Planck 2013 results. XV. CMB power spectra and likelihood

¹ APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France
² Aalto University Metsähovi Radio Observatory, Metsähovintie 114, 02540 Kylmälä, Finland
³ African Institute for Mathematical Sciences, 6-8 Melrose Road, 7945 Muizenberg, Cape Town, South Africa
⁴ Agenzia Spaziale Italiana Science Data Center, via del Politecnico snc, 00133 Roma, Italy
⁵ Agenzia Spaziale Italiana, Viale Liegi 26, 00198 Roma, Italy
⁶ Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, UK
⁷ Astrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, 4000 Durban, South Africa
⁸ Atacama Large Millimeter/submillimeter Array, ALMA Santiago Central Offices, Alonso de Cordova 3107, Vitacura, Casilla 763 0355 Santiago, Chile
⁹ CITA, University of Toronto, 60 St. George St., Toronto, ON M5S 3H8, Canada
¹⁰ CNRS, IRAP, 9 Av. colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France
¹¹ California Institute of Technology, Pasadena, California, USA
¹² Centre for Theoretical Cosmology, DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
¹³ Centro de Estudios de Física del Cosmos de Aragón (CEFCA), Plaza San Juan, 1, planta 2, 44001 Teruel, Spain
¹⁴ Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, California, USA
¹⁵ Consejo Superior de Investigaciones Científicas (CSIC), 28006 Madrid, Spain
¹⁶ DSM/Irfu/SPP, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
¹⁷ DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
¹⁸ Département de Physique Théorique, Université de Genève, 24 Quai E. Ansermet, 1211 Genève 4, Switzerland
¹⁹ Departamento de Física Fundamental, Facultad de Ciencias, Universidad de Salamanca, 37008 Salamanca, Spain
²⁰ Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo s/n, Oviedo, Spain
²¹ Department of Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, Ontario, Canada
²² Department of Astrophysics/IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
²³ Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California, USA
²⁴ Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada
²⁵ Department of Physics and Astronomy, Dana and David Dornsife College of Letter, Arts and Sciences, University of Southern California, Los Angeles CA 90089, USA
²⁶ Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
²⁷ Department of Physics, Florida State University, Keen Physics Building, 77 Chieftan Way, Tallahassee, Florida, USA
²⁸ Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, 00014 Helsinki, Finland
²⁹ Department of Physics, Princeton University, Princeton, New Jersey, USA
³⁰ Department of Physics, University of California, Berkeley, California, USA
³¹ Department of Physics, University of California, One Shields Avenue, Davis, California, USA
³² Department of Physics, University of California, Santa Barbara, California, USA
³³ Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, USA
³⁴ Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi di Padova, via Marzolo 8, 35131 Padova, Italy
³⁵ Dipartimento di Fisica e Astronomia, Università degli Studi di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
³⁶ Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, via Saragat 1, 44122 Ferrara, Italy
³⁷ Dipartimento di Fisica, Università La Sapienza, P.le A. Moro 2, 000185 Roma, Italy
³⁸ Dipartimento di Fisica, Università degli Studi di Milano, via Celoria, 16, 20133 Milano, Italy
³⁹ Dipartimento di Fisica, Università degli Studi di Trieste, via A. Valerio 2, 34127 Trieste, Italy
⁴⁰ Dipartimento di Fisica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, 00133 Roma, Italy
⁴¹ Dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, 000133 Roma, Italy
⁴² Discovery Center, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
⁴³ Dpto. Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain
⁴⁴ European Southern Observatory, ESO Vitacura, Alonso de Cordova 3107, Vitacura, Casilla 19001 Santiago, Chile
⁴⁵ European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo, s/n, Urbanización Villafranca del Castillo, Villanueva de la Cañada, Madrid, Spain
⁴⁶ European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
⁴⁷ Finnish Centre for Astronomy with ESO (FINCA), University of Turku, Väisäläntie 20, 21500, Piikkiö, Finland
⁴⁸ Haverford College Astronomy Department, 370 Lancaster Avenue, Haverford, Pennsylvania, USA
⁴⁹ Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, 00014 Helsinki, Finland
⁵⁰ INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy
⁵¹ INAF – Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monte Porzio Catone, Italy
⁵² INAF – Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, 34131 Trieste, Italy
⁵³ INAF Istituto di Radioastronomia, via P. Gobetti 101, 40129 Bologna, Italy
⁵⁴ INAF/IASF Bologna, via Gobetti 101, 40129 Bologna, Italy
⁵⁵ INAF/IASF Milano, via E. Bassini 15, 23011 Milano, Italy
⁵⁶ INFN, Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
⁵⁷ INFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185 Roma, Italy
⁵⁸ IPAG: Institut de Planétologie et d’Astrophysique de Grenoble, Université Joseph Fourier, Grenoble 1/CNRS-INSU, UMR 5274, Grenoble, 38041, France
⁵⁹ ISDC Data Centre for Astrophysics, University of Geneva, ch. d’Ecogia 16, Versoix, Switzerland
⁶⁰ IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India
⁶¹ Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, UK
⁶² Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA
⁶³ Institut Néel, CNRS, Université Joseph Fourier Grenoble I, 25 rue des Martyrs, 38042 Grenoble, France
⁶⁴ Institut Universitaire de France, 103 bd Saint-Michel, 75005 Paris, France
⁶⁵ Institut d’Astrophysique Spatiale, CNRS (UMR 8617) Université Paris-Sud 11, Bâtiment 121, 91405 Orsay, France
⁶⁶ Institut d’Astrophysique de Paris, CNRS (UMR 7095), 98bis Bd Arago, 75014 Paris, France
⁶⁷ Institute for Space Sciences, 077125 Bucharest-Magurale, Romania
⁶⁸ Institute of Astronomy and Astrophysics, Academia Sinica, 10617 Taipei, Taiwan
⁶⁹ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
⁷⁰ Institute of Mathematics and Physics, Centre for Cosmology, Particle Physics and Phenomenology, Louvain University, 1348 Louvain-la-Neuve, Belgium
⁷¹ Institute of Theoretical Astrophysics, University of Oslo, Blindern, 0315 Oslo, Norway
⁷² Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, 38205 La Laguna, Tenerife, Spain
⁷³ Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, 39005 Santander, Spain
⁷⁴ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, USA
⁷⁵ Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
⁷⁶ Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
⁷⁷ LAL, Université Paris-Sud, CNRS/IN2P3, 91898 Orsay, France
⁷⁸ LERMA, CNRS, Observatoire de Paris, 61 Avenue de l’Observatoire, 75014 Paris, France
⁷⁹ Laboratoire AIM, IRFU/Service d’Astrophysique – CEA/DSM – CNRS – Université Paris Diderot, Bât. 709, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
⁸⁰ Laboratoire Traitement et Communication de l’Information, CNRS (UMR 5141) and Télécom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France
⁸¹ Laboratoire de Physique Subatomique et de Cosmologie, Université Joseph Fourier Grenoble I, CNRS/IN2P3, Institut National Polytechnique de Grenoble, 53 rue des Martyrs, 38026 Grenoble Cedex, France
⁸² Laboratoire de Physique Théorique, Université Paris-Sud 11 & CNRS, Bâtiment 210, 91405 Orsay, France
⁸³ Lawrence Berkeley National Laboratory, Berkeley, California, USA
⁸⁴ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany
⁸⁵ McGill Physics, Ernest Rutherford Physics Building, McGill University, 3600 rue University, Montréal, QC, H3A 2T8, Canada
⁸⁶ MilliLab, VTT Technical Research Centre of Finland, Tietotie 3, 02044 Espoo, Finland
⁸⁷ National University of Ireland, Department of Experimental Physics, Maynooth, Co. Kildare, Ireland
⁸⁸ Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
⁸⁹ Observational Cosmology, Mail Stop 367-17, California Institute of Technology, Pasadena CA 91125, USA
⁹⁰ Optical Science Laboratory, University College London, Gower Street, London, UK
⁹¹ SB-ITP-LPPC, EPFL, 1015 Lausanne, Switzerland
⁹² SISSA, Astrophysics Sector, via Bonomea 265, 34136 Trieste, Italy
⁹³ School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK
⁹⁴ School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK
⁹⁵ Space Sciences Laboratory, University of California, Berkeley, California, USA
⁹⁶ Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, 369167 Karachai-Cherkessian Republic, Russia
⁹⁷ Stanford University, Dept of Physics, Varian Physics Bldg, 382 via Pueblo Mall, Stanford, California, USA
⁹⁸ Sub-Department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
⁹⁹ Theory Division, PH-TH, CERN, 1211, Geneva 23, Switzerland
¹⁰⁰ UPMC Univ. Paris 06, UMR7095, 98bis Boulevard Arago, 75014 Paris, France
¹⁰¹ Université de Toulouse, UPS-OMP, IRAP, 31028 Toulouse Cedex 4, France
¹⁰² University of Granada, Departamento de Física Teórica y del Cosmos, Facultad de Ciencias, 18071 Granada, Spain
¹⁰³ Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland

Received: 26 March 2013
Accepted: 5 May 2014

Abstract

This paper presents the Planck 2013 likelihood, a complete statistical description of the two-point correlation function of the CMB temperature fluctuations that accounts for all known relevant uncertainties, both instrumental and astrophysical in nature. We use this likelihood to derive our best estimate of the CMB angular power spectrum from Planck over three decades in multipole moment, ℓ, covering 2 ≤ ℓ ≤ 2500. The main source of uncertainty at ℓ ≲ 1500 is cosmic variance. Uncertainties in small-scale foreground modelling and instrumental noise dominate the error budget at higher ℓs. For ℓ < 50, our likelihood exploits all Planck frequency channels from 30 to 353 GHz, separating the cosmological CMB signal from diffuse Galactic foregrounds through a physically motivated Bayesian component separation technique. At ℓ ≥ 50, we employ a correlated Gaussian likelihood approximation based on a fine-grained set of angular cross-spectra derived from multiple detector combinations between the 100, 143, and 217 GHz frequency channels, marginalising over power spectrum foreground templates. We validate our likelihood through an extensive suite of consistency tests, and assess the impact of residual foreground and instrumental uncertainties on the final cosmological parameters. We find good internal agreement among the high-ℓ cross-spectra with residuals below a few μK² at ℓ ≲ 1000, in agreement with estimated calibration uncertainties. We compare our results with foreground-cleaned CMB maps derived from all Planck frequencies, as well as with cross-spectra derived from the 70 GHz Planck map, and find broad agreement in terms of spectrum residuals and cosmological parameters. We further show that the best-fit ΛCDM cosmology is in excellent agreement with preliminary PlanckEE and TE polarisation spectra. We find that the standard ΛCDM cosmology is well constrained by Planck from the measurements at ℓ ≲ 1500. One specific example is the spectral index of scalar perturbations, for which we report a 5.4σ deviation from scale invariance, n_s = 1. Increasing the multipole range beyond ℓ ≃ 1500 does not increase our accuracy for the ΛCDM parameters, but instead allows us to study extensions beyond the standard model. We find no indication of significant departures from the ΛCDM framework. Finally, we report a tension between the Planck best-fit ΛCDM model and the low-ℓ spectrum in the form of a power deficit of 5–10% at ℓ ≲ 40, with a statistical significance of 2.5–3σ. Without a theoretically motivated model for this power deficit, we do not elaborate further on its cosmological implications, but note that this is our most puzzling finding in an otherwise remarkably consistent data set.