Planck 2015 results

XIV. Dark energy and modified gravity

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 Aix Marseille Université, Centre de Physique Théorique, 163 Avenue de Luminy, 13288 Marseille, France
⁷ Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, UK
⁸ Astrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, 4000 Durban, South Africa
⁹ CGEE, SCS Qd 9, Lote C, Torre C, 4° andar, Ed. Parque Cidade Corporate, CEP 70308-200 Brasília, DF, Brazil
¹⁰ CITA, University of Toronto, 60 St.George St., Toronto, ON M5S 3H8, Canada
¹¹ CNRS, IRAP, 9 Av. colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France
¹² CRANN, Trinity College, Dublin, Ireland
¹³ California Institute of Technology, Pasadena, CA 91195 California, USA
¹⁴ Centre for Theoretical Cosmology, DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
¹⁵ Centro de Estudios de Física del Cosmos de Aragón (CEFCA), Plaza San Juan, 1, planta 2, 44001 Teruel, Spain
¹⁶ Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, CA 91195 California, USA
¹⁷ Consejo Superior de Investigaciones Científicas (CSIC), Madrid, Spain
¹⁸ DSM/Irfu/SPP, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
¹⁹ DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
²⁰ Département de Physique Théorique, Université de Genève, 24 Quai E. Ansermet, 1211 Genève 4, Switzerland
²¹ Departamento de Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain
²² Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo s/n, 33003 Oviedo, Spain
²³ Department of Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, Ontario, Canada
²⁴ Department of Astrophysics/IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
²⁵ Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada
²⁶ Department of Physics and Astronomy, Dana and David Dornsife College of Letter, Arts and Sciences, University of Southern California, Los Angeles, CA 90089, USA
²⁷ Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
²⁸ Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK
²⁹ Department of Physics, Florida State University, Keen Physics Building, 77 Chieftan Way, Tallahassee, Florida, USA
³⁰ Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, 00100 Helsinki, Finland
³¹ Department of Physics, Princeton University, NJ 08544 Princeton, New Jersey, USA
³² Department of Physics, University of California, Berkeley, CA 91195 California, USA
³³ Department of Physics, University of California, Santa Barbara, CA 91195 California, USA
³⁴ Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois, USA
³⁵ Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi di Padova, via Marzolo 8, 35131 Padova, 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, Roma, Italy
³⁸ Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, Milano, Italy
³⁹ Dipartimento di Fisica, Università degli Studi di Trieste, via A. Valerio 2, Trieste, Italy
⁴⁰ Dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, Roma, Italy
⁴¹ Discovery Center, Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark
⁴² Discovery Center, Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, Copenhagen, Denmark
⁴³ European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo, s/n, Urbanización Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
⁴⁴ European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
⁴⁵ Gran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’ Aquila, Italy
⁴⁶ HGSFP and University of Heidelberg, Theoretical Physics Department, Philosophenweg 16, 69120 Heidelberg, Germany
⁴⁷ Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, 00100 Helsinki, Finland
⁴⁸ INAF–Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, Italy
⁴⁹ INAF–Osservatorio Astronomico di Roma, via di Frascati 33, Monte Porzio Catone, Italy
⁵⁰ INAF–Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, Trieste, Italy
⁵¹ INAF/IASF Bologna, via Gobetti 101, Bologna, Italy
⁵² INAF/IASF Milano, via E. Bassini 15, Milano, Italy
⁵³ INFN, Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
⁵⁴ INFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185 Roma, Italy
⁵⁵ INFN, Sezione di Roma 2, Università di Roma Tor Vergata, via della Ricerca Scientifica 1, 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, Grenoble, France
⁶² Institut Universitaire de France, 103 Bd Saint-Michel, 75005 Paris, France
⁶³ Institut d’Astrophysique Spatiale, CNRS, Univ. Paris-Sud, Université Paris-Saclay, Bât. 121, 91405 Orsay Cedex, France
⁶⁴ Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bis Boulevard Arago, 75014 Paris, France
⁶⁵ Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, 52056 Aachen, Germany
⁶⁶ Institute for Space Sciences, 077125 Bucharest-Magurale, Romania
⁶⁷ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
⁶⁸ Institute of Theoretical Astrophysics, University of Oslo, Blindern, 0316 Oslo, Norway
⁶⁹ Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, La Laguna, Tenerife, Spain
⁷⁰ Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, Santander, Spain
⁷¹ Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, 35131 Padova, Italy
⁷² Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, USA
⁷³ Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
⁷⁴ Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
⁷⁵ Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
⁷⁶ Kazan Federal University, 18 Kremlyovskaya St., 420008 Kazan, Russia
⁷⁷ LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
⁷⁸ LERMA, CNRS, Observatoire de Paris, 61 Avenue de l’Observatoire, 75000 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 Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, 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, CA 94720 California, USA
⁸⁴ Lebedev Physical Institute of the Russian Academy of Sciences, Astro Space Centre, 84/32 Profsoyuznaya st., Moscow, GSP-7, 117997, Russia
⁸⁵ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany
⁸⁶ McGill Physics, Ernest Rutherford Physics Building, McGill University, 3600 rue University, Montréal, QC, H3A 2T8, Canada
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, 2100 Copenhagen, Denmark
⁹¹ Nordita (Nordic Institute for Theoretical Physics), Roslagstullsbacken 23, 106 91 Stockholm, Sweden
⁹² Optical Science Laboratory, University College London, Gower Street, WC1E 6 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 Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, 117997 Moscow, Russia
⁹⁹ Space Sciences Laboratory, University of California, Berkeley, California, USA
¹⁰⁰ Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, 369167 Karachai-Cherkessian Republic, Russia
¹⁰¹ Stanford University, Dept of Physics, Varian Physics Bldg, 382 via Pueblo Mall, Stanford, California, USA
¹⁰² Sub-Department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
¹⁰³ The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, 106 91 Stockholm, Sweden
¹⁰⁴ Theory Division, PH-TH, CERN, 1211, Geneva 23, Switzerland
¹⁰⁵ UPMC Univ. Paris 06, UMR7095, 98 bis Boulevard Arago, 75014 Paris, France
¹⁰⁶ Universität Heidelberg, Institut für Theoretische Astrophysik, Philosophenweg 12, 69120 Heidelberg, Germany
¹⁰⁷ Université de Toulouse, UPS-OMP, IRAP, 31028 Toulouse Cedex 4, France
¹⁰⁸ Universities Space Research Association, Stratospheric Observatory for Infrared Astronomy, MS 232-11, Moffett Field, CA 94035, USA
¹⁰⁹ University of Granada, Departamento de Física Teórica y del Cosmos, Facultad de Ciencias, 18010 Granada, Spain
¹¹⁰ University of Granada, Instituto Carlos I de Física Teórica y Computacional, 18010 Granada, Spain
¹¹¹ University of Heidelberg, Institute for Theoretical Physics, Philosophenweg 16, 69120 Heidelberg, Germany
¹¹² Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland

^⋆ Corresponding author: Valeria Pettorino, e-mail: v.pettorino@thphys.uni-heidelberg.de

Received: 5 February 2015
Accepted: 19 February 2016

Abstract

We study the implications of Planck data for models of dark energy (DE) and modified gravity (MG) beyond the standard cosmological constant scenario. We start with cases where the DE only directly affects the background evolution, considering Taylor expansions of the equation of state w(a), as well as principal component analysis and parameterizations related to the potential of a minimally coupled DE scalar field. When estimating the density of DE at early times, we significantly improve present constraints and find that it has to be below ~2% (at 95% confidence) of the critical density, even when forced to play a role for z < 50 only. We then move to general parameterizations of the DE or MG perturbations that encompass both effective field theories and the phenomenology of gravitational potentials in MG models. Lastly, we test a range of specific models, such as k-essence, f(R) theories, and coupled DE. In addition to the latest Planck data, for our main analyses, we use background constraints from baryonic acoustic oscillations, type-Ia supernovae, and local measurements of the Hubble constant. We further show the impact of measurements of the cosmological perturbations, such as redshift-space distortions and weak gravitational lensing. These additional probes are important tools for testing MG models and for breaking degeneracies that are still present in the combination of Planck and background data sets. All results that include only background parameterizations (expansion of the equation of state, early DE, general potentials in minimally-coupled scalar fields or principal component analysis) are in agreement with ΛCDM. When testing models that also change perturbations (even when the background is fixed to ΛCDM), some tensions appear in a few scenarios: the maximum one found is ~2σ for Planck TT+lowP when parameterizing observables related to the gravitational potentials with a chosen time dependence; the tension increases to, at most, 3σ when external data sets are included. It however disappears when including CMB lensing.