11institutetext: Dipartimento di Fisica e Astronomia ”Augusto Righi” - Alma
Mater Studiorum Università di Bologna, via Piero Gobetti 93/2, 40129 Bologna,
Italy 22institutetext: INAF-Osservatorio di Astrofisica e Scienza dello Spazio
di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy 33institutetext:
INFN-Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
44institutetext: INAF-Osservatorio Astronomico di Padova, Via
dell’Osservatorio 5, 35122 Padova, Italy 55institutetext: Dipartimento di
Fisica e Astronomia ”G. Galilei”, Università di Padova, Via Marzolo 8, 35131
Padova, Italy 66institutetext: Université Paris-Saclay, Université Paris Cité,
CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France 77institutetext: Department of
Physics ”E. Pancini”, University Federico II, Via Cinthia 6, 80126, Napoli,
Italy 88institutetext: INAF-Osservatorio Astronomico di Capodimonte, Via
Moiariello 16, 80131 Napoli, Italy 99institutetext: INFN section of Naples,
Via Cinthia 6, 80126, Napoli, Italy 1010institutetext: Istituto Nazionale di
Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy
1111institutetext: Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex
4, France 1212institutetext: INAF-Osservatorio Astronomico di Trieste, Via G.
B. Tiepolo 11, 34143 Trieste, Italy 1313institutetext: IFPU, Institute for
Fundamental Physics of the Universe, via Beirut 2, 34151 Trieste, Italy
1414institutetext: Université Paris-Saclay, CNRS, Institut d’astrophysique
spatiale, 91405, Orsay, France 1515institutetext: Institute of Cosmology and
Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK
1616institutetext: INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122
Milano, Italy 1717institutetext: Dipartimento di Fisica e Astronomia,
Università di Bologna, Via Gobetti 93/2, 40129 Bologna, Italy
1818institutetext: Max Planck Institute for Extraterrestrial Physics,
Giessenbachstr. 1, 85748 Garching, Germany 1919institutetext: Universitäts-
Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität
München, Scheinerstrasse 1, 81679 München, Germany 2020institutetext: INAF-
Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino Torinese
(TO), Italy 2121institutetext: Dipartimento di Fisica, Università di Genova,
Via Dodecaneso 33, 16146, Genova, Italy 2222institutetext: INFN-Sezione di
Genova, Via Dodecaneso 33, 16146, Genova, Italy 2323institutetext: Instituto
de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das
Estrelas, PT4150-762 Porto, Portugal 2424institutetext: Dipartimento di
Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino, Italy
2525institutetext: INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino,
Italy 2626institutetext: INAF-IASF Milano, Via Alfonso Corti 12, 20133 Milano,
Italy 2727institutetext: Institut de Física d’Altes Energies (IFAE), The
Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra
(Barcelona), Spain 2828institutetext: Port d’Informació Científica, Campus
UAB, C. Albareda s/n, 08193 Bellaterra (Barcelona), Spain 2929institutetext:
Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen
University, 52056 Aachen, Germany 3030institutetext: Institute of Space
Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona,
Spain 3131institutetext: Institut d’Estudis Espacials de Catalunya (IEEC),
Carrer Gran Capitá 2-4, 08034 Barcelona, Spain 3232institutetext: INAF-
Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone,
Italy 3333institutetext: Dipartimento di Fisica e Astronomia ”Augusto Righi” -
Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 6/2, 40127
Bologna, Italy 3434institutetext: Institute for Astronomy, University of
Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
3535institutetext: Jodrell Bank Centre for Astrophysics, Department of Physics
and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK
3636institutetext: European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044
Frascati, Roma, Italy 3737institutetext: ESAC/ESA, Camino Bajo del Castillo,
s/n., Urb. Villafranca del Castillo, 28692 Villanueva de la Cañada, Madrid,
Spain 3838institutetext: University of Lyon, Univ Claude Bernard Lyon 1,
CNRS/IN2P3, IP2I Lyon, UMR 5822, 69622 Villeurbanne, France 3939institutetext:
Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique Fédérale
de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland
4040institutetext: UCB Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, 4 rue Enrico Fermi,
69622 Villeurbanne, France 4141institutetext: Departamento de Física,
Faculdade de Ciências, Universidade de Lisboa, Edifício C8, Campo Grande,
PT1749-016 Lisboa, Portugal 4242institutetext: Instituto de Astrofísica e
Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo
Grande, 1749-016 Lisboa, Portugal 4343institutetext: Department of Astronomy,
University of Geneva, ch. d’Ecogia 16, 1290 Versoix, Switzerland
4444institutetext: INAF-Istituto di Astrofisica e Planetologia Spaziali, via
del Fosso del Cavaliere, 100, 00100 Roma, Italy 4545institutetext: Department
of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK
4646institutetext: INFN-Padova, Via Marzolo 8, 35131 Padova, Italy
4747institutetext: Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB,
Carrer de Can Magrans, s/n Cerdanyola del Vallés, 08193 Barcelona, Spain
4848institutetext: School of Physics, HH Wills Physics Laboratory, University
of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK 4949institutetext: University
Observatory, Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr.
1, 81679 Munich, Germany 5050institutetext: Institute of Theoretical
Astrophysics, University of Oslo, P.O. Box 1029 Blindern, 0315 Oslo, Norway
5151institutetext: Department of Physics, Lancaster University, Lancaster, LA1
4YB, UK 5252institutetext: von Hoerner & Sulger GmbH, SchloßPlatz 8, 68723
Schwetzingen, Germany 5353institutetext: Technical University of Denmark,
Elektrovej 327, 2800 Kgs. Lyngby, Denmark 5454institutetext: Cosmic Dawn
Center (DAWN), Denmark 5555institutetext: Institut d’Astrophysique de Paris,
UMR 7095, CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris,
France 5656institutetext: Max-Planck-Institut für Astronomie, Königstuhl 17,
69117 Heidelberg, Germany 5757institutetext: Aix-Marseille Université,
CNRS/IN2P3, CPPM, Marseille, France 5858institutetext: Jet Propulsion
Laboratory, California Institute of Technology, 4800 Oak Grove Drive,
Pasadena, CA, 91109, USA 5959institutetext: AIM, CEA, CNRS, Université Paris-
Saclay, Université de Paris, 91191 Gif-sur-Yvette, France 6060institutetext:
Université de Genève, Département de Physique Théorique and Centre for
Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
6161institutetext: Department of Physics, P.O. Box 64, 00014 University of
Helsinki, Finland 6262institutetext: Helsinki Institute of Physics, Gustaf
Hällströmin katu 2, University of Helsinki, Helsinki, Finland
6363institutetext: NOVA optical infrared instrumentation group at ASTRON, Oude
Hoogeveensedijk 4, 7991PD, Dwingeloo, The Netherlands 6464institutetext:
Universität Bonn, Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121
Bonn, Germany 6565institutetext: Aix-Marseille Université, CNRS, CNES, LAM,
Marseille, France 6666institutetext: Department of Physics, Institute for
Computational Cosmology, Durham University, South Road, DH1 3LE, UK
6767institutetext: University of Applied Sciences and Arts of Northwestern
Switzerland, School of Engineering, 5210 Windisch, Switzerland
6868institutetext: Institut d’Astrophysique de Paris, 98bis Boulevard Arago,
75014, Paris, France 6969institutetext: European Space Agency/ESTEC,
Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands 7070institutetext: Department
of Physics and Astronomy, University of Aarhus, Ny Munkegade 120, DK-8000
Aarhus C, Denmark 7171institutetext: Université Paris-Saclay, Université Paris
Cité, CEA, CNRS, Astrophysique, Instrumentation et Modélisation Paris-Saclay,
91191 Gif-sur-Yvette, France 7272institutetext: Space Science Data Center,
Italian Space Agency, via del Politecnico snc, 00133 Roma, Italy
7373institutetext: Centre National d’Etudes Spatiales – Centre spatial de
Toulouse, 18 avenue Edouard Belin, 31401 Toulouse Cedex 9, France
7474institutetext: Institute of Space Science, Str. Atomistilor, nr. 409
Măgurele, Ilfov, 077125, Romania 7575institutetext: Instituto de Astrofísica
de Canarias, Calle Vía Láctea s/n, 38204, San Cristóbal de La Laguna,
Tenerife, Spain 7676institutetext: Departamento de Astrofísica, Universidad de
La Laguna, 38206, La Laguna, Tenerife, Spain 7777institutetext: Departamento
de Física, FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile
7878institutetext: Satlantis, University Science Park, Sede Bld 48940, Leioa-
Bilbao, Spain 7979institutetext: Centre for Electronic Imaging, Open
University, Walton Hall, Milton Keynes, MK7 6AA, UK 8080institutetext: Centro
de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT),
Avenida Complutense 40, 28040 Madrid, Spain 8181institutetext: Infrared
Processing and Analysis Center, California Institute of Technology, Pasadena,
CA 91125, USA 8282institutetext: Instituto de Astrofísica e Ciências do
Espaço, Faculdade de Ciências, Universidade de Lisboa, Tapada da Ajuda,
1349-018 Lisboa, Portugal 8383institutetext: Universidad Politécnica de
Cartagena, Departamento de Electrónica y Tecnología de Computadoras, Plaza del
Hospital 1, 30202 Cartagena, Spain 8484institutetext: Institut de Recherche en
Astrophysique et Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES,
14 Av. Edouard Belin, 31400 Toulouse, France 8585institutetext: Kapteyn
Astronomical Institute, University of Groningen, PO Box 800, 9700 AV
Groningen, The Netherlands 8686institutetext: INFN-Bologna, Via Irnerio 46,
40126 Bologna, Italy 8787institutetext: Department of Mathematics and Physics
E. De Giorgi, University of Salento, Via per Arnesano, CP-I93, 73100, Lecce,
Italy 8888institutetext: INAF-Sezione di Lecce, c/o Dipartimento Matematica e
Fisica, Via per Arnesano, 73100, Lecce, Italy 8989institutetext: INFN, Sezione
di Lecce, Via per Arnesano, CP-193, 73100, Lecce, Italy 9090institutetext:
Institut für Theoretische Physik, University of Heidelberg, Philosophenweg 16,
69120 Heidelberg, Germany 9191institutetext: Université St Joseph; Faculty of
Sciences, Beirut, Lebanon 9292institutetext: Junia, EPA department, 41 Bd
Vauban, 59800 Lille, France 9393institutetext: SISSA, International School for
Advanced Studies, Via Bonomea 265, 34136 Trieste TS, Italy 9494institutetext:
INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste TS, Italy
9595institutetext: ICSC - Centro Nazionale di Ricerca in High Performance
Computing, Big Data e Quantum Computing, Via Magnanelli 2, Bologna, Italy
9696institutetext: Instituto de Física Teórica UAM-CSIC, Campus de
Cantoblanco, 28049 Madrid, Spain 9797institutetext: CERCA/ISO, Department of
Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH
44106, USA 9898institutetext: Laboratoire Univers et Théorie, Observatoire de
Paris, Université PSL, Université Paris Cité, CNRS, 92190 Meudon, France
9999institutetext: Dipartimento di Fisica e Scienze della Terra, Università
degli Studi di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy
100100institutetext: Istituto Nazionale di Fisica Nucleare, Sezione di
Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy 101101institutetext:
Dipartimento di Fisica - Sezione di Astronomia, Università di Trieste, Via
Tiepolo 11, 34131 Trieste, Italy 102102institutetext: NASA Ames Research
Center, Moffett Field, CA 94035, USA 103103institutetext: Kavli Institute for
Particle Astrophysics & Cosmology (KIPAC), Stanford University, Stanford, CA
94305, USA 104104institutetext: Bay Area Environmental Research Institute,
Moffett Field, California 94035, USA 105105institutetext: Minnesota Institute
for Astrophysics, University of Minnesota, 116 Church St SE, Minneapolis, MN
55455, USA 106106institutetext: INAF, Istituto di Radioastronomia, Via Piero
Gobetti 101, 40129 Bologna, Italy 107107institutetext: Institute Lorentz,
Leiden University, PO Box 9506, Leiden 2300 RA, The Netherlands
108108institutetext: Institute for Astronomy, University of Hawaii, 2680
Woodlawn Drive, Honolulu, HI 96822, USA 109109institutetext: Department of
Physics & Astronomy, University of California Irvine, Irvine CA 92697, USA
110110institutetext: Department of Astronomy & Physics and Institute for
Computational Astrophysics, Saint Mary’s University, 923 Robie Street,
Halifax, Nova Scotia, B3H 3C3, Canada 111111institutetext: Departamento Física
Aplicada, Universidad Politécnica de Cartagena, Campus Muralla del Mar, 30202
Cartagena, Murcia, Spain 112112institutetext: Université Paris Cité, CNRS,
Astroparticule et Cosmologie, 75013 Paris, France 113113institutetext:
Department of Computer Science, Aalto University, PO Box 15400, Espoo, FI-00
076, Finland 114114institutetext: NRC Herzberg, 5071 West Saanich Rd,
Victoria, BC V9E 2E7, Canada 115115institutetext: Ruhr University Bochum,
Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German
Centre for Cosmological Lensing (GCCL), 44780 Bochum, Germany
116116institutetext: Instituto de Astrofísica de Canarias (IAC); Departamento
de Astrofísica, Universidad de La Laguna (ULL), 38200, La Laguna, Tenerife,
Spain 117117institutetext: Université PSL, Observatoire de Paris, Sorbonne
Université, CNRS, LERMA, 75014, Paris, France 118118institutetext: Université
Paris-Cité, 5 Rue Thomas Mann, 75013, Paris, France 119119institutetext: Univ.
Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53, Avenue des Martyrs, 38000,
Grenoble, France 120120institutetext: Department of Physics and Astronomy,
Vesilinnantie 5, 20014 University of Turku, Finland 121121institutetext: Serco
for European Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion
Villafranca del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
122122institutetext: Centre for Astrophysics & Supercomputing, Swinburne
University of Technology, Victoria 3122, Australia 123123institutetext: ARC
Centre of Excellence for Dark Matter Particle Physics, Melbourne, Australia
124124institutetext: Department of Physics and Helsinki Institute of Physics,
Gustaf Hällströmin katu 2, 00014 University of Helsinki, Finland
125125institutetext: Oskar Klein Centre for Cosmoparticle Physics, Department
of Physics, Stockholm University, Stockholm, SE-106 91, Sweden
126126institutetext: Astrophysics Group, Blackett Laboratory, Imperial College
London, London SW7 2AZ, UK 127127institutetext: Centre de Calcul de
l’IN2P3/CNRS, 21 avenue Pierre de Coubertin 69627 Villeurbanne Cedex, France
128128institutetext: Dipartimento di Fisica, Sapienza Università di Roma,
Piazzale Aldo Moro 2, 00185 Roma, Italy 129129institutetext: INFN-Sezione di
Roma, Piazzale Aldo Moro, 2 - c/o Dipartimento di Fisica, Edificio G. Marconi,
00185 Roma, Italy 130130institutetext: Centro de Astrofísica da Universidade
do Porto, Rua das Estrelas, 4150-762 Porto, Portugal 131131institutetext:
Zentrum für Astronomie, Universität Heidelberg, Philosophenweg 12, 69120
Heidelberg, Germany 132132institutetext: Dipartimento di Fisica, Università di
Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma, Italy
133133institutetext: INFN, Sezione di Roma 2, Via della Ricerca Scientifica 1,
Roma, Italy 134134institutetext: Institute for Computational Science,
University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
135135institutetext: Mullard Space Science Laboratory, University College
London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK 136136institutetext:
Department of Physics and Astronomy, University of California, Davis, CA
95616, USA 137137institutetext: Department of Astrophysical Sciences, Peyton
Hall, Princeton University, Princeton, NJ 08544, USA 138138institutetext:
Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen,
Denmark 139139institutetext: Cosmic Dawn Center (DAWN)

#  Euclid preparation. XXXVII.

Galaxy colour selections with Euclid and ground photometry for cluster weak-
lensing analyses

Euclid Collaboration: G. F. Lesci0000-0002-4607-2830 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII<EMAIL_ADDRESS>M. Sereno0000-0003-0302-0325
Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. M. Radovich0000-0002-3585-866X Euclid
preparation. XXXVII.Euclid preparation. XXXVII. G.
Castignani0000-0001-6831-0687 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. L.
Bisigello0000-0003-0492-4924 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. F.
Marulli0000-0002-8850-0303 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII. L.
Moscardini0000-0002-3473-6716 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII. L. Baumont0000-0002-1518-0150
Euclid preparation. XXXVII.Euclid preparation. XXXVII. G.
Covone0000-0002-2553-096X Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII. S. Farrens0000-0002-9594-9387
Euclid preparation. XXXVII.Euclid preparation. XXXVII. C.
Giocoli0000-0002-9590-7961 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. L.
Ingoglia0000-0002-7587-0997 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. S. Miranda La Hera Euclid preparation. XXXVII.Euclid preparation.
XXXVII. M. Vannier Euclid preparation. XXXVII.Euclid preparation. XXXVII. A.
Biviano0000-0002-0857-0732 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. S. Maurogordato
Euclid preparation. XXXVII.Euclid preparation. XXXVII. N. Aghanim Euclid
preparation. XXXVII.Euclid preparation. XXXVII. A. Amara Euclid preparation.
XXXVII.Euclid preparation. XXXVII. S. Andreon0000-0002-2041-8784 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. N.
Auricchio0000-0003-4444-8651 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. M. Baldi0000-0003-4145-1943 Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. S. Bardelli
Euclid preparation. XXXVII.Euclid preparation. XXXVII. R.
Bender0000-0001-7179-0626 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. C. Bodendorf
Euclid preparation. XXXVII.Euclid preparation. XXXVII. D. Bonino Euclid
preparation. XXXVII.Euclid preparation. XXXVII. E.
Branchini0000-0002-0808-6908 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. M.
Brescia0000-0001-9506-5680 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII. J.
Brinchmann0000-0003-4359-8797 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. S. Camera0000-0003-3399-3574 Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. V.
Capobianco0000-0002-3309-7692 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. C. Carbone0000-0003-0125-3563 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. J. Carretero0000-0002-3130-0204 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. S. Casas0000-0002-4751-5138 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. F. J. Castander0000-0001-7316-4573 Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. M. Castellano0000-0001-9875-8263 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. S. Cavuoti0000-0002-3787-4196
Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. A. Cimatti Euclid preparation.
XXXVII.Euclid preparation. XXXVII. G. Congedo0000-0003-2508-0046 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. C. J. Conselice Euclid
preparation. XXXVII.Euclid preparation. XXXVII. L. Conversi0000-0002-6710-8476
Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. Y. Copin0000-0002-5317-7518 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. L. Corcione0000-0002-6497-5881
Euclid preparation. XXXVII.Euclid preparation. XXXVII. F.
Courbin0000-0003-0758-6510 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. H. M. Courtois0000-0003-0509-1776 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. A. Da Silva0000-0002-6385-1609 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. H. Degaudenzi0000-0002-5887-6799 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. A. M. Di Giorgio Euclid preparation.
XXXVII.Euclid preparation. XXXVII. J. Dinis Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. F.
Dubath0000-0002-6533-2810 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. C. A. J. Duncan Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. X. Dupac Euclid
preparation. XXXVII.Euclid preparation. XXXVII. S. Dusini0000-0002-1128-0664
Euclid preparation. XXXVII.Euclid preparation. XXXVII. M. Farina Euclid
preparation. XXXVII.Euclid preparation. XXXVII. S. Ferriol Euclid preparation.
XXXVII.Euclid preparation. XXXVII. P. Fosalba0000-0002-1510-5214 Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. S. Fotopoulou Euclid preparation.
XXXVII.Euclid preparation. XXXVII. M. Frailis0000-0002-7400-2135 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. E.
Franceschi0000-0002-0585-6591 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. P. Franzetti Euclid preparation. XXXVII.Euclid preparation. XXXVII. M.
Fumana0000-0001-6787-5950 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. S. Galeotta0000-0002-3748-5115 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. B. Garilli0000-0001-7455-8750 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. B. Gillis0000-0002-4478-1270 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. A. Grazian0000-0002-5688-0663
Euclid preparation. XXXVII.Euclid preparation. XXXVII. F. Grupp Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. S. V. H. Haugan0000-0001-9648-7260 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. I. Hook0000-0002-2960-978X
Euclid preparation. XXXVII.Euclid preparation. XXXVII. F. Hormuth Euclid
preparation. XXXVII.Euclid preparation. XXXVII. A.
Hornstrup0000-0002-3363-0936 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. P. Hudelot
Euclid preparation. XXXVII.Euclid preparation. XXXVII. K.
Jahnke0000-0003-3804-2137 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. M. Kümmel0000-0003-2791-2117 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. S. Kermiche0000-0002-0302-5735 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. A. Kiessling0000-0002-2590-1273 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. M.
Kilbinger0000-0001-9513-7138 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. B. Kubik Euclid preparation. XXXVII.Euclid preparation. XXXVII. M.
Kunz0000-0002-3052-7394 Euclid preparation. XXXVII.Euclid preparation. XXXVII.
H. Kurki-Suonio0000-0002-4618-3063 Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. S.
Ligori0000-0003-4172-4606 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. P. B. Lilje0000-0003-4324-7794 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. V. Lindholm0000-0003-2317-5471 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. I. Lloro Euclid preparation. XXXVII.Euclid preparation.
XXXVII. E. Maiorano0000-0003-2593-4355 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. O. Mansutti0000-0001-5758-4658 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. O. Marggraf0000-0001-7242-3852 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. K. Markovic0000-0001-6764-073X
Euclid preparation. XXXVII.Euclid preparation. XXXVII. N.
Martinet0000-0003-2786-7790 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. R. Massey0000-0002-6085-3780 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. E. Medinaceli0000-0002-4040-7783 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. M. Melchior Euclid preparation.
XXXVII.Euclid preparation. XXXVII. Y. Mellier Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. M. Meneghetti0000-0003-1225-7084 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. E. Merlin0000-0001-6870-8900 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. G. Meylan Euclid preparation. XXXVII.Euclid
preparation. XXXVII. M. Moresco0000-0002-7616-7136 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. E. Munari0000-0002-1751-5946 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. R. Nakajima Euclid preparation.
XXXVII.Euclid preparation. XXXVII. S.-M. Niemi Euclid preparation.
XXXVII.Euclid preparation. XXXVII. C. Padilla0000-0001-7951-0166 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. S. Paltani Euclid preparation.
XXXVII.Euclid preparation. XXXVII. F. Pasian Euclid preparation. XXXVII.Euclid
preparation. XXXVII. K. Pedersen Euclid preparation. XXXVII.Euclid
preparation. XXXVII. V. Pettorino Euclid preparation. XXXVII.Euclid
preparation. XXXVII. S. Pires Euclid preparation. XXXVII.Euclid preparation.
XXXVII. G. Polenta0000-0003-4067-9196 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. M. Poncet Euclid preparation. XXXVII.Euclid preparation.
XXXVII. L. A. Popa Euclid preparation. XXXVII.Euclid preparation. XXXVII. L.
Pozzetti0000-0001-7085-0412 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. F. Raison0000-0002-7819-6918 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. R. Rebolo Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. A.
Renzi0000-0001-9856-1970 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. J. Rhodes Euclid
preparation. XXXVII.Euclid preparation. XXXVII. G. Riccio Euclid preparation.
XXXVII.Euclid preparation. XXXVII. E. Romelli0000-0003-3069-9222 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. M.
Roncarelli0000-0001-9587-7822 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. E. Rossetti Euclid preparation. XXXVII.Euclid preparation. XXXVII. R.
Saglia0000-0003-0378-7032 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. D.
Sapone0000-0001-7089-4503 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. B. Sartoris Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. M.
Schirmer0000-0003-2568-9994 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. P. Schneider0000-0001-8561-2679 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. A. Secroun0000-0003-0505-3710 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. G. Seidel0000-0003-2907-353X Euclid
preparation. XXXVII.Euclid preparation. XXXVII. S. Serrano0000-0002-0211-2861
Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. C. Sirignano0000-0002-0995-7146 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. G. Sirri0000-0003-2626-2853 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. J. Skottfelt0000-0003-1310-8283 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. L. Stanco0000-0002-9706-5104
Euclid preparation. XXXVII.Euclid preparation. XXXVII. J.-L.
Starck0000-0003-2177-7794 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. P. Tallada-Crespí0000-0002-1336-8328 Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. A.
N. Taylor Euclid preparation. XXXVII.Euclid preparation. XXXVII. H. I.
Teplitz0000-0002-7064-5424 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. I. Tereno Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII. R. Toledo-
Moreo0000-0002-2997-4859 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. F. Torradeflot0000-0003-1160-1517 Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. I.
Tutusaus0000-0002-3199-0399 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. E. A. Valentijn Euclid preparation. XXXVII.Euclid preparation. XXXVII.
L. Valenziano0000-0002-1170-0104 Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. T.
Vassallo0000-0001-6512-6358 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. A.
Veropalumbo0000-0003-2387-1194 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. Y.
Wang0000-0002-4749-2984 Euclid preparation. XXXVII.Euclid preparation. XXXVII.
J. Weller Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII. A. Zacchei0000-0003-0396-1192
Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. G. Zamorani0000-0002-2318-301X Euclid
preparation. XXXVII.Euclid preparation. XXXVII. J. Zoubian Euclid preparation.
XXXVII.Euclid preparation. XXXVII. E. Zucca0000-0002-5845-8132 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. M.
Bolzonella0000-0003-3278-4607 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. E. Bozzo Euclid preparation. XXXVII.Euclid preparation. XXXVII. C.
Colodro-Conde Euclid preparation. XXXVII.Euclid preparation. XXXVII. D. Di
Ferdinando Euclid preparation. XXXVII.Euclid preparation. XXXVII. J. Graciá-
Carpio Euclid preparation. XXXVII.Euclid preparation. XXXVII. S. Marcin Euclid
preparation. XXXVII.Euclid preparation. XXXVII. N. Mauri0000-0001-8196-1548
Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. C. Neissner0000-0001-8524-4968 Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. A. A. Nucita Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. Z.
Sakr0000-0002-4823-3757 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII. V. Scottez Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. M. Tenti0000-0002-4254-5901 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. M. Viel0000-0002-2642-5707 Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. M. Wiesmann
Euclid preparation. XXXVII.Euclid preparation. XXXVII. Y.
Akrami0000-0002-2407-7956 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. S.
Anselmi0000-0002-3579-9583 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII. C.
Baccigalupi0000-0002-8211-1630 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. M. Ballardini0000-0003-4481-3559 Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. S. Borgani0000-0001-6151-6439 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. A. S.
Borlaff0000-0003-3249-4431 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII. S. Bruton Euclid preparation.
XXXVII.Euclid preparation. XXXVII. C. Burigana0000-0002-3005-5796 Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. R. Cabanac0000-0001-6679-2600 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. A. Calabro0000-0003-2536-1614
Euclid preparation. XXXVII.Euclid preparation. XXXVII. A. Cappi Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. C. S. Carvalho Euclid preparation.
XXXVII.Euclid preparation. XXXVII. T. Castro0000-0002-6292-3228 Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. G.
Cañas-Herrera0000-0003-2796-2149 Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. K.
C. Chambers0000-0001-6965-7789 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. A. R. Cooray0000-0002-3892-0190 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. J. Coupon Euclid preparation. XXXVII.Euclid preparation.
XXXVII. O. Cucciati0000-0002-9336-7551 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. S. Davini Euclid preparation. XXXVII.Euclid preparation.
XXXVII. S. de la Torre Euclid preparation. XXXVII.Euclid preparation. XXXVII.
G. De Lucia0000-0002-6220-9104 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. G. Desprez Euclid preparation. XXXVII.Euclid preparation. XXXVII. S.
Di Domizio0000-0003-2863-5895 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. H.
Dole0000-0002-9767-3839 Euclid preparation. XXXVII.Euclid preparation. XXXVII.
A. Díaz-Sánchez0000-0003-0748-4768 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. J. A. Escartin Vigo Euclid preparation. XXXVII.Euclid
preparation. XXXVII. S. Escoffier0000-0002-2847-7498 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. I. Ferrero0000-0002-1295-1132 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. F. Finelli0000-0002-6694-3269
Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. L. Gabarra Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. K. Ganga0000-0001-8159-8208 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. J. García-Bellido0000-0002-9370-8360 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. F.
Giacomini0000-0002-3129-2814 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. G. Gozaliasl0000-0002-0236-919X Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. S.
Gwyn0000-0001-8221-8406 Euclid preparation. XXXVII.Euclid preparation. XXXVII.
H. Hildebrandt0000-0002-9814-3338 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. M. Huertas-Company0000-0002-1416-8483 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. A. Jimenez
Muñoz0009-0004-5252-185X Euclid preparation. XXXVII.Euclid preparation.
XXXVII. J. J. E. Kajava0000-0002-3010-8333 Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. V.
Kansal Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII. C. C. Kirkpatrick Euclid
preparation. XXXVII.Euclid preparation. XXXVII. L. Legrand0000-0003-0610-5252
Euclid preparation. XXXVII.Euclid preparation. XXXVII. A.
Loureiro0000-0002-4371-0876 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. J. Macias-
Perez0000-0002-5385-2763 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. M. Magliocchetti0000-0001-9158-4838 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. G. Mainetti Euclid preparation. XXXVII.Euclid
preparation. XXXVII. R. Maoli0000-0002-6065-3025 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. M. Martinelli0000-0002-6943-7732 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. C. J. A. P. Martins0000-0002-4886-9261 Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. S. Matthew Euclid preparation.
XXXVII.Euclid preparation. XXXVII. M. Maturi0000-0002-3517-2422 Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. L. Maurin0000-0002-8406-0857 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. R. B.
Metcalf0000-0003-3167-2574 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. M. Migliaccio
Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII. P. Monaco0000-0003-2083-7564 Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. G.
Morgante Euclid preparation. XXXVII.Euclid preparation. XXXVII. S.
Nadathur0000-0001-9070-3102 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. L. Patrizii Euclid preparation. XXXVII.Euclid preparation. XXXVII. A.
Pezzotta Euclid preparation. XXXVII.Euclid preparation. XXXVII. C. Porciani
Euclid preparation. XXXVII.Euclid preparation. XXXVII. D.
Potter0000-0002-0757-5195 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. M. Pöntinen0000-0001-5442-2530 Euclid preparation. XXXVII.Euclid
preparation. XXXVII. P. Reimberg0000-0003-3410-0280 Euclid preparation.
XXXVII.Euclid preparation. XXXVII. P.-F. Rocci Euclid preparation.
XXXVII.Euclid preparation. XXXVII. A. G. Sánchez0000-0003-1198-831X Euclid
preparation. XXXVII.Euclid preparation. XXXVII. A.
Schneider0000-0001-7055-8104 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. M. Schultheis Euclid preparation. XXXVII.Euclid preparation. XXXVII.
E. Sefusatti0000-0003-0473-1567 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII. P. Simon Euclid preparation.
XXXVII.Euclid preparation. XXXVII. A. Spurio Mancini0000-0001-5698-0990 Euclid
preparation. XXXVII.Euclid preparation. XXXVII. S. A.
Stanford0000-0003-0122-0841 Euclid preparation. XXXVII.Euclid preparation.
XXXVII. J. Steinwagner Euclid preparation. XXXVII.Euclid preparation. XXXVII.
G. Testera Euclid preparation. XXXVII.Euclid preparation. XXXVII. R. Teyssier
Euclid preparation. XXXVII.Euclid preparation. XXXVII. S. Toft Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII. S. Tosi0000-0002-7275-9193 Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII.Euclid
preparation. XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. A.
Troja0000-0003-0239-4595 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. M. Tucci Euclid
preparation. XXXVII.Euclid preparation. XXXVII. J.
Valiviita0000-0001-6225-3693 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.Euclid preparation. XXXVII.Euclid preparation. XXXVII. D.
Vergani0000-0003-0898-2216 Euclid preparation. XXXVII.Euclid preparation.
XXXVII.

(Received –; accepted –)

###### Abstract

Aims.  We derived galaxy colour selections from Euclid and ground-based
photometry, aiming to accurately define background galaxy samples in cluster
weak-lensing analyses. These selections have been implemented in the Euclid
data analysis pipelines for galaxy clusters.

Methods.  Given any set of photometric bands, we developed a method for the
calibration of optimal galaxy colour selections that maximises the selection
completeness, given a threshold on purity. Such colour selections are
expressed as a function of the lens redshift.

Results.  We calibrated galaxy selections using simulated ground-based $griz$
and Euclid $Y_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ photometry. Both selections produce a purity
higher than $97\%$. The $griz$ selection completeness ranges from 30% to
$84\%$ in the lens redshift range $z_{\rm l}\in[0.2,0.8]$. With the full
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection, the completeness improves by up to
$25$ percentage points, and the $z_{\rm l}$ range extends up to $z_{\rm
l}=1.5$. The calibrated colour selections are stable to changes in the sample
limiting magnitudes and redshift, and the selection based on $griz$ bands
provides excellent results on real external datasets. Furthermore, the
calibrated selections provide stable results using alternative photometric
aperture definitions obtained from different ground-based telescopes. The
$griz$ selection is also purer at high redshift and more complete at low
redshift compared to colour selections found in the literature. We find
excellent agreement in terms of purity and completeness between the analysis
of an independent, simulated Euclid galaxy catalogue and our calibration
sample, except for galaxies at high redshifts, for which we obtain up to 50
percent points higher completeness. The combination of colour and photo-$z$
selections applied to simulated Euclid data yields up to 95% completeness,
while the purity decreases down to 92% at high $z_{\rm l}$. We show that the
calibrated colour selections provide robust results even when observations
from a single band are missing from the ground-based data. Finally, we show
that colour selections do not disrupt the shear calibration for stage III
surveys. The first Euclid data releases will provide further insights into the
impact of background selections on the shear calibration.

###### Key Words.:

Galaxies: statistics – Galaxies: photometry – Galaxies: distances and
redshifts – Galaxies: clusters: general – Cosmology: observations – large-
scale structure of Universe

##  1 Introduction

In the last decade, galaxy clusters have proven to be excellent probes for
cosmological analyses (see, e.g. Mantz et al. 2015; Sereno et al. 2015; Planck
Collaboration XXIV 2016; Costanzi et al. 2019; Marulli et al. 2021; Lesci et
al. 2022), also allowing for the investigation of dark matter interaction
models (Peter et al. 2013; Robertson et al. 2017; Eckert et al. 2022) and gas
astrophysics (Vazza et al. 2017; CHEX-MATE Collaboration 2021; Zhu et al.
2021; Sereno et al. 2021). As galaxy clusters are dominated by dark matter,
the functional form of their matter density profiles can be derived from
$N$-body dark-matter-only simulations (Navarro et al. 1997; Baltz et al. 2009;
Diemer & Kravtsov 2014). This allows one to estimate the mass of observed
clusters, which is essential for both astrophysical and cosmological studies
(Teyssier et al. 2011; Pratt et al. 2019). Currently, weak gravitational
lensing is one of the most reliable methods to accurately and precisely
measure cluster masses (Okabe et al. 2010; Hoekstra et al. 2012; Melchior et
al. 2015; Sereno et al. 2017; Stern et al. 2019; Schrabback et al. 2021;
Zohren et al. 2022). Consequently, weak-lensing cluster mass estimates are
widely used in current photometric galaxy surveys, such as the Kilo Degree
Survey (KiDS; Kuijken et al. 2019; Bellagamba et al. 2019), the Dark Energy
Survey (DES; Abbott et al. 2020; Sevilla-Noarbe et al. 2021), and the Hyper
Suprime-Cam survey (HSC; Medezinski et al. 2018; Li et al. 2022).
An accurate selection of lensed background galaxies is crucial to derive a
reliable cluster weak-lensing signal. Including the contribution from
foreground and cluster member galaxies may significantly dilute the weak-
lensing signal (Broadhurst et al. 2005; Medezinski et al. 2007; Sifón et al.
2015; McClintock et al. 2019). For example, background selections with 90%
purity dilute the cluster reduced shear measurements by 10% (see, e.g.
Dietrich et al. 2019), in the absence of intrinsic alignments (Heymans &
Heavens 2003). Highly pure background selections are required to properly
account for this effect in weak-lensing measurements, in order to minimise the
variance in the selection purity. Selection incompleteness, instead, impacts
the weak-lensing noise and, in turn, the signal-to-noise ratio (S/N), which
depends on the density of background sources along with the intrinsic
ellipticity dispersion and measurement noise (see, e.g. Schrabback et al.
2018; Umetsu 2020). The effect of low background densities can be partially
mitigated by increasing the size of the cluster-centric radial bins used in
the analysis, or through the stacking of the weak-lensing signal of cluster
ensembles.
Background selections based on the galaxy photometric redshift (photo-$z$)
posteriors are commonly used in the literature (Gruen et al. 2014; Applegate
et al. 2014; Melchior et al. 2017; Sereno et al. 2017; Bellagamba et al.
2019), as well as galaxy colour selections (Medezinski et al. 2010; Oguri et
al. 2012; Medezinski et al. 2018; Klein et al. 2019). These selections can
also be combined to significantly improve the background sample completeness
and, in turn, the weak-lensing S/N. In fact, colour selections have been
demonstrated to help identify galaxies with poorly defined photometric
redshifts that would not have been classified as background sources through
photo-$z$ selection alone (Covone et al. 2014; Sereno et al. 2017; Bellagamba
et al. 2019).
The aim of this paper is to develop a method to obtain optimal colour
selections, namely with a maximal completeness given a threshold on purity,
given any set of photometric filters. We provide, for the first time, colour
selections expressed as a continuous function of the lens limiting redshift.
This allows for a finer background definition compared to colour selections
found in the literature (Medezinski et al. 2010; Oguri et al. 2012; Medezinski
et al. 2018), implying a significant improvement in the weak-lensing source
statistics. In view of Euclid and stage IV surveys, we derived colour
selections on simulated data. We exploited the galaxy catalogue developed by
Bisigello et al. (2020), hereafter referred to as B20, and extended by Euclid
Collaboration: Bisigello et al. (2023), which includes simulated Sloan Digital
Sky Survey (SDSS; Gunn et al. 1998) $griz$ magnitudes and simulated Euclid
observations in the $Y_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ bands. In addition, we tested the efficiency of
these colour selections on real public external data and on simulations,
combining them with photo-$z$ selections.
This paper is part of a series presenting and discussing mass measurements of
galaxy clusters using the Euclid combined clusters and weak-lensing pipeline
COMB-CL. COMB-CL forms part of the global Euclid data processing pipeline and
is responsible for measuring weak-lensing shear profiles and masses for
photometrically detected clusters. A comprehensive description of the code
structure and methods employed by COMB-CL will be presented in a forthcoming
paper, but a brief overview of the pipeline can be found in the appendix of
Euclid Collaboration: Sereno et al. (2023). The galaxy colour selections
presented in this paper are already implemented in COMB-CL.
The paper is organised as follows. In Sect. 2, we describe the dataset used
for the calibration of galaxy colour selections, and in Sect. 3 we detail a
general method to derive optimal colour selections. In Sect. 4, we show the
selections obtained for $griz$ and $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ filter sets,
validating them on external datasets. In Sect. 5 we compare the $griz$
selection calibrated in this work with selections from the literature.
Finally, in Sect. 6, we draw our conclusions.

##  2 Calibration sample

Figure 1: Example of an uncalibrated selection in the $(r-z)$ \- $(g-i)$
colour-colour space. The grey dots represent the selected galaxy colours.
Galaxies within the octagonal hatched region are excluded by applying Eq. (3).
Specifically, $(r-z)$ and $(g-i)$ correspond to $x$ and $y$ in Eq. (3),
respectively.

We based our analysis on the photometric catalogue developed by B20 and
extended by Euclid Collaboration: Bisigello et al. (2023). This catalogue
contains simulated Euclid $I_{\scriptscriptstyle\rm E}Y_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ aperture
magnitudes111$I_{\scriptscriptstyle\rm E}$ band observations are supplied by
the Euclid Visible Imager (VIS; Cropper et al. 2016), while
$Y_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ photometry is provided by the Near-Infrared
Spectrometer and Photometer (NISP; Euclid Collaboration: Schirmer et al.
2022)., covering the spectral range 5500–20 000 Å, along with the Canada-
France Imaging Survey (CFIS; Ibata et al. 2017) $u$ band, for the galaxies
contained in the COSMOS catalogue by Laigle et al. (2016, COSMOS15).
Specifically, such photometry is based on $None$ fixed-aperture magnitudes.
Despite the $u$ band already being present in COSMOS15, B20 derived it using
the same approach adopted for the other filters in order to avoid colour
biases. B20 verified that this provides results that are consistent with the
observed fluxes. Simulated SDSS $griz$ magnitudes, spanning the wavelength
range 4000–11 000 Å, are also provided in the catalogue, since observations in
similar filters, such as those in Vera C. Rubin Observatory (Rubin/LSST;
Ivezic et al. 2008) and DES, will be available to complement Euclid
observations (Euclid Collaboration: Pocino et al. 2021; Euclid Collaboration:
Scaramella et al. 2022). Corrections for photometric offsets due to flux
outside the fixed-aperture, systematic offsets, and Galactic extinction, as
suggested in Laigle et al. (2016), have been applied. B20 derive simulated
magnitudes through two alternative approaches. The first is a linear
interpolation of the 30 medium-band and broad-band filters available in the
COSMOS15 catalogue, based on the effective wavelength of the filters. The
second approach is based on the best theoretical template that describes the
spectral energy distribution (SED) of each galaxy, assuming the COSMOS15
redshifts as the ground truth. The SED fitting is performed based on COSMOS15
bands and the template resulting in the minimum $\chi^{2}$ is used to predict
the expected fluxes. We refer to B20 for the details of the SED templates
used, based on the model by Bruzual & Charlot (2003). The expected fluxes are
then randomised 10 times considering a Gaussian distribution centred on the
true flux and with standard deviation equal to the expected photometric
uncertainities, scaled considering the depths listed in Table 1 of Euclid
Collaboration: Bisigello et al. (2023). In this process, the
$I_{\scriptscriptstyle\rm E}Y_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ magnitude errors
expected for the Euclid Wide Survey are considered. Despite the fact that the
$griz$ photometry is based on SDSS filter transmissions, the corresponding
uncertainties are based on depths that are consistent with those of DES and
the Ultraviolet Near-Infrared Optical Northern Survey (UNIONS).222UNIONS is
carried out with the Subaru Telescope (Iye et al. 2004), the Canada-France-
Hawaii Telescope (CFHT; Gwyn 2012), and the Panoramic Survey Telescope and
Rapid Response System (Pan-STARRS; Chambers et al. 2016). More information at
https://www.skysurvey.cc/news/. The $ugriz$ photometry provided by LSST is
expected to go from 1 to 2.5 magnitudes deeper at the end of the Euclid
mission, depending on the photometric filter. Throughout this paper, we focus
on the magnitudes derived from the best theoretical SED templates, as these
estimates better reproduce absorption and emission lines that are not covered
by COSMOS15 bands. We neglect $u$ magnitudes since, due to the low $u$-band
throughput, a 5$\sigma$ depth of 25.6 mag will only be reached after 10 years
of LSST observations333https://www.lsst.org/scientists/keynumbers. In
addition, the $u$ band is not available in DES wide fields. We emphasise that
the B20 catalogue contains all the galaxies present in the COSMOS15 sample,
which is deeper than the shear samples derived from current surveys (see, e.g.
Giblin et al. 2021; Gatti et al. 2021) and expected from the Euclid Wide
Survey (Euclid Collaboration: Scaramella et al. 2022). As we discuss in the
following, the colour selections calibrated in this study yield robust results
against alternative magnitude cuts, including those that reproduce the
selections adopted in current and Euclid cosmic shear analyses.

##  3 Method

ptSet of photometric bands $N_{\rm cond}$ colour conditions, based on Eq. (3)
Measure $\mathcal{C}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ and
$\mathcal{P}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$, with $i=1,\ldots,N_{\rm
cond}$ ptCalibration samplept$z_{\rm l}$ points Find $\boldsymbol{p}$ for
which $\mathcal{C}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ and
$\mathcal{P}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ are above the
corresponding thresholds ptThresholds on $\mathcal{C}_{i}^{nf}$ and
$\mathcal{P}_{i}^{nf}$ Derive optimal colour conditions, maximising
$\mathcal{C}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ $\forall i,z_{\rm l}$
Combine the optimal conditions, deriving $\mathcal{C}^{nf}_{\rm tot}(z_{{\rm
l}})$, $\mathcal{P}^{nf}_{\rm tot}(z_{{\rm l}})$, $\mathcal{F}^{nf}_{\rm
tot}(z_{{\rm l}})$ Exclude the $z_{\rm l}$ points for which
$\mathcal{C}^{nf}_{\rm tot}(z_{{\rm l}})=\mathcal{F}^{nf}_{\rm tot}(z_{{\rm
l}})=100\%$ Subset of optimal conditions minimising $\mathcal{S}^{nf}$ in Eq.
(7) Fit $\boldsymbol{p}$ as a function of $z_{\rm l}$ ptCalibrated optimal
colour selection Figure 2: Flowchart summarising the calibration process
described in Sect. 3. Round red rectangles represent the start and end points
of the calibration process. Grey rectangles represent processing steps, while
blue trapezoids correspond to the inputs.

In order to find a set of optimal galaxy colour-redshift relations that
maximises the selection completeness given a threshold on the foreground
contamination, we considered the colours given by any combination of
photometric bands. This includes bands that are not adjacent in wavelength.
Thus, for each colour-colour space, given a redshift lower limit, $z_{\rm l}$,
corresponding to the lens redshift, we considered the following set of
conditions,

$\displaystyle x>c_{1}\,\lor\,$ $\displaystyle x<c_{2}\,\lor\,$ $\displaystyle
y>c_{3}\,\lor\,$ $\displaystyle y<c_{4}\,\lor\,$ $\displaystyle
x>s_{1}y+c_{5}\,\lor\,$ $\displaystyle x>s_{2}y+c_{6}\,\lor\,$ $\displaystyle
x<s_{3}y+c_{7}\,\lor\,$ $\displaystyle x<s_{4}y+c_{8},$ (1)

where $\lor$ is the logical ‘or’ operator, $x$ and $y$ are two different
colours, and $c_{i}$ and $s_{i}$ are colour selection parameters.
Specifically, $c_{1},\ldots,c_{8}\in(-\infty,+\infty)$, $s_{1}$ and
$s_{3}\in(0,+\infty)$, while $s_{2}$ and $s_{4}\in(-\infty,0)$. The edges of
the aforementioned parameter ranges are excluded, and Eq. (3) defines an
irregular octagon that contains the foreground galaxies, as we show in Fig. 1.
As we shall see, since we only select the colour conditions that satisfy given
requirements, not all the sides of the irregular octagon may be considered. In
addition, since we considered the conditions in Eq. (3) as independent, the
$c_{1},\ldots,c_{8}$ and $s_{1},\ldots,s_{4}$ parameters are not related to
each other. In particular, for each condition in Eq. (3), we derived the
completeness,

$\mathcal{C}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p}):=\frac{N_{{\rm
sel},i}(z_{\rm g}>z_{\rm l}\,|\,\boldsymbol{p})}{N_{\rm tot}(z_{\rm g}>z_{\rm
l})}\,,$ (2)

and the purity,

$\mathcal{P}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p}):=\frac{N_{{\rm
sel},i}(z_{\rm g}>z_{\rm l}\,|\,\boldsymbol{p})}{N_{{\rm sel},i}(z_{\rm g}\geq
0\,|\,\boldsymbol{p})}\,,$ (3)

where $i$ is the $i$th colour condition index, $z_{\rm g}$ is the galaxy
redshift, $\boldsymbol{p}$ is the set of colour condition parameters, $N_{{\rm
sel},i}$ is the number of galaxies selected with the $i$th colour condition,
$N_{\rm tot}$ is the total number of galaxies in the calibration sample, while
the $nf$ superscript represents quantities derived from colour conditions not
fitted as a function of $z_{\rm l}$. As we shall see, we do not adopt any
superscripts for the quantities derived from fitted colour conditions. In Eqs.
(2) and (3), we have $i=1...\,N_{\rm cond}$, where $N_{\rm cond}$ is the
number of all possible colour conditions, given Eq. (3), expressed as

$N_{\rm cond}=8\,\frac{N_{\rm col}!}{(N_{\rm col}-2)!\,2!},$ (4)

where $N_{\rm col}$ is the number of colours, given by

$N_{\rm col}=\frac{N_{\rm band}!}{(N_{\rm band}-2)!\,2!},$ (5)

where $N_{\rm band}$ is the number of photometric bands.
We set requirements on completeness and purity to be satisfied by each colour
condition in Eq. (3). Specifically, for a given $z_{\rm l}$, we selected the
colour conditions having at least one $\boldsymbol{p}$ set providing
$\mathcal{C}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ and
$\mathcal{P}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ larger than their
corresponding thresholds. We remark that $\boldsymbol{p}$ does not explicitly
depend on $z_{\rm l}$ at this stage, and that $z_{\rm l}$ values are
arbitrarily sampled. Setting a threshold on $\mathcal{C}_{i}^{nf}(z_{\rm
l}\,|\,\boldsymbol{p})$ is important for excluding colour conditions that do
not significantly contribute to the total completeness, and that may appear as
optimal only due to statistical fluctuations. Thus, the threshold on
$\mathcal{C}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ is meant to be low
compared to that on $\mathcal{P}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$.
Indeed, as we shall detail in Sect. 4.6, impurities in the background
selection imply systematic uncertainties in galaxy cluster reduced shear
measurements. Highly pure selections are required to properly account for this
effect, in order to minimise the scatter in purity. We discuss the choice of
the thresholds on $\mathcal{C}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ and
$\mathcal{P}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ in greater detail in
Sect. 4.1. For each colour condition in Eq. (3), with parameter values for
which the conditions on $\mathcal{C}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$
and $\mathcal{P}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ are satisfied, we
selected the $\boldsymbol{p}$ set providing the highest completeness at a
given $z_{\rm l}$. In this way, we derived the set of optimal colour
conditions maximising the selection completeness, given the chosen threshold
on purity.
We note that the maximum $z_{\rm l}$ of the calibrated colour selections
depends on the $\mathcal{C}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ and
$\mathcal{P}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})$ limits, while the minimum
$z_{\rm l}$ is derived by excluding the $z_{\rm l}$ points for which
$\mathcal{C}^{nf}_{\rm tot}(z_{{\rm l}})=\mathcal{F}^{nf}_{\rm tot}(z_{{\rm
l}})=100\%$. Here, $\mathcal{C}^{nf}_{\rm tot}$ and $\mathcal{F}^{nf}_{\rm
tot}$ are the completeness and the foreground failure rate given by the full
set of optimal colour conditions, respectively. For simplicity, we drop the
dependence on $\boldsymbol{p}$ in the text. The foreground failure rate is
defined as follows:

$\mathcal{F}_{\rm tot}^{nf}(z_{\rm l}):=\frac{N_{{\rm sel}}(z_{\rm g}<z_{\rm
l})}{N_{\rm tot}(z_{\rm g}<z_{\rm l})}=\frac{N_{{\rm sel}}(z_{\rm g}>z_{\rm
l})}{N_{\rm tot}(z_{\rm g}<z_{\rm l})}\,\frac{1-\mathcal{P}_{\rm
tot}^{nf}(z_{\rm l})}{\mathcal{P}_{\rm tot}^{nf}(z_{\rm l})}\,,$ (6)

where $N_{{\rm sel}}$ is the number of galaxies selected with all the optimal
colour conditions, given a condition on $z_{\rm g}$, and $\mathcal{P}_{\rm
tot}^{nf}(z_{\rm l})$ is the purity given by the full set of optimal
conditions. On the right-hand side of Eq. (6), derived from Eqs. (2) and (3),
we can see that $\mathcal{F}_{\rm tot}^{nf}(z_{\rm l})$ diminishes with
increasing $z_{\rm l}$ if high lower limits on purity are chosen. We stress
that $\mathcal{F}_{\rm tot}^{nf}(z_{\rm l})\leq 1$ by definition.

Figure 3: Selection completeness (top panel), purity (middle panel), and
foreground failure rate (bottom panel), derived from subsets of optimal colour
conditions not fitted as a function of $z_{\rm l}$, for the case of $griz$
photometry. The solid black lines represent the selection given by the full
set of optimal colour conditions, while the dashed lines show the selection at
different steps of the iterative process detailed in Sect. 3, given by subsets
of optimal conditions.

Figure 4: Values of $s$ (left panel) and $c$ (right panel) parameters, from
Eq. (3), as a function of $z_{\rm l}$ for the colour condition quoted in the
left panel legend. The black dots represent the optimal values of $s$ and $c$,
while the blue curves represent the polynomial fits.

In the selection process described above, some colour conditions may be
redundant. Thus, we iteratively searched for an optimal subset of colour
conditions to find the minimum number of conditions sufficient to
approximately reproduce the required completeness. Specifically, at each step
of this iterative process, we computed the following quantity:

$\mathcal{S}^{nf}=\sum_{j=1}^{N}\mathcal{C}^{nf}_{\rm tot}(z_{{\rm
l},j})-\mathcal{C}^{nf}(z_{{\rm l},j}),$ (7)

where $N$ is the number of $z_{\rm l}$ points, $\mathcal{C}^{nf}_{\rm
tot}(z_{{\rm l},j})$ is the completeness given by all optimal conditions,
while $\mathcal{C}^{nf}(z_{{\rm l},j})$ is the completeness given by a subset
of optimal conditions, computed at the $j$th $z_{\rm l}$ value. As the first
step of this iterative process, we found the optimal colour condition
minimising $\mathcal{S}^{nf}$. Then, at each iteration, we added the colour
condition that, combined with the conditions selected in the previous steps,
minimises $\mathcal{S}^{nf}$. We repeated this process until
$\mathcal{S}^{nf}$ was lower than a given tolerance. We remark that the
logical operator between colour conditions is $\lor$.
Lastly, we applied a nonlinear least squares analysis to find the best fit to
the $\boldsymbol{p}$ parameters as a function of $z_{\rm l}$ for the subset of
optimal colour conditions. We chose the fitting formulae which best reproduce
the $z_{\rm l}$ dependence, namely polynomials, while aiming at minimising the
number of free parameters in the fit.
In Fig. 2 we show a flowchart summarising the calibration process described in
this section. In Fig. 4 we show an example of the iterative process detailed
above, while Fig. 4 displays an example of parameter dependence on $z_{\rm
l}$. Hereafter, we refer to the completeness, purity, and foreground failure
rate, derived from sets of fitted colour conditions, as $\mathcal{C}(z_{\rm
l})$, $\mathcal{P}(z_{\rm l})$, and $\mathcal{F}(z_{\rm l})$, respectively.
For better clarity, in Table 1 we summarise the symbols referring to the
completeness functions introduced in this section.

##  4 Results

Table 1: Description of the completeness functions introduced in Sect. 3.
Analogous descriptions hold for purity and foreground failure rate.

Symbol | Description
---|---
$C_{i}^{nf}$ |  Completeness of the $i$th colour condition, given a set of sampled parameters.
$C_{\rm tot}^{nf}$ |  Completeness derived through the combination of all the optimal colour conditions.
$C^{nf}$ |  Completeness given by the combination of a subset of optimal colour conditions.
$C$ |  Completeness obtained from a subset of optimal colour conditions fitted as a function of $z_{\rm l}$.
444The $nf$ superscript represents quantities derived from colour conditions
not fitted as a function of $z_{\rm l}$. Optimal colour conditions satisfy the
thresholds on purity and completeness, and provide maximal completeness.

###  4.1 Calibration of colour selections

Figure 5: Summary of the results on the colour selection optimisation, based
on the B20 galaxy sample. Top panels: Selection completeness (left panel),
purity (central panel), and foreground failure rate (right panel), as a
function of $z_{\rm l}$. The dashed lines represent the selections derived
from the full sets of optimal conditions not fitted as a function of $z_{\rm
l}$, in the case of ground-only (red), Euclid-only (green), and the
combination of ground-based and Euclid bands (grey). The solid lines represent
the selections obtained from the subsets of optimal conditions, with
parameters fitted as a function of $z_{\rm l}$, in the case of ground-only
(blue) and for the combination of ground-based and Euclid bands (black).
Bottom panels: In the left panel, $\mathcal{S}$ and $\mathcal{S}^{nf}$ are
shown as a function of the iteration number. For the ground-based selection,
using $griz$ filters, $\mathcal{S}$ and $\mathcal{S}^{nf}$ are represented by
solid blue and dashed red lines, respectively. For the selection derived from
the combination of ground-based and Euclid filters, namely
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$, $\mathcal{S}$ and $\mathcal{S}^{nf}$ are
represented by solid black and dashed grey lines, respectively. In the right
panel, the difference between $\mathcal{C}^{nf}_{\rm tot}$ and $\mathcal{C}$
is shown, for the $griz$ (blue lines) and $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ (black lines)
selections.

Figure 6: Results from the fitted colour selections derived in Sect. 4.1,
assuming the alternative magnitude and redshift selections described in Sect.
4.2. From left to right: completeness, purity, foreground failure rate, and
background density as a function of $z_{\rm l}$. The assumed $z_{\rm l}$
precision is $\delta z_{\rm l}=10^{-3}$. Top panels: efficiency of the $griz$
selection, detailed in Table 2, applied to the B20 catalogue with $({\rm
S/N})_{\rm E}>3$ and $i\leq i_{\rm max}$ (blue solid lines), to its subsample
including galaxies with $i<24.4$ (red dotted lines), to the case with $i<23.4$
(green dash-dotted lines), and to the sample with $z_{\rm g}<1.5$ (orange
dashed lines). Bottom panels: efficiency of the $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ selection, detailed
in Table 3, applied to the B20 catalogue with $({\rm S/N})_{\rm E}>3$ and
$i\leq i_{\rm max}$ (black solid lines), to its subsample with $i<23.4$ (green
dash-dotted lines), and to the subsample with $({\rm S/N})_{\rm E}>10$
(magenta dashed lines).

By applying the methods detailed in Sect. 3 and adopting the B20 calibration
sample described in Sect. 2, we calibrated galaxy colour selections using
ground-based and Euclid photometry, namely SDSS $griz$ and Euclid
$Y_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ filters, respectively. These selections are
implemented in COMB-CL, and will be available for weak-lensing analyses of
galaxy clusters. We considered the following cases: ground-only, Euclid-only,
and the combination of ground-based and Euclid photometry. For the cases
including Euclid photometry, we adopted an ${\rm S/N}$ threshold for Euclid
near-infrared observations of $({\rm S/N})_{\rm E}>3$, which corresponds to
$Y_{\scriptscriptstyle\rm E}<24.85$, $J_{\scriptscriptstyle\rm E}<25.05$, and
$H_{\scriptscriptstyle\rm E}<24.95$ (Euclid Collaboration: Scaramella et al.
2022). In addition, we considered $z_{\rm l}$ points in the range $z_{\rm
l}\in[0.1,2.5]$, assuming a precision of $\delta z_{\rm l}=0.1$ for the
sampling. To derive the full set of optimal colour conditions, we imposed
$\mathcal{C}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})>10\%$ for the $i$th colour
condition. For the ground-only and Euclid-only cases, we imposed that the
purity of each colour condition is $\mathcal{P}_{i}^{nf}(z_{\rm
l}\,|\,\boldsymbol{p})>99\%$. We adopted a more restrictive threshold on
purity for the combination of ground-based and Euclid photometry,
corresponding to $\mathcal{P}_{i}^{nf}(z_{\rm l}\,|\,\boldsymbol{p})>99.7\%$.
This threshold is chosen as the larger number of colour combinations leads to
a higher summation of impurities. We obtained $\mathcal{P}^{nf}(z_{\rm
l})>97\%$ for any $z_{\rm l}$, when combining all the optimal colour
conditions, as shown in Fig. 5. As we shall discuss in Sect. 4.3, the purity
derived from different real datasets is stable, showing sub-percent changes,
on average.
The $\mathcal{F}^{nf}(z_{\rm l})$ decrease with increasing $z_{\rm l}$, shown
in Fig. 5, is expected, as discussed in Sect. 3. In addition, for any
combination of photometric bands, we found $\mathcal{C}^{nf}_{\rm tot}(z_{\rm
l})=\mathcal{F}^{nf}_{\rm tot}(z_{\rm l})=100\%$ for $z_{\rm l}=0.1$.
Consequently, we set $z_{\rm l}=0.2$ as the minimum lens redshift for the
calibrated colour selections. As shown in Fig. 5, from $griz$ photometry we
derived a selection within $z_{\rm l}\in[0.2,0.8]$, with 84% completeness at
$z_{\rm l}=0.2$, decreasing to 29% at $z_{\rm l}=0.8$. In the Euclid-only
case, namely $Y_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ and $I_{\scriptscriptstyle\rm E}$ bands,
results are not competitive with those derived from $griz$ photometry. On the
other hand, by combining ground-based and Euclid photometry, the completeness
significantly increases in the $z_{\rm l}$ range covered by the $griz$
selection, by up to 25 percent points. Also the $z_{\rm l}$ range of the
selection is significantly extended compared to the $griz$ case, corresponding
to $z_{\rm l}\in[0.2,1.5]$. Specifically, in this case we exclude the Euclid
$I_{\scriptscriptstyle\rm E}$ band, as it covers a large wavelength interval,
namely $\sim\,$5000–10 000 $\AA$, corresponding to the wavelength range
already covered by $griz$ photometry. Furthermore, the use of very broad
photometric bands is not the most optimal choice for calibrating galaxy colour
selections, which share similarities with photo-$z$ estimates.
We excluded any possible redundant colour condition, as detailed in Sect. 3.
In Table 2 we show the subset of optimal colour conditions for the ground-only
case, namely $griz$ photometry, along with the corresponding parameter fits.
The first condition quoted in Table 2 corresponds to the one derived in the
first step of the iterative process described in Sect. 3. This is analogous
for the subsequent conditions. We remark that the quoted conditions have
different ranges of validity in $z_{\rm l}$. Analogous information is listed
in Table 3 for the combination of ground-based and Euclid photometry,
corresponding to $grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ filters. We neglected the optimisation and
parameter fitting for the Euclid-only case, as we have already shown that it
does not provide competitive completeness values.
In Fig. 5 we show the results for the selections obtained from the subsets of
optimal conditions, with parameters fitted as a function of $z_{\rm l}$. For
both $griz$ and $grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ photometry, such fitted selections well
reproduce those given by the full sets of optimal conditions. To quantify the
goodness of the colour condition parameter fits, we defined a parameter
analogous to $\mathcal{S}^{nf}$ in Eq. (7), namely $\mathcal{S}$. This
parameter quantifies the difference between $\mathcal{C}^{nf}_{\rm tot}$, that
is the completeness given by the full set of optimal conditions not fitted as
a function of $z_{\rm l}$, and $\mathcal{C}$, which is the completeness given
by the subset of optimal colour conditions fitted as a function of $z_{\rm
l}$. As shown in Fig. 5, $\mathcal{S}$ does not perfectly match
$\mathcal{S}^{nf}$, for both $griz$ and $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ selections. This is
due to the fact that the $c_{1},\ldots,c_{8}$, $s_{1},\ldots,s_{4}$ parameters
in Eq. (3) do not always show a simple dependence on $z_{\rm l}$. Despite the
fact that better parameter fits could be achieved by adopting an arbitrarily
high order polynomial as the model, we set a 4th order polynomial as the
highest-degree functional form for describing these parameters (see Tables 2
and 3). As shown in Fig. 5, $\mathcal{C}$ is underestimated by at most 4
percent points. We verified that adding further conditions to these
selections, that is, lowering the $\mathcal{S}$ threshold down to $0$,
provides sub-percent level improvements in the selection completeness, on
average. We remark that, in order to derive colour selections not defined in
$z_{\rm l}$ bins, the final selection completeness is slightly degraded
compared to $\mathcal{C}^{nf}_{\rm tot}$ for some $z_{\rm l}$ values. In
realistic cluster weak-lensing analyses, however, we expect this to
statistically increase the galaxy background completeness. When colour
selections are defined on finite sets of $z_{\rm l}$ points, the background
galaxies are excluded based on the $z_{\rm l}$ precision adopted in the colour
selection calibration.

###  4.2 Dependence on magnitude and redshift selections

Figure 7: Results of the application of the fitted colour selection based on
$griz$ photometry, reported in Table 2, to the datasets introduced in Sect.
4.3. From left to right: completeness, purity, foreground failure rate, and
background density as a function of $z_{\rm l}$. The assumed $z_{\rm l}$
precision is $\delta z_{\rm l}=10^{-3}$. The $griz$ selection is applied to
the B20 catalogue with magnitude limits corresponding to those used in the
calibration process (blue solid lines), to the full depth Moutard et al.
(2016) catalogue (green solid lines), and to the Weaver et al. (2022)
catalogue with HSC Kron, $None$ and $None$ aperture magnitudes (solid red,
dashed orange and dotted black lines, respectively), for which we imposed
$i<25$.

To verify the robustness of the $griz$ selection with respect to alternative
magnitude cuts, we applied the selection $i<24.4$, corresponding to the peak
value of the $i$ magnitude distribution in the B20 catalogue. We also
investigate the selection for the subsample with $i<23.4$, which is a
threshold similar to the DES $i$ band limit (Sevilla-Noarbe et al. 2021). In
both cases, we derived higher $\mathcal{P}(z_{\rm l})$ and lower
$\mathcal{F}(z_{\rm l})$, compared to what we found from the calibration
sample used in Sect. 4.1, namely the one with $({\rm S/N})_{\rm E}>3$ and
$i\leq i_{\rm max}$, where $i_{\rm max}=24.9$ is the maximum $i$ magnitude in
the sample (see Fig. 6). In the case with $i<24.4$, $\mathcal{C}(z_{\rm l})$
is close to that from the calibration sample, while for $i<23.4$ we derived
higher completeness, on average. In addition, as the bulk of the redshift
distribution in the calibration sample, described in Sect. 2, extends up to
$z_{\rm g}\sim 4$, we applied the $griz$ selection to the galaxy sample with
redshift $z_{\rm g}<1.5$, $({\rm S/N})_{\rm E}>3$, and $i\leq i_{\rm max}$. In
Fig. 6, we can see that this redshift limit provides $\mathcal{F}(z_{\rm l})$
values that are identical to those derived from the calibration sample, which
is expected since $\mathcal{F}(z_{\rm l})$ does not depend on the maximum
redshift of the sample, while the completeness increases by up to $10$ percent
points and the purity is at most 1 percent point lower. We note that the
computation of $\mathcal{C}(z_{\rm l})$ and $\mathcal{F}(z_{\rm l})$ is made
relative to the sample under consideration. In other words, they refer to
galaxy populations defined by given magnitude and redshift limits. We measured
the aforementioned colour selections by assuming a $z_{\rm l}$ precision of
$\delta z_{\rm l}=10^{-3}$. This $\delta z_{\rm l}$ value is one order of
magnitude lower (i.e. one order of magnitude higher precision) than the
typical galaxy cluster photometric redshift uncertainty in current surveys
(see, e.g. Rykoff et al. 2016; Maturi et al. 2019) and Euclid (Euclid
Collaboration: Adam et al. 2019). Consequently, the $\delta z_{\rm l}=10^{-3}$
precision ensures the reliability of the colour condition fits for galaxy
cluster background selections. We remark that we assumed $\delta z_{\rm
l}=0.1$ for the $z_{\rm l}$ sampling in the calibration process.
In Fig. 6 we show the efficiency of the $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ selection, computed
by adopting $\delta z_{\rm l}=10^{-3}$, applied to the B20 calibration sample,
with $({\rm S/N})_{\rm E}>3$ and $i\leq i_{\rm max}$. We found analogous
selections from the subsample with $i<23.4$ and from the one with $({\rm
S/N})_{\rm E}>10$. Specifically, in both cases, we derived higher
$\mathcal{P}(z_{\rm l})$ and lower $\mathcal{F}(z_{\rm l})$, in agreement with
what we found from the $griz$ selection. In addition, the increase in the
minimum Euclid ${\rm S/N}$ does not significantly change the completeness,
while the $i<23.4$ limit decreases $\mathcal{C}(z_{\rm l})$ by at most $18$
percent points. As we obtained excellent $\mathcal{P}(z_{\rm l})$ and
$\mathcal{F}(z_{\rm l})$ estimates from these tests, we conclude that both
$griz$ and $grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selections are stable and reliable with respect
to changes in the sample limiting magnitude and redshift. In addition, we note
that brighter galaxy samples provide lower foreground contamination. This is
expected, as faint galaxies have more scattered colour-redshift relations.
In Fig. 6 we show the density of background galaxies, $n_{\rm b}(z_{\rm l})$,
defined as the number of selected galaxies with $z_{\rm g}>z_{\rm l}$ per
square arcmin. For both $griz$ and $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ selections, $n_{\rm
b}(z_{\rm l})=16$ arcmin-2 at $z_{\rm l}=0.2$ for $i\leq i_{\rm max}$ and
$({\rm S/N})_{\rm E}>3$, decreasing with increasing $z_{\rm l}$. In both
colour selections, the $i<23.4$ limit implies the largest decrease in $n_{\rm
b}(z_{\rm l})$, providing $n_{\rm b}(z_{\rm l})<7$ arcmin-2. In addition, for
the $griz$ selection, the $i<24.4$ and $z_{\rm g}<1.5$ limits provide
consistent results on $n_{\rm b}(z_{\rm l})$, showing a difference of at most
3 arcmin-2 compared to that derived from the calibration sample. With regard
to the $grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection, the $({\rm S/N})_{\rm E}>10$ limit
implies a decrease in $n_{\rm b}(z_{\rm l})$ of up to 5 arcmin-2 at low
$z_{\rm l}$, while $n_{\rm b}(z_{\rm l})$ becomes compatible with that derived
from the calibration sample for $z_{\rm l}>1$.

###  4.3 $griz$ selection validation on real data

Figure 8: Application of the calibrated colour selections to the Flagship
simulated sample described in Sect. 4.4. From left to right: completeness,
purity, foreground failure rate, and background density as a function of
$z_{\rm l}$, from the fitted colour selections based on $griz$ and
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ bands, adopting 5$\sigma$ magnitude limits. The
assumed $z_{\rm l}$ precision is $\delta z_{\rm l}=10^{-3}$. The solid blue
and solid black lines represent the $griz$ and $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ selections,
respectively, applied to the B20 catalogue. The dashed red and dashed green
curves represent the $griz$ and $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ selections,
respectively, applied to the Flagship v2.1 catalogue (Euclid Collaboration in
prep.).

To further assess the reliability of the $griz$ colour selection detailed in
Sect. 4.1, we applied it to external datasets obtained from real observations.
In particular, we considered the VIMOS Public Extragalactic Redshift Survey
(VIPERS; Guzzo et al. 2014) Multi-Lambda Survey (VMLS) photometric catalogue
by Moutard et al. (2016), including Canada-France-Hawaii Telescope Legacy
Survey (CFHTLS; Hudelot et al. 2012) $griz$ Kron aperture magnitudes (Kron
1980). This catalogue covers $22$ deg2 and provides reliable photometric
redshifts for more than one million galaxies with a typical accuracy of
$\sigma_{z}\leq 0.04$, and a fraction of catastrophic failures lower than
$2\%$ down to $i\sim 23$. These statistics are based on VIPERS data,
complemented with the most secure redshifts selected from other spectroscopic
surveys. We remind that in VIPERS a colour-colour pre-selection was employed
to enhance the effective sampling of the VIMOS spectrograph. Nevertheless, the
VIPERS selection does not introduce any significant colour bias above z $\sim
0.6$ (Guzzo et al. 2014). In addition, as we shall see in the following, the
selection completeness and purity obtained from the VMLS dataset do not
exhibit remarkable deviations from those obtained from other galaxy samples.
In Fig. 7 we can see that, by applying the $griz$ selection to the VMLS
sample, we derived higher $\mathcal{P}(z_{\rm l})$ and lower
$\mathcal{F}(z_{\rm l})$ compared to what we found from the B20 catalogue, on
average. This agrees with what we found in Sect. 4.2, as the Moutard et al.
(2016) catalogue is shallower than the B20 sample. For the same reason,
$n_{\rm b}(z_{\rm l})$ is $3$ arcmin-2 lower, on average. In addition, the
completeness is up to $8$ percent points higher for $z_{\rm l}<0.6$, becoming
lower for higher $z_{\rm l}$ values.
We also applied the $griz$ selection to the COSMOS CLASSIC catalogue by Weaver
et al. (2022, COSMOS20), which reaches the same photometric redshift precision
as COSMOS15, namely $\sigma_{z}/(1+z)=0.007$, at almost one magnitude deeper.
We considered $griz$ Kron, $None$ and $None$ aperture magnitudes from HSC. In
addition, we selected galaxies with a photometric redshift derived from at
least 30 bands, and with $i<25$, in order to consider a sample with highly
reliable redshift estimates. By adopting more complex selection criteria,
which may involve galaxies with photometric redshifts derived from a shared
set of photometric bands, we do not expect remarkable differences in the
results. Similar results for the cases with Kron, $None$ and $None$ aperture
magnitudes are shown in Fig. 7. Compared to what we derived from the B20
sample, the completeness is similar, with the largest differences at $z_{\rm
l}>0.6$. In addition, $\mathcal{F}(z_{\rm l})$ is lower and
$\mathcal{P}(z_{\rm l})$ is higher for any $z_{\rm l}$. For Kron and $None$
aperture magnitudes, $n_{\rm b}(z_{\rm l})$ is slightly higher compared to
that obtained from the B20 sample, on average. Lower $n_{\rm b}(z_{\rm l})$
values show up for the $None$ aperture magnitudes, which is expected as we
applied the same magnitude limit for each photometric aperture definition.
Indeed, for these tests we did not include aperture correction terms. Lastly,
comparing the purity derived from the COSMOS20 and VMLS samples, we note that
for $z_{\rm l}>0.3$ the differences are below 1 percent point, on average.
Thus, we conclude that the $griz$ selection provides robust and reliable
results on real data.

###  4.4 Validation on Flagship v2.1

We tested the colour selections calibrated in Sect. 4.1 on the Euclid Flagship
galaxy catalogue v2.1.10 (Euclid Collaboration in prep.), which is currently
the best simulated Euclid galaxy catalogue available. This catalogue is based
on an $N$-body simulation with around 4 trillion particles with mass $m_{\rm
p}\sim 10^{9}$ $h^{-1}$M⊙. A flat $\Lambda$ cold dark matter ($\Lambda$CDM)
cosmological model was assumed, with matter density parameter $\Omega_{\rm
m}=0.319$, baryon density parameter $\Omega_{\rm b}=0.049$, dark energy
density parameter $\Omega_{\Lambda}=0.681$, scalar spectral index $n_{\rm
s}=0.96$, Hubble parameter $h=H_{0}/(100\,\text{ km s${}^{-1}$
Mpc${}^{-1}$})=0.67$, and standard deviation of linear density fluctuations on
8 $h^{-1}$Mpc scales $\sigma_{8}=0.83$. The haloes were identified using
Rockstar (Behroozi et al. 2013), and then populated with a halo occupation
distribution model which was calibrated to reproduce observables such as
clustering statistics as a function of galaxy luminosity. The galaxy SED
templates used are the COSMOS templates from Ilbert et al. (2009), based on
the models by Bruzual & Charlot (2003) and Polletta et al. (2007). In
addition, galaxy photo-$z$ probability distribution functions, namely
$p(z_{\rm g})$, are included in Flagship, derived through a Nearest Neighbours
Photometric Redshifts (NNPZ) pipeline (Euclid Collaboration: Desprez et al.
2020).
From the Flagship catalogue, we extracted a lightcone within ${\rm
RA}\in[$$,$$]$ and ${\rm Dec}\in[$$,$$]$, considering the galaxies in the
whole redshift range covered by the simulation, namely $z_{\rm g}\in[0,3]$.
Specifically, $z_{\rm g}$ is the galaxy true redshift, and we verified that
the contribution of peculiar velocities does not significantly change the
results. We focused on $None$ aperture LSST $ugrizy$ and Euclid
$I_{\scriptscriptstyle\rm E}Y_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ photometry, as the
simulated fluxes estimated for other ground-based surveys do not account for
observational noise. Specifically, the photometric noise takes into account
the depth expected in the southern hemisphere at the time of the third data
release (DR3) for the Euclid Wide Survey. The LSST and Euclid $10\sigma$
magnitude limits, which are proxies for extended sources, correspond to
$u<24.4$, $g<25.6$, $r<25.7$, $i<25.0$, $z<24.3$, $y<23.1$,
$I_{\scriptscriptstyle\rm E}<25$, $Y_{\scriptscriptstyle\rm E}<23.5$,
$J_{\scriptscriptstyle\rm E}<23.5$, and $H_{\scriptscriptstyle\rm E}<23.5$.
The fluxes we considered are not reddened due to Milky Way extinction,
consistent with the analyses performed in the previous sections.

Figure 9: Comparison of colour and photo-$z$ selections. From left to right:
completeness, purity, foreground failure rate, and background density as a
function of $z_{\rm l}$, obtained from Flagship v2.1. The solid black lines
represent the $grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection. The dashed green lines show the
combination, through the $\lor$ logical operator, of
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ and photo-$z$ selection (Eq. 9). The dashed red
lines represent the photo-$z$ selection, while the dotted black lines
represent the selection based on the $p(z_{\rm g})$ mode.

In Fig. 8, we show the application of $griz$ and $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ selections to
Flagship. For this test, we assumed 5$\sigma$ magnitude cuts for LSST $ugrizy$
and Euclid $I_{\scriptscriptstyle\rm E}Y_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ bands. In addition,
we show results from the B20 sample in Fig. 8, for which we assumed 5$\sigma$
magnitude cuts rescaled from the 10$\sigma$ limits listed in Euclid
Collaboration: Bisigello et al. (2023, Table 1). We found that $n_{\rm
b}(z_{\rm l})$ derived from Flagship agrees with that obtained from the B20
sample. The largest differences, of about 1 arcmin-2, arise when the
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection is applied. We note that $n_{\rm
b}(z_{\rm l})\sim 0$ for $z_{\rm l}\sim 1.5$, implying that lenses at these
values of $z_{\rm l}$ may not exhibit significant weak-lensing signals.
Nevertheless, we verified that $n_{\rm b}(z_{\rm l})$ is enhanced at any
$z_{\rm l}$ when the selection defined for Euclid weak-lensing analyses
(Laureijs et al. 2011; Euclid Collaboration: Scaramella et al. 2022) is
assumed. This selection consists in a 10$\sigma$ cut in the
$I_{\scriptscriptstyle\rm E}$ band, corresponding to $I_{\scriptscriptstyle\rm
E}<25$ for a $None$ aperture, yielding a galaxy density of around 39 arcmin-2
when applied to the Flagship dataset. In fact, in this case $n_{\rm b}(z_{\rm
l})$ ranges from 30 arcmin-2 at low $z_{\rm l}$ to 3 arcmin-2 at $z_{\rm
l}=1.5$.
As shown in Fig. 8, on average we obtained higher $\mathcal{P}(z_{\rm l})$ and
lower $\mathcal{F}(z_{\rm l})$ for $z_{\rm l}<1$ from Flagship, compared to
what we derived from the B20 sample. For the $griz$ selection case,
$\mathcal{C}(z_{\rm l})$ agrees with that derived from the B20 sample, with
the largest differences, of up to 16 percent points, at $z_{\rm l}\sim 0.5$.
Larger differences in $\mathcal{C}(z_{\rm l})$ are obtained from the
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection. From Flagship we obtained
$\mathcal{C}(z_{\rm l})$ up to $10$ percent points larger for $z_{\rm l}<0.6$,
and up to $50$ percent points larger for higher $z_{\rm l}$. We verified that
this discrepancy in the completeness, in the case of the
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection, is not significantly attenuated
through the assumption of 3$\sigma$ and 10$\sigma$ magnitude limits on both
B20 and Flagship catalogues. Analogous results were obtained by assuming
limits corresponding to the magnitude distribution peaks derived from the B20
catalogue, namely $g<24.9$, $r<24.6$, $i<24.3$, $z<24.1$,
$Y_{\scriptscriptstyle\rm E}<23.8$, $J_{\scriptscriptstyle\rm E}<23.6$, and
$H_{\scriptscriptstyle\rm E}<23.5$. Moreover, we verified that the
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection completeness does not remarkably vary
by assuming the Euclid weak-lensing selection defined above, namely
$I_{\scriptscriptstyle\rm E}<25$. Similar results are obtained by considering
the photometric errors expected for the DR2 of the Euclid Wide Survey,
assuming the corresponding 3$\sigma$, 5$\sigma$, and 10$\sigma$ magnitude
cuts. For each of the alternative magnitude cuts discussed in this section, we
found that the $grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection yields a purity up to 3 percent
points higher at $z_{\rm l}>1.2$ when it is applied to the B20 catalogue,
compared to what is derived from Flagship. At $z_{\rm l}<1.2$, instead, the
purity obtained from B20 is 1 percent point lower, on average. Furthermore,
the alternative magnitude cuts do not remarkably impact the selection purity
at any $z_{\rm l}$.
We additionally adopted SDSS fluxes, which do not include photometric noise,
in place of LSST fluxes in Flagship. In this case, the completeness is up to
35 percent points larger than that derived from the B20 sample, and the purity
approaches 100% for $z_{\rm l}>1$, which is similar to what we derived from
the B20 sample (see Fig. 8). Thus, the selection based on SDSS photometry is
less complete and purer compared to that obtained from LSST magnitudes.
Differences in the completeness derived from the Flagship and B20 samples may
originate from distinct assumptions on the physical properties of the
galaxies, such as dust extinction, stellar age, nebular emission lines, or on
the assumed intrinsic spectral energy distributions. This could be indicated
by a different fraction of star forming galaxies in the two samples. Following
B20, galaxies are classified as star forming if the following condition is
satisfied,

$\log_{10}({\rm sSFR}/{\rm yr^{-1}})>-10.5,$ (8)

where sSFR is the specific star formation rate, derived from the best SED
template in the catalogue by B20. We verified that, for $z_{\rm g}>1$, the
fraction of star forming galaxies in Flagship is consistent within 1 percent
point with that derived from the catalogue by B20. Thus, we conclude that the
completeness differences between the Flagship and B20 samples are not due to
different star forming galaxy populations. We also verified that the
$\log_{10}({\rm sSFR}/{\rm yr^{-1}})$ distributions derived from the two
datasets are compatible, having peaks at $\sim-8.13$ and $\sim-8.35$ in B20
and Flagship, respectively. The agreement of these peak values is well within
1$\sigma$ of the $\log_{10}({\rm sSFR}/{\rm yr^{-1}})$ distributions. We will
be able to further investigate such completeness differences through the
analysis of the first data release of the Euclid Deep Survey.

###  4.5 Comparison with photo-$z$ selections

To compare the colour selections derived in this work to selections based on
the galaxy $p(z_{\rm g})$, commonly referred to as photo-$z$ selections, we
analysed the Flagship sample described in Sect. 4.4. We considered only the
galaxies with a $p(z_{\rm g})$ estimate obtained with the NNPZ pipeline
(Euclid Collaboration: Desprez et al. 2020). The NNPZ photo-$z$s are designed
to work well for galaxies that are expected to be used in core Euclid weak-
lensing science, namely with 5$\sigma$ limits on the $I_{\scriptscriptstyle\rm
E}$ band. Thus we imposed $I_{\scriptscriptstyle\rm E}<25.75$, along with
5$\sigma$ limits on the $Y_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ bands, namely $Y_{\scriptscriptstyle\rm
E}<24.25$, $J_{\scriptscriptstyle\rm E}<24.25$, $H_{\scriptscriptstyle\rm
E}<24.25$. Specifically, we adopted the following photo-$z$ selection,

$z_{\rm g}^{\rm min}>z_{\rm l}\,,$ (9)

where $z_{\rm g}^{\rm min}$ is the minimum of the interval containing 95% of
the probability around the first mode of $p(z_{\rm g})$, namely
$\overline{z}_{\rm g}$. We chose $z_{\rm g}^{\rm min}$ in order to derive
$\mathcal{P}(z_{\rm l})$ values which are compatible with those obtained from
colour selections. We verified that adding a condition on the width of
$p(z_{\rm g})$ in Eq. (9) does not impact the results. Specifically, for the
latter test, we considered the additional condition
$\mathcal{A}>\mathcal{A}_{\rm min}$, where $\mathcal{A}$ is the integrated
probability around $\overline{z}_{\rm g}$, computed within the redshift
points, which are the closest to $\overline{z}_{\rm g}$, having an associated
probability of $0.2p(\overline{z}_{\rm g})$. We verified that imposing
$\mathcal{A}_{\rm min}=0$ or $\mathcal{A}_{\rm min}=0.8$ leads to compatible
purity values with sub-percent differences on average. However,
$\mathcal{A}_{\rm min}=0.8$ lowers the photo-$z$ selection completeness by
around 20 percent points at all $z_{\rm l}$. Consequently, we assumed
$\mathcal{A}_{\rm min}=0$.
To perform a fair comparison of colour and photo-$z$ selections, we considered
only the $grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ colour selection in this section. This is
because photo-$z$s in Flagship were derived from the combination of ground-
based and Euclid photometry. In Fig. 9, we show that the
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection provides, on average, a completeness
15 percent points lower than that of the photo-$z$ selection, with similar
contamination. By combining $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ and photo-$z$
selections, through the logical operator $\lor$, the completeness increases by
up to 10 percent points with respect to the case of photo-$z$ selection alone,
amounting to $\mathcal{C}(z_{\rm l})\sim 95\%$ for $z_{\rm l}<1.4$. These
preliminary tests confirm the importance of the combination of colour and
photo-$z$ selections, as it leads to significantly more complete background
galaxy samples. We also remark that increasing the selection completeness is
key to reduce biases in the shear calibration parameters due to background
selections, as we shall detail in Sect. 4.6. The analysis of Euclid data will
allow for a detailed investigation of the optimal photo-$z$ selections for
galaxy cluster weak-lensing analyses, outlining the synergies with colour
selections. For example, colour selections applied to Euclid data could
provide more robust background samples for massive or nearby galaxy clusters,
as derived by Medezinski et al. (2018). Leveraging colour selections also
serves as a valuable cross-validation method for addressing the effect of
unknown systematic uncertainties in photo-$z$ estimates. Lastly, Fig. 9 shows
the selection based only on the first mode of $p(z_{\rm g})$. Specifically, in
this case we selected the galaxies with $\overline{z}_{\rm g}>z_{\rm l}$.
Despite $\mathcal{C}(z_{\rm l})>90\%$ at all $z_{\rm l}$, the purity is up to
$10$ percent points lower than that obtained from the
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection.

###  4.6 Impact on shear measurements

Figure 10: Colour selection results, obtained from the B20 catalogue, in case
of missing $z$ (solid red), $i$ (solid green), $r$ (dashed orange), and $g$
(dashed grey) bands. The blue curves represent the results from the $griz$ and
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selections reported in Tables 2 and 3. Top
panels, from left to right: completeness, purity, foreground failure rate, and
background density are shown, in the case of ground-only photometry. Middle
panels: colour selections from the combination of ground-based and Euclid
photometry. The plot structure is analogous to that of top panels. Bottom
panels: difference between $\mathcal{C}^{nf}_{\rm tot}$ and $\mathcal{C}$, for
ground-only observations (left panel) and for the combination of ground-based
and Euclid photometry (right panel).

In cluster weak-lensing analyses, the inclusion of foreground sources in the
shear measurements may significantly dilute the signal (Broadhurst et al.
2005; Medezinski et al. 2007; Sifón et al. 2015; McClintock et al. 2019). As
discussed in the previous sections, the calibrated colour selections provide
$\mathcal{P}(z_{\rm l})<1$. To assess the impact of impurities on shear
measurements, we can express the cluster reduced tangential shear unaffected
by contamination as follows (Dietrich et al. 2019):

$g_{\rm t,true}(z_{\rm l})=\frac{g_{\rm t}(z_{\rm l})}{\mathcal{P}(z_{\rm
l})}\,,$ (10)

where $g_{\rm t}(z_{\rm l})$ is the measured cluster reduced tangential shear
at redshift $z_{\rm l}$. As the calibrated colour selections yield
$\mathcal{P}(z_{\rm l})>0.97$, we expect at most a 3% bias on the reduced
tangential shear. In addition, as discussed in Sect. 4.3, $\mathcal{P}(z_{\rm
l})$ derived from different observed datasets with only ground-based
photometry shows a scatter below 1 percent point. This scatter in
$\mathcal{P}(z_{\rm l})$ is lower than the systematic uncertainty on galaxy
shape measurements for stage III surveys, as we shall discuss in the
following. We remark that $\mathcal{P}(z_{\rm l})$ is derived from reference
fields, while galaxy clusters are overdense compared to the cosmic mean. Thus,
contamination from cluster galaxies must be properly accounted for in Eq. (10)
(see, e.g. Gruen et al. 2014; Dietrich et al. 2019). Nevertheless, such
contamination is consistent with zero in the typical cluster-centric radial
range adopted for mass calibration, namely at radii larger than 300
$h^{-1}$kpc (see, e.g. Medezinski et al. 2018; Bellagamba et al. 2019).
Furthermore, galaxy shear calibration is usually statistically derived, based
on observed and simulated galaxy samples. Nevertheless, through galaxy cluster
background selections, some galaxy populations may be systematically excluded.
This may invalidate the statistical estimate of the shape multiplicative bias,
namely $m$, depending on the shear measurement technique and on the actual
properties of the data (Heymans et al. 2012; Miller et al. 2013; Hildebrandt
et al. 2016).
The typical uncertainty on $m$ found for stage III surveys ranges in the
interval $\delta m\in[1\times 10^{-2},3\times 10^{-2}]$ (see, e.g. Jarvis et
al. 2016; Melchior et al. 2017; Giblin et al. 2021). To assess the impact of
colour selections on $m$, we considered the shape catalogues of Heymans et al.
(2012), based on CFHTLS, and of Mandelbaum et al. (2018), based on the HSC
Subaru Strategic Program (HSC-SSP; Miyazaki et al. 2018; Aihara et al. 2018).
Throughout this section, we adopted a lens redshift of $z_{\rm l}=0.5$. By
applying the $griz$ selection calibrated in this work, we derived a shift in
the mean shear multiplicative bias of $\Delta m=7\times 10^{-3}$ in CFHTLS and
of $\Delta m=-2\times 10^{-3}$ in HSC-SSP. In addition, the Oguri et al.
(2012) and Medezinski et al. (2018) colour selections provide $\Delta
m=-3\times 10^{-3}$ and $\Delta m=-1\times 10^{-2}$ from CFHTLS, respectively,
while from HSC-SSP we obtained $\Delta m=-5\times 10^{-3}$ and $\Delta
m=-7\times 10^{-3}$, respectively. Thus, galaxy population differences due to
colour cuts provide systematic effects that are within the typical $m$
uncertainty in stage III surveys. By combining colour and photo-$z$
selections, we expect $\Delta m$ to become closer to zero. In Euclid-like
surveys, shear has to be calibrated within an accuracy of $2\times 10^{-3}$
(Cropper et al. 2013). As we discussed in Sect. 4.5, the combination of
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ and photo-$z$ selections leads to 90%
background completeness in the Euclid Wide Survey, on average; thus, we may
expect the bias on $m$ to be subdominant with respect to the required shear
accuracy. Indeed, let us assume that 90% of galaxies, selected through the
combination of $grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ and photo-$z$ selections, have an average $m$
similar to that derived from stage III surveys, namely $\langle
m\rangle=0.01$. We assume that the remaining 10% of galaxies have a very
biased $m$, namely $\langle m\rangle=0.02$, compared to the selected
population. This would imply a systematic error of $\Delta m=10^{-3}$ in the
average $m$ of the selected population. We will delve deeper into these
variations in $m$ by examining the first data releases of the Euclid surveys.

###  4.7 Selection efficiency with missing bands

Figure 11: Comparison of the $griz$ selection described in Sect. 4.1 with
literature colour selections. From left to right: completeness, purity,
foreground failure rate, and background density, derived from the B20
catalogue. The blue solid lines represent the $griz$ selection derived in this
work. The red curves refer to the Medezinski et al. (2010) selection, where
the solid lines are given by Eqs. (5) – (5), while the dashed lines are given
by Eq. (5). The green curves represent the results from the Oguri et al.
(2012) selection, where the solid lines are given by Eqs. (13) – (16), while
the dashed lines are given by Eqs. (13) – (15). The grey curves refer to the
Medezinski et al. (2018) selection, where the solid lines are given by Eqs.
(5) – (5), while the dashed lines are given by Eq. (5).

In this work, we derived colour selections based on $griz$ and
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ photometry. In some cases, however, the full
ground-based $griz$ photometry may be not available. For example, the DES Year
3 galaxy shape catalogue was not based on $g$ band (Gatti et al. 2021), due to
issues in the point spread function estimation (Jarvis et al. 2021). Thus, we
investigated the efficiency of $griz$ and $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ selections in the
case of a missing band, based on the B20 calibration sample described in Sect.
2. In performing this test, we excluded the colour conditions in Tables 2 and
3 containing the chosen missing bands. In Fig. 10 we show that, in the case of
ground-only observations, the absence of the $r$ band implies the largest
completeness decrease, providing $\mathcal{C}(z_{\rm l})<60\%$. In addition,
the $z_{\rm l}$ range is substantially reduced, corresponding to $z_{\rm
l}\in[0.2,0.6]$. Also the absence of $i$ and $z$ bands implies a reduction of
the maximum $z_{\rm l}$ for the ground-based selection, corresponding to
$z_{\rm l}=0.7$ and $z_{\rm l}=0.6$, respectively, and a completeness decrease
of up to 10 and 20 percent points, respectively. On average, a 20 percent
point drop in completeness is found in absence of $g$ band photometry.
Nevertheless, in the latter case the $z_{\rm l}$ range is not reduced. We
remark that the considered samples differ from case to case, as they contain
only galaxies with photometry available in the required bands.
In Fig. 10 we show the effect of missing photometric bands on the combination
of ground-based and Euclid observations. In this case, the lack of $r$ band
does not imply changes in $\mathcal{C}(z_{\rm l})$ for $z_{\rm l}>1$. In the
absence of $i$ band, $\mathcal{C}(z_{\rm l})$ significantly decreases for
$z_{\rm l}\gtrsim 0.7$, being below $30\%$, while the $z_{\rm l}$ range is not
reduced. A $z_{\rm l}$ range reduction is obtained in the case of missing $z$
or $g$ bands, as we derived $z_{\rm l}\in[0.2,1.1]$ and $z_{\rm
l}\in[0.2,1.3]$, respectively. On average, in the case of the combination of
ground-based and Euclid observations, the largest completeness decrease is
caused by the lack of the $g$ band.
In this section, we defined colour selections with missing $g$, $r$, $i$, or
$z$ band, as subsets of the colour conditions defining the $griz$ and
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selections. In order to assess the difference
between the selections defined by such subsets and those that would be derived
from the colour selection calibration described in Sect. 3, we compute
$\mathcal{C}^{nf}_{\rm tot}$ for each case. In Fig. 10, we show the difference
between $\mathcal{C}^{nf}_{\rm tot}$ and $\mathcal{C}$, the latter being
derived by subsets of the colour conditions defining $griz$ and
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selections. In the ground-only case, the lack
of $r$ band provides the largest $\mathcal{C}$ underestimation, as
$\mathcal{C}^{nf}_{\rm tot}-\mathcal{C}\sim 15$ percent points for $z_{\rm
l}\in[0.3,0.5]$. Nevertheless, in case of other missing bands, the average
$\mathcal{C}^{nf}_{\rm tot}-\mathcal{C}$ is close to 0. The same holds for the
combination of ground-based and Euclid photometry. We conclude that $griz$ and
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selections provide robust results in the case
of a missing band, except for ground-only observations without the $r$ band,
for which a dedicated calibration might be needed.

##  5 Comparison with literature ground-based selections

Figure 12: Comparison of colour selections in the $(r-i)$ \- $(g-r)$ colour-
colour space. The solid grey contours indicate the 68%, 95%, and 99% galaxy
number density in the calibration sample, described in Sect. 2. The blue
shaded areas represent the regions excluded by applying the $griz$ selection
calibrated in Sect. 4.1. Completeness and purity of the selections are
reported in the legends. For the $griz$ selection, $\mathcal{C}$ and
$\mathcal{P}$ are computed by considering the colour conditions in Tab. 2
which are defined in the $(r-i)$ \- $(g-r)$ space. Top panel: the red hatched
area shows the region excluded through the Medezinski et al. (2010) selection,
and $z_{\rm l}=0.26$ is assumed. Bottom panel: the green hatched area shows
the region excluded through the Oguri et al. (2012) selection, and $z_{\rm
l}=0.6$ is assumed. Figure 13: From left to right: number of selected galaxies
as a function of $z_{\rm g}$ assuming $z_{\rm l}=0.26$, $z_{\rm l}=0.6$, and
$z_{\rm l}=0.8$. The $z_{\rm l}$ values are represented by vertical black
dashed lines. The blue histograms represent the $griz$ selection calibrated in
this work. The galaxy redshift distributions derived with the Medezinski et
al. (2010), Oguri et al. (2012), and Medezinski et al. (2018) selections are
represented by red, green, and grey hatched histograms, respectively.

Based on the B20 sample considered in Sect. 4.1, we compared our $griz$ colour
selection to those derived by Medezinski et al. (2010), Oguri et al. (2012),
and Medezinski et al. (2018), which are also implemented in COMB-CL. As
detailed below, for each of these selections, we considered two versions. One
includes all the colour conditions provided by the corresponding authors,
while the other comprises only a subsample of such conditions, providing lower
foreground contamination. COMB-CL includes both versions of each colour
selection.
Medezinski et al. (2010) derived colour selections for three massive clusters,
identified through deep Subaru imaging, by maximising their weak-lensing
signal. COMB-CL provides the selection calibrated for the A1703 cluster at
redshift $z_{\rm l}\sim 0.26$, as this is the one based on $gri$ photometry.
This selection is expressed as follows,

$\displaystyle\big{[}\,$ $\displaystyle(g-r)<2.17\,(r-i)-0.37\,\,\land$
$\displaystyle(g-r)<-0.6\,(r-i)+1.85\,\,\land\,\,(r-i)>0.3\,\big{]}\,\,\lor$
(11) $\displaystyle\big{[}\,$
$\displaystyle(g-r)<-0.4\,(r-i)+0.47\,\,\land\,\,(r-i)<0.3\,\big{]}\,\,\lor\,\,$
$\displaystyle(r-i)<-0.06\,,$ (12)

where $\land$ is the logical ‘and’ operator. Notably, Eq. (5) defines the red
background sample, namely the galaxies redder than cluster galaxies, while Eq.
(5) defines the blue background sample. In Fig. 11, we can see that this
selection provides $\mathcal{C}(z_{\rm l})=60\%$, which is $20$ percent points
lower than that provided by the $griz$ selection calibrated in this work. In
addition, from the Medezinski et al. (2010) selection we derived
$\mathcal{P}(z_{\rm l})>96\%$, which is slightly lower than that obtained from
the $griz$ selection discussed in Sect. 4.1. Foreground contamination can be
attenuated by considering the red sample selection only, namely Eq. (5), as
shown in Fig. 11. In this case, however, the completeness is lowered by 20
percent points. In Fig. 12 (upper panel), we show a comparison between the
selection by Medezinski et al. (2010), namely Eqs. (5) and (5), and our $griz$
selection in the $(r-i)$ \- $(g-r)$ colour-colour space, by assuming $z_{\rm
l}=0.26$. Within this colour-colour space, we obtained $\mathcal{C}(z_{\rm
l})=55\%$ and $\mathcal{P}(z_{\rm l})=98\%$ from the $griz$ selection, while
through the Medezinski et al. (2010) selection we found $\mathcal{C}(z_{\rm
l})=62\%$ and $\mathcal{P}(z_{\rm l})=95\%$. We remark that the full set of
colour conditions defining the $griz$ selection yields 80% completeness for
$z_{\rm l}=0.26$, and that a calibration based on $gri$ bands only would yield
larger completeness values in the $(r-i)$ \- $(g-r)$ space (see Sect. 4.7). In
addition, Fig. 12 shows that the $griz$ selection extends the selected region
defined by Eq. (5), thus enhancing the red background sample compared to
Medezinski et al. (2010). On the other hand, the $griz$ selection shows a more
conservative definition of the blue background sample, compared to Eq. (5).
Oguri et al. (2012) calibrated a selection based on $gri$ photometry from the
COSMOS catalogue by Ilbert et al. (2009), providing reliable results for
lenses at redshift $z_{\rm l}\lesssim 0.7$. This selection is expressed as

$\displaystyle(g-r)<0.3\,\,\lor$ (13) $\displaystyle(r-i)>1.3\,\,\lor$ (14)
$\displaystyle(r-i)>(g-r)\,\,\lor$ (15)
$\displaystyle(g-r)>1\,\,\land\,\,(r-i)<0.4\,(g-r)-0.5.$ (16)

The inclusion of Eq. (16) does not provide significant improvement in the
completeness, causing a lower selection purity (Covone et al. 2014). In fact,
Fig. 11 shows that the selection including Eqs. (13) – (16) provides sub-
percent improvements in $\mathcal{C}(z_{\rm l})$, compared to the selection
including Eqs. (13) – (15) only, while $\mathcal{P}(z_{\rm l})$ and
$\mathcal{F}(z_{\rm l})$ are up to 1 percent point lower and higher,
respectively. With respect to the $griz$ selection calibrated in this work,
the Oguri et al. (2012) selection provides a purity which is up to $5$ percent
points lower. This explains the higher completeness values for $z_{\rm
l}>0.6$. For lower $z_{\rm l}$, the Oguri et al. (2012) selection provides a
completeness up to 35 percent points lower, which is expected as the selection
by Oguri et al. (2012) was calibrated for clusters at $z_{\rm l}\sim 0.7$.
Similarly to the comparison with the Medezinski et al. (2010) selection
discussed above, in Fig. 12 (lower panel) we compare the $griz$ and Oguri et
al. (2012) selections in the $(r-i)$ \- $(g-r)$ colour-colour space, assuming
$z_{\rm l}=0.6$. We obtained $\mathcal{C}(z_{\rm l})=27\%$ and
$\mathcal{P}(z_{\rm l})=98\%$ from the $griz$ selection, while with the Oguri
et al. (2012) selection we found $\mathcal{C}(z_{\rm l})=60\%$ and
$\mathcal{P}(z_{\rm l})=94\%$. With respect to what we found by comparing
$griz$ and Medezinski et al. (2010) selections at $z_{\rm l}=0.26$, the
decrease in completeness due to a purity enhancement is much larger at $z_{\rm
l}=0.6$. This depends on the overlap of the foreground and background galaxy
distributions in the $(r-i)$ \- $(g-r)$ space. In addition, we remark that the
calibration process presented in Sect. 3 excludes redundant colour conditions.
This may partially explain the 27% completeness found in the case of the
$griz$ selection.
Medezinski et al. (2018) calibrated a colour selection based on HSC
observations, including $griz$ bands, following an approach similar to
Medezinski et al. (2010). This colour selection is expressed as

$\displaystyle\Bigg{[}\,$ $\displaystyle(g-i)<2.276\,(r-z)-0.152+a(z_{\rm
l})\,\,\land$ $\displaystyle(g-i)<\frac{1}{2.276}\,(r-z)+b(z_{\rm
l})\left(1+\frac{1}{2.276^{2}}\right)-\frac{0.152}{2.276^{2}}\,\,\land$
$\displaystyle(r-z)>0.5\,\,\land$ $\displaystyle z>21\,\Bigg{]}\,\,\lor$ (17)
$\displaystyle\Bigg{\\{}\,$ $\displaystyle(r-z)<-0.0248\,z+1.604+c(z_{\rm
l})\,\,\lor$ $\displaystyle\Bigg{[}\,(g-i)<\frac{1}{2.276}\,(r-z)+d(z_{\rm
l})\left(1+\frac{1}{2.276^{2}}\right)-\frac{0.152}{2.276^{2}}\,\,\land$
$\displaystyle(g-i)<4\,\Bigg{]}\,\,\lor$ $\displaystyle(r-z)<0.5\,\,\lor$
$\displaystyle z>22\,\Bigg{\\}}\,,$ (18)

where

$a(z_{\rm l})=\begin{cases}-0.7\,\,\,\text{if $z_{\rm l}<0.4$}\\\
-0.8\,\,\,\text{if $z_{\rm l}\geq 0.4$}\end{cases},$ (19) $b(z_{\rm
l})=\begin{cases}4.0\,\,\,\text{if $z_{\rm l}<0.4$}\\\ 1.7\,\,\,\text{if
$z_{\rm l}\geq 0.4$}\end{cases},$ (20) $c(z_{\rm
l})=\begin{cases}-0.8\,\,\,\text{if $z_{\rm l}<0.4$}\\\ -0.9\,\,\,\text{if
$z_{\rm l}\geq 0.4$}\end{cases},$ (21) $d(z_{\rm
l})=\begin{cases}0.5\,\,\,\text{if $z_{\rm l}<0.4$}\\\ 0.3\,\,\,\text{if
$z_{\rm l}\geq 0.4$}\end{cases}.$ (22)

Similarly to Medezinski et al. (2010), Eqs. (5) and (5) define the red and
blue background samples, respectively. This selection provides a much larger
amount of contamination compared to that derived in this work, reaching
$\mathcal{P}(z_{\rm l})<90\%$ for $z_{\rm l}>0.7$, as shown in Fig. 11.
However, by considering the red sample selection only, the purity improves by
up to $6$ percent points. In the latter case, compared to the $griz$ selection
detailed in Table 2, $\mathcal{C}(z_{\rm l})$ is up to $50$ percent points
lower for $z_{\rm l}<0.7$. For higher $z_{\rm l}$, the lower purity from the
Medezinski et al. (2018) selection allows for higher completeness values.
In Fig. 13 we show the redshift distributions of the galaxies selected through
the $griz$, Medezinski et al. (2010), Oguri et al. (2012), and Medezinski et
al. (2018) selections, assuming different $z_{\rm l}$ values. At $z_{\rm
l}=0.26$, the $griz$ selection shows a larger number of galaxies, of the order
of 103, which are correctly identified as background objects with $z_{\rm
g}<0.6$, compared to Medezinski et al. (2010). This results in the larger
completeness of the $griz$ selection shown in Fig. 11. At $z_{\rm l}=0.6$,
where the $griz$ and Oguri et al. (2012) selections have the same completeness
(see Fig. 11), the $griz$ selection is less complete at $z_{\rm g}>1.5$ and
more complete at lower redshifts, compared to Oguri et al. (2012). At $z_{\rm
l}=0.8$, where the $griz$ selection is remarkably purer than that by
Medezinski et al. (2018), a notable incompleteness of the $griz$ selection is
evident at any $z_{\rm g}$, compared to Medezinski et al. (2018). In fact, for
the case of the $griz$ selection, Fig. 13 shows that the number of rejected
galaxies at high redshift increases with the number of excluded foreground
galaxies. This reflects the overlapping of foreground and background galaxy
distributions in the considered colour-colour spaces.

##  6 Discussion and conclusions

We developed a method to derive optimal galaxy colour selections for cluster
weak-lensing analyses, given any set of photometric bands. To this aim, we
considered all the available colour-colour combinations. Based on the galaxy
catalogue by B20, we calibrated selections based on ground-based $griz$ and
Euclid $Y_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ bands, with purity higher than $97\%$.
Specifically, we showed that the $griz$ selection provides a completeness
between 30% and 84%, in the lens limiting redshift range $z_{\rm
l}\in[0.2,0.8]$. The inclusion of Euclid $Y_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ bands, leading to a
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection, improves the completeness by up to
$25$ percent points in this $z_{\rm l}$ range, allowing for a galaxy selection
up to $z_{\rm l}=1.5$. In addition, for the first time in the literature, we
expressed such selections as a continuous function of $z_{\rm l}$. In the
following, we summarise the main results obtained from the tests presented in
Sects. 4 and 5.

  * •

The calibrated colour selections, described in Sect. 4.1, are stable with
respect to changes in the sample limiting magnitudes and redshift.

  * •

By applying the $griz$ selection to the VMLS catalogue by Moutard et al.
(2016) and to the COSMOS20 catalogue by Weaver et al. (2022), we derived
completeness and purity estimates that are consistent with those obtained from
the calibration sample by B20. Consequently, the calibrated selections provide
stable results by assuming alternative photometric aperture definitions,
obtained from different ground-based telescopes.

  * •

The application of $griz$ and $grizY_{\scriptscriptstyle\rm
E}J_{\scriptscriptstyle\rm E}H_{\scriptscriptstyle\rm E}$ selections to the
simulated Euclid Flagship galaxy catalogue v2.1.10 (Euclid Collaboration in
prep.) provided a purity of around $99\%$, on average, which is higher than
that obtained from the B20 catalogue. The completeness from the Flagship and
B20 samples is compatible in the $griz$ selection case, while the
$grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection provides up to $50$ percent points
higher completeness from Flagship. We verified that this discrepancy does not
depend on magnitude limits. In addition, we found no significant differences
in the star forming galaxy fraction from the Flagship and B20 samples. A
calibration of the $grizY_{\scriptscriptstyle\rm E}J_{\scriptscriptstyle\rm
E}H_{\scriptscriptstyle\rm E}$ selection based on the Euclid Deep Survey will
allow for a more thorough investigation of these completeness differences.

  * •

Based on the Flagship sample, we combined the calibrated colour selections
with photo-$z$ selections based on the $p(z_{\rm g})$ shape. We showed that in
this case the completeness is up to $95\%$.

  * •

We found no significant systematic effects on the multiplicative shear bias
due to colour selections for stage III surveys. The first Euclid data releases
will provide further insights into the influence of background selections on
this bias.

  * •

The calibrated colour selections provide robust results in the case of a
missing band from ground-based observations, except for those without the $r$
band, for which a dedicated calibration might be needed.

  * •

Compared to the ground-based colour selections provided by Medezinski et al.
(2010), Oguri et al. (2012), and Medezinski et al. (2018), the $griz$
selection derived in this work is purer at high redshift and more complete at
low redshift.

One potential enhancement to the calibration presented in this work could
entail the inclusion of a magnitude dependence in the colour cuts. This might
mitigate the impact of large photometric scatter at faint magnitudes (see,
e.g. Schrabback et al. 2021). In addition, enhancing the set of photometric
bands in the calibration sample, for example by including the LSST $y$ band,
could remarkably improve the effectiveness of the colour selections. The
performance of colour selections could further improve through dedicated
calibration samples. Ongoing spectroscopic programmes are specifically
designed to calibrate the relationship between galaxy colours and redshifts to
match the depth of the Euclid survey (Euclid Collaboration: Saglia et al.
2022).
Furthermore, the colour selections presented in this work could improve the
shear calibration in cluster weak-lensing analyses. The lensing signal from
galaxy clusters differs from that of large scale structure in ways that affect
both shear and photometric measurements. The dense cluster environment causes
increased blending among light sources, due to both galaxy blends (Simet &
Mandelbaum 2015; Everett et al. 2022) and the presence of diffuse intra-
cluster light (Gruen et al. 2019; Kluge et al. 2020; Sampaio-Santos et al.
2021). In addition, cluster lines of sight exhibit characteristically stronger
shear, especially at small scales (McClintock et al. 2019; Ingoglia et al.
2022). These effects can lead to distinct biases in shear measurements
compared to those obtained from calibrations primarily designed for cosmic
shear analyses. Through the combination of colour and photo-$z$ selections,
cluster shear calibration and mass bias can be assessed based on dedicated,
multi-band cluster image simulations (see, e.g. Hernández-Martín et al. 2020).

###### Acknowledgements.

The Euclid Consortium acknowledges the European Space Agency and a number of
agencies and institutes that have supported the development of Euclid, in
particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian
Science Policy, the Canadian Euclid Consortium, the French Centre National
d’Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish
Space Research Institute, the Fundação para a Ciência e a Tecnologia, the
Ministerio de Ciencia e Innovación, the National Aeronautics and Space
Administration, the National Astronomical Observatory of Japan, the
Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the
Romanian Space Agency, the State Secretariat for Education, Research and
Innovation (SERI) at the Swiss Space Office (SSO), and the United Kingdom
Space Agency. A complete and detailed list is available on the Euclid web site
(http://www.euclid-ec.org)
We thank W. Hartley for his valuable advice, which remarkably enhanced the
quality of this work. GC and LM acknowledge the support from the grant ASI
n.2018-23-HH.0. LM and FM acknowledge the financial contribution from the
grant PRIN-MUR 2022 20227RNLY3 “The concordance cosmological model: stress-
tests with galaxy clusters” supported by Next Generation EU. LB acknowledges
financial support from PRIN-MIUR $2017-20173$ML3WW_001. MS acknowledges
financial contributions from contract ASI-INAF n.2017-14-H.0, contract INAF
mainstream project 1.05.01.86.10, and INAF Theory Grant 2023: Gravitational
lensing detection of matter distribution at galaxy cluster boundaries and
beyond (1.05.23.06.17). GC thanks the support from INAF Theory Grant 2022:
Illuminating Dark Matter using Weak Lensing by Cluster Satellites, PI: Carlo
Giocoli.
This work has made use of CosmoHub (Carretero et al. 2017; Tallada et al.
2020). CosmoHub has been developed by the Port d’Informació Científica (PIC),
maintained through a collaboration of the Institut de Física d’Altes Energies
(IFAE) and the Centro de Investigaciones Energéticas, Medioambientales y
Tecnológicas (CIEMAT) and the Institute of Space Sciences (CSIC & IEEC), and
was partially funded by the ”Plan Estatal de Investigación Científica y
Técnica y de Innovación” program of the Spanish government.

## References

  * Abbott et al. (2020) Abbott, T. M. C., Aguena, M., Alarcon, A., et al. 2020, Phys. Rev. D, 102, 023509
  * Aihara et al. (2018) Aihara, H., Armstrong, R., Bickerton, S., et al. 2018, Publ. Astron. Soc. Japan, 70, S8
  * Applegate et al. (2014) Applegate, D. E., von der Linden, A., Kelly, P. L., et al. 2014, MNRAS, 439, 48
  * Baltz et al. (2009) Baltz, E. A., Marshall, P., & Oguri, M. 2009, J. Cosm. Astro-Particle Phys., 2009, 015
  * Behroozi et al. (2013) Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013, ApJ, 762, 109
  * Bellagamba et al. (2019) Bellagamba, F., Sereno, M., Roncarelli, M., et al. 2019, MNRAS, 484, 1598
  * Bisigello et al. (2020) Bisigello, L., Kuchner, U., Conselice, C. J., et al. 2020, MNRAS, 494, 2337
  * Broadhurst et al. (2005) Broadhurst, T., Takada, M., Umetsu, K., et al. 2005, ApJ Lett., 619, L143