11institutetext: Code 665, NASA/GSFC, 8800 Greenbelt Road, Greenbelt, MD
20771, USA 22institutetext: SSAI, Lanham, MD 20706, USA 33institutetext:
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
44institutetext: Center for Space Sciences and Technology, University of
Maryland, Baltimore County, Baltimore, MD 21250, USA 55institutetext: Center
for Research and Exploration in Space Science and Technology, NASA/GSFC,
Greenbelt, MD 20771, USA 66institutetext: Center for Astrophysics — Harvard &
Smithsonian, 60 Garden St., Cambridge, MA 02138, USA 77institutetext:
Departamento de Física Fundamental. Universidad de Salamanca.Plaza de la
Merced s/n. 37008 Salamanca, Spain 88institutetext: INAF-Osservatorio
Astronomico di Roma, Via Frascati 33, 00078 Monteporzio Catone, Italy
99institutetext: INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c/o
Dipartimento di Fisica, Edificio G. Marconi, 00185 Roma, Italy
1010institutetext: Department of Astrophysical Sciences, Peyton Hall,
Princeton University, Princeton, NJ 08544, USA 1111institutetext: ESAC/ESA,
Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo, 28692
Villanueva de la Cañada, Madrid, Spain 1212institutetext: School of
Mathematics and Physics, University of Surrey, Guildford, Surrey, GU2 7XH, UK
1313institutetext: INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122
Milano, Italy 1414institutetext: INAF-Osservatorio di Astrofisica e Scienza
dello Spazio di Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy
1515institutetext: Dipartimento di Fisica e Astronomia, Università di Bologna,
Via Gobetti 93/2, 40129 Bologna, Italy 1616institutetext: INFN-Sezione di
Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy 1717institutetext: Max
Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748
Garching, Germany 1818institutetext: Universitäts-Sternwarte München, Fakultät
für Physik, Ludwig-Maximilians-Universität München, Scheinerstrasse 1, 81679
München, Germany 1919institutetext: Dipartimento di Fisica, Università di
Genova, Via Dodecaneso 33, 16146, Genova, Italy 2020institutetext: INFN-
Sezione di Genova, Via Dodecaneso 33, 16146, Genova, Italy 2121institutetext:
Department of Physics ”E. Pancini”, University Federico II, Via Cinthia 6,
80126, Napoli, Italy 2222institutetext: INAF-Osservatorio Astronomico di
Capodimonte, Via Moiariello 16, 80131 Napoli, Italy 2323institutetext: INFN
section of Naples, Via Cinthia 6, 80126, Napoli, Italy 2424institutetext:
Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP,
Rua das Estrelas, PT4150-762 Porto, Portugal 2525institutetext: Dipartimento
di Fisica, Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino,
Italy 2626institutetext: INFN-Sezione di Torino, Via P. Giuria 1, 10125
Torino, Italy 2727institutetext: INAF-Osservatorio Astrofisico di Torino, Via
Osservatorio 20, 10025 Pino Torinese (TO), Italy 2828institutetext: INAF-IASF
Milano, Via Alfonso Corti 12, 20133 Milano, Italy 2929institutetext: Institut
de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and
Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain 3030institutetext:
Port d’Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra
(Barcelona), Spain 3131institutetext: Institute for Theoretical Particle
Physics and Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany
3232institutetext: Dipartimento di Fisica e Astronomia ”Augusto Righi” - Alma
Mater Studiorum Università di Bologna, Viale Berti Pichat 6/2, 40127 Bologna,
Italy 3333institutetext: Institute for Astronomy, University of Edinburgh,
Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK 3434institutetext:
Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester M13 9PL, UK
3535institutetext: European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044
Frascati, Roma, Italy 3636institutetext: University of Lyon, Univ Claude
Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, 69622 Villeurbanne, France
3737institutetext: Institute of Physics, Laboratory of Astrophysics, Ecole
Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290
Versoix, Switzerland 3838institutetext: UCB Lyon 1, CNRS/IN2P3, IUF, IP2I
Lyon, 4 rue Enrico Fermi, 69622 Villeurbanne, France 3939institutetext:
Departamento de Física, Faculdade de Ciências, Universidade de Lisboa,
Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal 4040institutetext:
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal
4141institutetext: Department of Astronomy, University of Geneva, ch. d’Ecogia
16, 1290 Versoix, Switzerland 4242institutetext: INAF-Istituto di Astrofisica
e Planetologia Spaziali, via del Fosso del Cavaliere, 100, 00100 Roma, Italy
4343institutetext: INFN-Padova, Via Marzolo 8, 35131 Padova, Italy
4444institutetext: Univ Claude Bernard Lyon 1, CNRS, IP2I Lyon, UMR 5822,
69622 Villeurbanne, France 4545institutetext: Université Paris-Saclay,
Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France
4646institutetext: INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo
11, 34143 Trieste, Italy 4747institutetext: Istituto Nazionale di Fisica
Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy
4848institutetext: INAF-Osservatorio Astronomico di Padova, Via
dell’Osservatorio 5, 35122 Padova, Italy 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: Max-Planck-Institut für Astronomie,
Königstuhl 17, 69117 Heidelberg, Germany 5656institutetext: Department of
Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu 2, 00014
University of Helsinki, Finland 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:
Dipartimento di Fisica ”Aldo Pontremoli”, Università degli Studi di Milano,
Via Celoria 16, 20133 Milano, Italy 6565institutetext: INFN-Sezione di Milano,
Via Celoria 16, 20133 Milano, Italy 6666institutetext: Universität Bonn,
Argelander-Institut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany
6767institutetext: Aix-Marseille Université, CNRS, CNES, LAM, Marseille,
France 6868institutetext: Dipartimento di Fisica e Astronomia ”Augusto Righi”
- Alma Mater Studiorum Università di Bologna, via Piero Gobetti 93/2, 40129
Bologna, Italy 6969institutetext: Department of Physics, Institute for
Computational Cosmology, Durham University, South Road, DH1 3LE, UK
7070institutetext: Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex
4, France 7171institutetext: Institut d’Astrophysique de Paris, UMR 7095,
CNRS, and Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France
7272institutetext: Université Paris Cité, CNRS, Astroparticule et Cosmologie,
75013 Paris, France 7373institutetext: Institut d’Astrophysique de Paris,
98bis Boulevard Arago, 75014, Paris, France 7474institutetext: European Space
Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
7575institutetext: Department of Physics and Astronomy, University of Aarhus,
Ny Munkegade 120, DK-8000 Aarhus C, Denmark 7676institutetext: Centre for
Astrophysics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
7777institutetext: Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario N2L 3G1, Canada 7878institutetext: Perimeter
Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
7979institutetext: Space Science Data Center, Italian Space Agency, via del
Politecnico snc, 00133 Roma, Italy 8080institutetext: Centre National d’Etudes
Spatiales – Centre spatial de Toulouse, 18 avenue Edouard Belin, 31401
Toulouse Cedex 9, France 8181institutetext: Institute of Space Science, Str.
Atomistilor, nr. 409 Măgurele, Ilfov, 077125, Romania 8282institutetext:
Dipartimento di Fisica e Astronomia ”G. Galilei”, Università di Padova, Via
Marzolo 8, 35131 Padova, Italy 8383institutetext: Departamento de Física,
FCFM, Universidad de Chile, Blanco Encalada 2008, Santiago, Chile
8484institutetext: Universität Innsbruck, Institut für Astro- und
Teilchenphysik, Technikerstr. 25/8, 6020 Innsbruck, Austria 8585institutetext:
Institut d’Estudis Espacials de Catalunya (IEEC), Carrer Gran Capitá 2-4,
08034 Barcelona, Spain 8686institutetext: Institute of Space Sciences (ICE,
CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain
8787institutetext: Satlantis, University Science Park, Sede Bld 48940, Leioa-
Bilbao, Spain 8888institutetext: Centro de Investigaciones Energéticas,
Medioambientales y Tecnológicas (CIEMAT), Avenida Complutense 40, 28040
Madrid, Spain 8989institutetext: Infrared Processing and Analysis Center,
California Institute of Technology, Pasadena, CA 91125, USA 9090institutetext:
Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal
9191institutetext: Universidad Politécnica de Cartagena, Departamento de
Electrónica y Tecnología de Computadoras, Plaza del Hospital 1, 30202
Cartagena, Spain 9292institutetext: Institut de Recherche en Astrophysique et
Planétologie (IRAP), Université de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard
Belin, 31400 Toulouse, France 9393institutetext: INFN-Bologna, Via Irnerio 46,
40126 Bologna, Italy 9494institutetext: IFPU, Institute for Fundamental
Physics of the Universe, via Beirut 2, 34151 Trieste, Italy 9595institutetext:
INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129 Bologna, Italy
9696institutetext: Instituto de Astrofísica de Canarias, Calle Vía Láctea s/n,
38204, San Cristóbal de La Laguna, Tenerife, Spain 9797institutetext: Center
for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 10010, New
York, NY, USA 9898institutetext: School of Physics and Astronomy, Cardiff
University, The Parade, Cardiff, CF24 3AA, UK 9999institutetext: Centre de
Calcul de l’IN2P3/CNRS, 21 avenue Pierre de Coubertin 69627 Villeurbanne
Cedex, France 100100institutetext: Institut für Theoretische Physik,
University of Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
101101institutetext: Université St Joseph; Faculty of Sciences, Beirut,
Lebanon 102102institutetext: Junia, EPA department, 41 Bd Vauban, 59800 Lille,
France 103103institutetext: SISSA, International School for Advanced Studies,
Via Bonomea 265, 34136 Trieste TS, Italy 104104institutetext: INFN, Sezione di
Trieste, Via Valerio 2, 34127 Trieste TS, Italy 105105institutetext: ICSC -
Centro Nazionale di Ricerca in High Performance Computing, Big Data e Quantum
Computing, Via Magnanelli 2, Bologna, Italy 106106institutetext: Instituto de
Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain
107107institutetext: CERCA/ISO, Department of Physics, Case Western Reserve
University, 10900 Euclid Avenue, Cleveland, OH 44106, USA 108108institutetext:
Laboratoire Univers et Théorie, Observatoire de Paris, Université PSL,
Université Paris Cité, CNRS, 92190 Meudon, France 109109institutetext:
Dipartimento di Fisica e Scienze della Terra, Università degli Studi di
Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy 110110institutetext:
Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe
Saragat 1, 44122 Ferrara, Italy 111111institutetext: Dipartimento di Fisica -
Sezione di Astronomia, Università di Trieste, Via Tiepolo 11, 34131 Trieste,
Italy 112112institutetext: NASA Ames Research Center, Moffett Field, CA 94035,
USA 113113institutetext: Kavli Institute for Particle Astrophysics & Cosmology
(KIPAC), Stanford University, Stanford, CA 94305, USA 114114institutetext: Bay
Area Environmental Research Institute, Moffett Field, California 94035, USA
115115institutetext: Minnesota Institute for Astrophysics, University of
Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA 116116institutetext:
Institute Lorentz, Leiden University, PO Box 9506, Leiden 2300 RA, The
Netherlands 117117institutetext: Institute for Astronomy, University of
Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA 118118institutetext:
Department of Astronomy & Physics and Institute for Computational
Astrophysics, Saint Mary’s University, 923 Robie Street, Halifax, Nova Scotia,
B3H 3C3, Canada 119119institutetext: Université Paris-Saclay, CNRS, Institut
d’astrophysique spatiale, 91405, Orsay, France 120120institutetext:
Departamento Física Aplicada, Universidad Politécnica de Cartagena, Campus
Muralla del Mar, 30202 Cartagena, Murcia, Spain 121121institutetext:
Department of Physics, Oxford University, Keble Road, Oxford OX1 3RH, UK
122122institutetext: CEA Saclay, DFR/IRFU, Service d’Astrophysique, Bat. 709,
91191 Gif-sur-Yvette, France 123123institutetext: Institute of Cosmology and
Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK
124124institutetext: Department of Computer Science, Aalto University, PO Box
15400, Espoo, FI-00 076, Finland 125125institutetext: Ruhr University Bochum,
Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), German
Centre for Cosmological Lensing (GCCL), 44780 Bochum, Germany
126126institutetext: Department of Physics and Astronomy, Vesilinnantie 5,
20014 University of Turku, Finland 127127institutetext: Serco for European
Space Agency (ESA), Camino bajo del Castillo, s/n, Urbanizacion Villafranca
del Castillo, Villanueva de la Cañada, 28692 Madrid, Spain
128128institutetext: ARC Centre of Excellence for Dark Matter Particle
Physics, Melbourne, Australia 129129institutetext: Centre for Astrophysics &
Supercomputing, Swinburne University of Technology, Victoria 3122, Australia
130130institutetext: W.M. Keck Observatory, 65-1120 Mamalahoa Hwy, Kamuela,
HI, USA 131131institutetext: ICTP South American Institute for Fundamental
Research, Instituto de Física Teórica, Universidade Estadual Paulista, São
Paulo, Brazil 132132institutetext: Oskar Klein Centre for Cosmoparticle
Physics, Department of Physics, Stockholm University, Stockholm, SE-106 91,
Sweden 133133institutetext: Astrophysics Group, Blackett Laboratory, Imperial
College London, London SW7 2AZ, UK 134134institutetext: INAF-Osservatorio
Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Firenze, Italy
135135institutetext: Dipartimento di Fisica, Sapienza Università di Roma,
Piazzale Aldo Moro 2, 00185 Roma, Italy 136136institutetext: Centro de
Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762 Porto,
Portugal 137137institutetext: Dipartimento di Fisica, Università di Roma Tor
Vergata, Via della Ricerca Scientifica 1, Roma, Italy 138138institutetext:
INFN, Sezione di Roma 2, Via della Ricerca Scientifica 1, Roma, Italy
139139institutetext: Institute of Astronomy, University of Cambridge,
Madingley Road, Cambridge CB3 0HA, UK 140140institutetext: Institute for
Computational Science, University of Zurich, Winterthurerstrasse 190, 8057
Zurich, Switzerland 141141institutetext: Theoretical astrophysics, Department
of Physics and Astronomy, Uppsala University, Box 515, 751 20 Uppsala, Sweden
142142institutetext: Niels Bohr Institute, University of Copenhagen, Jagtvej
128, 2200 Copenhagen, Denmark 143143institutetext: Cosmic Dawn Center (DAWN)
144144institutetext: Center for Cosmology and Particle Physics, Department of
Physics, New York University, New York, NY 10003, USA 145145institutetext:
Technische Universitat Dresden, Institut für Kern- und Teilchenphysik,
Zellescher Weg 19, 01069 Dresden, Germany 146146institutetext: Deutsches
Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany

#  Euclid preparation.

The Near-IR Background Dipole Experiment with Euclid

Euclid Collaboration: A. Kashlinsky<EMAIL_ADDRESS>Euclid
preparation.Euclid preparation.Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation. R. G. Arendt Euclid preparation.Euclid
preparation.Euclid preparation.Euclid preparation.Euclid preparation.Euclid
preparation. M. L. N. Ashby Euclid preparation.Euclid preparation. F. Atrio-
Barandela Euclid preparation.Euclid preparation. R. Scaramella Euclid
preparation.Euclid preparation.Euclid preparation.Euclid preparation. M. A.
Strauss Euclid preparation.Euclid preparation. B. Altieri Euclid
preparation.Euclid preparation. A. Amara Euclid preparation.Euclid
preparation. S. Andreon Euclid preparation.Euclid preparation. N. Auricchio
Euclid preparation.Euclid preparation. M. Baldi Euclid preparation.Euclid
preparation.Euclid preparation.Euclid preparation.Euclid preparation.Euclid
preparation. S. Bardelli Euclid preparation.Euclid preparation. R. Bender
Euclid preparation.Euclid preparation.Euclid preparation.Euclid preparation.
C. Bodendorf Euclid preparation.Euclid preparation. E. Branchini Euclid
preparation.Euclid preparation.Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation. M. Brescia Euclid preparation.Euclid
preparation.Euclid preparation.Euclid preparation.Euclid preparation.Euclid
preparation. J. Brinchmann Euclid preparation.Euclid preparation. S. Camera
Euclid preparation.Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation.Euclid preparation. V. Capobianco Euclid
preparation.Euclid preparation. C. Carbone Euclid preparation.Euclid
preparation. J. Carretero Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation. S. Casas Euclid preparation.Euclid
preparation. M. Castellano Euclid preparation.Euclid preparation. S. Cavuoti
Euclid preparation.Euclid preparation.Euclid preparation.Euclid preparation.
A. Cimatti Euclid preparation.Euclid preparation. G. Congedo Euclid
preparation.Euclid preparation. C. J. Conselice Euclid preparation.Euclid
preparation. L. Conversi Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation. Y. Copin Euclid preparation.Euclid
preparation. L. Corcione Euclid preparation.Euclid preparation. F. Courbin
Euclid preparation.Euclid preparation. H. M. Courtois Euclid
preparation.Euclid preparation. A. Da Silva Euclid preparation.Euclid
preparation.Euclid preparation.Euclid preparation. H. Degaudenzi Euclid
preparation.Euclid preparation. A. M. Di Giorgio Euclid preparation.Euclid
preparation. J. Dinis Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation. F. Dubath Euclid preparation.Euclid
preparation. X. Dupac Euclid preparation.Euclid preparation. S. Dusini Euclid
preparation.Euclid preparation. A. Ealet Euclid preparation.Euclid
preparation. M. Farina Euclid preparation.Euclid preparation. S. Farrens
Euclid preparation.Euclid preparation. S. Ferriol Euclid preparation.Euclid
preparation. M. Frailis Euclid preparation.Euclid preparation. E. Franceschi
Euclid preparation.Euclid preparation. S. Galeotta Euclid preparation.Euclid
preparation. B. Gillis Euclid preparation.Euclid preparation. C. Giocoli
Euclid preparation.Euclid preparation.Euclid preparation.Euclid preparation.
A. Grazian Euclid preparation.Euclid preparation. F. Grupp Euclid
preparation.Euclid preparation.Euclid preparation.Euclid preparation. S. V. H.
Haugan Euclid preparation.Euclid preparation. I. Hook Euclid
preparation.Euclid preparation. F. Hormuth Euclid preparation.Euclid
preparation. A. Hornstrup Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation. K. Jahnke Euclid preparation.Euclid
preparation. E. Keihänen Euclid preparation.Euclid preparation. S. Kermiche
Euclid preparation.Euclid preparation. A. Kiessling Euclid preparation.Euclid
preparation. M. Kilbinger Euclid preparation.Euclid preparation. B. Kubik
Euclid preparation.Euclid preparation. M. Kunz Euclid preparation.Euclid
preparation. H. Kurki-Suonio Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation. S. Ligori Euclid preparation.Euclid
preparation. P. B. Lilje Euclid preparation.Euclid preparation. V. Lindholm
Euclid preparation.Euclid preparation.Euclid preparation.Euclid preparation.
I. Lloro Euclid preparation.Euclid preparation. D. Maino Euclid
preparation.Euclid preparation.Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation. E. Maiorano Euclid preparation.Euclid
preparation. O. Mansutti Euclid preparation.Euclid preparation. O. Marggraf
Euclid preparation.Euclid preparation. K. Markovic Euclid preparation.Euclid
preparation. N. Martinet Euclid preparation.Euclid preparation. F. Marulli
Euclid preparation.Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation.Euclid preparation. R. Massey Euclid
preparation.Euclid preparation. S. Maurogordato Euclid preparation.Euclid
preparation. H. J. McCracken Euclid preparation.Euclid preparation. E.
Medinaceli Euclid preparation.Euclid preparation. S. Mei Euclid
preparation.Euclid preparation. Y. Mellier Euclid preparation.Euclid
preparation.Euclid preparation.Euclid preparation. M. Meneghetti Euclid
preparation.Euclid preparation.Euclid preparation.Euclid preparation. G.
Meylan Euclid preparation.Euclid preparation. M. Moresco Euclid
preparation.Euclid preparation.Euclid preparation.Euclid preparation. L.
Moscardini Euclid preparation.Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation.Euclid preparation. E. Munari Euclid
preparation.Euclid preparation. S.-M. Niemi Euclid preparation.Euclid
preparation. C. Padilla Euclid preparation.Euclid preparation. S. Paltani
Euclid preparation.Euclid preparation. F. Pasian Euclid preparation.Euclid
preparation. K. Pedersen Euclid preparation.Euclid preparation. W. J. Percival
Euclid preparation.Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation.Euclid preparation. S. Pires Euclid
preparation.Euclid preparation. G. Polenta Euclid preparation.Euclid
preparation. M. Poncet Euclid preparation.Euclid preparation. L. A. Popa
Euclid preparation.Euclid preparation. F. Raison Euclid preparation.Euclid
preparation. A. Renzi Euclid preparation.Euclid preparation.Euclid
preparation.Euclid preparation. J. Rhodes Euclid preparation.Euclid
preparation. G. Riccio Euclid preparation.Euclid preparation. E. Romelli
Euclid preparation.Euclid preparation. M. Roncarelli Euclid preparation.Euclid
preparation. E. Rossetti Euclid preparation.Euclid preparation. R. Saglia
Euclid preparation.Euclid preparation.Euclid preparation.Euclid preparation.
D. Sapone Euclid preparation.Euclid preparation. B. Sartoris Euclid
preparation.Euclid preparation.Euclid preparation.Euclid preparation. M.
Schirmer Euclid preparation.Euclid preparation. P. Schneider Euclid
preparation.Euclid preparation. T. Schrabback Euclid preparation.Euclid
preparation. A. Secroun Euclid preparation.Euclid preparation. G. Seidel
Euclid preparation.Euclid preparation. M. Seiffert Euclid preparation.Euclid
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Verifying the fully kinematic nature of the long-known cosmic microwave
background (CMB) dipole is of fundamental importance in cosmology. In the
standard cosmological model with the Friedman–Lemaitre–Robertson–Walker (FLRW)
metric from the inflationary expansion the CMB dipole should be entirely
kinematic. Any non-kinematic CMB dipole component would thus reflect the
preinflationary structure of spacetime probing the extent of the FLRW
applicability. Cosmic backgrounds from galaxies after the matter-radiation
decoupling, should have kinematic dipole component identical in velocity with
the CMB kinematic dipole. Comparing the two can lead to isolating the CMB non-
kinematic dipole. It was recently proposed that such measurement can be done
using the near-IR cosmic infrared background (CIB) measured with the currently
operating Euclid telescope, and later with Roman. The proposed method
reconstructs the resolved CIB, the Integrated Galaxy Light (IGL), from
Euclid’s Wide Survey and probes its dipole, with a kinematic component
amplified over that of the CMB by the Compton–Getting effect. The
amplification coupled with the extensive galaxy samples forming the IGL would
determine the CIB dipole with an overwhelming signal-to-noise ratio, isolating
its direction to sub-degree accuracy. We develop details of the method for the
Euclid’s Wide Survey in four bands spanning 0.6 to 2
$\mathrm{\SIUnitSymbolMicro m}$. We isolate the systematic and other
uncertainties and present methodologies to minimize them, after confining the
sample to the magnitude range with negligible IGL/CIB dipole from galaxy
clustering. These include the required star–galaxy separation, accounting for
the extinction correction dipole using the method newly developed here
achieving total separation, accounting for the Earth’s orbital motion and
other systematic effects. Finally, we apply the developed methodology to the
simulated Euclid galaxy catalogs testing successfully the upcoming
applications. With the presented techniques one would indeed measure the
IGL/CIB dipole from Euclid’s Wide Survey with high precision probing the non-
kinematic CMB dipole.

###### Key Words.:

Cosmology: cosmic background radiation, Infrared: diffuse background,
Cosmology: inflation, Cosmology: large-scale structure of Universe, Cosmology:
observations, Cosmology: early Universe

###### Contents

  1. 1 Motivation
  2. 2 On importance of cosmic background dipoles
    1. 2.1 On the intrinsic CMB dipole
    2. 2.2 The Compton–Getting effect and dipole for cosmic backgrounds from galaxies
  3. 3 Probing the near-IR background dipole in the Euclid Wide Survey
  4. 4 Required development
    1. 4.1 Procedure
    2. 4.2 Euclid Wide Survey galaxy samples
    3. 4.3 Selecting the optimal magnitude range $[m_{0},m_{1}]$
      1. 4.3.1 Galaxy counts
      2. 4.3.2 Star–galaxy separation
      3. 4.3.3 Dipole contribution from clustering: $m_{0}$
      4. 4.3.4 Choice of $m_{1}$
      5. 4.3.5 The overall magnitude range required here
    4. 4.4 Understanding details of extinction in the measurement
      1. 4.4.1 Extinction
      2. 4.4.2 Managing the extinction contributions
    5. 4.5 Accounting for the Earth’s orbital motion
    6. 4.6 Photometry and magnitudes
    7. 4.7 Systematic corrections
    8. 4.8 Photometric zero points
  5. 5 Applying to the upcoming Euclid Wide Survey
    1. 5.1 Evaluating overall statistical uncertainties
    2. 5.2 Uncertainties on the IGL/CIB kinematic dipole after extinction corrections
    3. 5.3 Reducing systematic amplification uncertainties
  6. 6 Summing up
  7. A The Compton–Getting effect for cosmic backgrounds

##  1 Motivation

The cosmic microwave background (CMB) dipole is the oldest known CMB
anisotropy of $\delta T_{\rm CMB}=3.35$ mK, or $\delta T_{\rm CMB}/T_{\rm
CMB}=1.23\times 10^{-3}$, measured with the unprecedented precision of a
signal-to-noise ratio of ${\rm S/N}\gtrsim 200$ (Kogut et al. 1993; Fixsen et
al. 1994). It is conventionally interpreted as being entirely of kinematic
origin due to the Solar System moving at velocity $V_{\rm CMB}=370$ km s-1 in
the Galactic direction of $(l,b)_{\rm CMB}=(263\aas@@fstack{\circ}85\pm
0\aas@@fstack{\circ}1,48\aas@@fstack{\circ}25\pm 0\aas@@fstack{\circ}04)$.

The fully kinematic origin of the CMB dipole is further motivated
theoretically by the fact that any curvature perturbations on superhorizon
scales leave zero dipole because the density gradient associated with them is
exactly cancelled by that from their gravitational potential (Turner 1991).
However, already prior to the development of inflationary cosmology there were
suggestions that the CMB dipole may be, even if in part, primordial (King &
Ellis 1973; Matzner 1980). Within the inflationary cosmology, which posits the
non-Friedmann–Lemaitre–Robertson–Walker (FLRW) metric on sufficiently large
scales due to the primeval (preinflationary) structure of space-time (Turner
1991; Grishchuk 1992; Kashlinsky et al. 1994; Das et al. 2021), such
possibility can arise from isocurvature perturbations induced by the latter
(Turner 1991) and/or from entanglement of our Universe with other superhorizon
domains of the Multiverse (Mersini-Houghton & Holman 2009). Hence,
establishing the nature of the CMB dipole is a problem of fundamental
importance in cosmology.

Despite the overwhelming preference of the kinematic CMB dipole
interpretation, there have been longstanding observational claims to the
contrary (Gunn 1988). Comparing the gravity dipole with peculiar velocity
measurements (Villumsen & Strauss 1987) indicates an offset (Gunn 1988;
Erdoǧdu et al. 2006; Kocevski & Ebeling 2006; Lavaux et al. 2010; Wiltshire et
al. 2013) broadly buttressed by other peculiar velocity data (Mathewson et al.
1992; Lauer & Postman 1992; Ma et al. 2011; Colin et al. 2019). There appears
a “dark flow” of galaxy clusters in the analysis of the cumulative kinematic
Sunyaev–Zeldovich effect extending to at least $\sim$1 Gpc in both WMAP and
Planck data (Kashlinsky et al. 2008, 2009; Atrio-Barandela et al. 2010;
Kashlinsky et al. 2010, 2012a; Atrio-Barandela 2013; Atrio-Barandela et al.
2015), which is generally consistent with the radio (Nodland & Ralston 1997;
Jain & Ralston 1999; Singal 2011) and WISE (Secrest et al. 2021) source count
dipoles and the anisotropy in X-ray scaling relations (Migkas et al. 2020).
See reviews by Kashlinsky et al. (2012b); Aluri et al. (2023). All of these
assertions have achieved only a limited significance of ${\rm S/N}\sim 3$–$5$,
with the subsequently significant directional uncertainty, and are debated.

It is important to establish observationally the fully kinematic nature of the
CMB dipole and whether the homogeneity in the Universe as reflected in the
FLRW metric models is adequate to describe what we observe. Since any
curvature perturbations must have zero dipole at last scattering such probe
would be fundamental to cosmology with the non-kinematic CMB dipole component
potentially providing a probe of the primordial preinflationary structure of
spacetime. To this end a technique has been proposed recently by Kashlinsky &
Atrio-Barandela (2022) to be applied to the Euclid Wide Survey to probe the
dipole of the resolved part of the CIB, the IGL, at an overwhelming ${\rm
S/N}$ thereby settling the issue of the origin of the CMB dipole.

Here we develop the detailed methodology for this experiment we call NIRBADE
(Near IR BAckground Dipole Experiment) dedicated to measuring, at high ${\rm
S/N}$, the (amplified) CIB dipole from the Euclid Wide Survey. In Sect. 2 we
discuss the different physics governing CMB and CIB dipoles, pointing out how
at the Euclid-covered wavelengths the expected kinematic CIB dipole will be
significantly amplified over that of the CMB. Section 3 sums up the details of
the Euclid Wide Survey and their application to NIRBADE following Kashlinsky &
Atrio-Barandela (2022). Section 4 is devoted to the required development to
achieve the NIRBADE goal covering the overall pipeline. These topics include
isolating the needed magnitude range here (AB magnituudes are used throughout
this paper), developing the methodology to successfully isolate the dipole
from Galactic extinction, and accounting for the Earth’s orbital motion. Here,
we also discuss a slew of less critical, but still important items, such as
photometry, before moving on to quantifying the overall uncertainties expected
in the pipeline. Section 5 then applies the development here to the simulated
Euclid catalog to demonstrate how comparing the measured CIB dipole with the
well known CMB dipole will isolate any non-kinematic CMB dipole component down
to interestingly low levels. We sum up the prospects for NIRBADE with Euclid
in Sect. 6.

More specifically the outline of the developmental part of the study is as
follows:

  * •

The procedure of the measurement with the required steps to be implemented
here has been designed in Sect. 4.1. The procedure requires successfully
finessing the various items that are subsequently outlined, discussed, and
resolved.

  * •

In the following Sect. 4.2 we present the pre-launch plan of the Euclid Wide
Survey coverage that we use in the computations here. Now that the mission is
at L2, the details of the survey may be altered, so this is given as an
example used for development in finalizing the details of the methodology. The
methodology developed here will be applied to the actual observed coverage.

  * •

We identify the aspects required for selecting galaxies from the Euclid Wide
Survey for this measurement in Sect. 4.3 – Eq. (9) for VIS and Eq. (10) for
NISP. Throughout we used, in the absence of the forthcoming Euclid data, the
observed galaxy counts presented in Sect. 4.3.1 for JWST measurements
(Windhorst et al. 2023) and, when needed, the HRK reconstruction (Helgason et
al. 2012). The range of galaxy magnitudes required to sufficiently reduce the
clustering dipole component is isolated in Sect. 4.3.5. The prospects of the
star–galaxy separation desired in the experiment are given in Sect. 4.3.2.

  * •

Section 4.4 discusses how the extinction, using the SFD template (Schlegel et
al. 1998), can affect the measurement and design a method to isolate the
contribution due to extinction corrections from that of the IGL/CIB. The
method is applicable at small extinction corrections $A\ll 1$.

  * •

The needed corrections, for the high-precision measurement, from the effects
of the Earth’s orbital motion are then discussed in Sect. 4.5. It is shown how
the corrections will be incorporated into the designed pipeline.

  * •

The potential systematic effects, and how to correct for them are considered
in Sect. 4.7 followed by the requirements on the photometric accuracy and zero
points, etc. in Sect. 4.8.

  * •

Section 5 then shows the application of the developed methodology to the
forthcoming Euclid Wide Survey data. In Sect. 5.1 we evaluate the statistical
uncertainties after each year of the Euclid observation. In Sect. 5.2 we apply
the method developed here to correct for extinction using a simulated catalog
for Euclid with available spectral colors, to isolate the contribution from
extinction if the need arises in the actual data to finalize the high-
precision determination of the IGL/CIB dipole from the Euclid Wide Survey.
Section 5.3 discusses and quantifies the identified systematic corrections
when converting the measured IGL/CIB dipole into the equivalent velocity,
which affect all the velocity components equally thereby being of relevance to
its amplitude, and not direction.

Such an experiment can, and must, also be done with Roman (formerly WFIRST;
Akeson et al. 2019), which would require a separate and significantly
different preparation.

##  2 On importance of cosmic background dipoles

Here we discuss the different physics governing the CMB and CIB dipoles and
why and how the CIB kinematic dipole is amplified over that of the CMB.

###  2.1 On the intrinsic CMB dipole

The CMB as observed today originates at the last scattering, which occurred at
the cosmic epochs corresponding to redshift $z\simeq 10^{3}$. Its structure
from the quadrupole term ($\ell=2$) to higher-order in $\ell$ multipoles is in
very good general agreement with predictions of inflation. The latter posits
that the observed Universe originated from a small smooth patch, with the
underlying FLRW metric, of the size of or smaller than the horizon scale at
the start of inflation, which then quickly inflated to encompass scales well
beyond the current cosmological horizon (Kazanas 1980; Guth 1981). At the same
time, on sufficiently large scales the preinflationary spacetime could have
preserved its original structure, assumed generally to be inhomogeneous
(Turner 1991; Grishchuk 1992; Kashlinsky et al. 1994). Such preinflationary
structures, currently on superhorizon scales, could leave CMB signatures via
the Grishchuk–Zeldovich effect (Grishchuk & Zeldovich 1978). The smallness of
the measured CMB quadrupole (the relative value of $Q\sim 2\times 10^{-6}$)
indicates that preinflationary structures in spacetime were pushed during
inflation to scales currently $\gtrsim Q^{-1/2}cH_{0}^{-1}\sim
10^{3}cH_{0}^{-1}$ (Turner 1991; Kashlinsky et al. 1994).

However, as was shown by Turner (1991), the Grishchuk–Zeldovich effect does
not produce an observable dipole anisotropy in any superhorizon modes from
curvature perturbations because at the last scattering the linear gradient
associated with them is cancelled exactly by the corresponding dipole
anisotropy from their gravitational potential term. This points to the unique
importance of probing the fully kinematic nature of the CMB dipole where any
non-kinematic dipole would arise from, within the inflationary paradigm, the
preinflationary structure of spacetime and potentially provide new information
on the details of inflation and the applicability limits of the FLRW metric.

This differentiates the CMB dipole from the dipole components of the cosmic
backgrounds, discussed next, which are produced by sources that formed well
after decoupling.

###  2.2 The Compton–Getting effect and dipole for cosmic backgrounds from
galaxies

Cosmic backgrounds produced by luminous sources that formed at $z\ll 10^{3}$
are subject to a different physics and their spatial distribution is imprinted
during the inflationary period and later modified by the standard
gravitational evolution during radiation-dominated era. In addition if the
Solar System moves with respect to the frame defined by distant sources
producing the background with mean intensity $\bar{I}_{\nu}$ at frequency
$\nu$, it would have a dipole in the Sun’s rest frame

$\boldsymbol{d}_{\nu}=(3-\alpha_{\nu,\infty})\frac{\boldsymbol{V}}{c}\bar{I}_{\nu,\infty}\,,$
(1)

where

$\alpha_{\nu}=\frac{\partial{\ln I_{\nu}}}{\partial{\ln\nu}}$ (2)

and the subscript $\infty$ implies that the background intensity $I_{\nu}$
comes from integrating over the entire range of fluxes/magnitudes of the
contributing sources. This is known as the Compton–Getting (Compton & Getting
1935) effect for cosmic rays (e.g. Gleeson & Axford 1968). Equation (1)
follows since photons emitted at frequency $\nu_{0}$ from a source moving at
velocity $V\ll c$ forming angle $\Theta$ toward the apex of motion will be
received by the observer at rest at frequency
$\nu=\nu_{0}[1+({V}/{c})\cos\Theta]$ and the Lorentz transformation requires
that $I_{\nu}/\nu^{3}$ remains invariant (Peebles & Wilkinson 1968). Hence the
observer at rest will see the direction-dependent specific intensity
$I_{\nu}=(\nu/\nu_{0})^{3}I_{\nu/[1+({V}/{c})\cos\Theta]}$; see Appendix. The
spectral index of the Rayleigh-Jeans spectrum $\alpha_{\nu}^{\rm RJ}=2$
describes the CMB at mm wavelengths.

If the CMB is the rest frame of the Universe then
$\boldsymbol{V}=\boldsymbol{V}_{\rm CMB}$ for any cosmic background that
originates from galaxies. Otherwise, the non-zero non-kinematic part of the
CMB dipole would be likely to indicate the existence of superhorizon
deviations from the FLRW metric, possibly due to the primordial
(preinflationary) structure of spacetime.

At wavelengths where cosmic backgrounds from galaxies have $\alpha_{\nu}\ll
2$, the amplitude of their kinematic dipole in $I_{\nu}$ is amplified. This is
the case for CIB (Kashlinsky 2005) and is also the case at high energies
[X-ray (Fabian & Warwick 1979) and $\gamma$-ray (Maoz 1994; Kashlinsky et al.
2024) backgrounds, and cosmic rays (Kachelrieß & Serpico 2006)]. However, at
infrared wavelengths significant pollution to the CIB dipole would come from
dust emission and reflection by the Galaxy (cirrus) and the Solar System
(zodiacal light) as discussed in Kashlinsky & Atrio-Barandela (2022).

##  3 Probing the near-IR background dipole in the Euclid Wide Survey

The Euclid satellite was successfully launched on July 1, 2023 to the L2
orbit. The photometric bands covered by Euclid are shown in Fig. 1. The
unresolved CIB dipole at the Euclid bands will be subject to significant
contributions from Galactic and Solar System foregrounds, but the foreground
dipole contributions can be excluded efficiently by considering the CIB from
resolved galaxies.

Figure 1: Total sensitivity of Euclid’s photometric and spectroscopic bands.
(credit:ECSURV/J.-C. Cuillandre) Figure 2: Left: The mean IGL flux. Right: The
IGL dipole per Eqs. (2) and (3) and assuming $V=370$ km s-1. Black plus signs
correspond to the entire range of magnitudes. Black, blue, green, and red
circles correspond to IGL from galaxies at $m\geq 18,19,20,21$. The IGL is
integrated over the JWST latest counts (Windhorst et al. 2023) at the marked
central wavelengths.

To overcome the obstacles due to the otherwise dominant at near-IR foreground
dipoles Kashlinsky & Atrio-Barandela (2022) proposed to use the all-sky part
of the background, known as IGL (Integrated Galaxy Light), reconstructed from
resolved galaxies in the Euclid Wide Survey (Laureijs et al. 2011),

$I_{\nu}(\theta,\phi)=10^{-0.4A_{\nu}(\theta,\phi)}S_{0}\int_{m_{0}}^{m_{1}}10^{-0.4m}\left[\frac{{\rm
d}N_{\nu}(\theta,\phi)}{{\rm d}m}\right]{\rm d}m\,,$ (3)

where $S_{0}=3631$ Jy and $A_{\nu}$ is the magnitude extinction in the
direction $(\theta,\phi)$. The above expression is a short-hand for the actual
procedure outlined in Sec. 4.1, Eq. (7) which requires no source counts
determination. The Euclid Wide Survey galaxy samples will be corrected for
extinction, so strictly speaking $A_{\nu}$ should be interpreted as the
magnitude correction remaining after the extinction correction; more on this
will be presented later. The IGL is evaluated over a suitably selected
$m_{0}\sim 18$–$21$ required to remove the galaxy clustering dipole and
$m_{1}$ imposed by the sensitivity limits of the Wide Survey, which is also
below the expected magnitudes of the new populations expected to be present in
the CIB source-subtracted anisotropies (Kashlinsky et al. 2018).

As discussed in Kashlinsky & Atrio-Barandela (2022) at the Euclid VIS and NISP
bands $\alpha_{\nu}\sim-1$, so from an all-sky catalog of $N_{\rm gal}$
galaxies and for a fixed direction, one would reach the statistical signal-to-
noise ratio in the measured IGL dipole amplitude, $d_{\nu}/\langle
I_{\nu}\rangle$, of

${\rm S/N}\sim 160\left(\frac{3-\alpha_{\nu}}{4}\right)\left(\frac{V}{V_{\rm
CMB}}\right)\left(\frac{N_{\rm gal}}{10^{9}}\right)^{1/2}\,.$ (4)

An all-sky CMB dipole measured with a signal-to-noise ratio of ${\rm S/N}$
will have its direction probed with directional accuracy of (Fixsen &
Kashlinsky 2011)

$\Delta\Theta_{\rm dipole}\sim\sqrt{2}({\rm S/N})^{-1}\;{\rm radian}\,.$ (5)

This demonstrates that the directional uncertainty, say $\Delta\Theta_{\rm
dipole}\lesssim 1^{\circ}$, needed to decisively probe the alignment requires
${\rm S/N}\gtrsim 80$. The statistical significance will depend on the actual
dipole amplitude, direction and region of the sky observed by Euclid. For a
partial sky coverage the above order of magnitude estimates will be reduced
since the three dipole components $(X,Y,Z)$ will have different errors (Atrio-
Barandela et al. 2010; Kashlinsky & Atrio-Barandela 2022). A discussion of the
error budget is deferred to Sect. 5.1.

Equations (4) and (5) demonstrate why the to-date probes of the kinematic
nature of the CMB dipole discussed in sec. 1, which reach ${\rm S/N}\simeq
4$–$5$ by utilizing the cumulative kinematic Sunyaev-Zeldovich (Sunyaev &
Zeldovich 1980) effect (Kashlinsky & Atrio-Barandela 2000) or the relativistic
aberration (Ellis & Baldwin 1984), have poor directional accuracy of
$\Delta\Theta_{\rm dipole}\sim 15^{\circ}$–$20^{\circ}$ and hence are
insufficient to test, in addition to the dipole amplitude and its convergence
with distance, the consistency of the dipole directions. Both will be achieved
with NIRBADE as outlined below.

Figure 2 (left) shows the IGL reconstructed from integrating over magnitudes
exceeding some fiducial $m_{0}$ (see caption) using observed galaxy counts
from Figs. 9 and 10 of the JWST counts data by Windhorst et al. (2023) at the
wavelengths similar to the Euclid bands. The right panel of the figure shows
the expected dipole amplitudes evaluated with Eq. (2) for $V=V_{\rm CMB}=370$
km s-1. Plus signs mark the entire range out to the Euclid Wide Survey limit
of $m_{1}=24$. Circles mark the results for $m_{0}\leq m\leq m_{1}$ for
$m_{0}=18$, 19, 20, and 21 (black, blue, green, and red) and $m_{1}=25$. The
four Euclid filters are shown per Euclid Collaboration: Schirmer et al.
(2022). Later we will discuss the selections of $(m_{0},m_{1})$ required
specifically for this measurement.

##  4 Required development

If the CMB dipole is entirely kinematic, the expected CIB dipole components in
the Galactic coordinate system $(X,Y,Z)$ would be

$\boldsymbol{d}_{\nu}=5.2\times
10^{-3}\left(\frac{3-\alpha_{\nu}}{4}\right)(-0.07,-0.66,0.75)\langle
I_{\nu}\rangle\,,$ (6)

with the $X$-component being by far the smallest, contributing just a few
percent to the net dipole, and the $Z$-component being the largest, but close
in amplitude to the $Y$-component. Hence, if the CMB dipole is purely
kinematic, the IGL/CIB dipole, after correcting for the Earth motion, should
lie almost entirely in the $(Y,Z)$ plane with nearly equal amplitude $Y$ and
$Z$ components.

We are aiming to measure the IGL dipole of dimensionless amplitude of $\simeq
0.5\%$ in each or any of the Euclid’s four bands: $I_{\scriptscriptstyle\rm
E}$, $Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$, and
$H_{\scriptscriptstyle\rm E}$. Two points make this promising: 1) the IGL
dipole is amplified by the Compton–Getting effect, and 2) the noise is
substantially decreased by the large number of galaxies the Euclid Wide Survey
will have.

Below are the items to discuss in order to get this measurement done, and at
high precision. In what follows the sought dipole signal, Eq. (1), is denoted
with a lower case $\boldsymbol{d}$ and the nuisance dipoles with a capital
case $\boldsymbol{D}$.

In the absence of the Euclid data we will use here, for the bulk of estimates,
the formulation per Eq. (3) inputting the latest JWST counts data from
Windhorst et al. (2023), which are consistent with the reconstruction from
Helgason et al. (2012) used originally by Kashlinsky & Atrio-Barandela (2022).

Depending on the context throughout this section we will work with both the
absolute CIB dipole amplitude ($d_{\nu}$ in MJy sr-1 equivalent to $\delta T$
in mK for the CMB) and its relative amplitude ($d_{\nu}/I_{\nu}$ equivalent to
$\delta T/T$ for the CMB). The former would be useful when e.g discussing the
measurability and overcoming the Galactic components while the latter is
useful when estimating the extragalactic non-kinematic terms and converting to
velocity.

###  4.1 Procedure

The procedure required to apply the method of Kashlinsky & Atrio-Barandela
(2022) to probe the kinematic component of the IGL/CIB dipole would go through
the following steps:

  1. 1.

We will subdivide the Euclid sky coverage into areas, ${\cal A}$, centered on
Galactic coordinates $(l,b)$. ${\cal A}$ could be the size of each FOV (0.5
deg2) or larger.

  2. 2.

We will collect the photometry on all galaxies in each ${\cal A}$, with care
to exclude Galactic stellar sources from the sample.

  3. 3.

We will apply extinction corrections in each band and/or test for
contributions from the extinction effects on the dipole by band, latitude etc.
Then a method developed below to eliminate the extinction induced dipole will
be applied.

  4. 4.

We will identify, in each of the four Euclid photometric bands, the uniform
upper magnitude limit, $m_{1}$, that can be applied to all selected regions
${\cal A}$. This would be one of the important criteria for selecting the sky
for this measurement.

  5. 5.

We will select a lower magnitude limit, $m_{0}$, to ensure the IGL dipole from
galaxy clustering is sufficiently negligible. We may choose the same $m_{0}$
for each bands or leave it band-dependent, provided the clustering dipole
contribution is negligible in all four Euclid bands.

  6. 6.

We will compute the net IGL flux from the selected galaxies as

$I_{\nu}(l,b)=\frac{1}{{\cal A}}S_{0}\sum_{m_{0}\leq m\leq
m_{1}}10^{-0.4m_{\nu}}$ (7)

over ${\cal A}$ and do this for the entire sky, or a selected part of it. Here
$S_{0}=$3631$$ Jy.

  7. 7.

We will evaluate the IGL dipole, $d_{\nu}$, over the selected Euclid sky in
each of the four bands of frequency $\nu$ after dividing the galaxy sample by
color to eliminate extinction.

  8. 8.

We will eliminate the dipole contribution from extinction from a subsample of
galaxies with selected IGL spectral index, $\alpha_{\nu}$, and isolate the
kinematic IGL dipole part.

  9. 9.

We will compute the dipole error.

  10. 10.

We will evaluate the other systematics discussed below.

  11. 11.

We will translate into the effective velocity via the refined estimation, for
each galaxy subsample, of the IGL spectral index $\alpha_{\nu}$ and the
Compton–Getting amplification using

$V=(3-\alpha_{\nu})^{-1}\left(\frac{d_{\nu}}{\langle
I_{\nu}\rangle}\right)c\,.$ (8)

We use HEALPix remove_dipole routine (Górski et al. 2005) in the computations
throughout the paper.

###  4.2 Euclid Wide Survey galaxy samples

The Euclid Wide Survey aims to cover most of the best parts of the
extragalactic sky in terms of extinction and star density. An area larger than
$14\,000$ deg2 is expected to be covered with a single visit (four exposures
via three dithers). In each visit imaging data are acquired over 0.53 deg2 for
a wide visible band, $I_{\scriptscriptstyle\rm E}$ (sampling
$0\aas@@fstack{\prime\prime}1$), and three near infrared bands
($Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$, and
$H_{\scriptscriptstyle\rm E}$), where the sampling is
$0\aas@@fstack{\prime\prime}3$.

In Fig. 3 the latest planned sky coverage is shown. There are three main
contiguous areas that are covered [the fourth, that was presented in Fig. 45
of Euclid Collaboration: Scaramella et al. (2022), is now greatly reduced
because of the lack of timely ground based photometry]. Grey regions denote
unobserved areas due to the presence of extremely bright stars.

Figure 3: Expected coverage of the sky year by year of the wide survey in
ecliptic coordinates. The ecliptic poles are also shown in a different
projection. Circles along the ecliptic denote planet avoidance regions circa
2029 Sep 27, near the end of the survey. (credit: ECSURV/J.Dinis)

We focus in this subsection on the galaxy number density in $H$ band, which is
the one least affected by extinction. Deep galaxy counts in $i$ and $K$ bands
have been given by studies of the COSMOS field (Laigle et al. 2016; Weaver et
al. 2022). Here we are mostly concerned with the intermediate range of
magnitudes $20\leq m_{H}\leq 24$, which is appropriate to get a uniform sample
from the Euclid Wide Survey.

These literature estimates, however, are affected by cosmic variance (Abbott &
Wise 1984): the COSMOS area covers only 2 deg2 and therefore is different from
the average value taken on much larger areas. Moreover, we will need to work
with several sub areas of the wide survey because of cuts to get subsamples
and different epochs of increasing coverage.

Figure 4: Distribution of simulated counts per Euclid field (half square
degree).

Therefore we derive, for the time being, the impact of cosmic variance on the
$H_{\scriptscriptstyle\rm E}$ counts from the Euclid Flagship simulation. From
the large N-body simulation, the Flagship catalogue of many observables was
derived. Of particular importance is the color-color relation and photo-$z$
distribution obtained by imposing spectral energy distributions to the halos
identified as galaxies in the simulation. Therefore the parent spatial
halo/galaxy distribution is clustered and so the catalogue 2D sample has an
intrinsic angular correlation, which causes the distribution in cells to
deviate from the simple Poisson distribution. How large the deviation is would
be a function of both the limiting magnitude and the area considered.

In Fig. 4 we show how cosmic variance (Abbott & Wise 1984) affects the counts
in a single Euclid field: the standard deviation, $\sigma$, is ${\sim}\,14$
times larger than the simple Poisson one due to clustering of the sources
(Abbott & Wise 1984). We also show the counts from the COSMOS2020 catalog
(Weaver et al. 2022). At this scale the $\sigma$ is still ${\sim}\,5\%$ of the
average, but in simply increasing the basic area considered, this ratio will
greatly decrease.

The total expected number density of galaxies is ${\sim}\,1.8\times 10^{5}$
deg-2, which yields a total of $>2.5$ billion objects from the whole survey.

###  4.3 Selecting the optimal magnitude range $[m_{0},m_{1}]$

####  4.3.1 Galaxy counts

Figure 5: Galaxy counts per ${\rm d}m$, plotted $\propto{\rm d}I_{\nu}/{\rm
d}m$, at the four sets of Euclid-related wavelengths. Black lines show the
JWST counts at 0.88, 1.02, 1.25, and 1.63 $\mathrm{\SIUnitSymbolMicro m}$. HRK
reconstructions are displayed with red colors, going to the 0.8
$\mathrm{\SIUnitSymbolMicro m}$ at the VIS-related end. The spreads in the
reconstructions between the LFE (low-faint-end) and HFE (high-faint-end)
limits of the extrapolation allowed for the Schechter-type luminosity function
(Helgason et al. 2012) are shown with pink shades. Dashed vertical lines mark
the range of magnitudes identified for this study in Eq. (10). The JWST counts
are filter-transformed to either ground-based VISTA filters or Spitzer filters
at the long wavelengths (S. Tompkins, R. Windhorst, private communication).
HRK reconstructions are shown at the wavelengths of 0.8, 1.05, 1.25, and 1.63
$\mathrm{\SIUnitSymbolMicro m}$.

Throughout this discussion we will need the numbers for total galaxies
expected to be available from the Euclid Wide Survey in the given magnitude
range at the appropriate wavelengths. Such information is available from the
recent JWST counts (Windhorst et al. 2023) and we will be using also the HRK
reconstruction (Helgason et al. 2012) used in the pre-JWST era by Kashlinsky &
Atrio-Barandela (2022); we will use both intermittently in our numerical
estimates. Figure 5 shows the comparison at Euclid-related wavelengths of the
HRK reconstruction (red) and the JWST counts for the Euclid bands. Figure 6
shows the same for the longer bands adjacent to NISP, which will be used later
in Sect. 5. The VIS-related numbers are shown at 0.88
$\mathrm{\SIUnitSymbolMicro m}$ for JWST data and 0.8
$\mathrm{\SIUnitSymbolMicro m}$ for the HRK reconstruction mimicking the fits
to the broad $I_{\scriptscriptstyle\rm E}$ band. The overall comparison shows
good consistency, within the uncertainties, between the HRK reconstruction and
JWST data, indicating that the former can be used for the power estimates
below.

Figure 6: Same as Fig. 5, except at the wavelengths longward of, but adjacent
to the NISP bands, which are used for refinements in $\alpha_{\nu}$ (e.g. Fig.
2 and Sect. 5). The JWST counts are filter-transformed to either ground-based
VISTA filters or Spitzer filters at the long near-IR wavelengths (S. Tompkins,
R. Windhorst, private communication). HRK reconstruction is shown at the
wavelengths of 2.2, 3.6, and 4.5 $\mathrm{\SIUnitSymbolMicro m}$.

The quantities, $10^{0.4(12.5-m)}{\rm d}N/{\rm d}m$, plotted on the vertical
axis in the figures directly reflect the IGL/CIB produced by galaxies in the
${\rm d}m$ range of $m$. The value of 1 deg-2 on the vertical axis of Figs. 5
and 6 corresponds to ${\rm d}I_{\nu}/{\rm d}m=1.2\times 10^{-4}$ MJy sr-1.

####  4.3.2 Star–galaxy separation

Galactic stars need to be excluded from the Euclid source counts when
constructing the IGL. At wavelengths from 1 to 5 $\mu$m prior studies indicate
that stars outnumber galaxies at $m\lesssim 18$ (e.g. Ashby et al. 2013;
Windhorst et al. 2022, 2023). Thus, star–galaxy separation is essential if
$m_{0}<18$ and still important for sources with $m>18$.

Figure 7: Top: The SKY model evaluated at $K$ band for stars with $m>20.0$.
Intensities are in units of MJy sr-1 and are masked at $|b|<20\degr$ (left)
and for Year 1 of Euclid (right). Middle: The monopole amplitude of the SKY
model as a function of wavelength and the bright magnitude limit. Bottom: The
dipole amplitude of the SKY model as a function of wavelength and the bright
magnitude limit. For the $|b|<20\degr$ mask (left) only the dominant
$X$-component of the dipole is plotted. For the Year 1 mask (right) the total
dipole is shown in black, and the $X$-component is shown in red. At these
magnitude cuts and for these masks, the brightness and dipole of the model are
dominated by the halo component, which is a spherical distribution of stars,
peaked towards the Galactic center ($+X$).

To assess the possible dipole arising from Galactic stars, if they are
incompletely excluded from catalogs used to construct the IGL, we evaluated
the SKY model (Wainscoat et al. 1992; Cohen 1993, 1994, 1995) as implemented
by Arendt et al. (1998), at a variety of wavelengths, and with cuts imposed to
exclude stars brighter than chosen magnitude limits. Figure 7 shows the sky
brightness predicted by the SKY model, with masking applied generically for
$|b|<20\degr$ (left column) and specifically for Euclid Year 1 (right column).
We ran the HEALPix routine remove_dipole on these masked models. The second
row of Fig. 7 shows the derived monopoles. For the $|b|<20\degr$ mask, the
third row shows the $X$-components of the dipoles ($Y$\- and $Z$-components
are orders of magnitude smaller). For the Year 1 mask, the third row shows the
sum of all three dipole components (black), and the $X$-components only (red).
For this masking, the $X$-component is only dominant at fainter magnitudes
because the $X$ direction (towards the Galactic bulge) is not well sampled.
The dipole amplitudes confirm that if stars are not excluded efficiently to
faint magnitudes, then they may contaminate the IGL with a significant dipole.
To probe the IGL dipole at the levels from Fig. 2, we need to eliminate either
$>99\%$ of the stars or choose sufficiently faint $m_{0}$, while keeping
enough galaxies to ensure good ${\rm S/N}$. Figure 10 shows that even if
strict magnitude cuts are needed to exclude stars, there should be sufficient
numbers of galaxies.

At the high latitudes of the Wide Survey, Gaia DR3 thoroughly samples the
stellar disk populations and reaches into the Galactic halo. On the basis of
Gaia DR3 proper motions, it will be possible to reliably exclude Galactic
stars to Gaia’s $G\sim 21.4$ mag or $\sim$9 $\mu$Jy (Vallenari et al. 2022).
Stars will only be a minority of all Euclid detections fainter than this
limit. The Euclid pipeline will provide a flag in the final MER catalog
indicating whether a detected source is a Galactic star, with a 1% error rate.
Thus by combining Gaia and standard pipeline products it will be possible to
reduce the level of stellar contamination by the required amount. In addition,
star contamination can be entirely eliminated if one restricts the galaxy
sample to the one that will be used for WL measures, that is objects with
$I_{\scriptscriptstyle\rm E}$ size larger than $1.2\,{\rm PSF_{FWHM}}$
(Laureijs et al. 2011).

To estimate the extinction using different galaxy subsets as proposed here
(Sect. 4.4.2), the subsets must be drawn from the same area of the sky such
that the dipole due to extinction, $\boldsymbol{D}_{A}$, is unchanged, but
there is no requirement that the subsets be complete in terms of source
morphology. So while including some stars in the IGL calculation would
generate systematic errors, there is no systematic error if the exclusion of
Galactic stars is conservative and some galaxies are excluded because they are
mistaken for stars.

####  4.3.3 Dipole contribution from clustering: $m_{0}$

The lower limit on the magnitude of galaxies selected for this measurement is
dictated by the requirement that the contribution to the probed dipole from
their clustering is sufficiently lower than the one from the Compton–Getting
effect produced by our motion, which is expected to be
${\sim}\,(4.75$–$5.75)\times 10^{-3}(V/V_{\rm CMB})$ as displayed later in
Fig. 23. We have evaluated the dimensionless amplitude, $\sqrt{C_{1}}/\langle
I_{\nu}\rangle$, of the clustering dipole using the HRK reconstruction as
described in Kashlinsky & Atrio-Barandela (2022), which is shown in Fig. 8.
The figure shows that for that term to be comfortably below the
Compton–Getting terms one would want to select galaxies at $m_{0}\gtrsim 19$
in the VIS sample and $m_{0}\gtrsim 18$ for the NISP galaxies.

Figure 8: Dimensionless dipole from clustering evaluated from the HRK
reconstruction as described in Kashlinsky & Atrio-Barandela (2022). The left
panel includes the three photometric bands used by HRK that are covered by the
wide $I_{\scriptscriptstyle\rm E}$ channel. Solid lines correspond to
$m_{1}=24.5$ for $r$, $z$, and $i$, and $m_{1}=24$ for
$Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$, and
$H_{\scriptscriptstyle\rm E}$ bands, and dotted lines are for 0.5 magnitude
fainter. This is to be compared to ${\sim}\,(4.75$–$5.75)\times
10^{-3}(V/V_{\rm CMB})$ as displayed later in Fig. 23.

In real situation, post-launch we will compute the power from the dipole-
subtracted IGL maps, then extrapolate from higher (say, $\ell\gtrsim 10$–$20$)
harmonics to $\ell=1$ using (after verifying) the Harrison–Zeldovich spectrum
($C_{\ell}\propto\ell$). For now we already have 1/8 of the sky from
Flagship2.1 (via CosmoHub) simulations. Uniformity of $m_{0}$ across the sky
also can be tested via the uniformity of the shot-noise power component in
source-subtracted CIB achieved from the source-subtracted CIB studies with
Euclid (Kashlinsky et al. 2018).

####  4.3.4 Choice of $m_{1}$

Figure 9: Contours show the decimal log of the expected number
($J_{\scriptscriptstyle\rm E}$ band) of galaxies [from Fig. 11 of Windhorst et
al. (2023)] as a function of $m_{1}$ and area, for $m_{0}=18$. The dots mark
the nominal $m_{1}=24$ and survey area expected after each addition year of
the Wide Survey.

Euclid Collaboration: Scaramella et al. (2022) show that the Euclid Wide
Survey is expected to reach its intended limiting magnitudes of
$I_{\scriptscriptstyle\rm E}=24.5$ ($10\sigma$ extended source), and
$Y_{\scriptscriptstyle\rm E},\ J_{\scriptscriptstyle\rm E},\
H_{\scriptscriptstyle\rm E}=24$ ($5\sigma$ point source) (Laureijs et al.
2011). So we will assume these values for $m_{1}$. In practice, a brighter
limit for $m_{1}$ may be helpful for better source flux accuracy and
potentially more reliable star galaxy separation. Conversely, it should be
possible to choose a fainter limit for $m_{1}$ (up to ${\sim}\,1$ mag), though
at the cost of limiting the analysis to a smaller fraction of the sky. These
more sensitive regions (due to low zodiacal and Galactic foregrounds) will be
covered in the earlier years of the survey. However in general, given the
relatively shallow slope of galaxy counts at $m_{1}\sim 24$, it is better to
choose larger rather than deeper areas in order to maximize the number of
sources used for computing the IGL, and maximize the ${\rm S/N}$ of its dipole
measurement. The expected galaxy counts as a function of survey area and
$m_{1}$ are shown in Fig. 9.

####  4.3.5 The overall magnitude range required here

Thus we concentrate on galaxies in the magnitude range of

$19\leq m\leq 24.5\hskip 56.9055pt{\rm VIS}\,,$ (9) $18\leq m\leq 24\hskip
56.9055pt{\rm NISP}\,.$ (10)

In the following sections we will select galaxies from the available
simulation catalog according to Eqs. (9) and (10).

Figure 10 shows the number of galaxies expected around the required magnitude
range using the JWST counts from Windhorst et al. (2023). Given that the
statistical uncertainty is $\propto 1/\sqrt{N_{\rm gal}}$, we expect to have
only minor variations in the uncertainty as the magnitude range is refined if
necessary.

Figure 10: Number of galaxies after Year 1 of $2550$ deg2 coverage for
$m_{0}\leq m\leq m_{1}$. Circles correspond to $m_{0}=18$ and diamonds to
$m_{0}=22$. Filled/open symbols correspond to $m_{1}=25/24.5$. Red asterisks
show the numbers for the range in Eqs. (9) and (10) using the JWST counts
(Windhorst et al. 2023).

###  4.4 Understanding details of extinction in the measurement

####  4.4.1 Extinction

Extinction from dust in our own Galaxy would imprint apparent structure on an
otherwise isotropic extragalactic background, whether directly measured as the
CIB or reconstructed from observed galaxy brightnesses. Since the extinction
is most simply a function of Galactic latitude, the strongest effect is
expected in the quadrupole. One typically achieves $\sigma_{A}/A\sim 0.1$ per
band with photometric measurements and $\sigma_{A}/A\sim 0.01$ with
spectroscopic. However the Galactic ISM is highly and irregularly structured,
so a dipole component to the extinction will be present as well. Due to
Galactic structure the effects are expected to be smallest for the
$Z$-component (e.g. Gibelyou & Huterer 2012). The extinction dipole goes as
$\lambda^{-2}$ from $I_{\scriptscriptstyle\rm E}$ to $H_{\scriptscriptstyle\rm
E}$ in the opposite trend than that of the IGL.

Figure 11 shows maps of Galactic reddening, $E(B-V)$ from (Schlegel et al.
1998, hereafter SFD). The reddening map is masked by the cumulative coverage
of the Euclid Wide Survey for each of the six years of the mission. Each of
these masked images is fit for monopole and dipole components using the
HEALPix routine remove_dipole, and the resulting amplitudes (in magnitudes) of
each component are listed. Figure 12 (left) shows the the reddening converted
to extinction [using $A_{V}=R_{V}\ E(B-V)$] and plotted as a function of
latitude. At a fixed latitude there can be large variations in extinction.
However relatively low extinction regions may be found at latitudes as low as
$|b|\approx 40\degr$. Figure 12 (right) shows the total area of the sky that
has extinction lower than a given $A_{V}$ (i.e. below a horizontal line in the
left panel), or equivalently the area where the 100 $\mu$m emission is below
$I_{\rm 100\mu m}$, as $E(B-V)\approx 0.016\ I_{\rm 100\mu m}$. The 100 $\mu$m
results are shown for binning of the DIRBE measurements at three different
angular scales.

Figure 11: Cumulative coverage of the SFD $E(B-V)$ reddening map in Galactic
coordinates. $A_{V}\sim 3.1E(B-V)$ and $A_{\rm near-IR}\sim E(B-V)$. For each
year, annotations list the area of sky covered, the monopole, $\langle
E(B-V)\rangle$, and the three dipole components.

Maps of reddening or extinction are detailed but are subject to systematic
errors. Most commonly used ones originate in observations of far-IR emission
(e.g. Schlegel et al. 1998). Models are required to convert the emission to a
dust column density, then into a reddening $E(B-V)$, then into extinction
$A_{V}$, and finally to extinction at the desired wavelength $A_{\lambda}$.
Factors that influence these steps are dust temperature and composition, the
ratio of total to selective extinction $R_{V}$, and the reddening law
$A_{\lambda}$/$A_{V}$ (see Fig. 13). All of these are known to vary as a
function of line of sight, but it is common (sometimes necessary) to simply
adopt standard mean values for $R_{V}$ and the reddening law. This means that
there will be some imprint of extinction even on an IGL background that is
constructed from extinction corrected source magnitudes. The imprint will be
that of the errors in the extinction correction. The errors, $\sigma_{A_{V}}$,
will not necessarily have the same pattern on the sky as the extinction
itself, i.e. $\sigma_{A_{V}}\neq\epsilon A_{V}$, where $\epsilon\ll 1$.

The example of the SFD extinction maps used here shows that extinction
introduces a non-negligible diffuse dipole component, although it may differ
in some detail from other Galactic extinction maps (e.g. Schlafly & Finkbeiner
2011; Planck Collaboration et al. 2014; Delchambre et al. 2023) and their
wavelength dependence (e.g. Predehl & Schmitt 1995; Draine 2011). In what
follows we introduce methodology to remove the dipole contribution from
extinction (in the limit of low extinction, $A_{V}\ll 1$) independent of any
estimated extinction map.

Figure 12: Left: $A_{V}$ as a function of Galactic latitude derived by
converting SFD $E(B-V)$ results using $R_{V}=3.1$. The color excess
(reddening) is $E(B-V)=(B-V)_{\rm observed}-(B-V)_{\rm intrinsic}$ and
$R_{V}=A_{V}/E(B-V)$. Right: The cumulative area where the SFD extinction is
less than $A_{V}$ or where the COBE/DIRBE 100 $\mu$m intensity (shown at 3
resolutions) is less than $I_{100\ \mu{\rm m}}$.  Figure 13: Left: Galactic
extinction laws as reported by Draine (2011) (red $\circ$), Cardelli et al.
(1989) (black $\times$), Rieke & Lebofsky (1985) (green $\square$), and Savage
& Mathis (1979) (blue $+$). The dotted and dashed lines are the Cardelli et
al. (1989) evaluated for $R_{V}=2.75$ and 5.30, respectively. Right: The same
extinction laws are plotted normalized to the Draine (2011) extinction law to
show that differences of several percent exist, and that variations of
$\gtrsim 10\%$ can arise from $R_{V}$ variations.

####  4.4.2 Managing the extinction contributions

While the Galactic extinction can interfere with probing the intrinsic IGL
dipole, its interference can be removed with the method proposed in this
section. Let us say that the extinction correction, $A_{\nu}\ll 1$, is known
to within $\sigma_{A}$. Then, after the extinction correction the flux in Eq.
(7) becomes:

$\tilde{I}_{\nu}(l,b)=\left[1-0.4\,(\ln{10})\left(\frac{\sigma_{A}}{A}\right)A_{\nu}\right]I_{\nu}(l,b)\equiv
I_{\nu}(l,b)+\Delta I_{\nu}(l,b).$ (11)

If no extinction corrections are made then one should read $\sigma_{A}=A$.
This leads to the extinction uncertainty contribution to the measured IGL of

$\Delta I_{\nu}(l,b)=\pm
0.92\left(\frac{\sigma_{A}}{A}\right)A\,I_{\nu}(m_{0},m_{1}){|_{(l,b)}}\,.$
(12)

We now write $A\equiv\langle A\rangle+\delta A,I\equiv\langle I\rangle+\delta
I$. Assuming that the extinction map in Fig. 12 has a dipole
$\boldsymbol{D}_{A}$ we can write taking $\sigma_{A}/A=$const across the sky
leads to the following uncertainty in the IGL dipole, Eq. (12), from
extinction corrections

$\begin{split}\tilde{\boldsymbol{d}}_{\nu}=\leavevmode\nobreak\
&\boldsymbol{d}_{\nu}\left[1-0.92\left(\frac{\sigma_{A}}{A}\right)\langle
A_{\nu}\rangle\right]\\\
&-0.92\left(\frac{\sigma_{A}}{A}\right)\boldsymbol{D}_{A}\langle
I_{\nu}\rangle+O[{\rm dipole}(\delta A\,\delta I)]\,.\end{split}$ (13)

Here $D_{A}$ is the dipole of the extinction map to be evaluated from the
selected sky region in Fig. 12 (left). The spherical harmonic expansion is
defined via: $I_{\nu}=\sum_{\ell m}d_{\ell m}Y_{\ell m}\;;\;A_{\nu}=\sum_{\ell
m}D_{\ell m}Y_{\ell m}$. Over the full sky higher-$\ell$ harmonics do not
couple to lower-$\ell$ ones. The extinction component of the dipole due to Eq.
(12) decreases with wavelength ($\propto\lambda^{-2-\alpha_{\nu}}$) and, very
generally, has a very different dependence on $\lambda$ from that of the IGL,
$d_{\nu}$, shown in Fig. 2. This should enable component separation in Eq.
(11).

If Eq. (13) is valid then the task of minimizing the extinction contribution
to the probed IGL dipole is reduced to: 1) finding a region large enough to
contain many galaxies [Table 1 in Kashlinsky & Atrio-Barandela (2022) shows
that $N_{\rm gal}\gtrsim$1000$$–$2000$ deg2 could be enough] that has small
$\langle A_{\nu}\rangle$ and 2) where $D_{A}$ is sufficiently close to zero.
For the Euclid bands the dimensionless IGL dipole is
$(3-\alpha_{\nu})V/c\simeq 0.5\%$ if the measured CMB dipole is entirely
kinematic (if not, then it would be ${\simeq}\,0.5\%V/V_{\rm CMB})$. Hence in
the absence of further corrections discussed below, one would want to select
the sky regions with $D_{A}/\langle A\rangle\lesssim 0.2(\sigma_{A}/A)^{-1}$
in $I_{\scriptscriptstyle\rm E}$, and significantly more relaxed in
$H_{\scriptscriptstyle\rm E}$, in order to probe, in each of the bands, the
IGL dipole with the expected dimensionless amplitude of ${\simeq}\,0.5\%$ per
Fig. 23. The above hinges on three assumptions: 1) $D(\delta I\delta
A)\ll\langle I\rangle D_{A},\langle A\rangle d_{\nu}$, 2) $\langle
A\rangle,A\ll 1$, and 3) the coupling with higher $\ell$-order terms can be
neglected (for now). Choosing a region where $D_{A}$ is minimal can be
accomplished by selecting regions of the sky where $E(B-V)$ is roughly
constant. Figure 11 illustrates that while the total sky observed by Euclid
after any year contains a dipole, selection of an area of sky with near
constant $E(B-V)$ can reduce the dipole amplitude by roughly an order of
magnitude.

However, a more efficient technique would work as follows: Fig. 12 shows that
the additive term, when multiplying $\boldsymbol{d}_{\nu}$ in the RHS of Eq.
(13), is $0.92\langle A_{\nu}\rangle\ll 1$ and can be neglected. It further
affects only the precise conversion of the dipole into the velocity amplitude
(at less than a few percent level), not its direction. Assuming that the last
term on the RHS is negligible, a firmer way to eliminate the extinction terms
in Eq. (13) would be to 1) divide the large Euclid galaxy sample into two
groups with very different spectral/morphological properties and 2) use one to
eliminate the extinction dipole in the other. For example, if we define a
subsample “a” with, say $\alpha_{\nu}^{\rm a}\gtrsim 1$, and magnitude range
$m_{0}^{\rm a}<m<m_{1}^{\rm a}$, one would get for its dipole

$\frac{\tilde{\boldsymbol{d}}_{\nu}^{\rm a}}{\langle I_{\nu}^{\rm
a}\rangle}=(3-\alpha_{\nu}^{\rm
a})\frac{\boldsymbol{V}}{c}-0.92\boldsymbol{D}_{A}(\lambda)\,,$ (14)

and similarly for the main sample “b”. Then subtracting the two would lead to
the residual wavelength-dependent dipole from extinction corrections
determined as

$\begin{split}\boldsymbol{D}_{A}(\lambda)=&\frac{(0.92)^{-1}}{(\alpha_{\nu}^{\rm
b}-\alpha_{\nu}^{\rm a})}\\\ &\times\left[(3-\alpha_{\nu}^{\rm
a})\frac{\tilde{\boldsymbol{d}}_{\nu}^{\rm b}}{\langle I_{\nu}^{\rm
b}(m_{0}^{\rm b},m_{1}^{\rm b})\rangle}-(3-\alpha_{\nu}^{\rm
b})\frac{\tilde{\boldsymbol{d}}_{\nu}^{\rm a}}{\langle I_{\nu}^{\rm
a}(m_{0}^{\rm a},m_{1}^{\rm a})\rangle}\right]\,.\end{split}$ (15)

Whereas the wavelength-independent Compton–Getting velocity term is given by

$(\alpha_{\nu}^{\rm b}-\alpha_{\nu}^{\rm
a})\frac{\boldsymbol{V}}{c}=\boldsymbol{\Delta}_{\rm
ab}\equiv-[\boldsymbol{{\cal U}}_{\rm a}-\boldsymbol{{\cal U}}_{\rm b}]\,,$
(16)

where we further defined

$\boldsymbol{{\cal U}}_{a,b}\equiv\frac{\tilde{\boldsymbol{d}}_{\nu}^{\rm
a,b}}{\langle I_{\nu}^{\rm a,b}(m_{0}^{\rm a,b},m_{1}^{\rm a,b})\rangle}$ (17)

to be used later. Note that the rms in Eq. (16) depends on the relative
difference in $\alpha_{\nu}$ for the subsamples, rather than the absolute
Compton–Getting amplification, $(3-\alpha_{\nu})$.

The values of $\boldsymbol{D}_{A}$ determined empirically here will help
verify the accuracy of the Euclid extinction corrections. The errors resulting
from this procedure are discussed in the next section using simulated Euclid
catalogs. Naively speaking the statistical error on $D_{A}$ determined here
from sample “b” would be $\sigma_{D_{A}}\sim D_{A}({N^{b}_{\rm gal})}^{-0.5}$.

We will have plenty of galaxies to do this per Table 1 of Kashlinsky & Atrio-
Barandela (2022) and we want to choose subsample galaxies with, say,
$I_{\scriptscriptstyle\rm E}-Y_{\scriptscriptstyle\rm E}<0$,
$Y_{\scriptscriptstyle\rm E}-J_{\scriptscriptstyle\rm E}<0$,
$J_{\scriptscriptstyle\rm E}-H_{\scriptscriptstyle\rm E}<0$ (in the observer
frame) so that their IGL has positive $\alpha_{\nu}$, ideally closer to
$\alpha_{\nu}^{\rm RJ}=2$ and $m_{0}$ faint enough to ensure that the dipole
from the clustering component is small. Bisigello et al. (2020) show in their
Figs. 6 and 7 that one can choose a substantial subsample of such galaxies.
Since $I_{\nu}/\nu^{3}$ is Lorentz-invariant for each subsample its
$\alpha_{\nu}$ is independent. Hence we can select subsamples with a
significantly positive $\alpha_{\nu}$ from $I_{\scriptscriptstyle\rm E}$ to
$H_{\scriptscriptstyle\rm E}$ bands and the main sample from galaxies with
$\alpha_{\nu}$ substantially negative to compensate for the reduction in the
overall $N_{\rm gal}$. Then we would choose the optimal $(m_{0}^{\rm
sub},m_{1}^{\rm sub},m_{0},m_{1})$ to enable sufficient reduction in the
dipole from clustering. Then use Eq. (16) and propagate the errors; with this
in mind choose areas where $\tilde{d}_{\nu}$ (and $D_{A}$) are minimal. The
resultant velocity, Eq. (16), must be the same when derived at all bands and
the absorption dipole, $\boldsymbol{D}_{A}$ must point in the same direction
and have the corresponding wavelength dependence decreasing with $\lambda$.
This technique can be straightforwardly generalized to more subsamples with
sufficiently different $\alpha_{\nu}$.

###  4.5 Accounting for the Earth’s orbital motion

The overall cosmological information was not yet available when the
Compton–Getting effect was introduced almost 100 years ago as a way to probe
the orbital motion of the Earth from cosmic ray observations (Compton &
Getting 1935). The motivation is reversed here, but the well-known now Earth’s
orbital motion is important to account for in any precision measurement.

Dipole measurements having ${\rm S/N}\gtrsim 10$ will be sensitive to (require
correction for) the Earth’s (or spacecraft’s) orbital motion around the Sun at
$V_{\oplus}\sim 30$ km s-1. Mapping the data in a coordinate system that
rotates to keep the Sun fixed [i.e. $(l_{\rm ecl}-l_{\sun},b_{\rm ecl})$
rather than $(l_{\rm ecl},b_{\rm ecl})$], can be used to determine the “Solar
anisotropy” (e.g. Abbasi et al. 2012). In these coordinates the dipole can be
fit, and then its brightness can be subtracted from each of the observations
as mapped into a standard fixed coordinate system. In principle
$(V_{\oplus}/c)\cos\Theta$ is already known, but this fitting provides an
empirical measure of $\langle I\rangle$, which is needed for the subtraction.
For Euclid this dipole is poorly sampled in Year 1 because all observations
are towards the ecliptic poles, perpendicular to the Earth’s orbital motion
dipole. However, for the same reason, the effect of this dipole will be
correspondingly small in the Year 1 data. As the years progress and lower
ecliptic latitudes are observed, sensitivity to the Solar anisotropy dipole
increases. In this fixed-Sun coordinate system, the cosmological kinematic
dipole on the sky will be a noise term that averages down for Euclid as more
sky is sampled.

A more ideal solution is to modify dipole fitting routines (e.g. HEALPix
remove_dipole) to include input specifying the time (or Solar elongation) of
observation for each pixel and then subtract off the “Solar anisotropy” dipole
as part of the fitting. In this case the fitting does not involve any
additional free parameters, since the time (or Solar elongation) of the
observations is a known quantity.

###  4.6 Photometry and magnitudes

Because galaxies are extended sources without cleanly defined edges, there are
multiple ways to define the magnitude of a galaxy. Standard techniques include
using aperture photometry within a radius set by the shape of the galaxy
profile (e.g., Petrosian photometry), and fitting a generic profile shape to
galaxies to measure something akin to a total magnitude [such as the CModel
magnitudes developed by the Sloan Digital Sky Survey (Abazajian et al. 2004),
and being used by the Hyper Suprime-Cam survey and the Rubin Observatory
(Bosch et al. 2018)]. Such measurements will have Malmquist-like biases (i.e.,
where more faint galaxies scatter into the sample at the faint magnitude limit
than the converse), which will depend on the sources of noise. If those
sources of noise are not symmetric across the sky, this could imprint a false
dipole moment into the inferred magnitude range. Given that the Euclid Wide
Survey will have quite uniform coverage, and the location of Euclid at L2
means that the imaging depth should have little temporal variation, we
anticipate that this will be a subdominant effect in the IGL dipole
measurement, but we will have to measure it. The sky background of the
measurements will not be uniform across the sky, adding to the noise of the
galaxy photometry, and possibly leading to additional systematic errors. This
varying sky background is dominated by zodiacal light, with a possible
contribution at low Galactic latitudes from reflected starlight from diffuse
dust in the Milky Way (optical cirrus), although the latter is again likely to
be subdominant.

The availability of multi-band measurements of our galaxies allows us to carry
out the measurements of the dipole from multiple independently defined
subsamples, which should allow us to test the robustness of the results to
many of the systematic errors we have described in this paper. In particular,
we envision measuring the dipole on subsets of galaxies divided in the
following ways:

  * •

Dividing up the galaxies into different regions of color-color space, selected
to identify galaxies at different (photometric) redshifts and different
physical properties, and measuring the dipole for each;

  * •

Dividing the galaxies into bins of magnitude;

  * •

Dividing the sky by the season or year in which each patch was observed, to
test for systematic biases in photometric zeropoints with time or position of
the spacecraft;

  * •

Dividing the galaxy sample by the morphology of the galaxy, as measured, e.g.,
by Sérsic index.

Seeing consistent dipoles among all these divisions of the data will give us
confidence in the robustness of the results, and will allow us to constrain
any residual systematic effects.

###  4.7 Systematic corrections

The dipole from the configuration by Eq. (7) will have additional contribution
arising due to the Galaxy participating with the bulk motion $\Delta
m=-(2.5\operatorname{log_{10}}{\rm e})[1-\beta(m)]\frac{V}{c}\cos\Theta$ for
CIB sources with SED $f_{\nu}\propto\nu^{-\beta}$ (Ellis & Baldwin 1984; Itoh
et al. 2010). The correction to the dipole from this effect can be evaluated
by modifying the upper and lower limits on the integration in Eq. (3). Since
$V\ll c$ the dipole due to Eq. (1) will be modified by the $m_{0}$, $m_{1}$
variation to

$\boldsymbol{d}_{\nu}(m_{0}<m<m_{1})=[(3-\alpha_{\nu,m_{0}<m<m_{1}})+{\rm
d}\alpha_{\nu}]\frac{\boldsymbol{V}}{c}\langle
I_{\nu}(m_{0}<m<m_{1})\rangle\,,$ (18)

with

${\rm
d}\alpha_{\nu}=Q_{\nu}(m_{1})[1-\langle\beta(m_{1})\rangle]-Q_{\nu}(m_{0})[1-\langle\beta(m_{0})\rangle]\,,$
(19)

where we have defined

$Q_{\nu}(m)\equiv\frac{\frac{{\rm d}I_{\nu}}{{\rm d}m}|_{m}}{\langle
I_{\nu}(m_{0}\\!\\!<\\!\\!m\\!\\!<\\!\\!m_{1})\rangle}\,.$ (20)

It was argued that the correction $d\alpha_{\nu}$ in general is small
(Kashlinsky & Atrio-Barandela 2022), but here it must be evaluated explicitly
for the Euclid configurations. This correction will affect only the
(systematic) conversion of the measured IGL dipole into the effective velocity
amplitude.

Figure 14 shows the values of $Q_{\nu}(m_{0})$ and $Q_{\nu}(m_{1})$ for the
range of magnitudes defined by Eqs. (9) and (10) at the four Euclid bands with
the vertical axis deliberately plotted to compare with the expected near-IR
Compton–Getting amplification of $(3-\alpha_{\nu})\simeq 4.5$–$5.5$ at these
bands. We used the latest JWST observations of galaxy counts shown in Fig. 5.
The expected overall contribution, $\Delta\alpha_{\nu}$, after accounting for
the SED/color terms, $\bar{\beta}$, will be incorporated from Euclid
simulations in the next section.

Figure 14: The values of $Q_{\nu}(m_{0})$ (Eq. 20) vs. $\lambda$ are shown
with filled circles and $Q_{\nu}(m_{1})$ with open circles at the Euclid four
bands using the observed JWST galaxy counts from Windhorst et al. (2023). The
values of $(m_{0},m_{1})$ were selected per Eqs. (9) and (10). The displayed
vertical range reflects the Compton–Getting amplifications,
$(3-\alpha_{\nu})\sim 4$–$5.5$, reachable here.

The Euclid Wide Survey will have measurements at an effective $\lambda$,
within the four filters, from which we will need to reconstruct the genuine
high-precision $I_{\nu}$, Eq. (3) and its derivative, $\alpha_{\nu}$ together
with their uncertainties. This is discussed and quantified later in Sect. 5.3.
To aid with this we will have photometric $z$ for the entire sample and
spectroscopic $z$ for many of them. In particular, as discussed in Euclid
Collaboration: Scaramella et al. (2022); Laureijs et al. (2011) we would
expect upward of $5\times 10^{6}$ galaxies with spectroscopic redshifts
already in Year 1 providing a good sample for such estimates.

###  4.8 Photometric zero points

A precision measurement of the intrinsic dipole in galaxy counts requires
precision photometric calibration over the full survey footprint. Euclid’s
photometric calibration is good, but is not perfect, and any dipole component
in the error in photometric calibration will translate directly into a false
dipole signature. In particular, the dipole moment of a calibration error
$\Delta m_{\rm dipole}$ will behave exactly like that described in Eq. (18),
and the (false) bulk motion $V_{\rm false}$ inferred from this dipole is given
by

$\Delta m_{\rm dipole}=-(2.5\operatorname{log_{10}}{\rm
e})[1-\beta(m)]\frac{V_{\rm false}}{c}.$ (21)

For reasonable values of $\beta$, a $V_{\rm false}$ of 300 km/s (i.e., the
amplitude of the signal we are trying to measure) would correspond to a 0.5%
$\Delta m_{\rm dipole}$.

The Euclid Wide Survey strategy is described in detail in Euclid
Collaboration: Scaramella et al. (2022), and the calibration of the
photometric system is described in Euclid Collaboration: Schirmer et al.
(2022). Most of the roughly $15\,000$ deg2 of sky will be observed only once,
meaning that strategies that use overlaps to tie photometric calibration
together (e.g. Padmanabhan et al. 2008; Burke et al. 2018) will be of only
limited use. Rather, every 25–35 days, Euclid will observe a self-calibration
field within its continuous viewing zone near the north ecliptic pole, to
redetermine the photometric calibration of the bands. Section 6.3.3 of Euclid
Collaboration: Schirmer et al. (2022) predicts that temporal changes in the
photometric calibration will be determined to an accuracy of 1–2 milli-mag,
although the formal requirement on the calibration is far looser, 1.5% for the
NISP instrument. It is less clear what the dipole moment across the sky of
this calibration error will be. If the calibration error is uncorrelated
across the sky in each field that is observed, the dipole will be of order the
error per independent calibrated patch, divided by the square root of the
number of patches. In this context, the number of patches is determined by how
often the calibration is checked. This means $\sim 12$ patches per year of
survey operations. This would suggest a residual dipole error due to
calibration uncertainties that is small relative to the expected signal.
However, if the calibration errors are position-dependent in some way (e.g.,
somehow matching the scanning pattern of Euclid, or dependent on the angular
separation of any given field from the self-calibration field), the systematic
error on our measurement may be considerably larger, and we will need to work
with the Euclid calibration team to explore the systematic calibration errors.

We anticipate that any calibration errors, if they are due to, e.g., time-
dependent changes in Euclid’s throughput, will be correlated between Euclid’s
different bands. This means that comparisons of the inferred dipole between
bands is unlikely to be a panacea to such effects. Similarly, calibration
problems will be likely to affect the photometry of different galaxy
populations in the same way, so splitting galaxies by, e.g., measured color,
will not be informative.

##  5 Applying to the upcoming Euclid Wide Survey

###  5.1 Evaluating overall statistical uncertainties

Figure 15:  Variation of the Poisson errors over the cumulative sky coverage
of Fig. 11 as a function of mission years. Black, green and red correspond to
the $X$, $Y$ and $Z$ direction cosines. The blue straight line shows $\langle
X_{i}^{2}\rangle^{-1/2}=\sqrt{3}$, the value for a full sky coverage.

The statistical (Poisson) errors on each dipole component are a function of
the number density of galaxies and the area of the sky covered by the data. To
compute these statistical uncertainties for the magnitude ranges of Eqs. (9)
and (10) we used the number density of galaxies from the JWST counts, denoted
by red asterisks in Fig. 10. The number densities of galaxies on an area of
$2550$ deg2, the size of the observed region in the first year of integration
were $N_{\rm gal}=[313,270,313,362]\times 10^{6}$ for the Euclid
$I_{\scriptscriptstyle\rm E}$, $Y_{\scriptscriptstyle\rm E}$,
$J_{\scriptscriptstyle\rm E}$, and $H_{\scriptscriptstyle\rm E}$ filters,
respectively. The uncertainty in each component of the dipole $d_{\nu,i}$ is

$\sigma_{i}=\frac{1}{\sqrt{\langle X_{i}^{2}\rangle}}N_{\rm gal}^{-1/2}\,,$
(22)

where the index $i=(X,Y,Z)$ denotes the components of the dipole, $N_{\rm
gal}$ is the total number of galaxies and $\langle X_{i}^{2}\rangle$ is the
square of the $i$th component of the direction cosine averaged over the
observed region (Atrio-Barandela et al. 2010; Kashlinsky & Atrio-Barandela
2022).

In Fig. 15 we plot $\langle X_{i}^{2}\rangle^{-1/2}=\sigma_{i}N_{\rm
gal}^{1/2}$, which measures the variation of the Poisson errors due to the
increment of the sky coverage by mission years shown in Fig. 11. Black, green
and red dots show this magnitude for the $X$, $Y$, and $Z$ direction cosines,
respectively. The blue solid line shows the same magnitude for a full sky
coverage. In the first two years, the mission will observe preferentially
close to the ecliptic poles and the $Y$ and $Z$ components are reasonably well
measured. At the end of the mission, the $Z$ component will have the smallest
error bar since the areas near the Galactic poles will be the regions best
observed by Euclid. A different scanning strategy will lead to different
errors. Notice that, while the error on $Z$ is smaller than $\sqrt{3}$, the
other two components are measured with an error larger than $\sqrt{3}$ and the
error on the dipole amplitude is, as expected, larger than for a full sky
coverage; see Atrio-Barandela et al. (2010) for extensive discussion.

Figure 16:  Regions of 68% (dark grey), 95% (middle grey) and 99.75% (light
grey) confidence level measured from the region of the sky after 1 to 6 years
of observations with the $I_{\scriptscriptstyle\rm E}$ band. The left panel
shows the uncertainty of the dipole amplitude, with the horizontal thick line
indicating the expected amplitude for a velocity of $V=370$ km s-1. The right
panel shows the uncertainty on the CMB dipole direction determination. The
dipole is assumed to be in the direction of the Solar dipole,
$(l,b)=(263\aas@@fstack{\circ}85,48\aas@@fstack{\circ}25)$ in Galactic
coordinates.  Figure 17:  Same as in Fig. 16 for the NISP filters. In the top
three panels the uncertainty of the dipole amplitude is represented, with the
horizontal thick line showing the expected amplitude. The lower panels show
the uncertainty on the CMB dipole direction. Left, middle and right panels
correspond to the $Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$
and $H_{\scriptscriptstyle\rm E}$ NISP Euclid filters, as indicated.

In Figs. 16 and 17 we present the expected confidence contours on the dipole
amplitude and direction for the VIS and NISP Euclid Wide Survey data obtained
from the Poisson errors $\sigma_{i}$. We assumed the dipole is in the
direction of the CMB Solar dipole
$(l,b)=(263\aas@@fstack{\circ}85,48\aas@@fstack{\circ}25)$. For each filter
and year of observation we generated $10^{5}$ Gaussian distributed random
errors around the CMB measured components $(d_{\nu,X},d_{\nu,Y},d_{\nu,Z})$.
We computed the random dipole amplitudes and their angular separations with
respect to the CMB dipole direction. The confidence levels shown in Fig. 16
were defined as the regions that enclose the 68%, 95%, and 99.75% of all
simulated amplitudes and directions. The left panel displays the contours in
the amplitude and the right panel in the direction. The darkest and lightest
colors correspond to the 1 and 3 $\sigma$ contours, respectively. The
horizontal line in the left panel represents the dipole amplitude with the
Solar System moving at $V=V_{\rm CMB}=370$ km s-1 with respect to the CMB
frame. In Fig. 17, the top three panels represent the dipole amplitude and the
bottom three panels the dipole direction and their uncertainties with the same
notation than in Fig. 16. The left, center and right panels correspond to the
NISP filters $Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$, and
$H_{\scriptscriptstyle\rm E}$, as indicated. The $(3-\alpha_{\nu})(V/c)$
coefficients used to estimate the statistical significance are given later by
Eq. (33) in Sect. 5.3 below.

If the CMB dipole is entirely kinematic the statistical significance will be
dominated by the $Z$ component as shown in Fig. 15 due to a larger number of
observations close to the Galactic poles. As the figure shows, for the assumed
cumulative sky coverage, the $Y$ component is more efficiently probed in the
first two years, when the observations are closer to the ecliptic poles. The
$X$ component will always be ill sampled and since its value is close to zero,
its ${\rm S/N}$ will be always negligible.

The number of galaxies given separately by Eqs. (9) and (10) and the
amplification factor with respect to the CMB dipole are different for each of
the Euclid bands although the overall ${\rm S/N}$ is very similar for the
three NISP filters. The measured amplitude and direction on each of the
filters must be consistent with the uncertainties given in Figs. 16 and 17.
Larger differences will be an indication of systematics from extinction
correction, star contamination, etc., if remaining in the data. For instance,
extinction will be a stronger contaminant for $I_{\scriptscriptstyle\rm E}$
than for the $H_{\scriptscriptstyle\rm E}$ filter. In the following section we
use a galaxy catalog obtained from the Euclid Flagship Mock Galaxy Catalogue
to apply the methodology developed in the previous section for eliminating
extinction dipole contributions remaining in the Euclid Wide Survey.

The high precision measurement of the IGL/CIB dipole would allow subdividing
the galaxy sample by narrow magnitude bins within the range of Eqs. (9) and
(10) in order to probe any dependence of the velocity on these parameters.
Likewise, given the expected large sample of the Euclid Wide Survey galaxies
with spectroscopic redshifts, one can divide the galaxies by $z$ in order to
probe the behavior of $V$ in different $z$-shells.

###  5.2 Uncertainties on the IGL/CIB kinematic dipole after extinction
corrections

To incorporate the proposed above method for removing residual dipole from
extinction corrections we used the Euclid Flagship Mock Galaxy Catalogue
(version 2.1.10) (Castander et al., in prep.) This catalog contains photometry
in the Euclid bands (and many other UV – near-IR bands), and has major
emphasis on modelling the clustering and shapes of galaxies in the range
$0<z<3$, as needed for dark energy studies. This catalog does include Galactic
extinction and could be used to calculate $\boldsymbol{D}_{A}$, but its
${\sim}\,4.8$ billion sources are distributed only over ${\sim}\,$5000$$ deg2
in the general direction of the north Galactic pole. We used CosmoHub
(Carretero et al. 2017; Tallada et al. 2020) to download a 1/128th fraction of
the catalog (${\sim}\,38$ million galaxies) including photometry (with
extinction applied) at (Subaru) $b$, (Euclid) $I_{\scriptscriptstyle\rm E}$,
$Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$,
$H_{\scriptscriptstyle\rm E}$, and (WISE) W1 and W2 bands. Ancillary
parameters downloaded were the galactic coordinates $(l,b)$, the value of the
color excess [$E(B-V)$], and the redshift ($z$) of each source. The downloaded
catalog contains 38 million galaxies, a factor of ${\sim}\,1.6$ larger than
expected from the observed galaxy counts and the HRK reconstruction. This is a
known issue that was communicated to Euclid by K. Helgason (private
communication), but has no consequences for our goal here of testing the
separation of galaxies by the resultant $\alpha_{\nu}$ (the logarithmic slope
of the CIB with $\nu$). The catalog provides individual apparent galaxy
fluxes, $F$, in each band in units of erg Hz-1 s-1 cm-2, which were also
converted into AB magnitudes as $m=-2.5\operatorname{log_{10}}F-48.6$. We then
select a conservative subset of galaxies satisfying simultaneously both Eqs.
(9) and (10).

From the downloaded catalog we removed galaxies at VIS/NISP bands
$m<m_{0}=19/18$ and $m>m_{1}=24.5/24$. From the overall sample covering total
area $A$ sr at $m_{0}\leq m\leq m_{1}$ we computed the net CIB flux density in
MJy sr-1 as $I_{\nu}=\sum F_{i}/A$ at each frequency $\nu$ and used the data
to evaluate its logarithmic derivative $\alpha_{\nu}$ between 0.4 and 5
$\mathrm{\SIUnitSymbolMicro m}$ for each subsequently selected subsample
aiming to divide into at least two groups in Eq. (16). As discussed in the
previous section this achieves two goals: 1) verifying the same
$\boldsymbol{V}(\alpha)$ at each $\alpha_{\nu}$ as well as $\nu$ for every
pair of sample+subsample, and 2) refining the overall $\boldsymbol{V}$ via
$\chi^{2}$ from finer sample binning with each having more uniform $\alpha$.

We now turn to several specific examples of binning by the effective to apply
the method to isolate and separate the extinction term from the kinematic
IGL/CIB dipole. We define the effective slope between two adjacent Euclid
wavelengths (1 and 2) for each Flagship catalog source contributing individual
fluxes $F_{1,2}$ to IGL as

$\beta_{1\rightarrow
2}\equiv\frac{\ln(F_{1}/F_{2})}{\ln(\lambda_{1}/\lambda_{2})}\,.$ (23)

In the first example we select all sources in the 2 individual subsamples
where each color satisfies: 1)

$[\beta_{I_{\scriptscriptstyle\rm E}\rightarrow Y_{\scriptscriptstyle\rm
E}},\beta_{Y_{\scriptscriptstyle\rm E}\rightarrow J_{\scriptscriptstyle\rm
E}},\beta_{J_{\scriptscriptstyle\rm E}\rightarrow H_{\scriptscriptstyle\rm
E}}]\leq\beta$ (24)

for the extinction, and 2)

$[\beta_{I_{\scriptscriptstyle\rm E}\rightarrow Y_{\scriptscriptstyle\rm
E}},\beta_{Y_{\scriptscriptstyle\rm E}\rightarrow J_{\scriptscriptstyle\rm
E}},\beta_{J_{\scriptscriptstyle\rm E}\rightarrow H_{\scriptscriptstyle\rm
E}}]\geq\beta$ (25)

for the IGL dipole.

Figure 18: Blue asterisks mark selection according to Eq. (24), and red
circles according to Eq. (25). The left panel shows the fraction of the
Flagship catalog galaxies vs. $\beta$ according to Eqs. (24) and (25). Symbol
size increases with the sample size. The right panel shows the resultant
$\alpha_{\nu}$ vs. wavelength for the two subsamples following the notations
in the left panel. The panel demonstrates the separability when Eq. (16) is
applied at the marked Euclid bands of $I_{\scriptscriptstyle\rm E}$,
$Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$, and
$H_{\scriptscriptstyle\rm E}$.

Figure 18 (left) shows the fraction of galaxies vs. $\beta$ for each category
in this case marked with red asterisks for Eq. (24) and blue circles according
to Eq. (25). The plot shows that one could select samples of sufficient size
in order to achieve high statistical accuracy when separating extinction
contributions. The right panel of the figure shows the corresponding
$\alpha_{\nu}$ for the IGL from sources in each subsample. This demonstrates a
clear difference between the Compton–Getting dipole amplification in the two
subsample as required for good separation according to Eq. (16). The figure
illustrates the desirable separability of IGL by $\alpha_{\nu}$ and show that
one robustly recovers
$\Delta\alpha_{\nu}\equiv|\alpha_{\nu}^{a}-\alpha_{\nu}^{b}|\gtrsim 2$ at the
Euclid four bands, which are marked in black ($I_{\scriptscriptstyle\rm E}$),
blue ($Y_{\scriptscriptstyle\rm E}$), green ($J_{\scriptscriptstyle\rm E}$),
and red ($H_{\scriptscriptstyle\rm E}$).

In Figs. 19–21 we present similar examples using alternate criteria for
subsample selection. Subsample selections for Fig. 19 include only NISP colors
(omitting the bluest, $I_{\scriptscriptstyle\rm E}$, band):

$\displaystyle[\beta_{Y_{\scriptscriptstyle\rm E}\rightarrow
J_{\scriptscriptstyle\rm E}},\beta_{J_{\scriptscriptstyle\rm E}\rightarrow
H_{\scriptscriptstyle\rm E}}]$ $\displaystyle\leq$ $\displaystyle\beta\,,$
(26) $\displaystyle[\beta_{Y_{\scriptscriptstyle\rm E}\rightarrow
J_{\scriptscriptstyle\rm E}},\beta_{J_{\scriptscriptstyle\rm E}\rightarrow
H_{\scriptscriptstyle\rm E}}]$ $\displaystyle\geq$ $\displaystyle\beta\,.$
(27)

Figure 19: Blue asterisks mark selection according to Eq. (27), and red
circles according to Eq. (27). The left panel shows the fraction of the
Flagship catalog galaxies vs. $\beta$ according to Eqs. (27) and (27). Symbol
size increases with the sample size. The right panel shows the resultant
$\alpha_{\nu}$ vs. wavelength for the two subsamples following the notations
in the left panel. The panel demonstrates the separability when Eq. (16) is
applied at the marked Euclid bands of $I_{\scriptscriptstyle\rm E}$,
$Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$, and
$H_{\scriptscriptstyle\rm E}$.

Subsample selections for Fig. 20 omit the reddest, $H_{\scriptscriptstyle\rm
E}$, band:

$\displaystyle[\beta_{I_{\scriptscriptstyle\rm E}\rightarrow
Y_{\scriptscriptstyle\rm E}},\beta_{Y_{\scriptscriptstyle\rm E}\rightarrow
J_{\scriptscriptstyle\rm E}}]$ $\displaystyle\leq$ $\displaystyle\beta\,,$
(28) $\displaystyle[\beta_{I_{\scriptscriptstyle\rm E}\rightarrow
Y_{\scriptscriptstyle\rm E}},\beta_{Y_{\scriptscriptstyle\rm E}\rightarrow
J_{\scriptscriptstyle\rm E}}]$ $\displaystyle\geq$ $\displaystyle\beta\,.$
(29)

Figure 20: Blue asterisks mark selection according to Eq. (29), and red
circles according to Eq. (29). The left panel shows the fraction of the
Flagship catalog galaxies vs. $\beta$ according to Eqs. (29) and (29). Symbol
size increases with the sample size. The right panel shows the resultant
$\alpha_{\nu}$ vs. wavelength for the two subsamples following the notations
in the left panel. The panel demonstrates the separability when Eq. (16) is
applied at the marked Euclid bands of $I_{\scriptscriptstyle\rm E}$,
$Y_{\scriptscriptstyle\rm E}$, $J_{\scriptscriptstyle\rm E}$, and
$H_{\scriptscriptstyle\rm E}$.

For Fig. 21, we select all sources in the two individual subsamples where the
$1\rightarrow 2$ color for each of the three pairs ($I_{\scriptscriptstyle\rm
E}$,$Y_{\scriptscriptstyle\rm E}$), ($Y_{\scriptscriptstyle\rm
E}$,$J_{\scriptscriptstyle\rm E}$), and ($J_{\scriptscriptstyle\rm
E}$,$H_{\scriptscriptstyle\rm E}$) separately satisfies

$\displaystyle\beta_{\rm 1\rightarrow 2}$ $\displaystyle\leq$
$\displaystyle{\rm median}(\beta_{\rm 1\rightarrow 2})\,,$ (30)
$\displaystyle\beta_{\rm 1\rightarrow 2}$ $\displaystyle\geq$
$\displaystyle{\rm median}(\beta_{\rm 1\rightarrow 2})\,.$ (31)

Figure 21: Blue asterisks mark selection according to Eq. (31), and red
circles according to Eq. (31). The left panel shows the fraction of the
Flagship catalog galaxies vs. $\beta$ according to Eqs. (31) and (31). Symbol
size increases from ($I_{\scriptscriptstyle\rm E}$,$Y_{\scriptscriptstyle\rm
E}$), through ($Y_{\scriptscriptstyle\rm E}$,$J_{\scriptscriptstyle\rm E}$),
to ($J_{\scriptscriptstyle\rm E}$,$H_{\scriptscriptstyle\rm E}$). The right
panel shows the resultant $\alpha_{\nu}$ vs. wavelength for the two subsamples
following the notations in the left panel. The panel demonstrates the
separability when Eq. (16) is applied at the marked Euclid bands of
$I_{\scriptscriptstyle\rm E}$, $Y_{\scriptscriptstyle\rm E}$,
$J_{\scriptscriptstyle\rm E}$, and $H_{\scriptscriptstyle\rm E}$.

Now we can solve for $\boldsymbol{D}_{A}$ and $\boldsymbol{V}$ in Eqs. (15)
and (16) and determine their uncertainties. Equation (16) is sensitive to the
difference in $\alpha$’s between the “(a,b)” subsamples. Hence, to optimize
the ${\rm S/N}$ for Eq. (16) we need to select 1) a more amplified subsample
“a” (with more negative $\alpha_{\nu}^{a}$), then 2) select subsample “b” with
much less negative $\alpha_{\nu}^{b}$, so ${\cal U}_{\rm b}^{2}\ll{\cal
U}_{\rm a}^{2}$ (squares are since the uncertainties add in quadrature), while
3) keeping enough galaxies in the subsamples, so that 4) the final signal-to-
noise ratio

$({\rm S/N})_{\Delta_{\rm ab}}\simeq({\rm
S/N})\left[\frac{|\Delta\alpha_{\nu}|}{(3-\alpha_{\nu}^{\rm
total})}\;\;\sqrt{\frac{f_{\rm a}f_{\rm b}}{f_{\rm a}+f_{\rm b}}}\;\right]$
(32)

will still be high enough. In this expression, $f_{\rm a}$ and $f_{\rm b}$ are
the fractions of ${\rm a}$ and ${\rm b}$ galaxies to the total number of
galaxies with $\alpha_{\nu}^{\rm a,b}$ being the spectral indices of their
respective IGLs. Equation (32) shows that significant ${\rm S/N}$ can be
achieved by dividing the galaxy population into distinct samples with very
different $\alpha_{\nu}$ and the figures in this section show that several of
these samples are possible. We note that this procedure will be required only
if we determine, from the actual data, that the extinction dipole, Eq. (15),
is significant in all the Euclid bands. If it turns out negligible, the method
for correcting for the remaining extinction proposed here will not be required
and we will proceed with the measurement per Eqs. (7) and (8) directly.

Figure 22: Statistical significance of the three dipole components and the
dipole amplitude measured using the $I_{\scriptscriptstyle\rm E}$ data and
assuming the CMB dipole template, Eq. (6). Black, green, red full circles
correspond to the $X$, $Y$, and $Z$ dipole components; blue open circles to
the overall dipole amplitude $d$. The left panel (a) shows the result after
one year of integration and the right panel (b) at the end of the 6-year
mission. The fraction of the total number of galaxies in samples a and b are,
from left to right, $f_{\rm a}=[0.5,0.2,0.1,0.2,0.2,0.2,0.15,0.1,0.2]$ and
$f_{\rm b}=[0.5,0.3,0.3,0.25,0.2,0.2,0.2,0.2,0.1]$, respectively. For other
configurations the ${\rm S/N}$ will increase according to Eq. (32).

If the extinction dipole, Eq. (15), turns out to be important we will proceed
as outlined in this section. This discussion suggests many possibilities to
optimize the measurement after isolating the extinction. Equation (32) shows
that this is achievable with the ${\rm S/N}$ loss of a factor of ${\sim}\,2$
if one concentrates on galaxy subsamples with $|\Delta\alpha_{\nu}|\sim 2$–3
while keeping the bulk of galaxies in both samples, so that $f_{\rm a},f_{\rm
b}\sim 0.3$–0.5. Additionally, the measurement in the four Euclid bands with
significantly varying extinction levels may lead to further clarity, since the
$H_{\scriptscriptstyle\rm E}$ channel will have an order of magnitude lower
extinction levels than $I_{\scriptscriptstyle\rm E}$.

As the worst-case scenario in terms of the extinction contribution we consider
the application of the presented formalism to the $I_{\scriptscriptstyle\rm
E}$ band. Figure 22 shows with full circles the ${\rm S/N}$ of the three
IGL/CIB dipole components $(X,Y,Z)$ and the overall IGL/CIB dipole amplitude
$d$ with open circles, for the first year and after six years of observations
using the data of the $I_{\scriptscriptstyle\rm E}$ band alone and assuming
$\boldsymbol{V}=\boldsymbol{V}_{\rm CMB}$. Since the dipole amplitude follows
a $\chi^{2}$ distribution, we define the uncertainty as half the width of the
interval enclosing the 68% confidence level. As indicated in Sect. 5.1 the
best measured component is always $Z$ since Euclid will be observing
preferentially around the Galactic poles, while the $Y$ component is better
determined in the first two years of the mission, when the observations take
place near the ecliptic poles. For the assumed dipole direction to be
coincidental with the CMB dipole, the amplitude of the $X$ component is
negligible so its statistical significance is always small; this would change
if the observed CMB dipole has a non-kinematic component.

Dividing the galaxy sample into two equally sized subsamples with
$|\Delta\alpha_{\nu}|\gtrsim 2$ we can measure in $I_{\scriptscriptstyle\rm
E}$ alone the amplitude of the IGL/CIB dipole with ${\rm S/N}\gtrsim 15$ after
the first year and $\gtrsim 30$ at the end of the mission. After six years of
observation, the $Z$ dipole component is determined about a factor of two
better than $Y$, dominating the statistical significance of the dipole
amplitude, that is only a few percent better than $Z$. Depending on the value
of $|\Delta\alpha_{\nu}|$ four or six different samples could be constructed
and optimized from the actual data, so the statistical significance could
additionally increase. Similar results hold for the NISP filters, resulting in
a further increment of a factor of $2$. However, the statistical significance
would be ${\rm S/N}>100$ if the extinction dipole is small at least at the
longest wavelengths and does not need to be subtracted off from the data.
Comparison of the measured IGL/CIB dipole subtracting and not subtracting the
extinction dipole will further indicate the importance of this component at
each frequency. Furthermore, juxtaposition between different frequencies will
be a measure of systematic uncertainties. Large differences would indicate
that extinction and/or other systematic uncertainties are present in the data
and if such differences do not exist it will provide a strong vindication of
the final result for the IGL/CIB dipole.

If the post-extinction-correction IGL/CIB maps at the different Euclid bands
are found to be uncorrelated (e.g. the extinction dipoles, Eq. (15) are found
to be widely different) one can potentially gain an improvement of up to a
factor of 2 in the ${\rm S/N}$ over that shown in Fig. 22.

To conclude this section, we have shown explicitly the many combinations that
can be selected by color between the various Euclid bands, in order to isolate
the remaining extinction contribution and achieve the required IGL/CIB dipole
measurement. Of course, when the data arrive they will be dissected in more
possible ways to fine-tune and optimize the measurement, and still more
combinations of colors may be considered. The formalism developed here for
isolating and removing the extinction contributions to the CIB dipole is
mathematically precise and independent of any particular extinction model with
the SFD maps used merely as an example of what the extinction dipole may look
like.

###  5.3 Reducing systematic amplification uncertainties

Finally we concentrate on the systematic corrections needed to translate
accurately the IGL/CIB dipole, for a well determined direction, into the
corresponding velocity amplitude. The corrections below affect all 3
components of the velocity vector equally, leaving the direction intact. We
will reconstruct the IGL/CIB from counts, evaluate as accurately as possible
the value of $\alpha_{\nu}$ across the Euclid spectrum, probe its dipole
components and then deduce the effective velocity accounting for the
systematics below.

Our task is to translate the measured CIB dipole into the equivalent velocity
and compare to $V_{\rm CMB}=370$ km s-1 in the precisely known direction
$(l,b)_{\rm CMB}=(263\aas@@fstack{\circ}85\pm
0\aas@@fstack{\circ}1,48\aas@@fstack{\circ}25\pm 0\aas@@fstack{\circ}04)$
(Hinshaw et al. 2009). For precision measurement we must translate the
measured dipole amplitude into the equivalent velocity with required accuracy
when selecting galaxies with $m\geq m_{0}$ (to eliminate the contribution to
dipole from galaxy clustering). Figure 23 shows the dimensionless
Compton–Getting amplified IGL dipole amplitude for $V=370$ km s-1 expected for
Flagship 2.1 galaxies satisfying simultaneously Eqs. (9) and (10).
Specifically, if the CMB dipole is purely kinematic, one would recover at the
four Euclid bands, $I_{\scriptscriptstyle\rm E}$, $Y_{\scriptscriptstyle\rm
E}$, $J_{\scriptscriptstyle\rm E}$, and $H_{\scriptscriptstyle\rm E}$

$(3-\alpha_{\nu})\frac{V_{\rm CMB}}{c}=[6.4,5.7,5.3,5.0]\times 10^{-3}\,.$
(33)

Figure 23: The solid line shows the dimensionless IGL/CIB dipole from the
Flagship2.1 catalog in the configuration used here combining the configuration
given by Eqs. (9) and (10). The figure assumes that $V=V_{\rm CMB}$, with the
CMB dimensionless dipole marked with the thick horizontal line. The Euclid
filters are shown with the filled circles showing the values at their nominal
wavelengths. The Roman F213 filter covering photons just outside of the
$H_{\scriptscriptstyle\rm E}$ NISP band, JWST F210M and F277W filters also
near and longward of the $H_{\scriptscriptstyle\rm E}$ band, and Rubin’s $g$,
$r$, $i$, and $z$ filters are also shown. With this additional coverage the
value of $\alpha_{\nu}$ can be evaluated to good accuracy.

The systematic correction, $\Delta\alpha_{\nu}$ discussed in Sect. 4, if
uncorrected for would lead to velocity amplitude difference of $\Delta
V/V\simeq d\alpha_{\nu}/(3-\alpha_{\nu})$, or

$\frac{\Delta
V}{V}=\frac{Q_{\nu}(m_{1})[1-\langle\beta(m_{1})\rangle]-Q_{\nu}(m_{0})[1-\langle\beta(m_{0})\rangle]}{(3-\alpha_{\nu})}\,.$
(34)

For the Flagship2.1 catalog the relative magnitude of this systematic
correction, $\Delta V/V$, was evaluated to be $[0.04,0.05,0.07,0.17]$ in the
[$I_{\scriptscriptstyle\rm E}$, $Y_{\scriptscriptstyle\rm E}$,
$J_{\scriptscriptstyle\rm E}$, $H_{\scriptscriptstyle\rm E}$ ] bands and is
less than or comparable in magnitude to the correction due to Earth’s orbital
velocity. It does not affect the direction of the dipole and when converting
the measured IGL/CIB dipole into the equivalent amplitude for the velocity the
magnitude of $d\alpha_{\nu}$ will be evaluated for each band directly from the
Euclid galaxy data and incorporated into the overall amplification per Eqs.
(18) and (34). Moreover, we can gauge the effects of $\alpha_{\nu}$ by
considering the galaxy SED estimated by the photometric redshift fitting.

The derivative $\alpha_{\nu}$ may not be highly precise given the sparsity of
points from Euclid’s four bands alone, particularly at
$I_{\scriptscriptstyle\rm E}$ and $H_{\scriptscriptstyle\rm E}$. To more
accurately compute $\alpha_{\nu}$, Eq. (2), in each of the four Euclid bands
it would be useful to assemble a subset of the Euclid Wide Survey galaxies at
wavelengths shorter than $I_{\scriptscriptstyle\rm E}$ and longer than
$H_{\scriptscriptstyle\rm E}$ as e.g. shown in the figures in Sect. 5.2. From
such a subsample we would evaluate the net IGL over the range of Euclid-
selected magnitudes, Eqs. (9) and (10), and then evaluate its logarithmic
derivative at $I_{\scriptscriptstyle\rm E}$ and $H_{\scriptscriptstyle\rm E}$.
This can be achieved by using $ugriz$ data from various ground-based surveys
that are providing complementary short wavelength data (see Euclid
Collaboration: Scaramella et al. 2022), and using past Spitzer and future
Roman data at longer wavelengths. The additional Roman data will be
particularly useful here for accurately evaluating the Compton–Getting
amplification at the $H_{\scriptscriptstyle\rm E}$ band. The data will be
collected, even if over smaller area as currently envisaged, at F184 and F213
bands to magnitudes much deeper than the Euclid Wide Survey resulting in very
rich sample of galaxies in the range of Eqs. (9) and (10) as discussed in
Akeson et al. (2019).

The compilation of multiband galaxy photometry in the COSMOS field by the
COSMOS2020 team (Weaver et al. 2022) can be used to measure typical values for
$\alpha_{\nu}$ in the $I_{\scriptscriptstyle\rm E}$ and
$H_{\scriptscriptstyle\rm E}$ bands. The COSMOS2020 compilation includes very
deep multiband $YJHK_{s}$ imaging reaching $m>25$, $grizy$ photometry to 26–27
mag, and 3.6 and 4.5 $\mu$m to $m>25.5$ covering the 2 deg2 COSMOS field. The
COSMOS $K_{s}$ imaging has been homogenized and is considerably deeper than
the 24 mag requirement over the full field. All the COSMOS2020 imaging is
registered astrometrically to Gaia precision. Given the combination of
sensitivity and precision available for the publicly available COSMOS2020
catalog, we will easily be able to construct galaxy counts for of order
$200\,000$ $J$-detected galaxies reaching 24 mag in the $J$, $K_{s}$, 3.6
$\mu$m, and 4.5 $\mu$m bands (Fig. 11 of Weaver et al. 2022). We can go
similarly deep blueward of the $I_{\scriptscriptstyle\rm E}$ band, allowing us
to make a high-precision estimate of $\alpha_{\nu}$ in the Euclid
$H_{\scriptscriptstyle\rm E}$ and $I_{\scriptscriptstyle\rm E}$ bands. The
data from Spitzer at 3.6 nad 4.5 $\mathrm{\SIUnitSymbolMicro m}$ from e.g.
Ashby et al. (2013, 2018) would be further useful in refining the
high(er)-precision evaluation of $\alpha_{\nu}$.

It would be sufficient to collect photometric data shortward and longward of
the four Euclid bands for only a small fraction of the Euclid galaxies in the
range covered by Eqs. (9) and (10) to robustly probe the IGL and evaluate its
$\alpha_{\nu}$ at the Euclid filters $I_{\scriptscriptstyle\rm E}$ and
$H_{\scriptscriptstyle\rm E}$. The additional filters particularly useful here
are shown in Fig. 23. Longward of $H_{\scriptscriptstyle\rm E}$ is the Roman
F213 filter which probes emissions just outside of 2
$\mathrm{\SIUnitSymbolMicro m}$. The required magnitude coverage needed here
is well within the Roman planned program currently scheduled to begin in late
2027, which nominally goes much deeper than Euclid. Moreover the ongoing and