Issue 
A&A
Volume 594, October 2016
Planck 2015 results



Article Number  A11  
Number of page(s)  99  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201526926  
Published online  20 September 2016 
Planck 2015 results
XI. CMB power spectra, likelihoods, and robustness of parameters
^{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 and Dept of Radio Science and Engineering, PO Box 13000, 00076 Aalto, Finland
^{3} African Institute for Mathematical Sciences, 68 Melrose Road, Muizenberg 7945, Cape Town, South Africa
^{4} Agenzia Spaziale Italiana Science Data Center, via del Politecnico snc, 00133 Roma, Italy
^{5} AixMarseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388 Marseille, France
^{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} CGEE, SCS Qd 9, Lote C, Torre C, 4° andar, Ed. Parque Cidade Corporate, CEP 70308200, Brasília, DF, Brazil
^{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} CRANN, Trinity College, Dublin, Ireland
^{12} California Institute of Technology, Pasadena, California, USA
^{13} Centre for Theoretical Cosmology, DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
^{14} Centro de Estudios de Física del Cosmos de Aragón (CEFCA), Plaza San Juan, 1, planta 2, 44001 Teruel, Spain
^{15} Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, California, USA
^{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 Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain
^{20} Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo s/n, 33007 Oviedo, Spain
^{21} Department of Astrophysics/IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
^{22} Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada
^{23} Department of Physics and Astronomy, Dana and David Dornsife College of Letter, Arts and Sciences, University of Southern California, Los Angeles, CA 90089, USA
^{24} Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
^{25} Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK
^{26} Department of Physics, Florida State University, Keen Physics Building, 77 Chieftan Way, Tallahassee, Florida, USA
^{27} Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, 00014 Helsinki, Finland
^{28} Department of Physics, Princeton University, Princeton, NJ 08544, USA
^{29} Department of Physics, University of California, One Shields Avenue, Davis, CA 95616, USA
^{30} Department of Physics, University of California, Santa Barbara, CA 93106, USA
^{31} Department of Physics, University of Illinois at UrbanaChampaign, 1110 West Green Street, Urbana, Illinois, USA
^{32} Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi di Padova, via Marzolo 8, 35131 Padova, Italy
^{33} Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, via Saragat 1, 44122 Ferrara, Italy
^{34} Dipartimento di Fisica, Università La Sapienza, P.le A. Moro 2, 00185 Roma, Italy
^{35} Dipartimento di Fisica, Università degli Studi di Milano, via Celoria, 16, 20133 Milano, Italy
^{36} Dipartimento di Fisica, Università degli Studi di Trieste, via A. Valerio 2, 34127 Trieste, Italy
^{37} Dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, 00133 Roma, Italy
^{38} Discovery Center, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
^{39} Discovery Center, Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen, Denmark
^{40} European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo, s/n, Urbanización Villafranca del Castillo, 28691 Villanueva de la Cañada, Madrid, Spain
^{41} European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
^{42} Gran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’ Aquila, Italy
^{43} HGSFP and University of Heidelberg, Theoretical Physics Department, Philosophenweg 16, 69120 Heidelberg, Germany
^{44} Haverford College Astronomy Department, 370 Lancaster Avenue, Haverford, PA 19041, USA
^{45} Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, 00014 Helsinki, Finland
^{46} INAF–Osservatorio Astrofisico di Catania, via S. Sofia 78, Catania, Italy
^{47} INAF–Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy
^{48} INAF– Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monte Porzio Catone, Italy
^{49} INAF–Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, 40127 Trieste, Italy
^{50} INAF/IASF Bologna, via Gobetti 101, 40129 Bologna, Italy
^{51} INAF/IASF Milano, via E. Bassini 15, 20133 Milano, Italy
^{52} INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
^{53} INFN, Sezione di Ferrara, via Saragat 1, 44122 Ferrara, Italy
^{54} INFN, Sezione di Roma 1, Università di Roma Sapienza, P.le Aldo Moro 2, 00185 Roma, Italy
^{55} INFN, Sezione di Roma 2, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, 00185 Roma, Italy
^{56} INFN/National Institute for Nuclear Physics, via Valerio 2, 34127 Trieste, Italy
^{57} IPAG: Institut de Planétologie et d’Astrophysique de Grenoble, Université Grenoble Alpes, IPAG; CNRS, IPAG, 38000 Grenoble, France
^{58} IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, 411 007 Pune, India
^{59} Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, UK
^{60} Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA
^{61} Institut Néel, CNRS, Université Joseph Fourier Grenoble I, 25 rue des Martyrs, 38042 Grenoble, France
^{62} Institut Universitaire de France, 103 bd SaintMichel, 75005 Paris, France
^{63} Institut d’Astrophysique Spatiale, CNRS, Univ. ParisSud, Université ParisSaclay, Bât. 121, 91405 Orsay Cedex, France
^{64} Institut d’Astrophysique de Paris, CNRS (UMR 7095), 98bis boulevard Arago, 75014 Paris, France
^{65} Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, 52056 Aachen, Germany
^{66} Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
^{67} Institute of Theoretical Astrophysics, University of Oslo, Blindern, 0371 Oslo, Norway
^{68} Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, La Laguna, 38205 Tenerife, Spain
^{69} Instituto de Física de Cantabria (CSICUniversidad de Cantabria), Avda. de los Castros s/n, 93005 Santander, Spain
^{70} Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, 35131 Padova, Italy
^{71} Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 31109, USA
^{72} Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
^{73} Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
^{74} Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
^{75} Kazan Federal University, 18 Kremlyovskaya St., 420008 Kazan, Russia
^{76} LAL, Université ParisSud, CNRS/IN2P3, 91898 Orsay, France
^{77} LERMA, CNRS, Observatoire de Paris, 61 avenue de l’Observatoire, 75014 Paris, France
^{78} Laboratoire AIM, IRFU/Service d’Astrophysique – CEA/DSM – CNRS – Université Paris Diderot, Bât. 709, CEASaclay, 91191 GifsurYvette Cedex, France
^{79} Laboratoire Traitement et Communication de l’Information, CNRS (UMR 5141) and Télécom ParisTech, 46 rue Barrault 75634 Paris Cedex 13, France
^{80} Laboratoire de Physique Subatomique et Cosmologie, Université GrenobleAlpes, CNRS/IN2P3, 53 rue des Martyrs, 38026 Grenoble Cedex, France
^{81} Laboratoire de Physique Théorique, Université ParisSud 11 & CNRS, Bâtiment 210, 91405 Orsay, France
^{82} Lebedev Physical Institute of the Russian Academy of Sciences, Astro Space Centre, 84/32 Profsoyuznaya st., GSP7, 117997 Moscow, Russia
^{83} Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, 10617 Taipei, Taiwan
^{84} MaxPlanckInstitut für Astrophysik, KarlSchwarzschildStr. 1, 85741 Garching, Germany
^{85} National University of Ireland, Department of Experimental Physics, Maynooth, Co. Kildare, Ireland
^{86} Nicolaus Copernicus Astronomical Center, Bartycka 18, 00716 Warsaw, Poland
^{87} Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
^{88} Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 2100 Copenhagen, Denmark
^{89} Nordita (Nordic Institute for Theoretical Physics), Roslagstullsbacken 23, 106 91 Stockholm, Sweden
^{90} Optical Science Laboratory, University College London, Gower Street, London, UK
^{91} SISSA, Astrophysics Sector, via Bonomea 265, 34136 Trieste, Italy
^{92} SMARTEST Research Centre, Università degli Studi eCampus, Via Isimbardi 10, Novedrate (CO), 22060, Italy
^{93} School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK
^{94} Sorbonne UniversitéUPMC, UMR 7095, Institut d’Astrophysique de Paris, 98bis boulevard Arago, 75014 Paris, France
^{95} Space Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, 117997 Moscow, Russia
^{96} Space Sciences Laboratory, University of California, Berkeley, CA 94720, USA
^{97} Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, 369167 KarachaiCherkessian Republic, Russia
^{98} SubDepartment of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
^{99} Sydney Institute for Astronomy, School of Physics A28, University of Sydney, NSW 2006, Australia
^{100} The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, 106 91 Stockholm, Sweden
^{101} Theory Division, PHTH, CERN, 1211 Geneva 23, Switzerland
^{102} UPMC Univ. Paris 06, UMR 7095, 98bis boulevard Arago, 75014 Paris, France
^{103} Université de Toulouse, UPSOMP, IRAP, 31028 Toulouse Cedex 4, France
^{104} University Observatory, Ludwig Maximilian University of Munich, Scheinerstrasse 1, 81679 Munich, Germany
^{105} University of Granada, Departamento de Física Teórica y del Cosmos, Facultad de Ciencias, 18071 Granada, Spain
^{106} University of Granada, Instituto Carlos I de Física Teórica y Computacional, 18071 Granada, Spain
^{107} Warsaw University Observatory, Aleje Ujazdowskie 4, 00478 Warszawa, Poland
^{⋆}
Corresponding authors: F. R. Bouchet, email: bouchet@iap.fr
Received: 9 July 2015
Accepted: 18 May 2016
This paper presents the Planck 2015 likelihoods, statistical descriptions of the 2point correlationfunctions of the cosmic microwave background (CMB) temperature and polarization fluctuations that account for relevant uncertainties, both instrumental and astrophysical in nature. They are based on the same hybrid approach used for the previous release, i.e., a pixelbased likelihood at low multipoles (ℓ< 30) and a Gaussian approximation to the distribution of crosspower spectra at higher multipoles. The main improvements are the use of more and better processed data and of Planck polarization information, along with more detailed models of foregrounds and instrumental uncertainties. The increased redundancy brought by more than doubling the amount of data analysed enables further consistency checks and enhanced immunity to systematic effects. It also improves the constraining power of Planck, in particular with regard to smallscale foreground properties. Progress in the modelling of foreground emission enables the retention of a larger fraction of the sky to determine the properties of the CMB, which also contributes to the enhanced precision of the spectra. Improvements in data processing and instrumental modelling further reduce uncertainties. Extensive tests establish the robustness and accuracy of the likelihood results, from temperature alone, from polarization alone, and from their combination. For temperature, we also perform a full likelihood analysis of realistic endtoend simulations of the instrumental response to the sky, which were fed into the actual data processing pipeline; this does not reveal biases from residual lowlevel instrumental systematics. Even with the increase in precision and robustness, the ΛCDM cosmological model continues to offer a very good fit to the Planck data. The slope of the primordial scalar fluctuations, n_{s}, is confirmed smaller than unity at more than 5σ from Planck alone. We further validate the robustness of the likelihood results against specific extensions to the baseline cosmology, which are particularly sensitive to data at high multipoles. For instance, the effective number of neutrino species remains compatible with the canonical value of 3.046. For this first detailed analysis of Planck polarization spectra, we concentrate at high multipoles on the E modes, leaving the analysis of the weaker B modes to future work. At low multipoles we use temperature maps at all Planck frequencies along with a subset of polarization data. These data take advantage of Planck’s wide frequency coverage to improve the separation of CMB and foreground emission. Within the baseline ΛCDM cosmology this requires τ = 0.078 ± 0.019 for the reionization optical depth, which is significantly lower than estimates without the use of highfrequency data for explicit monitoring of dust emission. At high multipoles we detect residual systematic errors in E polarization, typically at the μK^{2} level; we therefore choose to retain temperature information alone for high multipoles as the recommended baseline, in particular for testing nonminimal models. Nevertheless, the highmultipole polarization spectra from Planck are already good enough to enable a separate highprecision determination of the parameters of the ΛCDM model, showing consistency with those established independently from temperature information alone.
Key words: cosmic background radiation / cosmological parameters / cosmology: observations / methods: data analysis / methods: statistical
© ESO, 2016
1. Introduction
This paper presents the angular power spectra of the cosmic microwave background (CMB) and the related likelihood functions, calculated from Planck^{1} 2015 data, which consists of intensity maps from the full mission, along with a subset of the polarization data.
The CMB power spectra contain all of the information available if the CMB is statistically isotropic and distributed as a multivariate Gaussian. For realistic data, these must be augmented with models of instrumental noise, of other instrumental systematic effects, and of contamination from astrophysical foregrounds.
The power spectra are, in turn, uniquely determined by the underlying cosmological model and its parameters. In temperature, the power spectrum has been measured over large fractions of the sky by the Cosmic Background Explorer (COBE; Wright et al. 1996) and the Wilkinson Microwave Anistropy Probe (WMAP; Bennett et al. 2013), and in smaller regions by a host of balloon and groundbased telescopes (e.g., Netterfield et al. 1997; Hanany et al. 2000; Grainge et al. 2003; Pearson et al. 2003; Tristram et al. 2005b; Jones et al. 2006; Reichardt et al. 2009; Fowler et al. 2010; Das et al. 2011, 2014; Keisler et al. 2011; Story et al. 2013). The Planck 2013 power spectrum and likelihood were discussed in Planck Collaboration XV (2014, hereafter Like13.
The distribution of temperature and polarization on the sky is further affected by gravitational lensing by the inhomogeneous mass distribution along the line of sight between the last scattering surface and the observer. This introduces correlations between large and small scales, which can be estimated by computing the expected contribution of lensing to the 4point function (i.e., the trispectrum). This can in turn be used to determine the power spectrum of the lensing potential, as is done in Planck Collaboration XV (2016) for this Planck release, and to further constrain the cosmological parameters via a separate likelihood function (Planck Collaboration XIII 2016).
Over the last decade, CMB intensity (temperature) has been augmented by linear polarization data (e.g., Kovac et al. 2002; Kogut et al. 2003; Sievers et al. 2007; Dunkley et al. 2009; Pryke et al. 2009; Araujo et al. 2012; Polarbear Collaboration 2014). Because linear polarization is given by both an amplitude and direction, it can, in turn, be decomposed into two coordinateindependent quantities, each with a different dependence on the cosmology (e.g., Seljak 1997; Kamionkowski et al. 1997; Zaldarriaga & Seljak 1997). One, the socalled E mode, is determined by much the same physics as the intensity, and therefore enables an independent measurement of the background cosmology, as well as a determination of some new parameters (e.g., the reionization optical depth). The other polarization observable, the B mode, is only sourced at early times by gravitational radiation, as produced, for example, during an inflationary epoch. The E and B components are also conventionally taken to be isotropic Gaussian random fields, with only E expected to be correlated with intensity. Thus we expect to be able to measure four independent power spectra, namely the three autospectra ${\mathit{C}}_{\mathit{\ell}}^{\mathit{TT}}$, ${\mathit{C}}_{\mathit{\ell}}^{\mathit{EE}}$, and ${\mathit{C}}_{\mathit{\ell}}^{\mathit{BB}}$, along with the crossspectrum ${\mathit{C}}_{\mathit{\ell}}^{\mathit{TE}}$.
Estimating these spectra from the likelihood requires cleaned and calibrated maps for all Planck detectors, along with a quantitative description of their noise properties. The required data processing is discussed in Planck Collaboration II (2016), Planck Collaboration III (2016), Planck Collaboration IV (2016), Planck Collaboration V (2016), and Planck Collaboration VIII (2016) for the lowfrequency instrument (LFI; 30, 44, and 70 GHz) and Planck Collaboration VII (2016) and Planck Collaboration VIII (2016) for the highfrequency instrument (HFI; 100, 143, 217, 353, 585, and 857 GHz). Although the CMB is brightest over 70–217 GHz, the full range of Planck frequencies is crucial to distinguish between the cosmological component and sources of astrophysical foreground emission, present in even the cleanest regions of sky. We therefore use measurements from those Planck bands dominated by such emission as a template to model the foreground in the bands where the CMB is most significant.
This paper presents the ${\mathit{C}}_{\mathit{\ell}}^{\mathit{TT}}$, ${\mathit{C}}_{\mathit{\ell}}^{\mathit{EE}}$, and ${\mathit{C}}_{\mathit{\ell}}^{\mathit{TE}}$ spectra, likelihood functions, and basic cosmological parameters from the Planck 2015 release. A complete analysis in the context of an extended ΛCDM cosmology of these and other results from Planck regarding the lensing power spectrum results, as well as constraints from other observations, is given in Planck Collaboration XIII (2016). Wider extensions to the set of models are discussed in other Planck 2015 papers; for example, Planck Collaboration XIV (2016) examines specific models for the dark energy component and extensions to general relativity, and Planck Collaboration XX (2016) discusses inflationary models.
This paper shows that the contribution of highℓ systematic errors to the polarization spectra are at quite a low level (of the order of a few μK^{2}), therefore enabling an interesting comparison of the polarizationbased cosmological results with those derived from ${\mathit{C}}_{\mathit{\ell}}^{\mathit{TT}}$ alone. We therefore discuss the results for ${\mathit{C}}_{\mathit{\ell}}^{\mathit{TE}}$ and ${\mathit{C}}_{\mathit{\ell}}^{\mathit{EE}}$ at high multipoles. However, the technical difficulties involved with polarization measurements and subsequent data analysis, along with the inherently lower signaltonoise ratio (especially for B modes), thus require a careful understanding of the random noise and instrumental and astrophysical systematic effects. For this reason, at large angular scales (i.e., low multipoles ℓ) the baseline results use only a subset of Planck polarization data.
Because of these different sensitivities to systematic errors at different angular scales, as well as the increasingly Gaussian behaviour of the likelihood function at smaller angular scales, we adopt a hybrid approach to the likelihood calculation (Efstathiou 2004, 2006), splitting between a direct calculation of the likelihood on large scales and the use of pseudospectral estimates at smaller scales, as we did for the previous release.
The plan of the paper reflects this hybrid approach along with the importance of internal tests and crossvalidation. In Sect. 2, we present the lowmultipole (ℓ< 30) likelihood and its validation. At these large scales, we compute the likelihood function directly in pixel space; the temperature map is obtained by a Gibbs sampling approach in the context of a parameterized foreground model, while the polarized maps are cleaned of foregrounds by a template removal technique.
In Sect. 3, we introduce the highmultipole (ℓ ≥ 30) likelihood and present its main results. At these smaller scales, we employ a pseudoC_{ℓ} approach, beginning with a numerical spherical harmonic transform of the fullsky map, debiased and deconvolved to account for the mask and noise.
Likelihood codes and datasets.
Section 4 is devoted to the detailed assessment of this highℓ likelihood. One technical difference between Like13 and the present work is the move from the CamSpec code to Plik for highℓ results as well as the released software (Planck Collaboration 2015). The main reason for this change is that the structure of Plik allows more finegrained tests on the polarization spectra for individual detectors or subsets of detectors. We are able to compare the effect of different cuts on Planck and external data, as well as using methods that take different approaches to estimate the maximumlikelihood spectra from the input maps; these illustrate the small impact of differences in methodology and data preparation, which are difficult to assess otherwise.
We then combine the low and highℓ algorithms to form the full Planck likelihood in Sect. 5, assessing there the choice of ℓ = 30 for the hybridization scale and establishing the basic cosmological results from Planck 2015 data alone.
Finally, in Sect. 6 we conclude. A series of Appendices discusses sky masks and gives more detail on the individual likelihood codes, both the released version and a series of other codes used to validate the overall methodology.
To help distinguish the many different likelihood codes, which are functions of different parameters and use different input data, Table 1 summarizes the designations used throughout the text.
2. Lowmultipole likelihood
At low multipoles, the current Planck release implements a standard joint pixelbased likelihood including both temperature and polarization for multipoles ℓ ≤ 29. Throughout this paper, we denote this likelihood “lowTEB”, while “lowP” denotes the polarization part of this likelihood. For temperature, the formalism uses the CMB maps cleaned with Commander (Eriksen et al. 2004, 2008) maps, while for polarization we use the 70 GHz LFI maps and explicitly marginalize over the 30 GHz and 353 GHz maps taken as tracers of synchrotron and dust emission, respectively (see Sect. 2.3), accounting in both cases for the induced noise covariance in the likelihood.
This approach is somewhat different from the Planck 2013 lowℓ likelihood. As described in Like13, this comprised two nearly independent components, covering temperature and polarization information, respectively. The temperature likelihood employed a BlackwellRao estimator (Chu et al. 2005) at ℓ ≤ 49, averaging over Monte Carlo samples drawn from the exact power spectrum posterior using Commander . For polarization, we had adopted the pixelbased 9year WMAP polarization likelihood, covering multipoles ℓ ≤ 23 (Bennett et al. 2013).
The main advantage of the exact joint approach now employed is mathematical rigour and consistency to higher ℓ, while the main disadvantage is a slightly higher computational expense due to the higher pixel resolution required to extend the calculation to ℓ = 29 in polarization. However, after implementation of the ShermanMorrisonWoodbury formula to reduce computational costs (see Appendix B.1), the two approaches perform similarly, both with respect to speed and accuracy, and our choice is primarily a matter of implementational convenience and flexibility, rather than actual results or performance.
2.1. Statistical description and algorithm
We start by reviewing the general CMB likelihood formalism for the analysis of temperature and polarization at low ℓ, as described for instance by Tegmark & de OliveiraCosta (2001), Page et al. (2007), and in Like13. We begin with maps of the three Stokes parameters { T,Q,U } for the observed CMB intensity and linear polarization in some set of HEALPix ^{2} (Górski et al. 2005) pixels on the sky. In order to use multipoles ℓ ≤ ℓ_{cut} = 29 in the likelihood, we adopt a HEALPix resolution of N_{side} = 16 which has 3072 pixels (of area 13.6 deg^{2}) per map; this accommodates multipoles up to ℓ_{max} = 3N_{side}−1 = 47, and, considering separate maps of T, Q, and U, corresponds to a maximum of N_{pix} = 3 × 3072 = 9216 pixels in any given calculation, not accounting for any masking.
After component separation, the data vector may be modelled as a sum of cosmological CMB signal and instrumental noise, m^{X} = s^{X} + n^{X}, where s is assumed to be a set of statistically isotropic and Gaussiandistributed random fields on the sky, indexed by pixel or sphericalharmonic indices (ℓm), with X = { T,E,B } selecting the appropriate intensity or polarization component. The signal fields s^{X} have auto and crosspower spectra ${\mathit{C}}_{\mathit{\ell}}^{\mathit{XY}}$ and a pixelspace covariance matrix $\mathrm{S}\mathrm{\left(}{\mathit{C}}_{\mathit{\ell}}\mathrm{\right)}\mathrm{=}\sum _{\mathit{\ell}\mathrm{=}\mathrm{2}}^{{\mathit{\ell}}_{\mathrm{max}}}\sum _{\mathit{XY}}{\mathit{C}}_{\mathit{\ell}}^{\mathit{XY}}{\mathrm{P}}_{\mathit{\ell}}^{\mathit{XY}}\mathit{.}$(1)Here we restrict the spectra to XY = { TT,EE,BB,TE }, with N_{side} = 16 pixelization, and ${\mathrm{P}}_{\mathit{\ell}}^{\mathit{XY}}$ is a beamweighted sum over (associated) Legendre polynomials. For temperature, the explicit expression is $\mathrm{(}{\mathrm{P}}_{\mathit{\ell}}^{\mathit{TT}}{\mathrm{)}}_{\mathit{i,j}}\mathrm{=}\frac{\mathrm{2}\mathit{\ell}\mathrm{+}\mathrm{1}}{\mathrm{4}\mathit{\pi}}\hspace{0.17em}{\mathit{B}}_{\mathit{\ell}}^{\mathrm{2}}\hspace{0.17em}{\mathit{P}}_{\mathit{\ell}}\mathrm{(}{{n\u0302}}_{\mathit{i}}\mathrm{\xb7}\hspace{0.17em}{{n\u0302}}_{\mathit{j}}\mathrm{)}\mathit{,}$(2)where $\stackrel{\u02c6}{{n}}{}_{\mathit{i}}$ is a unit vector pointing towards pixel i, B_{ℓ} is the product of the instrumental beam Legendre transform and the HEALPix pixel window, and P_{ℓ} is the Legendre polynomial of order ℓ; for corresponding polarization components, see, e.g., Tegmark & de OliveiraCosta (2001). The instrumental noise is also assumed to be Gaussian distributed, with a covariance matrix N that depends on the Planck detector sensitivity and scanning strategy, and the full data covariance is therefore M = S + N. With these definitions, the full likelihood expression reads $\mathrm{\mathcal{L}}\mathrm{\left(}{\mathit{C}}_{\mathit{\ell}}\mathrm{\right)}\mathrm{=}\mathrm{\mathcal{P}}\mathrm{\left(}{m}\mathrm{\right}{\mathit{C}}_{\mathit{\ell}}\mathrm{)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\mathit{\pi}\mathrm{}\mathrm{M}{\mathrm{}}^{\mathrm{1}\mathit{/}\mathrm{2}}}\mathrm{exp}\left(\mathrm{}\frac{\mathrm{1}}{\mathrm{2}}{{m}}^{\mathrm{T}}\hspace{0.17em}{\mathrm{M}}^{1}{m}\right)\mathit{,}$(3)where the conditional probability defines the likelihood ℒ(C_{ℓ}).
The computational cost of this expression is driven by the presence of the matrix inverse and determinant operations, both of which scale computationally as $\mathrm{\mathcal{O}}\mathrm{\left(}{\mathit{N}}_{\mathrm{pix}}^{\mathrm{3}}\mathrm{\right)}$. For this reason, the direct approach is only computationally feasible at large angular scales, where the number of pixels is low. In practice, we only analyse multipoles below or equal to ℓ_{cut} = 29 with this formalism, requiring maps with N_{side} = 16. Multipoles between ℓ_{cut} + 1 and ℓ_{max} are fixed to the bestfit ΛCDM spectrum when calculating S. This division between varying and fixed multipoles speeds up the evaluation of Eq. (3) through the ShermanMorrisonWoodbury formula and the related matrix determinant lemma, as described in Appendix B.1. This results in an orderofmagnitude speedup compared to the bruteforce computation.
2.2. Lowℓ temperature map and mask
Next, we consider the various data inputs that are required to evaluate the likelihood in Eq. (3), and we start our discussion with the temperature component. As in 2013, we employ the Commander algorithm for component separation. This is a Bayesian Monte Carlo method that either samples from or maximizes a global posterior defined by some explicit parametric data model and a set of priors. The data model adopted for the Planck 2015 analysis is described in detail in Planck Collaboration X (2016), and reads ${{s}}_{\mathit{\nu}}\mathrm{\left(}\mathit{\theta}\mathrm{\right)}\mathrm{=}{\mathit{g}}_{\mathit{\nu}}\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{{\mathit{N}}_{\mathrm{comp}}}{\mathrm{F}}_{\mathit{\nu}}^{\mathit{i}}\mathrm{\left(}{\mathit{\beta}}_{\mathit{i}}\mathit{,}{\mathrm{\Delta}}_{\mathit{\nu}}\mathrm{\right)}\hspace{0.17em}{{a}}_{\mathit{i}}\mathrm{+}\sum _{\mathit{j}\mathrm{=}\mathrm{1}}^{{\mathit{N}}_{\mathrm{template}}}{\mathit{T}}_{\mathit{\nu}}^{\mathit{j}}{{b}}_{\mathit{j}}^{\mathit{\nu}}\mathit{,}$(4)where θ denotes the full set of unknown parameters determining the signal at frequency ν. The first sum runs over N_{comp} independent astrophysical components including the CMB itself; a_{i} is the corresponding amplitude map for each component at some given reference frequency; β_{i} is a general set of spectral parameters for the same component; g_{ν} is a multiplicative calibration factor for frequency ν; Δ_{ν} is a linear correction of the bandpass central frequency; and the function ${\mathrm{F}}_{\mathit{\nu}}^{\mathit{i}}\mathrm{\left(}{\mathit{\beta}}_{\mathit{i}}\mathit{,}{\mathrm{\Delta}}_{\mathit{\nu}}\mathrm{\right)}$ gives the frequency dependence for component i (which can vary pixelbypixel and is hence most generally an N_{pix} × N_{pix} matrix). In the second sum, ${\mathit{T}}_{\mathit{\nu}}^{\mathit{j}}$ is one of a set of N_{template} correction template amplitudes, accounting for known effects such as monopole, dipole, or zodiacal light, with template maps ${{b}}_{\mathit{j}}^{\mathit{\nu}}$.
In 2013, only Planck observations between 30 and 353 GHz were employed in the corresponding fit. In the updated analysis, we broaden the frequency range considerably, by including the Planck 545 and 857 GHz channels, the 9year WMAP observations between 23 and 94 GHz (Bennett et al. 2013), and the Haslam et al. (1982) 408 MHz survey. We can then separate the lowfrequency foregrounds into separate synchrotron, freefree, and spinningdust components, as well as to constrain the thermal dust temperature pixelbypixel. In addition, in the updated analysis we employ individual detector and detectorset maps rather than coadded frequency maps, and this gives stronger constraints on both line emission (primarily CO) processes and bandpass measurement uncertainties. For a comprehensive discussion of all these results, we refer the interested reader to Planck Collaboration X (2016).
For the purposes of the present paper, the critical output from this process is the maximumposterior CMB temperature sky map, shown in the top panel of Fig. 1. This map is natively produced at an angular resolution of 1deg FWHM, determined by the instrumental beams of the WMAP 23 GHz and 408 MHz frequency channels. In addition, the Commander analysis provides a direct goodnessoffit measure per pixel in the form of the χ^{2} map shown in Planck Collaboration X (2016, Fig. 22). Thresholding this χ^{2} map results in a confidence mask that may be used for likelihood analysis, and the corresponding masked region is indicated in the top panel of Fig. 1 by a gray boundary. Both the map and mask are downgraded from their native HEALPix N_{side} = 256 pixel resolution to N_{side} = 16 before insertion into the likelihood code, and the map is additionally smoothed to an effective angular resolution of 440′ FWHM.
Fig. 1 Top: Commander CMB temperature map derived from the Planck 2015, 9year WMAP, and 408 MHz Haslam et al. observations, as described in Planck Collaboration X (2016). The gray boundary indicates the 2015 likelihood temperature mask, covering a total of 7% of the sky. The masked area has been filled with a constrained Gaussian realization. Middle: difference between the 2015 and 2013 Commander temperature maps. The masked region indicates the 2013 likelihood mask, removing 13% of the sky. Bottom: comparison of the 2013 (gray) and 2015 (black) temperature likelihood masks. 
Fig. 2 Top: comparison of the Planck 2013 (blue points) and 2015 (red points) posteriormaximum lowℓ temperature power spectra, as derived with Commander . Error bars indicate asymmetric marginal posterior 68% confidence regions. For reference, we also show the final 9year WMAP temperature spectrum in light gray points, as presented by Bennett et al. (2013); note that the error bars indicate symmetric Fisher uncertainties in this case. The dashed lines show the bestfit ΛCDM spectra derived from the respective data sets, including highmultipole and polarization information. Middle: difference between the 2015 and 2013 maximumposterior power spectra (solid black line). The gray shows the same difference after scaling the 2013 spectrum up by 2.4%. Dotted lines indicate the expected ± 1σ confidence region, accounting only for the sky fraction difference. Bottom: reduction in marginal error bars between the 2013 and 2015 temperature spectra; see main text for explicit definition. The dotted line shows the reduction expected from increased sky fraction alone. 
The middle panel of Fig. 1 shows the difference between the Planck 2015 and 2013 Commander maximumposterior maps, where the gray region now corresponds to the 2013 confidence mask. Overall, there are largescale differences at the 10 μK level at high Galactic latitudes, while at low Galactic latitudes there are a nonnegligible number of pixels that saturate the colour scale of ± 25 μK. These differences are well understood. First, the most striking red and blue largescale features at high latitudes are dominated by destriping errors in our 2013 analysis, due to bandpass mismatch in a few frequency channels effectively behaving as correlated noise during map making. As discussed in section 3 of Planck Collaboration X (2016) and illustrated in Fig. 2 therein, the most significant outliers have been removed from the updated 2015 analysis, and, consequently, the pattern is clearly visible from the difference map in Fig. 1. Second, the differences near the Galactic plane and close to the mask boundary are dominated by negative CO residuals near the Fan region, at Galactic coordinates (l,b) ≈ (110deg,20deg); by negative freefree residuals near the Gum nebula at (l,b) ≈ (260deg,15deg); and by thermal dust residuals along the plane. Such differences are expected because of the wider frequency coverage and improved foreground model in the new fit. In addition, the updated model also includes the thermal SunyaevZeldovich (SZ) effect near the Coma and Virgo clusters in the northern hemisphere, and this may be seen as a roughly circular patch near the Galactic north pole.
Overall, the additional frequency range provided by the WMAP and 408 MHz observations improves the component separation, and combining these data sets makes more sky effectively available for CMB analysis. The bottom panel of Fig. 1 compares the two χ^{2}based confidence masks. In total, 7% of the sky is removed by the 2015 confidence mask, compared with 13% in the 2013 version.
The top panel in Fig. 2 compares the marginal posterior lowℓ power spectrum, D_{ℓ} ≡ C_{ℓ}ℓ(ℓ + 1) / (2π), derived from the updated map and mask using the BlackwellRao estimator (Chu et al. 2005) with the corresponding 2013 spectrum (Like13). The middle panel shows their difference. The dotted lines indicate the expected variation between the two spectra, σ_{ℓ}, accounting only for their different sky fractions^{3}. From this, we can compute ${\mathit{\chi}}^{\mathrm{2}}\mathrm{=}\sum _{\mathit{\ell}\mathrm{=}\mathrm{2}}^{\mathrm{29}}{\left(\frac{{\mathit{D}}_{\mathit{\ell}}^{\mathrm{2015}}\mathrm{}{\mathit{D}}_{\mathit{\ell}}^{\mathrm{2013}}}{{\mathit{\sigma}}_{\mathit{\ell}}}\right)}^{\mathrm{2}}\mathit{,}$(5)and we find this to be 21.2 for the current data set. With 28 degrees of freedom, and assuming both Gaussianity and statistical independence between multipoles, this corresponds formally to a probabilitytoexceed (PTE) of 82%. According to these tests, the observed differences are consistent with random fluctuations due to increased sky fraction alone.
As discussed in Planck Collaboration I (2016), the absolute calibration of the Planck sky maps has been critically reassessed in the new release. The net outcome of this process was an effective recalibration of +1.2% in map domain, or +2.4% in terms of power spectra. The gray line in the middle panel of Fig. 2 shows the same difference as discussed above, but after rescaling the 2013 spectrum up by 2.4%. At the precision offered by these largescale observations, the difference is small, and either calibration factor is consistent with expectations.
Finally, the bottom panel compares the size of the statistical error bars of the two spectra, in the form of ${\mathit{r}}_{\mathit{\ell}}\mathrm{\equiv}\frac{{\mathrm{(}{\mathit{\sigma}}_{\mathit{\ell}}^{\mathrm{l}}\mathrm{+}{{\mathit{\sigma}}_{\mathit{\ell}}^{\mathrm{u}}}^{\mathrm{)}}}_{\mathrm{2013}}}{{\mathrm{(}{\mathit{\sigma}}_{\mathit{\ell}}^{\mathrm{l}}\mathrm{+}{{\mathit{\sigma}}_{\mathit{\ell}}^{\mathrm{u}}}^{\mathrm{)}}}_{\mathrm{2015}}}\mathrm{}\mathrm{1}\mathit{,}$(6)where ${\mathit{\sigma}}_{\mathit{\ell}}^{\mathrm{u}}$ and ${\mathit{\sigma}}_{\mathit{\ell}}^{\mathrm{l}}$ denote upper and lower asymmetric 68% error bars, respectively. Thus, this quantity measures the decrease in error bars between the 2013 and 2015 spectra, averaged over the upper and lower uncertainties. Averaging over 1000 ideal simulations and multipoles between ℓ = 2 and 29, we find that the expected change in the error bar due to sky fraction alone is 7%, in good agreement with the real data. Note that because the net uncertainty of a given multipole is dominated by cosmic variance, its magnitude depends on the actual power spectrum value. Thus, multipoles with a positive power difference between 2015 and 2013 tend to have a smaller uncertainty reduction than points with a negative power difference. Indeed, some multipoles have a negative uncertainty reduction because of this effect.
For detailed discussions and higherorder statistical analyses of the new Commander CMB temperature map, we refer the interested reader to Planck Collaboration X (2016) and Planck Collaboration XVI (2016).
2.3. 70 GHz polarization lowresolution solution
The likelihood in polarization uses only a subset of the full Planck polarization data, chosen to have wellcharacterized noise properties and negligible contribution from foreground contamination and unaccountedfor systematic errors. Specifically, we use data from the 70 GHz channel of the LFI instrument, for the full mission except for Surveys 2 and 4, which are conservatively removed because they stand as 3σ outliers in surveybased null tests (Planck Collaboration II 2016). While the reason for this behaviour is not completely understood, it is likely related to the fact that these two surveys exhibit the deepest minimum in the dipole modulation amplitude (Planck Collaboration II 2016; Planck Collaboration IV 2016), leading to an increased vulnerability to gain uncertainties and to contamination from diffuse polarized foregrounds.
To account for foreground contamination, the PlanckQ and U 70 GHz maps are cleaned using 30 GHz maps to generate a template for lowfrequency foreground contamination, and 353 GHz maps to generate a template for polarized dust emission (Planck Collaboration Int. XIX 2015; Planck Collaboration Int. XXX 2016; Planck Collaboration IX 2016). Linear polarization maps are downgraded from high resolution to N_{side} = 16 employing an inversenoiseweighted averaging procedure, without applying any smoothing (Planck Collaboration VI 2016).
The final cleaned Q and U maps, shown in Fig. 3, retain a fraction f_{sky} = 0.46 of the sky, masking out the Galactic plane and the “spur regions” to the north and south of the Galactic centre.
Fig. 3 Foregroundcleaned, 70 GHz Q (top) and U (bottom) maps used for the lowℓ polarization part of the likelihood. Each of the maps covers 46% of the sky. 
At multipoles ℓ< 30, we model the likelihood assuming that the maps follow a Gaussian distribution with known covariance, as in Eq. (3). For polarization, however, we use foregroundcleaned maps, explicitly taking into account the induced increase in variance through an effective noise correlation matrix.
To clean the 70 GHz Q and U maps we use a templatefitting procedure. Restricting m to the Q and U maps (i.e., m ≡ [Q,U]) we write ${m}\mathrm{=}\frac{\mathrm{1}}{\mathrm{1}\mathrm{}\mathit{\alpha}\mathrm{}\mathit{\beta}}\left({{m}}_{\mathrm{70}}\mathrm{}\mathit{\alpha}{{m}}_{\mathrm{30}}\mathrm{}\mathit{\beta}{{m}}_{\mathrm{353}}\right)\mathit{,}$(7)where m_{70}, m_{30}, and m_{353} are bandpasscorrected versions of the 70, 30, and 353 GHz maps (Planck Collaboration III 2016; Planck Collaboration VII 2016), and α and β are the scaling coefficients for synchrotron and dust emission, respectively. The latter can be estimated by minimizing the quantity ${\mathit{\chi}}^{\mathrm{2}}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{}\mathit{\alpha}\mathrm{}\mathit{\beta}{\mathrm{)}}^{\mathrm{2}}{{m}}^{\mathrm{T}}{\mathrm{C}}_{\mathrm{S}\mathrm{+}\mathrm{N}}^{1}{m}\mathit{,}$(8)where ${\mathrm{C}}_{\mathrm{S}\mathrm{+}\mathrm{N}}\mathrm{\equiv}\mathrm{(}\mathrm{1}\mathrm{}\mathit{\alpha}\mathrm{}\mathit{\beta}{\mathrm{)}}^{\mathrm{2}}\mathrm{\u27e8}{m}{{m}}^{\mathrm{T}}\mathrm{\u27e9}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{}\mathit{\alpha}\mathrm{}\mathit{\beta}{\mathrm{)}}^{\mathrm{2}}\mathrm{S}\mathrm{\left(}{\mathit{C}}_{\mathit{\ell}}\mathrm{\right)}\mathrm{+}{\mathrm{N}}_{\mathrm{70}}\mathit{.}$(9)Here N_{70} is the pure polarization part of the 70 GHz noise covariance matrix^{4} (Planck Collaboration VI 2016), and C_{ℓ} is taken as the Planck 2015 fiducial model (Planck Collaboration XIII 2016). We have verified that using the Planck 2013 model has negligible impact on the results described below. Minimization of the quantity in Eq. (8) using the form of the covariance matrix given in Eq. (9) is numerically demanding, since it would require inversion of the covariance matrix at every step of the minimization procedure. However, the signaltonoise ratio in the 70 GHz maps is relatively low, and we may neglect the dependence on the α and β of the covariance matrix in Eq. (8) using instead: ${\mathrm{C}}_{\mathrm{S}\mathrm{+}\mathrm{N}}\mathrm{=}\mathrm{S}\mathrm{\left(}{\mathit{C}}_{\mathit{\ell}}\mathrm{\right)}\mathrm{+}{\mathrm{N}}_{\mathrm{70}}\mathit{,}$(10)so that the matrix needs to be inverted only once. We have verified for a test case that accounting for the dependence on the scaling parameters in the covariance matrix yields consistent results. We find α = 0.063 and β = 0.0077, with 3σ uncertainties δ_{α} ≡ 3σ_{α} = 0.025 and δ_{β} ≡ 3σ_{β} = 0.0022. The bestfit values quoted correspond to a polarization mask using 46% of the sky and correspond to spectral indexes (with 2σ errors) n_{synch} = −3.16 ± 0.40 and n_{dust} = 1.50 ± 0.16, for synchrotron and dust emission respectively (see Planck Collaboration X 2016, for a definition of the foreground spectral indexes). To select the cosmological analysis mask, the following scheme is employed. We scale to 70 GHz both m_{30} and m_{353}, assuming fiducial spectral indexes n_{synch} = −3.2 and n_{dust} = 1.6, respectively. In this process, we do not include bandpass correction templates. From either rescaled template we compute the polarized intensities $\mathit{P}\mathrm{=}\sqrt{{\mathit{Q}}^{\mathrm{2}}\mathrm{+}{\mathit{U}}^{\mathrm{2}}}$ and sum them. We clip the resulting template at equally spaced thresholds to generate a set of 24 masks, with unmasked fractions in the range from 30% to 80% of the sky. Finally, for each mask, we estimate the bestfit scalings and evaluate the probability to exceed, $\mathrm{\mathcal{P}}\mathrm{(}{\mathit{\chi}}^{\mathrm{2}}\mathit{>}{\mathit{\chi}}_{\mathrm{0}}^{\mathrm{2}}\mathrm{)}$, where ${\mathit{\chi}}_{\mathrm{0}}^{\mathrm{2}}$ is the value achieved by minimizing Eq. (8). The f_{sky} = 43% processing mask is chosen as the tightest mask (i.e., the one with the greatest f_{sky}) satisfying the requirement (see Fig. 4). We use a slightly smaller mask (f_{sky} = 46%) for the cosmological analysis, which is referred to as the R1.50 mask in what follows.
Fig. 4 Upper panels: estimated bestfit scaling coefficients for synchrotron (α) and dust (β), for several masks, whose sky fractions are displayed along the bottom horizontal axis (see text). Lower panel: the probability to exceed, $\mathrm{\mathcal{P}}\mathrm{(}{\mathit{\chi}}^{\mathrm{2}}\mathit{>}{\mathit{\chi}}_{\mathrm{0}}^{\mathrm{2}}\mathrm{)}$. The red symbols identify the mask from which the final scalings are estimated, but note how the latter are roughly stable over the range of sky fractions. Choosing such a large “processing” mask ensures that the associated errors are conservative. 
We define the final polarization noise covariance matrix used in Eq. (3) as $\mathrm{N}\mathrm{=}\frac{\mathrm{1}}{\mathrm{(}\mathrm{1}\mathrm{}\mathit{\alpha}\mathrm{}\mathit{\beta}{\mathrm{)}}^{\mathrm{2}}}\mathrm{(}{\mathrm{N}}_{\mathrm{70}}\mathrm{+}{\mathit{\delta}}_{\mathit{\alpha}}^{\mathrm{2}}{{m}}_{\mathrm{30}}{{m}}_{\mathrm{30}}^{\mathrm{T}}\mathrm{+}{\mathit{\delta}}_{\mathit{\beta}}^{\mathrm{2}}{{m}}_{\mathrm{353}}{{{m}}_{\mathrm{353}}^{\mathrm{T}}}^{\mathrm{)}}\mathit{.}$(11)We use 3σ uncertainties, δ_{α} and δ_{β}, to define the covariance matrix, conservatively increasing the errors due to foreground estimation. We have verified that the external (column to row) products involving the foreground templates are subdominant corrections. We do not include further correction terms arising from the bandpass leakage error budget since they are completely negligible. Intrinsic noise from the templates also proved negligible.
2.4. LowℓPlanck power spectra and parameters
We use the foregroundcleaned Q and U maps derived in the previous section along with the Commander temperature map to derive angular power spectra. For the polarization part, we use the noise covariance matrix given in Eq. (11), while assuming only 1 μ diagonal regularization noise for temperature. Consistently, a white noise realization of the corresponding variance is added to the Commander map. By adding regularization noise, we ensure that the noise covariance matrix is numerically well conditioned.
For power spectra, we employ the BolPol code (Gruppuso et al. 2009), an implementation of the quadratic maximum likelihood (QML) power spectrum estimator (Tegmark 1997; Tegmark & de OliveiraCosta 2001). Figure 5 presents all five polarized power spectra. The errors shown in the plot are derived from the Fisher matrix. In the case of EE and TE we plot the Planck 2013 bestfit power spectrum model, which has an optical depth τ = 0.089, as derived from lowℓ WMAP9 polarization maps, along with the Planck 2015 best model, which has τ = 0.067 as discussed below^{5}. Since the EE power spectral amplitude scales with τ as τ^{2} (and TE as τ), the 2015 model exhibits a markedly lower reionization bump, which is a better description of Planck data. There is a 2.7σ outlier in the EE spectrum at ℓ = 9, not unexpected given the number of lowℓ multipole estimates involved.
Fig. 5 Polarized QML spectra from foregroundcleaned maps. Shown are the 2013 Planck bestfit model (τ = 0.089, dotdashed) and the 2015 model (τ = 0.067, dashed), as well as the 70 GHz noise bias computed from Eq. (11) (blue dotted). 
To estimate cosmological parameters, we couple the machinery described in Sect. 2.1 to cosmomc ^{6} (Lewis & Bridle 2002). We fix all parameters that are not sampled to their Planck 2015 ΛCDM bestfit value (Planck Collaboration XIII 2016) and concentrate on those that have the greatest effect at low ℓ: the reionization optical depth τ, the scalar amplitude A_{s}, and the tensortoscalar ratio r. Results are shown in Table 2 for the combinations (τ,A_{s}) and (τ,A_{s},r).
Parameters estimated from the lowℓ likelihood.
It is interesting to disentangle the cosmological information provided by lowℓ polarization from that derived from temperature. Lowℓ temperature mainly contains information on the combination A_{s}e^{− 2τ}, at least at multipoles corresponding to angular scales smaller than the scale subtended by the horizon at reionization (which itself depends on τ). The lowest temperature multipoles, however, are directly sensitive to A_{s}. On the other hand, largescale polarization is sensitive to the combination A_{s}τ^{2}. Thus, neither lowℓ temperature nor polarization can separately constrain τ and A_{s}. Combining temperature and polarization breaks the degeneracies and puts tighter constraints on these parameters.
In order to disentangle the temperature and polarization contributions to the constraints, we consider four versions of the lowresolution likelihood.
 1.
The standard version described above, which considers the fullset of T, Q, and U maps, along with their covariance matrix, and is sensitive to the TT, TE, EE, and BB spectra.
 2.
A temperatureonly version, which considers the temperature map and its regularization noise covariance matrix. It is only sensitive to TT.
 3.
A polarizationonly version, considering only the Q and U maps and the QQ, QU, and UU blocks of the covariance matrix. This is sensitive to the EE and BB spectra.
 4.
A mixed temperaturepolarization version, which uses the previous polarizationonly likelihood but multiplies it by the temperatureonly likelihood. This is different from the standard T,Q,U version in that it assumes vanishing temperaturepolarization correlations.
The posteriors derived from these four likelihood versions are displayed in Fig. 6. These plots show how temperature and polarization nicely combine to break the degeneracies and provide joint constraints on the two parameters. The degeneracy directions for cases (2) and (3) are as expected from the discussion above; the degeneracy in case (2) flattens for increasing values of τ because for such values the scale corresponding to the horizon at reionization is pulled forward to ℓ> 30. By construction, the posterior for case 4 must be equal to the product of the temperatureonly (2) and polarizationonly (3) posteriors. This is indeed the case at the level of the twodimensional posterior (see lower right panel of Fig. 6). It is not immediately evident in the onedimensional distributions because this property does not survive the final marginalization over the nonGaussian shape of the temperatureonly posterior. It is also apparent from Fig. 6 that EE and BB alone do not constrain τ. This is to be expected, and is due to the inverse degeneracy of τ with A_{s}, which is almost completely unconstrained without temperature information, and not to the lack of EE signal. By assuming a sharp prior 10^{9}A_{s}e^{− 2τ} = 1.88, corresponding to the best estimate obtained when also folding in the highℓ temperature information (Planck Collaboration XIII 2016), the polarizationonly analysis yields $\mathit{\tau}\mathrm{=}\mathrm{0.05}{\mathrm{1}}_{0.020}^{\mathrm{+}\mathrm{0.022}}$ (red dashed curve in Fig. 6). The latter bound does not differ much from having A_{s} constrained by including TT in the analysis, which yields $\mathit{\tau}\mathrm{=}\mathrm{0.05}{\mathrm{4}}_{0.021}^{\mathrm{+}\mathrm{0.023}}$ (green curves). Finally, the inclusion of nonvanishing temperaturepolarization correlations (blue curves) increases the significance of the τ detection at τ = 0.067 ± 0.023. We have also performed a threeparameter fit, considering τ, A_{s}, and r for all four likelihood versions described above, finding consistent results.
Fig. 6 Likelihoods for parameters from lowℓ data. Panels 1–3: onedimensional posteriors for log [10^{10}A_{s}], τ, and A_{s}e^{− 2τ} for the several subblocks of the likelihood, for cases 1 (blue), 2 (black), 3 (red), and 4 (green) – see text for definitions; dashed red is the same as case 3 but imposes a sharp prior 10^{9}A_{s}e^{− 2τ} = 1.88. Panel 4: twodimensional posterior for log [10^{10}A_{s}] and τ for the same data combinations; shading indicates the 68% and 95% confidence regions. 
2.5. Consistency analysis
Several tests have been carried out to validate the 2015 lowℓ likelihood. Mapbased validation and simple spectral tests are discussed extensively in Planck Collaboration IX (2016) for temperature, and in Planck Collaboration II (2016) for Planck 70 GHz polarization. We focus here on tests based on QML and likelihood analyses, respectively employing spectral estimates and cosmological parameters as benchmarks.
We first consider QML spectral estimates C_{ℓ} derived using BolPol . To test their consistency, we consider the following quantity: ${\mathit{\chi}}_{\mathrm{h}}^{\mathrm{2}}\mathrm{=}\sum _{\mathit{\ell}\mathrm{=}\mathrm{2}}^{{\mathit{\ell}}_{\mathrm{max}}}\mathrm{(}{\mathit{C}}_{\mathit{\ell}}\mathrm{}{\mathit{C}}_{\mathit{\ell}}^{\mathrm{th}}\mathrm{)}\hspace{0.17em}{\mathrm{M}}_{\mathit{\ell}{\mathit{\ell}}^{\mathrm{\prime}}}^{1}\hspace{0.17em}\mathrm{(}{\mathit{C}}_{\mathit{\ell}}\mathrm{}{\mathit{C}}_{\mathit{\ell}}^{\mathrm{th}}\mathrm{)}\mathit{,}$(12)where ${\mathrm{M}}_{\mathit{\ell}{\mathit{\ell}}^{\mathrm{\prime}}}\mathrm{=}\mathrm{\u27e8}\mathrm{(}{\mathit{C}}_{\mathit{\ell}}\mathrm{}{\mathit{C}}_{\mathit{\ell}}^{\mathrm{th}}\mathrm{)}\mathrm{(}{\mathit{C}}_{{\mathit{\ell}}^{\mathrm{\prime}}}\mathrm{}{\mathit{C}}_{{\mathit{\ell}}^{\mathrm{\prime}}}^{\mathrm{th}}\mathrm{)}\mathrm{\u27e9}$, ${\mathit{C}}_{\mathit{\ell}}^{\mathrm{th}}$ represents the fiducial Planck 2015 ΛCDM model, and the average is taken over 1000 signal and noise simulations. The latter were generated using the noise covariance matrix given in Eq. (11). We also use the simulations to sample the empirical distribution for ${\mathit{\chi}}_{\mathrm{h}}^{\mathrm{2}}$, considering both ℓ_{max} = 12 (shown in Fig. 7, along with the corresponding values obtained from the data) and ℓ_{max} = 30, for each of the six CMB polarized spectra. We report in Table 3 the empirical probability of observing a value of ${\mathit{\chi}}_{\mathrm{h}}^{\mathrm{2}}$ greater than for the data (hereafter, PTE). This test supports the hypothesis that the observed polarized spectra are consistent with Planck’s bestfit cosmological model and the propagated instrumental uncertainties. We verified that the low PTE values obtained for TE are related to the unusually high (but not intrinsically anomalous) estimates 9 ≤ ℓ ≤ 11, a range that does not contribute significantly to constraining τ. For spectra involving B, the fiducial model is null, making this, in fact, a null test, probing instrumental characteristics and data processing independent of any cosmological assumptions.
Fig. 7 Empirical distribution of ${\mathit{\chi}}_{\mathrm{h}}^{\mathrm{2}}$ derived from 1000 simulations, for the case ℓ_{max} = 12 (see text). Vertical bars reindicate the observed values. 
Empirical probability of observing a value of ${\mathit{\chi}}_{\mathit{h}}^{\mathrm{2}}$ greater than that calculated from the data.
In order to test the likelihood module, we first perform a 45deg rotation of the reference frame. This leaves the T map unaltered, while sending Q → −U and U → Q (and, hence, E → −B and B → E). The subblocks of the noise covariance matrix are rotated accordingly. We should not be able to detect a τ signal under these circumstances. Results are shown in Fig. 8 for all the full TQU and the TT+EE+BB subblock likelihoods presented in the previous section. Indeed, rotating polarization reduces only slightly the constraining power in τ for the TT+EE+BB case, suggesting the presence of comparable power in the latter two. On the other hand, τ is not detected at all when rotating the full T,Q,U set, which includes TE and TB. We interpret these results as further evidence that the TE signal is relevant for constraining τ, a result that cannot be reproduced by substituting TB for TE. These findings appear consistent with the visual impression of the lowℓ spectra of Fig. 5. We have also verified that our results stand when r is sampled.
Fig. 8 Posterior for τ for both rotated and unrotated likelihoods. The definition and colour convention of the datasets shown are the same as in the previous section (see Fig. 6), while solid and dashed lines distinguish the unrotated and rotated likelihood, respectively. 
As a final test of the 2015 Planck lowℓ likelihood, we perform a full endtoend Monte Carlo validation of its polarization part. For this, we use 1000 signal and noise full focal plane (FFP8) simulated maps (Planck Collaboration XII 2016), whose resolution has been downgraded to N_{side} = 16 using the same procedure as that applied to the data. We make use of a custommade simulation set for the Planck 70 GHz channel, which does not include Surveys 2 and 4. For each simulation, we perform the foregroundsubtraction procedure described in Sect. 2.3 above, deriving foregroundcleaned maps and covariance matrices, which we use to feed the lowℓ likelihood. As above, we sample only log [10^{10}A_{s}] and τ, with all other parameters kept to their Planck bestfit fiducial values. We consider two sets of polarized foreground simulations, with and without the instrumental bandpass mismatch at 30 and 70 GHz. To emphasize the impact of bandpass mismatch, we do not attempt to correct the polarization maps for bandpass leakage. This choice marks a difference from what is done to real data, where the correction is performed (Planck Collaboration II 2016); thus, the simulations that include the bandpass mismatch effect should be considered as a worstcase scenario. This notwithstanding, the impact of bandpass mismatch on estimated parameters is very small, as shown in Fig. 9 and detailed in Table 4. Even without accounting for bandpass mismatch, the bias is at most 1 / 10 of the final 1σ error estimated from real data posteriors. The Monte Carlo analysis also enables us to validate the (Bayesian) confidence intervals estimated by cosmomc on data by comparing their empirical counterparts observed from the simulations. We find excellent agreement (see Table 4).
Fig. 9 Empirical distribution of the mean estimated values for log [10^{10}A_{s}] (top) and τ (bottom), derived from 1000 FFP8 simulations (see text). For each simulation, we perform a full endtoend run, including foreground cleaning and parameter estimation. Blue bars refer to simulations that do not include the instrumental bandpass mismatch, while red bars do. The violet bars flag the overlapping area, while the vertical black lines show the input parameters. We note that the (uncorrected) bandpass mismatch effect hardly changes the estimated parameters. 
The validation described above only addresses the limited number of instrumental systematic effects that are modelled in the FFP8 simulations, i.e., the bandpass mismatch. Other systematics may in principle affect the measurement of polarization at large angular scales. To address this issue, we have carried out a detailed analysis to quantify the possible impact of LFIspecific instrumental effects in the 70 GHz map (see Planck Collaboration III 2016, for details). Here we just report the main conclusion of that analysis, which estimates the final bias on τ due to all known instrumental systematics to be at most 0.005, i.e., about 0.25σ, well below the final error budget.
Statistics for the empirical distribution of estimated cosmological parameters from the FFP8 simulations.
Scalings for synchrotron (α) and dust (β) obtained for WMAP, when WMAP K band and Planck 353 GHz data are used as templates.
2.6. Comparison with WMAP9 polarization cleaned with Planck 353 GHz
In Like13, we attempted to clean the WMAP9 low resolution maps using a preliminary version of Planck 353 GHz polarization. This resulted in an approximately 1σ shift towards lower values of τ, providing the first evidence based on CMB observations that the WMAP bestfit value for the optical depth may have been biased high. We repeat the analysis here with the 2015 Planck products. We employ the procedure described in Bennett et al. (2013), which is similar to that described above for Planck 2015. However, in contrast to the Planck 70 GHz foreground cleaning, we do not attempt to optimize the foreground mask based on a goodnessoffit analysis, but stick to the processing and analysis masks made available by the WMAP team. WMAP’s P06 mask is significantly smaller than the 70 GHz mask used in the Planck likelihood, leaving 73.4% of the sky. Specifically we minimize the quadratic form of Eq. (8), separately for the Ka, Q, and V channels from the WMAP9 release, but using WMAP9’s own K channel as a synchrotron tracer rather than Planck 30 GHz^{7}. The purpose of the latter choice is to minimize the differences with respect to WMAP’s own analysis. However, unlike the WMAP9 native likelihood products, which operate at N_{side} = 8 in polarization, we use N_{side} = 16 in Q and U, for consistency with the Planck analysis. The scalings we find are consistent with those from WMAP (Bennett et al. 2013) for α in both Ka and Q. However, we find less good agreement for the higherfrequency V channel, where our scaling is roughly 25% lower than that reported in WMAP’s own analysis^{8}. We combine the three cleaned channels in a noiseweighted average to obtain a threeband map and an associated covariance matrix.
We evaluate the consistency of the lowfrequency WMAP and Planck 70 GHz lowℓ maps. Restricting the analysis to the intersection of the WMAP P06 and Planck R1.50 masks (f_{sky} = 45.3%), we evaluate halfsum and halfdifference Q and U maps. We then compute the quantity ${\mathit{\chi}}_{\mathrm{sd}}^{\mathrm{2}}\mathrm{=}{{m}}^{\mathrm{T}}{\mathrm{N}}^{1}{m}$ where m is either the halfsum or the halfdifference [Q,U] combination and N is the corresponding noise covariance matrix. Assuming that ${\mathit{\chi}}_{\mathrm{sd}}^{\mathrm{2}}$ is χ^{2} distributed with 2786 degrees of freedom we find a $\mathrm{PTE}\mathrm{(}{\mathit{\chi}}^{\mathrm{2}}\mathit{>}{\mathit{\chi}}_{\mathrm{sd}}^{\mathrm{2}}\mathrm{)}\mathrm{=}\mathrm{1.3}\mathrm{\times}{\mathrm{10}}^{5}$ (reduced χ^{2} = 1.116) for the halfsum, and PTE = 0.84 (reduced χ^{2} = 0.973) for the halfdifference. This strongly suggests that the latter is consistent with the assumed noise, and that the common signal present in the halfsum map is wiped out in the difference.
We also produce noiseweighted sums of the lowfrequency WMAP and Planck 70 GHz lowresolution Q and U maps, evaluated in the union of the WMAP P06 and Planck R1.50 masks (f_{sky} = 73.8%). We compute BolPol spectra for the noiseweighted sum and halfdifference combinations. These EE, TE, and BB spectra are shown in Fig. 10 and are evaluated in the intersection of the P06 and R1.50 masks. The spectra also support the hypothesis that there is a common signal between the two experiments in the typical multipole range of the reionization bump. In fact, considering multipoles up to ℓ_{max} = 12 we find an empirical PTE for the spectra of the halfdifference map of 6.8% for EE and 9.5% for TE, derived from the analysis of 10000 simulated noise maps. Under the same hypothesis, but considering the noiseweighted sum, the PTE for EE drops to 0.8%, while that for TE is below the resolution allowed by the simulation set (PTE < 0.1%). The BB spectrum, on the other hand, is compatible with a null signal in both the noiseweighted sum map (PTE = 47.5%) and the halfdifference map (PTE = 36.6%).
Fig. 10 BolPol spectra for the noiseweighted sum (black) and halfdifference (red) WMAP and Planck combinations. The temperature map employed is always the Commander map described in Sect. 2.2 above. The fiducial model shown has τ = 0.065. 
We use the Planck and WMAP map combinations to perform parameter estimates from lowℓ data only. We show here results from sampling log [10^{10}A_{s}], τ, and the tensortoscalar ratio r, with all other parameters kept to the Planck 2015 best fit (the case with r = 0 produces similar results). Figure 11 shows the posterior probability for τ for several Planck and WMAP combinations. They are all consistent, except the Planck and WMAP halfdifference case, which yields a null detection for τ – as it should. As above, we always employ the Commander map in temperature. Table 6 gives the mean values for the sampled parameters, and for the derived parameters z_{re} (mean redshift of reionization) and A_{s}e^{− 2τ}. Results from a joint analysis of the WMAPbased lowℓ polarization likelihoods presented here and the Planck highℓ likelihood are discussed in Sect. 5.7.1.
Fig. 11 Posterior probabilities for τ from the WMAP (cleaned with Planck 353 GHz as a dust template) and Planck combinations listed in the legend. Results are presented for the noiseweighted sum both in the union and the intersection of the two analysis masks. The halfdifference map is consistent with a null detection, as expected. 
Selected parameters estimated from the lowℓ likelihood, for Planck, WMAP and their noiseweighted combination.
3. Highmultipole likelihood
At high multipoles (ℓ> 29), as in Like13, we use a likelihood function based on pseudoC_{ℓ}s calculated from Planck HFI data, as well as further parameters describing the contribution of foreground astrophysical emission and instrumental effects (e.g., calibration, beams). Aside from the data themselves, the main advances over 2013 include the use of highℓ polarization information along with more detailed models of foregrounds and instrumental effects.
Section 3.1 introduces the highℓ statistical description, Sect. 3.2 describes the data we use, Sects. 3.3 and 3.4 describe foreground and instrumental modelling, and Sect. 3.5 describes the covariance matrix between multipoles and spectra. Section 3.6 validates the overall approach on realistic simulations, while Sect. 3.7 addresses the question of the potential impact of lowlevel instrumental systematics imperfectly corrected by the DPC processing. The reference results generated with the high multipole likelihood are described in Sect. 3.8. A detailed assessment of these results is presented in Sect. 4.
3.1. Statistical description
Assuming a Gaussian distribution for the CMB temperature anisotropies and polarization, all of the statistical information contained in the Planck maps can be compressed into the likelihood of the temperature and polarization auto and crosspower spectra. In the case of a perfect CMB observation of the full sky (with spatially uniform noise and isotropic beamsmearing), we know the joint distribution of the empirical temperature and polarization power spectra and can build an exact likelihood, which takes the simple form of an inverse Wishart distribution, uncorrelated between multipoles. For a single power spectrum (i.e., ignoring polarization and temperature crossspectra between detectors) the likelihood for each multipole ℓ simplifies to an inverse χ^{2} distribution with 2ℓ + 1 degrees of freedom. At high enough ℓ, the central limit theorem ensures that the shape of the likelihood is very close to that of a Gaussian distributed variable. This remains true for the inverse Wishart generalization to multiple spectra, where, for each ℓ, the shape of the joint spectra and crossspectra likelihood approaches that of a correlated Gaussian (Hamimeche & Lewis 2008; Elsner & Wandelt 2012). In the simple fullsky case, the correlations are easy to compute (Hamimeche & Lewis 2008), and only depend on the theoretical CMB TT, TE, and EE spectra. For small excursions around a fiducial cosmology, as is the case here given the constraining power of the Planck data, one can show that computing the covariance matrix at a fiducial model is sufficient (Hamimeche & Lewis 2008).
The data, however, differ from the idealized case. In particular, foreground astrophysical processes contribute to the temperature and polarization maps. As we see in Sect. 3.3, the main foregrounds in the frequency range we use are emission from dust in our Galaxy, the clustered and Poisson contributions from the cosmic infrared background (CIB), and radio point sources. Depending on the scale and frequency, foreground emission can be a significant contribution to the data, or even exceed the CMB. This is particularly true for dust near the Galactic plane, and for the strongest point sources. We excise the most contaminated regions of the sky (see Sect. 3.2.2). The remaining foreground contamination is taken into account in our model, using the fact that CMB and foregrounds have different emission laws; this enables them to be separated while estimating parameters.
Foregrounds also violate the Gaussian approximation assumed above. The dust distribution, in particular, is clearly nonGaussian. Following Like13, however, we assume that outside the masked regions we can neglect nonGaussian features and assume that, as for the CMB, all the relevant statistical information about the foregrounds is encoded in the spatial power spectra. This assumption is verified to be sufficient for our purposes in Sect. 3.6, where we assess the accuracy of the cosmological parameter constraints in realistic Monte Carlo simulations that include databased (nonGaussian) foregrounds.
Cutting out the foregroundcontaminated regions from our maps biases the empirical power spectrum estimates. We debias them using the PolSpice ^{9} algorithm (Chon et al. 2004) and, following Like13, we take the correlation between multipoles induced by the mask and debiasing into account when computing our covariance matrix. The maskedsky covariance matrix is computed using the equations in Like13, which are extended to the case of polarization in Appendix C.1.1. Those equations also take into account the inhomogeneous distribution of coloured noise on the sky using a heuristic approach. The approximation of the covariance matrix that can be obtained from those equations is only valid for some specific mask properties, and for high enough multipoles. In particular, as discussed in Appendix C.1.4, correlations induced by point sources cannot be faithfully described in our approximation. Similarly, Monte Carlo simulations have shown that our analytic approximation loses accuracy around ℓ = 30. We correct for both of those effects using empirical estimates from Monte Carlo simulations. The computation of the covariance matrix requires knowledge of both the CMB and foreground power spectra, as well as the map characteristics (beams, noise, sky coverage). The CMB and foreground power spectra are obtained iteratively from previous, less accurate versions of the likelihood.
At this stage, we would thus construct our likelihood approximation by compressing all of the individual Planck detector data into maskcorrected (pseudo) crossspectra, and build a grand likelihood using these spectra and the corresponding analytical covariance matrix: $\mathrm{}\mathrm{ln}\mathrm{\mathcal{L}}\mathrm{\left(}{C\u0302}\mathrm{\right}{C}\mathrm{\left(}\mathit{\theta}\mathrm{\right)}\mathrm{)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{\left[{C\u0302}\mathrm{}{C}\mathrm{\left(}\mathit{\theta}\mathrm{\right)}\right]}^{\mathrm{T}}{\mathrm{C}}^{1}\left[{C\u0302}\mathrm{}{C}\mathrm{\left(}\mathit{\theta}\mathrm{\right)}\right]\mathrm{+}\mathrm{const}\mathit{.}\mathit{,}$(13)where Ĉ is the data vector, C(θ) is the model with parameters θ, and C is the covariance matrix. This formalism enables us to separately marginalize over or condition upon different components of the model vector, separately treating cases such as individual frequencydependent spectra, or temperature and polarization spectra. Obviously, Planck maps at different frequencies have different constraining powers on the underlying CMB, and following Like13 we use this to impose and assess various cuts to keep only the most relevant data.
We therefore consider only the three best CMB Planck channels, i.e., 100 GHz, 143 GHz, and 217 GHz, in the multipole range where they have significant CMB contributions and low enough foreground contamination after masking; we therefore did not directly include the adjacent channels at 70 GHz and 350 GHz in the analysis. In particular, including the 70 GHz data would not bring much at large scales where the results are already cosmic variance limited, and would entail additional complexity in foreground modelling (synchrotron at large scales, additional radio sources excisions at small scales). The cuts in multipole ranges is be described in detail in Sect. 3.2.4. Further, in order to achieve a significant reduction in the covariance matrix size (and computation time), we compress the data vector (and accordingly the covariance matrix), both by coadding the individual detectors for each frequency and by binning the combined power spectra. We also coadd the two different TE and ET interfrequency crossspectra into a single TE spectrum for each pair of frequencies. This compression is lossless in the case without foregrounds. The exact content of the data vector is discussed in Sect. 3.2.
The model vector C(θ) must represent the content of the data vector. It can be written schematically as $\begin{array}{ccc}& & {{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}}_{\mathit{\ell}}\mathrm{\left(}\mathit{\theta}\mathrm{\right)}\mathrm{=}{{\mathit{M}}_{\mathit{ZW,\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{inst}}\mathrm{\right)}{{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{ZW,}\mathrm{sky}}}_{\mathit{\ell}}\mathrm{\left(}\mathit{\theta}\mathrm{\right)}\mathrm{+}{{\mathit{N}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{inst}}\mathrm{\right)}\mathit{,}\\ & & \end{array}$(14)where ${{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}}_{\mathit{\ell}}\mathrm{\left(}\mathit{\theta}\mathrm{\right)}$ is the element of the model vector corresponding to the multipole ℓ of the XY crossspectra (X and Y being either T or E) between the pair of frequencies ν and ν′. This element of the model originates from the sum of the microwave emission of the sky, i.e., the CMB (C^{ZW,cmb}_{ℓ}(θ)) which does not depend of the pair of frequencies (all maps are in units of K_{cmb}), and foreground (${{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{ZW,}\mathrm{fg}}}_{\mathit{\ell}}\mathrm{\left(}\mathit{\theta}\mathrm{\right)}$). Section 3.3 describes the foreground modelling. The mixing matrix ${{\mathit{M}}_{\mathit{ZW,\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{inst}}\mathrm{\right)}$ accounts for imperfect calibration, imperfect beam correction, and possible leakage between temperature and polarization. It does depend on the pair of frequencies and can depend on the multipole^{10} when accounting for imperfect beams and leakages. Finally, the noise term ${{\mathit{N}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{inst}}\mathrm{\right)}$ accounts for the possible correlated noise in the XY crossspectra for the pair of frequencies ν × ν′. Sections 3.2.3 and 3.4 describe our instrument model.
3.2. Data
The data vector Ĉ in the likelihood equation (Eq. (13)) is constructed from concatenated temperature and polarization components, ${C\u0302}\mathrm{=}\mathrm{\left(}{{C\u0302}}^{\mathit{TT}}\mathit{,}{{C\u0302}}^{\mathit{EE}}\mathit{,}{{C\u0302}}^{\mathit{TE}}\mathrm{\right)}\mathit{,}$(15)which, in turn, comprise the following frequencyaveraged spectra: $\begin{array}{ccc}{{C\u0302}}^{\mathit{TT}}& \mathrm{=}& \mathrm{(}{{C\u0302}}_{\mathrm{100}\mathrm{\times}\mathrm{100}}^{\mathit{TT}}\mathit{,}{{C\u0302}}_{\mathrm{143}\mathrm{\times}\mathrm{143}}^{\mathit{TT}}\mathit{,}{{C\u0302}}_{\mathrm{143}\mathrm{\times}\mathrm{217}}^{\mathit{TT}}\mathit{,}{{{C\u0302}}_{\mathrm{217}\mathrm{\times}\mathrm{217}}^{\mathit{TT}}}^{\mathrm{)}}\\ {{C\u0302}}^{\mathit{EE}}& \mathrm{=}& \mathrm{(}{{C\u0302}}_{\mathrm{100}\mathrm{\times}\mathrm{100}}^{\mathit{EE}}\mathit{,}{{C\u0302}}_{\mathrm{100}\mathrm{\times}\mathrm{143}}^{\mathit{EE}}\mathit{,}{{C\u0302}}_{\mathrm{100}\mathrm{\times}\mathrm{217}}^{\mathit{EE}}\mathit{,}{{C\u0302}}_{\mathrm{143}\mathrm{\times}\mathrm{143}}^{\mathit{EE}}\mathit{,}{{C\u0302}}_{\mathrm{143}\mathrm{\times}\mathrm{217}}^{\mathit{EE}}\mathit{,}{{{C\u0302}}_{\mathrm{217}\mathrm{\times}\mathrm{217}}^{\mathit{EE}}}^{\mathrm{)}}\\ {{C\u0302}}^{\mathit{TE}}& \mathrm{=}& \mathrm{(}{{C\u0302}}_{\mathrm{100}\mathrm{\times}\mathrm{100}}^{\mathit{TE}}\mathit{,}{{C\u0302}}_{\mathrm{100}\mathrm{\times}\mathrm{143}}^{\mathit{TE}}\mathit{,}{{C\u0302}}_{\mathrm{100}\mathrm{\times}\mathrm{217}}^{\mathit{TE}}\mathit{,}{{C\u0302}}_{\mathrm{143}\mathrm{\times}\mathrm{143}}^{\mathit{TE}}\mathit{,}{{C\u0302}}_{\mathrm{143}\mathrm{\times}\mathrm{217}}^{\mathit{TE}}\mathit{,}{{{C\u0302}}_{\mathrm{217}\mathrm{\times}\mathrm{217}}^{\mathit{TE}}}^{\mathrm{)}}\mathit{.}\end{array}$The TT data selection is very similar to Like13. We still discard the 100 × 143 and 100 × 217 crossspectra in their entirety. They contain little extra information about the CMB, as they are strongly correlated with the high S/N maps at 143 and 217 GHz. Including them, in fact, would only give information about the foreground contributions in these crossspectra, at the expense of a larger covariance matrix with increased condition number. In TE and EE, however, the situation is different since the overall S/N is significantly lower for all spectra, so a foreground model of comparatively low complexity can be used and it is beneficial to retain all the available crossspectra.
Detector sets used to make the maps for this analysis.
We obtain cross power spectra at the frequencies ν × ν′ using weighted averages of the individual beamdeconvolved, maskcorrected halfmission (HM) map power spectra, ${\mathit{C\u0302}\begin{array}{c}\mathit{XY}\\ \mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}\end{array}}_{\mathit{\ell}}\mathrm{=}\sum _{\mathrm{\left(}\mathit{i,j}\mathrm{\right)}\mathrm{\in}\mathrm{\left(}\mathit{\nu ,}{\mathit{\nu}}^{\mathrm{\prime}}\mathrm{\right)}}{{\mathit{w}}_{\mathit{i,j}}^{\mathit{XY}}}_{\mathit{\ell}}\mathrm{\times}{\mathit{C\u0302}\begin{array}{c}\mathit{XY}\\ \mathit{i,j}\end{array}}_{\mathit{\ell}}\mathit{,}$(19)where XY ∈ { TT,TE,EE }, and ${{\mathit{w}}_{\mathit{i,j}}^{\mathit{XY}}}_{\mathit{\ell}}$ is the multipoledependent inversevariance weight for the detectorset map combination (i,j), derived from its covariance matrix (see Sect. 3.5). For XY = TE, we further add the ET power spectra of the same frequency combination to the sum of Eq. (19); i.e., the average includes the correlation of temperature information from detectorset i and polarization information of detectorset j and vice versa.
We construct the Planck highmultipole likelihood solely from the HFI channels at 100, 143, and 217 GHz. These perform best as they have high S/N combined with manageably low foreground contamination. As in Like13, we only employ 70 GHz LFI data for crosschecks (in the highℓ regime), while the HFI 353 GHz and 545 GHz maps are used to determine the dust model.
3.2.1. Detector combinations
Table 7 summarizes the main characteristics of individual HFI detector sets used in the construction of the likelihood function. As discussed in Sect. 3.1, the likelihood does not use the crossspectra from individual detectorset maps; instead, we first combine all those contributing at each frequency to form weighted averages. As in 2013, we disregard all autopowerspectra as the precision required to remove their noise bias is difficult to attain and even small residuals may hamper a robust inference of cosmological parameters (Like13).
In 2015, the additional data available from fullmission observations enables us to construct nearly independent fullsky maps from the first and the second halves of the mission duration. We constructed crossspectra by crosscorrelating the two halfmission maps, ignoring the halfmission autospectra at the expense of a very small increase in the uncertainties. This differs from the procedure used in 2013, when we estimated crossspectra between detectors or detectorsets, and has the advantage of minimizing possible contributions from systematic effects that are correlated in the time domain.
The main motivation for this change from 2013 is that the correlated noise between detectors (at the same or different frequencies) is no longer small enough to be neglected (see Sect. 3.4.4). And while the correction for the “feature” around ℓ = 1800, which was (correctly) attributed to residual ^{4}HeJT cooler lines in 2013 (Planck Collaboration VI 2014), has been improved in the 2015 TOI processing pipeline (Planck Collaboration VII 2016), crossspectra between the two halfmission periods can help to suppress timedependent systematics, as argued by Spergel et al. (2015). Still, in order to enable further consistency checks, we also build a likelihood based on crossspectra between fullmission detectorset maps, applying a correction for the effect of correlated noise. The result illustrates that not much sensitivity is lost with halfmission crossspectra (see the whisker labelled “DS” in Figs. 35, 36, and C.10).
3.2.2. Masks
Temperature and polarization masks are used to discard areas of the sky that are strongly contaminated by foreground emission. The choice of masks is a tradeoff between maximizing the sky coverage to minimize sample variance, and the complexity and potentially insufficient accuracy of the foreground model needed in order to deal with regions of stronger foreground emission. The masks combine a Galactic mask, excluding mostly low Galacticlatitude regions, and a pointsource mask. We aim to maximize the sky fraction with demonstrably robust results (see Sect. 4.1.2 for such a test).
Temperature masks are obtained by merging the apodized Galactic, CO, and pointsource masks described in Appendix A. In polarization, as discussed in Planck Collaboration Int. XXX (2016), even at 100 GHz foregrounds are dominated by the dust emission, so for polarization analysis we employ the same apodized Galactic masks as we use for temperature, because they are also effective in reducing fluctuations in polarized dust emission at the relatively small scales covered by the highℓ likelihood (contrary to the large Galactic scales), but we do not include a compactsource mask because polarized emission from extragalactic foregrounds is negligible at the frequencies of interest (Naess et al. 2014; Crites et al. 2015).
Masks used for the highℓ analysis.
Table 8 lists the masks used in the likelihood at each frequency channel. We refer throughout to the masks by explicitly indicating the percentage of the sky they retain: T66, T57, T47 for temperature and P70, P50, P41 for polarization. G70, G60, G50, and G41 denote the apodized Galactic masks. As noted above, the apodized P70, P50, and P41 polarization masks are identical to the G70, G50, and G41 Galactic masks.
The Galactic masks are obtained by thresholding the smoothed, CMBcleaned 353 GHz map at different levels to obtain different sky coverage. All of the Galactic masks are apodized with a 4.̊71 FWHM (σ = 2°) Gaussian window function to localize the mask power in multipole space. In order to adapt to the different relative strengths of signal, noise, and foregrounds, we use different sky coverage for temperature and polarization, ranging in effective sky fraction from 41% to 70% depending on the frequency. The Galactic masks are shown in Fig. 12.
Fig. 12 Top: apodized Galactic masks: G41 (blue), G50 (purple), G60 (red), and G70 (orange); these are identical to the polarization masks P41 (used at 217 GHz), P50 (143 GHz), P70 (100 GHz). Bottom: extragalacticobject masks for 217 GHz (purple), 143 GHz (red), and 100 GHz (orange); the CO mask is shown in yellow. 
For temperature we use the G70, G60, and G50 Galactic masks at (respectively) 100 GHz, 143 GHz, and 217 GHz. For the first release of Planck cosmological data (Planck Collaboration XI 2016) we made more conservative choices of masks than in this paper (f_{sky} = 49%,31%, and 31% at, respectively, 100, 143, and 217 GHz, to be compared to f_{sky} = 66%,57%, and 47%). Admitting more sky into the analysis requires a thorough assessment of the robustness of the foreground modelling, and in particular of the Galactic dust model (see Sect. 3.3). When retaining more sky close to the Galactic plane at 100 GHz, maps start to show contamination by CO emission that also needs to be masked. This was not the case in the Planck 2013 analysis. We therefore build a CO mask as described in Appendix A. Once we apply this mask, the residual foreground at 100 GHz is consistent with dust and there is no evidence for other anisotropic foreground components, as shown by the doubledifference spectra between the 100 GHz band and the 143 GHz band where there is no CO line (Sect. 3.3.1). We also use the CO mask at 217 GHz, although we expect it to have a smaller impact since at this frequency CO emission is fainter and the applied Galactic cut wider. The extragalactic “point” source masks in fact include both point sources and extended objects; they are used only with the temperature maps. Unlike in 2013, we use a different source mask for each frequency, taking into account different source selection and beam sizes (see Appendix A). Both the CO and the extragalactic object masks are apodized with a 30′ FWHM Gaussian window function. The different extragalactic masks, as well as the CO mask, are shown in Fig. 12. The resulting mask combinations for temperature are shown in Fig. 13.
Fig. 13 Top to bottom: temperature masks for 100 GHz (T66), 143 GHz (T57) and 217 GHz (T47). The colour scheme is the same as in Fig. 12. 
3.2.3. Beam and transfer functions
The response to a point source is given by the combination of the optical response of the Planck telescope and feedhorns (the optical beam) with the detector time response and electronic transfer function (whose effects are partially removed during the TOI processing). This response pattern is referred to as the “scanning beam”. It is measured on planet transits (Planck Collaboration VIII 2016). However, the value in any pixel resulting from the mapmaking operation comes from a sum over many different elements of the timeline, each of which has hit the pixel in a different location and from a different direction. Furthermore, combined maps are weighted sums of individual detectors. All of these result in an “effective beam” window function encoding the multiplicative effect on the angular power spectrum. We note that beam noncircularity and the nonuniform scanning of the sky create differences between auto and crossdetector beam window functions (Planck Collaboration VII 2014).
In the likelihood analysis, we correct for this by using the effective beam window function corresponding to each specific spectrum; the window functions are calculated with the QuickBeam pipeline, except for one of the alternative analyses (Xfaster ) which relied on the FEBeCoP window functions (see Planck Collaboration VII 2016; Planck Collaboration VII 2014, and references therein for details of these two codes). In Sect. 3.4.3 we discuss the model of their uncertainties.
3.2.4. Multipole range
Following the approach taken in Like13, we use specifically tailored multipole ranges for each frequencypair spectrum. In general, we exclude multipoles where either the S/N is too low for the data to contribute significant constraints on the CMB, or the level of foreground contamination is so high that the foreground contribution to the power spectra cannot be modelled sufficiently accurately; high foreground contamination would also require us to consider possible nonGaussian terms in the estimation of the likelihood covariance matrix. We impose the same ℓ cuts for the detectorset and halfmission likelihoods for comparison, and we exclude the ℓ> 1200 range for the 100 × 100 spectra, where the correlated noise correction is rather uncertain.
Fig. 14 Unbinned S/N per frequency for TT (solid blue, for those detector combinations used in the estimate of the TT spectrum), EE (solid red), and TE (solid green). The horizontal orange line corresponds to S/N = 1. The dashed lines indicate the S/N in a cosmicvariancelimited case, obtained by forcing the instrumental noise terms to zero when calculating the power spectrum covariance matrix. The dotted lines indicate the cosmicvariancelimited case computed with the approximate formula of Eq. (20). 
Fig. 15 Planck power spectra (not yet corrected for foregrounds) and data selection. The coloured tick marks indicate the ℓrange of the crossspectra included in the Planck likelihood. Although not used in the highℓ likelihood, the 70 GHz spectra at ℓ> 29 illustrate the consistency of the data. The grey line indicates the bestfit Planck 2015 spectrum. The TE and EE plots have a logarithmic horizontal scale for ℓ< 30. 
Figure 14 shows the unbinned S/N per frequency for TT, EE, and TE, where the signal is given by the frequencydependent CMB and foreground power spectra, while the noise term contains contributions from cosmic variance and instrumental noise and is given by the diagonal elements of the powerspectrum covariance matrix. The figure also shows the S/N assuming only cosmic variance (CV) in the noise term, obtained either by a full calculation of the covariance matrix with instrumental noise set to zero, or using the approximation $\begin{array}{ccc}& & {\mathit{\sigma}}_{\mathit{CV}}^{\mathrm{\left\{}\mathit{TT,EE}\mathrm{\right\}}}\mathrm{=}\sqrt{\left(\frac{\mathrm{2}}{\mathrm{(}\mathrm{2}\mathit{\ell}\mathrm{+}\mathrm{1}\mathrm{)}{\mathit{f}}_{\mathrm{sky}}}\right){\mathrm{(}{{\mathit{C}}_{\mathit{\ell}}^{\mathrm{\left\{}\mathit{TT,EE}\mathrm{\right\}}}}^{\mathrm{)}}}^{\mathrm{2}}}\\ & & {\mathit{\sigma}}_{\mathit{CV}}^{\mathit{TE}}\mathrm{=}\sqrt{\left(\frac{\mathrm{2}}{\mathrm{(}\mathrm{2}\mathit{\ell}\mathrm{+}\mathrm{1}\mathrm{)}{\mathit{f}}_{\mathrm{sky}}}\right)\frac{{\mathrm{(}{{\mathit{C}}_{\mathit{\ell}}^{\mathit{TE}}}^{\mathrm{)}}}^{\mathrm{2}}\mathrm{+}{\mathit{C}}_{\mathit{\ell}}^{\mathit{TT}}{\mathit{C}}_{\mathit{\ell}}^{\mathit{EE}}}{\mathrm{2}}}\mathrm{\xb7}\end{array}$(20)(see e.g. Percival & Brown 2006).
This figure illustrates that the multipole cuts we apply ensure that the  S/N  ≳ 1. The TT multipole cuts are similar to those adopted in Like13. While otherwise similar to the 2013 likelihood, the revised treatment of dust in the foreground model enables the retention of multipoles ℓ< 500 of the 143 × 217 and 217 × 217 GHz TT spectra. As discussed in detail in Sect. 3.3.1, we are now marginalizing over a free amplitude parameter of the dust template, which was held constant for the 2013 release. Furthermore, the greater sky coverage at 100 GHz maximizes its weight at low ℓ, so that the best estimate of the CMB signal on large scales is dominated by 100 GHz data. We do not detect noticeable parameter shifts when removing or including multipoles at ℓ< 500. See Sect. 4.1 for an indepth analysis of the impact of different choices of multipole ranges on cosmological parameters.
Multipole cuts for the Plik temperature and polarization spectra at high ℓ.
Fig. 16 The relative weights of each frequency crossspectrum in the TT (top), TE (middle) and EE (bottom) bestfit solution. Sharp jumps are due to the multipole selection. Weights are normalized to sum to one. 
For TE and EE we are more conservative, and cut the low S/N 100 GHz data at small scales (ℓ> 1000), and the possibly dustcontaminated 217 GHz at large scales (ℓ< 500). Only the 143 × 143TE and EE spectra cover the full multipole range, restricted to ℓ< 2000. Retaining more multipoles would require more indepth modelling of residual systematic effects, which is left to future work. All the cuts are summarized in Table 9 and shown in Fig. 15.
Figure 14 also shows that each of the TT frequency power spectra is cosmicvariance dominated in a wide interval of multipoles. In particular, if we define as cosmicvariance dominated the ranges of multipoles where cosmic variance contributes more than half of the total variance, we find that the 100 × 100 GHz spectrum is cosmicvariance dominated at ℓ ≲ 1156, the 143 × 143 GHz at ℓ ≲ 1528, the 143 × 217 GHz at ℓ ≲ 1607, and the 217 × 217 GHz at ℓ ≲ 1566. To determine these ranges, we calculated the ratio of cosmic to total variance, where the cosmic variance is obtained from the diagonal elements of the covariance matrix after setting the instrumental noise to zero. Furthermore, we find that each of the TE frequency power spectra is cosmicvariance limited in some limited ranges of multipoles, below ℓ ≲ 150 (ℓ ≲ 50 for the 100 × 100)^{11}, in the range ℓ ≈ 250−450 and additionally in the range ℓ ≈ 650−700 only for the 100 × 143 GHz and the 143 × 217 GHz power spectra.
Finally, when we coadd the foregroundcleaned frequency spectra to provide the CMB spectra (see Appendix C.4), we find that the CMB TT power spectrum is cosmicvariance dominated at ℓ ≲ 1586, while TE is cosmicvariance dominated at ℓ ≲ 158 and ℓ ≈ 257−464.
Due to the different masks, multipole ranges, noise levels, and to a lesser extent differing foreground contamination, each crossspectrum ends up contributing differently as a function of scale to the best CMB solution. The determination of the mixing weights is described in Appendix C.4. Figure 16 presents the resulting (relative) weights of each crossspectra. In temperature, the 100 × 100 spectrum dominates the solution until ℓ ≈ 800, when the solution becomes driven by the 143 × 143 up to ℓ ≈ 1400. The 143 × 217 and 217 × 217 provide the solution for the higher multipoles. In polarization, the 100 × 143 dominates the solution until ℓ ≈ 800 (with an equal contribution from 100 × 100 until ℓ ≈ 400 in TE only) while the higher ℓ range is dominated by the 143 × 217 contribution. Not surprisingly, the weights of the higher frequencies tend to increase with ℓ.
3.2.5. Binning
The 2013 baseline likelihood used unbinned temperature power spectra. For this release, we include polarization, which substantially increases the size of the numerical task. The 2015 likelihood therefore uses binned power spectra by default, downsizing the covariance matrix and speeding up likelihood computations. Indeed, even with the multipolerange cut just described, the unbinned data vector has around 23 000 elements, two thirds of which correspond to TE and EE. For some specific purposes (e.g., searching for oscillatory features in the TT spectrum or testing χ^{2} statistics) we also produce an unbinned likelihood.
The spectra are binned into bins of width Δℓ = 5 for 30 ≤ ℓ ≤ 99, Δℓ = 9 for 100 ≤ ℓ ≤ 1503, Δℓ = 17 for 1504 ≤ ℓ ≤ 2013, and Δℓ = 33 for 2014 ≤ ℓ ≤ 2508, with a weighting of the C_{ℓ} proportional to ℓ(ℓ + 1) over the bin widths, ${\mathit{C}}_{\mathrm{b}}\mathrm{=}\sum _{\mathit{\ell}\mathrm{=}{\mathit{\ell}}_{\mathit{b}}^{\mathrm{min}}}^{{\mathit{\ell}}_{\mathit{b}}^{\mathrm{max}}}{\mathit{w}}_{\mathit{b}}^{\mathit{\ell}}{\mathit{C}}_{\mathit{\ell}}\mathit{,}\mathrm{with}{\mathit{w}}_{\mathit{b}}^{\mathit{\ell}}\mathrm{=}\frac{\mathit{\ell}\mathrm{(}\mathit{\ell}\mathrm{+}\mathrm{1}\mathrm{)}}{{\sum}_{\mathit{\ell}\mathrm{=}{\mathit{\ell}}_{\mathit{b}}^{\mathrm{min}}}^{{\mathit{\ell}}_{\mathit{b}}^{\mathrm{max}}}\mathit{\ell}\mathrm{(}\mathit{\ell}\mathrm{+}\mathrm{1}\mathrm{)}}\mathrm{\xb7}$(21)The binwidths are odd numbers, since for approximately azimuthal masks we expect a nearly symmetrical correlation function around the central multipole. It is shown explicitly in Sect. 4.1 that the binning does not affect the determination of cosmological parameters in ΛCDMtype models, which have smooth power spectra.
Fig. 17 Best foreground model in each of the crossspectra used for the temperature highℓ likelihood. The data corrected by the best theoretical CMB C_{ℓ} are shown in grey. The bottom panel of each plot shows the residual after foreground correction. The pink line shows the 1σ value from the diagonal of the covariance matrix (32% of the unbinned points are out of this range). 
3.3. Foreground modelling
Most of the foreground elements in the model parameter vector are similar to those in Like13. The main differences are in the dust templates, which have changed to accommodate the new masks. The TE and EE foreground model only takes into account the dust contribution and neglects any other Galactic polarized emission, in particular the synchrotron contamination. Nor do we mask out any extragalactic polarized foregrounds, as they have been found to be negligible by groundbased, smallscale experiments (Naess et al. 2014; Crites et al. 2015).
Figure 17 shows the foreground decomposition in temperature for each of the crossspectra combinations we use in the likelihood. The figure also shows the CMBcorrected data (i.e., data minus the bestfit ΛCDM CMB model) as well as the residuals after foreground correction. In each spectrum, dust dominates the lowℓ modes, while point sources dominate the smallest scales. For 217 × 217 and 143 × 217, the intermediate range has a significant CIB contribution. We note that for 100 × 100, even when including 66% of the sky, the dust contribution is almost negligible and the pointsource term is dominant well below ℓ = 500. The least foregroundcontaminated spectrum is 143 × 143. For comparison, Fig. 18 shows the full model, including the CMB. The foreground contribution is a small fraction of the total power at large scales.
Table 10 summarizes the parameters used for astrophysical foreground modelling and their associated priors.
Fig. 18 Best model (CMB and foreground) in each of the crossspectra used for the temperature highℓ likelihood. The small light grey points show the unbinned data point, and the dashed grey line show the square root of the noise contribution to the diagonal of the unbinned covariance matrix. 
Parameters used for astrophysical foregrounds and instrumental modelling.
3.3.1. Galactic dust emission
Galactic dust is the main foreground contribution at large scales and thus deserves close attention. This section describes how we model its power spectra. We express the dust contribution to the power spectrum calculated from map X at frequency ν and map Y at frequency ν′ as ${\mathrm{(}{{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY,}\mathrm{dust}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{=}{\mathit{A}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY,}\mathrm{dust}}\mathrm{\times}{\mathit{C}}_{\mathit{\ell}}^{\mathit{XY,}\mathrm{dust}}\mathit{,}$(22)where XY is one of TT, EE, or TE, and ${\mathit{C}}_{\mathit{\ell}}^{\mathit{XY,}\mathrm{dust}}$ is the template dust power spectrum, with corresponding amplitude ${\mathit{A}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY,}\mathrm{dust}}$. We assume that the dust power spectra have the same spatial dependence across frequencies and masks, so the dependence on sky fraction and frequency is entirely encoded in the amplitude parameter A. We do not try to enforce any a priori scaling with frequency, since using different masks at different frequencies makes determination of this scaling difficult. When both frequency maps ν and ν′ are used in the likelihood with the same mask, we simply assume that the amplitude parameter can be written as ${\mathit{A}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY,}\mathrm{dust}}\mathrm{=}{\mathit{a}}_{\mathit{\nu}}^{\mathit{XY,}\mathrm{dust}}\mathrm{\times}{\mathit{a}}_{{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY,}\mathrm{dust}}\mathit{.}$(23)This is clearly not exact when XY = TE and ν ≠ ν′. Similarly the multipoledependent weight used to combine TE and ET for different frequencies breaks the assumption of an invariant dust template. These approximations do not appear to be the limiting factor of the current analysis.
In contrast to the choice we made in 2013, when all Galactic contributions were fixed and a dust template had been explicitly subtracted from the data, we now fit for the amplitude of the dust contribution in each crossspectrum, in both temperature and polarization. This enables exploration of the possible degeneracy between the dust amplitude and cosmological parameters. A comparison of the two approaches is given in Sect. D.1 and Fig. D.2.
In the following, we describe how we build our template dust power spectrum from highfrequency data and evaluate the amplitude of the dust contamination at each frequency and for each mask.
As we shall see later in Sect. 4.1.2, the cosmological values recovered from TT likelihood explorations do not depend on the dust amplitude priors, as shown by the case “No gal. priors” in Fig. 35 and discussed in Sect. 4.1.2. The polarization case is discussed in Sect. C.3.5. Section 5.3 and Figs. 44 and 45 show the correlation between the dust and the cosmological or other foreground parameters. The dust amplitudes are found to be nearly uncorrelated with the cosmological parameters except for TE. However, the priors do help to break the degeneracies between foreground parameters, which are found to be much more correlated with the dust. In Appendix E we further show that our results are insensitive to broader changes in the dust model.
Galactic TT dust emission.
We use the 545 GHz power spectra as templates for Galactic dust spatial fluctuations. The 353 GHz detectors also have some sensitivity to dust, along with a significant contribution from the CMB, and hence any error in removing the CMB contribution at 353 GHz data translates into biases on our dust template. This is much less of an issue at 545 GHz, to the point where entirely ignoring the CMB contribution does not change our estimate of the template. Furthermore, estimates using 545 GHz maps tend to be more stable over a wider range of multipoles than those obtained from 353 GHz or 857 GHz maps.
We aggressively mask the contribution from point sources in order to minimize their residual, the approximately white spectrum of which is substantially correlated with the value of some cosmological parameters (see the discussion of parameter correlations in Sect. 5.3). The downside of this is that the pointsource masks remove some of the brightest Galactic regions that lie in regions not covered by our Galactic masks. This means that we cannot use the wellestablished powerlaw modelling advocated in Planck Collaboration XI (2014) and must instead compute an effective dust (residual) template.
All of the masks that we use in this section are combinations of the joint pointsource, extendedobject, and CO masks used for 100 GHz, 143 GHz, and 217 GHz with Galactic masks of various sizes. In the following discussion we refer only to the Galactic masks, but in all cases the masks contain the other components as well. The halfmission crossspectra at 545 GHz provide us with a good estimate of the largescale behaviour of the dust. Small angular scales, however, are sensitive to the CIB, with the intermediate range of scales dominated by the clustered part and the smallest scales by the Poisson distribution of infrared point sources. These last two terms are statistically isotropic, while the dust amplitude depends on the sky fraction. Assuming that the shapes of the dust power spectra outside the masks do not vary substantially as the sky fraction changes, we rely on mask differences to build a CIBcleaned template of the dust.
Figure 19 shows that this assumption is valid when changing the Galactic mask from G60 to G41. It shows that the 545 GHz crosshalfmission power spectrum can be well represented by the sum of a Galactic template, a CIB contribution, and a point source contribution. The Galactic template is obtained by computing the difference between the spectra obtained in the G60 and the G41 masks. This difference is fit to a simple analytic model ${\mathit{C}}_{\mathit{\ell}}^{\mathit{TT,}\mathrm{dust}}\mathrm{\propto}\mathrm{(}\mathrm{1}\mathrm{+}\mathit{h}\hspace{0.17em}{\mathit{\ell}}^{\mathit{k}}\hspace{0.17em}{\mathrm{e}}^{\mathrm{}\mathit{\ell}\mathit{/}\mathit{t}}\mathrm{)}\mathrm{\times}\mathrm{(}\mathit{\ell}\mathit{/}{\mathit{\ell}}_{\mathit{p}}{\mathrm{)}}^{\mathit{n}}\mathit{,}$(24)with h = 2.3 × 10^{11}, k = 5.05, t = 56, n = −2.63, and fixing ℓ_{p} = 200. The model behaves like a ${\mathit{C}}_{\mathit{\ell ,}\mathrm{dust}}^{\mathit{TT}}\mathrm{\propto}{\mathit{\ell}}^{2.63}$ power law at small scales, and has a bump around ℓ = 200. The CIB model we use is described in Sect. 3.3.2.
We can compare this template model with the dust content in each of the power spectra we use for the likelihood. Of course those power spectra are strongly dominated by the CMB, so, to reveal the dust content, one has to rely on the same trick that was used for 545 GHz. This however is not enough, since the CMB cosmic variance itself is significant compared to the dust contamination. We can build an estimate of the CMB cosmic variance by assuming that at 100 GHz the dust contamination is small enough that a mask difference gives us a good variance estimate.
Fig. 19 Dust model at 545 GHz. The dust template is based on the G60–G41 mask difference of the 545 GHz halfmission crossspectrum (blue line and circles, rescaled to the dust level in mask G60). Coloured diamonds display the difference between this model (rescaled in each case) and the cross halfmission spectra in the G41, G50, and G60 masks. The residuals are all in good agreement (less so at low ℓ, because of sample variance) and are well described by the CIB+point source prediction (orange line). Individual CIB and point sources contributions are shown as dashed and dotted orange lines. The red line is the sum of the dust model, CIB, and point sources for the G60 mask, and is in excellent agreement with the 545 GHz cross halfmission spectrum in G60 (red squares). In all cases, the spectra were computed by using different Galactic masks supplemented by the single combination of the 100 GHz, 143 GHz, and 217 GHz point sources, extended objects and CO masks. 
Fig. 20 Dust model versus data. In blue, the power spectrum of the double mask difference between 217 GHz and 100 GHz halfmission crossspectra in masks G60 and G41 (complemented by the joint masks for CO, extended objects, and point sources). In orange, the equivalent spectrum for 143 and 100 GHz. The mask difference enables us to remove the contribution from all the isotropic components (CMB, CIB, and point sources) in the mean. But simple mask differences are still affected by the difference of the CMB in the two masks due to cosmic variance. Removing the 100 GHz mask difference, which is dominated by the CMB, reduces the scatter significantly. The error bars are computed as the scatter in bins of size Δℓ = 50. The dust model (green) based on the 545 GHz data has been rescaled to the expected dust contamination in the 217 GHz mask difference using values from Table 11. The 143 GHz double mask difference is also rescaled to the level of the 217 GHz difference; i.e., it is multiplied by approximately 14. Different multipole bins are used for the 217 GHz and 143 GHz data to improve readability. 
Contamination level in each frequency, D_{ℓ = 200}.
Figure 20 shows the mask difference (corrected for cosmic variance) between G60 and G41 for the 217 GHz and 143 GHz halfmission crossspectra, as well as the dust model from Eq. (24). The dust model has been rescaled to the expected mask difference dust residual for the 217 GHz. The 143 GHz maskdifference has also been rescaled in a similar way. The ratio between the two is about 14. Rescaling factors are obtained from Table 11. Error bars are estimated based on the scatter in each bin. The agreement with the model is very good at 217 GHz, but less good at 143 GHz where the greater scatter is probably dominated at large scales by the chance correlation between CMB and dust (which, as we see in Eq. (25), varies as the square root of the dust contribution to the spectra), and at small scale by noise. We also tested these double differences for other masks, namely G50−G41 and G60−G50, and verified that the results are similar (i.e., general agreement although with substantial scatter).
Finally, we can estimate the level of the dust contamination in each of our frequency maps used for CMB analysis by computing their crossspectra with the 545 GHz halfmission maps. Assuming that all our maps m^{ν} have in common only the CMB and a variable amount of dust, and assuming that m^{545} = m^{cmb} + a^{545}m^{dust}, the crossspectra between each of our CMB frequencies maps and the 545 GHz map is $\begin{array}{ccc}{\mathrm{(}{{\mathit{C}}_{\mathrm{545}\mathrm{\times}\mathit{\nu}}^{\mathit{TT}}}^{\mathrm{)}}}_{\mathit{\ell}}& \mathrm{=}& {\mathit{C}}_{\mathit{\ell}}^{\mathit{TT,}\mathrm{cmb}}\mathrm{+}{\mathit{a}}_{\mathrm{545}}^{\mathit{TT,}\mathrm{dust}}{\mathit{a}}_{\mathit{\nu}}^{\mathit{TT,}\mathrm{dust}}\hspace{0.17em}{\mathit{C}}_{\mathit{\ell}}^{\mathit{TT,}\mathrm{dust}}\\ & & \end{array}$(25)where ${\mathit{C}}_{\mathit{\ell}}^{\mathrm{chance}}$ is the chance correlation between the CMB and dust distribution (which would vanish on average over many sky realizations). By using the 100 GHz spectrum as our CMB estimate and assuming that the chance correlation is small enough, one can measure the amount of dust in each frequency map by fitting the rescaling factor between the (CMB cleaned) 545 GHz spectrum and the cross frequency spectra. This approach is limited by the presence of CIB which has a slightly different emission law than the dust. We thus limit our fits to the multipoles ℓ< 1000 where the CIB is small compared to the dust and we ignore the emissionlaw differences.
Table 11 reports the results of those fits at each frequency, for each Galactic mask. The error range quoted corresponds to the error of the fits, taking into account the variations when changing the multipole range of the fit from 30 ≤ ℓ ≤ 1000 to 30 ≤ ℓ ≤ 500. The values reported correspond to the sum of the CIB and the dust contamination at ℓ = 200. The last column gives the estimate of the CIB contamination at the same multipole from the joint cosmology and foreground fit. From this table, the ratio of the dust contamination at map level between the 217 GHz and 100 GHz is around 7, while the ratio between the 217 GHz and 143 GHz is close to 3.7.
We derive our priors on the foreground amplitudes from this table, combining the 545 GHz fit with the estimated residual CIB contamination, to obtain the following values: (7 ± 2) μ for the 100 × 100 spectrum (G70); (9 ± 2) μ for 143 × 143 (G60); and (80 ± 20) μ for 217 × 217 (G50). Finally the 143 × 217 value is obtained by computing the geometrical average between the two auto spectra under the worst mask (G60), yielding (21 ± 8.5) μ.
Galactic TE and EE dust emission.
We evaluate the dust contribution in the TE and EE power spectra using the same method as for the temperature. However, instead of the 545 GHz data we use the maps at 353 GHz, our highest frequency with polarization information. At sufficently high sky fractions, the 353 GHz TE and EE power spectra are dominated by dust. As estimated in Planck Collaboration Int. XXX (2016), there is no other significant contribution from the Galaxy, even at 100 GHz. Following Planck Collaboration Int. XXX (2016), and since we do not mask any “pointsourcelike” region of strong emission, we can use a powerlaw model as a template for the polarized Galactic dust contribution. Enforcing a single power law for TE and EE and our different masks, we obtain an index of n = −2.4. We use the same crossspectrabased method to estimate the dust contamination. The dust contribution being smaller in polarization, removing the CMB from the 353 × 353 and the 353 × ν (with ν being one of 100, 143 or 217) is particularly important. Our two best CMB estimates in EE and TE being 100 and the 143 GHz, we checked that using any of 100 × 100, 143 × 143, or 100 × 143 does not change the estimates significantly. Table 12 gives the resulting values. As for the TT case, the crossfrequency, crossmasks estimates are obtained by computing the geometric average of the autofrequency contaminations under the smallest mask.
TE and EE dust contamination levels, D_{ℓ = 500}.
3.3.2. Extragalactic foregrounds
The extragalactic foreground model is similar to that of 2013 and in the following we describe the differences. Since we are neglecting any possible contribution in polarization from extragalactic foregrounds, we omit the TT index in the following descriptions of the foreground models. The amplitudes are expressed as at ℓ = 3000 so that, for any component, the template, ${\mathit{C}}_{\mathrm{3000}}^{\mathrm{FG}}$, satisfies ${\mathit{C}}_{\mathrm{3000}}^{\mathrm{FG}}\hspace{0.17em}{\mathrm{\mathcal{A}}}_{\mathrm{3000}}\mathrm{=}\mathrm{1}$ with .
The cosmic infrared background.
The CIB model has a number of differences from that used in Like13. First of all, it is now entirely parameterized by a single amplitude ${\mathrm{\mathcal{D}}}_{\mathrm{217}}^{\mathrm{CIB}}$ and a template ${\mathit{C}}_{\mathit{\ell}}^{\mathrm{CIB}}$: ${\mathrm{(}{{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathrm{CIB}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{=}{\mathit{a}}_{\mathit{\nu}}^{\mathrm{CIB}}{\mathit{a}}_{{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathrm{CIB}}{\mathit{C}}_{\mathit{\ell}}^{\mathrm{CIB}}\mathrm{\times}{\mathrm{\mathcal{D}}}_{\mathrm{217}}^{\mathrm{CIB}}\mathit{,}$(26)where the spectral coefficients ${\mathit{a}}_{\mathit{\nu}}^{\mathrm{CIB}}$ represent the CIB emission law normalized at ν = 217 GHz.
In 2013, the template was an effective powerlaw model with a variable index with expected value n = −1.37 (when including the “highL” data from ACT and SPT). We did not assume any emission law and fitted the 143 GHz and 217 GHz amplitude, along with their correlation coefficient. The Planck Collaboration has studied the CIB in detail in Planck Collaboration XXX (2014) and now proposes a oneplustwohalo model, which provides an accurate description of the Planck and IRAS CIB spectra from 3000 GHz down to 217 GHz. We extrapolate this model here, assuming it remains appropriate in describing the 143 GHz and 100 GHz data. The CIB emission law and template are computed following Planck Collaboration XXX (2014). The template power spectrum provided by this work has a very small frequency dependence that we ignore.
At small scales, ℓ> 2500, the slope of the template is similar to the power law used in Like13. At larger scales, however, the slope is much shallower. This is in line with the variation we observed in 2013 on the powerlaw index of our simple CIB model when changing the maximum multipole. The current template is shown as the green line in the TT foreground component plots in Fig. 17.
In 2013, the correlation between the 143 GHz and 217 GHz CIB spectra was fitted, favouring a high correlation, greater than 90% (when including the “highL” data). The present model yields a fully correlated CIB between 143 GHz and 217 GHz.
We now include the the CIB contribution at 100 GHz, which was ignored in 2013. Another difference with the 2013 model is that the parameter controlling the amplitude at 217 GHz now directly gives the amplitude in the actual 217 GHz Planck band at ℓ = 3000, i.e., it includes the colour correction. The ratio between the two is 1.33. The 2013 amplitude of the CIB contribution at ℓ = 3000 (including the highL data) was 66 ± 6.7 μK^{2}, while our best estimate for the present analysis is 63.9 ± 6.6 μK^{2} (PlanckTT + lowP).
Point sources.
At the likelihood level, we cannot differentiate between the radio and IRpoint sources. We thus describe their combined contribution by their total emissivity per frequency pair, ${\mathrm{(}{{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathrm{PS}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{=}{\mathrm{\mathcal{D}}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathrm{PS}}\mathit{/}{\mathrm{\mathcal{A}}}_{\mathrm{3000}}\mathit{,}$(27)where is the amplitude of the pointsource contribution in at ℓ = 3000. Contrary to 2013, we do not use a correlation parameter to represent the 143 × 217 pointsource contribution; instead we use a free amplitude parameter. This has the disadvantage of not preventing a possible unphysical solution. However, it simplifies the parameter optimization, and it is easier to understand in terms of contamination amplitude.
Kinetic SZ (kSZ).
We use the same model as in 2013. The kSZ emission is parameterized with a single amplitude and a fixed template from Trac et al. (2011), ${\mathrm{(}{{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathrm{kSZ}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{=}{\mathit{C}}_{\mathit{\ell}}^{\mathrm{kSZ}}\mathrm{\times}{\mathrm{\mathcal{D}}}^{\mathrm{kSZ}}\mathit{,}$(28)where is the kSZ contribution at ℓ = 3000.
Thermal SZ (tSZ).
Here again, we use the same model as in 2013. The tSZ emission is also parameterized by a single amplitude and a fixed template using the ϵ = 0.5 model from Efstathiou & Migliaccio (2012), ${\mathrm{(}{{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathrm{tSZ}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{=}{\mathit{a}}_{\mathit{\nu}}^{\mathrm{tSZ}}{\mathit{a}}_{{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathrm{tSZ}}{\mathit{C}}_{\mathit{\ell}}^{\mathrm{tSZ}}\mathrm{\times}{\mathrm{\mathcal{D}}}_{\mathrm{143}}^{\mathrm{tSZ}}\mathit{,}$(29)where ${\mathit{a}}_{\mathit{\nu}}^{\mathrm{tSZ}}$ is the thermal SunyaevZeldovich spectrum, normalized to ν_{0} = 143 GHz and corrected for the Planck bandpass colour corrections. Ignoring the bandpass correction, we recall that the tSZ spectrum is given by ${\mathit{a}}_{\mathit{\nu}}^{\mathrm{tSZ}}\mathrm{=}\frac{\mathit{f}\mathrm{\left(}\mathit{\nu}\mathrm{\right)}}{\mathit{f}\mathrm{\left(}{\mathit{\nu}}_{\mathrm{0}}\mathrm{\right)}}\mathit{,}\mathit{f}\mathrm{\left(}\mathit{\nu}\mathrm{\right)}\mathrm{=}\left(\mathit{x}\mathrm{coth}\left(\frac{\mathit{x}}{\mathrm{2}}\right)\mathrm{}\mathrm{4}\right)\mathit{,}\mathit{x}\mathrm{=}\frac{\mathit{h\nu}}{{\mathit{k}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{cmb}}}\mathrm{\xb7}$(30)
Thermal SZ × CIB correlation.
Following Like13 the crosscorrelation between the thermal SZ and the CIB, tSZ × CIB, is parameterized by a single correlation parameter, ξ, and a fixed template from Addison et al. (2012), $\begin{array}{ccc}\hspace{0.17em}\begin{array}{c}\begin{array}{c}\end{array}{\mathrm{(}{{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathrm{tSZ}\mathrm{\times}\mathrm{CIB}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{=}\mathit{\xi}\hspace{0.17em}\sqrt{{\mathrm{\mathcal{D}}}_{\mathrm{143}}^{\mathrm{tSZ}}\hspace{0.17em}{\mathrm{\mathcal{D}}}_{\mathrm{217}}^{\mathrm{CIB}}}\begin{array}{c}\end{array}\\ \begin{array}{c}\end{array}\u2001\mathrm{\times}\mathrm{(}{\mathit{a}}_{\mathit{\nu}}^{\mathrm{tSZ}}\hspace{0.17em}{\mathit{a}}_{{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathrm{CIB}}\mathrm{+}{\mathit{a}}_{{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathrm{tSZ}}\hspace{0.17em}{{\mathit{a}}_{\mathit{\nu}}^{\mathrm{CIB}}}^{\mathrm{)}}\\ \begin{array}{c}\end{array}\u2001\mathrm{\times}{\mathit{C}}_{\mathit{\ell}}^{\mathrm{tSZ}\mathrm{\times}\mathrm{CIB}}\mathit{,}\end{array}& & \end{array}$(31)where ${\mathit{a}}_{\mathit{\nu}}^{\mathrm{tSZ}}$ is the thermal SunyaevZeldovich spectrum, corrected for the Planck bandpass colour corrections and ${\mathit{a}}_{\mathit{\nu}}^{\mathrm{CIB}}$ is the CIB spectrum, rescaled at ν = 217 GHz as in the previous paragraphs.
SZ prior.
The kinetic SZ, the thermal SZ, and its correlation with the CIB are not constrained accurately by the Planck data alone. Besides, the tSZ×CIB level is highly correlated with the amplitude of the tSZ. In 2013, we reduced the degeneracy between those parameters and improved their determination by adding the ACT and SPT data. In 2015, we instead impose a Gaussian prior on the tSZ and kSZ amplitudes, inspired by the constraints set by these experiments. From a joint analysis of the Planck 2013 data with those from ACT and SPT, we obtain ${\mathrm{\mathcal{D}}}^{\mathrm{kSZ}}\mathrm{+}\mathrm{1.6}{\mathrm{\mathcal{D}}}^{\mathrm{tSZ}}\mathrm{=}\mathrm{(}\mathrm{9.5}\mathrm{\pm}\mathrm{3}\mathrm{)}\hspace{0.17em}\mathit{\mu}{\mathrm{K}}^{\mathrm{2}}\mathit{,}$(32)in excellent agreement with the estimates from Reichardt et al. (2012), once they are rescaled to the Planck frequencies (see Planck Collaboration XIII 2016, for a detailed discussion).
As can be seen in Fig. 17, the kSZ, tSZ, and tSZ×CIB correlations are always dominated by the dust, CIB, and pointsource contributions.
3.4. Instrumental modelling
The following sections describe the instrument modelling elements of the model vector, addressing the issues of calibration and beam uncertainties in Sects. 3.4.1−3.4.3, and describing the noise properties in Sect. 3.4.4. For convenience, Table 10 defines the symbol used for the calibration parameters and the priors later used for exploring them.
3.4.1. Power spectra calibration uncertainties
As in 2013, we allow for a small recalibration of the different frequency power spectra, in order to account for residual uncertainties in the map calibration process. The mixing matrix in the model vector from Eq. (14) can be rewritten as $\begin{array}{ccc}{\mathrm{(}{{\mathit{M}}_{\mathit{ZW,\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{inst}}\mathrm{\right)}& \mathrm{=}& {\mathit{G}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{calib}}\mathrm{\right)}{\mathrm{(}{{\mathit{M}}_{\mathit{ZW,\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY,}\mathrm{other}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{other}}\mathrm{\right)}\mathit{,}\\ {\mathit{G}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{calib}}\mathrm{\right)}& \mathrm{=}& \end{array}$(33)where ${\mathit{c}}_{\mathit{\nu}}^{\mathit{XX}}$ is the calibration parameter for the XX power spectrum at frequency ν, X being either T or E, and y_{P} is the overall Planck calibration. We ignore the ℓdependency of the weighting function between the TE and ET spectra at different frequencies that are added to form an effective crossfrequency TE crossspectrum. As in 2013, we use the TT at 143 GHz as our intercalibration reference, so that ${\mathit{c}}_{\mathrm{143}}^{\mathit{TT}}\mathrm{=}\mathrm{1}$.
We further allow for an overall Planck calibration uncertainty, whose variation is constrained by a tight Gaussian prior, $\begin{array}{ccc}{\mathit{y}}_{\mathrm{P}}\mathrm{=}\mathrm{1}\mathrm{\pm}\mathrm{0.0025.}& & \end{array}$(34)This prior corresponds to the estimated overall uncertainty, which is discussed in depth in Planck Collaboration I (2016).
The calibration parameters can be degenerate with the foreground parameters, in particular the point sources at high ℓ (for TT) and the Galaxy for 217 GHz at low ℓ. We thus proceed as in 2013, and measure the calibration refinement parameters on the large scales and on small sky fractions near the Galactic poles. We perform the same estimates on a range of Galactic masks (G20, G30, and G41) restricted to different maximum multipoles (up to ℓ = 1500). The fits are performed either by minimizing the scatter between the different frequency spectra, or by using the SMICA algorithm (see Planck Collaboration VI 2014, Sect. 7.3) with a freely varying CMB and generic foreground contribution. For the TT spectra, we obtained in both cases very similar recalibration estimates, from which we extracted the conservative Gaussian priors on recalibration factors, $\begin{array}{ccc}{\mathit{c}}_{\mathrm{100}}^{\mathit{TT}}& \mathrm{=}& \mathrm{0.999}\mathrm{\pm}\mathrm{0.001}\mathit{,}\\ {\mathit{c}}_{\mathrm{217}}^{\mathit{TT}}& \mathrm{=}& \mathrm{0.995}\mathrm{\pm}\mathrm{0.002.}\end{array}$These are compatible with estimates made at the map level, but on the whole sky; see Planck Collaboration VIII (2016).
3.4.2. Polarization efficiency and angular uncertainty
We now turn to the polarization recalibration case. The signal measured by an imperfect PSB is given by $\mathit{d}\mathrm{=}\mathit{G}\mathrm{(}\mathrm{1}\mathrm{+}\mathit{\gamma}\mathrm{)}\left[\mathit{I}\mathrm{+}\mathit{\rho}\mathrm{(}\mathrm{1}\mathrm{+}\mathit{\eta}\mathrm{)}\left(\mathit{Q}\mathrm{cos}\mathrm{2}\mathrm{(}\mathit{\phi}\mathrm{+}\mathit{\omega}\mathrm{)}\mathrm{+}\mathit{U}\mathrm{sin}\mathrm{2}\mathrm{(}\mathit{\phi}\mathrm{+}\mathit{\omega}\mathrm{)}\right)\right]\mathrm{+}\mathit{n,}$(37)where I, Q, and U are the Stokes parameters; n is the instrumental noise; G, ρ, and φ are the nominal photometric calibration factor, polar efficiency, and direction of polarization of the PSB; and γ, η, and ω are the (small) errors made on each of them (see, e.g., Jones et al. 2007). Due to these errors, the measured crosspower spectra of maps a and b are then contaminated by a spurious signal given by $\begin{array}{ccc}\mathrm{\Delta}{\mathit{C}}_{\mathit{\ell}}^{\mathit{TT}}& \mathrm{=}& \\ \mathrm{\Delta}{\mathit{C}}_{\mathit{\ell}}^{\mathit{TE}}& \mathrm{=}& \\ \mathrm{\Delta}{\mathit{C}}_{\mathit{\ell}}^{\mathit{EE}}& \mathrm{=}& \mathrm{(}{\mathit{\gamma}}_{\mathit{a}}\mathrm{+}{\mathit{\gamma}}_{\mathit{b}}\mathrm{+}{\mathit{\eta}}_{\mathit{a}}\mathrm{+}{\mathit{\eta}}_{\mathit{b}}\mathrm{}\mathrm{2}{\mathit{\omega}}_{\mathit{a}}^{\mathrm{2}}\mathrm{}\mathrm{2}{{\mathit{\omega}}_{\mathit{b}}^{\mathrm{2}}}^{\mathrm{)}}{\mathit{C}}_{\mathit{\ell}}^{\mathit{EE}}\\ & & \end{array}$where γ_{x}, η_{x}, and ω_{x}, for x = a,b, are the effective instrumental errors for each of the two frequencyaveraged maps. Preflight measurements of the HFI polarization efficiencies, ρ, had uncertainties  η_{x}  ≈ 0.3%, while the polarization angle of each PSB is known to  ω_{x}  ≈ 1deg (Rosset et al. 2010). Analysis of the 2015 maps shows the relative photometric calibration of each detector at 100 to 217 GHz to be known to about  γ_{x}  = 0.16% at worst, with an absolute orbital dipole calibration of about 0.2%, while analysis of the Crab Nebula observations showed the polarization uncertainties to be consistent with the preflight measurements (Planck Collaboration VIII 2016).
Assuming ${\mathit{C}}_{\mathit{\ell}}^{\mathit{BB}}$ to be negligible, and ignoring ω^{2} ≪  η  in Eq. (35), the Gaussian priors on γ and η for each frequencyaveraged polarized map would have rms of σ_{γ} = 2 × 10^{3} and σ_{η} = 3 × 10^{3}. Adding those uncertainties in quadrature, the autopower spectrum recalibration ${\mathit{c}}_{\mathit{\nu}}^{\mathit{EE}}$ introduced in Eq. (33) would be given, for an equalweight combination of n_{d} = 8 polarized detectors, by $\begin{array}{ccc}{\mathit{c}}_{\mathit{\nu}}^{\mathit{EE}}\mathrm{=}\mathrm{1}\mathrm{\pm}\mathrm{2}\sqrt{\frac{{\mathit{\sigma}}_{\mathit{\gamma}}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma}}_{\mathit{\eta}}^{\mathrm{2}}}{{\mathit{n}}_{\mathrm{d}}}}\mathrm{=}\mathrm{1}\mathrm{\pm}\mathrm{0.0025.}& & \end{array}$(39)The most accurate recalibration factors for TE and EE could therefore be somewhat different from TT. We found, though, that setting the EE recalibration parameter to unity or implementing those priors makes no difference with respect to cosmology; i.e., we recover the same cosmological parameters, with the same uncertainties. Thus, for the baseline explorations, we fixed the EE recalibration parameter to unity, ${\mathit{c}}_{\mathit{\nu}}^{\mathit{EE}}\mathrm{=}\mathrm{1}\mathit{,}$(40)and the uncertainty on TE comes only from the TT calibration parameter through Eq. (33).
We also explored the case of much looser priors, and found that bestfit calibration parameters deviate very significantly, and reach values of several percent (between 3% and 12% depending on the frequencies and on whether we fit the EE or TE case). This cannot be due to the instrumental uncertainties embodied in the prior. In the absence of an informative prior, this degree of freedom is used to minimize the differences between frequencies that stem from other effects, not included in the baseline modelling.
The next section introduces one such effect, the temperaturetopolarization leakage, which is due to combining detectors with different beams without accounting for it at the mapmaking stage (see Sect. 3.4.3). But anticipating the results of the analysis described in Appendix C.3.5, we note that when the calibration and leakage parameters are explored simultaneously without priors, they remain in clear tension with the priors (even if the level of recalibration decreases slightly, by typically 2%, showing the partial degeneracy between the two). In other words, when calibration and leakage parameters are both explored with their respective priors, there is evidence of residual unmodelled systematic effects in polarization – to which we will return.
3.4.3. Beam and transfer function uncertainties
The power spectra from map pairs are corrected by the corresponding effective beam window functions before being confronted with the data model. However, these window functions are not perfectly known, and we now discuss various related sources of errors and uncertainties, the impact of which on the reconstructed C_{ℓ}s is shown in Fig. 21.
Subpixel effects.
The first source of error, the socalled “subpixel” effect, discussed in detail in Like13, is a result of the Planck scanning strategy and mapmaking procedure. Scanning along rings with very low nutation levels can result 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 centre of the pixel. This effect, similar to the gravitational lensing of the CMB, has a nondiagonal influence 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. We showed in Like13 that including this effect had little impact on the cosmological parameters measured by Planck.
Masking effect.
A second source of error is the variation, from one sky pixel to another, of the effective beam width, which is averaged over all samples falling in that pixel. While all the HEALPix pixels have the same surface area, their shape – and therefore their moment of inertia (which drives the pixel window function) – depends on location, as shown in Fig. 22, and therefore makes the effective beam window function depend on the pixel mask considered. Of course the actual sampling of the pixels by Planck leads to individual moments of inertia slightly different from the intrinsic values shown here, but spotcheck comparisons of this semianalytical approach used by QuickBeam with numerical simulations of the actual scanning by FEBeCoP showed agreement at the 10^{3} level for ℓ< 2500 on the resulting pixel window functions for sky coverage varying from 40 to 100%.
In the various Galactic masks used here (Figs. 12–13) the contribution of the unmasked pixels to the total effective window function departs from the fullsky average (which is not included in the effective beam window functions), and we therefore expect a different effective transfer function for each mask. We ignored this dependence and mitigated its effect by using transfer functions computed with the Galactic mask G60 which retains an effective sky fraction (including the mask apodization) of f_{sky} = 60%, not too different from the sky fractions f_{sky} between 41 and 70% (see Sect. 3.2.2) used for computing the power spectra.
Fig. 21 Contribution of various beamwindowfunctionrelated errors and uncertainties to the C_{ℓ} relative error. In each panel, the grey histogram shows the relative statistical error on the Planck CMB TT binned power spectrum (for a bin width Δℓ = 30) divided by 10, while the vertical grey dashes delineate the range ℓ< 1800 that is most informative for base ΛCDM. Top: estimation of the error made by ignoring the subpixel effects for a fiducial C_{ℓ} including the CMB and CIB contributions. Middle: error due to the sky mask, for the Galactic masks used in the TT analysis. Bottom: current beam window function error model, shown at 1σ (solid lines) and 10σ (dotted lines). 
Figure 21 compares the impact of these two sources of uncertainty on the stated Planck statistical error bars for Δℓ = 30. It shows that, for ℓ< 1800 where most of the information on ΛCDM lies, the error on the TT power spectra introduced by the subpixel effect and by the skycoverage dependence are less than about 0.1%, and well below the statistical error bars of the binned C_{ℓ}. In the range 1800 ≤ ℓ ≤ 2500, which helps constrain oneparameter extensions to base ΛCDM (such as N_{eff}), the relative error can reach 0.4% (note as a comparison that the highℓ ACT experiment states a statistical error of about 3% on the bin 2340 ≤ ℓ ≤ 2540, Das et al. 2014). The bottom panel shows the Monte Carlo error model of the beam window functions, which provides negligible (ℓcoupled) uncertainties. Even if this model is somewhat optimistic, since it does not include the effect of the ADC nonlinearities and the colourcorrection effect of beam measurements on planets (Planck Collaboration VII 2016), we note that even expanding them by a factor of 10 keeps them within the statistical uncertainty of the power spectra.
Modelling the uncertainties.
As in the 2013 analysis, the beam uncertainty eigenmodes were determined from 100 (improved) Monte Carlo (MC) simulations of each planet observation used to measure the scanning beams, then processed through the same QuickBeam pipeline as the nominal beam to determine their effective angular transfer function B(ℓ). Thanks to the use of Saturn and Jupiter transits instead of the dimmer Mars used in 2013, the resulting uncertainties are now significantly smaller (Planck Collaboration VII 2016).
For each pair of frequency maps (and frequencyaveraged beams) used in the present analysis, a singularvalue decomposition (SVD) of the correlation matrix of 100 Monte Carlo based B(ℓ) realizations was performed over the ranges [0,ℓ_{max}] with ℓ_{max} = (2000,3000,3000) at (100, 143, 217 GHz), and the five leading modes were kept, as well as their covariance matrix (since the error modes do exhibit Gaussian statistics). We therefore have, for each pair of beams, five ℓdependent templates, each associated with a Gaussian amplitude centred on 0, and a covariance matrix coupling all of them.
Fig. 22 Map of the relative variations of the trace of the HEALPix pixel moment of inertia tensor at N_{side} = 2048 in Galactic coordinates. 
Including the beam uncertainties in the mixing matrix of Eq. (14) gives $\begin{array}{ccc}{\mathrm{(}{{\mathit{M}}_{\mathit{ZW,\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{inst}}\mathrm{\right)}& \mathrm{=}& {\mathrm{(}{{\mathit{M}}_{\mathit{ZW,\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY,}\mathrm{other}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{other}}\mathrm{\right)}{\mathrm{(}\mathrm{\Delta}{{\mathit{W}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{ZW}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{beam}}\mathrm{\right)}\mathit{,}\\ {\mathrm{(}\mathrm{\Delta}{{\mathit{W}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{ZW}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{beam}}\mathrm{\right)}& \mathrm{=}& \mathrm{exp}\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathrm{5}}\mathrm{2}\hspace{0.17em}{\mathit{\theta}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{ZW,i}}{\mathrm{(}{{\mathit{E}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{ZW,i}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathit{,}\end{array}$(41)where ${{}^{\mathrm{(}}\mathrm{\Delta}{{\mathit{W}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{ZW}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{beam}}\mathrm{\right)}$ stands for the beam error built from the eigenmodes ${{}^{\mathrm{\right(}}{\mathit{E}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{ZW,i}}^{\mathrm{\left)}}}_{\mathit{\ell}}$. The quadratic sum of the beam eigenmodes is shown in Fig. 21. This is much smaller (less than a percent) than the combined TT spectrum error bars. This contrasts with the 2013 case where the beam uncertainties were greater; for instance, for the 100, 143, and 217 GHz channel maps, the rms of the W(ℓ) = B(ℓ)^{2} uncertainties at ℓ = 1000 dropped from (61,23,20) × 10^{4} to (2.2,0.84,0.81) × 10^{4}, respectively. The fact that beam uncertainties are subdominant in the total error budget is even more pronounced in polarization, where noise is higher. We use the beam modes computed from temperature data, combined with appropriate weights when used as parameters affecting the TE and EE spectra.
As in 2013, instead of including the beam error in the vector model, we include its contribution to the covariance matrix, linearizing the vector model so that ${\mathrm{(}{{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{\left(}\mathit{\theta}\mathrm{\right)}\mathrm{=}{\mathrm{(}{{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{(}\mathit{\theta ,}{\mathit{\theta}}_{\mathrm{beam}}\mathrm{=}\mathrm{0}\mathrm{)}\mathrm{+}{\mathrm{(}\mathrm{\Delta}{{\mathit{W}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{ZW}}}^{\mathrm{)}}}_{\mathit{\ell}}\mathrm{\left(}{\mathit{\theta}}_{\mathrm{beam}}\mathrm{\right)}{\mathrm{(}{{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}}^{\mathrm{)}}}_{\mathit{\ell}}^{\mathrm{\ast}}\mathit{,}$(42)where ${{}^{\mathrm{\right(}}{\mathit{C}}_{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}^{\mathit{XY}}^{\mathrm{\left)}}}_{\mathit{\ell}}^{\mathrm{\ast}}$ is the fiducial spectrum XY for the pair of frequencies ν × ν′ obtained using the best cosmological and foreground model. We can then marginalize over the beam uncertainty, enlarging the covariance matrix to obtain ${\mathrm{C}}_{\mathrm{beam}\mathrm{marg}\mathit{.}}\mathrm{=}\mathrm{C}\mathrm{+}{{C}}^{\mathrm{\ast}}\mathrm{\u27e8}\mathrm{\Delta}{W}\mathrm{\Delta}{{W}}^{\mathrm{T}}\mathrm{\u27e9}{{C}}^{\mathrm{\ast}\mathrm{T}}\mathit{,}$(43)where ${\mathrm{\u27e8}}^{}\mathrm{\Delta}{W}\mathrm{\Delta}{{W}}^{\mathrm{T}}{\mathrm{\u27e9}}^{}$ is the Monte Carlo based covariance matrix, restricted to its first five eigenmodes.
In 2013, beam errors were marginalized for all the modes except the two greatest of the 100 × 100 spectrum. In the present release we instead marginalize over all modes in TT, TE, and EE. We also performed a test in which we estimated the amplitudes for all of the first five beam eigenmodes in TT, TE, and EE, and found no indication of any beam error contribution (see Sect. 4.1.3 and Fig. 35).
Temperaturetopolarization leakage.
Polarization measurements are differential by nature. Therefore any unaccounted discrepancy in combining polarized detectors can create some leakage from temperature to polarization (Hu et al. 2003). Sources of such discrepancies in the current HFI processing include, but are not limited to: differences in the scanning beams that are ignored during the mapmaking; differences in the noise level, because of the individual inverse noise weighting used in HFI; and differences in the number of valid samples.
For this release, we did not attempt to model and remove a priori the form and amplitude of this coupling between the measured TT, TE, and EE spectra; we rather estimate the residual effect by fitting a posteriori in the likelihood some flexible template of this coupling, parameterized by some new nuisance parameters that we now describe.
The temperaturetopolarization leakage due to beam mismatch is assumed to affect the spherical harmonic coefficients via $\begin{array}{ccc}{\mathit{a}}_{\mathit{\ell m}}^{\mathit{T}}& \mathrm{}\mathrm{\to}& {\mathit{a}}_{\mathit{\ell m}}^{\mathit{T}}\mathit{,}\\ {\mathit{a}}_{\mathit{\ell m}}^{\mathit{E}}& \mathrm{}\mathrm{\to}& \end{array}$and, for each map, the spurious polarization power spectrum ${\mathit{C}}_{\mathit{\ell}}^{\mathit{XY}}\mathrm{\equiv}{\sum}_{\mathit{m}}{\mathit{a}}_{\mathit{\ell m}}^{\mathit{X}}{\mathit{a}}_{\mathit{\ell m}}^{\mathit{Y}\mathrm{\ast}}\mathit{/}\mathrm{(}\mathrm{2}\mathit{\ell}\mathrm{+}\mathrm{1}\mathrm{)}$ is modelled as $\begin{array}{ccc}\mathrm{\Delta}{\mathit{C}}_{\mathit{\ell}}^{\mathit{TE}}& \mathrm{=}& \\ \mathrm{\Delta}{\mathit{C}}_{\mathit{\ell}}^{\mathit{EE}}& \mathrm{=}& \end{array}$Here ε_{ℓ} is a polynomial in multipole ℓ determined by the effective beam of the detectorassembly measuring the polarized signal. Considering an effective beam map $\mathit{b}\mathrm{\left(}\stackrel{\u02c6}{{n}}\mathrm{\right)}$ (rotated so that it is centred on the north pole), its spherical harmonic coefficients are defined as ${\mathit{b}}_{\mathit{\ell m}}\mathrm{\equiv}{}^{\mathrm{\int}}\mathrm{d}\stackrel{\u02c6}{{n}}\hspace{0.17em}\mathit{b}\mathrm{\left(}\stackrel{\u02c6}{{n}}\mathrm{\right)}\hspace{0.17em}{\mathit{Y}}_{\mathit{\ell m}}^{\mathrm{\ast}}\mathrm{\left(}\stackrel{\u02c6}{{n}}\mathrm{\right)}$. As a consequence of the Planck scanning strategy, pixels are visited approximately every six months, with a rotation of the focal plane by 180deg, and we expect b_{ℓm} to be dominated by even values of m, and especially the modes m = 2 and 4, which describe the beam ellipticity. As noted by, e.g., Souradeep & Ratra (2001) for elliptical Gaussian beams, the PlanckHFI beams for a detector d obey ${\mathit{b}}_{\mathit{\ell m}}^{\mathrm{\left(}\mathit{d}\mathrm{\right)}}\mathrm{\simeq}{\mathit{\beta}}_{\mathit{m}}^{\mathrm{\left(}\mathit{d}\mathrm{\right)}}{\mathit{\ell}}^{\mathit{m}}{\mathit{b}}_{\mathit{\ell}\mathrm{0}}^{\mathrm{\left(}\mathit{d}\mathrm{\right)}}\mathit{.}$(46)We therefore fit the spectra using a fourthorder polynomial $\mathit{\epsilon}\mathrm{\left(}\mathit{\ell}\mathrm{\right)}\mathrm{=}{\mathit{\epsilon}}_{\mathrm{0}}\mathrm{+}{\mathit{\epsilon}}_{\mathrm{2}}{\mathit{\ell}}^{\mathrm{2}}\mathrm{+}{\mathit{\epsilon}}_{\mathrm{4}}{\mathit{\ell}}^{\mathrm{4}}\mathit{,}$(47)treating the coefficients ε_{0}, ε_{2}, and ε_{4} as nuisance parameters in the MCMC analysis. Tests performed on detailed simulations of Planck observations with known mismatched beams have shown that Eqs. (42) and (47) describe the power leakage due to beam mismatch with an accuracy of about 20% in the ℓ range 100−2000.
Fig. 23 Best fit of the power spectrum leakage due to the beam mismatch for TE (Eq. (45a), upper panel) and EE (Eq. (45b), lower panel). In each case, we show the correction for individual crossspectra (coloured thin lines) and the coadded correction (black line). The individual crossspectra corrections are only shown in the range of multipoles where the data from each particular pair is used. The individual correction can be much higher than the coadded correction. The coadded correction is dominated by the best S/N pair for each multipole. For example, up to ℓ = 500, the TE coadded correction is dominated by the 100 × 143 contribution. The grey dashed lines show the TE and EE bestfit spectra rescaled by a factor of 20, to give an idea of the location of the model peaks. 
The equations above suggest that the same polynomial ε can describe the contamination of the TE and EE spectra for a given pair of detector sets. But in the current Plik analysis, the TE crossspectrum of two different maps a and b is the inversevarianceweighted average of the crossspectra T_{a}E_{b} and T_{b}E_{a}, while EE is simply E_{a}E_{b}. In addition, the temperature maps include the signal from SWBs, which is obviously not the case for the E maps. We therefore describe the TE and EE corrections by different ε parameters. Similarly, we treated the parameters for the EE crossfrequency spectra as being uncorrelated with the parameters for the autofrequency ones.
The leakage is driven by the discrepancy between the individual effective beams ${\mathit{b}}_{\mathit{\ell m}}^{\mathrm{\left(}\mathit{d}\mathrm{\right)}}$ making up a detector assembly, coupled with the details of the scanning strategy and relative weight of each detector. If we assumed a perfect knowledge of the beams, precise – but not necessarily accurate – numerical predictions of the leakage would be possible. However, we preferred to adopt a more conservative approach in which the leakage was free to vary over a range wide enough to enclose the true value. On the other hand, in order to limit the unphysical range of variations permitted by so many nuisance parameters, we need priors on the ε_{m} terms used in the Monte Carlo explorations. We assume Gaussian distributions of zero mean with a standard deviation σ_{m} representative of the dispersion found in simulations of the effect with realistic instrumental parameters. We found σ_{0} = 1 × 10^{5}, σ_{2} = 1.25 × 10^{8}, and σ_{4} = 2.7 × 10^{15}. This procedure ignores correlations between terms of different m, and is therefore likely substantially too permissive.
Another way of deriving the beam leakage would be to use a cosmological prior, i.e., by finding the best fit when holding the cosmological parameters fixed at their bestfit values for base ΛCDM. Figure 23 shows the result of this procedure for the crossfrequency pairs. The figure also shows the implied correction for the coadded spectra. This correction is dominated by the pair with the highest S/N at each multipole. The fact that different sets are used in different ℓranges leads to discontinuities in the correction template of the coadded spectrum. As can be seen in the figure, the coadded beamleakage correction, of order μK^{2}, is much smaller than the individual corrections, which partially compensate each other on average (but improve the agreement between the individual polarized crossfrequency spectra).
It is shown in Appendix C.3.5 that neither procedure is fully satisfactory. The cosmological prior leads to nuisance parameters that vastly exceed the values allowed by the physical priors, and the physical priors are clearly overly permissive (leaving the cosmological parameters unchanged but with doubled error bars for some parameters). In any case, the agreement between the different crossspectra remains much poorer in polarization than in temperature (see Sect. 4.4, Fig. 40, and Appendix C.3.5); they present oscillatory features similar to the ones produced by our beam leakage model, but the model is clearly not sufficient. For lack of a completely satisfactory global instrumental model, this correction is only illustrative and it is not used in the baseline likelihood.
3.4.4. Noise modelling
To predict the variance of the empirical power spectra, we need to model the noise properties of all maps used in the construction of the likelihood. As described in detail in Planck Collaboration VII (2016) and Planck Collaboration VIII (2016), the Planck HFI maps have complicated noise properties, with noise levels varying spatially and with correlations between neighbouring pixels along the scanning direction.
Fig. 24 Deviations from a white noise power spectrum induced by noise correlations. We show halfring difference power spectra for 100 GHz halfmission 1 maps (blue lines) of Stokes parameters I (top panel), Q (middle panel), and U (bottom panel). The bestfitting analytical model of the form Eq. (48) is overplotted in red. 
Fig. 25 Difference between auto and crossspectra for the 100 GHz halfmission maps, divided by the noise estimate from halfring difference maps (blue and green lines). Noise estimates derived from halfring difference maps are biased low. We fit the average of both halfmission curves (black line) with a power law model (red line). The analysis procedure is applied to the Stokes parameter maps I, Q, and U (top to bottom). All data power spectra are smoothed. 
For each channel, fullresolution noise variance maps are constructed during the mapmaking process (Planck Collaboration VIII 2016). They provide an approximation to the diagonal elements of the true n_{pix} × n_{pix} noise covariance matrix for Stokes parameters I (temperature only), or I, Q, and U (temperature and polarization). While it is possible to capture the anisotropic nature of the noise variance with these objects, noise correlations between pixels remain unmodelled. To include deviations from a whitenoise power spectrum, we therefore make use of halfring difference maps. Choosing the 100 GHz map of the first halfmission as an example, we show the scalar (spin0) power spectra of the three temperature and polarization maps in Fig. 24, rescaled by arbitrary constants. We find that the logarithm of the HFI noise power spectra as given by the halfring difference maps can be accurately parameterized using a fourthorder polynomial with an additional logarithmic term, $\mathrm{log}\mathrm{\left(}{\mathit{C}}_{\mathit{\ell}}^{\mathrm{HRD}}\mathrm{\right)}\mathrm{=}\sum _{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathrm{4}}{\mathit{\alpha}}_{\mathit{i}}\hspace{0.17em}{\mathit{\ell}}^{\mathit{i}}\mathrm{+}{\mathit{\alpha}}_{\mathrm{5}}\mathrm{log}\mathrm{(}\mathit{\ell}\mathrm{+}{\mathit{\alpha}}_{\mathrm{6}}\mathrm{)}\mathit{.}$(48)Since lowfrequency noise and processing steps like deglitching leave residual correlations between both halfring maps, noise estimates derived from their difference are biased low, at the percent level at highℓ (where it was first detected and understood, see Planck Collaboration VI 2014). We correct for this effect by comparing the difference of autopowerspectra and crossspectra (assumed to be free of noise bias) at a given frequency with the noise estimates obtained from halfring difference maps. As shown in Fig. 25, we use a a powerlaw model with free spectral index to fit the average of the ratios of the first and second halfmission results to the halfring difference spectrum, using the average to nullify chance correlations between signal and noise: ${\mathit{C}}_{\mathit{\ell}}^{\mathrm{bias}}\mathrm{=}{\mathit{\alpha}}_{\mathrm{0}}\hspace{0.17em}{\mathit{\ell}}^{{\mathit{\alpha}}_{\mathrm{1}}}\mathrm{+}{\mathit{\alpha}}_{\mathrm{2}}\mathit{.}$(49)At a multipole moment of ℓ = 1000, we obtain correction factors for the temperature noise estimate obtained from halfring difference maps of 9%, 10%, and 9% at 100, 143, and 217 GHz, respectively.
In summary, our HFI noise model is obtained as follows. For each map, we capture the anisotropic nature of the noise amplitude by using the diagonal elements of the pixelspace noise covariance matrix. The corresponding whitenoise power spectrum is then modulated in harmonic space using the product of the two smooth fitting functions given in Eqs. (48) and (49).
Fig. 26 Correlated noise model. In grey are shown the crossdetector TT spectra of the halfring difference maps. The black line show the same, smoothed by a Δℓ = 200 sliding average, while the blue data points are a Δℓ = 100 binned version of the grey line. Error bars simply reflect the scatter in each bin. The green line is the splinesmoothed version of the data that we use as our correlated noise template. 
Correlated noise between detectors.
If there is some correlation between the noise in the different cuts in our data, the trick of only forming effective frequencypair power spectra from crossspectra to avoid the noise biases fails. In 2013, we evaluated the amplitude of such correlated noise between different detsets. The correlation, if any, was found to be small, and we estimated its effect on the cosmological parameter fits to be negligible. As stated in Sect. 3.2.1, the situation is different for the 2015 data. Indeed, we now detect a small but significant correlated noise contribution between the detsets. This is the reason we change our choice of data to estimate the crossspectra, from detsets to halfmission maps. The correlated noise appears to be much less significant in the latter.
To estimate the amount of correlated noise in the data, we measured the crossspectra between the halfring difference maps of all the individual detsets. The crossspectra are then summed using the same inversevariance weighting that we used in 2013 to form the effective frequencypair spectra. Figure 26 shows the spectra for each frequency pair. All of these deviate significantly from zero. We build an effective correlated noise template by fitting a smoothing spline on a Δℓ = 200 sliding average of the data. Given the noise level in polarization, we did not investigate the possible contribution of correlated noise in EE and TE.
Section 4.1.1 shows that when these correlated noise templates are used, the results of the detsets likelihood are in excellent agreement with those based on the baseline, halfmission one.
3.5. Covariance matrix structure
The construction of a Gaussian approximation to the likelihood function requires building covariance matrices for the pseudopower spectra. Mathematically exact expressions exist, but they are prohibitively expensive to calculate numerically at Planck resolution (Wandelt et al. 2001); we thus follow the approach taken in Like13 and make use of analytical approximations (Hansen et al. 2002; Hinshaw et al. 2003; Efstathiou 2004; Challinor & Chon 2005).
For our baseline likelihood, we calculate covariance matrices for all 45 unique detector combinations that can be formed out of the six frequencyaveraged halfmission maps at 100, 143, and 217 GHz. To do so, we assume a fiducial power spectrum that includes the data variance induced by the CMB and all foreground components described in Sect. 3.3; this variance is computed assuming these components are Gaussiandistributed. The effect of this approximation regarding Galactic foregrounds is tested by means of simulations in Sect. 3.6. The fiducial model is taken from the bestfit cosmological and foreground parameters; since they only become available after a full exploration of the likelihood, we iteratively refine our initial guess. As discussed in Sect. 3.1, the data vector used in the likelihood function of Eq. (13) is constructed from frequencyaveraged power spectra. Following Like13, for each polarization combination, we therefore build averaged covariance matrices for the four frequencies ν_{1},ν_{2},ν_{3},ν_{4}, $\begin{array}{ccc}\mathrm{Var}\mathrm{\left(}\mathit{C\u0302}\begin{array}{c}\mathit{XY}{\mathit{\nu}}_{\mathrm{1}}\mathit{,}{\mathit{\nu}}_{\mathrm{2}}\\ \mathit{\ell}\end{array}\mathit{,}\mathit{C\u0302}\begin{array}{c}\mathit{ZW}{\mathit{\nu}}_{\mathrm{3}}\mathit{,}{\mathit{\nu}}_{\mathrm{4}}\\ {\mathit{\ell}}^{\mathrm{\prime}}\end{array}\mathrm{\right)}& \mathrm{=}& \begin{array}{c}\sum \\ \mathrm{\left(}\mathit{i,j}\mathrm{\right)}\mathrm{\in}\mathrm{\left(}{\mathit{\nu}}_{\mathrm{1}}\mathit{,}{\mathit{\nu}}_{\mathrm{2}}\mathrm{\right)}\mathrm{\left(}\mathit{p,q}\mathrm{\right)}\mathrm{\in}\mathrm{\left(}{\mathit{\nu}}_{\mathrm{3}}\mathit{,}{\mathit{\nu}}_{\mathrm{4}}\mathrm{\right)}\end{array}{\mathit{w}}_{\mathit{\ell}}^{\mathit{XY}\mathit{i,j}}{\mathit{w}}_{{\mathit{\ell}}^{\mathrm{\prime}}}^{\mathit{ZW}\mathit{p,q}}\\ & & \end{array}$(50)where X,Y,Z,W ∈ { T,E }, and w^{XYi,j} is the inversevariance weight for the combination (i,j), computed from ${\mathit{w}}_{\mathit{\ell}}^{\mathit{XY}\mathit{i,j}}\mathrm{\propto}\mathrm{1}\mathit{/}\mathrm{Var}\mathrm{(}\mathit{C\u0302}\begin{array}{c}\mathit{XY}\mathit{i,j}\\ \mathit{\ell}\end{array}\mathit{,}\mathit{C\u0302}{\begin{array}{c}\mathit{XY}\mathit{i,j}\\ \mathit{\ell}\end{array}}^{\mathrm{)}}\mathit{,}$(51)and normalized to unity. For the averaged XY = TE covariance (and likewise for ZW = TE), the sum in Eq. (50) must be taken over the additional permutation XY = ET. That is, the two cases where the temperature map of channel i is correlated with the polarization map of channel j and vice versa are combined into a single frequencyaveraged covariance matrix. These matrices are then combined to form the full covariance used in the likelihood, $\mathrm{C}\mathrm{=}\left(\begin{array}{c}\\ {\mathit{C}}^{\mathit{TTTT}}& {\mathit{C}}^{\mathit{TTEE}}& {\mathit{C}}^{\mathit{TTTE}}\\ {\mathit{C}}^{\mathit{EETT}}& {\mathit{C}}^{\mathit{EEEE}}& {\mathit{C}}^{\mathit{EETE}}\\ {\mathit{C}}^{\mathit{TETT}}& {\mathit{C}}^{\mathit{TEEE}}& {\mathit{C}}^{\mathit{TETE}}\end{array}\right)\mathit{,}$(52)where the individual polarization blocks are constructed from the frequencyaveraged covariance matrices of Eq. (50) (Like13).
Shifts of parameters over 300 TT simulations.
Appendix C.1.1 provides a summary of the equations used to compute temperature and polarization covariance matrices and presents a validation of the implementation through direct simulations. Let us note that, for the approximations used in the analytical computation of the covariance matrix to be precise, the mask power spectra have to decrease quickly with multipole moment ℓ; this requirement gives rise to the apodization scheme discussed in Sect. 3.2.2. In the presence of a pointsource mask, however, the condition may no longer be fulfilled, reducing the accuracy of the approximations assumed in the calculation of the covariance matrices. We discuss in Appendix C.1.4 the heuristic correction we developed to restore the accuracy, which is based on direct simulations of the effect.
3.6. FFP8 simulations
In order to validate the overall implementation and our approximations, we generated 300 simulated HFI halfmission map sets in the frequency range 100 to 217 GHz, which we analysed like the real data. For the CMB, we created realizations of the ΛCDM model with the bestfit parameters obtained in this paper. After convolving the CMB maps with beam and pixel window functions, we superimposed CIB, dust, and noise realizations from the FFP8 simulations (Planck Collaboration XII 2016) that capture both the correlation structure and anisotropy of foregrounds and noise. We then computed power spectra using the set of frequencydependent masks described in Sect. 3.2.2 and created the corresponding Plik TT likelihood. We modified the shape of the foreground spectra to fit the FFP8 simulations, but kept the parameterization used on the data. In the case of dust, we used priors similar to those used on data. Furthermore, in the following the dust amplitude parameter is named ${\mathrm{gal}}_{\mathrm{545}}^{\mathit{\nu}\mathrm{\times}{\mathit{\nu}}^{\mathrm{\prime}}}$. We then ran an MCMC sampler to derive the cosmological and foreground parameters posterior distributions for all dataset realizations.
Fig. 27 Plik parameter results on 300 simulations for the six baseline cosmological parameters, as well as the FFP8 CIB and Galactic dust amplitudes. The simulations include quite realistic CMB, noise, and foregrounds (see text). The distributions of inferred posterior mean parameters are centred around their input values with the expected scatter. Indeed the dotted red lines show the bestfit Gaussian for each distribution, with a mean shift, Δμ, and a departure Δσ from unit standard deviation given in the legend; both are close to zero. These best fits are thus very close to Gaussian distributions with zero shift and unit variance, which are displayed for reference as black lines. The legend gives the numerical value of Δμ and Δσ, as well as the pvalues of a KolmogorovSmirnov test of the histograms against a Gaussian distribution shifted from zero by Δμ and with standard deviation shifted from unity by Δσ. This confirms that the distributions are consistent with Gaussian distributions with zero mean and unit standard deviation, with a small offset of the mean. 
For each simulation, we computed the shift of the derived posterior mean parameters with respect to the input cosmology, normalized by their posterior widths σ_{post}. When a Gaussian prior with standard deviation σ_{prior} is used, we rescale σ_{post} by $\mathrm{[}\mathrm{1}\mathrm{}{\mathit{\sigma}}_{\mathrm{post}}^{\mathrm{2}}\mathit{/}{\mathit{\sigma}}_{\mathrm{prior}}^{\mathrm{2}}{\mathrm{]}}^{\mathrm{1}\mathit{/}\mathrm{2}}$; this is the case for τ and for the Galactic dust amplitudes ${\mathrm{gal}}_{\mathrm{545}}^{\mathit{\nu}}$ in the four crossfrequency channels used. In Fig. 27, we show histograms of the shifts we found for all 300 simulations for the six baseline cosmological parameters, as well as the FFP8 CIB and galactic dust amplitudes. As shown in the figure, we recover the input parameters with little bias and a scatter of the normalized parameter shifts around unity. The pvalues of the KolmogorovSmirnov test that we ran are given in the legend and we do not detect significant departures from normality. The average reduced χ^{2} for the histograms of Fig. 27 is equal to 1.02.
Table 13 (second column) compiles the average shifts of Fig. 27, but in order to gauge whether they are as small as expected for this number of simulations (assuming no bias), the shifts are expressed in units of the posterior width rescaled by $\mathrm{1}\mathit{/}\sqrt{\mathrm{300}}$. We note that the shift of the average is above one (scaled)σ in three cases out of a total of 11 parameters (68% of the Δs would be expected to lie within 1σ if the parameters were uncorrelated), with θ, n_{s}, and ${\mathrm{gal}}_{\mathrm{545}}^{\mathrm{217}}$ at the 1.7, 2.0, and 1.5 (scaled)σ level, respectively.
Before proceeding, let us note that an estimate (third column) of these shifts is obtained by simply computing the shift from a single likelihood using as input the average spectra of the 300 simulations. This effectively reduces cosmic variance and noise amplitude by a factor $\sqrt{\mathrm{300}}$ and, more importantly, it decreases the cost and length of the overall computation, enabling additional tests. These shift estimates are noted r_{A}. The table shows that significant improvement in the determination of n_{s} is obtained by removing lowℓ multipoles. Indeed, Cols. 4 and 5 of Table 13 show the variation of the shift when the ℓ_{min} of the highℓ likelihood is increased from 30 to 65 and 100. The shift in n_{s} is decreased by a factor two, while the decrease in the number of bins per crossfrequency spectrum is only reduced from 199 to 185 (having little impact on the size of the covariance matrix of cosmological parameters).
These changes with ℓ_{min} therefore trace the small biases back to the lowestℓ bins. It suggests that the Gaussian approximation used in the highℓ likelihood starts to become mildly inaccurate at ℓ = 30. Indeed, even if noticeable, this effect would contribute at most a 0.11σ bias on n_{s}. This is further confirmed by the lack of a detectable effect found in Sect. 5.1 when varying the hybridization scale in TT between Commander and Plik . However, the exclusion of lowℓ information degrades our ability to accurately reconstruct the foreground amplitudes ${\mathit{A}}_{\mathrm{CIB}}^{\mathrm{217}}$, ${\mathrm{gal}}_{\mathrm{545}}^{\mathrm{143}\mathrm{}\mathrm{217}}$, and ${\mathrm{gal}}_{\mathrm{545}}^{\mathrm{217}}$. Indeed, the dust spectral amplitudes in the 143 × 217 and 217 × 217 channels are highest at low multipoles, and the CIB spectrum in the range 30 ≤ ℓ ≤ 100 also adds substantial information.
In spite of this lowℓ tradeoff between an accurate determination of n_{s} on the one hand and ${\mathit{A}}_{\mathrm{CIB}}^{\mathrm{217}}$, ${\mathrm{gal}}_{\mathrm{545}}^{\mathrm{143}\mathrm{}\mathrm{217}}$, and ${\mathrm{gal}}_{\mathrm{545}}^{\mathrm{217}}$ on the other, we can conclude that the Plik implementation is behaving as expected and can be used for actual data analysis.
Appendix C.2 extends this conclusion to the joint Plik TT, EE, TE likelihood case.
3.7. Endtoend simulations
Endtoend parameter shifts for a single realization of CMB and foregrounds, along with five different noise realizations. Shifts are computed with respect to those obtained without noise and with instrumental effects turned off.
Endtoend parameter shifts for four different CMB realizations but comprising four pairs of realizations with the same noise realization with respect to those obtained without noise and with instrumental effects turned off.
While the previous section validated our methodology, our approximations, and the overall implementation, this does not yet give the sensitivity to residual systematic uncertainties undetected by data consistency checks. These are by their very nature very much more difficult to address realistically, since, when an effect is detected and sufficiently well understood, it can be modelled and is corrected for, in general at the TOIprocessing stage; only the uncertainty of the correction needs to be addressed. Still, HFI has developed a complete model of the instrument which contains all identified systematic effects and enables realistic simulation of the instrumental response. We have therefore generated a number of fullmission time streams which we have then processed with the DPC TOI processing pipeline in order to create map datasets as close to instrumental reality as we can in order to assess the possible impact of lowlevel residual instrumental systematics, the effects of which might have remained undetected otherwise.
In this section, we report on the shifts in the values of the cosmological and foreground parameters induced by these specific residual systematic effects, comparing the results of a TT likelihood analysis for two overlapping sets of five simulations:

1.
five simulations of maps at 100 GHz,143 GHz and 217 GHz, for asingle realization of the CMB and of the foregrounds but for fivedifferent realizations of the noise,

2.
five simulations of maps at 100 GHz, 143 GHz and 217 GHz, composed of four CMB realizations, two noise realizations, a single realization of the foregrounds, but forming four pairs of realizations having the same noise but different CMB.
These simulations sum to a total of eight distinct simulations and are numbered from 1 to 8 in Tables 14 and 15. To be more explicit, in the second set, among simulations numbered 4 to 8, simulations 4, 7 and 8 have different CMB, but the same noise as each other. Simulations 5 and 6 have different CMB, and the same noise as each other, but different from simulations 4, 7 and 8. Realizations 4 and 5, having the same CMB but different noise, are common to the two sets of five realizations.
Each of these have been performed twice, with the endtoend (instrument plus TOI processing) pipeline and noise contribution switched either on or off. Endtoend simulations are computationally very costly, (typically a week for each simulated mission dataset) and hence only a few realizations were generated).
As explained in Sect. 5.4 of Planck Collaboration VII (2016), the endtoend simulations are created by feeding the TOI processing pipeline with simulated data to evaluate and characterize the overall transfer function and the respective contribution of each individual effect on the determination of the cosmological parameters. Simulated TOIs are produced by applying the real mission scanning strategy to a realistic input sky specified by the Planck Sky Model (PSM; Delabrouille et al. 2013) containing a lensed CMB realization, galactic diffuse foregrounds, and the dipole components. To this skyscanned TOI, we add a whitenoise component, representing the phonon and photon noises. The very lowtemporalfrequency thermal drift seen in the real data is also added to the TOI. The noisy sky TOI is then convolved with the appropriate bolometer transfer functions. Another whitenoise component, representing Johnson noise and readout noise, is also added. Simulated cosmic rays using the measured glitch rates, amplitudes, and shapes are added to the TOI. This TOI is interpolated to the electronic HFI fastsampling frequency. It is then converted from analogue to digital using a simulated nonlinear analoguetodigital converter (ADC). Identified 4 K cooler spectral lines are added to the TOI. Both effects (ADC and 4 K lines) are derived from the measured inflight behaviour. The TOI finally goes through the data compression/decompression algorithm used for communication between the Planck satellite and Earth. The simulated TOI is then processed in the same way as the real mission data for cleaning and systematic error removal, calibration, destriping, and mapmaking.
Some limitations of the current endtoend approach follow. No pointing error is included, although previous (dedicated) simulations suggest that this has negligible effect. In addition, this effect was included in the dedicated simulations performed to assess the precision of the beam recovery procedure. The first step of the TOI processing is to correct the ADC nonlinearity (ADC NL). For the flight data, the ADC NL was determined by using HFI’s measured signal at the end of the HFI mission, with the instrument’s cooling system switched off and an instrument temperature equal to 4 K. This determination relied on supposing the signal to be perfect white noise and therefore to correspond to the distortions brought in by the ADC. In the current implementation, we assume perfect knowledge of ADC NL and 4 K lines. This is of course not true for the real data and future endtosimulations, accompanying Planck’s next data release, will improve our model of this effect. After ADC NL correction, the signal is converted to volts. Deglitching is then performed by flagging glitch heads and by using glitch tail timelines. This enables the creation of the thermal baseline which is used for signal demodulation. The thermal baseline and glitch tails are subtracted, the signal is converted to watts, and the 4 K lines are removed. The resulting signal is then deconvolved by the bolometer transfer functions. We do not include uncertainties in the glitch tail shape used in the deglitching procedure, i.e., the templates are the same for the simulations and the processing; but here again, previous studies suggest any difference is a small effect.
The analysis of these sets of endtoend simulations, and of their counterparts for which all instrumental effects are turned off, is performed similarly to that of the simulations described in Sect. 3.6. Angular power spectra for all crosshalfmissions and for all frequency combinations are computed using the Planck masks described in Sect. 3.2.2 and with the appropriate beam functions. Noise levels are evaluated as described in Sect. 3.4.4. Templates for galactic foregrounds (CO, freefree, synchrotron, thermal and spinning dust), the kinetic and thermal SZ effects, the cosmic infrared background, and radio and IR point sources are constructed based on the PSM input foreground maps. The covariance matrix is computed with the method outlined in Appendix C.1 with the aforementioned input CMB power spectrum, input foreground spectra, noise levels, beam functions and masks.
Fig. 28 Plik 2015 coadded TT, TE, and EE spectra. The blue points are for bins of Δℓ = 30, while the grey points are unbinned. The lower panels show the residuals with respect to the best fit Plik TT+tauprior ΛCDM model. The yellow lines show the 68% unbinned error bars. For TE and EE, we also show the bestfit beamleakage correction (green line; see text and Fig. 23). 
Fig. 29 Zoom in to various ℓ ranges of the HM coadded power spectra, together with the Plik TT+tauprior ΛCDM bestfit model (red line). We show the TT (top), TE (centre) and EE (bottom) power spectra. The lower panels in each plot show the residuals with respect to the bestfit model. 
All sets of power spectra and the inverse covariance matrix are then binned and used in the likelihood analysis performed using an MCMC sampler together with Plik and PICO in order to determine the best fit cosmological parameters. The shifts in cosmological parameter values induced by the imperfect correction of instrumental effets by the TOI processing pipeline are then computed for the endtoend simulations with respect to those obtained for the simulations without noise and with instrumental effects turned off, normalized by the endtoend simulations’ posterior widths. Comparing shifts computed in this way cancels out cosmic variance and chance correlations between the CMB and the foregrounds and are thus fully attributable to the instrument and to the noise, which cannot be disentangled, as well as to CMBnoise chance correlations. That is, those shifts probe directly the scatter and possible biases induced by residual systematics effects.
The mean and median shifts for the five simulations with a single CMB realization and a single foreground realization, but different noise realizations, are given in Table 14. In order to verify that these shifts are within expectations, we computed the shifts in cosmological parameters for 100 FFP8 simulations, each with identical CMB signal but different FFP8 noise, with respect to the cosmological parameters obtained for the CMB only, normalized by their posterior widths. The standard deviations of the resulting distributions are given in the column labelled “σ_{FFP8}” of Table 14 and can be compared with the shifts obtained for the five endtoend simulations. All shifts are within 1σ of the shifts expected from FFP8. In addition, there is no indication of any detectable bias. All shifts are thus compatible with scatter introduced by noise.
The shifts for the five realizations with four different CMB realizations, the same foregrounds, but comprising four pairs with the same noise realization, are given in Cols. 4 to 8 of Table 15. As mentioned at the beginning of this section, realizations numbered 4 and 5 are common to the sets of Tables 14 and 15. In the columns labelled “Δ_{5−6}” to “Δ_{7−8}”, we computed the absolute differences in the shifts within pairs of realizations having different CMB but the same foreground and noise realizations. We compare these differences to the standard deviations of the distributions of cosmological parameter shifts of 100 FFP8 simulations, varying the CMB but keeping the same FFP8 noise realization, with respect to the cosmological parameters of the corresponding CMB but without noise (column labelled “σ_{FFP8}”). These distributions quantify the impact of CMBnoise correlations on the determination of the cosmological parameters. The Table shows that among all Δ’s, 11 are within 1σ_{FFP8}, 7 are within 1 to 2σ_{FFP8}, 3 are within 2 to 3σ_{FFP8}, 2 are within 3 to 4σ_{FFP8}. 50% of the differences are within 1σ and 78% within 2σ. At the very worst, taking the example of Δ_{7−8} a cosmological parameter (θ) moves a total of 4σ, from −3σ to 1σ in units of σ_{FFP8} when the CMB is changed but the noise is left the same. This is rare but can be expected in a few percent of simulations. As in the case of the shifts listed in Table 14, there is thus no detectable bias, with all shifts compatible with those expected from FFP8.
In summary, we have detected no sign as yet of systematic biases of the cosmological parameters due to known lowlevel instrumental effects as corrected by the current HFI TOI processing pipeline. An increase in the significance of these tests is left for further work once the simulation chain is further optimized for more massive numerical work.
Fig. 30 Residuals of the coadded CMB TT power spectra, with respect to the Plik TT+tauprior bestfit model, in units of standard deviation. The three coloured bands (from the centre, yellow, orange, and red) represent the ± 1, ± 2, and ± 3σ regions. 
Fig. 31 Interfrequency foregroundcleaned TT power spectra differences, in μK^{2}. Each of the subpanels shows the difference, after foreground subtraction, between pairs of frequency power spectra (the spectrum named on the vertical axis minus the one named on the horizontal axis), in units of standard deviation. The coloured bands identify deviations that are smaller than one (yellow), two (orange), or three (red) standard deviations. We show the differences for both the HM power spectra (blue points) and the DS power spectra (light blue points) after correlated noise correction. Figure 41 displays the same quantities for the TE and EE spectra. 
3.8. Highmultipole reference results
This section describes the results obtained using the baseline Plik likelihood, in combination with a prior on the optical depth to reionization, τ = 0.07 ± 0.02 (referred to, in TT, as Plik TT+tauprior). The robustness and validation of these results (presented in Sect. 4) can therefore be assessed independently of any potential lowℓ anomaly, or hybridization issues. The full lowℓ + highℓ likelihood will be discussed in Sect. 5.
Figure 28 shows the highℓ coadded CMB spectra in TT, TE, and EE, and their residuals with respect to the bestfit ΛCDM model in TT (red line), both ℓbyℓ (grey points) and binned (blue circles). The blue error bars per bin are derived from the diagonal of the covariance matrix computed with the bestfit CMB as fiducial model. The bottom subpanels with residuals also show (yellow lines) the diagonal of the ℓbyℓ covariance matrix, which may be compared to the dispersion of the individual ℓ determinations. Parenthetically, it provides graphical evidence that TT is dominated by cosmic variance through ℓ ≈ 1600, while TE is cosmicvariance dominated at ℓ ≲ 160 and ℓ ≈ 260−460. The jumps in the polarization diagonalcovariance errorbars come from the variable ℓ ranges retained at different frequencies, which therefore vary the amount of data included discontinuously with ℓ. Figure 29 zooms in to five adjacent ℓ ranges on the coadded spectra to allow close inspection of the data distribution around the model.
More quantitatively, Table 16 shows the χ^{2} values with respect to the ΛCDM best fit to the Plik TT+tauprior data combination for the unbinned CMB coadded power spectra (obtained as described in Appendix C.4). The TT spectrum has a reduced χ^{2} of 1.03 for 2479 degrees of freedom, corresponding to a probability to exceed (PTE) of 17.2%; the base ΛCDM model is therefore in agreement with the coadded data. The bestfit ΛCDM model in TT also provides an excellent description of the coadded polarized spectra, with a PTE of 12.8% in TE and 34.6% in EE. This already suggests that extensions with, e.g., isocurvature modes can be severely constrained.
Despite this overall agreement, we note that the PTEs are not uniformly good for all crossfrequency spectra (see in particular the 100 × 100 and 100 × 217 in TE). This shows that the baseline instrumental model needs to include further effects to describe all of the data in detail, even if the averages over frequencies appear less affected. The green line in Fig. 28 (mostly visible in the $\mathrm{\Delta}{\mathit{C}}_{\mathit{\ell}}^{\mathit{EE}}$ plot) shows the bestfit leakage correction (shown on its own in Fig. 23), which is obtained when fixing the cosmology to the TTbased model. Let us recall, though, that this correction is for illustrative purposes only, and it is set to zero for all actual parameter searches. Indeed, we shall see that these leakage effects are not enough to bring all the data into full concordance with the model.
In more quantitative detail, Fig. 30 shows the binned (Δℓ = 100) residuals for the coadded CMB spectra in units of the standard deviation of each data point, (data − model)/error. For TT, we find the greatest deviations at ℓ ≈ 434 (−1.8σ), 464(2.7σ), 1214 (−2.1σ), and 1450 (−1.8,σ). At ℓ = 1754, where we previously reported a deficit due to the imperfect removal of the ^{4}HeJT cooler line (see Planck Collaboration XIII 2016, Sect. 3), there is now a less significant fluctuation, at the level of −1.4σ. The residuals in polarization show similar levels of discrepancy.
In order to assess whether these deviations are specific to one particular frequency channel or appear as a common signal in all the spectra, Fig. 31 shows foregroundcleaned TT power spectra differences across all frequencies, in units of standard deviations (details on how this is derived can be found in Appendix C.3.2). The agreement between TT spectra is clearly quite good. Figure 32 then shows the residuals per frequency for the TT power spectra with respect to the ΛCDM Plik TT+tauprior bestfit model (see also the zoomedin residual plots in Fig. C.5). The ℓ ≈ 434, 464, and 1214 deviations from the model appear to be common to all frequency channels, with differences between the frequencies smaller than 2σ. However, the deviation at ℓ ≈ 1450 is higher at 217 × 217 than in the other channels. In particular, the interfrequency differences (Fig. 31) between the 217 × 217 power spectrum and the 100 × 100, 143 × 143, and 143 × 217 ones show deviations at ℓ ≈ 1450 at the roughly 1.7, 2.6, and 3.4σ levels, respectively.
This interfrequency difference is due to a deficit in the residuals of the 217 × 217 channel of about −3.4σ in the bin centred at 1454 in Fig. 32. To better quantify this deviation, we also fit for a feature of the type cos^{2}((π/ 2)(ℓ−ℓ_{p}) / (Δℓ)), with maximum amplitude centred at ℓ_{p} = 1460, width Δℓ = 25 (we impose the feature to be zero at  ℓ−ℓ_{p}  > Δℓ) and with an independent amplitude in each frequency channel. At 217 × 217, we find an amplitude of (− 37.44 ± 9.5)μK^{2}, while in the other channels we find (− 15.0 ± 7.8)μK^{2} at 143 × 143 and (− 19.7 ± 7.9)μK^{2} at 143 × 217. This outlier seems to be at least in part due to chance correlation between the CMB and dust. Indeed, the amplitude of the feature in the different spectra is in rough agreement with the dust emission law. Moreover, the feature can also be found when varying the retained sky fraction in the galactic mask, again with an amplitude scaling compatible with a dust origin. We discuss below the impact on cosmological parameters, see the case “CUT ℓ = 1404−1504” in Fig. 35.
Finally we note that there is a deficit in the ℓ = 500−800 region (in particular between ℓ = 700 and 800) in the residuals of all the frequency spectra, roughly in correspondence with the position of the second and third peaks. Section 4.1 is dedicated to the study of these deviations and their impact on cosmological parameters. In spite of these marginally significant deviations from the model, the χ^{2} values shown in Table 16 indicate that the ΛCDM model is an acceptable fit to each of the unbinned individual frequency power spectra, with PTEs always in TT. We therefore proceed to examine the parameters of the bestfit model.
Fig. 32 Residuals in the halfmission TT power spectra after subtracting the Plik TT+tauprior ΛCDM bestfit model (blue points, except for those which differ by at least 2 or 3σ, which are coloured in orange or red, respectively). The light blue line shows the difference between the bestfit model obtained assuming a ΛCDM+A_{L} model and the ΛCDM bestfit baseline; the green line shows the difference of bestfit models using the ℓ_{max} = 999 likelihood (fixing the foregrounds to the baseline solution) minus the baseline bestfit (both in the ΛCDM framework); while the pink line is the same as the green one but for ℓ_{max} = 1404 instead of ℓ_{max} = 999; see text in Sect. 4.1. For the TE and EE spectra, see Fig. 40. 
Fig. 33 ΛCDM parameters posterior distribution for Plik TT+tauprior. The lower left triangle of the matrix displays how the constraints are modified when the information from one of the frequency channels is dropped. The upper right triangle displays how the constraints are modified when the information from multipoles ℓ greater or less than 1000 is dropped. All the results shown in this figure were obtained using the CAMB code. 
Fig. 34 TE (left) and EE (right) residuals conditioned on the TT spectrum (black line) with 1 and 2σ error bands. The blue points are the actual TE and EE residuals. We do not include any beamleakage correction here. 
The cosmological parameters of interest are summarized in Table 17. Let us note that the cosmological parameters inferred here are obtained using the same codes, priors, and assumptions as in Planck Collaboration XIII (2016), except for the fact that we use the much faster PICO (Fendt & Wandelt 2007a) code instead of CAMB when estimating cosmological parameters^{12} from TT,TE or TT, TE, EE using highℓPlanck data. Appendix C.5 establishes that the results obtained with the two codes only differ by small fractions of a standard deviation (less than 15% for most parameters, with a few more extreme deviations). However, we still use the CAMB code for results from EE alone, since in this case the parameter space explored is so wide that it includes regions outside the PICO interpolation region (see Appendix C.5 for further details).
Figure 33 shows the posterior distributions of each pair of parameters of the base ΛCDM model from Plik TT+tauprior. The upperright triangle compares the 1σ and 2σ contours for the full likelihood with those derived from only the ℓ< 1000 or the ℓ ≥ 1000 data. Section 4.1.6 addresses the question of whether the results from these different cases are consistent with what can be expected statistically. The lowerleft triangle further shows that the results are not driven by the data from a specific channel, i.e., dropping any of the 100, 143, or 217 GHz map data from the analysis does not lead to much change. The next section provides a quantitative analysis of this and other jackknife tests.
Goodnessoffit tests for the Plik temperature and polarization spectra at high ℓ.
Cosmological parameters used in this analysis.
We now turn to polarization results. Interfrequency comparisons and residuals for TE and EE spectra are analysed in detail in Sect. 4.4. Suffice it to say here that the results are less satisfactory than in TT, both in the consistency between frequency spectra and in the detailed χ^{2} results. This shows that the instrumental data model for polarization is less complete than for temperature, with residual effects at the μK^{2} level. The model thus needs to be further developed to take full advantage of the HFI data in polarization, given the level of noise achieved. We thus consider the highℓ polarized likelihood as a “beta” version. Despite these limitations, we include it in the product delivery, to allow external reproduction of the results, even though the tests that we show indicate that it should not be used when searching for weak deviations (at the μK^{2} level) from the baseline model.
Nevertheless, we generally find agreement between the TT, TE, and EE spectra. Figure 34 shows the TE, and EE residual spectra conditioned on TT, which are close to zero. This is particularly the case for TE below ℓ = 1000, which gives some confidence in the polarization model. Most of the data points for TE and EE lie in the ± 2σ range. As for all χ^{2}based evaluations, the interpretation of this result depends crucially on the quality of the error estimates, i.e., on the quality of our noise model (see Sect. 3.4.4). We further note that the agreement is consistent with the finding that unmodelled instrumental effects in polarization are at the μK^{2} level.
4. Assessment of the highmultipole likelihood
This section describes tests that we performed to assess the accuracy and robustness of the reference results of the highℓ likelihood that were presented above. First we establish the robustness of the TT results using Plik alone in Sect. 4.1 and with other likelihoods in Sect. 4.2. We verify in Sect. 4.3 that the amplitudes of the compactsource contributions derived at various frequencies are consistent with our current knowledge of source counts. We then summarize in Sect. 4.4 the results of the detailed tests of the robustness of the polarization results, which are expanded upon in Appendix C.3.5. The paper Planck Collaboration XVI (2016) examines the dependence of the power spectrum on angular direction.
4.1. TT robustness tests
Figure 35 shows the marginal mean and the 68% CL error bars for cosmological parameters calculated assuming different data choices, likelihoods, parameter combinations, and data combinations. The 31 cases shown assume a baseΛCDM framework, except when otherwise specified. The reference case uses the Plik TT+tauprior data combination. Figure 36 adds the specific results for the lensing parameter A_{L} (left) in a ΛCDM+ A_{L} framework and for the effective number of relativistic species N_{eff} (right) in a ΛCDM+ N_{eff} extended framework.
In both figures, the grey bands show the standard deviation of the parameter shifts relative to the baseline likelihood expected when using a subsample of the data (e.g., excising ℓranges or frequencies). Because the data sets used to make inferences about a model are changed, one would naturally expect the inferences themselves to change, simply because of the effects of noise and cosmic variance. The inferences could also be influenced by inadequacies in the model, deficiencies in the likelihood estimate, and systematic effects in the data. Indeed, one may compare posterior distributions from different data subsets with each other and with those from the full data set, in order to assess the overall plausibility of the analysis.
To this end it is useful to have some idea about the typical variation in posteriors that one would expect to see even in the ideal case of an appropriate model being used to fit data sets with correct likelihoods and no systematic errors. It can be shown (Gratton & Challinor, in prep.) that if Y is a subset of a data set X, and P_{X} and P_{Y} are vectors of the maximumlikelihood parameter values for the two data sets, then the sampling distribution of the differences of the parameter values is given by $\overline{)\mathrm{(}{{P}}_{\mathit{Y}}\mathrm{}{{P}}_{\mathit{X}}\mathrm{)}{\mathrm{(}{{P}}_{\mathit{Y}}\mathrm{}{{P}}_{\mathit{X}}\mathrm{)}}^{\mathrm{T}}}\mathrm{=}\mathrm{cov}\mathrm{\left(}{{P}}_{\mathit{Y}}\mathrm{\right)}\mathrm{}\mathrm{cov}\mathrm{\left(}{{P}}_{\mathit{X}}\mathrm{\right)}\mathit{,}$(53)i.e., the covariance of the differences is simply the difference of their covariances. Here the covariances are approximated by the inverses of the appropriate Fisher information matrices evaluated for the true model. One might thus expect the scatter in the modes of the posteriors to follow similarly, and to be able, if the parameters are wellconstrained by the data, to use covariances of the appropriate posteriors on the righthand side.
Fig. 35 Marginal mean and 68% CL error bars on cosmological parameters estimated with different data choices for the Plik likelihood, in comparison with results from alternate approaches or model. We assume a ΛCDM model and use variations of the Plik TT likelihood in most of the cases, in combination with a prior τ = 0.07 ± 0.02 (using neither lowℓ temperature nor polarization data). The “Plik TT+tauprior” case (black dot and thin horizontal black line) indicates the baseline (HM, ℓ_{min} = 30, ℓ_{max} = 2508), while the other cases are described in Sect. 4.1 (and 4.2, 5.6, E.4). The grey bands show the standard deviation of the expected parameter shift, for those cases where the data used is a subsample of the baseline likelihood (see Eq. (53)). All the results were run with PICO except for few ones that were run with CAMB , as indicated in the labels. 
4.1.1. Detset likelihood
We have verified (case “DS”) that the results obtained using the halfmission crossspectra likelihood are in agreement with those obtained using the detset (DS) crossspectra likelihood. As explained in Sect. 3.4.4, the main difficulty in using the DS likelihood is that the results might depend on the accuracy of the correlated noise correction. Reassuringly, we find that the results from the HM and DS likelihoods agree within 0.2σ. This is an important crosscheck, since we expect the two likelihoods to be sensitive to different kinds of temporal systematics. Direct differences of halfmission versus detsetbased TT crossfrequency spectra are compared in Fig. 31 (Fig. 41 shows similar plots for the TE and EE spectra.).
When using the detsets, we fit the calibration coefficients of the various detector sets with respect to a reference. The resulting bestfit values are very close to one^{13}, with the greatest calibration refinement being less than 0.2%, in line with the accuracy expected from the description of the data processing in Planck Collaboration VIII (2016). This verifies that the maps produced by the HFI DPC and used for the halfmissionbased likelihood come from the aggregation of wellcalibrated and consistent data.
4.1.2. Impact of Galactic mask and dust modelling
We have tested the robustness of our results with respect to our model of the Galactic dust contribution in various ways.
Galactic masks.
We have examined the impact of retaining a smaller fraction of the sky, less contaminated by Galactic emission. The baseline TT likelihood uses the G70, G60, and G50 masks (see Appendix A) at 100, 143, and 217 GHz, respectively. We have tested the effects of using G50, G41, and G41 (corresponding to ${\mathit{f}}_{\mathrm{sky}}^{\mathrm{noap}}\mathrm{=}\mathrm{0.60}$, 0.50, and 0.50 before apodization, case “M605050” in Fig. 35), and of the priors on the Galactic dust amplitudes relative to these masks described in Table 11. We find stable results as we vary these sky cuts, with the greatest shift in θ_{MC} of 0.5σ, compatible with the expected shift of 0.57σ calculated using Eq. (53). Going to higher sky fraction is more difficult. Indeed, the improvement in the parameter determination from increasing the sky fraction at 143 GHz and 217 GHz would be modest, as we would only gain information in the smallscale regime, which is not probed by 100 GHz. Increasing the sky fraction at 100 GHz is also more difficult because our estimates have shown that adding as little as 5% of the sky closer to the Galactic plane requires a change in the dust template and more than doubles the dust contamination at 100 GHz.
Amplitude priors.
We have tested the impact of not using any prior (i.e., using arbitrarily wide, uniform priors) on the Galactic dust amplitudes (case “No gal. priors” in Fig. 35). Again, cosmological results are stable, with the greatest shifts in ln(10^{10}A_{s}) of 0.23σ and in n_{s} of 0.20σ. The values of the dust amplitude parameters, however, do change, and their bestfit values increase by about 15 μK^{2} for all pairs of frequencies, while at the same time the error bars of the dust amplitude parameters increase very significantly. All of the amplitude levels obtained from the 545 GHz crosscorrelation are within 1σ of this result. The dust levels from this experiment are clearly unphysically high, requiring for the 100 × 100 pair. This level of dust contamination is clearly not allowed by the 545 × 100 crosscorrelation, demonstrating that the prior deduced from it is informative. Nevertheless, the fact that cosmological parameters are barely modified in this test indicates that the values of the dust amplitudes are only weakly correlated with those of the cosmological parameters, consistent with the results of Figs. 44 and 45 below, which show the parameter correlations quantitatively.
Galactic dust template slope.
We have allowed for a variation of the Galactic dust index n, defined in Eq. (24), from its default value n = −2.63, imposing a Gaussian prior of −2.63 ± 0.05 (“GALINDEX” case in Fig. 35). We find no shift in cosmological parameters (smaller than ~ 0.1σ) and recover a value for the index of n = −2.572 ± 0.038, consistent with our default choice.
Impact of ℓ ≲500 at 217 GHz.
We have analysed the impact of excising the first 500 multipoles (“LMIN=505 at 217 GHz” in Fig. 35) in the 143 × 217 and 217 × 217 spectra, where the Galactic dust contamination is the strongest. We find very good stability in the cosmological parameters, with the greatest change being a 0.16σ increase in n_{s}. This is compatible with the expectations estimated from Eq. (53) of 0.14σ. The inclusion of the first 500 multipoles at 217 GHz in the baseline Plik likelihood is one of the sources of the roughly 0.45σ difference in n_{s} observed when using the CamSpec code, since the latter excises that range of multipoles; for further discussion see Planck Collaboration XIII (2016, Table 1 and Sect. 3.1), as well as Sect. 4.2.
Fig. 36 Marginal mean and 68% CL error bars on the parameters A_{L} (left) and N_{eff} (right) in ΛCDM extensions, estimated with different data choices for the Plik TT likelihood in comparison with results from alternate approaches or model, combined with a Gaussian prior on τ = 0.07 ± 0.02 (i.e., neither lowℓ temperature nor polarization data). The “Plik TT+tauprior” case indicates the baseline (HM, ℓ_{min} = 30, ℓ_{max} = 2508), while the other cases are described in subsections of Sect. 4.1. The thin horizontal black line shows the baseline result and the thick dashed grey line displays the ΛCDM value (A_{L} = 1 and N_{eff} = 3.04). The grey bands show the standard deviation of the expected parameter shift, for those cases where the data used is a subsample of the baseline likelihood (see Eq. (53)). 
4.1.3. Impact of beam uncertainties
The case labelled “BEIG” in Fig. 35 corresponds to the exploration of beam eigenvalues with priors 10 times higher than indicated by the analysis of our MC simulation of beam uncertainties (which indicated by dotted lines in Fig. 21). This demonstrates that these beam uncertainties are so small in this data release that they do not contribute to the parameter posterior widths. They are therefore not enabled by default.
4.1.4. Interfrequency consistency and redundancy
We have tested the effect of estimating parameters while excluding one frequency channel at a time. In Figs. 33 and 35, the “no100” case shows the effect of excluding the 100 × 100 frequency spectrum, the “no143” of excluding the 143 × 143 and 143 × 217 spectra, and the “no217” of excluding the 143 × 217 and 217 × 217 spectra.
We obtain the greatest deviations in the “no217” case for ln(10^{10}A_{s}) and τ, which shift to lower values by 0.53σ and 0.47σ, about twice the expected shift calculated using Eq. (53), 0.25σ and 0.23σ respectively (in units of standard deviations of the “no217” case). The value of Ω_{c}h^{2} decreases by only −0.1σ. Figure 37 further shows the 217 × 217 spectrum conditioned on the 100 × 100 and 143 × 143 ones. This conditional deviates significantly in two places, at ℓ = 200 and ℓ = 1450. The ℓ = 1450 case was already discussed in Sect. 3.8 and is further analysed in Sect. 4.1.6. Around ℓ = 200, we see some excess scatter (both positive and negative) in the data around a jump between two consecutive bins of the conditional. This corresponds to the two bins around the first peak (one right before and the other almost at the location of the first peak), as can be seen in Fig. 28. All of the frequencies exhibit a similar behaviour (see Fig. 32); however, it is most pronounced in the 217 GHz case. This multipole region is also near the location of the bump in the effective dust model. The magnitude of this excess power in the model is not big enough or sharp enough to explain this excess scatter (see Fig. 17). Finally, note that the bestfit CMB solution at large scales is dominated by the 100 × 100 data, which are measured on a greater sky fraction (see Fig. 14).
This test shows that the parameters of the ΛCDM model do not rely on any specific frequency map, except for a weak pull of the higher resolution 217 GHz data towards higher values of both A_{s} and τ (but keeping A_{s}exp(−2τ) almost constant).
Fig. 37 217 × 217 spectrum conditioned on the joint result from the 100 × 100 and 143 × 143 spectra. The most extreme outliers are at ℓ = 200 and ℓ = 1450. 
4.1.5. Changes of parameters with ℓ_{min}
We have checked the stability of the results when changing ℓ_{min} from the baseline value of ℓ_{min} = 30 to ℓ_{min} = 50 and 100 (and ℓ_{min} = 1000, which is discussed in Sect. 4.1.6). These correspond to the cases labelled “LMIN 50” and “LMIN 100” in Fig. 35 (to be compared to the reference case “Plik TT+tauprior”). This check is important, since the Gaussian approximation assumed in the likelihood is bound to fail at very low ℓ (for further discussion, see Sect. 3.6).
The results are in good agreement, with shifts in parameters smaller than 0.2σ, well within expectations calculated from Eq. (53). This is also confirmed in Fig. 42, where the TT hybridization scale of the full likelihood is varied (i.e., the multipole where the lowℓ and highℓ likelihoods are joined).
4.1.6. Changes of parameters with ℓ_{max}
We have tested the stability of our results against changes in the maximum multipole ℓ_{max} considered in the analysis. We test the restriction to ℓ_{max} in the range ℓ_{max} = 999−2310, with the baseline likelihood having ℓ_{max} = 2508. For each frequency power spectrum we choose ${\mathit{\ell}}_{\mathrm{max}}^{\mathrm{freq}}\mathrm{=}\mathrm{min}\mathrm{\left(}{\mathit{\ell}}_{\mathrm{max}}\mathit{,}{\mathit{\ell}}_{\mathrm{max}}^{\mathrm{freq}\mathit{,}\hspace{0.17em}\mathrm{base}}\mathrm{\right)}$, where ${\mathit{\ell}}_{\mathrm{max}}^{\mathrm{freq}\mathit{,}\hspace{0.17em}\mathrm{base}}$ is the baseline ℓ_{max} at each frequency as reported in Table 16. The results shown in Fig. 35 use the same settings as the baseline likelihood (in particular, we leave the same nuisance parameters free to vary) and always use a prior on τ.
The results in Fig. 35 suggest there is a shift in the mean values of the parameters when using low ℓ_{max}; e.g., for ℓ_{max} = 999, ln(10^{10}A_{s}), τ, and Ω_{c}h^{2} are lower by 1.0, 0.8, and 0.8σ with respect to the baseline parameters. These parameters then converge to the baseline values for ℓ_{max} ≳ 1500. Following the arguments given earlier (Eq. (53)), when using these nested subsamples of the baseline data we expect shifts of the order of 0.5, 0.4, and 0.8σ respectively, in units of the standard deviation of the ℓ_{max} = 999 results. We further note that the value of θ for ℓ_{max} ≲ 1197 is lower compared to the baseline value. In particular, at ℓ_{max} = 1197, its value is 0.8σ low, while the expected shift is of the order of 0.7σ, in units of the standard deviation of the ℓ_{max} = 1197 results. The value of θ then rapidly converges to the baseline for ℓ_{max} ≳ 1300. Figure C.8 in Appendix C.3.3 also shows that these shifts are related to a change in the amplitude of the foreground parameters. In particular, the overall level of foregrounds at each frequency decreases with increasing ℓ_{max}, partially compensating for the increase in ln(10^{10}A_{s}) and Ω_{c}h^{2}. Although all these shifts are compatible with expectations within a factor of 2, we performed some further investigations in order to understand the origin of these changes. In the following, we provide a tentative explanation.
Difference of χ^{2} values between pairs of bestfit models in different ℓ −ranges for the coadded TT power spectrum.
Table 18 shows the difference in χ^{2} between the bestfit model obtained using ℓ_{max} = 999 (or ℓ_{max} = 1404) and the baseline Plik TT+tauprior bestfit solution in different multipole intervals. For this test, we ran the ℓ_{max} cases fixing the nuisance parameters to the baseline bestfit solution. This is required in order to be able to “predict” the power spectra at multipoles higher than ℓ_{max}, since otherwise the foreground parameters, which are only weakly constrained by the lowℓ likelihood, can converge to unreasonable values. We note that fixing the foregrounds has an impact on cosmological parameters, which can differ from the ones shown in Fig. 35 (see Appendix C.3.4 for a direct comparison). Nevertheless, since the overall behaviour with ℓ_{max} is similar, we use this simplified scenario to study the origin of the shifts.
The χ^{2} differences in Table 18 indicate that the cosmology obtained using ℓ_{max} = 999 is a better fit in the region between ℓ = 630 and 829. In particular, the low value of θ preferred by the ℓ_{max} = 999 data set shifts the position of the third peak to smaller scales. This enables a better fit to the low points at ℓ ≈ 700−850 (before the third peak), followed by the high points at ℓ ≈ 850−950 (after the third peak). This is also clear from the residuals and the green solid line in Fig. 32, which shows the difference in bestfit models between the ℓ_{max} = 999 case and the reference case. However, the values in Table 18 also show that the ℓ_{max} = 999 cosmology is disfavoured by the multipole region between ℓ ≈ 1330−1430, before the fifth peak. The ℓ_{max} = 999 model predicts too little power in this multipole range, which can be better fit if the position of the fifth peak moves to lower multipoles. As a consequence, θ shifts to higher values when including ℓ_{max} ≳ 1400.
Concerning the shifts in Ω_{c}h^{2}, A_{s} and τ, Fig. 35 shows that these parameters converge to the full baseline solution between ℓ_{max} = 1404 and ℓ_{max} = 1505. The Δχ^{2} values in Table 18 between the bestfit ℓ_{max} = 1404 case and the baseline suggest that the ℓ_{max} = 1404 cosmology is disfavoured by the multipole region ℓ = 1430−1530 (fifth peak), and – at somewhat lower significance – by the regions close to the fourth peak (ℓ ≈ 1130−1230) and the sixth peak (ℓ ≈ 1730−1829). The pink line in Fig. 32 shows the differences between the ℓ_{max} = 1404 bestfit model and the baseline, and it suggests that the ℓ_{max} = 1404 cosmology predicts an amplitude of the highℓ peaks that is too large.
This effect can be compensated by more lensing, which can be obtained with greater values of Ω_{c}h^{2} and ln(10^{10}A_{s}), as well as a greater value of τ to compensate for the increase in A_{s} in the normalization of the spectra, as observed when considering ℓ_{max} ≳ 1500. This also explains why the baseline (ℓ_{max} = 2508) bestfit solution prefers a value of the optical depth which is 0.8σ higher than the mean value of the Gaussian prior (τ = 0.07 ± 0.02), τ = 0.085 ± 0.018. In order to verify this interpretation, we performed the following test (using the CAMB code instead of PICO ). We fixed the theoretical lensing power spectrum to the bestfit parameters preferred by the ℓ_{max} = 1404 cosmology, and estimated cosmological parameters using the baseline likelihood. This is the “CAMB, FIX LENS” case in Fig. 35, which shows that cosmological parameters shift back to the values preferred at ℓ_{max} = 1404 (“CAMB, ℓ_{max} = 1404”) if they cannot alter the amount of lensing in the model.
Since the ℓ ≈ 1400−1500 region is also affected by the deficit at ℓ = 1450 (described in Sect. 3.8), we tested whether excising this multipole region from the baseline likelihood (with ℓ_{max} = 2508) has an impact on the determination of cosmological parameters. The results in Fig. 35 (case “CUT ℓ = 1404−1504”) show that the parameter shifts are at the level of 0.47, −0.29, 0.38, and 0.45σ on Ω_{b}h^{2}, Ω_{c}h^{2}, θ, and n_{s}, respectively (0.39, 0.09, 0.24, and 0.29σ expected from Eq. (53)), confirming that this multipole region has some impact on the parameters, although it cannot completely account for the shift between the ℓ_{max} ≈ 1400 case and the baseline.
We also estimated cosmological parameters including only multipoles ℓ> 1000 (“LMIN 1000” case), and compared them to the “LMAX 999” case^{14} (see also Appendix C.5). The twodimensional posterior distributions in Fig. 33 show the complementarity of the information from ℓ ≤ 999 and ℓ ≥ 1000, with degeneracy directions between pairs of parameters changing in these two multipole regimes. The ℓ_{min} = 1000 likelihood sets constraints on the amplitude of the spectra A_{s}e^{− 2τ} and on n_{s} that are almost a factor of 2 weaker than the ones obtained with the baseline likelihood, and somewhat higher than the ones obtained with ℓ_{max} = 999. The value of τ is thus more effectively determined by its prior and shifts downward by 0.59σ with respect to the baseline. The value of Ω_{c}h^{2} shifts upward by 1.7σ (cf. 0.8σ expected from Eq. (53)). Whether this change is just due to a statistical fluctuation is still a matter of investigation.
However, since parameter shifts are correlated, we evaluated whether the ensemble of the shifts in all cosmological parameters between the ℓ_{max} = 999 and ℓ_{min} = 1000 cases are compatible with statistical expectations. In order to do so, we computed the ${\mathit{\chi}}_{\mathrm{\Delta}}^{\mathrm{2}}$ statistic of the shift as ${\mathit{\chi}}_{\mathrm{\Delta}}^{\mathrm{2}}\mathrm{=}\sum _{\mathit{ij}}{\mathrm{\Delta}}_{\mathit{i}}{\mathrm{\Sigma}}_{\mathit{ij}}^{1}{\mathrm{\Delta}}_{\mathit{j}}\mathit{,}$(54)where Δ_{i} is the difference in bestfit value of the ith parameter between the ℓ_{max} = 999 and ℓ_{min} = 1000 cases and Σ is the covariance matrix of the expected shifts, calculated as the sum of the parameter covariance matrices obtained in each of the two cases, ignoring correlations between the two datasets. We include in this calculation the ΛCDM parameters (Ω_{b}h^{2}, Ω_{c}h^{2}, θ, n_{s}, A_{s}exp(−2τ)), excluding τ, since the constraints on this parameter are dominated by the same prior in both cases, and using A_{s}exp(−2τ) instead of ln(10^{10}A_{s}), since the latter is very correlated with τ and the TT power spectrum is mostly sensitive to the combination A_{s}exp(−2τ). Finally, we estimate the ${\mathit{\chi}}_{\mathrm{\Delta}}^{\mathrm{2}}$ both in the case where we leave the foregrounds free to vary or in the case where we fix them to the best fit of the baseline PlikTT + tauprior solution. Assuming that ${\mathit{\chi}}_{\mathrm{\Delta}}^{\mathrm{2}}$ has a χ^{2} distribution for 5 degrees of freedom, we find that the shifts observed in the data are consistent with simulations at the 1.2σ (1.1σ with fixed foregrounds) level for the case where we do not include the lowℓTT likelihood at ℓ< 30 to the ℓ_{max} = 999 case, and at the 1.5σ (1.4σ with fixed foregrounds) level for the case where we include the lowℓTT likelihood. We also find that the use of A_{s}exp(−2τ) instead of ln(10^{10}A_{s}) changes these significances only in the case where we include the lowℓTT likelihood to the ℓ_{max} = 999 case and leave the foregrounds free to vary, in which case we find consistency at the level of 1.8σ, in agreement with the findings of Addison et al. (2016; although in this case the use of ln(10^{10}A_{s}) and the exclusion of τ makes this test less indicative of the true significance of the shifts). In all cases, we do not find evidence for a discrepancy between the two datasets. A more precise and extended evaluation and discussions of these shifts, based on numerical simulations, will be presented in a future publication.
4.1.7. Impact of varying A_{L}
Figure 36 (left) displays the impact of various choices on the value of the lensing parameter A_{L} in the ΛCDM+A_{L} framework. The baseline likelihood prefers a value of A_{L} that is about 2σ greater than the physical value, A_{L} = 1. It is clear that this preference only arises when data with ℓ_{max} ≳ 1400 are included, and it is caused by the same effects as we proposed in Sect. 4.1.6 to explain the shifts in parameters at ℓ_{max} ≳ 1400 in the ΛCDM case. More lensing helps to fit the data in the ℓ ≈ 1300−1500 region, as indicated by the χ^{2} differences between the ΛCDM+A_{L} bestfit and the ΛCDM one in Table 18. This drives the value of A_{L} to 1.159 ± 0.090 with Plik TT+tauprior, 1.8σ higher than expected. The case “ΛCDM+A_{L}” of Fig. 35 also shows that opening up this unphysical degree of freedom shifts the other cosmological parameters at the 1σ level; e.g., Ω_{c}h^{2} and A_{s} shift closer to the values preferred in the ΛCDM case when using ℓ_{max} ≲ 1400. While in the ΛCDM case high values of these parameters allow increasing lensing, in the ΛCDM+A_{L} case this is already ensured by a high value of A_{L}, so Ω_{c}h^{2} and A_{s} can adopt values that better fit the ℓ ≲ 1400 range. When using Plik TT in combination with the lowTEB likelihood, the deviation increases to 2.4σ, A_{L} = 1.204 ± 0.086,^{15} due to the fact that more lensing allows smaller values of Ω_{c}h^{2} and A_{s} and a greater value of n_{s}, better fitting the deficit at ℓ ≈ 20 in the temperature power spectrum (see Planck Collaboration XIII 2016, Sect. 5.1.2 and Fig. 13).
4.1.8. Impact of varying N_{eff}
We have investigated the effect of opening up the N_{eff} degree of freedom in order to assess the robustness of the constraints on the ΛCDM extensions, which rely heavily on the highℓ tail of the data. Figure 36 (right) shows that N_{eff} departs from the standard 3.04 value by about 1σ when using Plik TT+tauprior, N_{eff} = 2.7 ± 0.33. The χ^{2} improvement for this model over ΛCDM is only Δχ^{2} = 1.5. We note that when the lowTEB likelihood (or alternatively, the lowℓTT likelihood plus the prior on τ) is used in combination with Plik TT, the value of N_{eff} shifts higher by about 1σ, N_{eff} = 3.09 ± 0.29. This shift is about a factor 2 more than the one expected from Eq. (53), 0.5σ, between the Plik TT+tauprior and Plik TT+tauprior+lowℓTT cases. This shift is due to the fact that the deficit at ℓ ≈ 20 is better fit by higher n_{s} and, as a consequence, an increase in N_{eff} helps decreasing the enhanced power at high ℓ.
Figure 36 also shows that, not surprisingly, the most extreme variations as compared to the reference case (less than 1σ) arise when the highresolution data are dropped (by reducing ℓ_{max} or by removing the 217 GHz channel), owing to the strong dependence of the N_{eff} constraints on the damping tail.
Having opened up this degree of freedom, the standard parameters are now about 1σ away (see case “ΛCDM+N_{eff}” of Fig. 35), and such a model would prefer quite a low value of H_{0}, which would then be at odds with priors derived from direct measurements (see Planck Collaboration XIII 2016, for an indepth analysis).
4.2. Intercomparison of likelihoods
In addition to the baseline highℓPlik likelihood, we have developed four other highℓ codes, CamSpec , Hillipop , Mspec , and Xfaster . CamSpec and Xfaster have been described in separate papers (Planck Collaboration XV 2014; Rocha et al. 2011), and brief descriptions of Mspec and Hillipop are given in Appendix D. These codes have been used to perform data consistency tests, to examine various analysis choices, and to crosscheck each other by comparing their results and ensuring that they are the same. In general, we find good agreement between the codes, with only minor differences in cosmological parameters.
The CamSpec , Hillipop , and Mspec codes are, like Plik , based on pseudoC_{ℓ} estimators and an analytic calculation of the covariance (Efstathiou 2004, 2006), with some differences in the approximations used to calculate this covariance. The Xfaster code (Rocha et al. 2011) is an an approximation to the iterative, maximum likelihood, quadratic bandpower estimator based on a diagonal approximation to the quadratic Fisher matrix estimator (Rocha et al. 2011, 2010), with noise bias estimated using difference maps, as described in Planck Collaboration IX (2016). For temperature, all of the codes use the same Galactic masks, but they differ in pointsource masking: Hillipop uses a mask based on a combination of S/N> 7 and cuts based on flux, while the others use the baseline S/N> 5 mask described in Appendix A. The codes also differ in foreground modelling, in the choice of data combinations, and in the ℓrange. For the comparison presented here, all make use of halfmission maps.
Fig. 38 Comparison of power spectra residuals from different highℓ likelihood codes. The figure shows “data/calib − FG −Plik _{CMB}”, where “data” stands for the empirical crossfrequency spectra, “FG” and “calib” are the bestfit foreground model and recalibration parameter for each individual code at that frequency, and the bestfit model Plik _{CMB} is subtracted for visual presentation. These plots thus show the difference in the amount of power each code attributes to the CMB. The power spectra are binned in bins of width Δℓ = 100. The yaxis scale changes at ℓ = 500 for TT and ℓ = 1000 for EE (vertical dashes). 
Fig. 39 Comparison of error bars from the different highℓ likelihood codes. The quantities plotted are the ratios of each code’s error bars to those from Plik , and are for bins of width Δℓ = 100. Results are shown only in the ℓ range common to Plik and the code being compared. 
Figure 38 shows a comparison of the power spectra residuals and error bars from each code, while Fig. E.5 in Appendix E.4 compares the combined spectra with the bestfit model. In temperature, the main feature visible in these plots is an overall nearly constant shift, up to 10 μK^{2} in some cases. This represents a real difference in the bestfit power each code attributes to foregrounds. For context, it is useful to note the statistical uncertainty on the foregrounds; for example, the 1σ error on the total foreground power at 217 GHz at ℓ = 1500 is 2.5 μK^{2} (calculated here with Mspec , but similar for the other codes). Shifts of this level do not lead to very large differences in cosmological parameters except in a few cases that we discuss.
For easier visual comparison of error bars, we show in Fig. 39 the ratios of each code’s error bars to those from Plik . These have been binned in bins of width Δℓ = 100, and are thus sensitive to the correlation structure of each code’s covariance matrix, up to 100 multipoles into the offdiagonal. For all the codes and for both temperature and polarization, the correlation between multipoles separated by more than Δℓ = 100 is less than 3%, so Fig. 39 contains the majority of the relevant information about each code’s covariance.
A few differences are visible, mostly at high frequency, when the 217 GHz data are used. First, the Hillipop error bars in TT for 143 × 217 become increasingly tighter than the other codes at ℓ> 1700. This is because Hillipop , unlike the other codes, gives nonzero weight to 143 × 217 spectra when both the 143 and the 217 GHz maps come from the same halfmission. This leads to a slight increase in power at high ℓ compared to Plik , as can be seen in Fig. 38. Conversely, the Hillipop error bars are slightly larger by a few percent at ℓ< 1700; however the source of this difference is not understood. Second, the Mspec error bars in temperature are increasingly tighter towards higher frequency, as compared to other codes; for 217 × 217, Mspec uncertainties are smaller by 5−10% for ℓ between 1000 and 2000. This arises from the Mspec mapbased Galactic cleaning procedure, which removes excess variance due to CMB–foreground correlations by subtracting a scaled 545 GHz map. However, for polarization, where one must necessarily clean with the noisier 353 GHz maps, the Mspec error bars for TE and EE become larger. CamSpec , which also performs a map cleaning for lowℓ polarization, switches to a powerspectrum cleaning at higher ℓ to mitigate this effect.
The differences in ΛCDM parameters from TT are shown in Table 19. Generally, parameters agree to within a fraction ofσ, but with some differences we discuss. One thing to keep in mind in interpreting this comparison is that these differences are not necessarily indicative of systematic errors. Some of the differences are expected due to statistical fluctuations because different codes weight the data differently.
One of the biggest differences with respect to the baseline code is in n_{s}, which is higher by about 0.45σ for CamSpec , with a related downward shift of A_{s}e^{− 2τ}. To put these shifts into perspective, we refer to the whisker plots of Figs. 35 and 36 which compare CamSpec TT results with Plik in the ΛCDM case (base and extended). A difference in n_{s} of about 0.16σ between Plik and CamSpec can be attributed to the inclusion in Plik of the first 500 multipoles for 143 × 217 and 217 × 217; these multipoles are excluded in CamSpec (see also Sect. 4.1.2). Indeed, cutting out those multipoles in Plik brings n_{s} closer by 0.16σ to the CamSpec value and slightly degrades the constraint on n_{s} compared to the full Plik result. Using Eq. (53), we see that the shift and degradation in constraining power are consistent with expectations. A similar 0.16σ shift can be attributed to different dust templates. CamSpec uses a steeper power law index (−2.7). Using the CamSpec template in Plik brings n_{s} closer to the CamSpec value. Allowing the power law index of the galactic template to vary when exploring cosmological parameters yields a slightly shallower slope (see Sect. 4.1.2). The slope of the dust template is mainly determined at relatively high ℓ, i.e., in the regime where it is hardest to determine the template accurately since the dust contribution is only a small fraction of the CIB and pointsource contributions (see the ℓ ≳ 1000 parts of Figs. 19 and 20). The remaining difference of 0.13σ arises from differences in data preparation (maps, calibration, binning) and covariance estimates. We therefore believe that a 0.2σ is a conservative upper bound of the systematic error in n_{s} associated with the uncertainties in the modelling of foregrounds, which is the biggest systematic uncertainty in TT.
A shift that is less well understood is the ≈ 1σ shift in A_{s}e^{− 2τ} between Plik and Hillipop . The preference for a lower amplitude from Hillipop is sourced by the lower power attributed to the CMB, seen in Fig. 38. With τ partially fixed by the prior, this implies lower A_{s} and hence a smaller lensing potential envelope, explaining the somewhat lower value of A_{L} found by Hillipop . Tests performed with the same code suggest that 1σ is too great a shift to be explained simply by the different foreground models, so some part of it must be due to the different data weighting; as can be seen in Fig. 39, Hillipop gives less weight to 500 ≲ ℓ ≲ 1500, and slightly more outside of this region.
This comparison also shows the stability of the results with respect to the Galactic cleaning procedure. Mspec and Plik use different procedures, yet their parameter estimates agree to better than 0.5σ (see Appendix D.1). But we note that the Plik –CamSpec differences are higher in the polarization case, and can reach 1σ, as can be judged from the whisker plot in polarization of Fig. C.10.
Comparison between the parameter estimates from different highℓ codes.
4.3. Consistency of Poisson amplitudes with source counts
The Poisson component of the foreground model is sourced by shotnoise from astrophysical sources. In this section we discuss the consistency between the measured Poisson amplitudes and other probes and models of the source populations from which they arise. The Poisson amplitude priors that we calculate are not used in the main analysis, because they improve uncertainties on the cosmological parameters by at most 10%, and only for a few extensions; instead they serve as a selfconsistency check.
This type of check was also performed in Like13, which we update here by:

1.
developing a new method for calculating these priors that isaccurate enough to give realistic uncertainties on Poissonpredictions (for the first time);

2.
including a comparison of more theoretical models;

3.
taking into account the 2015 pointsource masks.
In Like13 the Poisson power predictions were calculated via $\begin{array}{ccc}{\mathit{C}}_{\mathit{\ell}}\mathrm{=}{\mathrm{\int}}_{\mathrm{0}}^{{\mathit{S}}_{\mathrm{cut}}}\mathrm{d}\mathit{S}\hspace{0.17em}{\mathit{S}}^{\mathrm{2}}\frac{\mathrm{d}\mathit{N}}{\mathrm{d}\mathit{S}}\mathit{,}& & \end{array}$(55)where dN/ dS is the differential number count, S_{cut} is an effective fluxdensity cut above which sources are masked, and the integral was evaluated independently at each frequency. Although it is adequate for rough consistency checks, Eq. (55) ignores the facts that the 2013 pointsource mask was built from a union of sources detected at different frequencies, and that the Planck fluxdensity cut varies across the sky, and it also ignores the effect of Eddington bias. In order to accurately account for all of these effects, we now calculate the Poisson power as $\begin{array}{ccc}{\mathit{C}}_{\mathit{\ell}}^{\mathit{ij}}\mathrm{=}{\mathrm{\int}}_{\mathrm{0}}^{\mathrm{\infty}}\mathrm{d}{\mathit{S}}_{\mathrm{1}}\mathit{...}\mathrm{d}{\mathit{S}}_{\mathit{n}}{\mathit{S}}_{\mathit{i}}{\mathit{S}}_{\mathit{j}}\frac{\mathrm{d}\mathit{N}\mathrm{\left(}{\mathit{S}}_{\mathrm{1}}\mathit{,}\mathit{...}\mathit{,}{\mathit{S}}_{\mathit{n}}\mathrm{\right)}}{\mathrm{d}{\mathit{S}}_{\mathrm{1}}\mathit{...}\mathrm{d}{\mathit{S}}_{\mathit{n}}}\hspace{0.17em}\mathit{I}\mathrm{\left(}{\mathit{S}}_{\mathrm{1}}\mathit{,}\mathit{...}\mathit{,}{\mathit{S}}_{\mathit{n}}\mathrm{\right)}\mathit{,}& & \end{array}$(56)where the frequencies are labelled 1...n, the differential source count model, dN/ dS, is now a function of the flux densities at each frequency, and I(S_{1},...,S_{n}) is the joint “incompleteness” of our catalogue for the particular cut that was used to build the pointsource mask.
Priors on the Poisson amplitudes given a number of different pointsource masks and models.
The joint incompleteness was determined by injecting simulated point sources into the Planck sky maps, using the procedure described in Planck Collaboration XXVI (2016). The same pointsource detection pipelines that were used to produce the Second Planck Catalogue of Compact Sources (PCCS2) were run on the injected maps, producing an ensemble of simulated Planck sky catalogues with realistic detection characteristics. The joint incompleteness is defined as the probability that a source would not be included in the mask as a function of the source flux density, given the specific masking thresholds being considered. The raw incompleteness is a function of sky location, because the Planck noise varies across the sky. The incompleteness that appears in Eq. (56) is integrated over the region of the sky used in the analysis; the injection pipeline estimates this quantity by injecting sources only into these regions.
Equation (56) can be applied to any theoretical model which makes a prediction for the multifrequency dN/ dS. We have adopted the following models.

1.
For radio galaxies we have two models. The first is the Tucciet al. (2011) model, updated toinclude new sourcecount measurements from Mocanuet al. (2013). We also consider aphenomenological model that is a power law in flux density andfrequency, and assumes that the sources’ spectral indices areGaussiandistributed with mean and standard deviation σ_{α}; we use different values for and σ_{α} above and below 143 GHz. We shall refer to this second model as the “powerlaw” model, and the differential source counts are given by $\begin{array}{ccc}& & \frac{\mathrm{d}\mathit{N}\mathrm{\left(}{\mathit{S}}_{\mathrm{1}}\mathit{,}{\mathit{S}}_{\mathrm{2}}\mathit{,}{\mathit{S}}_{\mathrm{3}}\mathrm{\right)}}{\mathrm{d}{\mathit{S}}_{\mathrm{1}}\mathrm{d}{\mathit{S}}_{\mathrm{2}}\mathrm{d}{\mathit{S}}_{\mathrm{3}}}\mathrm{=}\frac{\mathit{A}\mathrm{(}{\mathit{S}}_{\mathrm{1}}{\mathit{S}}_{\mathrm{2}}{\mathit{S}}_{\mathrm{3}}{\mathrm{)}}^{\mathit{\gamma}\mathrm{}\mathrm{1}}}{\mathrm{2}\mathit{\pi}{\mathit{\sigma}}_{\mathrm{12}}{\mathit{\sigma}}_{\mathrm{23}}}\\ & & \u2001\u2001\u2001\mathrm{\times}\mathrm{exp}\left[\mathrm{}\frac{\mathrm{\left(}\mathit{\alpha}\mathrm{\right(}{\mathit{S}}_{\mathrm{1}}\mathit{,}{\mathit{S}}_{\mathrm{2}}\mathrm{)}\mathrm{}\mathit{\alpha \u0305}\mathrm{12}{\mathrm{)}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\sigma}}_{\mathrm{12}}^{\mathrm{2}}}\mathrm{}\frac{\mathrm{\left(}\mathit{\alpha}\mathrm{\right(}{\mathit{S}}_{\mathrm{2}}\mathit{,}{\mathit{S}}_{\mathrm{3}}\mathrm{)}\mathrm{}\mathit{\alpha \u0305}\mathrm{32}{\mathrm{)}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\sigma}}_{\mathrm{32}}^{\mathrm{2}}}\right]\mathit{,}\end{array}$(57)where labels 1−3 refer to Planck 100, 143, and 217 GHz and α(S_{i},S_{j}) = ln(S_{j}/S_{i}) / ln(ν_{j}/ν_{i}). Both radio models are excellent fits to the available sourcecount data, and we take the difference between them as an estimate of model uncertainty. With the powerlaw model we are additionally able to propagate uncertainties in the source count data to the final Poisson estimate via MCMC.

2.
For dusty galaxies we use the Béthermin et al. (2012) model, as in Planck Collaboration XXX (2014). The model is in good agreement with the number counts measured with te Spitzer Space Telescope and the Herschel Space Observatory. It also gives a reasonable CIB redshift distribution, which is important for crossspectra, and is a very good fit to CIB power spectra (see Béthermin et al. 2013). In contrast to the radiosource case, the major contribution to the dusty galaxy Poisson power arises from sources with flux densities well below the cuts; for example, we note that decreasing the fluxdensity cuts by a factor of 2 decreases the Poisson power by less than 1% at the relevant frequencies. In this case, Eq. (55) is a sufficient and more convenient approximation, and we make use of it when calculating Poisson levels for dusty galaxies.
We give predictions for Poisson levels for three different masks: (1) the 2013 pointsource mask, which was defined for sources detected at S/N> 5 at any frequency between 100 and 353 GHz; (2) the 2015 pointsource mask, which is frequencydependent and includes S/N> 5 sources detected only at each individual frequency (used by Plik , CamSpec , and Mspec in this work); and (3) the Hillipop mask, which is also frequencydependent and involves both a S/N cut and a fluxdensity cut^{16}.
Table 20 summarizes the main results of this section. Generally, we find good agreement between the priors from source counts and the posteriors from chains, with the priors being much more constraining. The exception to the good agreement is at 100 GHz where the prediction is lower than the measured value by around 4σ for the baseline 2015 mask and 6σ for the Hillipop mask. This is a sign either of a foreground modelling error or (perhaps more likely) of a residual unmodelled systematic in the data. We note that this disagreement was not present in Like13, where the Poisson amplitude at 100 GHz was found to be smaller. We also note that removing the relative calibration prior (Eq. (35)) or increasing the ℓ_{max} at 100 GHz by a few hundred reduces the tension in the Mspec results. In any case, it is unlikely to affect parameter estimates at all, since very little cosmological information comes from the multipole range at 100 GHz that constrains the Poisson amplitude.
4.4. TE and EE test results
4.4.1. Residuals per frequency and interfrequency differences
Figure 40 shows the residuals for each frequency and Fig. 41 shows the differences between frequencies of the TE and EE power spectra (the procedure is explained in Appendix C.3.2). The residuals are calculated with respect to the bestfit cosmology as preferred by Plik TT+tauprior, although we use the bestfit solution of the Plik TT, TE, EE+tauprior run to subtract the polarized Galactic dust contribution.
Fig. 40 Residual frequency power spectra after subtraction of the Plik TT+tauprior bestfit model. We clean Galactic dust from the spectra from using the bestfit solution of Plik TT, TE, EE+tauprior. The residuals are relative to the baseline HM power spectra (blue points, except for those that deviate by at least 2 or 3σ, which are shown in orange or red, respectively). The vertical dashed lines delimit the ℓ ranges retained in the likelihood. Upper: TE power spectra. Lower: EE power spectra. 
Fig. 41 Interfrequency foregroundcleaned powerspectra differences. Each panel shows the difference of two frequency power spectra, that indicated on the left axis minus that on the bottom axis, after subtracting foregrounds using the bestfit PlanckTT+lowP foreground solutions. Differences are shown for both the HM power spectra (dark blue) and the DS power spectra (light blue). 
The binned interfrequency residuals show deviations at the level of a few μK^{2} from the bestfit model. These deviations do not necessarily correspond to high values of the χ^{2} calculated on the unbinned data (see Table 16). This is because some of the deviations are relatively small for the unbinned data and correctly follow the expected χ^{2} distribution. However, if the deviations are biased (e.g., have the same sign) in some ℓ range, they can result in larger deviations (and large χ^{2}) after binning. Thus, the χ^{2} calculated on unbinned data is not always sufficient to identify these type of biases. We therefore also use a second quantity, χ, defined as the weighted linear sum of residuals, to diagnose biased multipole regions or frequency spectra: $\begin{array}{ccc}\mathit{\chi}\mathrm{=}{{w}}^{\mathrm{T}}\mathrm{(}{C\u0302}\mathrm{}{C}\mathrm{)}\u2001\mathrm{with}\u2001{w}\mathrm{=}\mathrm{(}\mathrm{diag}\hspace{0.17em}\mathrm{C}{\mathrm{)}}^{\mathrm{}\mathrm{1}\mathit{/}\mathrm{2}}\mathit{,}& & \end{array}$(58)where Ĉ is the unbinned vector of data in the multipole region or frequency spectrum of interest, C) is the corresponding model, and w is a vector of weights, equal to the inverse standard deviation evaluated from the diagonal of the corresponding covariance matrix C. The χ statistic is distributed as a Gaussian with zero mean and standard deviation equal to $\begin{array}{ccc}{\mathit{\sigma}}_{\mathit{\chi}}\mathrm{=}\sqrt{{{w}}^{\mathrm{T}}\mathrm{C}{w}}\mathit{.}& & \end{array}$(59)We then define the normalized χ_{norm} as the χ in units of standard deviation, ${\mathit{\chi}}_{\mathrm{norm}}\mathrm{=}\mathit{\chi}\mathit{/}{\mathit{\sigma}}_{\mathit{\chi}}\mathit{.}$(60)The χ_{norm} values that we obtain for different frequency power spectra are given in Table 16.
For EE, the worstbehaved spectra from the χ_{norm} point of view are 143 × 143 (3.7σ deviation) and 100 × 217 (−3.0σ), while from the χ^{2} point of view, the worst is 100 × 143 (PTE = 3.9%). For TE, the worst from the χ_{norm} point of view are 100 × 217 (5σ), 100 × 100 (3.7σ), and 143 × 143 (−2.2σ), while from the χ^{2} point of view the worst is 100 × 100 (PTE = 0.43%). The extreme deviations from the expected distributions show that the frequency spectra are not described very accurately by our data model. This is also clear from Fig. 41, which shows that there are differences of up to 5σ between pairs of foregroundcleaned spectra.
However, as the coadded residuals in Fig. 29 show, systematic effects in the different frequency spectra appear to average out, leaving relatively small residuals with respect to the Plik TT+tauprior bestfit cosmology. In other words, these effects appear not to be dominated by common modes between detector sets or across frequencies. This is also borne out by the good agreement between the data and the expected polarization power spectra conditioned on the temperature ones, as shown in the conditional plots of Fig. 34.
4.4.2. TE and EE robustness tests
For TE and EE, we ran tests of robustness similar to those applied earlier to TT. These are presented in Appendix C.3.5, and the main conclusions are the following. We find that the Plik cosmological results are affected by less than 1σ when using detset crossspectra instead of halfmission ones. This is also the case when we relax the dust amplitude priors, when we marginalize over beam uncertainties, or when we change ℓ_{min} or ℓ_{max}. The alternative CamSpec likelihood has larger shifts, but still smaller than 1σ in TE and 0.5σ in EE. However, we also see larger shifts (more than 2σ in TE) with Plik when some frequency channels are dropped; and, when they are varied, the beam leakage parameters adopt much higher values than expected from the prior, while still leaving some small discrepancies between individual crossspectra that have yet to be explained.
These results shows that our data model leaves residual instrumental systematic errors and is not yet sufficient to take advantage of the full potential of the HFI polarization information. Indeed, the current data model and likelihood code do not account satisfactorily for deviations at the μK^{2} level, even if they can be captured in part by our beam leakage modelling. Nevertheless, the results for the ΛCDM model obtained from the Plik TE+tauprior and Plik EE+tauprior runs are in good agreement with the results from Plik TT+tauprior (see Appendix C.3.6). This agreement between temperature and polarization results within ΛCDM is not a proof of the accuracy of the coadded polarization spectra and their data model, but rather a check of consistency at the μK^{2} level. This consistency is, of course, a very interesting result in itself. But this comparison of probes cannot yet be pushed further to check for the potential presence of a physical inconsistency within the base model that the data could in principle detect or constrain.
5. The full Planck spectra and likelihoods
This section discusses the results that are obtained by using the full Planck likelihood. Section 5.1 first addresses the question of robustness with respect to the choice of the hybridization scale (the multipole at which we transition from the lowℓ likelihood to the highℓ likelihood). Sections 5.2 and 5.3 then present the full results for the power spectra and the baseline cosmological parameters. Section 5.4 summarizes the full systematic error budget. Section 5.5 concentrates on the significance of the possibly anomalous structure around ℓ ≈ 20 in this new release. We then introduce in Sect. 5.6 a useful compressed Planck highℓ temperature and polarization CMBonly likelihood, Plik_lite , which, when applicable, enables faster parameter exploration. Finally, in Sect. 5.7, we compare the Planck 2015 results with the previous results from WMAP, ACT, and SPT.
5.1. Insensitivity to hybridization scale
Fig. 42 Marginal mean and 68% CL error bars on cosmological parameters estimated with different multipoles for the transition between the lowℓ and the highℓ likelihood. Here we use only the TT power spectra and a Gaussian prior on the optical depth τ = 0.07 ± 0.02, within the baseΛCDM model. “Plik TT+tauprior” refers to the case where we use the Plik highℓ likelihood only. 
Before we use the lowℓ and highℓ likelihoods together, we address the question of the hybridization scale, ℓ_{hyb}, at which we switch from one to the other (neglecting correlations between the two regimes, as we did and checked in Like13). To that end, we focus on the TT case and use a likelihood based on the BlackwellRao estimator and the Commander algorithm (Chu et al. 2005; Rudjord et al. 2009) as described in Sect. 2.2, since this likelihood can be used to much higher ℓ_{max} than the full pixelbased T,E,B one. For this test without polarization data, we assume the same τ = 0.07 ± 0.02 prior as before.
The whisker plot of Fig. 42 shows the marginal mean and the 68% CL error bars for baseΛCDM cosmological parameters when ℓ_{hyb} is varied from the baseline value of 30 (case “LOWL 30”) to ℓ_{hyb} =50, 100, 150, 200, and 250, and compared to the Plik TT+tauprior case. The difference between the “LOWL 30” and “Plik TT+tauprior” values shows the effect of the lowℓ dip at ℓ ≈ 20, which reaches 0.5σ on n_{s}. The plot shows that the effect of varying ℓ_{hyb} from 30 to 150 is a shift in n_{s} by less than 0.1σ. This is the result of the Gaussian approximation pushed to ℓ_{min} = 30, already discussed in the simulation section (Sect. 3.6). It would have been much too slow to run the full lowℓTEB likelihood with ℓ_{max} substantially greater than 30, and we decided against the only other option, to leave a gap in polarization between ℓ = 30 and the hybridization scale chosen in TT.
5.2. The Planck 2015 CMB spectra
The visual appearance of Planck 2015 CMB coadded spectra in TT, TE, and EE can be seen in Fig. 50. Goodnessoffit values can be found in Table E.1 of Appendix E. These differ somewhat from those given previously in Table 16 for Plik alone, because the inclusion of low ℓ in temperature brings in the ℓ ≈ 20 feature (see Sect. 5.5). Still, they remain acceptable, with PTEs all above 10% (16.8% for TT).
With this release, Planck now detects 36 extrema in total, consisting of 19 peaks and 17 troughs. Numerical values for the positions and amplitudes of these extrema may be found in Table E.2 of Appendix E.2, which also provides details of the steps taken to derive them. We provide in Appendix E.3 an alternate display of the correlation between temperature and (Emode) polarization by showing their Pearson correlation coefficient and their decorrelation angle versus scale (Figs. E.2 and E.3).
5.3. Planck 2015 model parameters
Figure 43 compares constraints on pairs of parameters as well as their individual marginals for the baseΛCDM model. The grey contours and lines correspond to the results of the 2013 release (Like13), which was based on TT and WMAP polarization at low ℓ (denoted by WP), using only the data from the nominal mission. The blue contours and lines are derived from the 2015 baseline likelihood, Plik TT+lowTEB (“PlanckTT+lowP” in the plot), while the red contours and line are obtained from the full Plik TT, EE, TE+lowTEB likelihood (“PlanckTT, TE, EE+lowP” in the plot, see Appendix E.1 for the relevant robustness tests). In most cases the 2015 constraints are in quite good agreement with the earlier constraints, with the exception of the normalization A_{s}, which is higher by about 2%, reflecting the 2015 correction of the Planck calibration which was indeed revised upward by about 2% in power. The figure also illustrates the consistency and further tightening of the parameter constraints brought by adding the Emode polarization at high ℓ. The numerical values of the Planck 2015 cosmological parameters for base ΛCDM are given in Table 21.
Fig. 43 ΛCDM parameter constraints. The grey contours show the 2013 constraints, which can be compared with the current ones, using either TT only at high ℓ (red) or the full likelihood (blue). Apart from further tightening, the main difference is in the amplitude, A_{s}, due to the overall calibration shift. 
As shown in Fig. 44, the degeneracies between foreground and calibration parameters generally do not affect the determination of the cosmological parameters. In the Plik TT+lowTEB case (top panel), the dust amplitudes appear to be nearly uncorrelated with the basic ΛCDM parameters. Similarly, the 100 and 217 GHz channel calibration is only relevant for the level of foreground emission. Cosmological parameters are, however, mildly correlated with the pointsource and kinetic SZ amplitudes. Correlations are strongest (up to 30%) for the baryon density (Ω_{b}h^{2}) and spectral index (n_{s}). We do not show correlations with the Planck calibration parameter (y_{P}), which is uncorrelated with all the other parameters except the amplitude of scalar fluctuations (A_{s}). The bottom panel shows the correlation for the Plik TE+lowTEB and Plik EE+lowTEB cases, which do not affect the cosmological parameters, except for 20% correlations in EE between the spectral index (n_{s}) and the dust contamination amplitude in the 100 and 143 GHz maps.
Constraints on the basic sixparameter ΛCDM model using Planck angular power spectra.
We also display in Fig. 45 the correlations between the foreground parameters and the cosmological parameters in the Plik TT+lowTEB case when exploring classical extensions to the ΛCDM model. While n_{run} seems reasonably insensitive to the foreground parameters, some extensions do exhibit a noticeable correlation, up to 40% in the case of Y_{He} and the pointsource level at 143 GHz.
Finally, we note that power spectra and parameters derived from CMB maps obtained by the componentseparation methods described in Planck Collaboration IX (2016) are generally consistent with those obtained here, at least when restricted to the ℓ< 2000 range in TT; this is detailed in Sect. E.4.
Fig. 44 Parameter correlations for Plik TT+lowTEB (top), Plik TE+lowTEB (bottom left), and Plik EE+lowTEB (bottom right). The degeneracies between foreground and calibration parameters do not strongly affect the determination of the cosmological parameters. In these figures the lower triangle gives the numerical values of the correlations in percent (with values below 10% printed at the smallest size), while the upper triangle represents the same values using a colour scale. 
Fig. 45 Parameter correlations for Plik TT+lowTEB, including some ΛCDM extensions. The leftmost column is identical to Fig. 44 and is repeated here to ease comparison. Including extensions to the ΛCDM model changes the correlations between the cosmological parameters, sometimes dramatically, as can be seen in the case of A_{L}. There is no correlation between the cosmological parameters (including the extensions) and the dust amplitude parameters. In most cases, the extensions are correlated with the remaining foreground parameters (and in particular with the pointsource amplitudes at 100 and 143 GHz, and with the level of CIB fluctuations) with a strength similar to those of the other cosmological parameters (i.e., less than 30%). Y_{He} exhibits a stronger sensitivity to the pointsource levels. 
5.4. Overall systematic error budget assessment
The tests presented throughout this paper and its appendices documented our numerous tests of the Planck likelihood code and its outputs. Here, we summarize those results and attempt to isolate the dominant sources of systematic uncertainty. This assessment is of course a difficult task. Indeed, all known systematics are normally corrected for, and when relevant, the uncertainty on the correction is included in the error budget and thus in the error bar we report. In that sense, except for a very few cases where we decided to leave a known uncertainty in the data, this section tries to deal with the more difficult task of evaluating the unknown uncertainty!
This section summarizes the contribution of the known systematic uncertainties along with these potential unknown unknowns, specifically highlighting both internal consistency tests based on comparing subsets of the data, along with those using endtoend instrumental simulations.
5.4.1. Lowℓ budget
The lowℓ likelihood has been validated using both internal consistency tests and simulationbased, tests. Here we summarize only the main result of the analysis, which has been set forth in Sect. 2 above.
A powerful consistency test of the polarization data, described in Sect. 2.4, is derived by rotating some of the likelihood components by π/ 4. Specifically, the rotation is applied to the data maps and only to the noise covariance matrix (the likelihood being a scalar function, applying the same rotation to the signal matrix as well would be equivalent to not performing the rotation). The net effect is a conversion of E → −B and B → E for the signal, but leaving unaffected the Gaussian noise described in the covariance matrix. Under these circumstances, we do not expect to pick up any reionization signal, since it would then be present in BB or TB: the operation should result in a null τ detection. This is precisely what happens (see the blue dashed curve in Fig. 8). It is of course possible – though unlikely – that systematics are only showing up in the E channel, leaving B modes unaffected. Indeed, this possibility is further challenged by the fact that we do not detect anomalies in any of the six polarized power spectra; as detailed in Fig. 7, they are consistent with a ΛCDM signal and noise as described by the final 70 GHz covariance matrix.
These tests are specific to Planck and aimed at validating the internal consistency of the datasets employed to build the likelihood. As a further measure of consistency, we have carried out a null test employing the WMAP data, detailed in Sect. 2.6. In brief, we have taken WMAP’s K_{a}, Q and V channels and cleaned them from any polarized foreground contributions using a technique analogous to the one used to clean the LFI 70 GHz maps, employing the Planck 353 GHz map to minimize any dust contribution, but relying on WMAP’s K channel to remove any synchrotron contribution. The resulting LFI 70 GHz and WMAP maps separately lead to compatible τ detections; their halfdifference noise estimates are compatible with their combined noise and do not exhibit a reionization signal, as shown in Fig. 11.
We learn from these tests that if the EE and TE signal we measure at 70 GHz is due to systematics, then these systematics should affect only the above spectra in such a way to mimic a genuine reionization signal, and one that is fully compatible in the maps with that present in (cleaned) WMAP data. This is extremely unlikely and conclude that Planck 70 GHz is dominated by a genuine contribution from the sky, compatible with a signal from cosmic reionization.
The tests described so far do not let us accurately quantify the magnitude of a possible systematic contribution, nor to exclude artefacts arising from the data pipeline itself and, specifically, from the foreground cleaning procedure. These can be only controlled through detailed endtoend tests, using the FFP8 simulations (Planck Collaboration XII 2016). As detailed in Sect. 2.5, we have performed endtoend validation with 1000 simulated frequency maps containing signal, noise, and foreground contributions as well as specific systematics effects, mimicking all the steps in the actual data pipeline. Propagating to cosmological parameters (τ and A_{s}, which are most relevant at low ℓ in the ΛCDM model), we detect no bias within the simulation error budget. The total impact of any unknown systematics on the final τ estimate is at most 0.1σ. This effectively rules out any detectable systematic contribution from the data pipeline or or from the instrumental effects considered in the FFP8 simulations. A complementary analysis has been performed in Planck Collaboration III (2016), including further systematic contributions not incorporated into FFP8. This study, which should be taken as a worstcase scenario, limits the possible contribution to final τ of all known systematics at 0.005, i.e., about 0.25σ. We conclude that we were unable to detect any systematic contribution to the 2015 Planckτ measurement as driven by low ℓ, and have limited it to well within our final statistical error budget.
Finally, since the submission of this paper, dedicated work on HFI data at low ℓ leads to a higherprecision determination of τ (Planck Collaboration Int. XLVI 2016) which is consistent with the one described in this paper. This latest work paves the way towards a future release of improved Planck likelihoods.
5.4.2. Highℓ budget
We now turn to the highℓ likelihood. The approximate statistical model from which we build the likelihood function may turn out to be an unfaithful representation of the data for three main reasons. First the equations describing the likelihood or the parameters of those equations can be inaccurate. They are, of course, since we are relying on approximations, but we expect that in the regime where they are used our approximations are good enough not to bias the best fit or strongly alter the estimation of error bars. We call such errors due to a breakdown of the approximations a “methodological systematic”. We may also lump into this any coding errors. Second, our data model must include a faithful description of the relation between the sky and the data analysed, i.e., one needs to describe the transfer function and/or additive biases due to the nonideal instrument and data processing. Again, an error in this model or in its parameters translates into possible errors that we call “instrumental systematics”. Finally, to recover the properties of the CMB, the contribution of astrophysical foregrounds must be correctly modelled and accounted for. Errors in this model or its parameters is denoted “astrophysical systematics”. When propagating each of these systematics to cosmological parameters, this is always within the framework of the ΛCDM model, as systematic effects can project differently into parameters depending on the details of the model.
We investigated the possibility of methodological systematics with massive MonteCarlo simulations. One of the main technical difficulties of the highℓ likelihood is the computation of the covariance of the band powers. Appendix C.1.3 describes how we validated the covariance matrix, through the use of MonteCarlo simulations, to better than a percent accuracy. This includes a firstorder correction for the excess scatter due to point source masks, which can induce a systematic error in the covariance reaching a maximum of around 10% near the first peak and the largest scales (ℓ< 50), somewhat lower (about 5% or less) at other scales. In Sect. 3.6 we propagated the effect of those possible methodological systematics to the cosmological parameters and found a 0.1σ systematic shift on n_{s}, when using the temperature data, which decreases when cutting the largest and most nonGaussian modes. This is further demonstrated on the data in Sect. 5.1 where we vary the hybridization scale. At this stage it is unclear whether this is a sign of a breakdown of the Gaussian approximation at those scales, or if it is the result of the limitations of our point source correction to the covariance matrix. We did not try to correct for this bias in the likelihood and we assess this 0.1σ effect on n_{s} to be the main contribution to the methodological systematics error budget.
Instrumental systematics are mainly assessed in three ways. First, given a foreground model, we estimate the consistency between frequencies and between the TT, EE and TE combinations at the spectrum and at the parameter level (removing some crossspectra). For TT, the agreement is excellent, with shifts between parameters that are always compatible with the extra cosmic variance due to the removal of data when compared to the baseline solution (see Figs. 31 and 42). TE and EE interfrequency tests reveal discrepancies between the different cross spectra that we assigned to leakage from temperature to polarization (see Fig. 40). In coadded spectra, these discrepancies tend to average out, leaving a fewμK^{2}level residual in the difference between the coadded TE and EE spectra and their theoretical predictions based on the TT parameters. Section 3.4.3 describes an effective model that succeeds in capturing some of that mismatch, in particular in TE. But as argued in Sect. 3.4.3 and Appendix C.3.5 one cannot, at this stage, use this model asis to correct for the leakage, or to infer the level of systematic it may induce on cosmological parameters, due to a lack of a good prior on the leakage model parameters. However, cosmological parameters deduced from the current polarization likelihoods are in perfect agreement with those calculated from the temperature, within the uncertainty allowed by our covariance. The second way we assess possible instrumental systematics is by comparing the detset (DS) and the halfmission (HM) results. As argued in Sect. 3.4.4, the DS cross spectra are known to be affected by a systematic noise correlation that we correct for. Ignoring any uncertainty in this correction (which is difficult to assess), the overall shift between the HM and DSbased parameters is of the order of 0.2σ (on ω_{b}) at most on TT (Sect. 4.1.1 and Fig. 35), similar in TE and slightly worse in EE, particularily for n_{s}. Since the uncertainty on the correlated noise correction is not propagated, those shifts are only upper bounds on possible instrumental systematics (at least those which would manifest differently in these two data cuts which are completely different as regards temporal systematics). Finally, in Sect. 3.7, we evaluate the propagation of all known instrumental effects to parameters. Due to the cost of the required massive endtoend simulations, this test can only reveal large deviations; no such instrumental systematic bias is detected in this test. To summarize, our instrumental systematics budget is at most 0.2σ in temperature, slightly higher in EE, and there is no sign of bias due to temperaturetopolarization leakage that would not be compatible with our covariance (within the ΛCDM framework).
Finally, we assess the contribution of astrophysical systematics. Given the prior findings on polarisation, we only discuss the case of temperature here. The uncertainty on the faithfulness of the astrophysical model is relatively high, and we know from the DS/HM comparison that our astrophysical components certainly absorb part of the correlated noise that is not entirely captured by our model. In that sense, the recovered astrophysical parameters may be a biased estimate of the real astrophysical foreground contribution (due to the flexibility of the model which may absorb residual instrumental systematics provided they are sufficiently small). At small scales, the dominant astrophysical component is the point source Poisson term. We checked in Sect. 4.3 that the recovered point source contributions are in general agreement with models of their expected level. This is much less the case at 100 GHz and we argued in Sect. 4.3 that, nonetheless, an error in the description of the Poisson term is unlikely to translate into a bias in the cosmological parameters, as the point source contribution is negligible at all scales where the 100 GHz spectrum dominates the CMB solution. At large scales, the dust is our strongest foreground. We checked in Fig. 35 the effect of either marginalizing out the slope of the dust spectrum or removing the amplitude priors (i.e., making them arbitrarly wide). When marginalizing over the slope, one recovers a value compatible with the one in our model (−2.57 ± 0.038 whereas our model uses −2.63) and the cosmological parameters do not change (Sect. 4.1.2). When comparing the baseline likelihood result to CamSpec which uses a slightly different template we find a 0.16σ systematic shift in n_{s} that can be attributed to the steeper dust template slope (−2.7) (Sect. 4.2). When ignoring the amplitude priors, a 0.2σ shift appears on n_{s} (and A_{s}, due to its correlation with n_{s}). However, in this case the level of dust contribution increases by about 20 μK^{2} in all spectra, which corresponds to more than doubling the 100 × 100 dust contribution. This level is completely ruled out by the 100 × 545 cross spectrum, which enables estimation of the dust contribution in the 100 GHz channel. The parameter shift can hence be attributed to a degeneracy between the dust model and the cosmological model broken by the prior on the amplitude parameters. We also use the fact that the dust distribution is anisotropic on the sky and evaluate the cosmological parameters on a smaller sky fraction. On TT there is no shift in the parameters that cannot be attributed to the greater cosmic variance on the smaller sky fraction. We are also making a simplifying assumption by describing the dust as a Gaussian field with a specific power spectrum. The numerical simulations (FFP9 and Endtoend) that include a realistic, anisotropic template for the dust contribution do not uncover any systematic effect due to that approximation. In the end, we believe that 0.2σ on n_{s} is a conservative upper bound of our astrophysical systematic bias on the cosmological parameters. There is, however, a possibility of a residual instrumental bias affecting foreground parameters (but not cosmology), but we cannot, at this stage, provide quantitative estimates.
To summarize, our systematic error budget consists of a 0.1σ methodology bias on n_{s} for TT, at most a 0.2σ instrumental bias on TT (on ω_{b}), TE and possibly a slightly greater one on EE. The fewμK^{2}level leakage residual in polarization does not appear to project onto biases on the ΛCDM parameters. We conservatively evalute our astrophysical bias to be 0.2σ on n_{s}. The astrophysical parameters might suffer from instrumental biases.
5.5. The lowℓ “anomaly”
In Like13 we noted that the Planck 2013 lowℓ temperature power spectrum exhibited a tension with the Planck bestfit model, which is mostly determined by highℓ information. In order to quantify such a tension, we performed a series of tests, concluding that the lowℓ power anomaly was mainly driven by multipoles between ℓ = 20 and 30, which happen to be systematically low with respect to the model. The effect was shown to be also present (although less pronounced) using WMAP data (again, see Like13 and Page et al. 2007). The statistical significance of this anomaly was found to be around 99%, with slight variations depending on the Planck CMB solution or the estimator considered. This anomaly has drawn significant attention as a potential tracer of new physics (e.g., Kitazawa & Sagnotti 2015, 2014; Dudas et al. 2012; see also Destri et al. 2008), so it is worth checking its status in the 2015 analysis.
We present here updated results from a selection of the tests performed in 2013. While in Like13 we only concentrated on temperature, we now also consider lowℓ polarization, which was not available as a Planck product in 2013. We first perform an analysis through the Hausman test (Polenta et al. 2005), modified as in Like13 for the statistic s_{1} = sup_{r}B(ℓ_{max},r), with ℓ_{max} = 29 and $\begin{array}{ccc}& & \\ & & {\mathit{H}}_{\mathit{\ell}}\mathrm{=}\frac{\stackrel{\u02c6}{{\mathit{C}}_{\mathit{\ell}}}\mathrm{}{\mathit{C}}_{\mathit{\ell}}}{\sqrt{\mathrm{Var}\hspace{0.17em}\stackrel{\u02c6}{{\mathit{C}}_{\mathit{\ell}}}}}\mathit{,}\end{array}$where 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, expressed in units of standard deviations, while taking the socalled “lookelsewhere effect” into account by maximizing s_{1} over multipole ranges. We use the same simulations as described in Sect. 2.3, which are based on FFP8, for the likelihood validation. We plot in Fig. 46 the empirical distribution for s_{1} in temperature and compare it to the value inferred from the PlanckCommander 2015 map described in Sect. 2 above. The significance for the Commander map has weakened from 0.7% in 2013 to 2.8% in 2015. This appears consistent with the changes between the 2013 and 2015 Commander power spectra shown in Fig. 2, where we can see that the estimates in the range 20 <ℓ< 30 were generally shifted upwards (and closer to the Planck bestfit model) due to revised calibration and improved analysis on a larger portion of the sky. We also report in the lower panel of Fig. 46 the same test for the EE power spectrum, finding that the observed Planck lowℓ polarization maps are anomalous only at the 7.7% level.
Fig. 46 Top: empirical distribution for the Hausman s_{1} statistic for TT derived from simulations; the vertical bar is the observed value for the PlanckCommander map. Bottom: the empirical distribution of s_{1} for EE and the Planck 70 GHz polarization maps described in Sect. 2. 
As a further test of the lowℓ and highℓPlanck constraints, we compare the estimate of the primordial amplitude A_{s} and the optical depth τ, first separately for low and high multipoles, and then jointly. Results are displayed in Fig. 47, showing that the ℓ< 30 and the ℓ ≥ 30 data posteriors in the primordial amplitude are separated by 2.6σ, where the standard deviation is computed as the square root of the sum of the variances of each posterior. We note that a similar separation exists for τ, but it is only significant at the 1.5σ level. Fixing the value of the highℓ parameters to the Planck 2013 bestfit model slightly increases the significance of the power anomaly, but has virtually no effect on τ. A joint analysis using all multipoles retrieves bestfit values in A_{s} and τ which are between the low and highℓ posteriors. This behaviour is confirmed when the Planck 2015 lensing likelihood (Planck Collaboration XV 2016) is used in place of lowℓ polarization.
Fig. 47 Joint estimates of primordial amplitude A_{s} and τ for the data sets indicated in the legend. For lowℓ estimates, all other parameters are fixed to the 2015 fiducial values, except for the dashed line, which uses the Planck 2013 fiducial. The PlanckTT+lowP estimates fall roughly half way between the low and highℓ only ones. 
Finally, we note a similar effect on N_{eff}, which, in the highℓ analysis with a τ prior is about 1σ off the canonical value of 3.04, but is right on top of the canonical value once the lowP and its ℓ = 20 dip is included.
5.6. Compressed CMBonly highℓ likelihood
We extend the Gibbs sampling scheme described in Dunkley et al. (2013) and Calabrese et al. (2013) to construct a compressed temperature and polarization Planck highℓ CMBonly likelihood, Plik_lite , estimating CMB bandpowers and the associated covariances after marginalizing over foreground contributions. Instead of using the full multifrequency likelihood to directly estimate cosmological parameters and nuisance parameters describing other foregrounds, we take the intermediate step of using the full likelihood to extract CMB temperature and polarization power spectra, marginalizing over possible Galactic and extragalactic contamination. In the process, a new covariance matrix is generated for the marginalized spectra, which therefore includes foreground uncertainty. We refer to Appendix C.6.2 for a description of the methodology and to Fig. C.12 for a comparison between the multifrequency data and the extracted CMBonly bandpowers for TT, TE, and EE.
By marginalizing over nuisance parameters in the spectrumestimation step, we decouple the primary CMB from nonCMB information. We use the extracted marginalized spectra and covariance matrix in a compressed, highℓ, CMBonly likelihood. No additional nuisance parameters, except the overall Planck calibration y_{P}, are then needed when estimating cosmology, so the convergence of the MCMC chains is significantly faster. To test the performance of this compressed likelihood, we compare results using both the full multifrequency likelihood and the CMBonly version, for the ΛCDM sixparameter model and for a set of six ΛCDM extensions.
We show in Appendix C.6.2 that the agreement between the results of the full likelihood and its compressed version is excellent, with consistency to better than 0.1σ for all parameters. We have therefore included this compressed likelihood, Plik_lite , in the Planck likelihood package that is available in the Planck Legacy Archive^{17}.
5.7. Planck and other CMB experiments
5.7.1. WMAP9
In Sect. 2.6 we presented the WMAP9based lowℓ polarization likelihood, which uses the Planck 353 GHz map as a dust tracer, as well as the Planck and WMAP9 combination. Results for these likelihoods are presented in Table 22, in conjunction with the Planck highℓ likelihood. Parameter results for the joint Planck and WMAP data set in the union mask are further discussed in Planck Collaboration XIII (2016) and Planck Collaboration XX (2016).
Selected parameters estimated from Planck, WMAP, and their noiseweighted combination in lowℓ polarization, assuming Planck in temperature at all multipoles.
Fig. 48 Comparison of Planck and WMAP9 CMB power spectra. Top: direct comparison. Noise spectra are derived from the halfring difference maps. Bottom: residuals with respect to the PlanckΛCDM bestfit model. The error bars do not include the cosmic variance contribution (but the (brown) 1σ contour lines for the Likelihood best fit model do). 
We now illustrate the state of agreement reached between the Planck 2015 data, in both the raw and likelihood processed form, and the final cosmological power spectra results from WMAP9. In 2013 we noted that the difference between WMAP9 and Planck data was mostly related to calibration, which is now resolved with the upward calibration shift in the Planck 2015 maps and spectra, as discussed in Planck Collaboration I (2016). This leads to the rather impressive agreement that has been reached between the two Planck instruments and WMAP9.
Figure 48 (top panel) shows all the spectra after correction for the effects of sky masking, with different masks used in the three cases of the Planck frequencymap spectra, the spectrum computed from the Planck likelihood, and the WMAP9 final spectrum. The Planck 70, 100, and 143 GHz spectra (which are shown as green, red, and blue points, respectively) were derived from the raw frequency maps (crossspectra of the halfring data splits for the signal, and spectra of the difference thereof for the noise estimates) on approximately 60% of the sky (with no apodization), where the sky cuts include the Galaxy mask, and a concatenation of the 70, 100, and 143 GHz pointsource masks.
The spectrum computed from the Planck likelihood (shown in black as both individual and binned C_{ℓ} values in Fig. 48) was described earlier in the paper. We recall that it was derived with no use of the 70 GHz data, but including the 217 GHz data. Importantly, since it illustrates the likelihood output, this spectrum has been corrected (in the spectral domain) for the residual effects of diffuse foreground emission, mostly in the lowℓ range, and for the collective effects of several components of discrete foreground emission (including tSZ, point sources, CIB, etc.). This spectrum effectively carries the information that drives the likelihood solution of the Planck 2015 bestfit CMB anisotropy, shown in brown. Our aim here is to show the conformity between this Planck 2015 solution and the raw Planck data (especially at 70 GHz) and the WMAP9 legacy spectrum.
The WMAP9 spectrum (shown in magenta as both individual and binned C_{ℓ} values) is the legacy product from the WMAP9 mission, and it represents the final results of the WMAP team’s efforts to clean the residual effects of foreground emission from the cosmological anisotropy spectrum.
All these spectra are binned the same way, starting at ℓ = 30 with Δℓ = 40 bins, and the errorbars represent the error on the mean within each bin. In the lowℓ range, especially near the first peak, the error calculation includes the cosmic variance contribution from the multipoles within each bin, which vastly exceeds any measurement errors (all the measurements shown here have high S/N over the first spectral peak), so we would expect good agreement between the errors derived for all the spectra in the completely signaldominated range of the data.
The figure shows how WMAP9 loses accuracy above ℓ ≈ 800 due to its inherent beam resolution and instrumental noise, and shows how the LFI 70 GHz data achieve improved fidelity over this range. HFI was designed to improve over both WMAP9 and LFI in both noise performance and angular resolution, and the gains achieved are clearly visible, even over the relatively modest range of ℓ shown here, in the tiny spread of the individual C_{ℓ} values of the Planck 2015 power spectrum. While the overall agreement of the various spectra, especially in the lowℓ range, is noticeable in this coarse plot, it is also clear that the Planck raw frequencymap spectra do show excess power over the Planck bestfit spectrum at the higher end of the ℓrange shown – the highest level at 70 GHz and the lowest at 143 GHz. This illustrates the effect of uncorrected discrete foreground residuals in the raw spectra.
A better view of these effects is seen in the bottom panel of Fig. 48. Here we plot the binned values from the top panel as deviations from the bestfit model. Naturally, the black bins of the likelihood output fit well, since they were derived jointly with the bestfit spectrum, while correcting for foreground residuals. The WMAP9 points show good agreement, given their errors, with the Planck 2015 best fit, and illustrate very tight control of the largescale residual foregrounds (at the lowℓ range of the figure); beyond ℓ ~ 600 the WMAP9 spectrum shows an increasing loss of fidelity. Planck raw 70, 100, and 143 GHz spectra show excess power in the lowest ℓ bin due to diffuse foreground residuals. The higherℓ range now shows more clearly the upward drift of power in the raw spectra, growing from 143 GHz to 70 GHz. This is consistent with the welldetermined integrated discrete foreground contributions to those spectra. As previously shown in Planck Collaboration XXXI (2014, Fig. 8), the unresolved discrete foreground power (computed with the same sky masks as used here) can be represented in the bin near ℓ = 800 as levels of approximately 40 μK^{2} at 70 GHz, 15 μK^{2} at 100 GHz, and 5 μK^{2} at 143 GHz, in good agreement with the present figure.
5.7.2. ACT and SPT
Planck temperature observations are complemented at finer scales by measurements from the groundbased Atacama Cosmology Telescope (ACT) and South Pole Telescope (SPT). The ACT and SPT highresolution data help Planck in separating the primordial cosmological signal from other Galactic and extragalactic emission, so as not to bias cosmological reconstructions in the dampingtail region of the spectrum. In 2013 we combined Planck with ACT (Das et al. 2014) and SPT (Reichardt et al. 2012) data in the multipole range 1000 <ℓ< 10 000, defining a common foreground model and extracting cosmological parameters from all the data sets. Our updated “highL” temperature data include ACT power spectra at 148 and 218 GHz (Das et al. 2014) with a revised binning (Calabrese et al. 2013) and final beam estimates (Hasselfield et al. 2013), and SPT measurements in the range 2000 <ℓ< 13 000 from the 2540 deg^{2} SPTSZ survey at 95, 150, and 220 GHz (George et al. 2015). However, in this new analysis, given the increased constraining power of the Planck fullmission data, we do not use ACT and SPT as primary data sets. Using the same ℓ cuts as the 2013 analysis (i.e., ACT data at 1000 <ℓ< 10 000 and SPT at ℓ> 2000) we only check for consistency and retain information on the nuisance foreground parameters that are not well constrained by Planck alone.
To assess the consistency between these data sets, we extend the Planck foreground model up to ℓ = 13 000 with additional nuisance parameters for ACT and SPT, as described in Planck Collaboration XIII (2016, Sect. 4). Fixing the cosmological parameters to the bestfit PlanckTT+lowP baseΛCDM model and varying the ACT and SPT foreground and calibration parameters, we find a reduced χ^{2} = 1.004 (PTE = 0.46), showing very good agreement between Planck and the highL data.
Fig. 49 CMBonly power spectra measured by Planck (blue), ACT (orange), and SPT (green). The bestfit PlanckTT+lowP ΛCDM model is shown by the grey solid line. ACT data at ℓ> 1000 and SPT data at ℓ> 2000 are marginalized CMB bandpowers from multifrequency spectra presented in Das et al. (2014) and George et al. (2015) as extracted in this work. Lower multipole ACT (500 <ℓ< 1000) and SPT (650 <ℓ< 3000) CMB power extracted by Calabrese et al. (2013) from multifrequency spectra presented in Das et al. (2014) and Story et al. (2013) are also shown. The binned values in the range 3000 <ℓ< 4000 appear higher than the unbinned bestfit line because of the binning (this is numerically confirmed by the residual plot in Planck Collaboration XIII 2016, Fig. 9). 
As described in Planck Collaboration XIII (2016), we then take a further step and extend the Gibbs technique presented in Dunkley et al. (2013) and Calabrese et al. (2013; and applied to Planck alone in Sect. 5.6) to extract independent CMBonly bandpowers from Planck, ACT, and SPT. The extracted CMB spectra are reported in Fig. 49. We also show ACT and SPT bandpowers at lower multipoles as extracted by Calabrese et al. (2013). This figure shows the state of the art of current CMB observations, with Planck covering the lowtohighmultipole range and ACT and SPT extending into the damping region. We consider the CMB to be negligible at ℓ> 4000 and note that these ACT and SPT bandpowers have an overall calibration uncertainty (2% for ACT and 1.2% for SPT).
The inclusion of ACT and SPT improves the fullmission Planck spectrum extraction presented in Sect. 5.6 only marginally. The main contribution of ACT and SPT is to constrain small components (e.g., the tSZ, kSZ, and tSZ×CIB) that are not well determined by Planck alone. However, those components are subdominant for Planck and are well described by the prior based on the 2013 Planck+highL solutions imposed in the Planckalone analysis. The CIB amplitude estimate improves by 40% when including ACT and SPT, but the CIB power is also reasonably well constrained by Planck alone. The main Planck contaminants are the Poisson sources, which are treated as independent and do not benefit from ACT and SPT. As a result, the errors on the extracted Planck spectrum are only slightly reduced, with little additional cosmological information added by including ACT and SPT for the baseline ΛCDM model (see also Planck Collaboration XIII 2016, Sect. 4).
6. Conclusions
Fig. 50 Planck 2015 CMB spectra, compared with the base ΛCDM fit to PlanckTT+lowP data (red line). The upper panels show the spectra and the lower panels the residuals. In all the panels, the horizontal scale changes from logarithmic to linear at the “hybridization” scale, ℓ = 29 (the division between the lowℓ and highℓ likelihoods). For the residuals, the vertical axis scale changes as well, as shown by different left and right axes. We show for TT and TE, but C_{ℓ} for EE, which also has different vertical scales at low and highℓ. 
The Planck 2015 angular power spectra of the cosmic microwave background derived in this paper are displayed in Fig. 50. These spectra in TT (top), TE (middle), and EE (bottom) are all quite consistent with the bestfit baseΛCDM model obtained from TT data alone (red lines). The horizontal axis is logarithmic at ℓ< 30, where the spectra are shown for individual multipoles, and linear at ℓ ≥ 30, where the data are binned. The error bars correspond to the diagonal elements of the covariance matrix. The lower panels display the residuals, the data being presented with different vertical axes, a larger one at left for the lowℓ part and a zoomedin axis at right for the highℓ part.
The 2015 Planck likelihood presented in this work is based on more temperature data than in the 2013 release, and on new polarization data. It benefits from several improvements in the processing of the raw data, and in the modelling of astrophysical foregrounds and instrumental noise. Apart from a revision of the overall calibration of the maps, discussed in Planck Collaboration I (2016), the most significant improvements are in the likelihood procedures:

(i)
a joint temperaturepolarization pixelbased likelihood atℓ ≤ 29, with more highfrequency information used for foreground removal, and smaller sky masks (Sects. 2.1 and 2.2);

(ii)
an improved Gaussian likelihood at ℓ ≥ 30 that includes a different strategy for estimating power spectra from datasubset crosscorrelations, using halfmission data instead of detector sets (which enables us to reduce the effect of correlated noise between detectors, see Sects. 3.2.1 and 3.4.3), and better foreground templates, especially for Galactic dust (Sect. 3.3.1) that lets us mask a smaller fraction of the sky (Sect. 3.2.2) and to retain largeangle temperature information from the 217 GHz map that was neglected in the 2013 release (Sect. 3.2.4).
We performed several consistency checks of the robustness of our likelihoodmaking process, by introducing more or less freedom and nuisance parameters in the modelling of foregrounds and instrumental noise, and by including different assumptions about the relative calibration uncertainties across frequency channels and about the beam window functions.
For temperature, the reconstructed CMB spectrum and error bars are remarkably insensitive to all these different assumptions. Our final highℓ temperature likelihood, referred to as “PlanckTT” marginalizes over 15 nuisance parameters (12 modelling the foregrounds, and 3 for calibration uncertainties). Additional nuisance parameters (in particular, those associated with beam uncertainties) were found to have a negligible impact, and can be kept fixed in the baseline likelihood. Detailed endtoend simulations of the instrumental response to the sky analysed like the real data did not uncover hidden lowlevel residual systematics.
For polarization, the situation is different. Variation of the assumptions leads to scattered results, with greater deviations than would be expected due to changes in the data subsets used, and at a level that is significant compared to the statistical error bars. This suggests that further systematic effects need to be either modelled or removed. In particular, our attempt to model calibration errors and temperaturetopolarization leakage suggests that the TE and EE power spectra are affected by systematics at a level of roughly 1 μK^{2}. Removal of polarization systematics at this level of precision requires further work, beyond the scope of this release. The 2015 highℓ polarized likelihoods, referred to as “Plik TE” and “Plik EE”, or “Plik TT, EE, TE” for the combined version, ignore these uncertain corrections. They only include 12 additional nuisance parameters accounting for polarized foregrounds. Although these likelihoods are distributed in the Planck Legacy Archive^{18}, we stick to the PlanckTT+lowP choice in the baseline analysis of this paper and the companion papers such as Planck Collaboration XIII (2016), Planck Collaboration XIV (2016), and Planck Collaboration XX (2016).
We developed internally several likelihood codes, exploring not only different assumptions about foregrounds and instrumental noise, but also different algorithms for building an approximate Gaussian highℓ likelihood (Sect. 4.2). We compared these codes to check the robustness of the results, and decided to release:

(i)
A baseline likelihood called Plik (available for TT, TE, EE, or combined observables), in which the data are binned in multipole space, with a binwidth increasing from Δℓ = 5 at ℓ ≈ 30 to Δℓ = 33 at ℓ ≈ 2500.

(ii)
An unbinned version which, although slower, is preferable when investigating models with sharp features in the power spectra.

(iii)
A simplified likelihood called Plik_lite in which the foreground templates and calibration errors are marginalized over, producing a marginalized spectrum and covariance matrix. This likelihood does not allow investigation of correlations between cosmological and foreground/instrumental parameters, but speeds up parameter extraction, having no nuisance parameters to marginalize over.
In this paper we have also presented an investigation of the measurement of cosmological parameters in the minimal sixparameter ΛCDM model and a few simple sevenparameter extensions, using both the new baseline Planck likelihood and several alternative likelihoods relying on different assumptions. The cosmological analysis of this paper does not replace the investigation of many extended cosmological models presented, e.g., in Planck Collaboration XIII (2016), Planck Collaboration XIV (2016), and Planck Collaboration XX (2016). However, the careful inspection of residuals presented here addresses two questions:

(i)
a priori, is there any indication that an alternative model toΛCDM could provide a significantly better fit?

(ii)
if there is such an indication, could it come from caveats in the likelihoodbuilding (imperfect data reduction, foreground templates or noise modelling) instead of new cosmological ingredients?
Since this work is entirely focused on the powerspectrum likelihood, it can only address these questions at the level of 2point statistics; for a discussion of higherorder statistics, see Planck Collaboration XVI (2016) and Planck Collaboration XVII (2016).
The most striking result of this work is the impressive consistency of different cosmological parameter extractions, performed with different versions of the Plik TT+tauprior or PlanckTT+lowP likelihoods, with several assumptions concerning: data processing (halfmission versus detector set correlations); sky masks and foreground templates; beam window functions; the use of two frequency channels instead of three; different cuts at low ℓ or high ℓ; a different choice for the multipole value at which we switch from the pixelbased to the Gaussian likelihood; different codes and algorithms; the inclusion of external data sets like WMAP9, ACT, or SPT; and the use of foregroundcleaned maps (instead of fitting the CMB+foreground map with a sum of different contributions). In all these cases, the bestfit parameter values drift by only a small amount, compatible with what one would expect on a statistical basis when some of the data are removed (with a few exceptions summarized below).
The cosmological results are stable when one uses the simplified Plik_lite likelihood. We checked this by comparing PlanckTT+lowP results from Plik and Plik_lite for ΛCDM, and for six examples of sevenparameter extended models.
Another striking result is that, despite evidence for small unsolved systematic effects in the highℓ polarization data, the cosmological parameters returned by the Plik TT, Plik TE, or Plik EE likelihoods (in combination with a τ prior or Planck lowP) are consistent with each other, and the residuals of the (frequency combined) TE and EE spectra after subtracting the temperature ΛCDM bestfit are consistent with zero. As has been emphasized in other Planck 2015 papers, this is a tremendous success for cosmology, and an additional proof of the predictive power of the standard cosmological model. It also suggests that the level of temperaturetopolarization leakage (and possibly other systematic effects) revealed by our consistency checks is low enough(on average over all frequencies) not to significantly bias parameter extraction, at least for the minimal cosmological model. We do not know yet whether this conclusion applies also to extended models, especially those in which the combination of temperature and polarization data has stronger constraining power than temperature data alone, e.g., dark matter annihilation (Planck Collaboration XIII 2016) or isocurvature modes (Planck Collaboration XX 2016). One should thus wait for a future Planck release before applying the Planck temperaturepluspolarization likelihood to such models. However, the fact that we observe a significant reduction in the error bars when including polarization data is very promising, since this reduction is expected to remain after the removal of systematic effects.
Careful inspection of residuals with respect to the bestfit ΛCDM model has revealed a list of anomalies in the Planck CMB power spectra, of which the most significant is still the lowℓ temperature anomaly in the range 20 ≤ ℓ ≤ 30, already discussed at length in the 2013 release. In this 2015 release, with more data and with better calibration, foreground modelling, and sky masks, its significance has decreased from the 0.7% to the 2.8% level for the TT spectrum (Sect. 5.5). This probability is still small (although not very small), and the feature remains unexplained. We have also investigated the EE spectrum, where the anomaly, if any, is significant only at the 7.7% level.
Other “anomalies” revealed by inspection of residuals (and of their dependence on the assumptions underlying the likelihood) are much less significant. There are a few bins in which the power in the TT, TE, or EE spectrum lies 2–3σ away from the bestfit ΛCDM prediction, but this is not statistically unlikely and we find acceptable probabilitytoexceed (PTE) levels. Nevertheless, in Sects. 3.8 and 4.1, we presented a careful investigation of these features, to see whether they could be caused by some imperfect modelling of the data. We noted that a deviation in the TT spectrum at ℓ ≈ 1450 is somewhat suspicious, since it is driven mostly by a single channel (217 GHz), and since it depends on the foregroundremoval method. But this deviation is too small to be worrisome (1.8σ with the baseline Plik likelihood). As in the 2013 release, the data at intermediate ℓ would be fitted slightly better by a model with more lensing than in the bestfit ΛCDM model (to reduce the peaktotrough contrast), but more lensing generically requires higher values of A_{s} and Ω_{c}h^{2} that are disfavoured by the rest of the data, in particular when highℓ information is included. This mild tension is illustrated by the preference for a value greater than unity for the unphysical parameter A_{L}, a conclusion that is stable against variations in the assumptions underlying the likelihoods. However, A_{L} is compatible with unity at the 1.8σ level when using the baseline PlanckTT likelihood with a conservative τ prior (to avoid the effect of the lowℓ dip), so what we see here could be the result of statistical fluctuations.
This absence of large residuals in the Planck 2015 temperature and polarization spectra further establishes the robustness of the ΛCDM model, even with about twice as much data as in the Planck 2013 release. This conclusion is supported by several companion papers, in which many nonminimal cosmological models are investigated but no significant evidence for extra physical ingredients is found. The ability of the temperature results to pass several demanding consistency tests, and the evidence of excellent agreement down to the μK^{2} level between the temperature and polarization data, represent an important milestone set by the Planck satellite. The Planck 2015 likelihoods are the best illustration to date of the predictive power of the minimal cosmological model, and, at the same time, the best tool for constraining interesting, physicallymotivated deviations from that model.
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 and led by Principal Investigators from France and Italy, telescope reflectors provided through a collaboration between ESA and a scientific consortium led and funded by Denmark, and additional contributions from NASA (USA).
These rms estimates were computed with the PolSpice powerspectrum estimator (Chon et al. 2004) by averaging over 1000 noiseless simulations.
We assume an ℓdiagonal mixing matrix here. This is not necessarily the case, as subpixel beam effects, for example, can induce mode couplings. As discussed in Sect. 3.4.3, those were estimated in Like13 and found to be negligible for temperature. They are not investigated further in this paper.
The definition of A_{L} differs in PICO and CAMB ; see Appendix C.5.
During the revision of this paper, we noticed that the ℓ> 1000 case explores regions of parameter space that are outside the optimal PICO interpolation region, as also remarked by Addison et al. (2016). This inaccuracy mainly affected this particular test for constraints on n_{s} and Ω_{b}h^{2}: the error bars for these parameters were underestimated by a factor of about 2 while the mean values were misestimated by about 0.8σ with respect to runs performed with CAMB . Nevertheless, we found that for all other parameters, and in all other likelihood tests presented in this section, this problem did not arise, since the explored parameter space was entirely contained in the PICO interpolation region so as to guarantee accurate results, as also detailed in Sect. C.5. Furthermore, this inaccuracy does not change any of the conclusions of this paper. We therefore decided to keep in Fig. 35 the results obtained with PICO but we have added results for the ℓ> 1000 case obtained with CAMB (case “CAMB, l_{min} = 1000”).
These results were obtained with the PICO code, and are thus close to but not identical to those obtained with CAMB and reported in Planck Collaboration XIII (2016).
We note that the Hillipop mask was constructed partly so that Eq. (55) would be an accurate approximation. We find that for the radio contribution is is accurate to 2%, or 1σ, and for the dust contribution it is essentially exact.
We use the routine process_mask of the HEALPix package to obtain a map of the distance of each pixel of the mask from the closest null pixel. We then use a smoothed version of the distance map to build the Gaussian apodization. The smoothing of the distance map is needed to avoid sharp edges in the final mask.
We remind the reader that, in this test, at each frequency we always use ${\mathit{\ell}}_{\mathrm{max}}^{\mathrm{freq}}\mathrm{=}\mathrm{min}\mathrm{\left(}{\mathit{\ell}}_{\mathrm{max}}\mathit{,}{\mathit{\ell}}_{\mathrm{max}}^{\mathrm{freq}\mathit{,}\hspace{0.17em}\mathrm{baseline}}\mathrm{\right)}$, with ${\mathit{\ell}}_{\mathrm{max}}^{\mathrm{freq}\mathit{,}\hspace{0.17em}\mathrm{baseline}}$ the baseline ℓ_{max} at each frequency as reported in Table 16 (e.g., in the ℓ_{max} = 1404 case, we still use the 100 × 100 power spectrum through ℓ = 1197).
We discovered late in the preparation of this paper that in some of the tests the prior for the 143 × 217TE dust contamination was set inaccurately, with an offset of −0.3 μK^{2} at ℓ = 500. With our cuts, this spectrum contributes only at ℓ> 500 where the dust contamination is already small compared to the signal. We verified that this has no impact on the cosmology and on our conclusions.
Therefore, e.g., $\stackrel{}{{{\mathit{C}}_{\mathit{\ell}}^{\mathrm{143}\mathrm{h}\mathrm{2}\mathrm{\times}\mathrm{217}\mathrm{h}\mathrm{1}}}_{\mathrm{\u02dc}}}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{d}}_{\mathrm{143}\mathrm{h}\mathrm{2}}\mathrm{+}{\mathit{d}}_{\mathrm{217}\mathrm{h}\mathrm{1}}\mathrm{)}\hspace{0.17em}{\mathit{C}}_{\mathit{\ell}}^{\mathrm{143}\mathrm{h}\mathrm{2}\mathrm{\times}\mathrm{217}\mathrm{h}\mathrm{1}}$.
Acknowledgments
The Planck Collaboration acknowledges the support of: ESA; CNES, and CNRS/INSUIN2P3INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MINECO, 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); ERC and PRACE (EU). A description of the Planck Collaboration and a list of its members, indicating which technical or scientific activities they have been involved in, can be found at http://www.cosmos.esa.int/web/planck/planckcollaboration.
We further 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 DPC infrastructure hosted at the Institut d’Astrophysique de Paris (France) and financially supported by CNES.
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Appendix A: Sky masks
This appendix provides details of the way we build sky masks for the highℓ likelihood. Since it is based on data at frequencies between 100 and 217 GHz, Galactic dust emission is the main diffuse foreground to minimize. We subtract the SMICA CMB temperature map (Planck Collaboration IX 2016) from the 353 GHz map and we adopt the resulting CMBsubtracted 353 GHz map as a tracer of dust. After smoothing the map with a 10deg Gaussian kernel, we threshold it to generate a sequence of masks with different sky coverage. Galactic masks obtained in this way are named B80 to B50, where the number gives the retained sky fraction f_{sky} in percent (Fig. A.1).
For the likelihood analysis, we aim to find a tradeoff between maximizing the sky coverage and having a simple, but reliable, foreground model of the data. The combination of masks and frequency channels retained is given in Table A.1. In order to get C_{ℓ}covariance matrices for the cosmological analysis that are accurate at the few percent level (cf. Sect. 3.5), we actually use apodized versions of the Galactic masks. The apodization corresponds to a Gaussian taper of width σ = 2deg^{19}. Apodized Galactic masks are also used for the polarization analysis. The effective sky fraction of an apodized mask is ${\mathit{f}}_{\mathrm{sky}}\mathrm{=}{\sum}_{\mathit{i}}{\mathit{w}}_{\mathit{i}}^{\mathrm{2}}{\mathrm{\Omega}}_{\mathit{i}}\mathit{/}\mathrm{\left(}\mathrm{4}\mathit{\pi}\mathrm{\right)}$, where w_{i} is the value of the mask in pixel i and Ω_{i} is the solid angle of the pixel.
All the HFI frequency channels, except 143 GHz, are also contaminated by CO emission from rotational transition lines. Here we are concerned with emission around 100 and 217 GHz, associated with the CO J = 1 → 0 and J = 2 → 1 lines, respectively. Most of the emission is concentrated near the Galactic plane and is therefore masked out by the Galactic dust masks. However, there are some emission regions at intermediate and low latitudes that are outside the quite small B80 mask we use at 100 GHz. We therefore create a mask specifically targeted at eliminating CO emission. The Type 3 CO map, part of the Planck 2013 product delivery (Planck Collaboration XIII 2014), is sensitive to lowintensity diffuse CO emission over the whole sky. It is a multiline map, derived using prior information on line ratios and a multifrequency component separation method. Of the three types of Planck CO maps, this has the highest S/N. We smooth this map with a σ = 120′ Gaussian and mask the sky wherever the CO line brightness exceeds 1K_{RJ}kms^{1}. The mask is shown in Fig. A.2, before apodization with a Gaussian taper of FWHM = 30′.
Finally, we include extragalactic objects in our temperature masks, both point sources and nearby extended galaxies. The nearby galaxies that are masked are listed in Table A.2, together with the corresponding cut radii. For point sources, we build conservative masks for 100, 143, and 217 GHz separately. At each frequency, we mask sources that are detected above S/N = 5 in the 2015 pointsource catalogue (Planck Collaboration XXVI 2016) with holes of radius three times the $\mathit{\sigma}\mathrm{=}\mathit{FWHM}\mathit{/}\sqrt{\mathrm{ln}\mathrm{8}}$ of the effective Gaussian beam at that frequency. We take the FWHM values from the elliptic Gaussian fits to the effective beams (Planck Collaboration XXVI 2016), i.e., FWHM values of , , and at 100, 143, and 217 GHz, respectively. We apodize these masks with a Gaussian taper of FWHM = 30′. As already noted, these masks are designed to reduce the contribution of diffuse and discrete Galactic and extragalactic foreground emission in the “raw” (halfmission and detset) frequency maps used for the baseline highℓ likelihood.
Fig. A.1 Unapodized Galactic masks B50, B60, B70, and B80, from orange to dark blue. 
Fig. A.2 Unapodized CO mask (f_{sky} = 87%). 
Galactic masks used for the highℓ analysis.
Masked nearby galaxies and corresponding cut radii.
The masks described in this appendix are used in the papers on cosmological parameters (Planck Collaboration XIII 2016), inflation (Planck Collaboration XX 2016), dark energy (Planck Collaboration XIV 2016), and primordial magnetic fields (Planck Collaboration XIX 2016), which are notable examples of the application of the highℓ likelihood. However, the masks differ from those adopted in some of the other Planck papers. For example, reconstructions of gravitational lensing (Planck Collaboration XV 2016) and integrated SachsWolfe effect (Planck Collaboration XXI 2016), constraints on isotropy and statistics (Planck Collaboration XVI 2016), and searches for primordial nonGaussianity (Planck Collaboration XVII 2016) mainly rely on the highresolution foregroundreduced CMB maps presented in Planck Collaboration IX (2016). Those maps have been derived by four componentseparation methods that combine data from different frequency channels to extract “cleaned” CMB maps. For each method, the corresponding confidence masks, for both temperature and polarization, remove regions of the sky where the CMB solution is not trusted. This is described in detail in Appendices A−D of Planck Collaboration IX (2016). The masks recommended for the analysis of foregroundreduced CMB maps are constructed as the unions of the confidence masks of all the four component separation methods. Their sky coverages are f_{sky} = 0.776 in temperature and f_{sky} = 0.774 in polarization. Since component separation mitigates the foreground contamination even at relatively low Galactic latitudes, those masks feature a thinner cut along the Galactic plane than the ones described in this appendix. Nevertheless, propagation of noise, beam, and extragalactic foreground uncertainties in foregroundcleaned CMB maps is more difficult, and this is the main reason why we do not employ them in the baseline highℓ likelihood. We also note that the recommended mask for temperature foregroundreduced maps has a greater number of compact object holes than the masks used here. This is due to the fact that some component separation techniques can introduce contamination of sources from a wider range of frequencies than the approach considered here for the highℓ power spectra. According to the tests provided in Sect. C.1.4, such masks would result in suboptimal performance of the analytic C_{ℓ}covariance matrices.
Appendix B: Lowℓ likelihood supplement
Appendix B.1: ShermanMorrisonWoodbury formula
In the Planck 2015 release we follow a pixelbased approach to the joint lowℓ likelihood (up to ℓ = 29) of T, Q, and U. This approach treats temperature and polarization maps consistently at HEALPix resolution N_{side} = 16, as opposed to the WMAP lowℓ likelihood, which incorporates polarization information from lowerresolution maps to save computational time (Page et al. 2007). The disadvantage of a consistentresolution, bruteforce approach lies in its computational cost (Like13), which may require massively parallel coding (and adequate hardware) in order to be competitive in execution time with the highℓ part of the CMB likelihood (see, e.g., Finelli et al. 2013 for one such implementation). Such a choice, however, would hamper the ease of code distribution across a community not necessarily specialized in massively parallel computing. Luckily, the ShermanMorrisonWoodbury formula and the related matrix determinant lemma provide a means to achieve good timing without resorting to supercomputers. To see how this works, rewrite the covariance matrix from Eq. (3) in a form that explicitly separates the C_{ℓ} to be varied from those that stay fixed at the reference model: $\begin{array}{ccc}\mathrm{M}& \mathrm{=}& \sum _{\mathit{XY}}\sum _{\mathit{\ell}\mathrm{=}\mathrm{2}}^{{\mathit{\ell}}_{\mathrm{cut}}}{\mathit{C}}_{\mathit{\ell}}^{\mathit{XY}}{\mathrm{P}}_{\mathit{\ell}}^{\mathit{XY}}\mathrm{+}\sum _{\mathit{XY}}\sum _{\mathit{\ell}\mathrm{=}{\mathit{\ell}}_{\mathrm{cut}}\mathrm{+}\mathrm{1}}^{{\mathit{\ell}}_{\mathrm{max}}}{\mathit{C}}_{\mathit{\ell}}^{\mathit{XY,}\mathrm{ref}}{\mathrm{P}}_{\mathit{\ell}}^{\mathit{XY}}\mathrm{+}\mathrm{N}\\ & & \mathrm{\equiv}\sum _{\mathit{XY}}\sum _{\mathit{\ell}\mathrm{=}\mathrm{2}}^{{\mathit{\ell}}_{\mathrm{cut}}}{\mathit{C}}_{\mathit{\ell}}^{\mathit{XY}}{\mathrm{P}}_{\mathit{\ell}}^{\mathit{XY}}\mathrm{+}{\mathrm{M}}_{\mathrm{0}}\mathit{,}\end{array}$where we have effectively redefined the fixed multipoles as “highℓ correlated noise”, as far as the varying lowℓ multipoles are concerned. Next, note that for fixed ℓ, ${\mathrm{P}}_{\mathit{\ell}}^{\mathit{TT}}$ has rank^{20}λ = 2ℓ + 1, and this matrix may therefore be decomposed as ${\mathrm{P}}_{\mathit{\ell}}^{\mathit{TT}}\mathrm{=}\mathrm{(}{\mathrm{V}}_{\mathit{\ell}}^{\mathit{TT}}{\mathrm{)}}^{\mathrm{T}}\hspace{0.17em}{\mathrm{A}}_{\mathit{\ell}}^{\mathit{TT}}\hspace{0.17em}{\mathrm{V}}_{\mathit{\ell}}^{\mathit{TT}}$, where ${\mathrm{A}}_{\mathit{\ell}}^{\mathit{TT}}$ and ${\mathrm{V}}_{\mathit{\ell}}^{\mathit{TT}}$ are (λ × λ) and (λ × N_{pix}) matrices, respectively, which depend only upon the unmasked pixel locations. A similar decomposition holds for the ${\mathrm{P}}_{\mathit{\ell}}^{\mathit{EE,BB}}$ matrices, while ${\mathrm{P}}_{\mathit{\ell}}^{\mathit{TE}}$ can be expanded in the $\mathrm{\left[}{\mathrm{V}}_{\mathit{\ell}}^{\mathit{TT}}\mathit{,}{\mathrm{V}}_{\mathit{\ell}}^{\mathit{EE}}\mathrm{\right]}$ basis for the corresponding ℓ. We can then write $\mathrm{M}\mathrm{=}{\mathrm{V}}^{\mathrm{T}}\mathrm{A}\mathrm{\left(}{\mathit{C}}_{\mathit{\ell}}\mathrm{\right)}\mathrm{V}\mathrm{+}{\mathrm{M}}_{\mathrm{0}}\mathit{,}$(B.3)where $\mathrm{V}\mathrm{=}\mathrm{\left[}{\mathrm{V}}_{\mathrm{2}}^{\mathit{TT}}\mathit{,}{\mathrm{V}}_{\mathrm{2}}^{\mathit{EE}}\mathit{,}{\mathrm{V}}_{\mathrm{2}}^{\mathit{BB}}\mathit{,}\mathit{...}{\mathrm{V}}_{{\mathit{\ell}}_{\mathrm{cut}}}^{\mathit{BB}}\mathrm{\right]}$ is an (n_{λ} × N_{pix}) matrix with n_{λ} = 3 [(ℓ_{cut} + 1)^{2}−4], and A(C_{ℓ}) is an (n_{λ} × n_{λ}) blockdiagonal matrix (accounting for four modes removed in monopole and dipole subtraction). Each ℓblock in the latter matrix reads $\left[\begin{array}{c}\\ {\mathit{C}}_{\mathit{\ell}}^{\mathit{TT}}{\mathrm{A}}_{\mathit{\ell}}^{\mathit{TT}}& {\mathit{C}}_{\mathit{\ell}}^{\mathit{TE}}{\mathrm{A}}_{\mathit{\ell}}^{\mathit{TE}}& \mathrm{0}\\ {\mathit{C}}_{\mathit{\ell}}^{\mathit{TE}}{\mathrm{A}}_{\mathit{\ell}}^{\mathit{TE}}& {\mathit{C}}_{\mathit{\ell}}^{\mathit{EE}}{\mathrm{A}}_{\mathit{\ell}}^{\mathit{EE}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& {\mathit{C}}_{\mathit{\ell}}^{\mathit{BB}}{\mathrm{A}}_{\mathit{\ell}}^{\mathit{BB}}\end{array}\right]\mathit{.}$(B.4)Finally, using the ShermanMorrisonWoodbury identity and the matrix determinant lemma, we can rewrite the inverse and determinant of M as $\begin{array}{ccc}& & {\mathrm{M}}^{1}\mathrm{=}\hspace{0.17em}{\mathrm{M}}_{\mathrm{0}}^{1}\mathrm{}{\mathrm{M}}_{\mathrm{0}}^{1}{\mathrm{V}}^{\mathrm{T}}\mathrm{(}{\mathrm{A}}^{1}\mathrm{+}\mathrm{V}{\mathrm{M}}_{\mathrm{0}}^{1}{\mathrm{V}}^{\mathrm{T}}{\mathrm{)}}^{1}\mathrm{V}{\mathrm{M}}_{\mathrm{0}}^{1}\\ & & \mathrm{\left}\mathrm{M}\mathrm{\right}\mathrm{=}\hspace{0.17em}\mathrm{\left}{\mathrm{M}}_{\mathrm{0}}\mathrm{\right}\hspace{0.17em}\mathrm{\left}\mathrm{A}\mathrm{\right}\hspace{0.17em}\mathrm{}{\mathrm{A}}^{1}\mathrm{+}\mathrm{V}{\mathrm{M}}_{\mathrm{0}}^{1}{\mathrm{V}}^{\mathrm{T}}\mathrm{}\mathit{.}\end{array}$Because neither V nor M_{0} depends on C_{ℓ}, all terms involving only their inverses, determinants, and products may be precomputed and stored. Evaluating the likelihood for a new set of C_{ℓ} then requires only the inverse and determinant of an (n_{λ} × n_{λ}) matrix, not an (N_{pix} × N_{pix}) matrix. For the current data selection, described in Sects. 2.2 and 2.3, we find n_{λ} = 2688, which is to be compared to N_{pix} = 6307, resulting in an orderofmagnitude speedup compared to the bruteforce computation.
Appendix B.2: Lollipop
We performed a complementary analysis of lowℓ polarization using the HFI data, in order to check the consistency with the LFIbased baseline result. The level of systematic residuals in the HFI maps at low ℓ is quite small, but comparable to the HFI noise (see Planck Collaboration VIII 2016), so these residuals should be either corrected, which is the goal of a future release, or accounted for by a complete analysis including parameters for all relevant systematic effects, which we cannot yet perform. Instead, we use Lollipop , a lowℓ polarized likelihood function based on crosspower spectra. The idea behind this approach is that the systematics are considerably reduced in crosscorrelation compared to autocorrelation.
At low multipoles and for incomplete sky coverage, the C_{ℓ} statistic is not simply distributed and is correlated between modes. Lollipop uses the approximation presented in Hamimeche & Lewis (2008), modified as described in Mangilli et al. (2015) to apply to crosspower spectra. We restrict ourselves to the onefield approximation to derive a likelihood function based only on the EE power spectrum at very low multipoles. The likelihood function of the C_{ℓ} given the data $\begin{array}{c}{\mathrm{\u02dc}}_{}\\ {\mathit{C}}_{\mathit{\ell}}\end{array}$ is then $\mathrm{}\mathrm{2}\mathrm{ln}\mathit{P}\mathrm{\left(}{\mathit{C}}_{\mathit{\ell}}\mathrm{\right}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{C}}_{\mathit{\ell}}\end{array}\mathrm{)}\mathrm{=}\sum _{\mathit{\ell}{\mathit{\ell}}^{\mathrm{\prime}}}\mathrm{[}{\mathit{X}}_{\mathit{g}}{\mathrm{]}}_{\mathit{\ell}}^{\mathrm{T}}\mathrm{\left[}{\mathit{M}}_{\mathit{f}}^{1}{\mathrm{]}}_{\mathit{\ell}{\mathit{\ell}}^{\mathrm{\prime}}}\mathrm{\right[}{\mathit{X}}_{\mathit{g}}{\mathrm{]}}_{{\mathit{\ell}}^{\mathrm{\prime}}}\mathit{,}$(B.7)with the variable ${\mathrm{\left[}{\mathit{X}}_{\mathit{g}}\mathrm{\right]}}_{\mathit{\ell}}\mathrm{=}\sqrt{{\mathit{C}}_{\mathit{\ell}}^{\mathit{f}}\mathrm{+}{\mathit{O}}_{\mathit{\ell}}}\hspace{0.17em}\hspace{0.17em}\mathit{g}\left(\frac{\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{C}}_{\mathit{\ell}}\end{array}\mathrm{+}{\mathit{O}}_{\mathit{\ell}}}{{\mathit{C}}_{\mathit{\ell}}\mathrm{+}{\mathit{O}}_{\mathit{\ell}}}\right)\hspace{0.17em}\hspace{0.17em}\sqrt{{\mathit{C}}_{\mathit{\ell}}^{\mathrm{fid}}\mathrm{+}{\mathit{O}}_{\mathit{\ell}}}\mathit{,}$(B.8)where $\mathit{g}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\sqrt{\mathrm{2}\mathrm{(}\mathit{x}\mathrm{}\mathrm{ln}\mathit{x}\mathrm{}\mathrm{1}\mathrm{)}}$, ${\mathit{C}}_{\mathit{\ell}}^{\mathrm{fid}}$ is a fiducial model and O_{ℓ} is the offset needed in the case of crossspectra. This likelihood has been tested on Monte Carlo simulations including both realistic signal and noise. In order to extract cosmological information on τ from the EE spectrum alone, we restrict the analysis to the crosscorrelation between the HFI 100 and 143 GHz maps, which exhibits the lowest variance.
At large angular scales, the HFI maps are contaminated by systematic residuals coming from temperaturetopolarization leakage (see Planck Collaboration VIII 2016). We used our best estimate of the Q and U maps at 100 and 143 GHz, which we correct for residual leakage coming from destriping uncertainties, calibration mismatch, and bandpass mismatch, using templates as described in Planck Collaboration VIII (2016). Even though the level of systematic effects is thereby significantly reduced, we still have residuals above the noise level in null tests at very low multipoles (). To mitigate the effect of this on the likelihood, we restrict the range of multipoles to ℓ = 5−20.
Crosspower spectra are computed on the cleanest 50% of the sky by using a pseudoC_{ℓ} estimate (Xpol , an extension to polarization of the code described in Tristram et al. 2005a). The mask corresponds to thresholding a map of the diffuse polarized Galactic dust at large scales. In addition, we also removed pixels where the intensity of diffuse Galactic dust and CO lines is strong. This ensures that bandpass leakage from dust and CO lines does not bias the polarization spectra (see Planck Collaboration VIII 2016).
We construct the C_{ℓ} correlation matrix using simulations including CMB signal and realistic inhomogeneous and correlated noise. In order to take into account the residual systematics, we derive the noise level from the estimated BB autospectrum where we neglect any possible cosmological signal. This overestimates the noise level and ensures conservative errors. However, this estimate assumes by construction a Gaussian noise contribution, which is not a full description of the residuals.
We then sample the reionization optical depth τ from the likelihood, with all other parameters fixed to the Planck 2015 bestfit values (Planck Collaboration XIII 2016). Without any other data, the degeneracy between A_{s} and τ is broken by fixing the amplitude of the first peak of the TT spectrum (directly related to A_{s}e^{− 2τ}) at ℓ = 200. The resulting distribution is plotted in Fig. B.1. The best fit is at $\mathit{\tau}\mathrm{=}\mathrm{0.06}{\mathrm{4}}_{0.016}^{\mathrm{+}\mathrm{0.015}}\mathit{,}\u2001{\mathit{z}}_{\mathrm{re}}\mathrm{=}\mathrm{8.}{\mathrm{7}}_{1.6}^{\mathrm{+}\mathrm{1.4}}\mathit{,}$(B.9)in agreement with the current Planck lowℓ baseline (see Table 2), even though this result only relies on the EE spectrum between ℓ = 5 and 20.
Fig. B.1 Distribution of the reionization optical depth τ using the Lollipop likelihood, based on the crosscorrelation of the 100 and 143 GHz channels. 
Appendix C: Highℓ baseline likelihood: Plik
In this appendix, we provide detailed information on the Plik baseline likelihood used at high ℓ. First we describe in Sect. C.1 the Plik covariance matrix, by providing the equations we have implemented, by giving results from some of the numerical tests we carried out, and by describing our procedure to deal with the excess variance (as compared to the prediction of our approximate analytical model) due to the point source mask. Section C.2 validates the overall Plik implementation with Monte Carlo simulations of the full mission. For reference, Sect. C.3 gives the results of a large body of validation and stability tests on the actual data, including polarization in particular. We also discuss the numerical agreement of the temperature and polarizationbased results on baseΛCDM parameters. Section C.4 describes how we calculate coadded CMB spectra from foregroundcleaned frequency power spectra. Section C.5 compares Plik cosmological results obtained using the PICO or CAMB codes. Finally, Sect. C.6 details how we marginalize over nuisance parameters to provide a fast but accurate CMBonly likelihood.
Appendix C.1: Covariance matrix
Appendix C.1.1: Structure of the covariance matrix
Here we summarize the mathematical formalism implemented to calculate the pseudopower spectrum covariance matrices for temperature and polarization.
In the following, the fiducial power spectra C_{ℓ} are assumed to be the smooth theory spectra multiplied by beam (b) and pixel window function (p) for detectors i and j, ${\mathit{C}}_{\mathit{\ell}}^{\mathit{i,j}}\mathrm{=}{\mathit{b}}_{\mathit{\ell}}^{\mathit{i}}\hspace{0.17em}{\mathit{b}}_{\mathit{\ell}}^{\mathit{j}}\hspace{0.17em}{\mathit{p}}_{\mathit{\ell}}^{\mathrm{2}}\mathrm{(}{\mathit{C}}_{\mathit{\ell}}^{\mathrm{CMB}}\mathrm{+}{\mathit{C}}_{\mathit{\ell}}^{\mathrm{FG}}\mathrm{(}{\mathit{f}}_{\mathit{i}}\mathit{,}{\mathit{f}}_{\mathit{j}}{\mathrm{\left)}}^{\mathrm{\right)}}\mathit{,}$(C.1)where the f_{k} denote the frequency dependence of the foreground contribution.
We now present the equations used to compute all the unique covariance matrix polarization blocks that can be formed from temperature and Emode polarization maps (Hansen et al. 2002; Hinshaw et al. 2003; Efstathiou 2004; Challinor & Chon 2005; Like13). They approximate the variance of the biased pseudopower spectrum coefficients, before correcting for the effects of pixel window function, beam, and mask.
TTTT block:
$\begin{array}{ccc}& & \begin{array}{c}\mathrm{Var}\end{array}\mathrm{\left(}\mathit{C\u0302}\begin{array}{c}\mathit{TT}\hspace{0.17em}\mathit{i,j}\\ \mathit{\ell}\end{array}\mathit{,}\mathit{C\u0302}\begin{array}{c}\mathit{TT}\hspace{0.17em}\mathit{p,q}\\ {\mathit{\ell}}^{\mathrm{\prime}}\end{array}\mathrm{\right)}\\ & & \mathrm{\approx}\sqrt{{\mathit{C}}_{\mathit{\ell}}^{\mathit{TT}\hspace{0.17em}\mathit{i,p}}{\mathit{C}}_{{\mathit{\ell}}^{\mathrm{\prime}}}^{\mathit{TT}\hspace{0.17em}\mathit{i,p}}{\mathit{C}}_{\mathit{\ell}}^{\mathit{TT}\hspace{0.17em}\mathit{j,q}}{\mathit{C}}_{{\mathit{\ell}}^{\mathrm{\prime}}}^{\mathit{TT}\hspace{0.17em}\mathit{j,q}}}{\mathrm{\Xi}}_{\mathit{TT}}^{\mathrm{\varnothing \varnothing}\mathit{,}\mathrm{\varnothing \varnothing}}{\mathrm{[}\mathrm{\left(}\mathit{i,p}{\mathrm{)}}^{\mathit{TT}}\mathit{,}\mathrm{\right(}\mathit{j,q}{\mathrm{)}}^{\mathit{TT}}\mathrm{]}}_{\mathit{\ell}{}^{}}\end{array}$