11institutetext: 1 INAF-Osservatorio di Astrofisica e Scienza dello Spazio di
Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy
2 INFN-Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
3 INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143
Trieste, Italy
4 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151
Trieste, Italy
5 Dipartimento di Fisica - Sezione di Astronomia, Universitá di Trieste, Via
Tiepolo 11, 34131 Trieste, Italy
6 INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste TS, Italy
7 Institut für Theoretische Astrophysik, Zentrum für Astronomie, Heidelberg
Universität, Albert-Ueberle-Str. 2, 69120, Heidelberg, Germany
8 Dipartimento di Fisica e Astronomia ”Augusto Righi” - Alma Mater Studiorum
Università di Bologna, via Piero Gobetti 93/2, 40129 Bologna, Italy
9 Departamento de Física Teórica, Facultad de Ciencias, Universidad Autónoma
de Madrid, 28049 Cantoblanco, Madrid, Spain
10 Centro de Investigación Avanzada en Física Fundamental (CIAFF), Facultad de
Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
11 Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
12 International Centre for Radio Astronomy Research, University of Western
Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
13 Laboratoire Univers et Théorie, Observatoire de Paris, Université PSL,
Université Paris Cité, CNRS, 92190 Meudon, France
14 AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, 91191 Gif-
sur-Yvette, France
15 Institut für Astro- und Teilchenphysik, Universität Innsbruck,
Technikerstr. 25/8, 6020 Innsbruck, Austria
16 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71,
53121 Bonn, Germany
17 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-
sur-Yvette, France
18 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405,
Orsay, France
19 Institut für Theoretische Physik, University of Heidelberg, Philosophenweg
16, 69120 Heidelberg, Germany
20 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748
Garching, Germany
21 INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino
Torinese (TO), Italy
22 Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, 16146,
Genova, Italy
23 INFN-Sezione di Roma Tre, Via della Vasca Navale 84, 00146, Roma, Italy
24 Department of Physics ”E. Pancini”, University Federico II, Via Cinthia 6,
80126, Napoli, Italy
25 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP,
Rua das Estrelas, PT4150-762 Porto, Portugal
26 Dipartimento di Fisica, Universitá degli Studi di Torino, Via P. Giuria 1,
10125 Torino, Italy
27 INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
28 INAF-IASF Milano, Via Alfonso Corti 12, 20133 Milano, Italy
29 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona), Spain
30 Port d’Informació Científica, Campus UAB, C. Albareda s/n, 08193 Bellaterra
(Barcelona), Spain
31 Institut d’Estudis Espacials de Catalunya (IEEC), Carrer Gran Capitá 2-4,
08034 Barcelona, Spain
32 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans,
s/n, 08193 Barcelona, Spain
33 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio
Catone, Italy
34 INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, 80131
Napoli, Italy
35 INFN section of Naples, Via Cinthia 6, 80126, Napoli, Italy
36 Centre National d’Etudes Spatiales, Toulouse, France
37 Institut national de physique nucléaire et de physique des particules, 3
rue Michel-Ange, 75794 Paris Cédex 16, France
38 Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy,
University of Manchester, Oxford Road, Manchester M13 9PL, UK
39 ESAC/ESA, Camino Bajo del Castillo, s/n., Urb. Villafranca del Castillo,
28692 Villanueva de la Cañada, Madrid, Spain
40 European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044 Frascati, Roma,
Italy
41 Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822,
69622, Villeurbanne, France
42 Institute of Physics, Laboratory of Astrophysics, Ecole Polytechnique
Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix,
Switzerland
43 Mullard Space Science Laboratory, University College London, Holmbury St
Mary, Dorking, Surrey RH5 6NT, UK
44 Departamento de Física, Faculdade de Ciências, Universidade de Lisboa,
Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal
45 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal
46 Department of Astronomy, University of Geneva, ch. d’Ecogia 16, 1290
Versoix, Switzerland
47 INFN-Padova, Via Marzolo 8, 35131 Padova, Italy
48 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, Astrophysique,
Instrumentation et Modélisation Paris-Saclay, 91191 Gif-sur-Yvette, France
49 INAF-Osservatorio Astronomico di Padova, Via dell’Osservatorio 5, 35122
Padova, Italy
50 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-
Universität München, Scheinerstrasse 1, 81679 München, Germany
51 Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029
Blindern, 0315 Oslo, Norway
52 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Drive, Pasadena, CA, 91109, USA
53 Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
54 Cosmic Dawn Center (DAWN), Denmark
55 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg,
Germany
56 Aix-Marseille Université, CNRS/IN2P3, CPPM, Marseille, France
57 Université de Genève, Département de Physique Théorique and Centre for
Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
58 Department of Physics, P.O. Box 64, 00014 University of Helsinki, Finland
59 Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of
Helsinki, Helsinki, Finland
60 NOVA optical infrared instrumentation group at ASTRON, Oude Hoogeveensedijk
4, 7991PD, Dwingeloo, The Netherlands
61 Department of Physics, Institute for Computational Cosmology, Durham
University, South Road, DH1 3LE, UK
62 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire
Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France
63 Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013 Paris,
France
64 European Space Agency/ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The
Netherlands
65 Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The
Netherlands
66 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700
AV Groningen, The Netherlands
67 Department of Physics and Astronomy, University of Aarhus, Ny Munkegade
120, DK-8000 Aarhus C, Denmark
68 Space Science Data Center, Italian Space Agency, via del Politecnico snc,
00133 Roma, Italy
69 Institute of Space Science, Bucharest, 077125, Romania
70 Dipartimento di Fisica e Astronomia ”G.Galilei”, Universitá di Padova, Via
Marzolo 8, 35131 Padova, Italy
71 Dipartimento di Fisica e Astronomia, Universitá di Bologna, Via Gobetti
93/2, 40129 Bologna, Italy
72 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008,
Santiago, Chile
73 Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB, Carrer de Can
Magrans, s/n Cerdanyola del Vallés, 08193 Barcelona, Spain
74 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain
75 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal
76 Universidad Politécnica de Cartagena, Departamento de Electrónica y
Tecnología de Computadoras, 30202 Cartagena, Spain
77 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université
de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France
78 Infrared Processing and Analysis Center, California Institute of
Technology, Pasadena, CA 91125, USA
79 INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122 Milano, Italy
80 Instituto de Astrofísica de Canarias, Calle Vía Láctea s/n, 38204, San
Cristóbal de La Laguna, Tenerife, Spain
81 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue,
10010, New York, NY, USA
82 School of Physics and Astronomy, Cardiff University, The Parade, Cardiff,
CF24 3AA, UK
83 INAF-Istituto di Astrofisica e Planetologia Spaziali, via del Fosso del
Cavaliere, 100, 00100 Roma, Italy
84 Ernst-Reuter-Str. 4e, 31224 Peine, Germany
85 Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin
katu 2, 00014 University of Helsinki, Finland
86 Dipartimento di Fisica e Astronomia ”Augusto Righi” - Alma Mater Studiorum
Universitá di Bologna, Viale Berti Pichat 6/2, 40127 Bologna, Italy
87 Institut d’Astrophysique de Paris, 98bis Boulevard Arago, 75014, Paris,
France
88 Junia, EPA department, 59000 Lille, France
89 INFN-Bologna, Via Irnerio 46, 40126 Bologna, Italy
90 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid,
Spain
91 CERCA/ISO, Department of Physics, Case Western Reserve University, 10900
Euclid Avenue, Cleveland, OH 44106, USA
92 Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL,
CNRS, Sorbonne Université, 75005 Paris, France
93 Observatoire de Paris, Université PSL, Sorbonne Université, LERMA, 750
Paris, France
94 Astrophysics Group, Blackett Laboratory, Imperial College London, London
SW7 2AZ, UK
95 SISSA, International School for Advanced Studies, Via Bonomea 265, 34136
Trieste TS, Italy
96 Dipartimento di Fisica e Scienze della Terra, Universitá degli Studi di
Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy
97 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe
Saragat 1, 44122 Ferrara, Italy
98 Institut de Physique Théorique, CEA, CNRS, Université Paris-Saclay 91191
Gif-sur-Yvette Cedex, France
99 Institut d’Astrophysique de Paris, UMR 7095, CNRS, and Sorbonne Université,
98 bis boulevard Arago, 75014 Paris, France
100 NASA Ames Research Center, Moffett Field, CA 94035, USA
101 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129 Bologna,
Italy
102 Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH
Aachen University, 52056 Aachen, Germany
103 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive,
Honolulu, HI 96822, USA
104 Department of Physics & Astronomy, University of California Irvine, Irvine
CA 92697, USA
105 University of Lyon, UCB Lyon 1, CNRS/IN2P3, IUF, IP2I Lyon, France
106 INFN-Sezione di Genova, Via Dodecaneso 33, 16146, Genova, Italy
107 Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France
108 Department of Astronomy & Physics and Institute for Computational
Astrophysics, Saint Mary’s University, 923 Robie Street, Halifax, Nova Scotia,
B3H 3C3, Canada
109 University Observatory, Faculty of Physics, Ludwig-Maximilians-
Universität, Scheinerstr. 1, 81679 Munich, Germany
110 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical
Institute (AIRUB), German Centre for Cosmological Lensing (GCCL), 44780
Bochum, Germany
111 Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
112 Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53, Avenue des
Martyrs, 38000, Grenoble, France
113 Department of Physics and Astronomy, University College London, Gower
Street, London WC1E 6BT, UK
114 Department of Physics and Astronomy, Vesilinnantie 5, 20014 University of
Turku, Finland
115 Centre de Calcul de l’IN2P3, 21 avenue Pierre de Coubertin 69627
Villeurbanne Cedex, France
116 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2,
00185 Roma, Italy
117 University of Applied Sciences and Arts of Northwestern Switzerland,
School of Engineering, 5210 Windisch, Switzerland
118 INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c/o Dipartimento di Fisica,
Edificio G. Marconi, 00185 Roma, Italy
119 Centro de Astrofísica da Universidade do Porto, Rua das Estrelas, 4150-762
Porto, Portugal
120 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth PO1 3FX, UK
121 Department of Mathematics and Physics E. De Giorgi, University of Salento,
Via per Arnesano, CP-I93, 73100, Lecce, Italy
122 INFN, Sezione di Lecce, Via per Arnesano, CP-193, 73100, Lecce, Italy
123 INAF-Sezione di Lecce, c/o Dipartimento Matematica e Fisica, Via per
Arnesano, 73100, Lecce, Italy
124 CEA Saclay, DFR/IRFU, Service d’Astrophysique, Bat. 709, 91191 Gif-sur-
Yvette, France
125 Institute for Computational Science, University of Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland
126 Université St Joseph; Faculty of Sciences, Beirut, Lebanon
127 Dipartimento di Fisica ”Aldo Pontremoli”, Universitá degli Studi di
Milano, Via Celoria 16, 20133 Milano, Italy

#  Euclid preparation. XXXII. Evaluating the weak lensing cluster mass biases
using the Three Hundred Project hydrodynamical simulations

Euclid Collaboration: C. Giocoli1,2<EMAIL_ADDRESS>M. Meneghetti1,2 E.
Rasia3,4 S. Borgani3,5,6,4 G. Despali7 G. F. Lesci8,1 F. Marulli8,1,2 L.
Moscardini8,1,2 M. Sereno1,2 W. Cui9,10,11 A. Knebe9,10,12 G. Yepes9,10 T.
Castro3,6,4 P.-S. Corasaniti13 S. Pires14 G. Castignani8,1 T. Schrabback15,16
G. W. Pratt17 A. M. C. Le Brun13 N. Aghanim18 L. Amendola19 N. Auricchio1 M.
Baldi8,1,2 C. Bodendorf20 D. Bonino21 E. Branchini22,23 M. Brescia24 J.
Brinchmann25 S. Camera26,27,21 V. Capobianco21 C. Carbone28 J. Carretero29,30
F. J. Castander31,32 M. Castellano33 S. Cavuoti34,35 R. Cledassou36,37 G.
Congedo11 C.J. Conselice38 L. Conversi39,40 Y. Copin41 L. Corcione21 F.
Courbin42 M. Cropper43 A. Da Silva44,45 H. Degaudenzi46 J. Dinis45,44 F.
Dubath46 X. Dupac39 S. Dusini47 S. Farrens48 S. Ferriol41 P. Fosalba32,31 M.
Frailis3 E. Franceschi1 M. Fumana28 S. Galeotta3 B. Garilli28 B. Gillis11 A.
Grazian49 F. Grupp20,50 S. V. H. Haugan51 W. Holmes52 A. Hornstrup53,54 K.
Jahnke55 M. Kümmel50 S. Kermiche56 M. Kilbinger48 M. Kunz57 H. Kurki-
Suonio58,59 S. Ligori21 P. B. Lilje51 I. Lloro60 E. Maiorano1 O. Mansutti3 O.
Marggraf16 K. Markovic52 R. Massey61 S. Maurogordato62 S. Mei63 E. Merlin33 G.
Meylan42 M. Moresco8,1 E. Munari3 S.-M. Niemi64 J. Nightingale61 T. Nutma65,66
C. Padilla29 S. Paltani46 F. Pasian3 K. Pedersen67 V. Pettorino48 G. Polenta68
M. Poncet36 L. A. Popa69 F. Raison20 A. Renzi70,47 J. Rhodes52 G. Riccio34 E.
Romelli3 M. Roncarelli1 E. Rossetti71 R. Saglia50,20 D. Sapone72 B.
Sartoris50,3 P. Schneider16 A. Secroun56 S. Serrano32,73 C. Sirignano70,47 G.
Sirri2 L. Stanco47 J.-L. Starck14 P. Tallada-Crespí74,30 A.N. Taylor11 I.
Tereno44,75 R. Toledo-Moreo76 F. Torradeflot74,30 I. Tutusaus77 E. A.
Valentijn66 L. Valenziano1,2 T. Vassallo3 Y. Wang78 J. Weller50,20 G.
Zamorani1 J. Zoubian56 S. Andreon79 S. Bardelli1 A. Boucaud63 E. Bozzo46 C.
Colodro-Conde80 D. Di Ferdinando2 G. Fabbian81,82 M. Farina83 H. Israel84 E.
Keihänen85 V. Lindholm58,59 N. Mauri86,2 C. Neissner29 M. Schirmer55 V.
Scottez87,88 M. Tenti89 E. Zucca1 Y. Akrami90,91,92,93,94 C.
Baccigalupi95,4,3,6 M. Ballardini96,97,1 F. Bernardeau98,99 A. Biviano3,4 A.
S. Borlaff100 C. Burigana96,101,89 R. Cabanac77 A. Cappi1,62 C. S. Carvalho75
S. Casas102 K. C. Chambers103 A. R. Cooray104 H.M. Courtois105 S. Davini106 S.
de la Torre107 G. De Lucia3 G. Desprez46,108 H. Dole18 J. A. Escartin20 S.
Escoffier56 I. Ferrero51 F. Finelli1,89 L. Gabarra70,47 K. Ganga63 J. Garcia-
Bellido90 K. George109 F. Giacomini2 G. Gozaliasl58 H. Hildebrandt110 I.
Hook111 A. Jimenez Muñoz112 B. Joachimi113 J. J. E. Kajava114 V. Kansal14 C.
C. Kirkpatrick85 L. Legrand57 A. Loureiro11,94 J. Macias-Perez112 M.
Magliocchetti83 G. Mainetti115 R. Maoli116,33 S. Marcin117 M. Martinelli33,118
N. Martinet107 C. J. A. P. Martins119,25 S. Matthew11 L. Maurin18 R. B.
Metcalf8,1 P. Monaco5,3,6,4 G. Morgante1 S. Nadathur120 A.A. Nucita121,122,123
L. Patrizii2 A. Peel42 J. Pollack124,63 V. Popa69 C. Porciani16 D. Potter125
M. Pöntinen58 P. Reimberg87 A.G. Sánchez20 Z. Sakr77,19,126 A. Schneider125 E.
Sefusatti3,6,4 A. Shulevski65,66 A. Spurio Mancini43 J. Stadel125 J.
Steinwagner20 J. Valiviita58,59 A. Veropalumbo127 M. Viel95,4,3,6 I. A.
Zinchenko50

The photometric catalogue of galaxy clusters extracted from ESA Euclid data is
expected to be very competitive for cosmological studies. Using state-of-the-
art hydrodynamical simulations, we present systematic analyses simulating the
expected weak lensing profiles from clusters in a variety of dynamic states
and at wide range of redshifts. In order to derive cluster masses, we use a
model consistent with the implementation within the Euclid Consortium of the
dedicated processing function and find that, when jointly modelling mass and
the concentration parameter of the Navarro–Frenk–White halo profile, the weak
lensing masses tend to be, on average, biased low by 5–10% with respect to the
true mass, up to $z=0.5$. Using a fixed value for the concentration
$c_{200}=3$, the mass bias is diminished below 5%, up to $z=0.7$, along with
its relative uncertainty. Simulating the weak lensing signal by projecting
along the directions of the axes of the moment of inertia tensor ellipsoid, we
find that orientation matters: when clusters are oriented along the major
axis, the lensing signal is boosted, and the recovered weak lensing mass is
correspondingly overestimated. Typically, the weak lensing mass bias of
individual clusters is modulated by the weak lensing signal-to-noise ratio,
related to the redshift evolution of the number of galaxies used for weak
lensing measurements: the negative mass bias tends to be larger toward higher
redshifts. However, when we use a fixed value of the concentration parameter,
the redshift evolution trend is reduced. These results provide a solid basis
for the weak-lensing mass calibration required by the cosmological application
of future cluster surveys from Euclid and _Rubin_.

###### Key Words.:

Galaxy Clusters, Gravitational Lensing, Photometric Galaxies

##  1 Introduction

The abundance and the spatial distribution of galaxy clusters as a function of
redshift represent important cosmological probes for future wide-field
surveys, and particularly for the ESA Euclid mission (Laureijs et al. 2011;
Euclid Collaboration: Scaramella et al. 2022). Cluster cosmological studies
(e.g., Allen et al. 2011) will complement the two main probes based on weak
lensing cosmic shear and galaxy clustering, improving the figure of merit of
the derived cosmological parameters (see e.g., Sartoris et al. 2016).

Galaxy clusters form close to the mass density peaks of the initial matter
density fluctuations and accrete mass through cosmic time as a consequence of
repeated merging events (Tormen 1998; Tormen et al. 2004; Giocoli et al.
2012b; Kravtsov & Borgani 2012). In the present day, clusters represent the
largest virialised structures in the Universe. Their structural properties
(Giocoli et al. 2008; Despali et al. 2014, 2017) are important tracers of
their assembly history and dynamical state. For example, a lower value of
their concentration parameter and a higher mass fraction in substructures are
typical of late-forming systems. On the contrary, fewer substructures and high
concentrations are ordinary for clusters that assemble most of their mass at
higher redshifts, which appear more regular and are rounder (Gao et al. 2004;
De Lucia et al. 2004; Giocoli et al. 2010a; Bonamigo et al. 2015; Mostoghiu et
al. 2019).

The ESA Euclid mission, complemented by the multiband photometric support from
the ground by various observational facilities will be able to identify galaxy
clusters using two complementary algorithms: AMICO and PZWav (Euclid
Collaboration: Adam et al. 2019). While the AMICO (Adaptive Matched Identifier
of Clustered Objects) algorithm (Bellagamba et al. 2011, 2018; Maturi et al.
2019) uses an enhanced matched filter method that looks for cluster candidates
by convolving the 3D galaxy distribution with a redshift-dependent filter,
PZWav (Gonzalez 2014) is a wavelet transform-based code that searches for
overdensities on fixed physical scales. The two methods have been extensively
studied thanks to dedicated activities and performance challenges (Euclid
Collaboration: Adam et al. 2019) that guarantee both methods have a purity and
completeness of at least $80$ per cent for systems with a mass larger than
$M_{200}=10^{14}\,M_{\odot}$111 The mass within the radius that encloses 200
times the critical density of the universe at a given redshift. and redshift
$z<2$. Being complementary, the matching procedures of the two detection
algorithms are expected to generate a highly pure and complete catalogue.

The use of clusters as a cosmological tool relies on the accuracy with which
we can recover their true mass (Pratt et al. 2019, for a review). The mass
enclosed within a given overdensity can be measured in numerical simulations
(Sheth & Tormen 1999; Springel et al. 2001; Tormen et al. 2004), and it is
used to build up analytical functions to describe the number density of haloes
as a function of mass and redshift (Tinker et al. 2008; Despali et al. 2016;
Klypin et al. 2016; Chua et al. 2017; Bocquet et al. 2020; Ondaro-Mallea et
al. 2022; Euclid Collaboration: Castro et al. 2022). However, from an
observational perspective, we have to rely on mass proxy estimates that could
be scattered and biased. Typically, these biases and scatters are due to the
simple analytical models used to characterise the complexity of the
astrophysical processes taking place in the cluster environments (Becker &
Kravtsov 2011; Grandis et al. 2021). In addition, the scatter in the cluster
richness 222Defined as the total number of member galaxies within the cluster
radius $R_{200}$, above a typical characteristic luminosity.-mass relation
could increase because of different feedback and quenching mechanisms taking
place in the highly-dense cluster environments.

The dynamics of cluster galaxies has been extensively used in multiple
dedicated observations to characterise the cluster mass (Biviano et al. 2006,
2013; Bocquet et al. 2015; Capasso et al. 2019). This method may suffer from
the presence of interlopers that may systematically bias the derived dynamical
mass. This limitation could be alleviated by removing clusters with
significant evidence of sub-structures and groups along the line of sight or
by selecting early-type galaxies as spectroscopic targets (Damsted et al.
2023).

The hydrostatic masses derived from X-ray observations and assuming a model-
based hydrostatic equilibrium are biased low with respect to the true mass by
around 20-25 per cent, as discussed in different studies based on
hydrodynamical simulations (Lau et al. 2009; Meneghetti et al. 2010b; Rasia et
al. 2012; Biffi et al. 2016; Ansarifard et al. 2020; Barnes et al. 2021;
Gianfagna et al. 2021). The inclusion of turbulent and non-thermal pressure
support in the model attenuates the discrepancies (Angelinelli et al. 2020;
Ettori & Eckert 2022), but at the price of dedicated high-resolution
multiwavelength follow-up observations of the cluster environment.

Gravitational lensing, particularly in the weak regime, is the primary method
to be used within the Euclid Collaboration to weigh clusters. Since lensing
does not rely on any assumption on the dynamical state of the various mass
components of the cluster, it provides, in principle, an unbiased estimate of
the projected matter density distribution (Bartelmann & Schneider 2001;
Bartelmann 2010). However, it suffers from deprojection effects when
converting the derived weak lensing mass to the actual three-dimensional one
(Meneghetti et al. 2008; Giocoli et al. 2012, 2014). Dedicated works have been
recently carried out, based on state-of-the-art hydrodynamical simulations, to
assess the reliability of the weak lensing mass from future wide-field surveys
(Grandis et al. 2019) and to quantify the effects of baryonic physics (Henson
et al. 2017; Grandis et al. 2021; Cromer et al. 2022). On the other hand,
Martizzi et al. (2014); Cusworth et al. (2014); Velliscig et al. (2014);
Bocquet et al. (2016a); Debackere et al. (2021); Castro et al. (2021) have
studied the impact of baryons on the halo mass function and their effect on
the cosmological parameter estimates using cluster counts. Typically, weak
lensing mass biases are, in principle, not problematic, as long estimated from
representative simulated cluster samples (e.g. Applegate et al. 2016;
Schrabback et al. 2018; Grandis et al. 2019; Sommer et al. 2022) and self-
consistently accounted for in cluster scaling relation and cosmology analyses
(e.g. Dietrich et al. 2019; Bocquet et al. 2019; Schrabback et al. 2021; Chiu
et al. 2022; Zohren et al. 2022).

Recent works by Debackere et al. (2022a, b) have shown the potential of weak
lensing aperture mass to recover the projected mass density distribution of
galaxy cluster regions. The method does not rely on any particular mass
density profile model but needs to be tuned and trained on large sample
cosmological simulations. This approach also requires us to rewrite all
likelihoods of the cluster cosmology pipeline in terms of projected quantities
instead of 3D masses (see also Giocoli et al. 2012).

Thanks to the high efficiency, the unprecedented combination of spatial
resolution and sensitivity of the VIS instrument (Cropper et al. 2016) onboard
the Euclid satellite, we are expected to reach a number density of sources for
weak lensing studies of 30 galaxies per square arcminute. This large number
density of sources will give us the possibility to recover weak lensing masses
for individual massive clusters. However, going toward high-redshift clusters
$z>0.6$, the dearth of available sources requires stacking their signals, for
well-defined richness and redshift bins, to increase the signal-to-noise and
the accuracy of the (stacked) mass.

In this paper, we use a large sample of cosmological hydrodynamic simulations
of clusters from The Three Hundred Collaboration (Cui et al. 2018), to make a
systematic assessment of the weak lensing mass bias of clusters. We study how
the modelling function parameters (truncation radius, concentration, etc.)
affect the recovered weak lensing mass; in addition, we investigate how the
mass bias depends on the true mass of the cluster, on its redshift, and,
particularly, on its orientation with respect to the line of sight, which
represents the largest source of intrinsic scattering at fixed cluster mass.
We would like to underline that the results here presented have been obtained
assuming a specific modelling of the feedback inside clusters, and they could
be sensitive to this choice. However, Grandis et al. (2021) have estimated
that different hydrodynamic solvers may have an impact on the weak lensing
mass bias below $5\%$. We have planned a future work to systematically assess
this.

The paper is organised as follows: in Sect. 2 we introduce the cluster
simulations summarising the cluster properties that we will use for this work;
in Sect. 3 we present our lensing pipeline and how we simulate the expected
signal from the ESA Euclid data; Sect. 4 discusses the results using our
modelling functions. In Sect. 5 we summarise and conclude.

##  2 The Three Hundred simulation data-set

In this work, we rely on dedicated weak lensing simulations of clusters
extracted from the re-simulated regions by the Three Hundred Collaboration
(Cui et al. 2018, 2022). The data set consists of 324 regions centred on the
most massive clusters ($M_{200}\gtrapprox 8\times 10^{14}\,h^{-1}\,M_{\odot}$)
identified at $z=0$ in the DM-only MDPL2 MultiDark simulation (Klypin et al.
2016), with a box-size of $1\,$Gpc on a side. The parent simulation was run
adopting the cosmological parameters as derived by the Planck mission (Planck
Collaboration 2016): $\Omega_{\rm m}=0.307$, $\Omega_{\rm b}=0.048$,
$\Omega_{\rm\Lambda}=0.693$, $h=0.678$, $\sigma_{8}=0.823$ and $n_{\rm
s}=0.96$. For each selected region, initial conditions with multiple levels of
mass refinements were generated using the GINNUNGAGAP
code333https://github.com/ginnungagapgroup/ginnungagap. Within the highest
resolution Lagrangian region – at least five times larger than the cluster
virial radius – particles are divided into dark matter (DM) and gas types
according to the considered cosmological baryon fraction: $m_{\rm
dm}=12.7\times 10^{8}\,h^{-1}\,M_{\odot}$ and $m_{\rm gas}=2.36\times
10^{8}\,h^{-1}\,M_{\odot}$, respectively.

The resolution outside this region is degraded to reduce the computational
cost with respect to the parent original simulation. It is worth mentioning
that each re-simulated region, with a typical radius of
$15\,h^{-1}\,\mathrm{Mpc}$ from the centre, may contain additional groups and
filaments not physically associated and not gravitationally bound to the
virialised cluster but important for total projected mass quantity studies.

Figure 1: Left panel: concentration-mass relation of the clusters at $z=0.22$.
The blue and red data points represent the concentration values computed using
the Prada et al. (2012, hereafter P12) and Springel et al. (2008b, hereafter
S08) relations, respectively. The various lines display different
concentration-mass relation models computed at $z=0.22$ . Right panel: ratio
between the measured concentration, using different methods compared with
respect to the Springel et al. (2008b) formalism. Green crosses and magenta
pluses display the case in which we compute the concentration by fitting the
differential logarithmic density profile outside 10, and
$100\,h^{-1}\,\mathrm{kpc}$, respectively.

The evolution of the particle distribution from the initial-conditions
($z=120$) until the present time was followed using GADGET-X, based on the
gravity solved GADGET-3 Tree-PM code. The code uses an improved smooth-
particle hydrodynamics (SPH) scheme (Beck et al. 2016) to follow the evolution
of the gas component with artificial thermal diffusion, time-dependent
artificial viscosity, high-order Wendland C4 interpolating kernel and wake-up
scheme. These improvements increase the SPH capability of following gas-
dynamical instabilities and mixing processes by better describing the
discontinuities and reducing the clumpiness instability of gas.

Figure 2: Spherical averaged density profiles of halo-3 and halo-4. In each
panel, the solid black curve displays the total matter density profile of the
full hydrodynamical runs, as considered in this work. For comparison, the
dashed grey curve shows the profile of the dark matter only runs. With solid
blue and dashed cyan lines, we exhibit the NFW profiles assuming
concentrations as computed using P12 and S08 formalisms, respectively. With
solid green and dashed magenta lines, we show the NFW profiles where
concentrations have been computed fitting the logarithm profile outside 10 and
100 $h^{-1}\,kpc$, respectively. Profiles extend up to the halo virial radius
$R_{200}$, as computed by AHF.

As described in more detail in Rasia et al. (2015), the simulations include
metallicity-dependent radiative cooling and the effect of a uniform time-
dependent UV background (Planelles et al. 2014). The sub-resolution star
formation model follows Springel & Hernquist (2003), and the metal production
from SN-II, SN-Ia, and asymptotic-giant-branch stars use the original recipe
by Tornatore et al. (2007). The active galactic nucleus feedbacks and
supermassive black hole accretions are modelled with the implementations
presented in Steinborn et al. (2015).

Figure 3: Convergence maps of the three considered random projections of a
cluster, namely the halo-4 at $z=0.22$. The left, central and right panels
show the projections along the $z$, $y$, and $x$ axis coordinates of the re-
simulated box. The source redshift of the map is fixed at $z_{\rm s}=3$. The
size of the field of view is $5\,\mathrm{Mpc}$ ($3.4\,h^{-1}\,\mathrm{Mpc}$)
on a side, $10\,\mathrm{Mpc}$ along the line-of-sight. The map resolution is
2048 pixels on a side.

At each simulation snapshot, the haloes were identified using AHF (Amiga Halo
Finder: Knollmann & Knebe 2009) which consistently accounts for DM, star, and
gas particles in finding and characterising halo properties. For each halo,
the algorithm defines $M_{200}$, using a spherical overdensity algorithm,
i.e., the mass within the radius $R_{200}$ which encloses 200 times the
critical comoving density of the universe $\rho_{\rm c}(z)$ at the
corresponding redshift,

$M_{200}=\dfrac{4\pi}{3}R_{200}^{3}\,200\;\rho_{\rm c}(z)\,.$ (1)

For each halo, we consider subhaloes whose centres lie within the
corresponding halo radius $R_{200}$. The subhalo properties are defined at the
truncation radius $R_{\rm t}$ (the outer edge of the system) and hence mass,
density profile, velocity dispersion, rotation curve are calculated using the
gravitationally bound particles inside this radius (Tormen et al. 1998). In
particular, for each halo in this work we make use of:

  * •

the mass fraction in substructures $f_{\rm sub}$ accounting for the full
subhalo hierarchy of subhaloes within subhaloes;

  * •

the halo virial circular velocity $V_{200}=\sqrt{\dfrac{GM_{200}}{R_{200}}}$;

  * •

the maximum circular velocity $V_{\rm max}$;

  * •

the radius $R_{\rm max}$ at which the maximum circular velocity is attained;

  * •

the centre-of-mass offset $x_{\rm off}$ – expressed in the unit of the halo
radius $R_{200}$, defined as the difference between the centre-of-mass and the
maximum density peak of the halo, which we denote as the cluster centre;

  * •

the virial ratio $\eta\equiv\left(2T-E_{\rm s}\right)/|W|$, where $T$
indicates the total kinetic energy, $E_{\rm s}$ the energy from surface
pressure and $W$ refers to the total potential energy;

  * •

the eigenvalues and the eigenvectors of the moment of inertia tensor.

The Navarro–Frenk–White (hereafter NFW) density profile (Navarro et al. 1996,
1997), depending on the host halo mass and concentration is defined as:

$\rho_{\rm NFW}(r|M_{200},c_{200})=\frac{\rho_{\rm s}}{(r/r_{\rm
s})(1+r/r_{\rm s})^{2}}$ (2)

where $\rho_{\rm s}$ is defined as:

$\rho_{\rm s}=\dfrac{M_{200}}{4\pi r_{\rm
s}^{3}}\dfrac{1}{\ln(1+c_{200})-c_{200}/(1+c_{200})}\;,$ (3)

and $r_{\rm s}=R_{200}/c_{200}$.

From the global halo quantities, we adopt two prescriptions to define the halo
concentration. Assuming that the halo density profile follows an NFW relation,
Springel et al. (2008a, hereafter S08) defines the halo concentration
numerically by solving

$\dfrac{200}{3}\dfrac{c_{200}^{3}}{\ln(1+c_{200})-c_{200}/(1+c_{200})}=14.426\left(\dfrac{V_{\rm
max}}{H(z)\,R_{\rm max}}\right)^{2}\,,$ (4)

where $H(z)$ represents the Hubble parameter at the corresponding redshift.

Following a simpler version of the density profile parametrisation, Prada et
al. (2012, hereafter P12) describe the halo concentration in terms of the
velocity ratio only as follows:

$\dfrac{V_{\rm
max}}{V_{200}}=\sqrt{\dfrac{0.216\;c_{200}}{\ln(1+c_{200})-c_{200}/(1+c_{200})}}\,.$
(5)

The two above definitions of concentration coincide if the halo is perfectly
defined by a spherical NFW profile.

Different post-processing analyses of numerical simulations have historically
adopted different concentration definitions. In this paragraph, we compare
models based on different concentration definitions employed in our study. The
left panel of Fig.1 displays the concentration-mass relation for the Three
Hundred clusters at $z=0.22$. The red diamonds and the blue-filled circles
refer to the concentration parameters computed using Eq. (4) and (5),
respectively. The figure also contains various predictions at the
corresponding redshift, obtained after modelling the results from numerical
simulations (Zhao et al. 2009; Meneghetti et al. 2014, hereafter Z09, M14,
respectively) or interpreting observational data (Sereno et al. 2017,
hereafter S17). In particular, for the Z09 model we adopt the Giocoli et al.
(2012b) formalism to follow the main halo mass accretion history back in time.
From the figure, we notice that the two adopted models for the concentration
predict a different value at fixed halo mass: at this redshift on average
Prada et al. (2012) underestimates the concentration parameter by
approximately $20\%$ with respect to Springel et al. (2008a). This is a
manifestation of the fact that the clusters, particularly in hydrodynamical
simulations, deviate from the perfect NFW profile for which the two models
would have the same concentration. In addition, the S08 model, written in
terms of the radius $R_{\rm max}$, tends to be more sensitive to baryonic
physics and adiabatic contraction than P12, which is parametrised only in
terms of velocities, the average relative difference between the two models
varies with redshift. In the right panel of Fig. 1, we display the ratio
between various concentration definitions with respect to the prediction by
Springel et al. (2008a). Green crosses and magenta plus signs correspond to
the concentration computed by fitting the logarithm of the total differential
density profile, computed by the AHF, outside 10 and 100 $h^{-1}$kpc from the
centre, respectively: binning procedures and reference models may impact the
concentration definition (Meneghetti & Rasia 2013). Density profiles are very
sensitive to baryonic effects at small radii. Higher concentration parameters
result from a steepening of the inner slope of the profile (Schaller et al.
2015b, a; Ahad et al. 2021; Jung et al. 2022) due to the adiabatic
contraction, whose effect is mitigated by the AGN feedback but not completely
suppressed (Rasia et al. 2013). Since this paper aims at studying the WL mass
reconstruction and the projected density profiles are not reliable within the
central 100 kpc – noticing that the corresponding concentrations are in good
agreement with P12. For the rest of this work, we will also use S08 as a
reference to give an idea of the theoretical uncertainties.

In Fig. 2, we show the spherical average density profiles of two clusters:
halo-3 (on the left) and halo-4 (on the right). In each panel, the black solid
and dashed grey curves refer to the total density profile of the full
hydrodynamical simulations and of the corresponding dark matter-only runs.
Profiles extend up to $R_{200}$ defined as the radius enclosing 200 times the
critical density. Solid blue and dashed cyan curves display the NFW profiles
using the concentration values as derived from the P12 and S08 relations,
respectively. With solid green and dashed magenta lines, we display the NFW
profiles where concentration has been derived modelling the logarithmic
density profile outside 10 and 100 $h^{-1}\,kpc$, respectively. From the
figure, we notice that while for halo-3 the solid blue and green lines
describe relatively well the total mass density profile of the cluster, for
halo-4 only the green line well represents the matter density distribution.
This could be due to the fact that while the halo-3 has an inner slope
$\mathrm{d}\log\rho/\mathrm{d}\log r$ close to -1, the cluster halo-4 has a
steeper slope toward the centre. We can also notice that the density in the
outskirts of the clusters is higher than expected from the NFW profile due to
the presence of substructures. Those differences reduce when looking at
projected quantities, highlighting that the NFW prescription is sufficient for
the analysis in this paper.

We would also like to underline that the extrapolation of those 3D
uncertainties into projected quantities is not straightforward and may require
some reference concentration-mass relation model to be adopted (Hoekstra et
al. 2013; Simet et al. 2017a, b; Kiiveri et al. 2021; Sommer et al. 2022). We
will describe how those impact the recovered weak lensing mass bias in the
next section.

##  3 Cluster weak lensing simulations

In order to build up the mass maps, we proceed as follows. From the AHF
catalogues, we read the positions of the halo centres in comoving units
($x_{\rm c},\,y_{\rm c},\,z_{\rm c}$) and then the particle positions with
associated masses and types from the corresponding snapshot file. Each
snapshot contains dark matter, gas, star, and black hole particles. For each
cluster, we consider three random lines of sight corresponding to the axes of
the simulation box, along which we project the cluster particles on a
perpendicular plane centred on the cluster centre. We consider particles in a
slice of depth $\pm 5\,\mathrm{Mpc}$ ($3.4\,h^{-1}\,\mathrm{Mpc}$) in front
and behind the cluster. The maps have a final size of $5\,\mathrm{Mpc}$ on a
side, resolved with $2048^{2}$ pixels and are produced using Py-SPHViewer (for
more details we refer to Benitez-Llambay 2015). The choice of the size of the
field of view is motivated by the fact that we are mainly interested in
modelling the projected matter density distribution of the main cluster
without being much affected by the additional source of uncertainty associated
to the large-scale matter density distribution along the line of sight
(Hoekstra 2001, 2003). This value also represents a reasonable compromise to
avoid influence due to low-resolution particles in the re-simulated box. In
addition, it is worth stressing that the scatter in the weak lensing mass bias
could be underestimated with respect to what is measured by Becker & Kravtsov
(2011), who used a larger line-of-sight integration.

The convergence $\kappa$ is obtained from the mass map by dividing the mass
per pixel by its associated area to obtain the surface density
$\Sigma(\boldsymbol{\theta})$ and by the critical surface density $\Sigma_{\rm
crit}$ (Bartelmann & Schneider 2001) that can be read as:

$\Sigma_{\rm crit}\equiv\dfrac{c^{2}}{4\pi G}\dfrac{D_{\rm s}}{D_{\rm l}D_{\rm
ls}}\,,$ (6)

where $D_{\rm l}$, $D_{\rm s}$, and $D_{\rm ls}$ are the observer-lens,
observer-source, and source-lens angular diameter distances, respectively; $c$
represents the speed of light and $G$ the universal gravitational constant.
The critical surface density is a function of the redshift of the lens and of
the source. Our reference weak lensing maps have been created assuming a fixed
source redshift $z_{\rm s}=3$.

In Fig. 3, we show three convergence maps of the same cluster, namely halo-4
at $z=0.22$. The left, central and right panels display the system projected
along the $z$, $y$, and $x$-axis of the comoving re-simulated box,
respectively. Since there is no particular selection with respect to the
triaxial properties of the cluster haloes, these projections can be seen as
random projections and as such we will name them as proj0, proj1, proj2,
respectively, in the following.

For each cluster, redshift, and projection, we generate weak lensing
convergence maps, specifically, we consider 9 redshifts from $z=0.12$ to
$z=0.98$ as listed in Table 1.

Table 1: List of the analysed simulation snapshots (first column) and the
corresponding redshifts (second column). The third column specifies the
average number of background galaxies used to construct the simulated weak
lensing shear profile as expected by the Euclid-ESA mission to lay beyond a
cluster at a given redshift. The fourth expresses the square root of the
average number of background galaxies rescaled by the ratio of angular
diameter distances, relevant for quantifying lensing detectability. The last
column shows the average cluster mass at the corresponding redshift, in the
unit of $10^{14}\,h^{-1}\,M_{\odot}$.

snap. | $z$ | $n_{\rm g}^{a}$ | $\sqrt{n_{\rm g}}\,D_{\rm ls}/D_{\rm s}^{b}$ | $\langle M_{200}\rangle^{c}$
---|---|---|---|---
123 | 0.12 | 30 | 5.04 | $7.85$
119 | 0.22 | 29 | 4.62 | $6.77$
115 | 0.33 | 28 | 4.20 | $5.77$
113 | 0.39 | 27 | 3.95 | $5.32$
110 | 0.49 | 26 | 3.60 | $4.66$
107 | 0.59 | 24 | 3.22 | $3.99$
104 | 0.70 | 22 | 2.65 | $3.38$
101 | 0.82 | 19 | 2.40 | $2.83$
098 | 0.94 | 17 | 2.07 | $2.38$

${}^{a}\,[arcmin^{-2}]$
${}^{b}\,[arcmin^{-1}]$
${}^{c}\,[10^{14}\,h^{-1}\,M_{\odot}]$

From the convergence $\kappa$, we define the lensing potential $\psi$, using
the two-dimensional Poisson equation as:

$\Delta_{\theta}\psi(\boldsymbol{\theta})=2\,\kappa(\boldsymbol{\theta})\,,$
(7)

that we numerically solve in Fourier space using the Fast Fourier Transform
(FFT) method. Since the FFT algorithm assumes periodic conditions on the
boundaries of the map, we zero-pad the exterior of the convergence map by
$0.5\,\mathrm{Mpc}$ before computing the discrete Fourier transform and then
remove the region from the final result.

From the lensing potential $\psi$, we can then define the two components
$(\gamma_{1},\gamma_{2})$ of the pseudo-vector shear, as:

$\displaystyle\gamma_{1}(\boldsymbol{\theta})$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\dfrac{\partial^{2}\psi(\boldsymbol{\theta})}{\partial
x^{2}}-\dfrac{\partial^{2}\psi(\boldsymbol{\theta})}{\partial
y^{2}}\right)\;,$ (8) $\displaystyle\gamma_{2}(\boldsymbol{\theta})$
$\displaystyle=$
$\displaystyle\dfrac{\partial^{2}\psi(\boldsymbol{\theta})}{\partial
x\,\partial y}\,,$ (9)

where $x$ and $y$ represent the two components of the vector
$\boldsymbol{\theta}$.

For each cluster, we use the shear maps to simulate an observed differential
surface mass density profile by randomly sampling the field of view with a
given number density of expected background sources.

In our analysis, we adopt a background density of sources for weak lensing
following the predicted distribution for the ESA Euclid wide-field survey
normalised to the total value of 30 galaxies per square arcmin (Laureijs et
al. 2011). In the third column of Table 1 we report the average number of
background sources beyond each considered cluster redshift, whose
corresponding snapshot number is displayed in the first column.

Figure 4: Top panel: Excess surface mass density profiles of the three random
projections for a cluster-size halo at $z_{\rm l}=0.22$, as displayed in Fig.
3. The blue, red and green data points show the circular averaged profiles of
the halo projected along the $z$, $y$, and $x$ axis of the re-simulation box.
The error bars on the data points account for both the dispersion of the
intrinsic ellipticity of background galaxies and the error associated with the
average measure in the considered radial bin. The blue solid line displays the
best-fit model to proj0, while the shaded region represents the model within
the $68\%$ of confidence. Bottom panel: relative difference of the best-fit
models assuming different values of the truncation radius $R_{\rm t}$ with
respect to the reference case with $t=3$.

The approach of randomly sampling the shear field has been chosen to mimic the
expected average profile of a cluster from a Euclid wide-field exposure. Given
the high number density of background sources, the profile has a much smaller
scatter with respect to the case in which we consider the complete shear
field. We tested this for the halo-4 at $z=0.22$ and $z=0.94$, generating
10,000 random shear field realisations, finding results consistent with our
reference case well within 1 $\sigma$ of the credibility regions of the
posteriors.

From the two components of the shear, we define the tangential shear
$\gamma_{\rm t}$ (Umetsu 2020) at each sampled point $\theta_{i}$ of the map,
as:

$\gamma_{\rm
t}(\theta_{i})=-\gamma_{1}(x_{i},y_{i})\cos(2\phi_{i})-\gamma_{2}(x_{i},y_{i})\sin(2\phi_{i})\,,$
(10)

where $(0,0)$ is the centre of the cluster by construction,
$\theta_{i}=\smash{(x_{i}^{2}+y_{i}^{2})^{1/2}}$ and
$\phi_{i}=\arctan\left(y_{i}/x_{i}\right)$. This gives us the possibility to
write the excess surface mass density – azimuthally averaging the measured
quantities – as:

$\Delta\Sigma(\theta)=\bar{\Sigma}(<\theta)-\Sigma(\theta)\equiv\Sigma_{\rm
crit}\gamma_{\rm t}(\theta)\,,$ (11)

where $\Sigma(\theta)$ represents the mass surface density of the lens at
distance $\theta$ from the putative cluster centre, and
$\bar{\Sigma}(<\theta)$ its mean within $\theta$.

The approach that we follow in this work aims at quantifying the weak lensing
mass bias associated with projection effects and its redshift evolution,
depending on the available number density of galaxies from which we can
measure the lensing signal induced by the interposed cluster projected mass
density distribution. In this way, we quantify the most optimistic weak
lensing mass bias that we expect from Euclid data. In this analysis, we do not
assume any uncertainty on the cluster centre, the lens and source redshifts,
all assumed to be known with infinite accuracy. Forthcoming analyses, within
the Euclid Collaboration, will be dedicated to study these systematics, and
the corresponding propagation, into the weak lensing mass biases, and
concentration that should be degenerate and more affected by the miscentring
(Giocoli et al. 2021; Lesci et al. 2022).

Figure 5: Left panel: posterior distributions of the recovered logarithm of
the mass $M_{\rm wl}$ and concentration $c_{\rm wl}$ obtained by fitting the
excess surface mass density profile of the three random projections of the
halo-4 at $z=0.22$. Right panel: posterior distributions obtained modelling
the excess surface mass density profile on the halo-4 when oriented along
proj0, assuming different truncation radii. The blue data displays the
reference case when assuming a truncation radius $R_{\rm t}=3\,R_{200}$, while
the red, pink, green and orange one exhibits the results when using
$R_{t}=t\,R_{200}$ with $t=1$, 2, 6, and 9, respectively. Dark and light-
shaded areas enclose $1$ and $3\,\sigma$ credibility regions, respectively. In
both panels the vertical dashed line marks the true mass of the system, while
the horizontal dotted and dashed ones indicate the concentrations computed
using the Springel et al. (2008a) and Prada et al. (2012) formalisms,
respectively.

To simulate the weak lensing signal of each cluster, we build the average
excess surface mass density by binning the measured $\Delta\Sigma(r_{i})$,
where $r_{i}\equiv D_{\rm l}\,\theta_{i}$, in $22$ logarithmically equispaced
intervals from 0.02 to $1.7\,h^{-1}\,\mathrm{Mpc}$ from the cluster centre. As
an example, in the top panel of Fig. 4, blue, red, and green data points
display the simulated $\Delta\Sigma$ profile for three random projections of
the cluster halo-4 at $z=0.22$, as shown in Fig. 3. In order to limit the
analysis to well-constrained shear estimates (depending on the angular binning
and on the source density), we consider only radial bins with at least 10
simulated data measures. This guarantees a reliable estimate of the average
signal and conservatively neglects bins close to the centre, where baryonic
effects tend to steepen the profile. However, we have tested that considering
radial bins with at least one galaxy does not alter the mass bias results of
our work. The colours correspond to those used for the panel frame of Fig. 3.
The corresponding error bars are computed as:

$\sigma_{\Delta\Sigma}=\sqrt{\sigma_{\langle\Delta\Sigma\rangle}^{2}+\Sigma_{\rm
crit}^{2}\dfrac{\sigma_{\rm e}^{2}}{n_{\rm
g}\;\pi\left(\theta_{2}^{2}-\theta_{1}^{2}\right)}}\,,$ (12)

where $\sigma_{\rm e}=0.3$ (Hoekstra et al. 2004, 2011; Kilbinger 2015; Euclid
Collaboration: Blanchard et al. 2020) is the dispersion of the shape of
background source galaxies, and $\theta_{1}$ and $\theta_{2}$ are the lower
and the upper bounds of the considered radial annulus. In our error budget, we
also include the error of the mean estimated excess surface mass density in
each radial interval $\sigma_{\langle\Delta\Sigma\rangle}=\sigma_{\rm
rms}/\sqrt{n_{\rm g,ring}}$ – with $\sigma_{\rm rms}$ representing the
$\Delta\Sigma$ root-mean-square:

$\sigma_{\rm rms}=\sqrt{\sum_{i=1}^{n_{\rm
g,ring}}\dfrac{\left(\Delta\Sigma_{i}-\langle\Delta\Sigma\rangle_{\rm
ring}\right)^{2}}{n_{\rm g,ring}}}\,$ (13)

$\langle\Delta\Sigma\rangle_{\rm ring}$ the average value, and $n_{\rm
g,ring}$ the number of galaxies in each ring – in order to account for: the
projected cluster triaxiality, correlated large-scale structures, and subhalo
contribution in each annulus (Gruen et al. 2015, 2011; Umetsu et al. 2020). It
is also worth mentioning that this term is negligible with respect to the one
that accounts for the intrinsic shape noise. From the figure, we notice that
while proj0 and proj2 are similar, proj1 shows a steepening toward the centre.
This highlights that depending on which projection, structural properties
recovered from weak lensing mass and concentration can be different from each
other and even in contrast with respect to the three-dimensional ones. In the
figure, we highlight that the data binning of the three projections have a
different minimum scale because we exclude the rings with less than 10
galaxies.

##  4 Analytical models & results

In order to recover the weak lensing mass and concentration, we model the data
of the excess surface mass density profile using projected analytical
relations. We use the same strategy that is under development within the
Euclid Consortium in the implementation of the dedicated processing function:
the total density profile is constructed considering the signal coming from
the central part of the cluster called the 1-halo term, and the one caused by
correlated large-scale structures, named the 2-halo term. Methods based on
non-parametric formalism (Clowe et al. 2004; Bradač et al. 2005, 2006; Diego
et al. 2007; Merten et al. 2009; Jauzac et al. 2012; Jullo et al. 2014;
Niemiec et al. 2020) may give more accurate and precise mass reconstructions,
but at the price of being slow and memory expensive, and in this way, hardly
applicable to the very large statistics of clusters expected from Euclid
photometric data (Sartoris et al. 2016; Euclid Collaboration: Adam et al.
2019).

For the cluster main halo, we adopt a smoothly-truncated NFW density profile
(BMO, Baltz et al. 2009), defined as:

$\rho_{\rm BMO}(r_{\rm 3D}|M_{200},c_{200},R_{\rm t})=\rho_{\rm NFW}(r_{\rm
3D}|M_{200},c_{200})\left(\frac{R_{\rm t}^{2}}{r_{\rm 3D}^{2}+R_{\rm
t}^{2}}\right)^{2}\,,$ (14)

with $R_{\rm t}=t\,R_{200}$ with $t$ defined as the truncation factor. For our
reference model, following the results by Oguri & Hamana (2011), Bellagamba et
al. (2019), and Giocoli et al. (2021) we will adopt a truncation radius
$R_{\rm t}=3\,R_{200}$. The total mass enclosed within $R_{200}$, i.e.,
$M_{200}$, can be thought of as the normalisation of the model and as a mass-
proxy of the true enclosed mass of the dark matter halo hosting the cluster
(Giocoli et al. 2012a). Writing $r_{\rm 3D}^{2}$ as the sum in quadrature
between the sky projected coordinate $D_{\rm d}\theta$ and the line-of-sight
$\zeta$ coordinate, and integrating along $\zeta$ we can write

$\Sigma_{\rm 1h}(\theta|M_{200},c_{200},R_{\rm t})=\int_{0}^{\infty}\rho_{\rm
BMO}(\theta,\zeta|M_{200},c_{200},R_{\rm t})\,\mathrm{d}\zeta\,.$ (15)

This projected parametrisation describes the smooth blending of the halo
boundary into the connected large-scale structures, regulated by the
truncation radius. However, at larger distances from the cluster centre, the
projected lensing signal starts to increase due to the large-scale structures
of the correlated and the uncorrelated matter along the line of sight: the
2-halo term, describing the asymptotic regime far from the halo centre. This
term can be written as a function of the integrated linear matter power
spectrum weighted by a Bessel function (Oguri & Takada 2011; Oguri & Hamana
2011; Sereno et al. 2017),

$\Delta\Sigma_{\rm 2h}(\theta,M_{200})\\!=\\!\int_{0}^{\infty}\frac{\ell\;{\rm
d}\ell}{2\pi}J_{2}(\ell\theta)\frac{\bar{\rho}_{\rm m}(z)\;b_{\rm
h}(M_{200};z_{\rm l})}{(1+z)^{3}D_{\rm l}^{2}(z)}P_{\rm lin,m}(k_{\ell};z_{\rm
l})\,,$ (16)

where $z_{\rm l}$ represents the cluster (lens) redshift, $J_{2}$ is the
Bessel function of second type, $k_{\ell}=\ell/[(1+z)D_{\rm l}(z)]$ indicates
the wave vector mode, $\bar{\rho}_{\rm m}$ is the background density, $P_{\rm
lin,m}$ refers to the linear matter power spectrum and $b_{\rm h}(M;z)$ is the
halo bias, for which we adopt the Tinker et al. (2010) model.

The total model can be read as follows:

$\Delta\Sigma(\theta)=\Delta\Sigma_{\rm 1h}(\theta|M_{200},c_{200},R_{\rm
t})+\Delta\Sigma_{\rm 2h}(\theta,M_{200})\,.$ (17)

It is worth noticing that the 2-halo term contribution at small radii is
expected to be negligible, while its contribution becomes important, with
respect to the 1-halo term, only at scales $\gtrsim 3\,h^{-1}\,\text{Mpc}$
(Giocoli et al. 2021; Ingoglia et al. 2022). In their Fig. 1, Giocoli et al.
(2021) show that for distances between 2 and 3 $h^{-1}\,\text{Mpc}$ from the
halo centre the profile is sensitive also to the truncation radius definition.
Given the size of the field-of-view of our maps, 3.4$\,h^{-1}\,$Mpc on a side,
the 2-halo term has a weak contribution, and we decided to ignore it in our
modelling function.

In the top panel of Fig. 4, the solid blue curve stands for the best-fit model
to the data points of proj0, associated to the median values of the posteriors
of $\operatorname{log_{10}}(M_{200})$ and $c_{200}$. The shaded region marks
the $1\,\sigma$ uncertainties of the recovered mass and concentration
parameters propagating in the modelling of the $\Delta\Sigma$ profile,
associated with the $16^{\rm th}$ and $84^{\rm th}$ percentiles of the
posterior distributions. In the bottom panel, we show the relative difference
of the best-fit models where adopting various values for the truncation radius
with respect to the reference case where we set $R_{\rm t}=3\,R_{200}$.

The left panel of Fig. 5 displays the posterior distributions of the recovered
mass, $\operatorname{log_{10}}(M_{200})$, and concentration, $c_{200}$,
obtained by modelling the three random projections of the halo-4 at $z_{\rm
l}=0.22$, (top panel of Fig. 4). In blue, red, and green, we show the
distribution for the projections proj0, proj1, and proj2, respectively, using
only the 1-halo term in our modelling function and fixing $R_{\rm
t}=3\,R_{200}$: we define this as the reference model.

We perform a Monte Carlo Markov Chain run assuming a Gaussian log-likelihood
between the model and the data as defined within our reference CBL
libraries444https://gitlab.com/federicomarulli/CosmoBolognaLib (Marulli et al.
2016) that can be read as:

$\mathcal{L}\propto\exp\left(-\dfrac{1}{2}\chi^{2}\right)\,,$ (18)

where

$\chi^{2}=\sum_{i}\left(\dfrac{\Delta\Sigma_{i}(\theta_{i})-\Delta\Sigma_{\rm
model}(\theta_{i})}{\sigma_{\Delta\Sigma_{i}}}\right)^{2}\,,$ (19)

summing on the number of radial bins. Note that we do not account for reduced
shear corrections both in the simulations and in the model; however, it is
worth mentioning that observationally this model absorbs the reduced shear
impact in the resulting mass bias. We set uniform priors for
$\operatorname{log_{10}}(M_{200}/[h^{-1}\,M_{\odot}])\in[12.5,\,16]$ and
$c_{200}\in[1,\;15]$, and let each MCMC chain run for $16\,000$ steps. In
cyan, purple, and gold, we show the posterior distributions obtained by also
adding the contribution of the 2-halo term in the modelling function, as
expressed by Eq. 16. Comparing the two, we notice that the results are in full
agreement with the analysis that does not include the 2-halo term, also for
high-redshift clusters. This guarantees that for each cluster and redshift the
data-set that we built by projecting the mass density distribution from the
simulations, takes a negligible contribution from correlated structures along
the line-of-sight, due to the limited size of the field-of-view
($3.4\,h^{-1}\,$Mpc), as discussed earlier.

In the right panel of Fig. 5, we examine the dependence of different
assumptions for the truncation radius on the recovered weak lensing halo mass
and concentration, as parametrised in Eq. 14 and inserted into Eq. 15. We show
the posterior distributions when assuming $t$ to be equal to 1 (red), 2
(pink), 3 (blue), 6 (green), and 9 (orange). From the figure, we notice that
the different choices result in a diverse recovered concentration: smaller
truncation radii prefer lower concentration parameters. When $t=1$ (relatively
unphysical case) and 2, the mass also tends to be underestimated. To guide the
reader, we underline that in both panels the dotted and dashed lines refer to
the true 3D halo mass and concentrations, adopting the Springel et al. (2008a)
and Prada et al. (2012) relations, respectively.

In Fig. 6, we show the weak lensing mass bias as a function of the cluster
true mass $M_{200}$, for systems at $z_{\rm l}=0.22$. The grey data points
display the ratio between the weak lensing mass and the true mass, modelling
the excess surface mass density profile with our reference model: 1-halo term
with a truncation radius parameter $t=3$. The error bars indicate the $16^{\rm
th}$ and $84^{\rm th}$ percentiles of the posterior distributions for each
considered cluster projection. The black solid line shows the moving average
of the data. Red, pink, green, and orange lines display the moving average
when considering the truncation radius equal to 1, 2, 6, and 9 times $R_{200}$
in our modelling function, respectively (the corresponding data points are not
displayed in the figure). From the figure, we notice that the average weak
lensing cluster mass bias depends on the value of the truncation radius
adopted in the modelling. In particular, assuming $t=1$ we underestimate the
mass by approximately $\sim 30\%$, larger $t$ values increase the weak lensing
mass and therefore decrease the mass bias. For our reference case with $t=3$,
the average mass bias is as low as $\sim 7\%$ with a standard deviation, shown
by the grey shaded region (see also Becker & Kravtsov 2011), thus highlighting
that triaxiality is the largest source of intrinsic scattering. The right
panel exhibits the distributions for the corresponding cases independently of
the halo mass $M_{200}$, while the top one shows the average relative
uncertainty for the weak lensing mass estimates as a function of 3D cluster
mass. The blue-filled circles report the results by Bahé et al. (2012,
hereafter B12) that model the weak lensing mass bias using tangential shear
profiles up to 2.2$\,h^{-1}\,$Mpc from the cluster centre (while we extend our
analysis out to 1.7 $\,h^{-1}\,$Mpc). Their value is larger than ours and the
authors are using only the 1-halo term, modelled with a pure NFW profile (we
adopt a truncated NFW using the BMO relation), without quantifying the 2-halo
contribution – that, in their case, may not be entirely negligible because of
a larger field of view. In addition, they use a dispersion parameter of the
intrinsic shape of background source galaxies $\sigma_{\rm e}=0.2$ while we
adopt a larger value $\sigma_{\rm e}=0.3$. All these factors may increase the
relative uncertainties in the recovered weak lensing masses.

Figure 6: Weak lensing mass bias as a function of the cluster true mass, for
all systems at $z_{\rm l}=0.22$, for different assumptions of the truncation
factor. The data points show the ratio between the recovered weak lensing mass
obtained by fitting the excess surface mass density profile using our
reference model – 1-halo term assuming a truncation radius $R_{\rm
t}=3\,R_{200}$ – and the true $M_{200}$ mass. The error bars correspond to the
$16^{\rm th}$ and $84^{\rm th}$ percentiles of the posterior distribution. The
black solid line represents the moving average of the data points. Red, pink,
green, and orange lines display the moving average of the data considering the
truncation radius parameter equal to $t=$1, 2, 6, and 9 in our modelling
function, respectively. The cyan dashed, and dotted lines point to 20 and 30%
of the scatter, respectively, while the magenta vertical line indicates the
minimum cluster mass – almost constant with redshift – expected to be detected
from the photometric catalogue as predicted by Sartoris et al. (2016). The
light and dark yellow bands mark $10\%$ and $20\%$ scatters, respectively. The
top subpanel displays the relative weak lensing mass uncertainty as a function
of the true cluster mass. The blue circles show the prediction by Bahé et al.
(2012, hereafter B12). The histograms in the right panel show the weak lensing
mass bias distributions over all cluster masses for the various considered
cases.

###  4.1 Mass bias and concentration

In the spirit of understanding how well we can recover the weak lensing mass
bias when assuming a concentration model, in Fig. 7 we show the posterior
distribution of the logarithm of the recovered cluster mass in the case of the
halo-4 when considering different assumptions for the concentration.
Generally, we see they have an impact on the recovered mass. In particular, if
we assume a fixed value equal to $c_{200}=3$, we have no bias on the recovered
mass for this specific cluster and projection. More relevant than mass biases
for individual realisations are average biases computed for representative
cluster samples, which show a similar dependence on the adopted concentration
mass relation (e.g Sommer et al. 2022).

The average weak lensing mass bias, as a function of the true cluster mass, is
displayed in Fig. 8. The black line shows our reference case. The solid green,
dashed cyan, and dark red lines correspond to the moving average when assuming
different concentration-mass relation models: Zhao et al. (2009), Meneghetti
et al. (2014) and Sereno et al. (2017), respectively. The orange and olive
curves display the results assuming a fixed value $c_{200}=3$ and $c_{200}=4$,
respectively. Referring to the P12 (S08) model the average concentration of
clusters at $z=0.22$ is $\langle c_{200}\rangle=4$ (5) (see the left panel of
Fig. 1). The corresponding coloured dotted lines display the linear
regressions:

$\left\langle\dfrac{M_{\rm
wl}}{M_{200}}\right\rangle=\alpha\,\left\langle\log\left(\dfrac{M_{200}}{M_{14.5}}\right)\right\rangle+\beta\,,$
(20)

where $M_{14.5}=10^{14.5}\,h^{-1}M_{\odot}$, for the reference and $c_{200}=3$
cases (see Tab. 3). The average values are computed assuming a step-size=0.1
and bin-size=0.3 in $\log\left(M_{200}\right)$.

From the top subpanel, it is interesting to notice that the average relative
scatter as a function of $M_{200}$ is reduced by approximately a factor of two
when assuming a deterministic model for the concentration, with respect to the
case in which we vary it inside the Monte Carlo analyses. The right panel
displays the distribution of the relative mass ratio.

Figure 7: Posterior distributions when recovering the logarithm of the cluster
mass for proj0 of the halo-4 at $z=0.22$. The black curve shows the reference
case where we model both the mass and the concentration, as in the top panel
of Fig. 5. The other histograms show the posteriors when assuming a
parametrisation of the concentration-mass relation; in particular, orange and
olive curves refer to the cases when assuming a constant value of $c_{200}=3$
and $c_{200}=4$, respectively. The vertical black dashed line marks the true
value of $M_{200}$ as computed by AHF. Figure 8: Weak lensing mass biases for
clusters at $z=0.22$; the points are the same as in Fig. 6. The different line
styles and coloured curves display the moving averages when assuming different
approaches when modelling the concentration. The black solid line refers to
the reference case where we vary both the mass and the concentration in our
MCMC analysis. The solid green, dotted cyan, and dark-red curves refer to the
case in which we assume a concentration-mass relation model, as in the figure
label. On the other side, the orange and olive ones assume a constant value of
$c_{200}=3$ and $c_{200}=4$, respectively. The coloured dotted lines display
the linear regressions to the moving average and the corresponding uncertainty
(not shown for the orange data to not overcrowd the figure). Top and left
panels show the mean relative mass uncertainties as a function of the true
cluster mass and the distributions of the weak lensing mass biases,
respectively.

Figure 9: Left panel: relaxation criteria for the clusters at $z=0.22$. In our
analysis, we assume the systems to be relaxed if their mass fraction in
substructures $f_{\rm sub}<0.1$ and their centre of mass offset $x_{\rm
off}<0.05\;R_{200}$ indicated by the white rectangular region on the bottom
left. Our relaxed (unrelaxed) clusters are circled in green (purple). For
comparison, we mark also the relaxation criterium by Cui et al. (2018):
$f_{\rm sub}<0.1$, $x_{\rm off}<0.04\;R_{200}$ and energy parameter
$0.85<\eta<1.15$ (blue-filled circle, otherwise gold-coloured). Right panel:
weak lensing mass bias as a function of the true mass of the cluster. Green
and magenta circled data points correspond to systems that are or are not
relaxed, respectively; the solid green and magenta show the moving average of
the two samples. For comparison, the dashed lines, accordingly coloured, show
the results using the relaxation criteria by Cui et al. (2018). The sub-panel
on the right displays the mass bias distributions for the various cluster
samples, while the top one is the average relative scatter in the weak lensing
mass estimates as a function of $M_{200}$.

###  4.2 Mass bias and relaxation criteria

The morphological properties of galaxy clusters depend on their mass assembly
history. Typically, clusters that experienced recent merging events may
contain a larger fraction of their mass in substructures and eventually a mass
centroid which is not coincident with the position of the density peak due to
the presence of multi-mass components. These properties manifest themselves in
a variety of observables at different wavelengths, going from X-ray to the
radio band. The projected mass recovered using weak lensing data is also
influenced by the level of dynamical relaxation of the lens. In this section,
we aim at quantifying these dependencies.

In the left panel of Fig. 9 we display the scatter plot of clusters at
$z=0.22$ for parameters $f_{\rm sub}$ and $x_{\rm off}$, that we use to define
the relaxation status of a system – see their definition in Sect. 2. We define
as relaxed systems those with $f_{\rm sub}<0.1$ and $x_{\rm
off}<0.05\,R_{200}$, i.e., the clusters lying in the lower left white
rectangle of the figure: 147 over 324 objects satisfy this condition. Our
relaxed clusters are green-circled, while unrelaxed systems are displayed
using magenta circles. In the figure, we also mark the relaxation criterium
adopted by Cui et al. (2018): $x_{\rm off}<0.04\,R_{200}$ (with a dotted
vertical line) and with virial ratio $0.85<\eta<1.15$. The latter condition is
portrayed in the figure, colouring the data points in blue or otherwise in
yellow. The relation criterion adopted by Cui et al. (2018) is satisfied by 91
systems. In the right panel, we display the weak lensing mass bias, as a
function of the cluster mass, for the relaxed (green) and the unrelaxed
(magenta) clusters. The green and magenta solid curves represent the moving
average of the two samples, respectively. The corresponding dashed curves
refer to the cases in which the relaxation criterion by Cui et al. (2018) is
adopted, which also uses information on the virial energy ratio. From the top
subpanel, we notice that the average relative uncertainty in the recovered
mass is lower for relaxed systems – even if the differences between the two
cases are, to some extent, small, as well as the relative mass bias has a
lower scatter than for unrelaxed clusters (right subpanel). This highlights
the fact that unrelaxed systems could not be well modelled by the single halo
model (Lee et al. 2023). As in previous figures, and to highlight the
differences between the relaxed and unrelaxed samples, the coloured dotted
lines display the linear regressions to the moving average and the
corresponding uncertainty. The best-fit parameters and the corresponding
uncertainties, as expressed in Eq. (20) for the relaxed and unrelaxed sample
at $z=0.22$ are shown in Tab. 2, which can be compared with the full sample as
in Tab. 3.

Table 2: Linear regression parameters, as in Eq. (20), for the relaxed and unrelaxed cluster sample at $z=0.22$.  | Slope $\alpha$ | Intercept $\beta$
---|---|---
relaxed | $0.1354\pm 0.029$ | $0.890\pm 0.596$
unrelaxed | $0.243\pm 0.033$ | $0.846\pm 0.681$

###  4.3 Mass bias along preferential directions

Figure 10: Left panel: weak lensing signal-to-noise distributions for the
sample of clusters at $z=0.22$. The grey-shaded histogram shows the
distribution for the clusters oriented along random projections, namely the
$x$, $y$, and $z$ cartesian axes of the re-stimulated box region. The solid
black curve represents the Kernel Density Estimate (KDE) of the grey discrete
histogram. The blue (dashed), red (dotted), and green (dot-dashed) curves
refer to the KDE of the distributions when systems are oriented along the
minor, intermediate, and major axis of the cluster ellipsoid, respectively.
Right panel: weak lensing derived mass bias as a function of the true mass of
the cluster. Blue (dashed), red (dotted), and green (dot-dashed) solid lines
display the moving average of the corresponding mass bias data points of the
preferential projections, minor, intermediate, and major axis, respectively.
The top and right sub-panels display the relative mass uncertainties as a
function of the true mass, and weak lensing mass bias distributions,
respectively.

Figure 11:  Top panel: average excess surface mass density profile averaging
six different projections of the same cluster (halo-4 at $z=0.22$): three
randoms and three preferentials. For comparison, solid blue, red, and green
curves show the individual profiles around the three random projections, while
dashed blue, orange and green when the cluster is oriented along the minor,
intermediate and major axis of the moment of inertia tensor ellipsoid. Bottom
panel: relative difference of the individual profiles with respect to the
averaged one.

The reliability with which the cluster mass can be recovered is also related
to the orientation of the moment of inertia tensor ellipsoid with respect to
the line-of-sight (Sereno & Zitrin 2012; Sereno et al. 2013; Despali et al.
2014, 2017; Sereno et al. 2018). Herbonnet et al. (2022) have shown that the
shape of the bright central galaxy can give unbiased information about the
shape of the total halo mass ellipsoid and that the recovered mass via weak
gravitational lensing is biased low (high) if the halo is oriented along the
minor (major) axis with respect to the observer. In this section, we quantify
in detail how the mass bias depends on the orientation of the mass tensor
ellipsoid with respect to the line of sight. Following Sereno et al. (2017)
and Umetsu et al. (2020), we can define the weak lensing signal-to-noise ratio
as:

$\dfrac{S}{N}=\dfrac{\dfrac{\sum_{i}\Delta\Sigma_{i}\,\sigma_{\Delta\Sigma_{i}}^{-2}}{\sum_{i}\sigma_{\Delta\Sigma_{i}}^{-2}}}{\left(\sum_{i}\sigma_{\Delta\Sigma_{i}}^{-2}\right)^{-1/2}}\,,$
(21)

where the variable $i$ runs on all the considered radial bins.

In the left panel of Fig. 10, we show the weak lensing signal-to-noise ratio
of clusters at $z=0.22$. The grey histogram shows the distribution of the
$S/N$ for all considered random projections, while the black curve displays
the Kernel Density Estimate (KDE) of the binned histogram. Blue (dashed), red
(dotted), and green (dot-dashed) curves refer to the distributions of the weak
lensing signal-to-noise ratios when cluster ellipsoids are oriented with
respect to the minor, intermediate, and major axis of the mass tensor
ellipsoid. The triaxial parameters used are those computed by the AHF
algorithm; we refer to Knollmann & Knebe (2009) and Cui et al. (2018) for more
details. The results confirm that orientation matters: weak lensing signal-to-
noise ratio distributions are shifted toward larger (lower) values when
clusters are oriented along the major (minor) axis of the mass tensor
ellipsoid with the line of sight. Projection effects also impact the weak
lensing mass bias and the scatter, as we can observe in the right panel of
Fig. 10. The mass bias, if clusters are preferentially selected either along
the line of sight or in the plane of the sky, is approximately $+25\%$ or
$-25\%$. From the top subpanel, we can also notice that the average relative
weak lensing mass uncertainty varies with the cluster ellipsoid orientation.

Before concluding this section, it is worth devoting a few words to cluster
detections and orientation biases. At fixed mass, when systems are oriented
along the major axis of the ellipsoid, they tend to have, on average, a more
compact and concentrated galaxy satellite distribution than when they are
oriented along the minor axis: this indicates that optical cluster finder
algorithms in observational data may suffer from orientation bias (Wu et al.
2022). We plan to examine and analyse this argument in more detail in a future
dedicated work.

Figure 12: Individual cluster weak lensing signal-to-noise ratio
distributions, for random projections and different redshifts, as expected in
the Euclid wide-field survey of a cluster population that is representative of
the simulated clusters studied here.

For example, in Fig. 11, the magenta solid curve shows the average excess
surface mass density profile of the 6 projections of cluster halo-4 at
redshift $z=0.22$. Solid red, blue, and green lines are the individual
profiles for three random projections. Dotted blue, orange, and green curves
show the profiles when the cluster is oriented along the minor, intermediate,
and major axis of the ellipsoid, with respect to the line of sight, as done
previously. Only radial bins with more than 10 galaxies are shown.

Figure 13: Average weak lensing mass bias as a function of the cluster halo
mass $M_{200}$ for different lens redshifts. We consider all clusters with
$M_{200}>10^{14}M_{\odot}$, above the minimum mass expected to be selected in
the photometric catalogue (Sartoris et al. 2016), relatively constant with
redshift. Right and top subpanels are the same as Fig. 6.

###  4.4 Reshift evolution of the mass bias

The expected source redshift distribution available for weak lensing
measurements tells us how many background sources can be used to probe the
total projected mass distribution of a cluster acting as a gravitational lens.
The reliability of the lensing measurement also depends on the intrinsic
magnitude of the sources and their projected distance with respect to the
cluster centre because of possible confusion with cluster member galaxies. The
weak lensing signal-to-noise ratio of individual clusters is also modulated by
the expected available number density of sources expected from the Euclid wide
survey, as we display in Fig. 12. Low-redshift systems have a larger weak
lensing signal-to-noise ratio than the high-redshift ones. In the figure, we
consider only the measurement of the three random projections per cluster
since we have already discussed in the previous section that the three
preferential ones need particular attention. We also underline that eventual
stacking procedures will shift the expected weak lensing signal-to-noise ratio
distributions toward larger values by a factor that is approximately equal to
the square root of the number of systems that are combined.

In Fig. 13 we exhibit the average weak lensing mass bias for the different
redshifts, considering the three random projections per cluster, and cluster
masses with $M_{200}>10^{14}M_{\odot}$. In our modelling analysis, we assume a
uniform range both for the logarithm of the mass and for the concentration
parameter. On average, higher redshift clusters tend to be more biased low
with respect to $M_{200}$, as a result of their lower signal-to-noise ratio.
Looking to the top subpanel, it is interesting to notice that the relative
error of the recovered weak lensing mass tends to be larger for systems at
higher redshifts, which is due to the lower background source densities and
average lensing efficiencies for those systems. In Tab. 3 we summarise the
values of the parameters describing the linear fitting functions and the
corresponding uncertainties, as in Eq. (20), at different redshifts, for the
reference modelling case, and assuming a constant concentration $c_{200}=3$.

Averaging over all cluster masses with $M_{200}>10^{14}M_{\odot}$, we display
in Fig. 14 the weak lensing mass bias as a function of the lens redshift.
Black circles show the results averaged over all random projections. We can
notice that higher redshift systems are more biased low with respect to the
low redshift ones. Blue crosses and green squares represent the redshift
evolution of the mass bias when the clusters are oriented along the minor or
major axis of the ellipsoid with respect to the line of sight, respectively.
Orange diamonds and olive pentagons refer to the case when in individual
random projections, we model only the logarithm of the cluster mass while
keeping the concentration fixed to $c_{200}=3$ and $c_{200}=4$, respectively.
Those values represent the typical concentration parameters of cluster size-
halo as highlighted in different numerical simulation analyses (Zhao et al.
2009; Giocoli et al. 2012b; Ludlow et al. 2012, 2016). Assuming a fixed
concentration $c_{200}=3$ in our modelling analysis of the weak lensing signal
of individual clusters, we can recover, on average, a weak lensing mass that
is biased by no more than 5% with respect to the true one, up to redshift
$z=0.7$ – positive for $z<0.4$ and negative at higher redshift. The error bars
associated with all data points correspond to the errors on the mean value.
The dotted lines, which differ more significantly from each other at $z>0.6$,
display the results for the whole cluster sample with no minimum mass cut.

Figure 14: Average cluster weak lensing mass bias as a function of the lens redshift, for clusters with $M_{200}>10^{14}M_{\odot}$. The various data points and colours refer to different ways of computing the cluster masses. Black circles display the case of random projections and modelling both the halo mass and concentration, orange diamonds, and olive-coloured pentagons consider the cases in which we assume a fixed concentration parameter – three and four, respectively – and we fit only the halo mass. Light-blue crosses, red triangles and green squares show the average mass bias for the three particular projections, along the minor, the intermediate, and the major axis, respectively. The dotted lines, which slightly deviated from the data points only at $z>0.6$, correspond to the results for the whole cluster sample, with no minimum mass cut.  Table 3: Linear regression parameters as in Eq. (20), at different redshifts for the reference and $c_{200}=3$ cases, considering random projections and for clusters more massive than $M_{200}>10^{14}M_{\odot}.$ snap. | $z$ | Slope $\alpha$ | Intercept $\beta$
---|---|---|---
reference | 0.12 | $0.090\pm 0.028$ | $0.883\pm 0.598$
$c_{200}=3$ | ” | $-0.017\pm 0.035$ | $1.053\pm 0.736$
| 0.22 | $0.183\pm 0.022$ | $0.869\pm 0.452$
| ” | $0.133\pm 0.027$ | $0.973\pm 0.567$
| 0.33 | $0.208\pm 0.023$ | $0.874\pm 0.473$
| ” | $0.196\pm 0.025$ | $0.966\pm 0.524$
| 0.39 | $0.215\pm 0.022$ | $0.873\pm 0.464$
| ” | $0.142\pm 0.027$ | $0.974\pm 0.549$
| 0.49 | $0.223\pm 0.022$ | $0.874\pm 0.453$
| ” | $0.064\pm 0.024$ | $0.985\pm 0.501$
| 0.59 | $0.273\pm 0.019$ | $0.862\pm 0.401$
| ” | $0.126\pm 0.024$ | $0.976\pm 0.486$
| 0.70 | $0.292\pm 0.024$ | $0.836\pm 0.488$
| ” | $0.172\pm 0.027$ | $0.951\pm 0.556$
| 0.82 | $0.279\pm 0.023$ | $0.811\pm 0.473$
| ” | $0.218\pm 0.028$ | $0.935\pm 0.564$
| 0.94 | $0.255\pm 0.024$ | $0.790\pm 0.498$
| ” | $0.314\pm 0.032$ | $0.920\pm 0.653$

The impact of the scatter and the redshift evolution of the cluster mass bias
will be of utter importance for cluster cosmology. Different future works have
already been planned and organised to further assess systematics and nuisance
parameters in the cosmological likelihood pipeline.

##  5 Summary & Conclusions

In this work, we have performed a systematic study of the weak lensing mass
bias using hydrodynamical simulations of clusters. We have simulated the
expected excess surface mass density profile from the expected number density
of background sources of the ESA Euclid wide-field survey, normalised to 30
galaxies per square arcminute. We have adopted a projected truncated NFW
profile to model the data. Our main results are summarised as follows:

  * •

on average individual weak lensing masses $M_{\rm wl}$ are typically lower by
$5\%$ than the true one, various projections of the same cluster may have
different recovered weak lensing masses differing by up to $30\%$: cluster
triaxiality represents the largest source of intrinsic scattering;

  * •

in our reference model, we have adopted a truncation radius of three times
$R_{200}$, a lower truncation radius returns a more biased weak-lensing mass:
from $t=1$ to $t=9$ the mass bias for clusters at $z=0.22$ ranges from $-30\%$
to a few percent;

  * •

in modelling the data set we have also investigated the impact of using a
concentration both dependent, in one case, and independent of the total mass
in another case, finding that those systematically impact the recovered weak
lensing mass (see e.g. also Sommer et al. 2022);

  * •

relaxed clusters, being better described by a 1-halo projected density profile
are less biased than the unrelaxed ones by approximately $3\%$;

  * •

clusters oriented along the major (minor) axis of the mass tensor ellipsoid
have their weak lensing mass overestimated (underestimated) by approximately
$+(-)$ $25\%$;

  * •

averaging over all masses, when also varying the concentration, the negative
mass bias tends to increase as a function of redshift; however, assuming a
fixed concentration, this effect is reduced. The increase of the negative mass
bias with redshift is due to the reduced $S/N$ of the weak lensing mass
constraints of high-redshift clusters. As demonstrated by Sommer et al. (2022)
this $S/N$ dependence can be avoided if the weak lensing mass scatter is
separated into an intrinsic component (e.g. due to variations in density
profiles and substructures) and a shape noise component (e.g. Bocquet et al.
2019; Chiu et al. 2022).

The use of galaxy clusters as cosmological probes relies on the accuracy and
precision of mass measurements. The future ESA Euclid mission will make use of
weak gravitational lensing to determine the projected total mass. In this
work, we have quantified the accuracy and bias effects by using dedicated
lensing simulations of clusters in a variety of dynamical states and at
different redshifts, based on a specific set of hydrodynamical simulations:
the Three Hundred clusters. While the mass bias depends on the considered
choice of the modelling function, the scatter is driven by the complex
triaxial structure of galaxy clusters. However, assuming a fixed value
$c_{200}=3$ for the concentration parameter, when modelling the data of
individual clusters, produces less unbiased results, at least for the set of
simulated clusters analysed in this work.

We would like to conclude underling that future works will be devoted to study
whether these calibrations are robust against variations in the astrophysical
processes included in the simulations, paving the way for future cluster
analyses and opening new perspectives for the cosmological exploitation of
galaxy clusters.

###### Acknowledgements.

CG, MM, and LM acknowledge support from the grant PRIN-MIUR 2017 WSCC32
ZOOMING, and the support from the grant ASI n.2018-23-HH.0. CG acknowledges
funding from the Italian National Institute of Astrophysics under the grant
”Bando PrIN 2019”, PI: Viola Allevato and from the HPC-Europa3 Transnational
Access programme HPC17VDILO. GC also thanks the support from INAF theory Grant
2022: Illuminating Dark Matter using Weak Lensing by Cluster Satellites, PI:
Carlo Giocoli. MM acknowledge financial support from PRIN-MIUR grant and
2020SKSTHZ, INAF “main-stream” 1.05.01.86.20: ”Deep and wide view of galaxy
clusters (P.I.: M. Nonino)” and INAF “main-stream” 1.05.01.86.31 ”The deepest
view of high-redshift galaxies and globular cluster precursors in the early
Universe” (P.I.: E. Vanzella). SB and MM acknowledge financial support from
the InDark INFN Grant. GY and AK would like to thank the Ministerio de Ciencia
e Innovación (Spain) for financial support under research grant
PID2021-122603NB-C21 The THREE HUNDRED simulations used in this work have been
performed in the MareNostrum Supercomputer at the Barcelona Supercomputing
Center, thanks to CPU time granted by the Red Española de Supercomputación. WC
is supported by the STFC AGP Grant ST/V000594/1 and the Atracción de Talento
Contract no. 2020-T1/TIC-19882 granted by the Comunidad de Madrid in Spain. He
also thanks the Ministerio de Ciencia e Innovación (Spain) for financial
support under Project grant PID2021-122603NB-C21. He further acknowledges the
science research grants from the China Manned Space Project with NO. CMS-
CSST-2021-A01 and CMS-CSST-2021-B01. We acknowledge the use of computational
resources from the parallel computing cluster of the Open Physics Hub
(https://site.unibo.it/openphysicshub/en) at the Physics and Astronomy
Department in Bologna. TS acknowledges support from FFG grant number 899537.
CJM acknowledges FCT and POCH/FSE (EC) support through Investigador FCT
Contract 2021.01214.CEECIND/CP1658/CT0001. The figures with the corner plots
displaying the posterior distributions have been produced using the package
pygtc developed by Bocquet & Carter (2016). The authors acknowledge the Euclid
Consortium, the European Space Agency, and a number of agencies and institutes
that have supported the development of Euclid, in particular the Academy of
Finland, the Agenzia Spaziale Italiana, the Belgian Science Policy, the
Canadian Euclid Consortium, the French Centre National d’Etudes Spatiales, the
Deutsches Zentrum für Luft- und Raumfahrt, the Danish Space Research
Institute, the Fundação para a Ciência e a Tecnologia, the Ministerio de
Ciencia e Innovación, the National Aeronautics and Space Administration, the
National Astronomical Observatory of Japan, the Netherlandse Onderzoekschool
Voor Astronomie, the Norwegian Space Agency, the Romanian Space Agency, the
State Secretariat for Education, Research and Innovation (SERI) at the Swiss
Space Office (SSO), and the United Kingdom Space Agency. A complete and
detailed list is available on the Euclid web site (http://www.euclid-ec.org).

## References

  * Ahad et al. (2021) Ahad, S. L., Bahé, Y. M., Hoekstra, H., van der Burg, R. F. J., & Muzzin, A. 2021, MNRAS, 504, 1999
  * Allen et al. (2011) Allen, S. W., Evrard, A. E., & Mantz, A. B. 2011, ARA&A, 49, 409
  * Angelinelli et al. (2020) Angelinelli, M., Vazza, F., Giocoli, C., et al. 2020, MNRAS, 495, 864
  * Ansarifard et al. (2020) Ansarifard, S., Rasia, E., Biffi, V., et al. 2020, A&A, 634, A113
  * Applegate et al. (2016) Applegate, D. E., Mantz, A., Allen, S. W., et al. 2016, MNRAS, 457, 1522
  * Bahé et al. (2012) Bahé, Y. M., McCarthy, I. G., & King, L. J. 2012, MNRAS, 421, 1073
  * Baltz et al. (2009) Baltz, E. A., Marshall, P., & Oguri, M. 2009, J. Cosmology Astropart. Phys., 2009, 015
  * Barnes et al. (2021) Barnes, D. J., Vogelsberger, M., Pearce, F. A., et al. 2021, MNRAS, 506, 2533
  * Bartelmann (2010) Bartelmann, M. 2010, Classical and Quantum Gravity, 27, 233001
  * Bartelmann & Schneider (2001) Bartelmann, M. & Schneider, P. 2001, Physics Reports, 340, 291
  * Beck et al. (2016) Beck, A. M., Dolag, K., & Donnert, J. M. F. 2016, MNRAS, 458, 2080
  * Becker & Kravtsov (2011) Becker, M. R. & Kravtsov, A. V. 2011, ApJ, 740, 25
  * Bellagamba et al. (2011) Bellagamba, F., Maturi, M., Hamana, T., et al. 2011, MNRAS, 413, 1145
  * Bellagamba et al. (2018) Bellagamba, F., Roncarelli, M., Maturi, M., & Moscardini, L. 2018, MNRAS, 473, 5221
  * Bellagamba et al. (2019) Bellagamba, F., Sereno, M., Roncarelli, M., et al. 2019, MNRAS, 484, 1598
  * Benitez-Llambay (2015) Benitez-Llambay, A. 2015, py-sphviewer: Py-SPHViewer v1.0.0
  * Biffi et al. (2016) Biffi, V., Borgani, S., Murante, G., et al. 2016, ApJ, 827, 112
  * Biviano et al. (2006) Biviano, A., Murante, G., Borgani, S., et al. 2006, A&A, 456, 23
  * Biviano et al. (2013) Biviano, A., Rosati, P., Balestra, I., et al. 2013, A&A, 558, A1
  * Bocquet & Carter (2016) Bocquet, S. & Carter, F. W. 2016, The Journal of Open Source Software, 1
  * Bocquet et al. (2019) Bocquet, S., Dietrich, J. P., Schrabback, T., et al. 2019, ApJ, 878, 55
  * Bocquet et al. (2020) Bocquet, S., Heitmann, K., Habib, S., et al. 2020, ApJ, 901, 5
  * Bocquet et al. (2016a) Bocquet, S., Saro, A., Dolag, K., & Mohr, J. J. 2016a, MNRAS, 456, 2361
  * Bocquet et al. (2015) Bocquet, S., Saro, A., Mohr, J. J., et al. 2015, ApJ, 799, 214
  * Bonamigo et al. (2015) Bonamigo, M., Despali, G., Limousin, M., et al. 2015, MNRAS, 449, 3171
  * Bradač et al. (2006) Bradač, M., Clowe, D., Gonzalez, A. H., et al. 2006, ApJ, 652, 937
  * Bradač et al. (2005) Bradač, M., Schneider, P., Lombardi, M., & Erben, T. 2005, A&A, 437, 39
  * Capasso et al. (2019) Capasso, R., Saro, A., Mohr, J. J., et al. 2019, MNRAS, 482, 1043
  * Castro et al. (2021) Castro, T., Borgani, S., Dolag, K., et al. 2021, MNRAS, 500, 2316
  * Chiu et al. (2022) Chiu, I. N., Ghirardini, V., Liu, A., et al. 2022, A&A, 661, A11
  * Chua et al. (2017) Chua, K. T. E., Pillepich, A., Rodriguez-Gomez, V., et al. 2017, MNRAS, 472, 4343
  * Clowe et al. (2004) Clowe, D., Gonzalez, A., & Markevitch, M. 2004, ApJ, 604, 596
  * Cromer et al. (2022) Cromer, D., Battaglia, N., Miyatake, H., & Simet, M. 2022, J. Cosmology Astropart. Phys., 2022, 034
  * Cropper et al. (2016) Cropper, M., Pottinger, S., Niemi, S., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9904, Space Telescopes and Instrumentation 2016: Optical, Infrared, and Millimeter Wave, ed. H. A. MacEwen, G. G. Fazio, M. Lystrup, N. Batalha, N. Siegler, & E. C. Tong, 99040Q
  * Cui et al. (2022) Cui, W., Dave, R., Knebe, A., et al. 2022, MNRAS, 514, 977
  * Cui et al. (2018) Cui, W., Knebe, A., Yepes, G., et al. 2018, MNRAS, 480, 2898
  * Cusworth et al. (2014) Cusworth, S. J., Kay, S. T., Battye, R. A., & Thomas, P. A. 2014, MNRAS, 439, 2485
  * Damsted et al. (2023) Damsted, S., Finoguenov, A., Clerc, N., et al. 2023, arXiv e-prints, arXiv:2307.08749
  * De Lucia et al. (2004) De Lucia, G., Kauffmann, G., Springel, V., et al. 2004, MNRAS, 348, 333
  * Debackere et al. (2022a) Debackere, S. N. B., Hoekstra, H., & Schaye, J. 2022a, MNRAS, 515, 6023
  * Debackere et al. (2022b) Debackere, S. N. B., Hoekstra, H., Schaye, J., Heitmann, K., & Habib, S. 2022b, MNRAS, 515, 3383
  * Debackere et al. (2021) Debackere, S. N. B., Schaye, J., & Hoekstra, H. 2021, MNRAS, 505, 593
  * Despali et al. (2016) Despali, G., Giocoli, C., Angulo, R. E., et al. 2016, MNRAS, 456, 2486
  * Despali et al. (2017) Despali, G., Giocoli, C., Bonamigo, M., Limousin, M., & Tormen, G. 2017, MNRAS, 466, 181
  * Despali et al. (2014) Despali, G., Giocoli, C., & Tormen, G. 2014, MNRAS, 443, 3208
  * Diego et al. (2007) Diego, J. M., Tegmark, M., Protopapas, P., & Sandvik, H. B. 2007, MNRAS, 375, 958
  * Dietrich et al. (2019) Dietrich, J. P., Bocquet, S., Schrabback, T., et al. 2019, MNRAS, 483, 2871
  * Ettori & Eckert (2022) Ettori, S. & Eckert, D. 2022, A&A, 657, L1
  * Euclid Collaboration: Adam et al. (2019) Euclid Collaboration: Adam, R., Vannier, M., et al. 2019, A&A, 627, A23
  * Euclid Collaboration: Blanchard et al. (2020) Euclid Collaboration: Blanchard, A., Camera, S., et al. 2020, A&A, 642, A191
  * Euclid Collaboration: Castro et al. (2022) Euclid Collaboration: Castro, T., Fumagalli, A., et al. 2022, arXiv:2208.02174
  * Euclid Collaboration: Scaramella et al. (2022) Euclid Collaboration: Scaramella, R., Amiaux, J., et al. 2022, A&A, 662, A112
  * Gao et al. (2004) Gao, L., White, S. D. M., Jenkins, A., Stoehr, F., & Springel, V. 2004, MNRAS, 355, 819
  * Gianfagna et al. (2021) Gianfagna, G., De Petris, M., Yepes, G., et al. 2021, MNRAS, 502, 5115
  * Giocoli et al. (2021) Giocoli, C., Marulli, F., Moscardini, L., et al. 2021, A&A, 653, A19
  * Giocoli et al. (2012a) Giocoli, C., Meneghetti, M., Bartelmann, M., Moscardini, L., & Boldrin, M. 2012a, MNRAS, 421, 3343
  * Giocoli et al. (2012) Giocoli, C., Meneghetti, M., Ettori, S., & Moscardini, L. 2012, MNRAS, 426, 1558
  * Giocoli et al. (2014) Giocoli, C., Meneghetti, M., Metcalf, R. B., Ettori, S., & Moscardini, L. 2014, MNRAS, 440, 1899
  * Giocoli et al. (2012b) Giocoli, C., Tormen, G., & Sheth, R. K. 2012b, MNRAS, 422, 185
  * Giocoli et al. (2010a) Giocoli, C., Tormen, G., Sheth, R. K., & van den Bosch, F. C. 2010a, MNRAS, 404, 502
  * Giocoli et al. (2008) Giocoli, C., Tormen, G., & van den Bosch, F. C. 2008, MNRAS, 386, 2135
  * Gonzalez (2014) Gonzalez, A. 2014, in Building the Euclid Cluster Survey - Scientific Program, proceedings of a conference held July 6-11 2014 at the Sexten Center for Astrophysics., 7
  * Grandis et al. (2021) Grandis, S., Bocquet, S., Mohr, J. J., Klein, M., & Dolag, K. 2021, MNRAS, 507, 5671
  * Grandis et al. (2019) Grandis, S., Mohr, J. J., Dietrich, J. P., et al. 2019, MNRAS, 488, 2041
  * Gruen et al. (2011) Gruen, D., Bernstein, G. M., Lam, T. Y., & Seitz, S. 2011, MNRAS, 416, 1392
  * Gruen et al. (2015) Gruen, D., Seitz, S., Becker, M. R., Friedrich, O., & Mana, A. 2015, MNRAS, 449, 4264
  * Henson et al. (2017) Henson, M. A., Barnes, D. J., Kay, S. T., McCarthy, I. G., & Schaye, J. 2017, MNRAS, 465, 3361
  * Herbonnet et al. (2022) Herbonnet, R., Crawford, A., Avestruz, C., et al. 2022, MNRAS, 513, 2178
  * Hoekstra (2001) Hoekstra, H. 2001, A&A, 370, 743
  * Hoekstra (2003) Hoekstra, H. 2003, MNRAS, 339, 1155
  * Hoekstra et al. (2013) Hoekstra, H., Bartelmann, M., Dahle, H., et al. 2013, Space Sci. Rev., 177, 75
  * Hoekstra et al. (2011) Hoekstra, H., Hartlap, J., Hilbert, S., & van Uitert, E. 2011, MNRAS, 412, 2095
  * Hoekstra et al. (2004) Hoekstra, H., Yee, H. K. C., & Gladders, M. D. 2004, ApJ, 606, 67
  * Ingoglia et al. (2022) Ingoglia, L., Covone, G., Sereno, M., et al. 2022, MNRAS, 511, 1484
  * Jauzac et al. (2012) Jauzac, M., Jullo, E., Kneib, J.-P., et al. 2012, MNRAS, 426, 3369
  * Jullo et al. (2014) Jullo, E., Pires, S., Jauzac, M., & Kneib, J.-P. 2014, MNRAS, 437, 3969
  * Jung et al. (2022) Jung, S. L., Rennehan, D., Saeedzadeh, V., et al. 2022, MNRAS, 515, 22
  * Kiiveri et al. (2021) Kiiveri, K., Gruen, D., Finoguenov, A., et al. 2021, MNRAS, 502, 1494
  * Kilbinger (2015) Kilbinger, M. 2015, Reports on Progress in Physics, 78, 086901
  * Klypin et al. (2016) Klypin, A., Yepes, G., Gottlöber, S., Prada, F., & Heß, S. 2016, MNRAS, 457, 4340
  * Knollmann & Knebe (2009) Knollmann, S. R. & Knebe, A. 2009, ApJS, 182, 608
  * Kravtsov & Borgani (2012) Kravtsov, A. V. & Borgani, S. 2012, ARA&A, 50, 353
  * Lau et al. (2009) Lau, E. T., Kravtsov, A. V., & Nagai, D. 2009, ApJ, 705, 1129
  * Laureijs et al. (2011) Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, arXiv:1110.3193
  * Lee et al. (2023) Lee, W., Cha, S., Jee, M. J., et al. 2023, ApJ, 945, 71
  * Lesci et al. (2022) Lesci, G. F., Marulli, F., Moscardini, L., et al. 2022, A&A, 659, A88
  * Ludlow et al. (2016) Ludlow, A. D., Bose, S., Angulo, R. E., et al. 2016, MNRAS, 460, 1214
  * Ludlow et al. (2012) Ludlow, A. D., Navarro, J. F., Li, M., et al. 2012, MNRAS, 427, 1322
  * Martizzi et al. (2014) Martizzi, D., Mohammed, I., Teyssier, R., & Moore, B. 2014, MNRAS, 440, 2290
  * Marulli et al. (2016) Marulli, F., Veropalumbo, A., & Moresco, M. 2016, Astronomy and Computing, 14, 35
  * Maturi et al. (2019) Maturi, M., Bellagamba, F., Radovich, M., et al. 2019, MNRAS, 485, 498
  * Meneghetti et al. (2008) Meneghetti, M., Melchior, P., Grazian, A., et al. 2008, A&A, 482, 403
  * Meneghetti & Rasia (2013) Meneghetti, M. & Rasia, E. 2013, arXiv:1303.6158
  * Meneghetti et al. (2010b) Meneghetti, M., Rasia, E., Merten, J., et al. 2010b, A&A, 514, A93
  * Meneghetti et al. (2014) Meneghetti, M., Rasia, E., Vega, J., et al. 2014, ApJ, 797, 34
  * Merten et al. (2009) Merten, J., Cacciato, M., Meneghetti, M., Mignone, C., & Bartelmann, M. 2009, A&A, 500, 681
  * Mostoghiu et al. (2019) Mostoghiu, R., Knebe, A., Cui, W., et al. 2019, MNRAS, 483, 3390
  * Navarro et al. (1996) Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563
  * Navarro et al. (1997) Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493
  * Niemiec et al. (2020) Niemiec, A., Jauzac, M., Jullo, E., et al. 2020, MNRAS, 493, 3331
  * Oguri & Hamana (2011) Oguri, M. & Hamana, T. 2011, MNRAS, 414, 1851
  * Oguri & Takada (2011) Oguri, M. & Takada, M. 2011, Phys. Rev. D, 83, 023008
  * Ondaro-Mallea et al. (2022) Ondaro-Mallea, L., Angulo, R. E., Zennaro, M., Contreras, S., & Aricò, G. 2022, MNRAS, 509, 6077
  * Planck Collaboration (2016) Planck Collaboration. 2016, Astron. Astrophys., 594, A13
  * Planelles et al. (2014) Planelles, S., Borgani, S., Fabjan, D., et al. 2014, MNRAS, 438, 195
  * Prada et al. (2012) Prada, F., Klypin, A. A., Cuesta, A. J., Betancort-Rijo, J. E., & Primack, J. 2012, MNRAS, 423, 3018
  * Pratt et al. (2019) Pratt, G. W., Arnaud, M., Biviano, A., et al. 2019, Space Sci. Rev., 215, 25
  * Rasia et al. (2013) Rasia, E., Borgani, S., Ettori, S., Mazzotta, P., & Meneghetti, M. 2013, ApJ, 776, 39
  * Rasia et al. (2015) Rasia, E., Borgani, S., Murante, G., et al. 2015, ApJ, 813, L17
  * Rasia et al. (2012) Rasia, E., Meneghetti, M., Martino, R., et al. 2012, New Journal of Physics, 14, 055018
  * Sartoris et al. (2016) Sartoris, B., Biviano, A., Fedeli, C., et al. 2016, MNRAS, 459, 1764
  * Schaller et al. (2015a) Schaller, M., Frenk, C. S., Bower, R. G., et al. 2015a, MNRAS, 451, 1247
  * Schaller et al. (2015b) Schaller, M., Frenk, C. S., Bower, R. G., et al. 2015b, MNRAS, 452, 343
  * Schrabback et al. (2018) Schrabback, T., Applegate, D., Dietrich, J. P., et al. 2018, MNRAS, 474, 2635
  * Schrabback et al. (2021) Schrabback, T., Bocquet, S., Sommer, M., et al. 2021, MNRAS, 505, 3923
  * Sereno et al. (2017) Sereno, M., Covone, G., Izzo, L., et al. 2017, MNRAS, 472, 1946
  * Sereno et al. (2013) Sereno, M., Ettori, S., Umetsu, K., & Baldi, A. 2013, MNRAS, 428, 2241
  * Sereno et al. (2018) Sereno, M., Umetsu, K., Ettori, S., et al. 2018, ApJ, 860, L4
  * Sereno & Zitrin (2012) Sereno, M. & Zitrin, A. 2012, MNRAS, 419, 3280
  * Sheth & Tormen (1999) Sheth, R. K. & Tormen, G. 1999, MNRAS, 308, 119
  * Simet et al. (2017a) Simet, M., Battaglia, N., Mandelbaum, R., & Seljak, U. 2017a, MNRAS, 466, 3663
  * Simet et al. (2017b) Simet, M., McClintock, T., Mandelbaum, R., et al. 2017b, MNRAS, 466, 3103
  * Sommer et al. (2022) Sommer, M. W., Schrabback, T., Applegate, D. E., et al. 2022, MNRAS, 509, 1127
  * Springel & Hernquist (2003) Springel, V. & Hernquist, L. 2003, MNRAS, 339, 289
  * Springel et al. (2008a) Springel, V., Wang, J., Vogelsberger, M., et al. 2008a, MNRAS, 391, 1685
  * Springel et al. (2008b) Springel, V., White, S. D. M., Frenk, C. S., et al. 2008b, Nature, 456, 73
  * Springel et al. (2001) Springel, V., White, S. D. M., Tormen, G., & Kauffmann, G. 2001, MNRAS, 328, 726
  * Steinborn et al. (2015) Steinborn, L. K., Dolag, K., Hirschmann, M., Prieto, M. A., & Remus, R.-S. 2015, MNRAS, 448, 1504
  * Tinker et al. (2008) Tinker, J., Kravtsov, A. V., Klypin, A., et al. 2008, ApJ, 688, 709
  * Tinker et al. (2010) Tinker, J. L., Robertson, B. E., Kravtsov, A. V., et al. 2010, ApJ, 724, 878
  * Tormen (1998) Tormen, G. 1998, MNRAS, 297, 648
  * Tormen et al. (1998) Tormen, G., Diaferio, A., & Syer, D. 1998, MNRAS, 299, 728
  * Tormen et al. (2004) Tormen, G., Moscardini, L., & Yoshida, N. 2004, MNRAS, 350, 1397
  * Tornatore et al. (2007) Tornatore, L., Borgani, S., Dolag, K., & Matteucci, F. 2007, MNRAS, 382, 1050
  * Umetsu (2020) Umetsu, K. 2020, A&A Rev., 28, 7
  * Umetsu et al. (2020) Umetsu, K., Sereno, M., Lieu, M., et al. 2020, ApJ, 890, 148
  * Velliscig et al. (2014) Velliscig, M., van Daalen, M. P., Schaye, J., et al. 2014, MNRAS, 442, 2641
  * Wu et al. (2022) Wu, H.-Y., Costanzi, M., To, C.-H., et al. 2022, MNRAS, 515, 4471
  * Zhao et al. (2009) Zhao, D. H., Jing, Y. P., Mo, H. J., & Bnörner, G. 2009, ApJ, 707, 354
  * Zohren et al. (2022) Zohren, H., Schrabback, T., Bocquet, S., et al. 2022, A&A, 668, A18