Planck intermediate results

XVI. Profile likelihoods for cosmological 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, 6-8 Melrose Road, 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, 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 Centro de Estudios de Física del Cosmos de Aragón (CEFCA), Plaza San Juan 1, planta 2, 44001 Teruel, Spain
¹³ Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, California, USA
¹⁴ Consejo Superior de Investigaciones Científicas (CSIC), Madrid, Spain
¹⁵ DSM/Irfu/SPP, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
¹⁶ DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
¹⁷ Département de Physique Théorique, Université de Genève, 24 Quai E. Ansermet, 1211 Genève 4, Switzerland
¹⁸ Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo s/n, Oviedo, Spain
¹⁹ Department of Astrophysics/IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
²⁰ Department of Physics and Astronomy, Dana and David Dornsife College of Letter, Arts and Sciences, University of Southern California, Los Angeles CA 90089, USA
²¹ Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
²² Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK
²³ Department of Physics, Florida State University, Keen Physics Building, 77 Chieftan Way, Tallahassee, Florida, USA
²⁴ Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, Helsinki, Finland
²⁵ Department of Physics, Princeton University, Princeton, New Jersey, USA
²⁶ Department of Physics, University of California, Berkeley, California, USA
²⁷ Department of Physics, University of California, One Shields Avenue, Davis, California, USA
²⁸ Department of Physics, University of California, Santa Barbara, California, USA
²⁹ Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, USA
³⁰ Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi diPadova, via Marzolo 8, 35131 Padova, Italy
³¹ Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, via Saragat 1, 44122 Ferrara, Italy
³² Dipartimento di Fisica, Università La Sapienza, P.le A. Moro 2, Roma, Italy
³³ Dipartimento di Fisica, Università degli Studi di Milano, via Celoria, 16, Milano, Italy
³⁴ Dipartimento di Fisica, Università degli Studi di Trieste, via A. Valerio 2, Trieste, Italy
³⁵ Dipartimento di Fisica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, Roma, Italy
³⁶ Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark
³⁷ Dpto. Astrofísica, Universidad de La Laguna (ULL), 38206, La Laguna, Tenerife, Spain
³⁸ European Southern Observatory, ESO Vitacura, Alonso de Cordova 3107, Vitacura, Casilla 19001, Santiago, Chile
³⁹ European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo s/n, Urbanización Villafranca del Castillo, Villanueva de la Cañada, Madrid, Spain
⁴⁰ European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
⁴¹ Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland
⁴² INAF – Osservatorio Astrofisico di Catania, via S. Sofia 78, Catania, Italy
⁴³ INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, Italy
⁴⁴ INAF – Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, Italy
⁴⁵ INAF – Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, Trieste, Italy
⁴⁶ INAF Istituto di Radioastronomia, via P. Gobetti 101, 40129 Bologna, Italy
⁴⁷ INAF/IASF Bologna, via Gobetti 101, Bologna, Italy
⁴⁸ INAF/IASF Milano, via E. Bassini 15, Milano, Italy
⁴⁹ INFN, Sezione di Bologna, via Irnerio 46, 40126, Bologna, Italy
⁵⁰ INFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185 Roma, Italy
⁵¹ INFN/National Institute for Nuclear Physics, via Valerio 2, 34127 Trieste, Italy
⁵² IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, 411 007 Pune, India
⁵³ Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, UK
⁵⁴ Infrared Processing and Analysis Center, California Institute of Technology, Pasadena CA 91125, USA
⁵⁵ Institut Universitaire de France, 103 bd Saint-Michel, 75005 Paris, France
⁵⁶ Institut d’Astrophysique Spatiale, CNRS (UMR 8617) Université Paris-Sud 11, Bâtiment 121, Orsay, France
⁵⁷ Institut d’Astrophysique de Paris, CNRS (UMR 7095), 98bis Boulevard Arago, 75014 Paris, France
⁵⁸ Institute for Space Sciences, Bucharest-Magurale, Romania
⁵⁹ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
⁶⁰ Institute of Theoretical Astrophysics, University of Oslo, Blindern, Oslo, Norway
⁶¹ Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, La Laguna, Tenerife, Spain
⁶² Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, Santander, Spain
⁶³ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, USA
⁶⁴ Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
⁶⁵ Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
⁶⁶ LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
⁶⁷ LERMA, CNRS, Observatoire de Paris, 61 Avenue de l’Observatoire, Paris, France
⁶⁸ Laboratoire AIM, IRFU/Service d’Astrophysique – CEA/DSM – CNRS – Université Paris Diderot, Bât. 709, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
⁶⁹ Laboratoire Traitement et Communication de l’Information, CNRS (UMR 5141) and Télécom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France
⁷⁰ 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
⁷¹ Laboratoire de Physique Théorique, Université Paris-Sud 11 & CNRS, Bâtiment 210, 91405 Orsay, France
⁷² Lawrence Berkeley National Laboratory, Berkeley, California, USA
⁷³ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany
⁷⁴ National University of Ireland, Department of Experimental Physics, Maynooth, Co. Kildare, Ireland
⁷⁵ Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark
⁷⁶ Optical Science Laboratory, University College London, Gower Street, London, UK
⁷⁷ SISSA, Astrophysics Sector, via Bonomea 265, 34136 Trieste, Italy
⁷⁸ SUPA, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
⁷⁹ School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK
⁸⁰ UPMC Univ. Paris 06, UMR 7095, 98bis BoulevardArago, 75014 Paris, France
⁸¹ Université de Toulouse, UPS-OMP, IRAP, 31028 Toulouse Cedex 4, France
⁸² University of Granada, Departamento de Física Teórica y del Cosmos, Facultad de Ciencias, Granada, Spain
⁸³ University of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, Spain
⁸⁴ Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland

Received: 7 November 2013
Accepted: 2 May 2014

Abstract

We explore the 2013 Planck likelihood function with a high-precision multi-dimensional minimizer (Minuit). This allows a refinement of the ΛCDM best-fit solution with respect to previously-released results, and the construction of frequentist confidence intervals using profile likelihoods. The agreement with the cosmological results from the Bayesian framework is excellent, demonstrating the robustness of the Planck results to the statistical methodology. We investigate the inclusion of neutrino masses, where more significant differences may appear due to the non-Gaussian nature of the posterior mass distribution. By applying the Feldman-Cousins prescription, we again obtain results very similar to those of the Bayesian methodology. However, the profile-likelihood analysis of the cosmic microwave background (CMB) combination (Planck+WP+highL) reveals a minimum well within the unphysical negative-mass region. We show that inclusion of the Planck CMB-lensing information regularizes this issue, and provide a robust frequentist upper limit ∑ m_ν ≤ 0.26 eV (95% confidence) from the CMB+lensing+BAO data combination.