Planck 2015 results

XVIII. Background geometry and topology of the Universe

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 Aix Marseille 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 KwaZulu-Natal, 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 70308-200, 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, 2 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 delCosmos de Aragón (CEFCA), Plaza San Juan, 1, planta 2, 44001 Teruel, Spain
¹⁵ Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, CA94720 California, USA
¹⁶ Consejo Superior de Investigaciones Científicas (CSIC), 28006 Madrid, Spain
¹⁷ DSM/Irfu/SPP, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
¹⁸ DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
¹⁹ Département de Physique Théorique, Université de Genève, 24 Quai E. Ansermet, 1211 Genève 4, Switzerland
²⁰ Departamento de Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain
²¹ Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo s/n, 33003 Oviedo, Spain
²² Department of Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, ONMSS Ontario, Canada
²³ Department of Astrophysics/IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
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, 00100 Helsinki, Finland
²⁹ Department of Physics, Princeton University, Princeton, New Jersey NJ 08544, USA
³⁰ Department of Physics, University of Alberta, 11322-89 Avenue, Edmonton, Alberta, T6G 2G7, Canada
³¹ Department of Physics, University of California, Santa Barbara, California CA 93106, USA
³² Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, USA
³³ Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi di Padova, via Marzolo 8, 35131 Padova, Italy
³⁴ Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, via Saragat 1, 44122 Ferrara, Italy
³⁵ Dipartimento di Fisica, Università La Sapienza, P.le A. Moro 2, Roma, Italy
³⁶ Dipartimento di Fisica, Università degli Studi di Milano, via Celoria, 16 Milano, Italy
³⁷ Dipartimento di Fisica, Università degli Studi di Trieste, via A. Valerio 2, Trieste, Italy
³⁸ Dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, Roma, Italy
³⁹ Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark
⁴⁰ Discovery Center, Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, Copenhagen, Denmark
⁴¹ European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo, s/n, Urbanización Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
⁴² European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
⁴³ Gran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’ Aquila, Italy
⁴⁴ HGSFP and University of Heidelberg, Theoretical Physics Department, Philosophenweg 16, 69120 Heidelberg, Germany
⁴⁵ Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland
⁴⁶ INAF−Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, Italy
⁴⁷ INAF−Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, Italy
⁴⁸ INAF−Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, Trieste, Italy
⁴⁹ INAF/IASF Bologna, via Gobetti 101, Bologna, Italy
⁵⁰ INAF/IASF Milano, via E. Bassini 15, Milano, Italy
⁵¹ INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
⁵² INFN, Sezione di Ferrara, via Saragat 1, 44122 Ferrara, Italy
⁵³ INFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185, Roma, Italy
⁵⁴ INFN, Sezione di Roma 2, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, Roma, Italy
⁵⁵ INFN/National Institute for Nuclear Physics, via Valerio 2, 34127 Trieste, Italy
⁵⁶ IPAG: Institut de Planétologie et d’Astrophysique de Grenoble, Université Grenoble Alpes, IPAG; CNRS, IPAG, 38000 Grenoble, France
⁵⁷ IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, 411 007 Pune, India
⁵⁸ Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, UK
⁵⁹ Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA
⁶⁰ Institut Néel, CNRS, Université Joseph Fourier Grenoble I, 25 rue des Martyrs, Grenoble, France
⁶¹ Institut Universitaire de France, 103 Bd Saint-Michel, 75005 Paris, France
⁶² Institut d’Astrophysique Spatiale, CNRS, Univ. Paris-Sud, Université Paris-Saclay, Bât. 121, 91405 Orsay Cedex, France
⁶³ Institut d’Astrophysique de Paris, CNRS (UMR 7095), 98 bis Boulevard Arago, 75014 Paris, France
⁶⁴ Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, 52056 Aachen, Germany
⁶⁵ Institute for Space Sciences, Bucharest-Magurale, 077125 Bucharest, Romania
⁶⁶ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
⁶⁷ Institute of Theoretical Astrophysics, University of Oslo, 1072 Blindern, Oslo, Norway
⁶⁸ Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, La Laguna, 38205 Tenerife, Spain
⁶⁹ Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, 39005 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, California, 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 Paris-Sud, CNRS/IN2P3, Orsay, France
⁷⁷ LERMA, CNRS, Observatoire de Paris, 61 Avenue de l’Observatoire, Paris, France
⁷⁸ Laboratoire AIM, IRFU/Service d’Astrophysique− CEA/DSM − CNRS − Université Paris Diderot, Bât. 709, CEA-Saclay, 1191 Gif-sur-Yvette 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 Grenoble-Alpes, CNRS/IN2P3, 53 rue des Martyrs, 38026 Grenoble Cedex, France
⁸¹ Laboratoire de Physique Théorique, Université Paris-Sud 11 & CNRS, Bâtiment 210, 91405 Orsay, France
⁸² Lawrence Berkeley National Laboratory, Berkeley, CA 94720 California, USA
⁸³ Lebedev Physical Institute of the Russian Academy of Sciences, Astro Space Centre, 84/32 Profsoyuznaya st., 117997 Moscow, Russia
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 Mullard Space Science Laboratory, University College London, Surrey RH5 6NT, UK
⁸⁷ National University of Ireland, Department of Experimental Physics, Maynooth, Co. Kildare, Ireland
⁸⁸ Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland
⁸⁹ Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark
⁹⁰ Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, Copenhagen, Denmark
⁹¹ Nordita (Nordic Institute for Theoretical Physics), Roslagstullsbacken 23, 106 91 Stockholm, Sweden
⁹² Optical Science Laboratory, University College London, Gower Street, WC 1E6 BT London, UK
⁹³ SISSA, Astrophysics Sector, via Bonomea 265, 34136 Trieste, Italy
⁹⁴ SMARTEST Research Centre, Università degli Studi e-Campus, via Isimbardi 10, 22060 Novedrate (CO), Italy
⁹⁵ School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK
⁹⁶ School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK
⁹⁷ Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, 75014 Paris, France
⁹⁸ Space Sciences Laboratory, University of California, Berkeley, CA 94720 California, USA
⁹⁹ Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, 369167 Karachai-Cherkessian Republic, Russia
¹⁰⁰ Stanford University, Dept of Physics, Varian Physics Bldg, 382 via Pueblo Mall, Stanford, California, USA
¹⁰¹ Sub-Department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
¹⁰² The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, 106 91 Stockholm, Sweden
¹⁰³ Theory Division, PH-TH, CERN, 1211 Geneva 23, Switzerland
¹⁰⁴ UPMC Univ. Paris 06, UMR7095, 98 bis Boulevard Arago, 75014 Paris, France
¹⁰⁵ Université de Toulouse, UPS-OMP, IRAP, 31028 Toulouse Cedex 4, France
¹⁰⁶ University of Granada, Departamento de Física Teórica y del Cosmos, Facultad de Ciencias, 18010 Granada, Spain
¹⁰⁷ University of Granada, Instituto Carlos I de Física Teórica y Computacional, Granada, Spain
¹⁰⁸ Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland

^⋆ Corresponding author: A. H. Jaffe, e-mail: a.jaffe@imperial.ac.uk

Received: 6 February 2015
Accepted: 10 April 2016

Abstract

Maps of cosmic microwave background (CMB) temperature and polarization from the 2015 release of Planck data provide the highestquality full-sky view of the surface of last scattering available to date. This enables us to detect possible departures from a globally isotropic cosmology. We present the first searches using CMB polarization for correlations induced by a possible non-trivial topology with a fundamental domain that intersects, or nearly intersects, the last-scattering surface (at comoving distance χ_rec), both via a direct scan for matched circular patterns at the intersections and by an optimal likelihood calculation for specific topologies. We specialize to flat spaces with cubic toroidal (T3) and slab (T1) topologies, finding that explicit searches for the latter are sensitive to other topologies with antipodal symmetry. These searches yield no detection of a compact topology with a scale below the diameter of the last-scattering surface. The limits on the radius ℛ_i of the largest sphere inscribed in the fundamental domain (at log-likelihood ratio Δlnℒ > −5 relative to a simply-connected flat Planck best-fit model) are: ℛ_i > 0.97 χ_rec for the T3 cubic torus; and ℛ_i > 0.56 χ_rec for the T1 slab. The limit for the T3 cubic torus from the matched-circles search is numerically equivalent, ℛ_i > 0.97 χ_rec at 99% confidence level from polarization data alone. We also perform a Bayesian search for an anisotropic global Bianchi VII_h geometry. In the non-physical setting, where the Bianchi cosmology is decoupled from the standard cosmology, Planck temperature data favour the inclusion of a Bianchi component with a Bayes factor of at least 2.3 units of log-evidence. However, the cosmological parameters that generate this pattern are in strong disagreement with those found from CMB anisotropy data alone. Fitting the induced polarization pattern for this model to the Planck data requires an amplitude of −0.10 ± 0.04 compared to the value of + 1 if the model were to be correct. In the physically motivated setting, where the Bianchi parameters are coupled and fitted simultaneously with the standard cosmological parameters, we find no evidence for a Bianchi VII_h cosmology and constrain the vorticity of such models to (ω/H)₀ < 7.6 × 10^-10 (95% CL).