Euclid: Fast two-point correlation function covariance through linear construction^⋆

¹ Department of Physics, PO Box 64 00014 University of Helsinki, Finland
e-mail: elina.keihanen@helsinki.fi
² Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu 2, 00014 University of Helsinki, Finland
³ Dipartimento di Fisica – Sezione di Astronomia, Universitá di Trieste, Via Tiepolo 11, 34131 Trieste, Italy
⁴ IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy
⁵ INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy
⁶ INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste TS, Italy
⁷ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild Str. 1, 85741 Garching, Germany
⁸ INAF-IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy
⁹ Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany
¹⁰ Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland
¹¹ Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK
¹² INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy
¹³ Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93/2, 40129 Bologna, Italy
¹⁴ INFN-Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
¹⁵ INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese (TO), Italy
¹⁶ Dipartimento di Fisica, Universitá degli studi di Genova, and INFN-Sezione di Genova, via Dodecaneso 33, 16146 Genova, Italy
¹⁷ INFN-Sezione di Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
¹⁸ INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli, Italy
¹⁹ Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal
²⁰ Dipartimento di Fisica, Universitá degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy
²¹ INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
²² Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain
²³ Port d’Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra (Barcelona), Spain
²⁴ INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy
²⁵ INFN section of Naples, Via Cinthia 6, 80126 Napoli, Italy
²⁶ Department of Physics “E. Pancini”, University Federico II, Via Cinthia 6, 80126 Napoli, Italy
²⁷ Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma Mater Studiorum Universitá di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
²⁸ INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy
²⁹ Centre National d’Etudes Spatiales, Toulouse, France
³⁰ Institut national de physique nucléaire et de physique des particules, 3 rue Michel-Ange, 75794 Paris Cedex 16, France
³¹ Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
³² European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma, Italy
³³ ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid, Spain
³⁴ Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, 69622 Villeurbanne, France
³⁵ Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK
³⁶ Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande, 1749-016 Lisboa, Portugal
³⁷ Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal
³⁸ Department of Astronomy, University of Geneva, ch. dÉcogia 16, 1290 Versoix, Switzerland
³⁹ Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405 Orsay, France
⁴⁰ Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK
⁴¹ INFN-Padova, Via Marzolo 8, 35131 Padova, Italy
⁴² AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, 91191 Gif-sur-Yvette, France
⁴³ Istituto Nazionale di Astrofisica (INAF) – Osservatorio di Astrofisica e Scienza dello Spazio (OAS), Via Gobetti 93/3, 40127 Bologna, Italy
⁴⁴ Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy
⁴⁵ INAF-Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, 35122 Padova, Italy
⁴⁶ Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany
⁴⁷ Dipartimento di Fisica “Aldo Pontremoli”, Universitá degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy
⁴⁸ INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy
⁴⁹ INFN-Sezione di Milano, Via Celoria 16, 20133 Milano, Italy
⁵⁰ Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern 0315 Oslo, Norway
⁵¹ Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands
⁵² Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
⁵³ von Hoerner & Sulger GmbH, SchloßPlatz 8, 68723 Schwetzingen, Germany
⁵⁴ Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
⁵⁵ Aix-Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
⁵⁶ Université de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, 1211 Genève 4, Switzerland
⁵⁷ NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk 4, 7991PD Dwingeloo, The Netherlands
⁵⁸ Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
⁵⁹ Department of Physics, Institute for Computational Cosmology, Durham University, South Road, DH1 3LE Durham, UK
⁶⁰ University of Applied Sciences and Arts of Northwestern Switzerland, School of Engineering, 5210 Windisch, Switzerland
⁶¹ INFN-Bologna, Via Irnerio 46, 40126 Bologna, Italy
⁶² Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland
⁶³ European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
⁶⁴ Department of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, 8000 Aarhus C, Denmark
⁶⁵ Université Paris-Saclay, Université Paris Cité, CEA, CNRS, Astrophysique, Instrumentation et Modélisation Paris-Saclay, 91191 Gif-sur-Yvette, France
⁶⁶ Space Science Data Center, Italian Space Agency, via del Politecnico snc, 00133 Roma, Italy
⁶⁷ Institute of Space Science, Bucharest 077125, Romania
⁶⁸ Dipartimento di Fisica e Astronomia “G.Galilei”, Universitá di Padova, Via Marzolo 8, 35131 Padova, Italy
⁶⁹ Aix-Marseille Univ, CNRS, CNES, LAM, Marseille, France
⁷⁰ Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain
⁷¹ Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal
⁷² Universidad Politécnica de Cartagena, Departamento de Electrónica y Tecnología de Computadoras, 30202 Cartagena, Spain
⁷³ Kapteyn Astronomical Institute, University of Groningen, PO Box 800 9700 AV Groningen, The Netherlands
⁷⁴ Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA

Received: 20 May 2022
Accepted: 27 June 2022

Abstract

We present a method for fast evaluation of the covariance matrix for a two-point galaxy correlation function (2PCF) measured with the Landy–Szalay estimator. The standard way of evaluating the covariance matrix consists in running the estimator on a large number of mock catalogs, and evaluating their sample covariance. With large random catalog sizes (random-to-data objects’ ratio M ≫ 1) the computational cost of the standard method is dominated by that of counting the data-random and random-random pairs, while the uncertainty of the estimate is dominated by that of data-data pairs. We present a method called Linear Construction (LC), where the covariance is estimated for small random catalogs with a size of M = 1 and M = 2, and the covariance for arbitrary M is constructed as a linear combination of the two. We show that the LC covariance estimate is unbiased. We validated the method with PINOCCHIO simulations in the range r = 20 − 200 h⁻¹ Mpc. With M = 50 and with 2 h⁻¹ Mpc bins, the theoretical speedup of the method is a factor of 14. We discuss the impact on the precision matrix and parameter estimation, and present a formula for the covariance of covariance.