Euclid preparation

LXXXIV. The flat-sky approximation for the clustering of Euclid’s photometric galaxies

¹ Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
² Korea Astronomy and Space Science Institute, 776 Daedeok-daero Yuseong-gu, Daejeon, 34055, Republic of Korea
³ Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy
⁴ INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
⁵ INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese (TO), Italy
⁶ Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Barcelona, Spain
⁷ Institut d’Estudis Espacials de Catalunya (IEEC), Edifici RDIT, Campus UPC, 08860 Castelldefels, Barcelona, Spain
⁸ Institut de Recherche en Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France
⁹ ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo 28692 Villanueva de la Cañada Madrid, Spain
¹⁰ School of Mathematics and Physics, University of Surrey, Guildford, Surrey, GU2 7XH, UK
¹¹ INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy
¹² INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy
¹³ IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy
¹⁴ INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy
¹⁵ INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste TS, Italy
¹⁶ SISSA, International School for Advanced Studies, Via Bonomea 265, 34136 Trieste TS, Italy
¹⁷ Dipartimento di Fisica e Astronomia, Università di Bologna, Via Gobetti 93/2, 40129 Bologna, Italy
¹⁸ INFN-Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
¹⁹ Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146 Genova, Italy
²⁰ INFN-Sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy
²¹ Department of Physics “E. Pancini”, University Federico II, Via Cinthia 6, 80126 Napoli, Italy
²² INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy
²³ European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
²⁴ Institute Lorentz, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands
²⁵ Leiden Observatory, Leiden University, Einsteinweg 55, 2333 CC Leiden, The Netherlands
²⁶ INAF-IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy
²⁷ INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy
²⁸ INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c/o Dipartimento di Fisica Edificio G. Marconi, 00185 Roma, Italy
²⁹ Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain
³⁰ Port d’Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra (Barcelona), Spain
³¹ Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany
³² INFN section of Naples, Via Cinthia 6, 80126 Napoli, Italy
³³ Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive Honolulu, HI, 96822, USA
³⁴ Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
³⁵ Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain
³⁶ Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill Edinburgh, EH9 3HJ, UK
³⁷ Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester, Oxford Road Manchester, M13 9PL, UK
³⁸ European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy
³⁹ Université Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, Villeurbanne, F-69100, France
⁴⁰ Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, 08028 Barcelona, Spain
⁴¹ Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig de Lluís Companys 23, 08010 Barcelona, Spain
⁴² UCB Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne, France
⁴³ Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal
⁴⁴ Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal
⁴⁵ Department of Astronomy, University of Geneva, ch. d’Ecogia 16, 1290 Versoix, Switzerland
⁴⁶ Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405 Orsay, France
⁴⁷ INFN-Padova, Via Marzolo 8, 35131 Padova, Italy
⁴⁸ Aix-Marseille Université, CNRS/IN2P3, CPPM, Marseille, France
⁴⁹ INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del Cavaliere 100, 00100 Roma, Italy
⁵⁰ Space Science Data Center, Italian Space Agency, via del Politecnico snc, 00133 Roma, Italy
⁵¹ INFN-Bologna, Via Irnerio 46, 40126 Bologna, Italy
⁵² University Observatory, LMU Faculty of Physics, Scheinerstrasse 1, 81679 Munich, Germany
⁵³ Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany
⁵⁴ INAF-Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, 35122 Padova, Italy
⁵⁵ Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany
⁵⁶ Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, 0315 Oslo, Norway
⁵⁷ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive Pasadena, CA, 91109, USA
⁵⁸ Felix Hormuth Engineering, Goethestr. 17, 69181 Leimen, Germany
⁵⁹ Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
⁶⁰ Cosmic Dawn Center (DAWN), Denmark
⁶¹ Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
⁶² NASA Goddard Space Flight Center, Greenbelt, MD, 20771, USA
⁶³ Department of Physics and Astronomy, University College London, Gower Street London, WC1E 6BT, UK
⁶⁴ Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu 2 University of Helsinki, 00014 Helsinki, Finland
⁶⁵ Department of Physics, P.O. Box 64 University of Helsinki, 00014 Helsinki, Finland
⁶⁶ Helsinki Institute of Physics, Gustaf Hällströmin katu 2 University of Helsinki, 00014 Helsinki, Finland
⁶⁷ Laboratoire d’etude de l’Univers et des phenomenes eXtremes, Observatoire de Paris, Université PSL, Sorbonne Université, CNRS, 92190 Meudon, France
⁶⁸ SKAO, Jodrell Bank, Lower Withington, Macclesfield, SK11 9FT, UK
⁶⁹ Centre de Calcul de l’IN2P3/CNRS, 21 avenue Pierre de Coubertin, 69627 Villeurbanne Cedex, France
⁷⁰ Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy
⁷¹ INFN-Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
⁷² University of Applied Sciences and Arts of Northwestern Switzerland, School of Computer Science, 5210 Windisch, Switzerland
⁷³ Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany
⁷⁴ Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France
⁷⁵ Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93/2, 40129 Bologna, Italy
⁷⁶ Department of Physics, Institute for Computational Cosmology, Durham University, South Road Durham, DH1 3LE, UK
⁷⁷ Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris, France
⁷⁸ CNRS-UCB International Research Laboratory, Centre Pierre Binétruy, IRL2007, CPB-IN2P3 Berkeley, USA
⁷⁹ Institut d’Astrophysique de Paris, 98bis Boulevard Arago, 75014 Paris, France
⁸⁰ Institut d’Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France
⁸¹ Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland
⁸² Telespazio UK S.L. for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
⁸³ Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191 Gif-sur-Yvette, France
⁸⁴ Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain
⁸⁵ DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 155, 2200 Copenhagen, Denmark
⁸⁶ Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
⁸⁷ Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
⁸⁸ Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada
⁸⁹ Centre National d’Etudes Spatiales – Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France
⁹⁰ Institute of Space Science, Str. Atomistilor nr. 409 Măgurele Ilfov, 077125, Romania
⁹¹ Consejo Superior de Investigaciones Cientificas, Calle Serrano 117, 28006 Madrid, Spain
⁹² Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife, Spain
⁹³ Dipartimento di Fisica e Astronomia “G. Galilei”, Università di Padova, Via Marzolo 8, 35131 Padova, Italy
⁹⁴ Institut für Theoretische Physik, University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
⁹⁵ Université St Joseph; Faculty of Sciences, Beirut, Lebanon
⁹⁶ Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008 Santiago, Chile
⁹⁷ Universität Innsbruck, Institut für Astro- und Teilchenphysik, Technikerstr. 25/8, 6020 Innsbruck, Austria
⁹⁸ Satlantis, University Science Park, Sede Bld 48940 Leioa-Bilbao, Spain
⁹⁹ Department of Physics, Royal Holloway, University of London, Surrey, TW20 0EX, UK
¹⁰⁰ Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal
¹⁰¹ Cosmic Dawn Center (DAWN)
¹⁰² Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen, Denmark
¹⁰³ Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, Plaza del Hospital 1, 30202 Cartagena, Spain
¹⁰⁴ Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA, 91125, USA
¹⁰⁵ Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy
¹⁰⁶ Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy
¹⁰⁷ INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129 Bologna, Italy
¹⁰⁸ Astronomical Observatory of the Autonomous Region of the Aosta Valley (OAVdA), Loc. Lignan 39, I-11020 Nus (Aosta Valley), Italy
¹⁰⁹ Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France
¹¹⁰ Department of Physics, Oxford University, Keble Road Oxford, OX1 3RH, UK
¹¹¹ Aurora Technology for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
¹¹² Zentrum für Astronomie, Universität Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany
¹¹³ ICL, Junia, Université Catholique de Lille, LITL, 59000 Lille, France
¹¹⁴ ICSC – Centro Nazionale di Ricerca in High Performance Computing, Big Data e Quantum Computing, Via Magnanelli 2 Bologna, Italy
¹¹⁵ Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain
¹¹⁶ CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue Cleveland, OH 44106, USA
¹¹⁷ Technical University of Munich, TUM School of Natural Sciences, Physics Department, James-Franck-Str. 1, 85748 Garching, Germany
¹¹⁸ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany
¹¹⁹ Laboratoire Univers et Théorie, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS, 92190 Meudon, France
¹²⁰ Departamento de Física Fundamental. Universidad de Salamanca, Plaza de la Merced s/n., 37008 Salamanca, Spain
¹²¹ Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, 67000 Strasbourg, France
¹²² Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa Chiba, 277-8583, Japan
¹²³ Dipartimento di Fisica – Sezione di Astronomia, Università di Trieste, Via Tiepolo 11, 34131 Trieste, Italy
¹²⁴ California Institute of Technology, 1200 E California Blvd Pasadena, CA, 91125, USA
¹²⁵ Department of Physics & Astronomy, University of California Irvine, Irvine, CA, 92697, USA
¹²⁶ Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands
¹²⁷ Departamento Física Aplicada, Universidad Politécnica de Cartagena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain
¹²⁸ Instituto de Física de Cantabria, Edificio Juan Jordá, Avenida de los Castros, 39005 Santander, Spain
¹²⁹ INFN, Sezione di Lecce, Via per Arnesano CP-193, 73100 Lecce, Italy
¹³⁰ Department of Mathematics and Physics E. De Giorgi, University of Salento, Via per Arnesano CP-I93, 73100 Lecce, Italy
¹³¹ INAF-Sezione di Lecce, c/o Dipartimento Matematica e Fisica, Via per Arnesano, 73100 Lecce, Italy
¹³² CEA Saclay, DFR/IRFU, Service d’Astrophysique Bat. 709, 91191 Gif-sur-Yvette, France
¹³³ Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK
¹³⁴ Department of Computer Science, Aalto University, PO Box 15400 Espoo, FI-00 076, Finland
¹³⁵ Instituto de Astrofísica de Canarias, E-38205 La Laguna; Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife, Spain
¹³⁶ Caltech/IPAC, 1200 E. California Blvd. Pasadena, CA, 91125, USA
¹³⁷ Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German Centre for Cosmological Lensing (GCCL), 44780 Bochum, Germany
¹³⁸ Department of Physics and Astronomy, Vesilinnantie 5 University of Turku, 20014 Turku, Finland
¹³⁹ Serco for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca del Castillo Villanueva de la Cañada, 28692 Madrid, Spain
¹⁴⁰ ARC Centre of Excellence for Dark Matter Particle Physics, Melbourne, Australia
¹⁴¹ Centre for Astrophysics & Supercomputing, Swinburne University of Technology, Hawthorn Victoria, 3122, Australia
¹⁴² Department of Physics and Astronomy, University of the Western Cape, Bellville, Cape Town, 7535, South Africa
¹⁴³ Université Libre de Bruxelles (ULB), Service de Physique Théorique CP225, Boulevard du Triophe, 1050 Bruxelles, Belgium
¹⁴⁴ DAMTP, Centre for Mathematical Sciences, Wilberforce Road Cambridge, CB3 0WA, UK
¹⁴⁵ Kavli Institute for Cosmology Cambridge, Madingley Road Cambridge, CB3 0HA, UK
¹⁴⁶ Department of Astrophysics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
¹⁴⁷ Department of Physics, Centre for Extragalactic Astronomy, Durham University, South Road Durham, DH1 3LE, UK
¹⁴⁸ IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette Cedex, France
¹⁴⁹ Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53 Avenue des Martyrs, 38000 Grenoble, France
¹⁵⁰ Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy
¹⁵¹ Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal
¹⁵² Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal
¹⁵³ HE Space 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 d’Abruzzo, Via Maggini, 64100 Teramo, Italy
¹⁵⁵ Theoretical astrophysics, Department of Physics and Astronomy, Uppsala University, Box 516, 751 37 Uppsala, Sweden
¹⁵⁶ Mathematical Institute, University of Leiden, Einsteinweg 55, 2333 CA Leiden, The Netherlands
¹⁵⁷ Institute of Astronomy, University of Cambridge, Madingley Road Cambridge, CB3 0HA, UK
¹⁵⁸ Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA
¹⁵⁹ Space physics and astronomy research unit, University of Oulu, Pentti Kaiteran katu 1, FI-90014 Oulu, Finland
¹⁶⁰ Institut de Physique Théorique, CEA, CNRS, Université Paris-Saclay, 91191 Gif-sur-Yvette Cedex, France
¹⁶¹ Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 10010 New York, NY, USA

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Received: 20 October 2025
Accepted: 22 November 2025

Abstract

We compared the performance of the flat-sky approximation and Limber approximation for the clustering analysis of the photometric galaxy catalogue of Euclid. We studied a 6-bin configuration, representing the first data release (DR1), and a 13-bin configuration, representing the third and final data release (DR3). We find that the Limber approximation is sufficiently accurate for the analysis of the wide bins of DR1. Instead, the 13 bins of DR3 cannot be modelled accurately with the Limber approximation. Instead, the flat-sky approximation is accurate to below 5% in recovering the angular power spectra of galaxy number counts in both cases and can be used to simplify the computation of the full power spectrum in harmonic space for the data analysis of DR3.