Euclid preparation

LXII. Simulations and non-linearities beyond Lambda cold dark matter. 1. Numerical methods and validation

¹ Department of Astrophysics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
² Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK
³ Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti 93/2, 40129 Bologna, Italy
⁴ INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy
⁵ INFN-Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
⁶ Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Muhlenberg 1, 14476 Potsdam-Golm, Germany
⁷ Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain
⁸ Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB, Carrer de Can Magrans, s/n Cerdanyola del Vallés, 08193 Barcelona, Spain
⁹ Laboratoire Univers et Théorie, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS, 92190 Meudon, France
¹⁰ Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway
¹¹ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
¹² Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany
¹³ Department of Physics, Institute for Computational Cosmology, Durham University, South Road, DH1 3LE, UK
¹⁴ Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK
¹⁵ INAF-IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy
¹⁶ Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy
¹⁷ Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
¹⁸ Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, UK
¹⁹ Institut für Theoretische Physik, University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
²⁰ Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France
²¹ Université St Joseph; Faculty of Sciences, Beirut, Lebanon
²² Institut de Physique Théorique, CEA, CNRS, Université Paris-Saclay 91191 Gif-sur-Yvette Cedex, France
²³ School of Mathematics and Physics, University of Surrey, Guildford, Surrey, GU2 7XH, UK
²⁴ INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy
²⁵ IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy
²⁶ INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy
²⁷ INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste TS, Italy
²⁸ SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste TS, Italy
²⁹ INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese (TO), Italy
³⁰ Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146, Genova, Italy
³¹ INFN-Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy
³² Department of Physics “E. Pancini”, University Federico II, Via Cinthia 6, 80126, Napoli, Italy
³³ INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy
³⁴ INFN section of Naples, Via Cinthia 6, 80126 Napoli, Italy
³⁵ Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal
³⁶ Faculdade de Ciências da Universidade do Porto, Rua do Campo de Alegre, 4150-007 Porto, Portugal
³⁷ Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France
³⁸ Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy
³⁹ INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
⁴⁰ INAF – Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy
⁴¹ INFN-Sezione di Roma, Piazzale Aldo Moro 2, c/o Dipartimento di Fisica, Edificio G. Marconi, 00185 Roma, Italy
⁴² Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain
⁴³ Port d’Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra (Barcelona), Spain
⁴⁴ Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany
⁴⁵ Institut d’Estudis Espacials de Catalunya (IEEC), Edifici RDIT, Campus UPC, 08860 Castelldefels, Barcelona, Spain
⁴⁶ Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
⁴⁷ Instituto de Astrofísica de Canarias, Calle Vía Láctea s/n, 38204, San Cristóbal de La Laguna, Tenerife, Spain
⁴⁸ European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy
⁴⁹ ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain
⁵⁰ Université Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, Villeurbanne 69100, France
⁵¹ Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland
⁵² Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, 08028 Barcelona, Spain
⁵³ Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig de Lluís Companys 23, 08010 Barcelona, Spain
⁵⁴ UCB Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne, France
⁵⁵ Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, 1749-016 Lisboa, Portugal
⁵⁶ Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal
⁵⁷ Department of Astronomy, University of Geneva, ch. d’Ecogia 16, 1290 Versoix, Switzerland
⁵⁸ Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405 Orsay, France
⁵⁹ INFN – Padova, Via Marzolo 8, 35131 Padova, Italy
⁶⁰ INAF – Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere, 100, 00100 Roma, Italy
⁶¹ Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France
⁶² FRACTAL S.L.N.E., calle Tulipán 2, Portal 13 1A, 28231, Las Rozas de Madrid, Spain
⁶³ INAF – Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, 35122 Padova, Italy
⁶⁴ Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany
⁶⁵ Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany
⁶⁶ Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy
⁶⁷ Felix Hormuth Engineering, Goethestr. 17, 69181 Leimen, Germany
⁶⁸ Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
⁶⁹ Cosmic Dawn Center (DAWN), Denmark
⁷⁰ Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
⁷¹ Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
⁷² NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
⁷³ Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
⁷⁴ Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu 2, 00014 University of Helsinki, Finland
⁷⁵ Aix-Marseille Université, CNRS/IN2P3, CPPM, Marseille, France
⁷⁶ Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland
⁷⁷ Department of Physics, PO Box 64, 00014 University of Helsinki, Finland
⁷⁸ Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland
⁷⁹ NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk 4, 7991PD Dwingeloo, The Netherlands
⁸⁰ Centre de Calcul de l’IN2P3/CNRS, 21 avenue Pierre de Coubertin 69627 Villeurbanne Cedex, France
⁸¹ Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany
⁸² Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93/2, 40129 Bologna, Italy
⁸³ Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France
⁸⁴ University of Applied Sciences and Arts of Northwestern Switzerland, School of Engineering, 5210 Windisch, Switzerland
⁸⁵ Institut d’Astrophysique de Paris, 98bis Boulevard Arago, 75014 Paris, France
⁸⁶ Institut d’Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France
⁸⁷ Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain
⁸⁸ European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
⁸⁹ DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 155, 2200 Copenhagen, Denmark
⁹⁰ Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
⁹¹ Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
⁹² Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
⁹³ Space Science Data Center, Italian Space Agency, via del Politecnico snc, 00133 Roma, Italy
⁹⁴ Centre National d’Etudes Spatiales – Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France
⁹⁵ Institute of Space Science, Str. Atomistilor, nr. 409 Măgurele, Ilfov, 077125, Romania
⁹⁶ Dipartimento di Fisica e Astronomia “G. Galilei”, Università di Padova, Via Marzolo 8, 35131 Padova, Italy
⁹⁷ Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile
⁹⁸ Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr. 25/8, 6020 Innsbruck, Austria
⁹⁹ Satlantis, University Science Park, Sede Bld 48940, Leioa-Bilbao, Spain
¹⁰⁰ Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal
¹⁰¹ Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, Plaza del Hospital 1, 30202 Cartagena, Spain
¹⁰² Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands
¹⁰³ INFN-Bologna, Via Irnerio 46, 40126 Bologna, Italy
¹⁰⁴ Dipartimento di Fisica, Università degli studi di Genova, and INFN-Sezione di Genova, via Dodecaneso 33, 16146 Genova, Italy
¹⁰⁵ Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA
¹⁰⁶ INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129 Bologna, Italy
¹⁰⁷ Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, 11020, Nus (Aosta Valley), Italy
¹⁰⁸ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
¹⁰⁹ School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, UK
¹¹⁰ Junia, EPA department, 41 Bd Vauban, 59800 Lille, France
¹¹¹ ICSC – Centro Nazionale di Ricerca in High Performance Computing, Big Data e Quantum Computing, Via Magnanelli 2, Bologna, Italy
¹¹² Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain
¹¹³ CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA
¹¹⁴ INFN-Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
¹¹⁵ Departamento de Física Fundamental. Universidad de Salamanca. Plaza de la Merced s/n, 37008 Salamanca, Spain
¹¹⁶ Departamento de Astrofísica, Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain
¹¹⁷ Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy
¹¹⁸ Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy
¹¹⁹ Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
¹²⁰ Ludwig-Maximilians-University, Schellingstrasse 4, 80799 Munich, Germany
¹²¹ Max-Planck-Institut für Physik, Boltzmannstr. 8, 85748 Garching, Germany
¹²² Dipartimento di Fisica – Sezione di Astronomia, Università di Trieste, Via Tiepolo 11, 34131 Trieste, Italy
¹²³ Minnesota Institute for Astrophysics, University of Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA
¹²⁴ Institute Lorentz, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands
¹²⁵ Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France
¹²⁶ Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
¹²⁷ Department of Physics & Astronomy, University of California Irvine, Irvine CA 92697, USA
¹²⁸ Department of Astronomy & Physics and Institute for Computational Astrophysics, Saint Mary’s University, 923 Robie Street, Halifax, Nova Scotia B3H 3C3, Canada
¹²⁹ Departamento Física Aplicada, Universidad Politécnica de Cartagena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain
¹³⁰ Instituto de Astrofísica de Canarias (IAC); Departamento de Astrofísica, Universidad de La Laguna (ULL), 38200, La Laguna, Tenerife, Spain
¹³¹ Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK
¹³² CEA Saclay, DFR/IRFU, Service d’Astrophysique, Bat. 709, 91191 Gif-sur-Yvette, France
¹³³ Department of Computer Science, Aalto University, PO Box 15400, Espoo 00 076, Finland
¹³⁴ Instituto de Astrofísica de Canarias, c/ Via Lactea s/n, La Laguna E-38200, Spain. Departamento de Astrofísica de la Universidad de La Laguna, Avda. Francisco Sanchez, La Laguna 38200, Spain
¹³⁵ Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing (GCCL), 44780 Bochum, Germany
¹³⁶ Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53, Avenue des Martyrs, 38000 Grenoble, France
¹³⁷ Department of Physics and Astronomy, Vesilinnantie 5, 20014 University of Turku, Finland
¹³⁸ Serco for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
¹³⁹ ARC Centre of Excellence for Dark Matter Particle Physics, Melbourne, Australia
¹⁴⁰ Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia
¹⁴¹ School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK
¹⁴² Department of Physics and Astronomy, University of the Western Cape, Bellville, Cape Town 7535, South Africa
¹⁴³ ICTP South American Institute for Fundamental Research, Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil
¹⁴⁴ IRFU, CEA, Université Paris-Saclay 91191 Gif-sur-Yvette Cedex, France
¹⁴⁵ Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, Stockholm 106 91, Sweden
¹⁴⁶ Astrophysics Group, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK
¹⁴⁷ INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy
¹⁴⁸ Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy
¹⁴⁹ Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal
¹⁵⁰ Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma, Italy
¹⁵¹ INFN, Sezione di Roma 2, Via della Ricerca Scientifica 1, Roma, Italy
¹⁵² HE Space for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
¹⁵³ Aurora Technology for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
¹⁵⁴ INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy, and INFN-Sezione di Genova, Via Dodecaneso 33, 16146, Genova, Italy
¹⁵⁵ Theoretical astrophysics, Department of Physics and Astronomy, Uppsala University, Box 515, 751 20 Uppsala, Sweden
¹⁵⁶ Department of Physics, Royal Holloway, University of London TW20 0EX, UK
¹⁵⁷ Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK
¹⁵⁸ Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA
¹⁵⁹ Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark
¹⁶⁰ Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY 10003, USA
¹⁶¹ Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA

^★ Corresponding author; julian.adamek@uzh.ch

Received: 9 September 2024
Accepted: 5 December 2024

Abstract

To constrain cosmological models beyond ACDM, the development of the Euclid analysis pipeline requires simulations that capture the non-linear phenomenology of such models. We present an overview of numerical methods and N-body simulation codes developed to study the non-linear regime of structure formation in alternative dark energy and modified gravity theories. We review a variety of numerical techniques and approximations employed in cosmological N-body simulations to model the complex phenomenology of scenarios beyond ACDM. This includes discussions on solving non-linear field equations, accounting for fifth forces, and implementing screening mechanisms. Furthermore, we conduct a code comparison exercise to assess the reliability and convergence of different simulation codes across a range of models. Our analysis demonstrates a high degree of agreement among the outputs of different simulation codes, typically within 2% for the predicted modification of the matter power spectrum and within 4% for the predicted modification of the halo mass function, although some approximations degrade accuracy a bit further. This provides confidence in current numerical methods of modelling cosmic structure formation beyond ACDM. We highlight recent advances made in simulating the non-linear scales of structure formation, which are essential for leveraging the full scientific potential of the forthcoming observational data from the Euclid mission.