galactic dust correction with the Plik likelihood. Rather than correcting the empirical spectra during a preprocessing step, the Plik likelihood implements an explicit oneparameter model that describes the dust contribution to the crossspectrum between detectors i and j, (10)Here (11)where the dust amplitude, A^{Dust}, is measured in units of μK^{2}, ν_{i} is the reference frequency for map i, ν_{0} is a reference frequency, which is taken to be 143 GHz, B(T,ν) is the emission law of a black body with temperature T, and the dust colourcorrection terms, , are computed by integrating the dust spectrum within the spectral band of each detector (set). We fix the frequency and angular scaling parameters to γ_{d} = 2.6, β_{d} = 1.6 and T_{d} = 18 K.
3.3. Poisson power from unresolved point sources
Unresolved galaxies contribute both shot noise and clustered power to the Planck maps. The Poisson contribution leads to a scale independent tem, C_{ℓ} = const. We model this power with a single amplitude parameter for each autospectrum (, , and ) and a cross correlation coefficient for each cross spectrum (). These quantities are not of primary interest for cosmological results, so to avoid modelling error we do not separate the power into that sourced by “dusty" or “radio" galaxies (i.e., with increasing or decreasing brightness with frequency, respectively) as is done in the analysis of the ACT and SPT power spectra (Dunkley et al. 2011, 2013; Reichardt et al. 2012). We also make no assumptions about their coherence between frequencies.
The Poisson power is related to the flux density dN/dS via (12)where we have explicitly introduced the Planck flux cut . Since Planck utilises a constant signaltonoise cut, and the Planck noise varies significantly across the sky, this flux cut has a spatial dependence. Although this does not alter the shape of the Poisson term, extra care must be taken when comparing results ^{4} with models of dN/dS. In Sect. 7.3, we explore the consistency between the Poisson power recovered from the Planck power spectrum analysis and predictions from source count measurements.
3.4. Clustered power from unresolved point sources
Unresolved galaxies also contribute power because they trace largescale structures. The mean flux from the radio galaxies is much smaller than that from the dusty galaxies, so only the dusty galaxies contribute a significant clustering term (Millea et al. 2012). The CIB clustering has been studied extensively, starting with Bond et al. (1986, 1991). Further theoretical investigation (Scott & White 1999; Haiman & Knox 2000) was stimulated by the detection of the infrared background in the COBE data (Puget et al. 1996; Fixsen et al. 1998), and the detection of bright “submillimetre” galaxies in SCUBA data (Hughes et al. 1998). Subsequently, the clustering has been detected at 160 microns (Lagache et al. 2007), at 250, 350 and 500 microns by the Balloonborne Large Aperture Submillimeter Telescope (BLAST, Viero et al. 2009; Hajian et al. 2012) and at 217 GHz by SPT and ACT (Hall et al. 2010; Dunkley et al. 2011). Recent Planck measurements of the CIB (Planck Collaboration XVIII 2011) have extended the measurements at 217 GHz, 353 GHz, and 545 GHz to larger scales, and recent Herschel measurements (Viero et al. 2013) have improved on the BLAST measurements and extended them to smaller angular scales.
Rather than attempt to establish a physical model of the CIB, we adopt in this analysis a phenomenological model that captures the CIB uncertainties for both Planck and highℓ experiments. Our baseline model for the clustered CIB component is a powerlaw spectrum with a free spectral index, with an amplitude at each frequency, and , and a crosscorrelation between frequencies, . We assume that the CIB clustering power at 100 GHz is negligible.
3.5. Unresolved SunyaevZeldovich effects
Based on analysis of ACT and SPT data, the thermal SunyaevZeldovich (tSZ) contribution is expected to contribute approximately at 100 GHz and at 143 GHz (Reichardt et al. 2012; Dunkley et al. 2013; Sievers et al. 2013). The kinetic SunyaevZeldovich (kSZ) effect is expected to have a similar, or smaller, contribution, with . In addition, theoretical arguments (Reichardt et al. 2012; Addison et al. 2012b) suggest that there should be a tSZ × CIB correlation that should contribute about the same order of magnitude as the kSZ term at 143 GHz.
For Planck, all of these SZ contributions are small in comparison to other unresolved foregrounds and are therefore poorly constrained by Planck data alone. Nevertheless, to eliminate biases in cosmological parameters (Millea et al. 2012; Zahn et al. 2005), we model their contributions, with appropriate constraints from higher resolution CMB experiments, using three templates.
Fig. 6
SZ templates at 143 GHz computed for a normalization of σ_{8} = 0.8. The tSZ templates are from the model of Efstathiou & Migliaccio (2012) for three values of the evolution parameter ϵ, ϵ = 0 (top), ϵ = 0.25 (middle) and ϵ = 0.5 (lower). The kinetic SZ template is from Trac et al. (2011). The tSZ × CIB cross correlation (143 × 143 GHz) is from the Addison et al. (2012b) template with parameters described in the text, and is negative for 143 × 143 GHz. 

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First, for the thermal SZ effect we adopt the family of templates described by Efstathiou & Migliaccio (2012). These are based on the Komatsu & Seljak (2002) model, but use the “universal” Xray electron pressure profile, P_{e}, of Arnaud et al. (2010) extrapolated to high redshift via (13)Here, ϵ describes departures from selfsimilar evolution, and a value of ϵ = 0.5, which is adopted as the default for parameter estimation purposes, provides a good match to the results from recent hydrodynamical numerical simulations incorporating feedback processes (Battaglia et al. 2010, 2012). Figure 6 shows the tSZ templates for three values of ϵ; the template shape is not particularly sensitive to ϵ. We treat the (dimensionless) normalization of the tSZ template at 143 GHz as an adjustable parameter, (14)This parameter fixes the amplitude at 100 GHz via the frequency dependence of the tSZ effect, (15)We neglect the tSZ at 217 GHz.
Second, for the kinetic SZ effect we adopt the template described by Trac et al. (2011), and as in Eq. (14) we treat the dimensionless amplitude of the template, A^{kSZ}, as a free parameter, (16)Third and finally, for the crosscorrelation between the thermal SZ component and the CIB we adopt the template described by Addison et al. (2012a). In this case, the amplitude is parametrised in terms of a single correlation coefficient, (17)These templates are plotted in Fig. 6, normalized to σ_{8} = 0.8 and with ξ = 1.0 using a fiducial CIB amplitude. Note that with these parameters, the tSZ × CIB crossspectrum approximately cancels the kSZ spectrum at 143 GHz.
As seen in Fig. 6, the SZ contributions are at the level of a few μK^{2}, which, although small, must be taken into account to assess interfrequency residuals. However, one can see that these templates have similar shapes at multipoles ≲2000, and therefore they cannot be disentangled using Planck data alone. On the other hand, higher resolution experiments can break this degeneracy, and as shown in Planck Collaboration XVI (2014), the combination of Planck, ACT, and SPT, better constrains the amplitude of the thermal SZ effect. The ACT and SPT data at 150 GHz can be fitted to high, subμK^{2}, accuracy without kSZ and tSZ x CIB templates, yet we expect a kSZ contribution of at least the amplitude shown in Fig. 6, and larger if we account for patchy reionisation (see, e.g., Knox 2003, and references therein) and references therein). This implies a cancellation of the kSZ and tSZ x CIB contributions at 150 GHz (Addison et al. 2012b), as discussed in greater detail in Planck Collaboration XVI (2014).
4. Consistency checks and combined crossspectra
The large number of cross spectra in a detectorbydetector power spectrum analysis allows for a number of internal consistency checks of the data. Within a frequency band, we expect to see exactly the same sky signals (primordial CMB, Galactic, and extragalactic foregrounds), and so any intrafrequency residuals reflects instrumental systematics, for example beam errors, “gain” fluctuations, and bandpass mismatch. In contrast, interfrequency residuals are harder to analyse because the sky signals vary with frequency. An accurate model of the unresolved foregrounds is therefore required to assess interfrequency residuals. Furthermore, as we show below, the scatter caused by chance CMB–foreground crosscorrelations can dominate the interfrequency residuals. For a precision experiment such as Planck, where the power spectra are expected to be signal dominated over a wide multipole range, intra and interfrequency residuals provide a powerful way of assessing possible systematic errors. It is essential that contributions of systematic errors to both types of residual are small enough that they have negligible impact on cosmological parameter analysis.
Figure 7 shows a selection of temperature crossspectra and estimates of the analytic covariance matrices, together with the bestfit cosmological model described in Sect. 5. Unresolved foregrounds have been subtracted using the bestfit foreground parameters of the model described in Sect. 3. The scatter varies substantially between crossspectra, reflecting differences in the instrument noise and effective resolution of different detector combinations. The analytic error model summarized in Appendix A.2 is indicated, modified by the nonwhite noise correction. This model provides an excellent description of the scatter seen in the data, over the full multipole range shown in plots, with an accuracy of a few percent or better.
4.1. Intrafrequency residuals
In this section we analyse the intrafrequency residuals at 143 and 217 GHz. There are N_{spec} = 10 crossspectra at 143 GHz and 15 crossspectra at 217 GHz^{5}. At each frequency, we solve for multiplicative (“effective” calibration) coefficients, y_{i}, that minimise (18)where (19)subject to the constraint that y_{1} = 1 (where i = 1 corresponds to detector 5 at 143 GHz and detector 1 at 217 GHz). Note that the power spectra in Eq. (18) and (19) are corrected for beam transfer functions. To minimise the possible impact of beam errors and noise, we restrict the sum in Eq. (18) to the multipole range 50 ≤ ℓ ≤ 500 where the spectra are signal dominated. Numerical values for the calibration coefficients are given in Table 3, using mask CL31. The calibration factors are insensitive to the choice of mask or multipole range.
Map calibration coefficients.
The results of Table 3 show that effective calibration factors of ~0.2% are quite typical for HFI maps, in the 100 − 217 GHz frequency range. These recalibrations are of the order of magnitude of the statistical errors of the calibrations on dipole (see Planck Collaboration VIII 2014, Table 2). Note that the data are corrected for individual bolometer time transfer functions (TTFs; Planck Collaboration VII 2014). For each detector, the TTF model is tuned to minimise survey differences and by construction normalized to unity at the spin frequency of the satellite (0.01666 Hz) to preserve the dipole calibration. The consistency of intrafrequency power spectrum residuals therefore provides a test of the consistency of the TTFs in addition to the beam transfer functions.
Figure 8 shows the remarkable consistency of the power spectra at each frequency. The upper panels show the spectra corrected for the beam and effective calibration, together with the mean cross spectra. The lower panels show the dispersion around the mean. Up to ℓ ≃ 700 the Δℓ = 31 binned crossspectra show a dispersion in their values typically less than expected from ~0.2% calibration variations. This excess scatter (compared to the noiseinduced expected scatter) has negligible impact on cosmological parameter analysis.
Fig. 7
A selection of cross spectra from individual detectors, with the best fit unresolved foreground model subtracted. The bestfit six parameter ΛCDM model is shown, with ±1σ errors determined from the diagonal components of the analytic covariance matrices. The lower panel in each plot shows the residuals with respect to the model. This shows that our covariance matrix captures well the transition between signal and noise dominance. 

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The residuals of the cross spectra in band averages of width Δℓ ~ 61 are shown in Fig. 9, before and after correction for the effective intrafrequency calibrations. The reduction in scatter after correction is evident at ℓ ≲ 500, and the residual scatter is consistent with instrument noise and beam errors. At 217 GHz, beam errors dominate over noise at multipoles ≲1000. There is no evidence that the excess scatter is caused by a small number of “anomalous” detectors.
Fig. 8
Cross spectra corrected for the beam and effective calibration, together with the mean power spectrum. Top: the 10 cross spectra at 143 GHz. Bottom: the 15 cross spectra at 217 GHz, with SWB × SWB spectra (magenta), SWB × 217ds1 (red), SWB × 217ds2 (green), 217ds1 × 217ds2 (purple). The SWB detector sets at 217 GHz are 217ds3, 217ds4, 217ds5 and 217ds6. The power spectra are distinguishable only at high multipoles where the data become noise dominated. The lower panels show the dispersion of the cross spectra around the mean, together with a ± 0.2% calibration error (orange line in the lower panels). 

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Fig. 9
Cross spectra for the 143 GHz (top) and 217 GHz (bottom) channels, as in Fig. 8, before correction for multiplicative intrafrequency calibration coefficients (above), and after correction (below). The colour coding is the same as in Fig. 8. 

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4.2. Interfrequency residuals
The results of the previous section show that the intrafrequency crossspectra between detector/detector sets are consistent to within a few μK^{2} at multipoles ℓ ≲ 1000. In a likelihood analysis, there is therefore little loss of information in compressing the power spectra for each distinct frequency combination, as opposed to retaining the spectra for each map pair. This compression greatly reduces the size of the data vector and its covariance matrix, and speeds up the likelihood computation at high multipoles. In this section we intercompare the residuals of these compressed power spectra.
One might naïvely expect that with accurate foreground modelling, the interfrequency residuals in the signal dominated regime should be reduced to levels comparable to those seen in the intrafrequency comparisons described in the previous section. This is incorrect. Figure 10 shows power spectrum differences between the cosmologically significant spectra for Planck at high multipoles (143 × 143, 143 × 217, 217 × 217). In this figure, which is independent of the cosmological model, the bestfit unresolved foreground model has been subtracted from each spectrum, and relative calibration factors have been applied. Residual beam, calibration and unresolved foreground errors would show up in this figure as largescale smooth residuals.
In fact, we see smallscale residuals at multipoles ℓ ≲ 800, which are larger than expected from instrument noise alone. For example, the 143 × 143 − 217 × 217 spectrum in the upper panel of Fig. 10 has a χ^{2} of 35.6 for the 21 points in the multipole range 100 ≤ ℓ ≤ 800, an approximately 2.3σ excess. (But note that this effect would become more significant on the full mission data.) This excess scatter arises from the chance CMBforeground crosscorrelations. Even if the foreground contamination is much smaller than the CMB, chance crosscorrelations can produce scatter in the interfrequency power spectra that dominates over instrument noise in the signal dominated regime. We develop a model for these CMBforeground crosscorrelations in Appendix C and show it can explain quantitatively the observed interfrequency scatter, and the scaling of this scatter with frequency, at multipoles ℓ ≲ 500.
At high enough multipoles, instrument noise, beam errors, and errors in foreground modelling dominate the interfrequency residuals. A complete analysis of interfrequency residuals therefore requires the full likelihood machinery and MCMC analysis to determine foreground, beam and calibration parameters. We therefore revisit the interfrequency residuals in the following sections.
Fig. 10
Differences between the 143 × 143, 143 × 217, and 217 × 217 cross spectra (plotted in bins of width δℓ ≈ 31). The bestfit model for unresolved foregrounds has been subtracted from each spectrum. The numbers list the dispersions over the multipole range 800 ≤ ℓ ≤ 1500. 

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5. Reference results of the highℓ likelihood
In this section we study the highℓCamSpec likelihood, and present the power spectrum and parameters derived from this baseline likelihood for the basic sixparameter ΛCDM model. In order to break the wellknown degeneracy between the optical depth, τ, and the scalar index of scalar perturbations, n_{s}, we adopt a Gaussian prior on τ (inspired from WMAP7 data, i.e., 0.088 ± 0.015, see Komatsu et al. 2011) instead of the lowℓ likelihood at ℓ < 50. We return to the global Planck results after introducing the lowℓ likelihood.
Fig. 11
Planck power spectra and data selection. The coloured tick marks indicate the ℓrange of the four crossspectra included in CamSpec (and computed with the same mask, see Table 4). Although not used, the 70 GHz and 143 × 353 GHz spectra demonstrate the consistency of the data. The dashed line indicates the bestfit Planck spectrum. 

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We choose separate masks for each frequency map to minimise Galactic foreground emission. First, since the HFI data are signaldominated at ℓ ≲ 500, and diffuse Galactic emission is low at 100 GHz outside the CL49 mask, there is little to be gained from analysing the dustier 217 GHz with the same region; it contains no new information about the primordial CMB. At higher multipoles, ℓ ≳ 500, we use the CL31 mask, optimizing the sky coverage while ensuring a low amplitude of smallscale Galactic emission relative to the isotropic unresolved foregrounds and the primordial CMB. In addition, we tune the multipole range for each frequency to mitigate Galactic foreground contamination and beam errors.
The choices of masks and angular ranges used in the highℓ likelihood are summarized in Table 4, together with basic χ^{2} statistics with respect to the minimal ΛCDM model per crossspectrum and combined. The 100 GHz crossspectrum is computed over the largest sky fraction, a total of 49% of the sky, and measures the largest scales. On the other hand, it has lowest resolution, and it is therefore only used for ℓ ≤ 1200. The 143 GHz crossspectrum has higher resolution, and is used for ℓ ≤ 2000. Finally, the 217 GHz crossspectrum has the highest resolution, but also the most Galactic dust contamination, and is therefore evaluated from only 31% of the sky, but including an angular range of 500 ≤ ℓ ≤ 2500.
Overview of crossspectra, multipole ranges and masks used in the CamSpec highℓ likelihood.
Given these masks and angular ranges, we compute the angular power spectra and covariance matrices, and construct the CamSpec likelihood. The angular power spectra for each frequency combination are shown in Fig. 11, and compared to spectra derived from the 70 GHz and 353 GHz Planck maps.
Overview of cosmological parameters used in this analysis, including symbols, the baseline values if fixed for the standard ΛCDM model, and their definition (see text for further details).
Overview of parameters describing astrophysical foreground modelling, instrumental calibration and beam uncertainties, including symbols, definitions, and prior ranges (see text for further details).
We use the likelihood to estimate six ΛCDM cosmological parameters, together with a set of 14 nuisance parameters (11 foreground parameters described in Sect. 3, two relative calibration parameters, and one beam error parameter^{6}, see Eq. (5) and Appendix A.7 for details). Tables 5 and 6 summarize these parameters and the associated priors^{7}. Apart from the beam eigenmode amplitude and calibration factors, we adopt uniform priors. To map out the corresponding posterior distributions we use the methods described in Planck Collaboration XVI (2014), and the resulting marginal distributions are shown in Fig. 12. Note that on the parameters A^{tSZ}, A^{kSZ} and we are using larger prior ranges as compared to Planck Collaboration XVI (2014).
Fig. 12
Marginal posterior distributions for the six cosmological (top two rows) and eleven nuisance parameters (lower four rows) estimated with the CamSpec likelihood. 

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Figure 12 shows the strong constraining power of the Planck data, but also highlights some of the limitations of a “Planck alone” analysis. The thermal SZ amplitude provides a good example; the distribution is broad, and the “best fit” value is excluded by the ACT and SPT high resolution CMB experiments (Reichardt et al. 2012). For the CIB amplitudes, the upper bound on e.g., is significantly weaker than the ACT and SPT constraints. To accurately estimate the foreground parameters at the ≲μK^{2} level, we need to supplement the Planck power spectra with temperature data from ACT and SPT, as described in Planck Collaboration XVI (2014). The fiducial model and foreground parameters used in the CamSpec likelihood are therefore derived from a joint Planck+ACT+SPT analysis and is not based on the parameters listed in Table 8. In the rest of this section, we use the parameters of Table 8, which is derived using only Planck data, to discuss interfrequency residuals.
Figure 13 shows the foreground residuals and total residuals after removing the bestfit foreground model for all spectra (including the 100 × 143 and 100 × 217 spectra, which are not used in the CamSpec likelihood). The first point to note here is that Planck has a limited ability to disentangle foregrounds. While the Planck data constrain the Poisson point source amplitudes at each frequency, as well as the CIB amplitude at 217 GHz (which dominates over the Poisson point source amplitude over much of the multipole range), they have only marginal sensitivity to the tSZ amplitude in the 100 × 100 spectrum, though the thermal SZ is strongly degenerate with the Poisson point source amplitude. The remaining foreground parameters are highly degenerate. For Planck alone, these minor foreground contributions combine to absorb interfrequency residuals.
Pairs of spectra are compared in Fig. 14, averaged over bands of width Δℓ = 31 below ℓ ≲ 2000 and wider bands above 2000. The error bars show the diagonals of the covariance matrices of these averages, but it is important to note that these points are correlated (neighbouring points are actually anticorrelated at the few percent level) even with bin widths as large as these because of the sky masks. This comparison shows that each of the spectra used in the CamSpec likelihood is consistent with the bestfit theoretical spectrum to high accuracy. In fact, each spectrum can be used to form a likelihood, and each gives a reduced χ^{2} close to unity (see Table 4). Thus, the six parameter ΛCDM model provides an excellent fit to the Planck highℓ power spectra at all frequencies between 100 and 217 GHz.
Figure 15 shows our maximum likelihood primary CMB spectrum, together with the bestfit theoretical spectrum. The residuals with respect to the model are shown in the lower panel. The error bars are computed from the diagonal components of the bandaveraged covariance matrix. The binning scheme is the same as in Fig. 14.
Finally, in Fig. 16 we zoom in on this spectrum in four multipole ranges using finer binning. The correlated fluctuations seen in this figure are maskinduced, and perfectly compatible with the six parameter ΛCDM model. Features such as the “bite” missing from the third peak at ℓ ~ 800 and the oscillatory features in the range 1300 ≲ ℓ ≲ 1500 are in excellent agreement with what we expect from our covariance matrices and from simulations; see Appendix A.4 for a few specific examples.
6. Accuracy assessment of the highℓ likelihoods
In this section we compare the power spectra and likelihoods derived using our two independent methods, and test these likelihoods using full Planck simulations.
6.1. Comparison of the Plik and CamSpec likelihoods
To allow a more direct comparison between the CamSpec and Plik likelihoods, we use the same frequency crossspectra for both codes in the following, i.e., we discard the 100 × 143 and 100 × 217 GHz frequency combinations from the default setting of the Plik likelihood. To achieve this, we modify the Plik likelihood to use the fiducial Gaussian approximation instead of the Kullback divergence. On the other hand, while we use the same multipole coverage, 100 ≤ ℓ ≤ 2500, and only one mask (CL39) for all crossspectra for Plik, we still use multipole ranges and masks as defined in Table 4 for CamSpec. In addition, we perform one Plik analysis with the CL49 mask, which matches the CamSpec mask at 100 GHz.
The left column of Fig. 17 shows the differences between the Plik power spectra, adopting the above validation settings, and the corresponding bestfit model. The right column shows the total spectra decomposed into cosmological and foreground components. The residuals are similar to those in Fig. 13 in the sense that they do not show any evidence of biases, except for some excess power in the 217 × 217 GHz spectra at small scales, where foreground modelling has the highest impact. At scales ℓ ≲ 1500 (ℓ ≲ 1000 for the 100 × 100 GHz cross spectra), the residuals are coherent between cross spectra as they are computed with the same Galactic mask, and the residuals are dominated by (CMBinduced) cosmic variance. At smaller scales (ℓ ≳ 1500) the residuals are dominated by noise and become uncorrelated.
In Fig. 18 we show the CMB power spectrum recovered by Plik estimated by removing the bestfit foreground amplitudes from each crossspectrum and computing their optimally weighted average, and the corresponding difference with respect to the bestfit ΛCDM model. The large scatter at low multipoles is expected, due to cosmic variance. The residual scatter at higher multipoles is at the ± 20 μK^{2} level, demonstrating the good fit provided by the sum of the ΛCDM and foreground models. These CMB residuals can be compared to the CamSpec inversecovariance weighted CMB residuals shown in Fig. 15, which are of the same order of magnitude. Thus, the Planck likelihood fit to the ΛCDM model is robust with respect to the detailed shape of the likelihood, as quantified in terms of power spectrum residuals.
Fig. 13
Foreground model over the full range of HFI cosmological frequency combinations. The upper panel in each plot shows the residual between the CamSpec measured power spectrum and the “bestfit” primary CMB power spectrum, i.e., the unresolved foreground residual for each frequency combination. The lower panels show the residuals after removing the bestfit foreground model. The lines in the upper panels show the various foreground components. Major foreground components are shown by the solid lines, colour coded as follows: total foreground spectrum (red); Poisson point sources (orange); CIB (blue); thermal SZ (green), Galactic dust (purple). Minor foreground components are shown by the dotted lines: kinetic SZ (green); tSZ × CIB cross correlation (purple). The 100 × 143 and 100 × 217 GHz spectra are not used in the CamSpec likelihood. Here we have assumed and . 

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Fig. 14
Comparison of pairs of foreground subtracted cross spectra, demonstrating consistency of the residuals with respect to the bestfit theoretical model. The red line in each of the upper panels shows the theoretical six parameter ΛCDM spectrum for the bestfit parameters listed in Table 8. The lower panels show the residuals with respect to this spectrum, together with error bars computed from the diagonal components of the covariance matrices of the band averages. The points here are bandaveraged in bins of width Δℓ ~ 31. 

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The ΛCDM parameter constraints derived from the two likelihoods are shown in Fig. 19, while Fig. 20 shows the foreground parameters. For the cosmological parameters, the agreement between the two likelihoods is excellent: when the CL49 mask is adopted for the Plik likelihood, which is also used by CamSpec at 100 GHz, all parameters agree to 0.2σ in terms of maximum posterior values. We also see that the widths of the distributions are quite similar, with the Plik ones slightly broader than the CamSpec ones. Significantly larger differences are seen for the foreground parameters.
These differences can be understood as follows: we use the CamSpec likelihood with Galactic mask CL49 for the 100 × 100 GHz spectra, to minimise the cosmic variance in the low to intermediate multipole range (ℓ ≲ 1200), taking advantage of the low level of Galactic emission in this channel. At higher multipoles we use the more conservative Galactic mask CL31 for both the 143 and 217 GHz channels, at the price of a higher variance. However, in the specific case of the ΛCDM model considered here, most of the constraints on cosmological parameters come from relatively modest multipoles, ℓ < 1500, rather than from the damping tail. This explains why when we repeat the Plik analysis, enlarging the sky area from Galactic mask CL39 to CL49, we find parameter distributions in good agreement with those of CamSpec.
Figure 20 shows the foreground parameters estimated from both Plik and CamSpec. We consider the case for Planck data alone, and with the inclusion of data from ACT and SPT. We also impose, for CamSpec, a Gaussian prior of 0.7 ± 0.2 on the CIB slope parameter, γ^{CIB}. We find that the upper bounds on the CIB amplitude at 143 GHz, , and on the SZ amplitudes (both thermal and kinetic, A^{tSZ} and A^{kSZ}) are in good agreement using Planck data alone, but we see differences (~1.5σ) in the CIB and Poisson amplitudes at 217 GHz (, ) as well as a difference in the CIB correlation coefficient, .
We understand this effect in the following way: CamSpec uses a more limited multipole range and a more conservative mask at 217 GHz than Plik, in order to minimise the Galactic emission in this channel. This enhances the degeneracy between the foreground parameters at 217 GHz, and enhances the sensitivity to possible deviations of the CIB power spectrum from the pure power law assumed here. This is artificially enhanced by normalizing the components at ℓ = 3000, which is more suitable for high resolution experiments than for Planck. Let us stress that, despite the disagreement of the precise decomposition into the different physical components, the sum of the foreground contributions at 217 GHz in Plik and CamSpec is in good agreement. And when ACT and SPT data are added, these differences are largely reduced, as shown in Fig. 20.
Are these differences important for cosmology? To address this question, we examine the covariance between cosmological and foreground parameters. Figure 21 shows the correlation matrix for all the estimated parameters. The basic ΛCDM parameters have wellknown correlations: the scalar spectral index, n_{s}, is anticorrelated with both the amplitude A_{s} and the dark matter density Ω_{c}h^{2}, which is itself correlated with the amplitude. Within the foreground parameters, there are strong correlations between the Galactic dust amplitude, the CIB amplitudes, and the SZ amplitude, as well as between point source amplitudes at different frequencies. These correlations result from the conservative foreground model adopted here, where all amplitudes of the CIB and Poisson contributions are left free to vary in each frequency pair; this choice results in small residuals in the fits, at the price of partial degeneracies between the foreground parameters.
Fig. 15
Top: Planck maximumlikelihood (primary) CMB spectrum compared with the bestfit six parameter ΛCDM spectrum. Bottom: power spectrum residuals with respect to the ΛCDM model. The error bars are computed from the diagonal elements of the bandaveraged covariance matrix, as given by Eq. (A.25), including contributions from foreground and beam transfer function errors. 

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Despite this conservative foreground model, there is a small correlation between cosmological and foreground parameters, the strongest effect being a 34% anticorrelation between the kinetic SZ amplitude and the scalar spectral index. The kinetic SZ power spectrum amplitude is positive, so marginalising over it affects the peak value of n_{s} despite the fact that A^{kSZ} is not significantly different from zero. The addition of smaller scale data from ACT and SPT helps to break this degeneracy. In addition, as can be seen in Fig. 20, the posterior distributions of CamSpec and Plik for A^{kSZ} are in good agreement, showing the stability of the cosmological parameters to the likelihood method.
In Sect. 7, we further explore the stability of the cosmological and foreground distributions to technical choices made in the likelihood and data selections.
6.2. Comparison to simulations
We now investigate the precision and accuracy with which cosmological parameters, and foreground parameters to a lesser extent, can be recovered from realistically simulated data. Here we compare the posterior distributions of cosmological and foreground parameters, together with calibration and beam error parameters, inferred using the Plik likelihood, with the input values of a set of simulations, referred to as “Full Focal Plane” (FFP6). The signals in these simulations are based on the “Planck Sky Model” (Delabrouille et al. 2013), which includes a detailed model of the astrophysical emission, both Galactic and extragalactic, at the Planck frequencies. The simulations also reproduce in detail the main instrumental systematic effects of Planck, including correlated timeline noise, instrumental pointing, flags, anisotropic detector beams, and spectral bandpasses. One thousand CMB and noise realisations were generated using the same foreground emission, and a hundred realisations were performed at the level of different detector sets. These simulations are described further in Planck Collaboration (2013).
In order to test for any methodological bias, we estimate cosmological parameters from 100 FFP6 simulations using the Plik likelihood. They consist of random realisations of CMB anisotropies and noise, superimposed with a single realisation of a frequencydependent foreground template. We assume the foreground power spectra to be known exactly, and include their additional power as a constant component in our model. We do not estimate the foreground parameters in this case. The goal of this test is to demonstrate the reliability of our analysis pipeline and explore the effects of the noise and CMBforeground chance correlations on parameter estimation.
To compare our results to the simulation inputs, we also estimate cosmological parameters from the 100 CMB realisations, without including noise or foregrounds. In the likelihood evaluations we downweight the highℓ part of the power spectrum as if there were noise at the level of the noise simulations. A direct comparison of the derived parameters from this CMBonly analysis and the full simulation (that contains noise and foreground emission) allows us to remove the scatter introduced by cosmic variance.
The result of the analysis is shown in Fig. 22, where we plot the distribution of the difference of the mean estimated parameters, between the CMBonly simulation and the noisy CMB simulation with foregrounds, in units of the standard deviation of each individual distribution. An unbiased pipeline will give a difference consistent with zero. Averaged over 100 simulations, only the bias on Ω_{c}h^{2} is statistically significant, but is <0.3σ. An interesting point to note from these histograms is that the distributions are rather wide, mostly around 0.5σ, while by construction, we only studied the effect of noise and CMBforeground chance correlations in this test.
The FFP6 simulations can also be used to assess the sensitivity to foreground modelling errors. In Fig. 23, we compare the posterior marginal distributions of cosmological parameters, estimated with different assumptions about the foreground model. The blue lines, which correspond to the analysis of CMBonly simulations (but accounting for the noise covariance in the likelihood) correspond to the idealised case where the foregrounds play no role, and where the noiseinduced variance has been averaged. The red, thick lines show results obtained when marginalising over the parameters of the model of foregrounds that is applied to the Planck data (see Sect. 3), with a fixed value of the CIB spectral index γ^{CIB}. Purple and grey lines show respectively the effect of leaving γ^{CIB} free when marginalising, or fixing it to the displaced value of 0.4, more than 2σ away from the peak posterior. The green lines show the effect of leaving γ^{Dust} (spectral index of the Galactic dust emission) free in the marginalisation.
The distributions are all in reasonable agreement with the input parameters of the simulation. In addition, we see that varying assumptions on the parameters of the foreground model (red, green, purple, and yellow lines) have negligible impact on the recovered cosmological parameters. Finally, the broadening of the posteriors between the CMBonly exploration and the full case (including noise random realisations and foregrounds) is expected, as the latter includes all the sources of variance, including CMBforeground and CMBnoise chance correlations.
It is worth noting that the FFP6 foreground simulations, based on extrapolations of existing observations, cannot be described by the simple foreground model used in the likelihood analyses. The upper bounds on biases introduced by a possible mismatch between the simulated foreground templates and the model used in the analysis, inferred from Fig. 22, should be representative of the Planck data analysis. The negligible impact of the various assumptions made on the foreground model parameters of Fig. 23 confirms that the cosmological parameter estimations are robust to details of the foreground model.
Fig. 16
Zoomin of regions of the Planck primary CMB power spectrum using finer bin widths (Δℓ = 15 for ℓ < 1000 and Δℓ = 7 for 1000 ≤ ℓ ≤ 2200. In the upper panels, the red lines show the bestfit ΛCDM spectrum, and the blue lines join the Planck data points. Error bars are computed as in Fig. 15. 

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Fig. 17
Left: residuals between the Planck power spectrum derived with Plik using “validation” settings (described in the text) and the Plik bestfit model. The light grey lines show residuals for individual detector pairs within each frequency combination. The blue lines show the inverse covariance weighted averages of the individual residuals, together with their errors computed from the covariance matrix. Right: decomposition of the total bestfitting model power spectra into CMB, combined thermal SZ and CIB, unresolved point sources, kinetic SZ, and Galactic dust. 

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7. Consistency checks
In this section we investigate the stability of the distributions of cosmological and foreground parameters. The technical choices made in constructing the highℓ likelihood fall into three broad categories. The first category covers internal parameter choices that leave the data selection unchanged. This includes choices such as the binning strategy, marginalising or not over calibration and beam errors, and the description of the noise model. The second category includes variations in the data selection, such as the multipole range used, and the choice of masks, or of spectra. The final category accounts for variation in the foreground model. We perform a suite of tests to investigate the impact of these choices on parameters. We use the Plik likelihood, and all tests are compared to the baseline Plik spectra. Most of the results can be summarized by “whisker plots”, which compare the main properties of the posterior distribution of the cosmological and foreground parameters. More detailed results are reported in Appendix D.1.
In this section we also compare our estimated cosmological parameters to those derived from spectra computed from the LFI 70 GHz channel. We additionally check the consistency of parameters with results obtained using the power spectrum of CMB maps derived by component separation methods (described in Planck Collaboration XII 2014) that use Planck data at all frequencies. This battery of tests demonstrates the stability of the inferred cosmological parameters.
A final test is to compare the predicted polarisation spectrum of the best fitting ΛCDM model with spectra measured from Planck. As discussed in Planck Collaboration I (2014) and Planck Collaboration VI (2014), the Planck polarisation data is not yet used in our cosmological analysis, as further tests must be performed, but the current results increase our confidence in the robustness of the highℓ temperature likelihood.
7.1. Impact of technical choices for fixed data selection
Here we consider three changes: (1) fixing the interfrequency calibration and beam errors to the bestfit values, rather than marginalising; (2) including subpixel effects; and (3) including noise correlation between detectors. Figure 24 shows the corresponding impact on parameters.
The effect of fixing the calibration and beam errors is negligible on most cosmological parameters within the six parameter ΛCDM model, with the exception of n_{s} where we see a 0.16σ shift. There is a bigger effect on the foreground parameters due to their partial degeneracy, and their subdominant contribution to the total power. There is only a small correlation between the cosmological parameters and the calibration coefficients, so marginalising or fixing their value has little impact on the cosmology.
The calibration coefficients are, however, strongly correlated with each other, in particular at 217 GHz, since they are also significantly correlated with e.g., the CIB amplitudes in the 217 × 217 and 217 × 143 GHz spectra. This is important to keep in mind when comparing the calibration estimates obtained here with those obtained from the CMB dipole in the HFI data processing paper (Planck Collaboration VI 2014). Nevertheless, as can be seen in Fig. 26, the peak posterior values of the relative calibration coefficients are found to differ from 1 at most at the few parts per thousand for all the 13 detectors sets involved, in agreement with the estimates of the calibration accuracy of the maps (Planck Collaboration VI 2014), although a wide flat prior has been applied on these coefficients. The same test applied on simulations with no beam or calibration errors shows how well this test is passed. This confirms that the deviations found at the 0.1% are significantly detected, and it is important to show that these deviations have little impact on the cosmology.
The estimated values of the beam errors do not imply that extra beam corrections are required. A comparison of the prior and posterior distributions suggests that we have quite conservatively estimated the uncertainties from the beam determination. We keep this conservative approach in CamSpec, in which we marginalise analytically over all beam eigenmodes except for the dominant 100 GHz mode, , which we sample directly.
As an extended test, we investigate the effect of possible errors in the beam transfer function when the helium abundance Y_{P} is also allowed to vary freely (i.e., without imposing constraints from Big Bang Nucleosynthesis), as it has a larger effect on the small scale spectrum. Varying this parameter leads to a substantial broadening of the posterior distributions for Ω_{b}h^{2}, θ_{MC}, and n_{s}, as can be seen in Fig. 27. We confirm that marginalising over calibration and beam errors has a small impact on all cosmological parameters, including Y_{P}. On the other hand, it has a somewhat larger impact on some of the foreground parameters (see Appendix D.1).
Fig. 18
Top: CMB power spectrum estimated using the Plik likelihood, after subtracting the bestfit foreground model. Bottom: residual difference (blue line) between the Plik spectrum and the bestfit ΛCDM model. The solid green line shows the difference between the bestfit models derived from the Plik and CamSpec likelihoods. 

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Fig. 19
Comparison of cosmological parameters estimated from the CamSpec (blue) and Plik (red for mask CL39; purple for mask CL49) likelihoods. All parameters agree to better than 0.2σ when CL49 is used for Plik, matching the sky area used by CamSpec at 100 GHz. 

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Fig. 20
Comparison of foreground parameters estimated with the CamSpec and Plik likelihoods. The purple (red) lines show the CamSpec (Plik) distributions using only Planck data, and the green (blue) lines show the CamSpec (Plik) results when additionally including ACT and SPT. 

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We then investigate the impact of two subdominant effects: the “subpixel effect” and the possible presence of a correlation in the noise between detectors or detector sets. The subpixel effect has a convolving effect on the power spectra that is similar to gravitational lensing of the CMB, but is purely a result of the Planck scanning strategy and the mapmaking procedure (Planck Collaboration VII 2014). The scanning strategy on rings with very low nutation levels results in the centroid of the samples being slightly shifted from the pixel centres; however, the mapmaking algorithm assigns the mean value of samples in the pixel to the position at the centre of the pixel. This has a nondiagonal effect on the power spectra, but the correction can be computed given the estimated power spectra for a given data selection, and recast into an additive, fixed component of the model covariance matrix.
The possible noise correlation between detectors may appear due to factors such as common residual thermal fluctuations, electronic chain noise, or cosmic ray showers. To build a model of this correlated component, we compute the crosspower spectra between detectors of difference maps that are free of signal. This procedure should capture all correlations on time scales shorter than half a ring’s observation. These estimates are noisy, so we compute an average amplitude of the correlation for ℓ ≥ 1000.
We find that the impact on cosmological parameters of both these effects is negligible, with less than a 0.1σ variation on the mean posterior values.
7.2. Impact of data selection
Here we consider three changes: (1) varying the angular range used in the likelihood; (2) varying the Galactic mask; and (3) discarding individual frequency channels. These are expected to result in changes in the parameter distributions due to the fact that we are changing the input data. The corresponding impact on parameter distributions are illustrated in Fig. 24.
We first vary the maximum and minimum multipole. Using ℓ_{max} = 1008 gives parameter distributions of similar width to those obtained by WMAP. We find that all basic ΛCDM cosmological parameters have converged by ℓ_{max} ≃ 1500, since no parameters are specifically sensitive to the damping tail. The convergence of the posteriors on foreground parameters is, as expected, slower when increasing ℓ_{max}, as most of them are dominant at small scales. The Galactic dust normalization decreases with ℓ_{max} due to its correlation with the CIB components. Similarly, changing ℓ_{min} from 100 to 50 has a negligible effect on cosmological parameters, and mostly affects the determination of the Galactic dust amplitude, which decreases for ℓ_{min} as it is better measured on large scales; its correlations with the CIB clustered and Poisson contributions explain the slight variations in the corresponding parameters , γ^{CIB}, and .
We then investigate the impact of varying the Galactic mask, from the most conservative (CL20) to the least conservative (CL49), for fixed multipole range 100 ≤ ℓ ≤ 2508. As expected, the errors decrease as the sky fraction increases. From CL20 to CL39, cosmological and foreground parameters are stable. The foreground parameters change significantly, however, when we use CL49 for all channels, showing that our foreground model is unable to properly fit the data: the clearest sign of this failure is the unphysically low value of the CIB spectral index (γ^{CIB}), indicating that our CIB component determination is getting contaminated by a dustlike component with a steeper angular power spectrum than the CIB, but shallower than that of our (single) Galactic dust component. At low Galactic latitudes, the presence of compact Galactic sources leads to a flatter angular power spectrum than that of (highlatitude) diffuse thermal dust, and this likely affects the CIB determination. This justifies the choice made in CamSpecto use a conservative masking strategy (CL31) for the 143 and 217 GHz channels.
Fig. 21
Correlation matrix between all the cosmological (top block), foreground (middle block), and derived (bottom block) parameters, estimated using the Plik likelihood. 

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Next, we remove one frequency channel at a time. Results change by less than 0.5σexcept when removing the 217 GHz channel. Removing the 217 GHz channel retains only 21 crossspectra out of the 78. As a finer test, we consider surveydifference tests, which were performed in the HFI data processing paper (Planck Collaboration VI 2014) at each frequency, for all combinations of the input maps used in the likelihood analysis (two at 100 GHz, five at 143 GHz, and six at 217 GHz), and for two survey differences: Survey 1 – Survey 2 and Survey 1 – Survey 3. It was found there that in addition to the 217ds1 × 217ds2 cross spectrum, only two other crossspectra failed this test, namely 2171 × 217ds2 and 2171 × 2173. We therefore consider here a data selection where only the (other) “best 217 GHz” spectra are retained in the analysis. Figures 24 and 25 show that this has no discernible influence on the determination of all basic ΛCDM cosmological parameters (but for a slight broadening of the posterior distribution), while the main foreground parameter change is a 1σ shift of the point source amplitude at 217 GHz.
We also note that since the public release of the data in March 2013, we have found evidence that the “ℓ ~ 1800 dip” in some 217 GHz detector crossspectra (see Fig. 17) is stronger in the first sixmonth survey than in subsequent surveys and that its amplitude may be reduced by additional data flagging targeted at EMIEMC interference from the 4 K cooler drive and readout electronics. This dip is therefore likely to be a (small) residual systematic effect in the data, but one, which has little impact on cosmological parameter determination in the standard model. Planck Collaboration XVI (2014) extends the verification to other cosmological models and data selections.
Fig. 22
Distribution of the difference between cosmological parameter mean posterior values, as estimated by Plik (assuming foreground spectra are known) from the (CMB + fixed foregrounds + noise) FFP6 simulations, and the corresponding posterior parameter mean values from CMB only simulations (based on the same CMB realisations as in the FFP6 simulations), in units of the FFP6 posterior standard deviations.This demonstrates the absence of methodological bias at the level of the intrinsic dispersion between realisations, while removing from the distributions the dominating scatter coming from the purely CMB induced cosmic variance. In these figures, the dominant source of scatter comes from the CMBforeground chance correlations in the signal dominated part of the spectrum, and from the noise scatter at large multipoles. 

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Fig. 23
Posterior marginal distributions of cosmological parameters obtained by Plik on a single FFP6 realisation, under different assumptions on the foreground model. Red thick lines correspond to the foreground model parametrisation described in Sect. 3 with fixed γ^{CIB}. Purple lines (resp. green, grey) show the effect of leaving γ^{CIB} free (resp. letting γ^{Dust} free, or fixing γ^{CIB} to 0.4, 2σ away from the peak posterior value). Finally, the blue lines correspond to the analysis made on CMBonly simulations. The input values of the parameters of the simulation are shown as dashed vertical lines. 

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7.3. Testing the extragalactic sources model
In this section we describe various tests of the validity of our model for the extragalactic sources. Further tests are also reported in Appendix B of Planck Collaboration XVI (2014).
7.3.1. Poisson power from extragalactic sources
Here we check that the Poisson power estimated in the likelihood, which comes from sources below Planck’s detection threshold, is roughly consistent with the level expected given number counts of detected galaxies. Figure 28 shows source counts from Planck (Planck Collaboration Int. VII 2013) at 100, 143, and 217GHz derived from the Planck Early Release Compact Source Catalogue (Planck Collaboration VII 2011; Planck Collaboration XIII 2011). At 143 and 217 GHz we also show the source counts from SPT as reported in Vieira et al. (2010) at 150 and 220 GHz, and from ACT (Marriage et al. 2011) at 150 GHz. The models of de Zotti et al. (2005) and Tucci et al. (2011) are also shown and are discussed in Planck Collaboration Int. VII (2013).
Planck Collaboration Int. VII (2013) use spectral information to separate the sources into “synchrotron” and “dusty” sources, and show that the counts at 100–217 GHz are dominated by synchrotron sources at flux densities above ~400 mJy. Vieira et al. (2010) performed a similar separation. The counts at 150 GHz are dominated by synchrotron sources at flux densities S> 10 mJy, but dusty galaxies contribute roughly equally at 220 GHz at flux densities ≲30 mJy (Vieira et al. 2010; Hall et al. 2010). The ACT counts have not been separated according to spectral type, but should be dominated by radio sources at these flux densities.
Fig. 24
Comparison of the distributions of cosmological parameters obtained with Plik in the reference case (left) with a set of validation test cases. The red line indicates the median and blue the mean, computed from the posterior histograms. The box shows the 68% confidence interval; the outer line the 95% interval. 

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Fig. 25
Comparison of the distributions of foreground model parameters obtained with Plik, as in Fig. 24. 

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Fig. 26
Estimated calibration and beam eigenmode parameters, compared to the priors, using (left) 100 to 217 GHz data, and (right) a single “FFP6” simulation where there was no calibration or beam errors. The shaded area shows the width of the prior imposed on the first beam eigenvalue. The individual SWB detectors (those denoted in the legend by their number rather than by a detset number) are not single out by this comparison. 

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Fig. 27
Posterior distributions for cosmological parameters obtained with Plik, varying the Helium abundance Y_{P} (red). We show the effect of marginalising over calibration and beam errors (blue), and the results obtained using the reference settings (purple), where Y_{P} is constrained by BBN. 

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Figure 28 show models fit to the counts using the function (20)where A, S_{0}, α, β, B, and S_{1} are free parameters. The bestfit values of these parameters are given in Table 7.
Given this model, and given the approximate flux cut applied to the Planck maps, the expected contribution of radio sources to the Planck power spectra, at flux densities smaller than 400, 350, and 225 ± 50 mJy at 100, 143, and 217 GHz, are 8.47 ± 1, 6.05 ± 0.8, and 3.10 ± 0.7 Jy^{2}/sr. The contribution of unresolved infrared galaxies to the power spectra is not negligible. They are expected to dominate at 217 GHz, even if they are subdominant in the Planck counts. Indeed, faint IR galaxies create a “bump” in the S^{5 / 2}dN/ dS distribution, below the detection limit of ACT or SPT. This bump is seen at higher frequencies, e.g., with the Herschel SPIRE instrument (see Planck Collaboration Int. VII 2013, for details).
This bump of infrared galaxies has not been measured at frequencies of 217 GHz and below. However, measurements with the AzTEC telescope at 1.1mm (270 GHz, Scott et al. 2012) can be used to extrapolate the counts down to 217 GHz. This leads to a predicted peak in the number counts (S^{5 / 2}dN/ dS) around 1.4 mJy at a level of 190 Jy^{1.5}/ sr, somewhat higher than the values in Hall et al. (2010). The corresponding contribution of infrared galaxies to the power spectrum is estimated in Planck Collaboration XVIII (2011) to be 16 Jy^{2}/ sr at 217 GHz but with significant uncertainty.
Summing the expected contributions from radio and IR galaxies, we estimate the following values for D_{3000} for Planck: 200, 75, and 120 μK^{2} at 100, 143, and 217 GHz respectively. These predictions are much less certain at 217 GHz due to the absence of infrared galaxy counts at this frequency.
7.3.2. Clustered power in CIB fluctuations
Here we check the consistency of the estimated clustered power in CIB fluctuations. As already noted, from the CamSpec and Plik likelihoods we find only an upper limit on the clustered CIB power at 143 GHz. With CamSpec we detect clustered power at 217 GHz with , and Poisson power with . This Poisson power is dominated by the CIB fluctuations. For the Plik likelihood, we have and . The sum of the CIB power at 217 GHz, and at pivot scale ℓ = 3000, is in the range ≈105–125 μK^{2}^{8}.
We compare this level to the measurements by the ACT and SPT experiments, which probe higher angular resolution. Fitting a common model to the ACT power spectra from Das et al. (2014), and the SPT spectra from Keisler et al. (2011); Reichardt et al. (2012), the analysis in Dunkley et al. (2013) finds for the CIB clustered component (and for the CIB Poisson component) at effective frequency 219.6 GHz. For SPT, the clustered level is (and Poisson ), also at an effective frequency of 219.6 GHz. This is consistent with the SPT analysis in Reichardt et al. (2012). These are estimated assuming γ^{CIB} = 0.8 and , and that the CIB emission can be modelled with frequency as a modified blackbody, following Addison et al. (2012a).
The total CIB signal seen by Planck, extrapolated to ℓ = 3000 scales, is therefore consistent with the ACT and SPT observations, but given the limited angular range of Planck, the clustered and Poisson part are degenerate. This motivates us to include the ACT and SPT data in many of our cosmological analyses.
When combining Planck, ACT and SPT data together, using the same foreground model (except for Poisson power, which depends on the respective flux cuts of the experiments), both CamSpec and Plik give , , , , and . These estimates of the Poisson power are in good agreement with the predictions given in Sect. 7.3.1 for the 100 and 143 GHz channels. In the latter, radio sources below the Planck flux cuts dominate the Poisson power, which can be reliably estimated from existing source counts measurements.
We also consider modifying our model for the clustered part of the CIB. There have been a wealth of CIB models (e.g., Knox et al. 2001; Amblard & Cooray 2007; Hall et al. 2010; Pénin et al. 2012; Addison et al. 2013; Xia et al. 2012), most assuming that the dust is a biased tracer of the dark matter distribution, but differing in their parametrisation of the dust emissivity and its evolution, and their treatment of the dark matter power spectrum. Recent papers (Planck Collaboration XVIII 2011; Addison et al. 2013) have shown that the addition of Planck CIB measurements, when combined with other small scale probes including SPT, ACT, BLAST, and Herschel, rule out models that assume the underlying dark matter power spectrum is linear.
Fig. 28
Number counts from Planck (filled circles, Planck Collaboration Int. VII 2013), ACT, and SPT (open squares) as described in the text, from 100 GHz to 217 GHz. The models from de Zotti et al. (2005, solid line) and Tucci et al. (2011, dots) are overplotted. The analytical fit from Eq. (20) and Table 7 is shown dashed, and shows a similar behaviour to the scaleddown de Zotti et al. (2005) model. 

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We therefore test a set of models that try to simultaneously fit the nonlinear spectrum with one or more template spectra. We consider fixing the scale dependence to a power law, either ℓ^{0.8} or ℓ^{0.6}, or extending the power law model to have a running of the index. The ℓ^{0.8} model has been used in Shirokoff et al. (2011); Reichardt et al. (2012); Addison et al. (2012a); Dunkley et al. (2013), while the ℓ^{0.6} more closely matches the CIB model of Addison et al. (2013). We find that a simple power law does not fit both the Planck and highℓ data sufficiently well, but that allowing the additional freedom of a running spectrum opens up the parameter space too much, with little improvement in goodness of fit, motivating our use of the varying γ^{CIB} model.
We also test a simple model using just a linear theory dark matter power spectrum, assuming that the nonlinear power can be absorbed into the Poisson term. This results in an estimate of the Poisson level at 217 GHz that is inconsistent with ACT and SPT, so we do not use this model. For all these models we test the effect on cosmological parameters, using the Hubble constant as a test case, and find the effect on parameters to be small. This is also investigated in Planck Collaboration XVI (2014).
7.4. Consistency of the Planck 70 GHz data
Figure 11 shows individual frequency spectra from 70 to 353 GHz. We only use data from the 100 to 217 GHz channels to form the highℓ likelihoods, but here we compare cosmological parameters derived from the Planck 70 GHz channel alone. Maps at 70 GHz are easier to characterise than the higher frequency channels in terms of instrumental and foreground properties, but the resolution and sensitivity are lower. The 70 GHz noise properties are in general well described by a simple three parameter model involving 1 /f and white noise contributions (Planck Collaboration II 2014). The 70 GHz channel also has the least diffuse foreground emission at large scales (Planck Collaboration XII 2014), and the extragalactic source contribution is dominated by radio galaxies whose emission is well known at these frequencies (Planck Collaboration XXVIII 2014). We adopt a Galactic plane cut leaving ~70% of the sky for the analysis (CS70 from Planck Collaboration XII 2014) to which we add a point source mask optimised for 70 GHz. In Appendix D.2 we describe how cosmological parameters are estimated from the 70 GHz channel, which are summarized in Fig. 24. Accounting for the lower sensitivity and angular resolution at 70 GHz, which translate into a narrower multipole range (ℓ < 1200), the parameter distributions are consistent with the reference values.
7.5. Consistency with power spectra of CMB maps obtained by component separation methods
Fig. 29
Top: CMB power spectrum, with bestfitting foreground model removed, compared to the CamSpec best fit model. Middle: the residuals with respect to this bestfit model. Bottom: residuals of the bestfit models from the mapbased likelihoods, with respect to the CamSpec best fit model. 

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Fig. 30
The variation of cosmological and foreground parameters estimated for the four CMB maps as function of ℓ_{max} = 1000, 1500, 2000, 2500, with ℓ_{min} = 70, compared to those obtained with the CamSpec and Plik likelihoods. Parameters are consistent to 1σ for ℓ_{max} ≤ 2000 

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The likelihoods we consider in this paper account for component separation by modelling the multifrequency data at the power spectrum level, to fully exploit the signal at the smallest scales probed by Planck. We can compare the results to those derived from an alternative approach, measuring the power spectrum of CMB maps estimated from component separation techniques. Here we present results obtained with four CMB maps, derived using methods referred to as CommanderRuler, SMICA, NILC, and SEVEM, described in detailed in the accompanying paper (Planck Collaboration XII 2014). These maps are weighted with the union of the confidence masks of the different component separation methods, covering 72% of the sky (mask U73 of Planck Collaboration XII 2014), with a further cosine apodisation of 30′. We compare their angular power spectra and cosmological parameters with those from CamSpec. To estimate the power spectra we use the XFaster method, an approximation to the iterative, maximum likelihood, quadratic band power estimator based on a diagonal approximation to the quadratic Fisher matrix estimator (Rocha et al. 2011, 2010). The noise bias is estimated using difference maps, as described in Planck Collaboration XII (2014). The resulting spectra are shown in Fig. 29, and agree well out to scales ℓ_{max} = 2000, even though the agreement is less striking for the CommanderRuler smallscale spectrum.
Fig. 31
Planck TE and EE polarisation spectra computed as described in the text, together with the polarisation spectra predicted from the sixparameter ΛCDM model, fit only to the Planck temperature data. 

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We then estimate cosmological parameters using a Gaussian correlated likelihood derived from these bandpowers. To model the residual extragalactic foregrounds, we consider two nuisance parameters: A_{ps}, the amplitude of a Poisson component, and A_{Cℓ}, the amplitude of a clustered component, scaling a term with shape D_{ℓ} ∝ ℓ^{0.8}. Figure 30 compares the parameters obtained as a function of ℓ_{max} for each method, compared to the values from the highℓ likelihoods. The results are consistent to 1σ or better for ℓ_{max} = 1500 and ℓ_{max} = 2000. Despite adopting a simple twoparameter model for the extragalactic foregrounds, the likelihood using a CMB map as input data works reasonably well, and may be further exploited with analysis of simulations, improved extragalactic foreground modelling, and the development of an error model.
7.6. Consistency with highℓ polarisation
Our final consistency test is illustrated in Fig. 31, showing the polarisation power spectra derived from the Planck data. Both the TE and EE crossspectra are shown, in bins of width of Δℓ = 40. These spectra are computed by performing a uniformly weighted average of all detector sets combinations at 70 × 100, 100 × 143, 100 × 217, and 143 × 217 GHz. We use the temperature beam window functions for beam deconvolution. For the analysis, we applied CL39 to the temperature maps, and discarded 60% of the polarisation data (i.e., f_{sky} = 0.4). Other than masking, no efforts have been made to subtract foreground contributions or take instrumental effects such as leakage into account. Despite the substantial masking applied, we see evidence for residual contributions of noncosmological origin. Besides demonstrating the potential of Planck to deliver high quality polarisation maps and spectra (with the limitations explained in Planck Collaboration VI 2014), the figure demonstrates the high level of consistency of these polarisation spectra between themselves, and with the prediction from the model fit using just the temperature spectrum shown in Fig. 15.
As discussed in Planck Collaboration VI (2014) and Planck Collaboration II (2014), at present, the HFI and LFI polarisation spectra at low multipoles are affected by systematic errors that cause biases, which will need to be accurately modelled or removed for the next Planck release. However, these systematics rapidly become unimportant at higher multipoles. While not yet fit for cosmological parameter analysis, the consistency at the level of a few μK of these Planck polarisation spectra adds to our confidence in the analysis of temperature data. It shows that within the ΛCDM framework, the cosmological parameters estimated from Planck temperature data are not strongly affected by the uncertainties in the modelling of unresolved foregrounds.
8. Lowℓ likelihood
At low multipoles (ℓ ≲ 50), the distribution of the estimated C_{ℓ}s is not well approximated by a Gaussian due to the limited degrees of freedom per ℓ (e.g., Efstathiou 2004). However, both the CMB signal, s, and instrumental noise, n, are individually nearly Gaussian distributed at the map level, provided that foreground emission and instrumental systematics effects are negligible (e.g., Planck Collaboration XXIII 2014; Planck Collaboration XXIV 2014), and the actually observed map, m = s + n, is therefore also nearly Gaussian distributed. Under this assumption, the CMB power spectrum likelihood is given by (21)where n is the number of observed pixels, M(C_{ℓ}) = C(C_{ℓ}) + N is the data covariance matrix, and C and N are the CMB and noise covariance matrices, respectively.
In the general case, the data vector m includes both temperature (T) and linear polarisation (Q, U) Stokes parameter maps. Pixels exhibiting high foreground contamination are removed by masking, such that the data vector is restricted to the subset of valid pixels, m = (T_{i1},T_{i2},...,T_{nT},Q_{j1},Q_{j2},...Q_{nP},U_{j1},U_{j2},...U_{nP}). The corresponding rows and columns are removed from M, effectively corresponding to marginalising over the masked region of the sky. Note that in general, n_{T} ≠ n_{P}, and the sets of indexes of temperature and polarisation measurements are different. We assume the same number of pixels in Q and U, although this is not a requirement.
The signal covariance matrix can be written symbolically as (22)where the signal correlations for the temperature component are explicitly given by (23)Here P_{ℓ} are the Legendre polynomials, and θ_{i1i2} is the angle between the centres of pixels i_{1} and i_{2}. Similar expressions are available for the polarisation correlations (e.g., Tegmark & de OliveiraCosta 2001). The effect of the (azimuthally symmetric) instrumental beam, b_{ℓ}, and pixel window function, w_{ℓ}, are encoded in .
The main problem with the likelihood expression given in Eq. (21) is its high computational cost. This is determined by the matrix inversion and determinant evaluations, both of which scale as with N = n_{T} + 2n_{P}. In practice, this approach is therefore limited to coarse pixelisations, N_{side} ≤ 16, which reliably only supports multipoles below ℓ ≲ 30. On the other hand, the Gaussian approximation adopted by the highℓ likelihood is not sufficiently accurate for the stringent requirements of Planck below ℓ ≲ 50. In the next section, we therefore describe a faster lowℓ likelihood estimator, based on Gibbs/MCMC sampling, which allows us to exploit the full range up to ℓ ≤ 50 with low computational cost, while additionally supporting physically motivated foreground marginalisation.
Page et al. (2007) pointed out that the temperature and polarisation parts of the likelihood can be separated and evaluated independently, under the assumption of negligible noise in temperature and in the temperaturepolarisation cross correlations (i.e., the TQ and TU blocks of the pixel level noise covariance matrices). Further assuming vanishing primordial B modes and TB correlations, the TE correlations can be accounted for by redefining the modified Q and U maps as where are spin weighted spherical harmonics and are the harmonic coefficients of the signal in the temperature map. One can show by direct substitution that these modified Q and U maps are free of temperature correlations. The polarisation likelihood can be then computed independently from the temperature likelihood and, possibly, at lower resolution to save computational expenses. We test this strategy in Sect. 8.2, and adopt it for the current release of the Planck likelihood.
8.1. Lowℓ temperature likelihood
As discussed above, we do not implement the likelihood expression given in Eq. (21) directly, due to its high computational cost and limited flexibility with respect to foreground modelling. Instead, we adopt the Gibbs sampling approach (Eriksen et al. 2004; Jewell et al. 2004; Wandelt et al. 2004), as implemented by the Commander code (Eriksen et al. 2008), which allows both for physically motivated component separation and accurate likelihood estimation. A similar Gibbs sampling method was used to estimate the lowℓ temperature likelihood for WMAP (Dunkley et al. 2009; Larson et al. 2011), although not simultaneously accounting for component separation.
8.1.1. Methodology
We start by generalizing the above data model to include both multifrequency observations and a set of foreground signal terms, (26)Here d_{ν} denotes the observed sky map at frequency ν, and denotes a specific foreground signal component. As above, the CMB field is assumed to be a Gaussian random field with power spectrum C_{ℓ}, and the noise is assumed Gaussian with covariance matrix N_{ν}. The foreground model can be adjusted as needed for a given data set, and a full description of the model relevant for Planck is presented in Planck Collaboration XII (2014). In short, this consists of a single lowfrequency foreground component (i.e., the sum of synchrotron, anomalous microwave emission, and freefree emission), a carbon monoxide (CO) component, and a thermal dust component, in addition to unknown monopole and dipole components at each frequency.
Given this data model, we map out the full posterior distribution, P(s,f^{i},C_{ℓ}  d), using a Monte Carlo sampling algorithm called Gibbs sampling. Directly drawing samples from P(s,f^{i},C_{ℓ}  d) is computationally prohibitive, but this algorithm achieves the same by iteratively sampling from each corresponding conditional distribution,
It is straightforward to show that P(s  f,C_{ℓ},d) is a multivariate Gaussian distribution, and P(C_{ℓ}  s,f^{i},d) is an inverse Gamma distribution. The foreground distribution, P(f  s,C_{ℓ},d), does not have a closed analytic form, but can easily be mapped out numerically (Eriksen et al. 2008). Thus, all three distributions are associated with simple textbook sampling algorithms.
For CMB power spectrum and likelihood estimation using Gibbs sampling, the crucial intermediate product from the above sampling process is the ensemble of CMB sky samples, s^{k}. Each individual sample corresponds to one possible CMB realisation consistent with the observed data. In the absence of sky cuts, foreground contamination and instrumental noise, this map is identical to the true sky. In that case, the likelihood as a function of C_{ℓ} is determined by cosmic variance alone, and given by an inverse gamma distribution, (27)Here we have introduced the realisation specific power spectrum, , where are the spherical harmonic coefficients of s^{k}. In the case of realistic data, we need to marginalise over uncertainties due to sky cuts, foregrounds, and instrumental noise. Hence, Eq. (27) is replaced by an average over all possible sampled CMB realisations, (28)This expression is known as the BlackwellRao estimator (Chu et al. 2005), and is guaranteed to converge to the exact likelihood as the number of samples, N_{samp}, increases. Note that the normalization factor in this expression is unknown, but since the likelihood function is only used to compare different models through an effective likelihood ratio, this factor is irrelevant for actual calculations.
8.1.2. Data selection and preprocessing
As described in Planck Collaboration XII (2014), we include Planck frequencies between 30 and 353 GHz in the lowℓ likelihood. Each frequency map is downgraded from its native resolution to a common resolution of 40′, and projected onto an N_{side} = 256 HEALPix grid. Uncorrelated Gaussian regularization noise is added to each frequency channel map, with an RMS proportional to the spatial mean of the instrumental noise of the corresponding channel, ⟨σ_{ν}⟩, conserving relative signaltonoise between channels. The regularization noise level at frequency ν is . The purpose of this is to make the results insensitive to unmodelled features at scales comparable to and beyond the smoothing scale of 40′, in addition to improve the convergence speed of the Gibbs sampler. The resulting signaltonoise is unity at ℓ ~ 400, and the additional uncertainty due to the regularization noise is less than 0.2 μK^{2} below ℓ = 50, and less than 1 μK^{2} below ℓ = 100.
To study the stability of the lowℓ likelihood with respect to sky fraction, we constructed a suite of five different masks, leaving between 81 and 100% of the sky for the analysis, and for completeness we include the WMAP KQ85 mask (75% of the sky), as a sixth case. These lowℓ masks are distinct from those employed for the highℓ likelihood, and are produced in a preliminary Commander fullsky analysis in which we estimate individual foreground components and residual χ^{2} values per pixel. These maps are thresholded at various levels to produce a useful range of sky fractions.
For each mask, we fit a twoparameter amplitude and tilt power spectrum model of the form , using the lowℓ likelihood between ℓ_{min} = 2 and ℓ_{max} = 49, where is the bestfit PlanckΛCDM spectrum, and ℓ_{0} = (ℓ_{min} + ℓ_{max}) / 2. The resulting distributions are shown in Fig. 32 for three masks, covering 77.4 (A; WMAP KQ85), 87.5 (B) and 97.2% (C) of the sky, respectively. The internal agreement is excellent, with parameters differing by less than 0.3σ between the very aggressive mask A and the conservative mask C. While any of these masks would establish an acceptable likelihood, we adopt Mask B as our fiducial mask for two reasons. On the one hand, the parameter uncertainties obtained with Mask B are only 4% larger than those obtained for the minimal Mask A, indicating that both nearly saturate the cosmic variance limit. On the other hand, analysis of realistic simulations indicate the presence of statistically significant map residuals near the Galactic plane that are accepted by Mask A, but rejected by Mask B (Planck Collaboration XII 2014). The latter therefore represents a good compromise between rejecting foreground residuals and maximizing statistical power.
Fig. 32
Twoparameter probability distribution for an amplitudetilt model constrained by the lowℓPlanck likelihood using three different masks. Angular scales between 2 ≤ ℓ ≤ 49 are included in this fit. 

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Fig. 33
Comparison of constraints on τ and A_{s} using the split temperaturepolarisation WMAP likelihood approach (dashed contours; Eqs. (24) and (25)) with those obtained with the exact bruteforce pixel likelihood (shaded contours; Eq. (21)). 

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We include 100 000 Gibbs samples in the likelihood estimator, ensuring excellent convergence characteristics for the BlackwellRao estimator for ℓ < 50.
8.2. Lowℓ polarisation likelihood
The present Planck data release includes only temperature data. In this release, we therefore supplement the Planck likelihood with the 9 year WMAP polarisation likelihood^{9} derived from the WMAP polarisation maps at 33, 41, and 61 GHz (Ka, Q, and V bands) (Page et al. 2007; Bennett et al. 2013). However, we introduce one modification to this pixelbased likelihood code, replacing the spherical harmonics coefficients of the temperature field, , in Eqs. (24) and (25) with those derived from the Planck temperature map derived by Commander, for which the Galactic plane has been replaced with a Gaussian constrained realisation.
Fig. 34
Top panel: temperature power spectra evaluated from downgraded Planck maps, estimated with Commander, NILC, SEVEM, or SMICA, and the 9 year WMAP ILC map, using the Bolpol quadratic estimator. The grey shaded area indicates the 1σ Fisher errors while the solid line shows the PlanckΛCDM best fit model. Bottom panel: power spectrum differences for each algorithm/data set relative to the Commander spectrum, estimated from the spectra shown in the panel above. The black lines show the expected 1σ uncertainty due to (regularization) noise. 

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In Fig. 33, we compare constraints on τ and A_{s} as derived with this split likelihood with those obtained through an exact bruteforce evaluation of Eq. (21), simultaneously including temperature and polarisation measurements at N_{side} = 16. The two methods produce almost indistinguishable results.
In Appendix E we assess the robustness of the WMAP polarisation likelihood with respect to dust contamination, by replacing the WMAP polarised dust template with the far more sensitive HFI 353 GHz polarisation map. We find that the optical depth to reionisation, τ, is reduced by about 0.5 − 1σ, depending on the template removal method adopted. However, since the Planck polarisation maps are not included from the current data release, we adopt the WMAP polarisation likelihood without further changes for now, and will return to this topic in the next data release.
8.3. Lowℓ power spectrum – consistency and robustness
Fig. 35
Top: comparison between the lowℓPlanck temperature power spectrum estimated by Commander and the 9year WMAP spectrum (Bennett et al. 2013). Error bars indicate 68% confidence regions. Bottom: difference between the WMAP and Planck lowℓ spectra. 

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Fig. 36
Probability distribution for a twoparameter (amplitude and tilt) model derived from the Planck (blue) and WMAP (red) lowℓ likelihoods, including angular scales between ℓ = 2 and 30. 

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In this section, we present the lowℓPlanck CMB temperature power spectrum derived using the Commander approach described above, and assess its robustness through comparisons with three alternative foregroundcleaned Planck CMB maps (NILC, SEVEM, and SMICA; Planck Collaboration XII 2014), as well as with the 9 year WMAP ILC temperature map (Bennett et al. 2013).
As a first consistency test, we compute the power spectrum from each map using Bolpol (Gruppuso et al. 2009), an implementation of the quadratic maximumlikelihood power spectrum estimator (Tegmark 1997). Each map is smoothed to an effective resolution of 329.81′ FWHM, to suppress aliasing from high multipoles (Keskitalo et al. 2010), and repixelised on an N_{side} = 32 HEALPix grid. Gaussian white noise with a variance of 4 μK^{2} is added to each map to regularise the noise covariance matrix.
Here we adopt the U78 common mask, defined in Planck Collaboration XII (2014), to exclude regions of high Galactic emission, leaving 78% of the sky for analysis. We remove the observed monopole and dipole in the mask. The resulting power spectra up to ℓ ≤ 64 are shown in the top panel of Fig. 34, while the bottom panel shows the power spectrum residuals of each map relative to the Commander map. Note that the same noise realisation was added to each map, and the regularization noise therefore contributes little in this plot. For the different internallyderived Planck maps, no residual spectrum exceeds ≲100μK^{2} and is typically ≲50μK^{2} at ℓ ≳ 10. The WMAP spectrum exhibits significantly larger residuals, and are typically of the order of ~100 μK^{2} at ℓ ≳ 30.
Figure 35 shows the Planck and WMAP temperature power spectra derived directly from the respective likelihood code, while Fig. 36 shows the corresponding constraints on the twoparameter amplitudetilt model employed in Sect. 8.1.2, including multipoles between ℓ = 2 and 30. Neglecting the minor differences in the masks adopted by the two codes, these power spectra and parameter constraints are largely dominated by cosmic variance, and one should therefore expect the two distributions to be almost identical. Instead, from Fig. 36 we see that the WMAP lowℓ spectrum is 2.5–3% higher than the Planck spectrum. For a detailed discussion of this discrepancy, including a comparison at higher ℓ, see Planck Collaboration XXXI (2014). Here we only note that the effect is robust with respect to foreground removal and power spectrum evaluation algorithms, and also point out that the effect at low multipoles is too large to be explained by uncertainties in the Planck transfer functions (Planck Collaboration II 2014; Planck Collaboration VII 2014) or calibration (Planck Collaboration V 2014; Planck Collaboration VIII 2014). Also note that the amplitude of the lowℓ spectrum relative to the Planck bestfit model, (q,n) = (1,0), derived including the full multipole range between 2 ≤ ℓ ≤ 2500, is somewhat low in Fig. 36, with a bestfit amplitude of q ~ 0.9. This observation is discussed and quantified in greater detail in Sect. 9.3.
9. The Planck CMB spectrum and likelihood
9.1. Hybridisation of low and highℓ likelihoods
The highℓ and lowℓ likelihoods introduced in Sects. 2 and 8 each describe only a part of the full Planck data set. To estimate cosmological parameters from all the angular scales probed by Planck, they must be combined into a single likelihood function that describes all multipoles from ℓ = 2 to 2500.
In principle, it is desirable to include as many multipoles as possible in the lowℓ likelihood, since it captures the full nonGaussian structure of the likelihood. The Gaussian approximation for the likelihood using pseudospectra also improves at higher multipole due to the increasing number of degrees of freedom (Efstathiou 2004). For Planck we adopt a transition multipole of ℓ_{trans} = 50, a compromise between obtaining robust convergence properties for the lowℓ likelihood, and ensuring that the Gaussian approximation holds for the highℓ likelihood (Hamimeche & Lewis 2009).
To combine the likelihoods, we must account for the weak correlations between the low and highℓ components. We consider three options:

1.
Sharp transition: the lowℓ likelihood ends at ℓ_{max} = 49; the highℓlikelihood starts at ℓ_{min} = 50; no correlations are accounted for.

2.
Gap: the lowℓ likelihood ends at ℓ_{max} = 32; the highℓ likelihood starts at ℓ_{min} = 50; no correlations are accounted for, but the gap is sufficiently wide that any correlations are negligible.

3.
Overlap with correction: the lowℓ likelihood ends at ℓ_{max} = 70; the highℓ likelihood starts at ℓ_{max} = 50; the doublecounting of the overlap region is accounted for by subtracting from the loglikelihood a contribution only including 50 ≤ ℓ ≤ 70 as evaluated by the Commander estimator. Under the assumption that no correlations extend from ℓ ≤ 50 to ℓ ≥ 70, this approach is exact).
We estimate cosmological parameters using all three methods, and find that the posterior means typically vary by <0.1σ. The largest variation is seen including the running of the spectral index of scalar perturbations, in which the posterior mean changes by 0.2σ. Further, all deviations at the 0.1–0.2σ level are seen for case 2 above, which excludes data compared to the other two; case 1 and 3 give nearly indistinguishable results. Since case 1 is implementationally simpler, and can be estimated more efficiently (see Sect. 8), we select this method, adopting a sharp transition at ℓ_{max} = 50.
9.2. The Planck power spectrum and ΛCDM constraints
Using the full Planck likelihood, we now present the final 2013 Planck CMB power spectrum. For this, we fix all nuisance parameters to their maximumlikelihood values. The resulting spectrum is shown in Fig. 37 together with the corresponding bestfit sixparameter ΛCDM model. The agreement between the observations and the model is excellent over most of the multipole range. Only at low ℓs is it possible to see a systematic offset in the form of a slight power deficit; this is addressed separately in the next section.
Fig. 37
The 2013 Planck CMB temperature angular power spectrum. The error bars include cosmic variance, whose magnitude is indicated by the green shaded area around the best fit model. The lowℓ values are plotted at 2, 3, 4, 5, 6, 7, 8, 9.5, 11.5, 13.5, 16, 19, 22.5, 27, 34.5, and 44.5. 

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Constraints on the basic sixparameter ΛCDM model using Planck data.
Table 8 provides a summary of the ΛCDM parameters derived using the methodology described in Planck Collaboration XVI (2014) from the Planck likelihood. Here we use the same prior ranges on all parameters as in Planck Collaboration XVI (2014). These are as in Table 5 and 6, except for , A^{tSZ}, and A^{kSZ}, which are modified to [0,20], [0,10], and [0,10] respectively. Results are given for Planck alone, and in combination with the lowℓ WMAP polarisation likelihood (Planck+WP). For each case, we report both posterior maximum and mean values. Uncertainties denote 68% confidence limits.
A detailed discussion of these results is presented in Planck Collaboration XVI (2014), including an analysis of extended cosmological models, and their compatibility with other astrophysical data sets. The bounds derived from Planck alone are significantly tighter than those from the 9 year WMAP data alone, and comparable or better than those inferred from WMAP combined with SPT and ACT observations. These new constraints provide a precision test of the ΛCDM model. In general, we find good agreement with results derived from other astrophysical data sets, although there are a few exceptions that are in moderate tension with Planck (Planck Collaboration XVI 2014).
Considering each of the six ΛCDM parameters in turn, we first note that Planck constrains the physical baryon density to Ω_{b}h^{2} = 0.02207 ± 0.00033, which is in remarkable agreement with standard BBN predictions based on a determination of the primordial abundance of deuterium, Ω_{b}h^{2} = 0.021 ± 0.001 (Iocco et al. 2009), but with a fractional uncertainty of 1.5 %, three times smaller than the BBN uncertainty. The physical density of dark matter is measured with a fractional uncertainty of 2.6%, providing new constraints on specific dark matter production scenarios. The single most precise parameter, however, is the angular size of the sound horizon at the lastscattering surface, θ_{MC}, which is measured with a fractional uncertainty of 0.065 % by Planck, improving on the combined WMAP, ACT, SPT, and SNLS constraint by a factor of two.
Next, given that no polarisation data are included in the current data release, it is remarkable that Planck alone constrains the optical depth to reionisation, τ, with a fractional error of 40%. This is made possible by Planck’s high angular resolution and sensitivity, which allows a high signaltonoise measurement of lensing in the smallscale CMB power spectrum. This in turn breaks the wellknown e degeneracy between τ and the amplitude of scalar perturbations, A_{s}. The fractional uncertainty on A_{s} from Planck alone is 7%.
Having sufficient power to measure τ from small angular scale temperature data, Planck naturally also provides very strong constraints on the spectral index of scalar perturbations, n_{s}, leading to a fractional uncertainty of 0.97 %. The scaleinvariant HarrisonZeldovich spectrum, n_{s} = 1, is ruled out at a significance of 4.1σ from the Planck temperature spectrum alone. The significance of this exclusion rises to 5.4σ when the lowℓ WMAP polarisation is included. The analyses presented in Planck Collaboration XVI (2014) and Planck Collaboration XXII (2014) show that the preference for a (red) tilted primordial spectrum remains very strong also within most extensions beyond the minimal ΛCDM model. The implications of this results for inflationary models are discussed in Planck Collaboration XXII (2014).
With our choice of cosmological parameters, the Hubble parameter, H_{0}, and the fractional density of the cosmological constant, Ω_{Λ} = 1 − Ω_{m}, are derived parameters. They are probed by CMB observations mainly through their impact on θ_{MC}, and, to lesser extent, by the impact of Ω_{Λ} on the latetime integrated SachsWolfe effect. Since θ_{MC} is accurately measured, a particular combination of H_{0} and Ω_{Λ} is very well constrained by Planck, although in a modeldependent way; θ_{MC} depend on other cosmological parameters, such as the spatial curvature radius, neutrino masses, the number of relativistic degrees of freedom, or a possible dark energy equation of state parameter.
The results reported in Table 8 rely on the assumption of a flat ΛCDM cosmology with three neutrino species, two of which are assumed to massless and one featuring a small mass m_{ν} = 0.06 eV, reflecting the lower bound on neutrino masses imposed by neutrino oscillation experiments. Under these assumptions, Planck finds preferred ranges for H_{0} and Ω_{Λ} that are lower than previous CMB experiments. For instance, Planck+WP gives H_{0} = 67.3 ± 1.2 km s^{1}Mpc^{1}, to be compared with 70.5 ± 1.6 km s^{1}Mpc^{1} for the combined WMAP9+eCMB data set presented by Hinshaw et al. (2013). The underlying cosmology in the two analyses is the same, excepted for the small neutrino mass introduced in our default ΛCDM model. However, if we assume all three neutrino species to be massless, our bestfit and mean values for H_{0} increase only by 0.6 km s^{1}Mpc^{1}. Thus, the tension is clearly driven by the data rather than by theoretical assumptions. Planck Collaboration XVI (2014) shows that our results for H_{0} and Ω_{Λ} are in very good agreement with Baryon Acoustic Oscillation data, but in moderate tension with other cosmological probes. For instance, our Planck+WP bounds on H_{0} disagree at the 2.5σ level with direct determinations of the Hubble parameter using Cepheid and supernovae (Riess et al. 1998) or quasar time delays (Suyu et al. 2013), as well as with the results of the Carnegie Hubble Program (Freedman et al. 2012). Our bounds on Ω_{Λ} are in a slight 2σ tension with the results of the SNLS supernovae collaboration (Conley et al. 2011; Sullivan et al. 2011), although in better agreement with the Union2.1 compilation (Suzuki et al. 2012). Our combined determination of σ_{8} and Ω_{m} shows larger tension with recent data based on cosmic shear or cluster count techniques. On the other hand the Planck bestfit ΛCDM model is in good agreement with the halo power spectrum derived from the luminous red galaxy catalogue of the Sloan Digital Sky Survey (Reid et al. 2010), especially when the analysis is restricted to linear scales.
9.3. Significance of the lowℓ tension with ΛCDM models
From the above discussion, it is clear that the ΛCDM framework provides an excellent model for most of the Planck data. However, as noted in Sect. 8.3 and seen in Fig. 37, the lowℓPlanck temperature power spectrum appears to be in some tension with the bestfit PlanckΛCDM model, which for Planck is almost exclusively determined by the smallscale spectrum. In this section we assess the significance and impact of this tension between low and high ℓs using three different statistical tests.
We start by applying a modified Hausman test (Polenta et al. 2005; Planck Collaboration II 2014) to the lowℓ spectra derived from the four foregroundcleaned Planck maps (Planck Collaboration XII 2014) and the 9year WMAP ILC map, using multipoles up to ℓ_{max} = 32. This test uses the statistic s_{1} = sup_{r}B(ℓ_{max},r), where
and Ĉ_{ℓ} and C_{ℓ} denote the observed and model power spectra, respectively. Intuitively, this statistic measures the relative bias between the observed spectrum and model, measured in units of standard deviations, while taking the socalled “lookelsewhere effect” by maximizing s_{1} over multipole ranges into account. We use realistic Planck “FFP6” simulations (Planck Collaboration I 2014) to derive the empirical distribution of s_{1} under the null hypothesis. Figure 38 compares the results obtained from the data with the simulation distribution, and Table 9 lists significances. As measured by this statistic, we see that a negative bias is found in the lowℓPlanck power spectrum relative to the ΛCDM model at the 99% confidence level.
For the WMAP ILC map the significance of the negative bias nominally decreases to 93%. This is consistent with the findings in Sect. 8.3, where it was shown that the WMAP temperature power spectrum is 2.5–3 % higher than the Planck spectrum at low ℓ’s. However, as discussed in Planck Collaboration XXXI (2014), a similar amplitude difference between the two experiments is also seen at smaller scales. Since the current test compares the observed WMAP data with the bestfit PlanckΛCDM model, the present test is not optimal for assessing internal consistency between low and high ℓs within the WMAP data.
Fig. 38
Results of the Hausman test applied to the temperature power spectrum for 2 ≤ ℓ ≤ 32. The black histogram shows the expected distribution, estimated with simulations, of the s_{1} test statistic. The vertical bars represent Planck CMB maps and the 9 year WMAP ILC map. Note that the statistic is indistinguishable for the NILC and Commander maps. 

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Results of the Hausman test applied to the temperature power spectrum for 2 ≤ ℓ ≤ 32.
Next, to obtain a quantitative measure of the relative power discrepancy between low and high ℓs, we fit the twoparameter amplitude–tilt power spectrum model (see Sect. 8.1.2) to the Planck data using the lowℓ likelihood restricted to various multipole ranges defined by 2 ≤ ℓ ≤ ℓ_{max}, where ℓ_{max} is allowed to vary. Thus, this measures the amplitude of the lowℓ spectrum relative to the bestfit PlanckΛCDM spectrum, which is driven by the smaller angular scales. Figure 39 shows the resulting constraints on the power spectrum amplitude, q, as a function of ℓ_{max}, after marginalising over the tilt, n. For comparison, we also show similar constraints derived using the lowℓ WMAP temperature likelihood up to ℓ = 30. The bestfit amplitude is q ~ 0.9 for ℓ_{max} = 20–35, different from unity at a statistical significance of 2–2.5σ by this measure. The WMAP spectrum shows a consistent behaviour, up to the same overall scaling factor of 2.5–3% between Planck and WMAP discussed above. We have verified that these results are insensitive to the (wellknown) low quadrupole moment by excluding C_{2} from the analysis; the large cosmic variance of this particular mode results in a low overall statistical weight in the fit.
Fig. 39
Power spectrum amplitude, q, relative to the bestfit Planck model as a function of ℓ_{max}, as measured by the lowℓPlanck and WMAP temperature likelihoods, respectively. Error bars indicate 68 and 95% confidence regions. 

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Finally, we assess the impact of the lowℓ power deficit on the ΛCDM model estimated using the Planck likelihood^{10} (augmented with the WMAP polarisation likelihood). We fit a lowℓ rescaling amplitude, A_{low} for ℓ <ℓ_{low} jointly with the ΛCDM parameters, i.e., for ℓ <ℓ_{low} and for ℓ ≥ ℓ_{low}. Figure 40 shows the resulting posterior distributions for A_{low} for ℓ_{low} = 32 (green) and ℓ_{low} = 49 (blue). The purple line shows the same when replacing the Planck lowℓ likelihood with the WMAP lowℓ likelihood (ℓ_{low} = 32). The corresponding bestfit values are A_{low} = 0.899 ± 0.046 (Planck; ℓ_{low} = 32), A_{low} = 0.953 ± 0.033 (Planck; ℓ_{low} = 49) and A_{low} = 0.953 ± 0.048 (WMAP; ℓ_{low} = 32), respectively. As already noted in Sect. 8.3, the observed deviations from unity are too large to be explained by the <1% uncertainties in the Planck transfer functions (Planck Collaboration II 2014; Planck Collaboration VI 2014).
Fig. 40
Posterior distributions for the lowℓ spectrum amplitude, A_{low}, estimated using the Planck likelihood, with ℓ_{low} = 32 (green) and ℓ_{low} = 49 (blue). The purple line show the distribution derived using the WMAP temperature likelihood with ℓ_{low} = 32. 

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In Fig. 41 we show the posterior distributions for Ω_{c}h^{2}, n_{s} and H_{0} after marginalising over A_{low} for ℓ_{low} = 49. (Adopting ℓ_{low} = 32 results in negligible differences for all parameters except A_{low}). Shifts of 0.6–1σ are observed compared to the reference model, A_{low} = 1. We note that H_{0}, which already has a “low” value (for a detailed discussion, see Planck Collaboration XVI 2014), prefers an even lower value when allowing a rescaling of the lowℓ spectrum. As a final test, we replace the entire lowℓ likelihood, both temperature and polarisation, with a Gaussian prior on the optical depth of reionisation, τ = 0.089 ± 0.014, matching the WMAP measurement (Hinshaw et al. 2013). The resulting posteriors are shown as purple lines in Fig. 41, and agree well with the case including a lowℓ scaling factor, but are, in fact, slightly further away from the reference model. Although not very significant in an absolute sense, these results do indicate that the highℓ likelihood is challenged in finding models that also fit the lowℓ power spectrum.
Fig. 41
Comparison of the posterior distributions for Ω_{c}h^{2}, n_{s}, and H_{0} for the default sixparameter ΛCDM model constrained by Planck (red); compared to the case when we allow a variable lowℓ power spectrum amplitude at ℓ ≤ 49 (blue); and when replacing the lowℓ temperature likelihood with a Gaussian prior on τ, the optical depth of reionisation (purple). The lower right panel shows the posterior distribution for the lowℓ amplitude, A_{low}. 

Open with DEXTER 
To summarize, we have phenomenologically quantified a tension between the lowℓ CMB power spectrum at ℓ ≲ 40 and the Planck bestfit ΛCDM model. Its significance varies between 2.5 and 3σ depending on the estimator used. The effect is seen in all four Planck foregroundcleaned CMB maps with little variation. It is also present in the 9year WMAP data, although an overall amplitude difference of 2.5–3% between the data sets complicate a direct comparison. To make further progress, one would seek to establish a physical model that predicts a lowℓ power deficit compared to high ℓ’s, and that may also predict other observable effects, which may be tested with cosmological data. Such a model may be related to the tentative detections of violations of statistical isotropy discussed in Planck Collaboration XXIII (2014), e.g., the low CMB temperature variance, the hemispherical power asymmetry, or the alignment between the quadrupole and octopole moments.
10. Discussion and conclusions
We have presented the Planck likelihood, which provides a detailed and accurate characterisation of the twopoint statistics of the CMB temperature field, accounting for all significant sources of uncertainty; statistical, instrumental, and astrophysical. This likelihood function allows us to present an estimate of the CMB temperature power spectrum that spans more than three decades in ℓ with unprecedented precision; a spectrum that saturates the cosmic variance limit at all scales ≳0.1°, nearly exhausting the information content of the temperature anisotropies, and, in fact, becoming limited by uncertainties due to astrophysical foreground modelling. This is precisely what was originally promised at the time when Planck was selected by ESA in March 1996.
On large angular scales, ℓ < 50, the Planck likelihood is based on a Gibbs sampling approach that allows joint CMB power spectrum and component separation analysis, while accurately marginalising over a physically motivated foreground model constrained by the 30–353 GHz Planck frequencies. On intermediate and small scales, the Planck likelihood employs a finegrained set of crossspectrum combinations among the 100, 143, and 217 GHz detector maps to constrain the highℓ CMB power spectrum, ensuring that no noise bias can compromise the results, while at the same time allowing for physical foreground modelling in terms of power spectrum templates. This emphasis on physical foreground modelling has made it possible to combine the full power of the Planck data with observations from higherℓ CMB experiments.
We have validated our results through an extensive suite of consistency and robustness analyses, propagating both instrumental and astrophysical uncertainties to final parameter estimates. Further, we have studied in detail the wellknown degeneracies that exist between the foreground and cosmological parameters at high ℓs when only including Planck observations, and shown that they have only a weak impact on cosmological conclusions.
On a more detailed level, we draw the following conclusions:

The consistency between power spectra measured within thedifferent frequency channels is remarkable. In thesignaldominated regime for single detectors, at ℓ ≲ 1000, thecrossspectra show an RMS dispersion of a few μK^{2} in multipolebands of δℓ = 31. This confirms the relative calibration of the 100, 143,and 217 GHz detectors to ~0.2%.

The differences, , between the 143 × 143, 143 × 217, and 217 × 217 crossspectra averaged over multipole bands of δℓ ≈ 31 have a dispersion over 800 ≤ ℓ ≤ 1500 of 9, 5, and 5 μK^{2}, respectively, after subtracting the bestfit foreground model. This dispersion is not primarily of instrumental origin, but can be predicted from a model of the chance correlations between foregrounds and CMB fluctuations.

At high ℓs, the power spectrum of the four foregroundcleaned CMB maps derived through component separation are consistent within their uncertainties. The cosmological parameters derived from these maps are consistent with those estimated by the Planck likelihood for ℓ ≲ 2000, despite very different foreground models.

At low ℓs, the power spectrum differences among the four foregroundcleaned CMB maps are below 50 μK^{2} for nearly every single multipole. Residuals with respect to the 9 year WMAP ILC map are slightly larger, typically 100 μK^{2} or more. A detailed comparison between Planck and WMAP reveals a systematic power spectrum amplitude difference at the 2–3% level that cannot be accounted for within the Planck instrumental error budget. This is consistent with the findings presented in Planck Collaboration XXXI (2014).

Parameters derived from the 70 GHz Planck frequency map are in excellent agreement with the reference results derived using the Planck likelihood; when the latter is limited to ℓ ≤ 1000, the agreement is even more striking. This confirms the strong internal consistency between the LFI and HFI instruments.

The bestfit ΛCDM model derived from the Planck likelihood predicts TE and EE spectra in good agreement with the measured polarisation signature over a broad range of frequencies (70 to 217 GHz) and multipoles (ℓ ≲ 1000). At 100, 143, and 217 GHz, the instrumental noise in the EE spectrum is at the μK^{2} level for ℓ ≲ 1000, and the visible (small) differences between the spectra suggests different levels of foreground contribution.

We report a tension between the Planck bestfit ΛCDM model and the lowℓ spectrum in the form of a power deficit of 5–10% at ℓ ≲ 40, with a statistical significance of 2.5–3σ. Thus, while the minimal ΛCDM model provides an outstanding fit for intermediate and small angular scales, this tension may suggest that the model is incomplete. In this respect, it is worth noting that other, but possibly related, anomalies have been reported in a companion paper studying statistical isotropy in the Planck sky maps at statistically significant levels.
In summary, we find that the majority of the Planck data can be described by a minimal sixparameter ΛCDM model with a very high degree of accuracy. Within this model the statistical uncertainties are dominated by astrophysical foreground modelling by scales of ℓ ≃ 1500. At lower ℓs, the unprecedented quality of the Planck data is such that the only fundamental limit is that we can only observe one CMB sky. In other words, Planck is cosmic variance dominated at ℓ ≲ 1500, extragalactic foreground dominated at ℓ ≳ 1500, and dominated nowhere by instrumental noise or systematic errors.
Using only Planck temperature data, we report a detection of n_{s}< 1 at more than 4σ confidence, which by itself is already a bit stronger than the limit derived from WMAP, SPT, ACT, and SNLS combined. Complementing the Planck observations with the 9 year WMAP polarisation data increases the significance further to 5.4σ. The multipole range above ℓ > 1500 is crucial for constraining possible extensions to the minimal ΛCDM model; for a detailed exploration of a wide range of such models, see Planck Collaboration XVI (2014) and Planck Collaboration XXII (2014). There we report some tensions among the CMB damping tail parameters, including Ω_{K}, n_{run}, and Y_{P}. However, none of these indicate significant departures from the ΛCDM framework.
In the near future, we will extend our analysis to produce a cosmic variance limited likelihood and power spectrum reaching to higher multipoles. To some extent, this will be achieved through more sophisticated astrophysical foreground modelling, and by exploiting additional frequency information. However, the two major steps forward will be, first, to include the Planck polarisation observations in the likelihood analysis, and, second, to exploit the full data set generated by the two Planck instruments. The amount of HFI data available for analysis is nearly double that, which is presented here, and the LFI instrument is still observing at the time of writing.
Planck (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 (in particular the lead countries France and Italy), with contributions from NASA (USA) and telescope reflectors provided by a collaboration between ESA and a scientific consortium led and funded by Denmark.
The scaling coefficient for the 143 GHz spectrum is (3.14 × 10^{5})^{2}, derived from the 7parameter fitting function of Eq. (A.46).
One must also account for the fact that these numbers correspond to the amplitude for a suitably averaged spectral band, which is approximately that of the map, and is described in detail in Planck Collaboration IX (2014).
We use the approximation θ_{MC} to the acoustic scale θ_{⋆} (the ratio of the comoving size of the horizon at the time of recombination, r_{S}, to the angular diameter distance at which we observe the fluctuations, D_{A}), which was introduced by Hu & Sugiyama (1996). θ_{MC} is commonly used, e.g., in CosmoMC, to speed up calculations; see also Kosowsky et al. (2002) for further details.
These effective beam transfer function represents the combined effect of the instrument and the processing chain. They are derived from the scanning beam, derived from the data obtained while scanning planets, which therefore describe the combined effect of the instrument and of the temporal flow processing (sometimes referred to the F(ℓ) term as in the original “master” paper by Hivon et al. 2002). We then derive the effective beam, which further accounts for the effect of mapmaking, which combines samples with scanning beams of several orientation in each pixel, according to the scanning strategy. This is described in detail in the beam and processing papers (Planck Collaboration IV 2014; Planck Collaboration VI 2014; Planck Collaboration VII 2014).
As described in the HFI Data Processing paper Planck Collaboration VI (2014), these difference maps provide an estimate of the noise level in the sum maps with an accuracy of about 1%.
Actually, as demonstrated in Fig. 5, the CIB dominates over Galactic dust emission over most of the area of mask0, but this is not a precise template for the CIB emission at cosmological channels: (a) because the spectrum of the CIB differs slightly from Galactic dust; (b) the CIB emission decorrelates from high to low frequencies because lower frequencies probe galaxies at higher redshifts.
Acknowledgments
The development of Planck has been supported by: ESA; CNES and CNRS/INSUIN2P3INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, JA and RES (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A description of the Planck Collaboration and a list of its members with the technical or scientific activities they have been involved into, can be found at http://www.sciops.esa.int/index.php?project=planck&page=Planck_Collaboration. We acknowledge the use of the CLASS Boltzmann code (Lesgourgues 2011) and the Monte Python package (Audren et al. 2013) in earlier stages of this work. The likelihood code and some of the validation work was built on the library pmclib from the CosmoPMC package (Kilbinger et al. 2011). This research used resources of the IN2P3 Computer Center (http://cc.in2p3.fr) as well as of the PlanckHFI data processing centre infrastructures hosted at the Institut d’Astrophysique de Paris (France) and financially supported by CNES.
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Appendix A: Highℓ likelihood details
Appendix A.1: Power spectra and the coupling matrix
We denote the pixel weight function for temperature by . The pseudospectra of Eq. (1) are constructed using the following: (A.1)where the sum is over the pixels in the map. For all of the methods presented in the paper the weighting per pixel is uniform, except for the apodisation of the different Galactic and point source masks referred to in the main text.
The coupling matrix appearing in Eq. (2) is given by Hivon et al. (2002): (A.2)where for the cross spectrum (i,j), is the power spectrum of the window function (A.3)
Appendix A.2: PseudoC_{ℓ} covariance matrices
For the case of narrow window functions and uncorrelated pixel noise , the covariance matrices can be approximated as (A.4)where Ξ is the matrix defined in Eq. (A.2). The window functions are given by:
where To avoid cumbersome notation, we have omitted indices from the theoretical spectra appearing in Eq. (A.4). In practice, these spectra include unresolved foreground contributions and are smoothed by the appropriate “beam” transfer functions b^{ij}, which actually describe the overall transfer function from the sky to the maps^{11}. In addition, these covariance matrices are corrected for the pixel window functions p_{ℓ} (i.e., covariance matrices such as are divided by ).
Finally, the Planck maps have correlated pixel noise. We adopt an approximate, heuristic modification of the formulae above to include this effect in the covariance matrices, based on the power spectra of halfring difference maps. This procedure is detailed in Appendix A.8.
Appendix A.3: Combining intrafrequency crossspectra
For Planck, the vector containing the power spectra, and its associated covariance matrix, are both large and so require substantial compression to make the computation of a highℓ likelihood fast enough for parameter estimation. As described in Appendix A.7, after we correct for the “effective” calibration factors for each individual detector set, the power spectra at each frequency are consistent to extremely high accuracy. Any remaining residuals have a negligible impact on the cosmological analysis. Thus, we combine the crossspectra from different detectors within a given frequency combination into a single power spectrum. We do not average across frequency combinations since the unresolved foregrounds depend on frequency. Further compression can be accomplished, if desired, only after unresolved foreground parameters have been determined.
We form the linear combination of individual crossspectra, for each multipole, in the following way: (A.10)Here the index k denotes the particular frequency crossspectrum combination (e.g., 100 × 100, 143 × 217), the coefficients y_{i} denote the multiplicative factors for each map, is the (isotropised) beam transfer function for the map combination ij, and p_{ℓ} is the isotropised pixel window function ^{12}. The coefficients α_{ij} are normalized so that (A.11)How can we determine the coefficients α_{ij}? A near optimal combination, , is given by solving (A.12)where is the block of the inverse covariance matrix appropriate to the spectrum combination k. If the covariance matrix accurately describes the data, the solution of Eq. (A.12) properly accounts for the correlations between the crossspectra. Solving Eq. (A.12) requires the inversion of a large matrix, so we adopt a simpler solution by weighting each estimate by the diagonal component of the relevant covariance matrix, e.g., (A.13)This has the effect of assigning each crossspectrum equal weight in the signal dominated regime and an inverse variance weighting in the noise dominated regime. This is the correct solution in the noise dominated regime. The analysis of intrafrequency residuals presented in Appendix A.7 shows that in the signal dominated regime we see excess variance (with no obvious dependence on the detector/detector set combination) compared to what we expect from instrument noise alone. This excess variance is small compared to the signal and is caused by residual beam errors, consistent with the beam eigenmode amplitudes discussed in Appendix A.7, that are not included in the covariance matrices. This is our justification for assigning roughly equal weight to the spectra in the signal dominated regime.
When we construct a likelihood from the combined estimates we construct the full covariance matrix including crosscorrelations between the various spectra. In this matrix, the crosscorrelations in the signal dominated regime are dominated by cosmic variance if different masks are used for different frequencies. If identical masks are used for all frequencies, the crosscorrelations in the signaldominated regime are dominated by the cross correlations between the CMB and unresolved foregrounds, which are included in the analytic covariance matrices and act as a regularizing contribution (see Appendix C).
Appendix A.4: Covariance matrix of combined spectra
Fig. A.1
Toy power spectra drawn from a Gaussian distribution with mean given by a ΛCDM power spectrum and a covariance given by the bandaveraged C_{ℓ} covariance matrix, (top), and the corresponding differences with respect to the input ΛCDM model. Note the apparent presence of “coherent oscillatory features” in the difference spectra. These are fully described by the power spectrum covariance matrix. To assess the statistical significance of apparently “unexpected features” in the power spectrum, it is critical to include all sources of correlations among different C_{ℓ}’s. 

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The estimates of the Planck crossspectra are linear combinations of the pseudoC_{ℓ} estimates, so their covariance matrices are given by e.g., (A.14)Analytic expressions for these covariance matrices have been given in Efstathiou (2004, 2006); Hamimeche & Lewis (2008), and are described in Appendix A.2. The covariance matrices are computed assuming a fixed fiducial theoretical model including an unresolved foreground model for each frequency combination. Typically, the unresolved foregrounds introduce corrections to the covariance matrices of a few percent in the transition region between signal and noise domination. In addition, we compute the fiducial model by applying appropriate beam functions b^{Tij} for each detector combination.
As discussed above, the number of coupling matrices required to compute expressions such as Eq. (A.14) scales as and so becomes prohibitively expensive as the number of crossspectra becomes large. However, most of these coupling matrices are similar, differing primarily in the amplitude of the noise levels and in minor respects such as a small number of missing pixels. We can therefore adopt the same masks and weightings for groups of crossspectra and compute coupling matrices only for distinct combinations. This dramatically reduces the computational burden. A similar approach was adopted by Lewis (2008) to analyse the WMAP 5year temperature maps.
It is also straightforward to calculate covariance matrices for differences between different averages. If we form two spectra averaging over different detector combinations then the covariance matrix of the difference is simply (A.17)As a pedagogical illustration of the importance of these correlations, we show in Fig. A.1 two toy power spectra drawn from a Gaussian distribution with mean given by a ΛCDM spectrum and covariance given by . That is, these spectra are not computed from a real sky map, but simply drawn directly from the C_{ℓ} error model, and therefore exclusively contain correlations modelled by the covariance matrix. The apparent “coherent oscillatory features” seen in the difference spectra (bottom panels) are therefore fully described by the CamSpec covariance matrix, accounting for correlated beam and foreground uncertainties, maskinduced coupling etc. The left panel shows a typical realisation, while the case in the right panel is selected as, visually speaking, one of the most “peculiar” within a relatively small set of simulatoins. When assessing the statistical significance of “unexpected features” in the real CMB spectrum, e.g., similar to those seen in Figs. 15 and 16, it is critical to account for these correlations.
Appendix A.5: The “fiducial Gaussian” approximation
We use a likelihood based on the socalled “fiducial Gaussian” approximation. Here we present our justification for this choice, based on an “expansion in covariance” of the exact likelihood in the exact fullsky, isotropic noise case.
Assuming the CMB, noise, and foregrounds are Gaussian, then the probability distribution for the a_{lm} coefficients of a collection of maps, given a model, is p = e^{− S}, where (A.18)up to a modelindependent normalization. Here is the matrix of empirical spectra at a given multipole, and Cℓ are their expectation values for the model in question.
Now, a key point to note is that theoretical power spectra typically differ from each other at each ℓ by less than they differ from the observed , because of cosmic variance and noise in the latter. So we are justified in expanding Eq. (A.18) about a reasonable fiducial model. Considering a single value of ℓ for simplicity, writing (A.19)we obtain (A.20)to second order in Δ. We may now complete the square in Δ after extracting a term that is small if the fiducial model is accurate. Up to terms independent of Δ, we have, to second order in Δ: (A.21)Here we have recombined the perturbation and the fiducial model back together, using Eq. (A.19), in the second term to obtain exactly the “fiducial Gaussian” likelihood. The first term is a correction to the fiducial Gaussian likelihood that is typically small if the fiducial model is accurate.
One can motivate neglecting this term by noticing that in its absence the approximate likelihood is unbiased (as the exact one is). One trades getting second derivatives exactly right in the vicinity of the fiducial model with getting the position, though not the depth, of the minimum right.
Vectorizing the distinct elements of C − Ĉ (following Appendix A of Hamimeche & Lewis 2009), and recognizing the coefficients as the inverse covariance matrix elements of the spectra under the fiducial model, we obtain (A.22)where is the fiducial covariance matrix of the spectra, and the upper indices run on the different pairs of frequencies.
This suggests an easy generalisation to the coupled cut sky pseudospectra, given our calculation of their covariances in Appendix A.2. We now replace the power spectra above with corresponding appropriate averages of detector crossspectra. With bold face now denoting spectra laid out as vectors, and the grand fiducial covariance, our final action is: (A.23)Another advantage of the “fiducial Gaussian” approximation is that instrumental uncertainties (calibration errors, beam errors, etc.) do not appear in the inverse covariance, but only in the expression of the theoretical spectra C^{T} in Eq. (A.23) above.
Note that if we fix the foreground model C^{Fk} for each spectrum k, together with the calibration coefficients and beam parameters, we can minimise the likelihood (Eq. (A.23)) with respect to a “bestfit” primary CMB spectrum. This “bestfit” spectrum is given by the solution of (A.24)where the c^{k} are spectrum effective calibration factors (see Appendix A.7). The covariance matrix of the estimates is given by the inverse of the Fisher matrix:(A.25)
Appendix A.6: Uncertainties on individual detector sets beams and calibrations
Let us consider two detectors (or detector sets) X and Y. Neglecting instrumental noise, the crossspectrum is related to the true one, , through (A.26)where is the effective beam window function. Note that because of the optical beam noncircularity and the Planck scanning strategy, when X ≠ Y, while W^{XY} = W^{YX} for any X and Y. In the ℓ range of interest, , so we denote , following the usual prescription for simple (circular) beam models. In what follows, we drop the XY pair superscript except when they are required for clarity.
Our analyses use the best estimated of the sky power spectrum, where the measured is corrected by a nominal effective window : (A.27)The ratio , which determines the uncertainty on the angular power spectrum due to the beam, is estimated using MonteCarlo simulations of planet transits.
We estimate and from the Monte Carlo simulations as
and compute the deviations around the mean (A.30)Since the relative dispersion of the simulated is small (less than 1%), the deviations are well approximated by (A.31)The matrix Δ then has n_{MC} rows and ℓ_{max} + 1 columns. Its Singular Value Decomposition (SVD) is given by (A.32)where M is an orthogonal n_{MC} × n_{MC} matrix (i.e., M^{T}M = MM^{T} = I_{nMC}), D is a diagonal matrix with n_{MC} nonnegative eigenvalues, and V is a matrix with ℓ_{max} + 1 rows whose n_{MC} columns are orthonormal vectors (i.e., V^{T}V = I_{nMC}). Here I_{nMC} is the identity matrix.
The covariance matrix of the beam deviations is defined as (A.33)from which we compute the eigenmode matrix (A.34)using the SVD of Δ. Most of the statistical content of Δ or C is limited to the first few modes n_{modes} with the largest eigenvalues. We therefore keep only the largest n_{modes} = 5 of the E matrix. The beam uncertainty for a given spectrum is then given by (A.35)where g is a vector of independant Gaussian variates of unit variance with n_{modes} elements, and E_{k}(ℓ) is the kth row of E.
This can be generalised to a set of spectra. Taking three pairs of detector sets a = { UV }, b = { XY } and c = { ZT }, one can write (A.36)and the covariance matrix is given by (A.37)The beam errors can therefore be correlated (and in fact are strongly so, see Planck Collaboration VII 2014). In the next Appendix, this general covariance matrix is used to derive the beam error eigenmodes of combined spectra for the CamSpec likelihood.
Appendix A.7: Calibration and beam uncertainties for the CamSpec likelihood
Four effective crossspectra are used in the CamSpec likelihood, each using an individuallyprescribed ℓrange. For each of the effective power spectra, all eligible mask and beamdeconvolved crossspectra are used, weighted according to (A.38)with p labelling the effective spectrum. As described in Appendix A.6, uncertainties in the determination of the HFI effective beams are described in terms of beam eigenmodes, E_{k}, and distributions of the coresponding eigenvalues. To propagate beam errors into the likelihood, we start by using the eigenvalues, along with the weights, to construct an appropriate covariance matrix for the effects of beam errors in : (A.39)where is the correlation between the ith eigenmode of the XY crossspectrum with the jth eigenmode of the ZW crossspectrum. The portion of this matrix corresponding to the ℓrange used in the likelihood is then extracted, and itself singularvaluedecomposed. We keep the first n_{effmodes} (typically five) eigenmodes , i = 1,...,n_{effmodes}, orthogonal over the ℓrange and normalized such that the sum of their outer product directly approximates the covariance: (A.40)The eigenmodes are illustrated in Fig. A.2.
Fig. A.2
Eigenmodes of the beam covariance matrix, shown for each crossspectrum used in the analysis. The largest five modes are shown for each crossspectrum. 

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Next we calculate a suitable covariance matrix between the eigenmodes. This requires the (nondiagonal in ℓ) intereffectivespectrum covariance matrices , given by: (A.41)Given these matrices, we can “stack” the effective spectra to form a data vector X and form a grand beamcovariance matrix ⟨ ⟨ XX^{T} ⟩ ⟩ _{beam}. X has length . Zeroextending each eigenmode at both ends, and arranging these into a matrix, we can form an n_{X} by n_{effmodes}·n_{eff} matrix of extended eigenmodes, E_{eff}, where n_{eff} = 4. Now we imagine approximating ⟨ ⟨ XX^{T} ⟩ ⟩ _{beam} as a correlated outer product of E_{eff}, (A.42)Requiring that the covariance be chosen to minimise the summed squareddifference between elements on the two sides yields: (A.43)The ’s and the associated covariance M_{eff} are then passed to the likelihood.
Noise estimates for the detector maps, applying the Galactic masks used in the CamSpec likelihood (“mask_3” for 100 GHz, retaining 58% of the sky, and “mask_1” for 143 and 217 GHz, retaining 37% of the sky, combined with an extragalactic point source mask).
Appendix A.8: Noise model of HFI detector sets
For strictly uncorrelated pixel noise , and pixel weights w_{i}, the noise power spectra for the temperature maps are: (A.44)with contribution to the pseudoC_{ℓ} estimates, for uncorrelated noise, of (A.45)Values for are listed in Table A.1. There is a significant dispersion in the noise properties of the two maps at 100 GHz. At 143 and 217 GHz, the PSB maps have significantly lower noise than the SWB maps by a factor of two, as expected.
Fig. A.3
Noise spectra computed from difference maps for a selection of detector sets: 100GHz (top), 143GHz (middle), and 217GHz(bottom). Green lines show the spectra for T, purple for Q and magenta points for U. The solid black lines show the modeled fits to the spectra (Eq. (A.46)), and the coloured horizonal lines show the whitenoise levels of Eq. (A.44). These are computed using the same masks as used in Table A.1. 

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The noise spectra for the Planck HFI maps are nonwhite. The following 7parameter function (A.46)is a flexible parameterization that provides accurate fits for all of the HFI channels. The first term on the left models the excess “1/f”like noise while the second term models the “bell shaped” noise spectrum at high multipoles introduced by time constant deconvolution applied to the timeordered data, and the lowpass filter designed to remove highfrequency noise due to demodulation. Estimates of the noise spectra can be computed from difference maps constructed from different halfring surveys^{13}. Examples of fits to noise spectra for the 143GHz and 217GHz channels are shown in Fig. A.3. Note that the 100GHz noise spectra are significantly nonwhite. At 143 and 217GHz, the deviations from white noise are smaller. Since these cross spectra contribute almost all of the weight in the likelihood at high multipoles, the modelling of nonwhite noise is not a critical factor in forming an accurate likelihood.
We adopt a heuristic approach to fold departures from white noise into the power spectrum covariance estimates. We define a set of noise weight functions, e.g., (A.47)Wherever a σ^{2} term appears in a covariance matrix, we multiply the appropriate coupling matrix by a factor (A.48)This heuristic approach can be partially justified by noting that for isotropic Gaussian noise over the full sky, the distribution of Ĉ_{ℓ} is given by the inverse Wishart distribution: (A.49)where (A.50)e.g., Percival & Brown (2006). In this special case, our heuristic correction is exact. Further justification of the accuracy of this heuristic approach comes from direct comparisons with numerical simulations incorporating nonwhite noise (see Sect. 6) and from the accurate agreement of covariance matrices with the ℓbyℓ scatter measured in all of the crossspectra used to form the likelihood.
Appendix B: Sky masks
Fig. B.1
Set of unapodised foreground masks, G22, G35, G45, G56, G65, PS96, which, once apodised, are used for the likelihood analyses. The Galactic (G22 to G65) masks are defined using a threshold of the 353 GHz Planck temperature map. The (lowerright) PS96 point source mask is based on the union of resolved sources in all channels from 100 GHz to 353 GHz. 

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Here we show the various masks, which are mentioned in this paper and summarized in Table 2. Figure B.1 shows the set of diffuse Galactic masks, which we obtained by applying a threshold to an ILCsubtracted 353 GHz temperature map. Figure B.2 shows the apodised masks, which we combined to form the set of masks used inthe likelihood analysis.
Fig. B.2
Apodised Galactic and point source masks, which we derived (see text) from their nonapodised version shown in Fig. B.1. From left to right and top to bottom, the panels show the GA34, GA38, GA54 and PSA82 masks, which are the ones we combine to use in the likelihood analysis. 

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Appendix C: Chance correlations and interfrequency consistency tests
Here we explicitly show that, even if the foreground contamination is much smaller than the CMB, chance crosscorrelations can produce scatter in the interfrequency power spectra that is large in the signal dominated regime. To see this, consider the case of two frequencies. Frequency 1 provides a faithful map of the CMB fluctuations. Frequency 2 contains a foreground component F. We therefore write the maps at the two frequencies as: with spherical transforms The power spectra of the two maps are therefore: and the difference between the power spectra is (C.7)If the CMB is uncorrelated with the foreground, the first term will average to zero over a large number of CMB realisations. But we observe only one realisation of the CMB, and so the crossterm will dominate the interfrequency residuals even if the foreground contamination is much lower than the CMB (C^{F} ≪ C^{CMB}). This is the origin of the excess scatter between the 143 and 217 GHz power spectra at low multipoles shown in Fig. 10.
Fig. C.1
Top: An ILC CMB map constructed from 100353 GHz maps. Middle: The ILC map added to the 857 GHz map, scaled to match the diffuse dust emission at 217 GHz. This map (and an equivalent at 143 GHz) is used as the CMB+dust template to assess CMB/foreground cross correlations. Bottom: The real 217 GHz map. 

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We construct a specific example of this. The upper map in Fig. C.1 shows an ILC map, estimating the CMB, generated from the 100, 143, 217, and 353 GHz maps. The map in the middle panel shows a “fake” 217 GHz map, i.e. the sum of the ILC map and the 857 GHz map scaled in amplitude to match dust emission at 217 GHz. The real 217 GHz map is shown in the lower panel of Fig. C.1. The “fake” 217 GHz map is evidently quite a good match to the real 217 GHz. By rescaling the 857 GHz map to estimate the dust emission at 143 GHz, we can generate a “fake” 143 GHz map in an analogous way. These “fake” maps each contain two components by construction, and so the interfrequency residuals from these maps will be dominated by the CMBforeground cross term in Eq. (C.7).
The 143 and 217 GHz power spectrum difference from these fake maps are compared to the 143 − 217 residuals of the real data in Fig. C.2. The magenta points show the same mask1mask0 doubledifference power spectrum between 217 and 143 GHz as shown in Fig. 3 (with mask0 ≡ G22∪PSA82 and mask1 ≡ G35∪PSA82). The only difference here is that the smoothed dust fit of Eq. (9) has been subtracted from the spectra so that the points scatter around zero. There are advantages to using the double difference because: (a) it is insensitive to calibration differences between frequencies; (b) the contrast between dust emission and other foregrounds (point sources/SZ) is stronger in the area of sky defined by mask1 – mask0 and so the double difference power spectrum should be closer to the results from the fake maps, which use only a dust template ^{14}. The solid green line shows the double difference power spectrum computed from the fake maps. The amplitude of the scatter from the fake maps and the real data are very similar. In fact, there is almost pointbypoint agreement between the results from the real data and the fake maps. This provides compelling evidence that the observed interfrequency scatter at low multipoles is dominated by the CMBforeground cross term in Eq. (C.7) rather than some mysterious systematic effect in the data.
The blue points in Fig. C.2 show the difference of the 217 and 143 GHz power spectra for mask1. The scatter at multipoles ≲100 is almost identical to the scatter of the purple points, but increases slightly at higher multipoles. This behaviour is expected and is caused by the additional foreground components (CIB/point sources/SZ), which become comparable in amplitude to Galactic dust at multipoles greater than a few hundred.
Scatter in doubledifferenced spectra.
Fig. C.2
Fake versus real data difference spectra. The magenta points show the double difference power spectrum of actual data (as in Fig. 3). The green line shows the same double difference spectrum computed from the ILC + 857 dust template maps described in the text. This model provides a good match to the magenta points both in amplitude and phasing. The blue points show the difference of the 217 and 143 power spectra for mask1, which adds CMB cosmic variance, illustrating why we use doubledifferenced spectra for this comparison. In all cases, the smoothed dust power spectrum model of Eq. (9) (with slightly adjusted amplitude to bring the residuals at these low multipoles close to zero) has been subtracted. 

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Fig. C.3
Power spectrum residuals between the summed crossspectra at three HFI frequencies. The top panel shows 100 × 100–143 × 143, the middle panel shows 100 × 100–217 × 217 and the bottom panel shows 143 × 143–217 × 217. A “best fit” model for unresolved foregrounds has been subtracted from the power spectrum at each frequency. The scatter in the multipole range 50 ≤ ℓ ≤ 500 is listed in each panel. 

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We would also expect a strong dependence of the interfrequency residuals with frequency at low multipoles. Diffuse Galactic emission rises steadily in amplitude from 100 GHz to 217 GHz and hence we would expect the interfrequency scatter to rise as we go up in frequency. This is what we see in the real data (shown in Fig. C.3). Since diffuse Galactic emission is well approximated by the 857 GHz map at all frequencies, we can predict the scatter seen in this figure by scaling 857 GHz to lower frequencies. For 143 − 217 GHz we observe a scatter of 16(μK)^{2} over the multipole range 50 ≤ ℓ ≤ 500. So, from the 857 GHz scalings to lower frequencies, we predict the scatter given in the Table C.1, which is in excellent agreement with the scatter seen in Fig. C.3.
Appendix D: Validity tests
Appendix D.1: Detailed validity checks
In this appendix we show (Figs. D.1–D.6) the distribution of the cosmological and foreground model parameters for the suite of tests described in Sect. 7. We also show in Figs. D.7 and D.8 the correlation matrix between these parameters and the calibration coefficients of each detector.
Fig. D.1
Impact on cosmological and foreground parameters of fixing the calibration and beam coefficients at their maximum posterior value (red), compared to marginalising over these nuisance parameters (blue). 

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Fig. D.2
Impact of removing the subpixel effect (blue) or the correlated noise between detector sets (purple), compared to the Plik reference case (red). 

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Fig. D.3
Impact on the cosmological and foreground parameters of varying the maximum multipole, ℓ_{max}, retained in the Plik likelihood. We consider ℓ_{max} = 1008 (blue), 1503 (purple), and 2013 (green) in addition to the reference ℓ_{max} = 2508 (red). 

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Fig. D.4
Impact of changing in the Plik likelihood the minimum multipole from ℓ_{min} = 100 (red, as in reference case) to ℓ_{min} = 50 (blue). 

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Fig. D.5
Impact of changing the Plik Galactic mask, increasing the sky area used from GA21 (blue), GA34 (purple), GA38 (red, reference), to the least conservative GA54 (green). All results use the 100 ≤ ℓ ≤ 2508 range. Note that the CamSpec likelihood uses the G35 mask for the 143 and 217 GHz channels, and G56 at 100 GHz, but with a restricted, composite multipole range. 

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Fig. D.6
Impact on parameters of removing one single frequency channel (i.e., all spectra with at least one frequency in the removed channel). Results are shown removing the 100 GHz (green), 143 GHz (purple), or 217 GHz (blue) channels, compared to the reference case (red). Where the 217 GHz channel is removed, the CIB spectral index is held fixed at γ^{CIB} = 0.6. 

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Fig. D.7
Covariance matrix between cosmological parameters and detector crosscalibration coefficients (the calibration of the 1435 detector is set to 1 to avoid an overall degeneracy with the total signal amplitude). 

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Fig. D.8
Covariance matrix between foreground parameters and detector crosscalibration coefficients, as in Fig. D.7. 

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Fig. D.9
Power spectrum from the Planck 70 GHz channel, before removal of unresolved sources. Both the autospectrum from the 70 GHz maps, and the weighted cross spectra from maps of the three 70 GHz horn pairs are shown. The bestfitting cosmological model from CamSpec, shown for comparison with the bestfitting source power added, gives a good fit to the data. 

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Appendix D.2: Cosmological parameters from Planck 70 GHz data
For this analysis we implement the pseudoC_{ℓ} method described in Hivon et al. (2002) extended to derive both auto and crosspower spectra from the 70 GHz maps (see, e.g. Polenta et al. 2005, for a comparison between the two estimators). The noise power spectrum and the covariance matrix are computed using 1000 realistic Planck simulations (FFP6, Planck Collaboration I 2014) of both signal and noise maps. The beam window functions are presented in Planck Collaboration IV (2014), and modecoupling kernels to correct for incomplete sky coverage are computed from formulae analogous to those in Appendix. A.1.
In Fig. D.9 we show the auto and crosspower spectra computed from the 70 GHz maps, where crossspectra are obtained by crosscorrelating maps from different pairs of horns (there are three such pairs in total). We use these to construct a likelihood at ℓ > 49 by assuming a Gaussian distribution for the bandpowers, and include the covariance matrix estimated from simulations. To estimate cosmological parameters we use this likelihood in combination with the Planck lowℓ likelihood. We marginalise over a single extragalactic foreground parameter, which is a Poisson term modelling unresolved residual point sources. Figure D.10 shows the resulting parameters, compared to those from CamSpec. Considering the different ℓ range contributing to the two analysis, the parameter distributions are consistent. In Fig. D.11, we show 70 GHz parameters for three choices of the maximum ℓ considered: ℓ_{max} = 800, 1000, and 1200. The latter two are consistent. Minor discrepancies are displayed at ℓ_{max} = 800, which can be explained since there is, in this range, no detection of the point source component. In the same figure, we show results using the Plik likelihood for ℓ_{max} = 1008, which is consistent with Planck 70 GHz over the same ℓ range.
Appendix D.3: Consistency of the Planck low resolution CMB maps
Here we extend the discussion presented in Sect. 8.3. In Fig. D.12 we show the power spectrum of the residual maps, relative to Commander, for NILC, SEVEM, and SMICA. The maximum discrepancy in the range ℓ ≲ 40 (the multipole where noise begins to become nonnegligible) is localised at the quadrupole and is less than 20μK^{2}, whereas for the range 3 ≤ ℓ ≲ 40 the differences are of order ≈5μK^{2}. Overall, the Planck maps are more in agreement among themselves than with WMAP, except perhaps at the quadrupole. The residual map between WMAP and Planck shows power spectrum residuals from ≈10μK^{2} up to ≈40μK^{2} to ℓ ≲ 40. These figures should be compared to the residual estimated from simulated foreground maps, shown to be ≲10μK^{2} at ℓ ≲ 70 in Planck Collaboration XII (2014).
We complement our results with those obtained from the “FFP6” simulations described in Planck Collaboration XII (2014). They consist of 1000 signal plus noise maps processed through each of the four component separation pipelines.
For each Monte Carlo realisation, we follow the same procedure as in the previous section, i.e. smoothing (FWHM = 329.81′) and binning the maps to N_{side} = 32. We apply this procedure to both the CMB input maps and the output maps derived by the four component separation algorithms. Again, a Gaussian white noise with a variance of 4μK^{2} is added, and the noise covariance matrix is corrected accordingly. Note that the additional white noise is taken into account not only for numerical regularization (to this extent its amplitude may well be lower), but principally because the output instrumental noise processed through component separation and downgraded to low resolution is far from being white. The additional white noise makes the detailed knowledge of the full noise covariance matrix unimportant.
For each realisation and for each component separation code, we compute the power spectrum of the processed map and of the input signal map, using a common mask. When estimating the power spectrum, the input CMB maps are regularised by adding a negligible Gaussian white noise with 0.1μK^{2} variance. We have checked explicitly that when the white noise is added to the component separated maps, we are able to recover the input power spectrum without bias up to ℓ ~ 60 for all the four component separation methods.
Moreover, we compute the power spectrum of the difference maps (output processed map minus input CMB) for each realisation, in order to evaluate the total amount of residual noise. Note that this is not only given by the added regularization noise, but also from the intrinsic noise, albeit small, that is present in the maps.
Figure D.13 shows the average and the 1σ levels of such noise residuals for each of the component separation solutions. We thus see, that the level of such total noise residuals is well below the difference plotted in Fig. D.12. Therefore, we argue that the existing differences between the codes are due to genuine foreground separation residuals.
Fig. D.10
Cosmological parameters derived from the 70 GHz maps (solid black) are compared to CamSpec results (red dashed). 

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Fig. D.11
Cosmological parameters derived from the 70 GHz maps for different values of the maximum multipole – ℓ_{max} = 800 (green dotted), ℓ_{max} = 1000 (blue dashed), and ℓ_{max} = 1200 (solid black) – are compared to Plik at ℓ_{max} = 1008 (pink dotdashed). 

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Fig. D.12
Power spectrum of the residual CMB maps, relative to Commander, for the NILC, SEVEM, and SMICA methods. The 1σ expected noise level is shown in black. 

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Fig. D.13
Estimated total residual noise (intrinsic and regularizing white noise) levels for each of the four Planck CMB maps: Commander, Nilc, Sevem, and Smica. Solid lines show the average and the hatched regions show the 68% CL. 

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Appendix E: Dust cleaning using Planck 353 GHz
The WMAP polarisation products are weighted combinations of Ka, Q, and V band 9 year maps. The WMAP analysis mitigates polarised foreground emission using template fitting for the synchrotron and dust emissions. As a template for synchrotron emission, the WMAP K band channel is used, and for dust a polarisation model is used to create a template map (Page et al. 2007). Here we assess the impact on the WMAP polarisation signal when this dust template is replaced by the Planck 353 GHz map, which provides a more direct tracer of the polarisation. Note that this is only for comparison; in all other analyses, we continue to use the WMAP polarisation products as released by the WMAP team (except using the Planck map as discussed in Sect. 8).
Foreground cleaned maps can be written as where m = (Q,U) are linear polarisation Stokes parameter maps, and the index i is for Ka, Q, and V bands. Here m_{synch} is the WMAP 9year K band map and for m_{dust} we use either the WMAP dust template or the Planck 353 GHz maps. For each frequency band, the scaling coefficients α_{i} and β_{i} are estimated by minimizing the χ^{2}: (E.1)where C is the covariance matrix. Following the WMAP analysis, we do not include the signal contribution in C and we only use the diagonal elements of the noise covariance matrix to estimate χ^{2}. Scaling coefficients are computed using two different WMAP masks: the “processing mask” that masks a narrow region in the plane of the Galaxy, and the more conservative “P06” mask used for power spectrum estimation and cosmological analysis.
The template coefficients at each channel are shown in Fig. E.1, estimated using the two different Galactic masks. At Q and V band the estimated coefficients are consistent for the two masks; at Ka band the 353 GHz map gives more stable results than the WMAP dust template. The coefficients using the P06 mask are more uncertain, however, as the residual dust signal outside the mask is low, especially for Ka band. We find that the preferred synchrotron coefficient, α, is slightly lower using the 353 GHz map, and the overall χ^{2}, shown in Table E.1, is slightly improved using the Planck dust map.
We now test the effect on cosmological parameters, in particular the optical depth to reionisation, using these two different templates. Using the Planck 353 GHz channel as the dust template, with coefficients estimated using the processing mask, lowers the best fit value of τ by about 1σ (see Fig. E.2). we find τ = 0.075 ± 0.013, compared with τ = 0.089 ± 0.013 using the WMAP dust model. This in turn has the effect of lowering A_{s}, from 3.088 ± 0.025 to 3.061 ± 0.025, but other ΛCDM parameters are not affected. We note though that using template coefficients estimated outside the P06 Galactic mask, the optical depth using the Planck template is lowered by only 0.5σ compared to the WMAP template, indicating some spatial dependence.We conclude that the impact on cosmological parameters from the choice of dust template is small, but not insignificant.
Still, the lowℓ polarisation from Planck has not been qualified for cosmological analyses yet, and we only used it to reject the hypothesis of a large effect.
Fig. E.1
1σ and 2σ contours for the template coefficient scalings estimated using the WMAP dust template (top) and the Planck 353 GHz map as a dust template (bottom), for the Ka, Q, and V bands. We compare the coefficients estimated using the WMAP “P06” mask, to this with the smaller WMAP “processing mask”. We also indicate the template values quoted in the WMAP paper (Bennett et al. 2013). The dust scaling values are very different for WMAP and Planck 353 GHz because the templates have different normalizations. 

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Fig. E.2
1D (left) and 2D (right) posterior probability for τ and the combination τ − A_{s}, for two different Galactic dust templates. These are computed using dust template coefficients estimated with the WMAP “processing” mask. The difference in τ is reduced to ~0.5σ if template coefficients are estimated outside the “P06” Galactic mask. 

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All Tables
Overview of crossspectra, multipole ranges and masks used in the CamSpec highℓ likelihood.
Overview of cosmological parameters used in this analysis, including symbols, the baseline values if fixed for the standard ΛCDM model, and their definition (see text for further details).
Overview of parameters describing astrophysical foreground modelling, instrumental calibration and beam uncertainties, including symbols, definitions, and prior ranges (see text for further details).
Results of the Hausman test applied to the temperature power spectrum for 2 ≤ ℓ ≤ 32.
Noise estimates for the detector maps, applying the Galactic masks used in the CamSpec likelihood (“mask_3” for 100 GHz, retaining 58% of the sky, and “mask_1” for 143 and 217 GHz, retaining 37% of the sky, combined with an extragalactic point source mask).
All Figures
Fig. 1
Covariance matrix blocks used in the likelihood, accounting for the correlations between crossspectra estimated from the 100, 143, and 217 GHz channels. 

Open with DEXTER  
In the text 
Fig. 2
Set of masks (CL31, CL39, CL49) used for the likelihood analyses, in Mollweide projection. 

Open with DEXTER  
In the text 
Fig. 3
Differences between power spectra evaluated from the union masks G22∪PSA82 and G35∪PSA82 (≡ mask0 and mask1 respectively in the legend), showing the presence of Galactic dust. For ℓ ≤ 1400 the spectra show the 217−143 “doubledifferenced” power spectrum, rescaled to correct for dust emission at 143 GHz. For ℓ > 1400 the 217 mask differenced power spectrum is plotted. The blue line shows the 857 GHz maskdifferenced power spectrum scaled to 217 GHz as described in the text. The green dotted line shows the bestfit of Eq. (9) scaled to 217 GHz. This is the model that we use to subtract dust emission at high multipoles from the 143 × 143, 143 × 217 and 217 × 217 spectra with suitably adjusted scaling factors. 

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In the text 
Fig. 4
857 GHz maskdifferenced power spectrum (points), interpreted as Galactic dust emission. The solid line shows the bestfit model defined by Eq. (9). 

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In the text 
Fig. 5
Top: 857 GHz power spectra for the four different masks defined in the text. The dotted lines show the bestfit model defined by Eq. (9) fit to ℓ ≤ 500, capturing the Galactic dust. An estimate of the CIB power spectrum is shown in orange points (Planck Collaboration XVIII 2011). Bottom: power spectra after subtracting the Galactic dust model. The horizontal purple line shows a constant amplitude of 7 × 10^{6}μK^{2}, which roughly describes the Planck CIB results. 

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In the text 
Fig. 6
SZ templates at 143 GHz computed for a normalization of σ_{8} = 0.8. The tSZ templates are from the model of Efstathiou & Migliaccio (2012) for three values of the evolution parameter ϵ, ϵ = 0 (top), ϵ = 0.25 (middle) and ϵ = 0.5 (lower). The kinetic SZ template is from Trac et al. (2011). The tSZ × CIB cross correlation (143 × 143 GHz) is from the Addison et al. (2012b) template with parameters described in the text, and is negative for 143 × 143 GHz. 

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In the text 
Fig. 7
A selection of cross spectra from individual detectors, with the best fit unresolved foreground model subtracted. The bestfit six parameter ΛCDM model is shown, with ±1σ errors determined from the diagonal components of the analytic covariance matrices. The lower panel in each plot shows the residuals with respect to the model. This shows that our covariance matrix captures well the transition between signal and noise dominance. 

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In the text 
Fig. 8
Cross spectra corrected for the beam and effective calibration, together with the mean power spectrum. Top: the 10 cross spectra at 143 GHz. Bottom: the 15 cross spectra at 217 GHz, with SWB × SWB spectra (magenta), SWB × 217ds1 (red), SWB × 217ds2 (green), 217ds1 × 217ds2 (purple). The SWB detector sets at 217 GHz are 217ds3, 217ds4, 217ds5 and 217ds6. The power spectra are distinguishable only at high multipoles where the data become noise dominated. The lower panels show the dispersion of the cross spectra around the mean, together with a ± 0.2% calibration error (orange line in the lower panels). 

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In the text 
Fig. 9
Cross spectra for the 143 GHz (top) and 217 GHz (bottom) channels, as in Fig. 8, before correction for multiplicative intrafrequency calibration coefficients (above), and after correction (below). The colour coding is the same as in Fig. 8. 

Open with DEXTER  
In the text 
Fig. 10
Differences between the 143 × 143, 143 × 217, and 217 × 217 cross spectra (plotted in bins of width δℓ ≈ 31). The bestfit model for unresolved foregrounds has been subtracted from each spectrum. The numbers list the dispersions over the multipole range 800 ≤ ℓ ≤ 1500. 

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In the text 
Fig. 11
Planck power spectra and data selection. The coloured tick marks indicate the ℓrange of the four crossspectra included in CamSpec (and computed with the same mask, see Table 4). Although not used, the 70 GHz and 143 × 353 GHz spectra demonstrate the consistency of the data. The dashed line indicates the bestfit Planck spectrum. 

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In the text 
Fig. 12
Marginal posterior distributions for the six cosmological (top two rows) and eleven nuisance parameters (lower four rows) estimated with the CamSpec likelihood. 

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In the text 
Fig. 13
Foreground model over the full range of HFI cosmological frequency combinations. The upper panel in each plot shows the residual between the CamSpec measured power spectrum and the “bestfit” primary CMB power spectrum, i.e., the unresolved foreground residual for each frequency combination. The lower panels show the residuals after removing the bestfit foreground model. The lines in the upper panels show the various foreground components. Major foreground components are shown by the solid lines, colour coded as follows: total foreground spectrum (red); Poisson point sources (orange); CIB (blue); thermal SZ (green), Galactic dust (purple). Minor foreground components are shown by the dotted lines: kinetic SZ (green); tSZ × CIB cross correlation (purple). The 100 × 143 and 100 × 217 GHz spectra are not used in the CamSpec likelihood. Here we have assumed and . 

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In the text 
Fig. 14
Comparison of pairs of foreground subtracted cross spectra, demonstrating consistency of the residuals with respect to the bestfit theoretical model. The red line in each of the upper panels shows the theoretical six parameter ΛCDM spectrum for the bestfit parameters listed in Table 8. The lower panels show the residuals with respect to this spectrum, together with error bars computed from the diagonal components of the covariance matrices of the band averages. The points here are bandaveraged in bins of width Δℓ ~ 31. 

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In the text 
Fig. 15
Top: Planck maximumlikelihood (primary) CMB spectrum compared with the bestfit six parameter ΛCDM spectrum. Bottom: power spectrum residuals with respect to the ΛCDM model. The error bars are computed from the diagonal elements of the bandaveraged covariance matrix, as given by Eq. (A.25), including contributions from foreground and beam transfer function errors. 

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In the text 
Fig. 16
Zoomin of regions of the Planck primary CMB power spectrum using finer bin widths (Δℓ = 15 for ℓ < 1000 and Δℓ = 7 for 1000 ≤ ℓ ≤ 2200. In the upper panels, the red lines show the bestfit ΛCDM spectrum, and the blue lines join the Planck data points. Error bars are computed as in Fig. 15. 

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In the text 
Fig. 17
Left: residuals between the Planck power spectrum derived with Plik using “validation” settings (described in the text) and the Plik bestfit model. The light grey lines show residuals for individual detector pairs within each frequency combination. The blue lines show the inverse covariance weighted averages of the individual residuals, together with their errors computed from the covariance matrix. Right: decomposition of the total bestfitting model power spectra into CMB, combined thermal SZ and CIB, unresolved point sources, kinetic SZ, and Galactic dust. 

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In the text 
Fig. 18
Top: CMB power spectrum estimated using the Plik likelihood, after subtracting the bestfit foreground model. Bottom: residual difference (blue line) between the Plik spectrum and the bestfit ΛCDM model. The solid green line shows the difference between the bestfit models derived from the Plik and CamSpec likelihoods. 

Open with DEXTER  
In the text 
Fig. 19
Comparison of cosmological parameters estimated from the CamSpec (blue) and Plik (red for mask CL39; purple for mask CL49) likelihoods. All parameters agree to better than 0.2σ when CL49 is used for Plik, matching the sky area used by CamSpec at 100 GHz. 

Open with DEXTER  
In the text 
Fig. 20
Comparison of foreground parameters estimated with the CamSpec and Plik likelihoods. The purple (red) lines show the CamSpec (Plik) distributions using only Planck data, and the green (blue) lines show the CamSpec (Plik) results when additionally including ACT and SPT. 

Open with DEXTER  
In the text 
Fig. 21
Correlation matrix between all the cosmological (top block), foreground (middle block), and derived (bottom block) parameters, estimated using the Plik likelihood. 

Open with DEXTER  
In the text 
Fig. 22
Distribution of the difference between cosmological parameter mean posterior values, as estimated by Plik (assuming foreground spectra are known) from the (CMB + fixed foregrounds + noise) FFP6 simulations, and the corresponding posterior parameter mean values from CMB only simulations (based on the same CMB realisations as in the FFP6 simulations), in units of the FFP6 posterior standard deviations.This demonstrates the absence of methodological bias at the level of the intrinsic dispersion between realisations, while removing from the distributions the dominating scatter coming from the purely CMB induced cosmic variance. In these figures, the dominant source of scatter comes from the CMBforeground chance correlations in the signal dominated part of the spectrum, and from the noise scatter at large multipoles. 

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In the text 
Fig. 23
Posterior marginal distributions of cosmological parameters obtained by Plik on a single FFP6 realisation, under different assumptions on the foreground model. Red thick lines correspond to the foreground model parametrisation described in Sect. 3 with fixed γ^{CIB}. Purple lines (resp. green, grey) show the effect of leaving γ^{CIB} free (resp. letting γ^{Dust} free, or fixing γ^{CIB} to 0.4, 2σ away from the peak posterior value). Finally, the blue lines correspond to the analysis made on CMBonly simulations. The input values of the parameters of the simulation are shown as dashed vertical lines. 

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In the text 
Fig. 24
Comparison of the distributions of cosmological parameters obtained with Plik in the reference case (left) with a set of validation test cases. The red line indicates the median and blue the mean, computed from the posterior histograms. The box shows the 68% confidence interval; the outer line the 95% interval. 

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In the text 
Fig. 25
Comparison of the distributions of foreground model parameters obtained with Plik, as in Fig. 24. 

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In the text 
Fig. 26
Estimated calibration and beam eigenmode parameters, compared to the priors, using (left) 100 to 217 GHz data, and (right) a single “FFP6” simulation where there was no calibration or beam errors. The shaded area shows the width of the prior imposed on the first beam eigenvalue. The individual SWB detectors (those denoted in the legend by their number rather than by a detset number) are not single out by this comparison. 

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In the text 
Fig. 27
Posterior distributions for cosmological parameters obtained with Plik, varying the Helium abundance Y_{P} (red). We show the effect of marginalising over calibration and beam errors (blue), and the results obtained using the reference settings (purple), where Y_{P} is constrained by BBN. 

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In the text 
Fig. 28
Number counts from Planck (filled circles, Planck Collaboration Int. VII 2013), ACT, and SPT (open squares) as described in the text, from 100 GHz to 217 GHz. The models from de Zotti et al. (2005, solid line) and Tucci et al. (2011, dots) are overplotted. The analytical fit from Eq. (20) and Table 7 is shown dashed, and shows a similar behaviour to the scaleddown de Zotti et al. (2005) model. 

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In the text 
Fig. 29
Top: CMB power spectrum, with bestfitting foreground model removed, compared to the CamSpec best fit model. Middle: the residuals with respect to this bestfit model. Bottom: residuals of the bestfit models from the mapbased likelihoods, with respect to the CamSpec best fit model. 

Open with DEXTER  
In the text 
Fig. 30
The variation of cosmological and foreground parameters estimated for the four CMB maps as function of ℓ_{max} = 1000, 1500, 2000, 2500, with ℓ_{min} = 70, compared to those obtained with the CamSpec and Plik likelihoods. Parameters are consistent to 1σ for ℓ_{max} ≤ 2000 

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In the text 
Fig. 31
Planck TE and EE polarisation spectra computed as described in the text, together with the polarisation spectra predicted from the sixparameter ΛCDM model, fit only to the Planck temperature data. 

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In the text 
Fig. 32
Twoparameter probability distribution for an amplitudetilt model constrained by the lowℓPlanck likelihood using three different masks. Angular scales between 2 ≤ ℓ ≤ 49 are included in this fit. 

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In the text 
Fig. 33
Comparison of constraints on τ and A_{s} using the split temperaturepolarisation WMAP likelihood approach (dashed contours; Eqs. (24) and (25)) with those obtained with the exact bruteforce pixel likelihood (shaded contours; Eq. (21)). 

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In the text 
Fig. 34
Top panel: temperature power spectra evaluated from downgraded Planck maps, estimated with Commander, NILC, SEVEM, or SMICA, and the 9 year WMAP ILC map, using the Bolpol quadratic estimator. The grey shaded area indicates the 1σ Fisher errors while the solid line shows the PlanckΛCDM best fit model. Bottom panel: power spectrum differences for each algorithm/data set relative to the Commander spectrum, estimated from the spectra shown in the panel above. The black lines show the expected 1σ uncertainty due to (regularization) noise. 

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In the text 
Fig. 35
Top: comparison between the lowℓPlanck temperature power spectrum estimated by Commander and the 9year WMAP spectrum (Bennett et al. 2013). Error bars indicate 68% confidence regions. Bottom: difference between the WMAP and Planck lowℓ spectra. 

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In the text 
Fig. 36
Probability distribution for a twoparameter (amplitude and tilt) model derived from the Planck (blue) and WMAP (red) lowℓ likelihoods, including angular scales between ℓ = 2 and 30. 

Open with DEXTER  
In the text 
Fig. 37
The 2013 Planck CMB temperature angular power spectrum. The error bars include cosmic variance, whose magnitude is indicated by the green shaded area around the best fit model. The lowℓ values are plotted at 2, 3, 4, 5, 6, 7, 8, 9.5, 11.5, 13.5, 16, 19, 22.5, 27, 34.5, and 44.5. 

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In the text 
Fig. 38
Results of the Hausman test applied to the temperature power spectrum for 2 ≤ ℓ ≤ 32. The black histogram shows the expected distribution, estimated with simulations, of the s_{1} test statistic. The vertical bars represent Planck CMB maps and the 9 year WMAP ILC map. Note that the statistic is indistinguishable for the NILC and Commander maps. 

Open with DEXTER  
In the text 
Fig. 39
Power spectrum amplitude, q, relative to the bestfit Planck model as a function of ℓ_{max}, as measured by the lowℓPlanck and WMAP temperature likelihoods, respectively. Error bars indicate 68 and 95% confidence regions. 

Open with DEXTER  
In the text 
Fig. 40
Posterior distributions for the lowℓ spectrum amplitude, A_{low}, estimated using the Planck likelihood, with ℓ_{low} = 32 (green) and ℓ_{low} = 49 (blue). The purple line show the distribution derived using the WMAP temperature likelihood with ℓ_{low} = 32. 

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In the text 
Fig. 41
Comparison of the posterior distributions for Ω_{c}h^{2}, n_{s}, and H_{0} for the default sixparameter ΛCDM model constrained by Planck (red); compared to the case when we allow a variable lowℓ power spectrum amplitude at ℓ ≤ 49 (blue); and when replacing the lowℓ temperature likelihood with a Gaussian prior on τ, the optical depth of reionisation (purple). The lower right panel shows the posterior distribution for the lowℓ amplitude, A_{low}. 

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In the text 
Fig. A.1
Toy power spectra drawn from a Gaussian distribution with mean given by a ΛCDM power spectrum and a covariance given by the bandaveraged C_{ℓ} covariance matrix, (top), and the corresponding differences with respect to the input ΛCDM model. Note the apparent presence of “coherent oscillatory features” in the difference spectra. These are fully described by the power spectrum covariance matrix. To assess the statistical significance of apparently “unexpected features” in the power spectrum, it is critical to include all sources of correlations among different C_{ℓ}’s. 

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In the text 
Fig. A.2
Eigenmodes of the beam covariance matrix, shown for each crossspectrum used in the analysis. The largest five modes are shown for each crossspectrum. 

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In the text 
Fig. A.3
Noise spectra computed from difference maps for a selection of detector sets: 100GHz (top), 143GHz (middle), and 217GHz(bottom). Green lines show the spectra for T, purple for Q and magenta points for U. The solid black lines show the modeled fits to the spectra (Eq. (A.46)), and the coloured horizonal lines show the whitenoise levels of Eq. (A.44). These are computed using the same masks as used in Table A.1. 

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In the text 
Fig. B.1
Set of unapodised foreground masks, G22, G35, G45, G56, G65, PS96, which, once apodised, are used for the likelihood analyses. The Galactic (G22 to G65) masks are defined using a threshold of the 353 GHz Planck temperature map. The (lowerright) PS96 point source mask is based on the union of resolved sources in all channels from 100 GHz to 353 GHz. 

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In the text 
Fig. B.2
Apodised Galactic and point source masks, which we derived (see text) from their nonapodised version shown in Fig. B.1. From left to right and top to bottom, the panels show the GA34, GA38, GA54 and PSA82 masks, which are the ones we combine to use in the likelihood analysis. 

Open with DEXTER  
In the text 
Fig. C.1
Top: An ILC CMB map constructed from 100353 GHz maps. Middle: The ILC map added to the 857 GHz map, scaled to match the diffuse dust emission at 217 GHz. This map (and an equivalent at 143 GHz) is used as the CMB+dust template to assess CMB/foreground cross correlations. Bottom: The real 217 GHz map. 

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In the text 
Fig. C.2
Fake versus real data difference spectra. The magenta points show the double difference power spectrum of actual data (as in Fig. 3). The green line shows the same double difference spectrum computed from the ILC + 857 dust template maps described in the text. This model provides a good match to the magenta points both in amplitude and phasing. The blue points show the difference of the 217 and 143 power spectra for mask1, which adds CMB cosmic variance, illustrating why we use doubledifferenced spectra for this comparison. In all cases, the smoothed dust power spectrum model of Eq. (9) (with slightly adjusted amplitude to bring the residuals at these low multipoles close to zero) has been subtracted. 

Open with DEXTER  
In the text 
Fig. C.3
Power spectrum residuals between the summed crossspectra at three HFI frequencies. The top panel shows 100 × 100–143 × 143, the middle panel shows 100 × 100–217 × 217 and the bottom panel shows 143 × 143–217 × 217. A “best fit” model for unresolved foregrounds has been subtracted from the power spectrum at each frequency. The scatter in the multipole range 50 ≤ ℓ ≤ 500 is listed in each panel. 

Open with DEXTER  
In the text 
Fig. D.1
Impact on cosmological and foreground parameters of fixing the calibration and beam coefficients at their maximum posterior value (red), compared to marginalising over these nuisance parameters (blue). 

Open with DEXTER  
In the text 
Fig. D.2
Impact of removing the subpixel effect (blue) or the correlated noise between detector sets (purple), compared to the Plik reference case (red). 

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In the text 
Fig. D.3
Impact on the cosmological and foreground parameters of varying the maximum multipole, ℓ_{max}, retained in the Plik likelihood. We consider ℓ_{max} = 1008 (blue), 1503 (purple), and 2013 (green) in addition to the reference ℓ_{max} = 2508 (red). 

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In the text 
Fig. D.4
Impact of changing in the Plik likelihood the minimum multipole from ℓ_{min} = 100 (red, as in reference case) to ℓ_{min} = 50 (blue). 

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In the text 
Fig. D.5
Impact of changing the Plik Galactic mask, increasing the sky area used from GA21 (blue), GA34 (purple), GA38 (red, reference), to the least conservative GA54 (green). All results use the 100 ≤ ℓ ≤ 2508 range. Note that the CamSpec likelihood uses the G35 mask for the 143 and 217 GHz channels, and G56 at 100 GHz, but with a restricted, composite multipole range. 

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In the text 
Fig. D.6
Impact on parameters of removing one single frequency channel (i.e., all spectra with at least one frequency in the removed channel). Results are shown removing the 100 GHz (green), 143 GHz (purple), or 217 GHz (blue) channels, compared to the reference case (red). Where the 217 GHz channel is removed, the CIB spectral index is held fixed at γ^{CIB} = 0.6. 

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In the text 
Fig. D.7
Covariance matrix between cosmological parameters and detector crosscalibration coefficients (the calibration of the 1435 detector is set to 1 to avoid an overall degeneracy with the total signal amplitude). 

Open with DEXTER  
In the text 
Fig. D.8
Covariance matrix between foreground parameters and detector crosscalibration coefficients, as in Fig. D.7. 

Open with DEXTER  
In the text 
Fig. D.9
Power spectrum from the Planck 70 GHz channel, before removal of unresolved sources. Both the autospectrum from the 70 GHz maps, and the weighted cross spectra from maps of the three 70 GHz horn pairs are shown. The bestfitting cosmological model from CamSpec, shown for comparison with the bestfitting source power added, gives a good fit to the data. 

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In the text 
Fig. D.10
Cosmological parameters derived from the 70 GHz maps (solid black) are compared to CamSpec results (red dashed). 

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In the text 
Fig. D.11
Cosmological parameters derived from the 70 GHz maps for different values of the maximum multipole – ℓ_{max} = 800 (green dotted), ℓ_{max} = 1000 (blue dashed), and ℓ_{max} = 1200 (solid black) – are compared to Plik at ℓ_{max} = 1008 (pink dotdashed). 

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In the text 
Fig. D.12
Power spectrum of the residual CMB maps, relative to Commander, for the NILC, SEVEM, and SMICA methods. The 1σ expected noise level is shown in black. 

Open with DEXTER  
In the text 
Fig. D.13
Estimated total residual noise (intrinsic and regularizing white noise) levels for each of the four Planck CMB maps: Commander, Nilc, Sevem, and Smica. Solid lines show the average and the hatched regions show the 68% CL. 

Open with DEXTER  
In the text 
Fig. E.1
1σ and 2σ contours for the template coefficient scalings estimated using the WMAP dust template (top) and the Planck 353 GHz map as a dust template (bottom), for the Ka, Q, and V bands. We compare the coefficients estimated using the WMAP “P06” mask, to this with the smaller WMAP “processing mask”. We also indicate the template values quoted in the WMAP paper (Bennett et al. 2013). The dust scaling values are very different for WMAP and Planck 353 GHz because the templates have different normalizations. 

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In the text 
Fig. E.2
1D (left) and 2D (right) posterior probability for τ and the combination τ − A_{s}, for two different Galactic dust templates. These are computed using dust template coefficients estimated with the WMAP “processing” mask. The difference in τ is reduced to ~0.5σ if template coefficients are estimated outside the “P06” Galactic mask. 

Open with DEXTER  
In the text 