Planck 2013 results. XXVI. 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, 7945 Cape Town, South Africa
⁴ Agenzia SpazialeItaliana 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 CITA, University of Toronto, 60 St. George St., Toronto, ON M5S 3H8, Canada
⁹ CNRS, IRAP, 9 Av. colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France
¹⁰ California Institute of Technology, Pasadena, California, USA
¹¹ Centre for Theoretical Cosmology, DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
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), 28037 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 Fundamental, Facultad de Ciencias, Universidad de Salamanca, 37008 Salamanca, Spain
¹⁹ Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo s/n, 33007 Oviedo, Spain
²⁰ Department of Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, Ontario, Canada
²¹ Department of Astrophysics/IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
²² Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California, USA
²³ Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada
²⁴ Department of Physics and Astronomy, Dana and David Dornsife College of Letter, Arts and Sciences, University of Southern California, Los Angeles, CA 90089, USA
²⁵ Department of Physics and Astronomy, University College London, London WC1E 6BT, 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, New Jersey, USA
²⁹ Department of Physics, University of Alberta, 11322-89 Avenue, Edmonton, Alberta, T6G 2G7, Canada
³⁰ Department of Physics, University of California, One Shields Avenue, Davis, California, USA
³¹ Department of Physics, University of California, Santa Barbara, California, 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, 00185 Roma, Italy
³⁶ Dipartimento di Fisica, Università degli Studi di Milano, via Celoria, 16, 20133 Milano, Italy
³⁷ Dipartimento di Fisica, Università degli Studi di Trieste, via A. Valerio 2, 34127 Trieste, Italy
³⁸ Dipartimento di Fisica, Università di Roma Tor Vergata, via della Ricerca Scientifica, 1, 00133 Roma, Italy
³⁹ Discovery Center, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
⁴⁰ Dpto. Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain
⁴¹ European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo, s/n, Urbanización Villafranca del Castillo, 28691 Villanueva de la Cañada, Madrid, Spain
⁴² European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
⁴³ Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, 00014 Helsinki, Finland
⁴⁴ INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 25122 Padova, Italy
⁴⁵ INAF – Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monte PorzioCatone, Italy
⁴⁶ INAF – Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, 34143 Trieste, Italy
⁴⁷ INAF Istituto di Radioastronomia, via P. Gobetti 101, 40129 Bologna, Italy
⁴⁸ INAF/IASF Bologna, via Gobetti 101, 40129 Bologna, Italy
⁴⁹ INAF/IASF Milano, via E. Bassini 15, 20133 Milano, Italy
⁵⁰ INFN, Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
⁵¹ INFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185 Roma, Italy
⁵² IPAG: Institut de Planétologie et d’Astrophysique de Grenoble, Université Joseph Fourier, Grenoble 1/CNRS-INSU, UMR 5274, 38041 Grenoble, France
⁵³ IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, 411 007 Pune, India
⁵⁴ Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, UK
⁵⁵ Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA
⁵⁶ Institut Néel, CNRS, Université Joseph Fourier Grenoble I, 25 rue des Martyrs, 38042 Grenoble, France
⁵⁷ Institut Universitaire de France, 103 bd Saint-Michel, 75005 Paris, France
⁵⁸ Institut d’Astrophysique Spatiale, CNRS (UMR 8617), Université Paris-Sud 11, Bâtiment 121, 91405 Orsay, France
⁵⁹ Institut d’Astrophysique de Paris, CNRS (UMR 7095), 98bis boulevard Arago, 75014 Paris, France
⁶⁰ Institute for Space Sciences, 077125 Bucharest-Magurale, Romania
⁶¹ Institute of Astronomy and Astrophysics, Academia Sinica, 106 Taipei, Taiwan
⁶² Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
⁶³ Institute of Theoretical Astrophysics, University of Oslo, Blindern, 0315 Oslo, Norway
⁶⁴ Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, La Laguna, 38200 Tenerife, Spain
⁶⁵ Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, 39005 Santander, Spain
⁶⁶ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, USA
⁶⁷ Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
⁶⁸ Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
⁶⁹ LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
⁷⁰ LERMA, CNRS, Observatoire de Paris, 61 Av. l’Observatoire, 75014 Paris, France
⁷¹ Laboratoire AIM, IRFU/Service d’Astrophysique – CEA/DSM – CNRS – Université Paris Diderot, Bât. 709, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
⁷² Laboratoire Traitement et Communication de l’Information, CNRS (UMR 5141) and Télécom ParisTech, 46 rue Barrault, 75634 Paris Cedex 13, France
⁷³ 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
⁷⁴ Laboratoire de Physique Théorique, Université Paris-Sud 11 & CNRS, Bât. 210, 91405 Orsay, France
⁷⁵ Lawrence Berkeley National Laboratory, Berkeley, California, USA
⁷⁶ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany
⁷⁷ McGill Physics, Ernest Rutherford Physics Building, McGill University, 3600 rue University, Montréal, QC, H3A 2T8, Canada
⁷⁸ MilliLab, VTT Technical Research Centre of Finland, Tietotie 3, 02044 Espoo, Finland
⁷⁹ Mullard Space Science Laboratory, University College London, Surrey RH5 6NT, UK
⁸⁰ National University of Ireland, Department of Experimental Physics, Maynooth, Co. Kildare, Ireland
⁸¹ Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
⁸² Observational Cosmology, Mail Stop 367-17, California Institute of Technology, Pasadena, CA, 91125, USA
⁸³ Optical Science Laboratory, University College London, Gower Street, London, UK
⁸⁴ SB-ITP-LPPC, EPFL, 1015, Lausanne, Switzerland
⁸⁵ SISSA, Astrophysics Sector, via Bonomea 265, 34136 Trieste, Italy
⁸⁶ School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK
⁸⁷ School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK
⁸⁸ Space Sciences Laboratory, University of California, Berkeley, California, USA
⁸⁹ Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, 369167 Karachai-Cherkessian Republic, Russia
⁹⁰ Stanford University, Dept of Physics, Varian Physics Bldg, 382 via Pueblo Mall, Stanford, California, USA
⁹¹ Sub-Department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
⁹² Theory Division, PH-TH, CERN, CH-1211, 23 Geneva, Switzerland
⁹³ UPMC Univ Paris 06, UMR7095, 98bis boulevard Arago, 75014 Paris, France
⁹⁴ Université de Toulouse, UPS-OMP, IRAP, 31028 Toulouse Cedex 4, France
⁹⁵ University of Granada, Departamento de Física Teórica y del Cosmos, Facultad de Ciencias, 18071 Granada, Spain
⁹⁶ Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland

Received: 22 March 2013
Accepted: 23 February 2014

Abstract

The new cosmic microwave background (CMB) temperature maps from Planck provide the highest-quality full-sky view of the surface of last scattering available to date. This allows us to detect possible departures from the standard model of a globally homogeneous and isotropic cosmology on the largest scales. We search for correlations induced by a possible non-trivial topology with a fundamental domain intersecting, or nearly intersecting, the last scattering surface (at comoving distance χ_rec), both via a direct search for matched circular patterns at the intersections and by an optimal likelihood search for specific topologies. For the latter we consider flat spaces with cubic toroidal (T3), equal-sided chimney (T2) and slab (T1) topologies, three multi-connected spaces of constant positive curvature (dodecahedral, truncated cube and octahedral) and two compact negative-curvature spaces. These searches yield no detection of the compact topology with the scale below the diameter of the last scattering surface. For most compact topologies studied the likelihood maximized over the orientation of the space relative to the observed map shows some preference for multi-connected models just larger than the diameter of the last scattering surface. Since this effect is also present in simulated realizations of isotropic maps, we interpret it as the inevitable alignment of mild anisotropic correlations with chance features in a single sky realization; such a feature can also be present, in milder form, when the likelihood is marginalized over orientations. Thus marginalized, the limits on the radius ℛ_i of the largest sphere inscribed in topological domain (at log-likelihood-ratio Δln ℒ > −5 relative to a simply-connected flat Planck best-fit model) are: in a flat Universe, ℛ_i> 0.92χ_rec for the T3 cubic torus; ℛ_i> 0.71χ_rec for the T2 chimney; ℛ_i> 0.50χ_rec for the T1 slab; and in a positively curved Universe, ℛ_i> 1.03χ_rec for the dodecahedral space; ℛ_i> 1.0χ_rec for the truncated cube; and ℛ_i> 0.89χ_rec for the octahedral space. The limit for a wider class of topologies, i.e., those predicting matching pairs of back-to-back circles, among them tori and the three spherical cases listed above, coming from the matched-circles search, is ℛ_i> 0.94χ_rec at 99% confidence level. Similar limits apply to a wide, although not exhaustive, range of topologies. 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 data favour the inclusion of a Bianchi component with a Bayes factor of at least 1.5 units of log-evidence. Indeed, the Bianchi pattern is quite efficient at accounting for some of the large-scale anomalies found in Planck data. However, the cosmological parameters that generate this pattern are in strong disagreement with those found from CMB anisotropy data alone. 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< 8.1 × 10^-10 (95% confidence level).