A&A, 684, A139 (2024)
https://doi.org/10.1051/0004-6361/202348743
c© The Authors 2024
Astronomy
&Astrophysics
Euclid preparation
XXXVII. Galaxy colour selections with Euclid and ground photometry for cluster
weak-lensing analyses
Euclid Collaboration: G. F. Lesci1,2 , M. Sereno2,3 , M. Radovich4 , G. Castignani1,2 , L. Bisigello5,4 , F. Marulli1,2,3 , L. Moscardini1,2,3 ,
L. Baumont6 , G. Covone7,8,9 , S. Farrens6 , C. Giocoli2,10 , L. Ingoglia1 , S. Miranda La Hera6, M. Vannier11, A. Biviano12,13 ,
S. Maurogordato11, N. Aghanim14, A. Amara15, S. Andreon16 , N. Auricchio2 , M. Baldi17,2,3 , S. Bardelli2, R. Bender18,19 , C. Bodendorf18,
D. Bonino20, E. Branchini21,22 , M. Brescia7,8,9 , J. Brinchmann23 , S. Camera24,25,20 , V. Capobianco20 , C. Carbone26 , J. Carretero27,28 ,
S. Casas29 , F. J. Castander30,31 , M. Castellano32 , S. Cavuoti8,9 , A. Cimatti33, G. Congedo34 , C. J. Conselice35, L. Conversi36,37 ,
Y. Copin38 , L. Corcione20 , F. Courbin39 , H. M. Courtois40 , A. Da Silva41,42 , H. Degaudenzi43 , A. M. Di Giorgio44, J. Dinis42,41,
F. Dubath43 , C. A. J. Duncan35,45, X. Dupac37, S. Dusini46 , M. Farina44, S. Ferriol38, P. Fosalba31,47 , S. Fotopoulou48, M. Frailis12 ,
E. Franceschi2 , P. Franzetti26, M. Fumana26 , S. Galeotta12 , B. Garilli26 , B. Gillis34 , A. Grazian4 , F. Grupp18,49, S. V. H. Haugan50 ,
I. Hook51 , F. Hormuth52, A. Hornstrup53,54 , P. Hudelot55, K. Jahnke56 , M. Kümmel19 , S. Kermiche57 , A. Kiessling58 , M. Kilbinger59 ,
B. Kubik38, M. Kunz60 , H. Kurki-Suonio61,62 , S. Ligori20 , P. B. Lilje50 , V. Lindholm61,62 , I. Lloro63, E. Maiorano2 , O. Mansutti12 ,
O. Marggraf64 , K. Markovic58 , N. Martinet65 , R. Massey66 , E. Medinaceli2 , M. Melchior67, Y. Mellier68,55, M. Meneghetti2,3 ,
E. Merlin32 , G. Meylan39, M. Moresco1,2 , E. Munari12 , R. Nakajima64, S.-M. Niemi69, C. Padilla27 , S. Paltani43, F. Pasian12, K. Pedersen70,
V. Pettorino71, S. Pires6, G. Polenta72 , M. Poncet73, L. A. Popa74, L. Pozzetti2 , F. Raison18 , R. Rebolo75,76, A. Renzi5,46 , J. Rhodes58,
G. Riccio8, E. Romelli12 , M. Roncarelli2 , E. Rossetti17, R. Saglia19,18 , D. Sapone77 , B. Sartoris19,12, M. Schirmer56 , P. Schneider64 ,
A. Secroun57 , G. Seidel56 , S. Serrano31,30,78 , C. Sirignano5,46 , G. Sirri3 , J. Skottfelt79 , L. Stanco46 , J.-L. Starck59 ,
P. Tallada-Crespí80,28 , A. N. Taylor34, H. I. Teplitz81 , I. Tereno41,82, R. Toledo-Moreo83 , F. Torradeflot28,80 , I. Tutusaus84 , E. A. Valentijn85,
L. Valenziano2,86 , T. Vassallo19,12 , A. Veropalumbo16,22 , Y. Wang81 , J. Weller19,18, A. Zacchei12,13 , G. Zamorani2 , J. Zoubian57,
E. Zucca2 , M. Bolzonella2 , E. Bozzo43, C. Colodro-Conde75, D. Di Ferdinando3, J. Graciá-Carpio18, S. Marcin67, N. Mauri33,3 ,
C. Neissner27,28 , A. A. Nucita87,88,89, Z. Sakr90,84,91 , V. Scottez68,92, M. Tenti3 , M. Viel13,12,93,94,95 , M. Wiesmann50, Y. Akrami96,97 ,
S. Anselmi5,46,98 , C. Baccigalupi93,12,94,13 , M. Ballardini99,100,2 , S. Borgani101,13,12,94 , A. S. Borlaff102,103,104 , S. Bruton105,
C. Burigana106,86 , R. Cabanac84 , A. Calabro32 , A. Cappi2,11, C. S. Carvalho82, T. Castro12,94,13,95 , G. Cañas-Herrera69,107 ,
K. C. Chambers108 , A. R. Cooray109 , J. Coupon43, O. Cucciati2 , S. Davini22, S. de la Torre65, G. De Lucia12 , G. Desprez110,
S. Di Domizio21,22 , H. Dole14 , A. Díaz-Sánchez111 , J. A. Escartin Vigo18, S. Escoffier57 , I. Ferrero50 , F. Finelli2,86 , L. Gabarra5,46,
K. Ganga112 , J. García-Bellido96 , F. Giacomini3 , G. Gozaliasl113,61 , S. Gwyn114 , H. Hildebrandt115 , M. Huertas-Company75,116,117,118 ,
A. Jimenez Muñoz119 , J. J. E. Kajava120,121 , V. Kansal122,123, C. C. Kirkpatrick124, L. Legrand60 , A. Loureiro125,126 , J. Macias-Perez119 ,
M. Magliocchetti44 , G. Mainetti127, R. Maoli128,32 , M. Martinelli32,129 , C. J. A. P. Martins130,23 , S. Matthew34, M. Maturi90,131 ,
L. Maurin14 , R. B. Metcalf1,2 , M. Migliaccio132,133, P. Monaco101,12,94,13 , G. Morgante2, S. Nadathur15 , L. Patrizii3, A. Pezzotta18,
C. Porciani64, D. Potter134 , M. Pöntinen61 , P. Reimberg68 , P.-F. Rocci14, A. G. Sánchez18 , A. Schneider134 , M. Schultheis11,
E. Sefusatti12,13,94 , P. Simon64, A. Spurio Mancini135 , S. A. Stanford136 , J. Steinwagner18, G. Testera22, R. Teyssier137, S. Toft54,138,
S. Tosi21,22,16 , A. Troja5,46 , M. Tucci43, J. Valiviita61,62 , and D. Vergani2
(Affiliations can be found after the references)
Received 27 November 2023 / Accepted 19 January 2024
ABSTRACT
Aims. We derived galaxy colour selections from Euclid and ground-based photometry, aiming to accurately define background galaxy samples in
cluster weak-lensing analyses. These selections have been implemented in the Euclid data analysis pipelines for galaxy clusters.
Methods. Given any set of photometric bands, we developed a method for the calibration of optimal galaxy colour selections that maximises the
selection completeness, given a threshold on purity. Such colour selections are expressed as a function of the lens redshift.
Results. We calibrated galaxy selections using simulated ground-based griz and Euclid YEJEHE photometry. Both selections produce a purity
higher than 97%. The griz selection completeness ranges from 30% to 84% in the lens redshift range zl ∈ [0.2, 0.8]. With the full grizYEJEHE
selection, the completeness improves by up to 25 percentage points, and the zl range extends up to zl = 1.5. The calibrated colour selections are
stable to changes in the sample limiting magnitudes and redshift, and the selection based on griz bands provides excellent results on real external
datasets. Furthermore, the calibrated selections provide stable results using alternative photometric aperture definitions obtained from different
ground-based telescopes. The griz selection is also purer at high redshift and more complete at low redshift compared to colour selections found
in the literature. We find excellent agreement in terms of purity and completeness between the analysis of an independent, simulated Euclid galaxy
catalogue and our calibration sample, except for galaxies at high redshifts, for which we obtain up to 50 percentage points higher completeness.
The combination of colour and photo-z selections applied to simulated Euclid data yields up to 95% completeness, while the purity decreases down
to 92% at high zl. We show that the calibrated colour selections provide robust results even when observations from a single band are missing
from the ground-based data. Finally, we show that colour selections do not disrupt the shear calibration for stage III surveys. The first Euclid data
releases will provide further insights into the impact of background selections on the shear calibration.
Key words. galaxies: clusters: general – galaxies: distances and redshifts – galaxies: photometry – galaxies: statistics – cosmology: observations –
large-scale structure of Universe
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.
A139, page 1 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
1. Introduction
In the last decade, galaxy clusters have proven to be excel-
lent probes for cosmological analyses (see, e.g. Mantz et al.
2015; Sereno et al. 2015; Planck Collaboration XXIV 2016;
Costanzi et al. 2019; Marulli et al. 2021; Lesci et al. 2022),
also allowing for the investigation of dark matter interaction
models (Peter et al. 2013; Robertson et al. 2017; Eckert et al.
2022) and gas astrophysics (Vazza et al. 2017; CHEX-MATE
Collaboration 2021; Zhu et al. 2021; Sereno et al. 2021). As
galaxy clusters are dominated by dark matter, the functional
form of their matter density profiles can be derived from N-body
dark-matter-only simulations (Navarro et al. 1997; Baltz et al.
2009; Diemer & Kravtsov 2014). This allows one to estimate the
mass of observed clusters, which is essential for both astrophys-
ical and cosmological studies (Teyssier et al. 2011; Pratt et al.
2019). Currently, weak gravitational lensing is one of the most
reliable methods to accurately and precisely measure cluster
masses (Okabe et al. 2010; Hoekstra et al. 2012; Melchior et al.
2015; Sereno et al. 2017; Stern et al. 2019; Schrabback et al.
2021; Zohren et al. 2022). Consequently, weak-lensing cluster
mass estimates are widely used in current photometric galaxy
surveys, such as the Kilo Degree Survey (KiDS; Kuijken et al.
2019; Bellagamba et al. 2019), the Dark Energy Survey (DES;
Abbott et al. 2020; Sevilla-Noarbe et al. 2021), and the Hyper
Suprime-Cam survey (HSC; Medezinski et al. 2018; Li et al.
2022).
An accurate selection of lensed background galaxies is cru-
cial to derive a reliable cluster weak-lensing signal. Includ-
ing the contribution from foreground and cluster mem-
ber galaxies may significantly dilute the weak-lensing sig-
nal (Broadhurst et al. 2005; Medezinski et al. 2007; Sifón et al.
2015; McClintock et al. 2019). For example, background selec-
tions with 90% purity dilute the cluster reduced shear measure-
ments by 10% (see, e.g. Dietrich et al. 2019), in the absence of
intrinsic alignments (Heymans & Heavens 2003). Highly pure
background selections are required to properly account for this
effect in weak-lensing measurements, in order to minimise
the variance in the selection purity. Selection incompleteness,
instead, impacts the weak-lensing noise and, in turn, the signal-
to-noise ratio (S/N), which depends on the density of back-
ground sources along with the intrinsic ellipticity dispersion and
measurement noise (see, e.g. Schrabback et al. 2018; Umetsu
2020). The effect of low background densities can be partially
mitigated by increasing the size of the cluster-centric radial bins
used in the analysis, or through the stacking of the weak-lensing
signal of cluster ensembles.
Background selections based on the galaxy photometric red-
shift (photo-z) posteriors are commonly used in the literature
(Gruen et al. 2014; Applegate et al. 2014; Melchior et al. 2017;
Sereno et al. 2017; Bellagamba et al. 2019), as well as galaxy
colour selections (Medezinski et al. 2010, 2018; Oguri et al.
2012; Klein et al. 2019). These selections can also be com-
bined to significantly improve the background sample com-
pleteness and, in turn, the weak-lensing S/N. In fact, colour
selections have been demonstrated to help identify galaxies
with poorly defined photometric redshifts that would not have
been classified as background sources through photo-z selection
alone (Covone et al. 2014; Sereno et al. 2017; Bellagamba et al.
2019).
The aim of this paper is to develop a method to obtain opti-
mal colour selections, namely with a maximal completeness
given a threshold on purity, given any set of photometric fil-
ters. We provide, for the first time, colour selections expressed
as a continuous function of the lens limiting redshift. This allows
for a finer background definition compared to colour selections
found in the literature (Medezinski et al. 2010, 2018; Oguri et al.
2012), implying a significant improvement in the weak-lensing
source statistics. In view of Euclid and stage IV surveys,
we derived colour selections on simulated data. We exploited
the galaxy catalogue developed by Bisigello et al. (2020, here-
after referred to as B20), and extended by Euclid Collaboration
(2023a), which includes simulated Sloan Digital Sky Survey
(SDSS; Gunn et al. 1998) griz magnitudes and simulated Euclid
observations in the YEJEHE bands. In addition, we tested the effi-
ciency of these colour selections on real public external data and
on simulations, combining them with photo-z selections.
This paper is part of a series presenting and discussing mass
measurements of galaxy clusters using the Euclid combined
clusters and weak-lensing pipeline COMB-CL. COMB-CL forms
part of the global Euclid data processing pipeline and is respon-
sible for measuring weak-lensing shear profiles and masses for
photometrically detected clusters. A comprehensive description
of the code structure and methods employed by COMB-CL will
be presented in a forthcoming paper, but a brief overview of the
pipeline can be found in the appendix of Euclid Collaboration
(2023b). The galaxy colour selections presented in this paper are
already implemented in COMB-CL.
The paper is organised as follows. In Sect. 2, we describe the
dataset used for the calibration of galaxy colour selections, and
in Sect. 3 we detail a general method to derive optimal colour
selections. In Sect. 4, we show the selections obtained for griz
and grizYEJEHE filter sets, validating them on external datasets.
In Sect. 5 we compare the griz selection calibrated in this work
with selections from the literature. Finally, in Sect. 6, we draw
our conclusions.
2. Calibration sample
We based our analysis on the photometric catalogue devel-
oped by B20 and extended by Euclid Collaboration (2023a).
This catalogue contains simulated Euclid IEYEJEHE aperture
magnitudes1, covering the spectral range 5500−20 000 Å, along
with the Canada-France Imaging Survey (CFIS; Ibata et al.
2017) u band, for the galaxies contained in the COSMOS cat-
alogue by Laigle et al. (2016, COSMOS15). Specifically, such
photometry is based on 3′′ fixed-aperture magnitudes. Despite
the u band already being present in COSMOS15, B20 derived it
using the same approach adopted for the other filters in order to
avoid colour biases. B20 verified that this provides results that
are consistent with the observed fluxes. Simulated SDSS griz
magnitudes, spanning the wavelength range 4000−11 000 Å, are
also provided in the catalogue, since observations in similar fil-
ters, such as those in Vera C. Rubin Observatory (Rubin/LSST;
Ivezic et al. 2008) and DES, will be available to comple-
ment Euclid observations (Euclid Collaboration 2021, 2022a).
Corrections for photometric offsets due to flux outside the fixed-
aperture, systematic offsets, and Galactic extinction, as sug-
gested in Laigle et al. (2016), have been applied. B20 derive
simulated magnitudes through two alternative approaches. The
first is a linear interpolation of the 30 medium-band and broad-
band filters available in the COSMOS15 catalogue, based on
the effective wavelength of the filters. The second approach is
1 IE band observations are supplied by the Euclid Visible Imager
(VIS; Cropper et al. 2016), while YEJEHE photometry is pro-
vided by the Near-Infrared Spectrometer and Photometer (NISP;
Euclid Collaboration 2022b).
A139, page 2 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
based on the best theoretical template that describes the spectral
energy distribution (SED) of each galaxy, assuming the COS-
MOS15 redshifts as the ground truth. The SED fitting is per-
formed based on COSMOS15 bands and the template resulting
in the minimum χ2 is used to predict the expected fluxes. We
refer to B20 for the details of the SED templates used, based
on the model by Bruzual & Charlot (2003). The expected fluxes
are then randomised 10 times considering a Gaussian distribu-
tion centred on the true flux and with standard deviation equal
to the expected photometric uncertainities, scaled considering
the depths listed in Table 1 of Euclid Collaboration (2023a).
In this process, the IEYEJEHE magnitude errors expected for
the Euclid Wide Survey are considered. Despite the fact that
the griz photometry is based on SDSS filter transmissions, the
corresponding uncertainties are based on depths that are con-
sistent with those of DES and the Ultraviolet Near-Infrared
Optical Northern Survey (UNIONS)2. The ugriz photometry
provided by LSST is expected to go from 1 to 2.5 mag deeper
at the end of the Euclid mission, depending on the photomet-
ric filter. Throughout this paper, we focus on the magnitudes
derived from the best theoretical SED templates, as these esti-
mates better reproduce absorption and emission lines that are not
covered by COSMOS15 bands. We neglect u magnitudes since,
due to the low u-band throughput, a 5σ depth of 25.6 mag will
only be reached after 10 years of LSST observations3. In addi-
tion, the u band is not available in DES wide fields. We empha-
sise that the B20 catalogue contains all the galaxies present in
the COSMOS15 sample, which is deeper than the shear sam-
ples derived from current surveys (see, e.g. Giblin et al. 2021;
Gatti et al. 2021) and expected from the Euclid Wide Survey
(Euclid Collaboration 2022a). As we discuss in the following,
the colour selections calibrated in this study yield robust results
against alternative magnitude cuts, including those that repro-
duce the selections adopted in current and Euclid cosmic shear
analyses.
3. Method
In order to find a set of optimal galaxy colour-redshift relations
that maximises the selection completeness given a threshold on
the foreground contamination, we considered the colours given
by any combination of photometric bands. This includes bands
that are not adjacent in wavelength. Thus, for each colour-colour
space, given a redshift lower limit, zl, corresponding to the lens
redshift, we considered the following set of conditions,
x > c1 ∨
x < c2 ∨
y > c3 ∨
y < c4 ∨
x > s1y + c5 ∨
x > s2y + c6 ∨
x < s3y + c7 ∨
x < s4y + c8, (1)
where ∨ is the logical “or” operator, x and y are two different
colours, and ci and si are colour selection parameters. Specifi-
2 UNIONS is carried out with the Subaru Telescope (Iye et al.
2004), the Canada-France-Hawaii Telescope (CFHT; Gwyn 2012), and
the Panoramic Survey Telescope and Rapid Response System (Pan-
STARRS; Chambers et al. 2016). More information at https://www.
skysurvey.cc/news/
3 https://www.lsst.org/scientists/keynumbers
Fig. 1. Example of an uncalibrated selection in the (r−z)−(g−i) colour-
colour space. The grey dots represent the selected galaxy colours.
Galaxies within the octagonal hatched region are excluded by apply-
ing Eq. (1). Specifically, (r − z) and (g − i) correspond to x and y in
Eq. (1), respectively.
cally, c1, . . . , c8 ∈ (−∞,+∞), s1 and s3 ∈ (0,+∞), while s2 and
s4 ∈ (−∞, 0). The edges of the aforementioned parameter ranges
are excluded, and Eq. (1) defines an irregular octagon that con-
tains the foreground galaxies, as we show in Fig. 1. As we shall
see, since we only select the colour conditions that satisfy given
requirements, not all the sides of the irregular octagon may be
considered. In addition, since we considered the conditions in
Eq. (1) as independent, the c1, . . . , c8 and s1, . . . , s4 parameters
are not related to each other. In particular, for each condition in
Eq. (1), we derived the completeness,
Cn fi (zl | p) :=
Nsel,i(zg > zl | p)
Ntot(zg > zl)
, (2)
and the purity,
Pn fi (zl | p) :=
Nsel,i(zg > zl | p)
Nsel,i(zg ≥ 0 | p) , (3)
where i is the ith colour condition index, zg is the galaxy redshift,
p is the set of colour condition parameters, Nsel,i is the number
of galaxies selected with the ith colour condition, Ntot is the total
number of galaxies in the calibration sample, while the n f super-
script represents quantities derived from colour conditions not
fitted as a function of zl. As we shall see, we do not adopt any
superscripts for the quantities derived from fitted colour condi-
tions. In Eqs. (2) and (3), we have i = 1 . . . Ncond, where Ncond
is the number of all possible colour conditions, given Eq. (1),
expressed as
Ncond = 8
Ncol!
(Ncol − 2)! 2! , (4)
where Ncol is the number of colours, given by
Ncol =
Nband!
(Nband − 2)! 2! , (5)
where Nband is the number of photometric bands.
We set requirements on completeness and purity to be satis-
fied by each colour condition in Eq. (1). Specifically, for a given
zl, we selected the colour conditions having at least one p set
A139, page 3 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
Set of photometric bands
Ncond colour
conditions, based on Eq. (1)
Measure Cn fi (zl | p) and
Pn fi (zl | p), with i = 1, . . . ,Ncond
Calibration sample
zl points
Find p for which Cn fi (zl | p)
and Pn fi (zl | p) are above the
corresponding thresholds
Thresholds on
Cn fi and Pn fi
Derive optimal colour
conditions, maximis-
ing Cn fi (zl | p) ∀i, zl
Combine the optimal
conditions, deriving
Cn ftot(zl), Pn ftot(zl), F n ftot (zl)
Exclude the zl points for which
Cn ftot(zl) = F n ftot (zl) = 100%
Subset of optimal conditions
minimising Sn f in Eq. (7)
Fit p as a function of zl
Calibrated optimal colour selection
Fig. 2. Flowchart summarising the calibration process described in
Sect. 3. Round red rectangles represent the start and end points of the
calibration process. Grey rectangles represent processing steps, while
blue trapezoids correspond to the inputs.
providing Cn fi (zl | p) and Pn fi (zl | p) larger than their correspond-
ing thresholds. We remark that p does not explicitly depend on
zl at this stage, and that zl values are arbitrarily sampled. Set-
ting a threshold on Cn fi (zl | p) is important for excluding colour
conditions that do not significantly contribute to the total com-
pleteness, and that may appear as optimal only due to statisti-
cal fluctuations. Thus, the threshold on Cn fi (zl | p) is meant to be
low compared to that on Pn fi (zl | p). Indeed, as we shall detail
in Sect. 4.6, impurities in the background selection imply sys-
tematic uncertainties in galaxy cluster reduced shear measure-
ments. Highly pure selections are required to properly account
for this effect, in order to minimise the scatter in purity. We dis-
cuss the choice of the thresholds on Cn fi (zl | p) and Pn fi (zl | p) in
greater detail in Sect. 4.1. For each colour condition in Eq. (1),
with parameter values for which the conditions on Cn fi (zl | p) and
Pn fi (zl | p) are satisfied, we selected the p set providing the high-
est completeness at a given zl. In this way, we derived the set of
optimal colour conditions maximising the selection complete-
ness, given the chosen threshold on purity.
We note that the maximum zl of the calibrated colour selec-
tions depends on the Cn fi (zl | p) and Pn fi (zl | p) limits, while the
minimum zl is derived by excluding the zl points for which
Cn ftot(zl) = F n ftot (zl) = 100%. Here, Cn ftot and F n ftot are the com-
pleteness and the foreground failure rate given by the full set of
optimal colour conditions, respectively. For simplicity, we drop
the dependence on p in the text. The foreground failure rate is
defined as follows:
F n ftot (zl) :=
Nsel(zg < zl)
Ntot(zg < zl)
=
Nsel(zg > zl)
Ntot(zg < zl)
1 − Pn ftot(zl)
Pn ftot(zl)
, (6)
where Nsel is the number of galaxies selected with all the opti-
mal colour conditions, given a condition on zg, and Pn ftot(zl) is the
purity given by the full set of optimal conditions. On the right-
hand side of Eq. (6), derived from Eqs. (2) and (3), we can see
that F n ftot (zl) diminishes with increasing zl if high lower limits on
purity are chosen. We stress that F n ftot (zl) ≤ 1 by definition.
In the selection process described above, some colour con-
ditions may be redundant. Thus, we iteratively searched for an
optimal subset of colour conditions to find the minimum number
of conditions sufficient to approximately reproduce the required
completeness. Specifically, at each step of this iterative process,
we computed the following quantity:
Sn f =
N∑
j=1
Cn ftot(zl, j) − Cn f (zl, j), (7)
where N is the number of zl points, Cn ftot(zl, j) is the completeness
given by all optimal conditions, while Cn f (zl, j) is the complete-
ness given by a subset of optimal conditions, computed at the
jth zl value. As the first step of this iterative process, we found
the optimal colour condition minimising Sn f . Then, at each iter-
ation, we added the colour condition that, combined with the
conditions selected in the previous steps, minimises Sn f . We
repeated this process until Sn f was lower than a given tolerance.
We remark that the logical operator between colour conditions
is ∨.
Lastly, we applied a nonlinear least squares analysis to find
the best fit to the p parameters as a function of zl for the sub-
set of optimal colour conditions. We chose the fitting formulae
which best reproduce the zl dependence, namely polynomials,
while aiming at minimising the number of free parameters in
the fit.
In Fig. 2 we show a flowchart summarising the calibration
process described in this section. In Fig. 3 we show an exam-
ple of the iterative process detailed above, while Fig. 4 displays
an example of parameter dependence on zl. Hereafter, we refer
to the completeness, purity, and foreground failure rate, derived
from sets of fitted colour conditions, as C(zl), P(zl), and F (zl),
respectively. For better clarity, in Table 1 we summarise the sym-
bols referring to the completeness functions introduced in this
section.
A139, page 4 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
20
30
40
50
60
70
80
90
nf
(z
l) 
[%
]
All conditions
Iteration 2
Iteration 5
Iteration 10
97.0
97.5
98.0
nf
(z
l) 
[%
]
0.2 0.3 0.4 0.5 0.6 0.7 0.8
zl
0
20
40
nf
(z
l) 
[%
]
Fig. 3. Selection completeness (top panel), purity (middle panel), and
foreground failure rate (bottom panel), derived from subsets of optimal
colour conditions not fitted as a function of zl, for the case of griz pho-
tometry. The solid black lines represent the selection given by the full
set of optimal colour conditions, while the dashed lines show the selec-
tion at different steps of the iterative process detailed in Sect. 3, given
by subsets of optimal conditions.
Table 1. Description of the completeness functions introduced in
Sect. 3.
Symbol Description
Cn fi
Completeness of the ith colour condition,
given a set of sampled parameters.
Cn ftot
Completeness derived through the combination
of all the optimal colour conditions.
Cn f Completeness given by the combinationof a subset of optimal colour conditions.
C Completeness obtained from a subset of optimalcolour conditions fitted as a function of zl.
Notes. Analogous descriptions hold for purity and foreground failure
rate. The n f superscript represents quantities derived from colour con-
ditions not fitted as a function of zl. Optimal colour conditions satisfy
the thresholds on purity and completeness, and provide maximal com-
pleteness.
4. Results
4.1. Calibration of colour selections
By applying the methods detailed in Sect. 3 and adopting the
B20 calibration sample described in Sect. 2, we calibrated
galaxy colour selections using ground-based and Euclid pho-
tometry, namely SDSS griz and Euclid YEJEHE filters, respec-
tively. These selections are implemented in COMB-CL, and will
be available for weak-lensing analyses of galaxy clusters. We
0.2 0.4 0.6 0.8
zl
1.2
1.3
1.4
1.5
1.6
1.7
s
(r i) < s (r z) + c
Fit
Optimisation
0.2 0.4 0.6 0.8
zl
1.1
1.0
0.9
0.8
0.7
c
Fig. 4. Values of s (left panel) and c (right panel) parameters, from
Eq. (1), as a function of zl for the colour condition quoted in the left
panel legend. The black dots represent the optimal values of s and c,
while the blue curves represent the polynomial fits.
considered the following cases: ground-only, Euclid-only, and
the combination of ground-based and Euclid photometry. For
the cases including Euclid photometry, we adopted an S/N
threshold for Euclid near-infrared observations of (S/N)E > 3,
which corresponds to YE < 24.85, JE < 25.05, and HE <
24.95 (Euclid Collaboration 2022a). In addition, we considered
zl points in the range zl ∈ [0.1, 2.5], assuming a precision of
δzl = 0.1 for the sampling. To derive the full set of optimal colour
conditions, we imposed Cn fi (zl | p) > 10% for the ith colour con-
dition. For the ground-only and Euclid-only cases, we imposed
that the purity of each colour condition is Pn fi (zl | p) > 99%.
We adopted a more restrictive threshold on purity for the com-
bination of ground-based and Euclid photometry, corresponding
to Pn fi (zl | p) > 99.7%. This threshold is chosen as the larger
number of colour combinations leads to a higher summation of
impurities. We obtained Pn f (zl) > 97% for any zl, when combin-
ing all the optimal colour conditions, as shown in Fig. 5. As we
shall discuss in Sect. 4.3, the purity derived from different real
datasets is stable, showing sub-percent changes, on average.
The F n f (zl) decrease with increasing zl, shown in Fig. 5, is
expected, as discussed in Sect. 3. In addition, for any combina-
tion of photometric bands, we found Cn ftot(zl) = F n ftot (zl) = 100%
for zl = 0.1. Consequently, we set zl = 0.2 as the minimum
lens redshift for the calibrated colour selections. As shown in
Fig. 5, from griz photometry we derived a selection within
zl ∈ [0.2, 0.8], with 84% completeness at zl = 0.2, decreasing
to 29% at zl = 0.8. In the Euclid-only case, namely YEJEHE and
IE bands, results are not competitive with those derived from griz
photometry. On the other hand, by combining ground-based and
Euclid photometry, the completeness significantly increases in
the zl range covered by the griz selection, by up to 25 percentage
points. Also the zl range of the selection is significantly extended
compared to the griz case, corresponding to zl ∈ [0.2, 1.5].
Specifically, in this case we exclude the Euclid IE band, as it cov-
ers a large wavelength interval, namely ∼5000−10 000 Å, corre-
sponding to the wavelength range already covered by griz pho-
tometry. Furthermore, the use of very broad photometric bands is
not the most optimal choice for calibrating galaxy colour selec-
tions, which share similarities with photo-z estimates.
We excluded any possible redundant colour condition, as
detailed in Sect. 3. In Table A.1 we show the subset of optimal
colour conditions for the ground-only case, namely griz photom-
etry, along with the corresponding parameter fits. The first con-
dition quoted in Table A.1 corresponds to the one derived in the
first step of the iterative process described in Sect. 3. This is anal-
ogous for the subsequent conditions. We remark that the quoted
A139, page 5 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
0.5 1.0 1.5
zl
20
40
60
80
Co
m
ple
te
ne
ss
 [%
]
Ground (subset, fit)
Ground (full set, measure)
Euclid (full set, measure)
Ground+Euclid (subset, fit)
Ground+Euclid (full set, measure)
0.5 1.0 1.5
zl
97.0
97.5
98.0
98.5
Pu
rit
y [
%
]
0.5 1.0 1.5
zl
0
10
20
30
40
50
60
Fo
re
gr
ou
nd
 fa
ilu
re
 ra
te
 [%
]
0 5 10 15 20 25 30
Iteration
0
50
100
150
200
250
300
350
nf
; 
griz, 
griz, nf
grizYEJEHE, 
grizYEJEHE, nf
0.2 0.4 0.6 0.8 1.0 1.2 1.4
zl
8
6
4
2
0
2
4
nf to
t(z
l)
(z
l) 
[%
]
griz
grizYEJEHE
Fig. 5. Summary of the results on the colour selection optimisation, based on the B20 galaxy sample. Top panels: selection completeness (left
panel), purity (central panel), and foreground failure rate (right panel), as a function of zl. The dashed lines represent the selections derived from
the full sets of optimal conditions not fitted as a function of zl, in the case of ground-only (red), Euclid-only (green), and the combination of
ground-based and Euclid bands (grey). The solid lines represent the selections obtained from the subsets of optimal conditions, with parameters
fitted as a function of zl, in the case of ground-only (blue) and for the combination of ground-based and Euclid bands (black). Bottom panels:
in the left panel, S and Sn f are shown as a function of the iteration number. For the ground-based selection, using griz filters, S and Sn f are
represented by solid blue and dashed red lines, respectively. For the selection derived from the combination of ground-based and Euclid filters,
namely grizYEJEHE, S and Sn f are represented by solid black and dashed grey lines, respectively. In the right panel, the difference between Cn ftot
and C is shown, for the griz (blue lines) and grizYEJEHE (black lines) selections.
conditions have different ranges of validity in zl. Analogous
information is listed in Table A.2 for the combination of ground-
based and Euclid photometry, corresponding to grizYEJEHE fil-
ters. We neglected the optimisation and parameter fitting for the
Euclid-only case, as we have already shown that it does not pro-
vide competitive completeness values.
In Fig. 5 we show the results for the selections obtained from
the subsets of optimal conditions, with parameters fitted as a
function of zl. For both griz and grizYEJEHE photometry, such
fitted selections well reproduce those given by the full sets of
optimal conditions. To quantify the goodness of the colour con-
dition parameter fits, we defined a parameter analogous to Sn f
in Eq. (7), namely S. This parameter quantifies the difference
between Cn ftot, that is the completeness given by the full set of
optimal conditions not fitted as a function of zl, and C, which is
the completeness given by the subset of optimal colour condi-
tions fitted as a function of zl. As shown in Fig. 5, S does not
perfectly match Sn f , for both griz and grizYEJEHE selections.
This is due to the fact that the c1, . . . , c8, s1, . . . , s4 parameters in
Eq. (1) do not always show a simple dependence on zl. Despite
the fact that better parameter fits could be achieved by adopt-
ing an arbitrarily high order polynomial as the model, we set a
4th order polynomial as the highest-degree functional form for
describing these parameters (see Tables A.1 and A.2). As shown
in Fig. 5, C is underestimated by at most 4 percentage points.
We verified that adding further conditions to these selections,
that is, lowering the S threshold down to 0, provides sub-percent
level improvements in the selection completeness, on average.
We remark that, in order to derive colour selections not defined
in zl bins, the final selection completeness is slightly degraded
compared to Cn ftot for some zl values. In realistic cluster weak-
lensing analyses, however, we expect this to statistically increase
the galaxy background completeness. When colour selections
are defined on finite sets of zl points, the background galaxies are
excluded based on the zl precision adopted in the colour selec-
tion calibration.
4.2. Dependence on magnitude and redshift selections
To verify the robustness of the griz selection with respect to
alternative magnitude cuts, we applied the selection i < 24.4,
corresponding to the peak value of the i magnitude distribution
in the B20 catalogue. We also investigate the selection for the
subsample with i < 23.4, which is a threshold similar to the
DES i band limit (Sevilla-Noarbe et al. 2021). In both cases,
we derived higher P(zl) and lower F (zl), compared to what we
found from the calibration sample used in Sect. 4.1, namely the
one with (S/N)E > 3 and i ≤ imax, where imax = 24.9 is the
maximum i magnitude in the sample (see Fig. 6). In the case
with i < 24.4, C(zl) is close to that from the calibration sample,
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Euclid Collaboration: A&A, 684, A139 (2024)
0.2 0.4 0.6 0.8
zl
20
40
60
80
(z
l) 
[%
]
griz, (S/N)E > 3, zl = 10 3
i imax i < 24.4 i < 23.4 i imax, zg < 1.5
0.2 0.4 0.6 0.8
zl
96
97
98
99
(z
l) 
[%
]
0.2 0.4 0.6 0.8
zl
0
10
20
30
40
50
(z
l) 
[%
]
0.2 0.4 0.6 0.8
zl
2.5
5.0
7.5
10.0
12.5
15.0
n b
(z
l) 
[a
rc
m
in
2 ]
0.5 1.0 1.5
zl
20
40
60
80
(z
l) 
[%
]
grizYEJEHE, zl = 10 3
i imax, (S/N)E > 3 i < 23.4, (S/N)E > 3 i imax, (S/N)E > 10
0.5 1.0 1.5
zl
97
98
99
100
(z
l) 
[%
]
0.5 1.0 1.5
zl
0
10
20
30
40
(z
l) 
[%
]
0.5 1.0 1.5
zl
0
5
10
15
n b
(z
l) 
[a
rc
m
in
2 ]
Fig. 6. Results from the fitted colour selections derived in Sect. 4.1, assuming the alternative magnitude and redshift selections described in
Sect. 4.2. From left to right: completeness, purity, foreground failure rate, and background density as a function of zl. The assumed zl precision is
δzl = 10−3. Top panels: efficiency of the griz selection, detailed in Table A.1, applied to the B20 catalogue with (S/N)E > 3 and i ≤ imax (blue solid
lines), to its subsample including galaxies with i < 24.4 (red dotted lines), to the case with i < 23.4 (green dash-dotted lines), and to the sample
with zg < 1.5 (orange dashed lines). Bottom panels: efficiency of the grizYEJEHE selection, detailed in Table A.2, applied to the B20 catalogue
with (S/N)E > 3 and i ≤ imax (black solid lines), to its subsample with i < 23.4 (green dash-dotted lines), and to the subsample with (S/N)E > 10
(magenta dashed lines).
while for i < 23.4 we derived higher completeness, on aver-
age. In addition, as the bulk of the redshift distribution in the
calibration sample, described in Sect. 2, extends up to zg ∼ 4,
we applied the griz selection to the galaxy sample with redshift
zg < 1.5, (S/N)E > 3, and i ≤ imax. In Fig. 6, we can see
that this redshift limit provides F (zl) values that are identical
to those derived from the calibration sample, which is expected
since F (zl) does not depend on the maximum redshift of the
sample, while the completeness increases by up to 10 percentage
points and the purity is at most 1 percentage point lower. We note
that the computation of C(zl) and F (zl) is made relative to the
sample under consideration. In other words, they refer to galaxy
populations defined by given magnitude and redshift limits. We
measured the aforementioned colour selections by assuming a zl
precision of δzl = 10−3. This δzl value is one order of magni-
tude lower (i.e. one order of magnitude higher precision) than
the typical galaxy cluster photometric redshift uncertainty in
current surveys (see, e.g. Rykoff et al. 2016; Maturi et al. 2019)
and Euclid (Euclid Collaboration 2019). Consequently, the δzl =
10−3 precision ensures the reliability of the colour condition
fits for galaxy cluster background selections. We remark that
we assumed δzl = 0.1 for the zl sampling in the calibration
process.
In Fig. 6 we show the efficiency of the grizYEJEHE selection,
computed by adopting δzl = 10−3, applied to the B20 calibration
sample, with (S/N)E > 3 and i ≤ imax. We found analogous
selections from the subsample with i < 23.4 and from the one
with (S/N)E > 10. Specifically, in both cases, we derived higher
P(zl) and lowerF (zl), in agreement with what we found from the
griz selection. In addition, the increase in the minimum Euclid
S/N does not significantly change the completeness, while the
i < 23.4 limit decreases C(zl) by at most 18 percentage points.
As we obtained excellent P(zl) and F (zl) estimates from these
tests, we conclude that both griz and grizYEJEHE selections are
stable and reliable with respect to changes in the sample lim-
iting magnitude and redshift. In addition, we note that brighter
galaxy samples provide lower foreground contamination. This is
expected, as faint galaxies have more scattered colour-redshift
relations.
In Fig. 6 we show the density of background galaxies, nb(zl),
defined as the number of selected galaxies with zg > zl per
square arcmin. For both griz and grizYEJEHE selections, nb(zl) =
16 arcmin−2 at zl = 0.2 for i ≤ imax and (S/N)E > 3, decreas-
ing with increasing zl. In both colour selections, the i < 23.4
limit implies the largest decrease in nb(zl), providing nb(zl) <
7 arcmin−2. In addition, for the griz selection, the i < 24.4 and
zg < 1.5 limits provide consistent results on nb(zl), showing a dif-
ference of at most 3 arcmin−2 compared to that derived from the
calibration sample. With regard to the grizYEJEHE selection, the
(S/N)E > 10 limit implies a decrease in nb(zl) of up to 5 arcmin−2
at low zl, while nb(zl) becomes compatible with that derived from
the calibration sample for zl > 1.
4.3. griz selection validation on real data
To further assess the reliability of the griz colour selection
detailed in Sect. 4.1, we applied it to external datasets obtained
from real observations. In particular, we considered the VIMOS
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Euclid Collaboration: A&A, 684, A139 (2024)
0.2 0.4 0.6 0.8
zl
20
40
60
80
(z
l) 
[%
]
griz, zl = 10 3
Bisigello+20 Moutard+16, AUTO COSMOS20, HSC, AUTO COSMOS20, HSC, 2'' COSMOS20, HSC, 3''
0.2 0.4 0.6 0.8
zl
97
98
99
(z
l) 
[%
]
0.2 0.4 0.6 0.8
zl
0
20
40
60
(z
l) 
[%
]
0.2 0.4 0.6 0.8
zl
2.5
5.0
7.5
10.0
12.5
15.0
17.5
n b
(z
l) 
[a
rc
m
in
2 ]
Fig. 7. Results of the application of the fitted colour selection based on griz photometry, reported in Table A.1, to the datasets introduced in
Sect. 4.3. From left to right: completeness, purity, foreground failure rate, and background density as a function of zl. The assumed zl precision is
δzl = 10−3. The griz selection is applied to the B20 catalogue with magnitude limits corresponding to those used in the calibration process (blue
solid lines), to the full depth Moutard et al. (2016) catalogue (green solid lines), and to the Weaver et al. (2022) catalogue with HSC Kron, 2′′ and
3′′ aperture magnitudes (solid red, dashed orange and dotted black lines, respectively), for which we imposed i < 25.
Public Extragalactic Redshift Survey (VIPERS; Guzzo et al.
2014) Multi-Lambda Survey (VMLS) photometric catalogue by
Moutard et al. (2016), including Canada-France-Hawaii Tele-
scope Legacy Survey (CFHTLS; Hudelot et al. 2012) griz Kron
aperture magnitudes (Kron 1980). This catalogue covers 22 deg2
and provides reliable photometric redshifts for more than one
million galaxies with a typical accuracy of σz ≤ 0.04, and a frac-
tion of catastrophic failures lower than 2% down to i ∼ 23. These
statistics are based on VIPERS data, complemented with the
most secure redshifts selected from other spectroscopic surveys.
We remind that in VIPERS a colour-colour pre-selection was
employed to enhance the effective sampling of the VIMOS spec-
trograph. Nevertheless, the VIPERS selection does not introduce
any significant colour bias above z ∼ 0.6 (Guzzo et al. 2014). In
addition, as we shall see in the following, the selection complete-
ness and purity obtained from the VMLS dataset do not exhibit
remarkable deviations from those obtained from other galaxy
samples. In Fig. 7 we can see that, by applying the griz selection
to the VMLS sample, we derived higher P(zl) and lower F (zl)
compared to what we found from the B20 catalogue, on average.
This agrees with what we found in Sect. 4.2, as the Moutard et al.
(2016) catalogue is shallower than the B20 sample. For the same
reason, nb(zl) is 3 arcmin−2 lower, on average. In addition, the
completeness is up to 8 percentage points higher for zl < 0.6,
becoming lower for higher zl values.
We also applied the griz selection to the COSMOS CLASSIC
catalogue by Weaver et al. (2022, COSMOS20), which reaches
the same photometric redshift precision as COSMOS15, namely
σz/(1 + z) = 0.007, at almost one magnitude deeper. We con-
sidered griz Kron, 2′′ and 3′′ aperture magnitudes from HSC.
In addition, we selected galaxies with a photometric redshift
derived from at least 30 bands, and with i < 25, in order to
consider a sample with highly reliable redshift estimates. By
adopting more complex selection criteria, which may involve
galaxies with photometric redshifts derived from a shared set of
photometric bands, we do not expect remarkable differences in
the results. Similar results for the cases with Kron, 2′′ and 3′′
aperture magnitudes are shown in Fig. 7. Compared to what we
derived from the B20 sample, the completeness is similar, with
the largest differences at zl > 0.6. In addition, F (zl) is lower and
P(zl) is higher for any zl. For Kron and 3′′ aperture magnitudes,
nb(zl) is slightly higher compared to that obtained from the B20
sample, on average. Lower nb(zl) values show up for the 2′′ aper-
ture magnitudes, which is expected as we applied the same mag-
nitude limit for each photometric aperture definition. Indeed, for
these tests we did not include aperture correction terms. Lastly,
comparing the purity derived from the COSMOS20 and VMLS
samples, we note that for zl > 0.3 the differences are below
1 percentage point, on average. Thus, we conclude that the griz
selection provides robust and reliable results on real data.
4.4. Validation on Flagship v2.1
We tested the colour selections calibrated in Sect. 4.1 on the
Euclid Flagship galaxy catalogue v2.1.10 (Euclid Collaboration,
in prep.), which is currently the best simulated Euclid galaxy cat-
alogue available. This catalogue is based on an N-body simula-
tion with around 4 trillion particles with mass mp ∼ 109 h−1 M.
A flat Λ cold dark matter (ΛCDM) cosmological model was
assumed, with matter density parameter Ωm = 0.319, baryon
density parameter Ωb = 0.049, dark energy density parameter
ΩΛ = 0.681, scalar spectral index ns = 0.96, Hubble parame-
ter h = H0/(100 km s−1 Mpc−1) = 0.67, and standard deviation of
linear density fluctuations on 8 h−1 Mpc scales σ8 = 0.83. The
haloes were identified using Rockstar (Behroozi et al. 2013),
and then populated with a halo occupation distribution model
which was calibrated to reproduce observables such as cluster-
ing statistics as a function of galaxy luminosity. The galaxy SED
templates used are the COSMOS templates from Ilbert et al.
(2009), based on the models by Bruzual & Charlot (2003) and
Polletta et al. (2007). In addition, galaxy photo-z probability
distribution functions, namely p(zg), are included in Flagship,
derived through a Nearest Neighbours Photometric Redshifts
(NNPZ) pipeline (Euclid Collaboration 2020).
From the Flagship catalogue, we extracted a lightcone within
RA ∈ [158◦, 160◦] and Dec ∈ [12◦, 15◦], considering the galax-
ies in the whole redshift range covered by the simulation, namely
zg ∈ [0, 3]. Specifically, zg is the galaxy true redshift, and we
verified that the contribution of peculiar velocities does not sig-
nificantly change the results. We focused on 2′′ aperture LSST
ugrizy and Euclid IEYEJEHE photometry, as the simulated fluxes
estimated for other ground-based surveys do not account for
observational noise. Specifically, the photometric noise takes
into account the depth expected in the southern hemisphere at the
time of the third data release (DR3) for the Euclid Wide Survey.
The LSST and Euclid 10σ magnitude limits, which are proxies
for extended sources, correspond to u < 24.4, g < 25.6, r < 25.7,
i < 25.0, z < 24.3, y < 23.1, IE < 25, YE < 23.5, JE < 23.5,
A139, page 8 of 19
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0.5 1.0 1.5
zl
20
40
60
80
(z
l) 
[%
]
zl = 10 3
griz, Bisigello+20 grizYEJEHE, Bisigello+20 griz, Flagship v2.1 grizYEJEHE, Flagship v2.1
0.5 1.0 1.5
zl
97
98
99
100
(z
l) 
[%
]
0.5 1.0 1.5
zl
0
10
20
30
40
(z
l) 
[%
]
0.5 1.0 1.5
zl
0
1
2
3
4
5
6
n b
(z
l) 
[a
rc
m
in
2 ]
Fig. 8. Application of the calibrated colour selections to the Flagship simulated sample described in Sect. 4.4. From left to right: completeness,
purity, foreground failure rate, and background density as a function of zl, from the fitted colour selections based on griz and grizYEJEHE bands,
adopting 5σ magnitude limits. The assumed zl precision is δzl = 10−3. The solid blue and solid black lines represent the griz and grizYEJEHE
selections, respectively, applied to the B20 catalogue. The dashed red and dashed green curves represent the griz and grizYEJEHE selections,
respectively, applied to the Flagship v2.1 catalogue (Euclid Collaboration, in prep.).
and HE < 23.5. The fluxes we considered are not reddened due
to Milky Way extinction, consistent with the analyses performed
in the previous sections.
In Fig. 8, we show the application of griz and grizYEJEHE
selections to Flagship. For this test, we assumed 5σ magnitude
cuts for LSST ugrizy and Euclid IEYEJEHE bands. In addition,
we show results from the B20 sample in Fig. 8, for which we
assumed 5σ magnitude cuts rescaled from the 10σ limits listed
in Euclid Collaboration (2023a, Table 1). We found that nb(zl)
derived from Flagship agrees with that obtained from the B20
sample. The largest differences, of about 1 arcmin−2, arise when
the grizYEJEHE selection is applied. We note that nb(zl) ∼ 0
for zl ∼ 1.5, implying that lenses at these values of zl may
not exhibit significant weak-lensing signals. Nevertheless, we
verified that nb(zl) is enhanced at any zl when the selection
defined for Euclid weak-lensing analyses (Laureijs et al. 2011;
Euclid Collaboration 2022a) is assumed. This selection consists
in a 10σ cut in the IE band, corresponding to IE < 25 for a 2′′
aperture, yielding a galaxy density of around 39 arcmin−2 when
applied to the Flagship dataset. In fact, in this case nb(zl) ranges
from 30 arcmin−2 at low zl to 3 arcmin−2 at zl = 1.5.
As shown in Fig. 8, on average we obtained higher P(zl)
and lower F (zl) for zl < 1 from Flagship, compared to what
we derived from the B20 sample. For the griz selection case,
C(zl) agrees with that derived from the B20 sample, with the
largest differences, of up to 16 percentage points, at zl ∼ 0.5.
Larger differences in C(zl) are obtained from the grizYEJEHE
selection. From Flagship we obtained C(zl) up to 10 percent-
age points larger for zl < 0.6, and up to 50 percentage points
larger for higher zl. We verified that this discrepancy in the com-
pleteness, in the case of the grizYEJEHE selection, is not signif-
icantly attenuated through the assumption of 3σ and 10σ mag-
nitude limits on both B20 and Flagship catalogues. Analogous
results were obtained by assuming limits corresponding to the
magnitude distribution peaks derived from the B20 catalogue,
namely g < 24.9, r < 24.6, i < 24.3, z < 24.1, YE < 23.8,
JE < 23.6, and HE < 23.5. Moreover, we verified that the
grizYEJEHE selection completeness does not remarkably vary
by assuming the Euclid weak-lensing selection defined above,
namely IE < 25. Similar results are obtained by considering the
photometric errors expected for the DR2 of the Euclid Wide Sur-
vey, assuming the corresponding 3σ, 5σ, and 10σ magnitude
cuts. For each of the alternative magnitude cuts discussed in this
section, we found that the grizYEJEHE selection yields a purity
up to 3 percentage points higher at zl > 1.2 when it is applied to
the B20 catalogue, compared to what is derived from Flagship.
At zl < 1.2, instead, the purity obtained from B20 is 1 percentage
point lower, on average. Furthermore, the alternative magnitude
cuts do not remarkably impact the selection purity at any zl.
We additionally adopted SDSS fluxes, which do not include
photometric noise, in place of LSST fluxes in Flagship. In this
case, the completeness is up to 35 percentage points larger than
that derived from the B20 sample, and the purity approaches
100% for zl > 1, which is similar to what we derived from the
B20 sample (see Fig. 8). Thus, the selection based on SDSS pho-
tometry is less complete and purer compared to that obtained
from LSST magnitudes.
Differences in the completeness derived from the Flagship
and B20 samples may originate from distinct assumptions on
the physical properties of the galaxies, such as dust extinction,
stellar age, nebular emission lines, or on the assumed intrinsic
spectral energy distributions. This could be indicated by a differ-
ent fraction of star forming galaxies in the two samples. Follow-
ing B20, galaxies are classified as star forming if the following
condition is satisfied,
log10(sSFR/yr
−1) > −10.5, (8)
where sSFR is the specific star formation rate, derived from the
best SED template in the catalogue by B20. We verified that,
for zg > 1, the fraction of star forming galaxies in Flagship is
consistent within 1 percentage point with that derived from the
catalogue by B20. Thus, we conclude that the completeness dif-
ferences between the Flagship and B20 samples are not due to
different star forming galaxy populations. We also verified that
the log10(sSFR/yr
−1) distributions derived from the two datasets
are compatible, having peaks at ∼−8.13 and ∼−8.35 in B20 and
Flagship, respectively. The agreement of these peak values is
well within 1σ of the log10(sSFR/yr
−1) distributions. We will be
able to further investigate such completeness differences through
the analysis of the first data release of the Euclid Deep Survey.
4.5. Comparison with photo-z selections
To compare the colour selections derived in this work to selec-
tions based on the galaxy p(zg), commonly referred to as photo-
z selections, we analysed the Flagship sample described in
Sect. 4.4. We considered only the galaxies with a p(zg) esti-
mate obtained with the NNPZ pipeline (Euclid Collaboration
2020). The NNPZ photo-zs are designed to work well for
A139, page 9 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
0.5 1.0 1.5
zl
50
60
70
80
90
100
(z
l) 
[%
]
grizYEJEHE
Colour selection photo-z selection Colour  photo-z selection p(zg) mode
0.5 1.0 1.5
zl
85.0
87.5
90.0
92.5
95.0
97.5
100.0
(z
l) 
[%
]
0.5 1.0 1.5
zl
0
5
10
15
20
25
(z
l) 
[%
]
0.5 1.0 1.5
zl
10
20
30
n b
(z
l) 
[a
rc
m
in
2 ]
Fig. 9. Comparison of colour and photo-z selections. From left to right: completeness, purity, foreground failure rate, and background density as a
function of zl, obtained from Flagship v2.1. The solid black lines represent the grizYEJEHE selection. The dashed green lines show the combination,
through the ∨ logical operator, of grizYEJEHE and photo-z selection (Eq. (9)). The dashed red lines represent the photo-z selection, while the dotted
black lines represent the selection based on the p(zg) mode.
galaxies that are expected to be used in core Euclid weak-lensing
science, namely with 5σ limits on the IE band. Thus we imposed
IE < 25.75, along with 5σ limits on the YEJEHE bands, namely
YE < 24.25, JE < 24.25, HE < 24.25. Specifically, we adopted
the following photo-z selection,
zming > zl, (9)
where zming is the minimum of the interval containing 95% of the
probability around the first mode of p(zg), namely zg. We chose
zming in order to derive P(zl) values which are compatible with
those obtained from colour selections. We verified that adding a
condition on the width of p(zg) in Eq. (9) does not impact the
results. Specifically, for the latter test, we considered the addi-
tional condition A > Amin, where A is the integrated probabil-
ity around zg, computed within the redshift points, which are the
closest to zg, having an associated probability of 0.2p(zg). We
verified that imposingAmin = 0 orAmin = 0.8 leads to compati-
ble purity values with sub-percent differences on average. How-
ever, Amin = 0.8 lowers the photo-z selection completeness by
around 20 percentage points at all zl. Consequently, we assumed
Amin = 0.
To perform a fair comparison of colour and photo-z selec-
tions, we considered only the grizYEJEHE colour selection in
this section. This is because photo-zs in Flagship were derived
from the combination of ground-based and Euclid photometry.
In Fig. 9, we show that the grizYEJEHE selection provides, on
average, a completeness 15 percentage points lower than that of
the photo-z selection, with similar contamination. By combining
grizYEJEHE and photo-z selections, through the logical opera-
tor ∨, the completeness increases by up to 10 percentage points
with respect to the case of photo-z selection alone, amounting to
C(zl) ∼ 95% for zl < 1.4. These preliminary tests confirm the
importance of the combination of colour and photo-z selections,
as it leads to significantly more complete background galaxy
samples. We also remark that increasing the selection complete-
ness is key to reduce biases in the shear calibration parameters
due to background selections, as we shall detail in Sect. 4.6.
The analysis of Euclid data will allow for a detailed investiga-
tion of the optimal photo-z selections for galaxy cluster weak-
lensing analyses, outlining the synergies with colour selections.
For example, colour selections applied to Euclid data could pro-
vide more robust background samples for massive or nearby
galaxy clusters, as derived by Medezinski et al. (2018). Lever-
aging colour selections also serves as a valuable cross-validation
method for addressing the effect of unknown systematic uncer-
tainties in photo-z estimates. Lastly, Fig. 9 shows the selection
based only on the first mode of p(zg). Specifically, in this case
we selected the galaxies with zg > zl. Despite C(zl) > 90% at
all zl, the purity is up to 10 percentage points lower than that
obtained from the grizYEJEHE selection.
4.6. Impact on shear measurements
In cluster weak-lensing analyses, the inclusion of foreground
sources in the shear measurements may significantly dilute
the signal (Broadhurst et al. 2005; Medezinski et al. 2007;
Sifón et al. 2015; McClintock et al. 2019). As discussed in
the previous sections, the calibrated colour selections provide
P(zl) < 1. To assess the impact of impurities on shear mea-
surements, we can express the cluster reduced tangential shear
unaffected by contamination as follows (Dietrich et al. 2019):
gt,true(zl) =
gt(zl)
P(zl) , (10)
where gt(zl) is the measured cluster reduced tangential shear at
redshift zl. As the calibrated colour selections yieldP(zl) > 0.97,
we expect at most a 3% bias on the reduced tangential shear. In
addition, as discussed in Sect. 4.3, P(zl) derived from different
observed datasets with only ground-based photometry shows a
scatter below 1 percentage point. This scatter in P(zl) is lower
than the systematic uncertainty on galaxy shape measurements
for stage III surveys, as we shall discuss in the following. We
remark that P(zl) is derived from reference fields, while galaxy
clusters are overdense compared to the cosmic mean. Thus, con-
tamination from cluster galaxies must be properly accounted for
in Eq. (10) (see, e.g. Gruen et al. 2014; Dietrich et al. 2019).
Nevertheless, such contamination is consistent with zero in
the typical cluster-centric radial range adopted for mass cal-
ibration, namely at radii larger than 300 h−1 kpc (see, e.g.
Medezinski et al. 2018; Bellagamba et al. 2019).
Furthermore, galaxy shear calibration is usually statistically
derived, based on observed and simulated galaxy samples. Nev-
ertheless, through galaxy cluster background selections, some
galaxy populations may be systematically excluded. This may
invalidate the statistical estimate of the shape multiplicative
bias, namely m, depending on the shear measurement technique
and on the actual properties of the data (Heymans et al. 2012;
Miller et al. 2013; Hildebrandt et al. 2016).
The typical uncertainty on m found for stage III sur-
veys ranges in the interval δm ∈ [1 × 10−2, 3 × 10−2] (see,
A139, page 10 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
0.2 0.4 0.6 0.8
zl
20
40
60
80
(z l
) [
%
]
Ground
0.2 0.4 0.6 0.8
zl
95.5
96.0
96.5
97.0
97.5
98.0
(z l
) [
%
]
All bands Missing z Missing i Missing r Missing g
0.2 0.4 0.6 0.8
zl
0
10
20
30
40
50
(z l
) [
%
]
0.2 0.4 0.6 0.8
zl
2.5
5.0
7.5
10.0
12.5
15.0
n b
(z l
) [
ar
cm
in
2 ]
0.5 1.0 1.5
zl
20
40
60
80
(z l
) [
%
]
Ground+Euclid
0.5 1.0 1.5
zl
97.0
97.5
98.0
98.5
(z l
) [
%
]
0.5 1.0 1.5
zl
0
10
20
30
40
(z l
) [
%
]
0.5 1.0 1.5
zl
0
5
10
15
n b
(z l
) [
ar
cm
in
2 ]
0.2 0.3 0.4 0.5 0.6 0.7 0.8
zl
15
10
5
0
5
10
nf to
t(z
l)
(z
l) 
[%
]
Ground
0.2 0.4 0.6 0.8 1.0 1.2 1.4
zl
Ground+Euclid
Fig. 10. Colour selection results, obtained from the B20 catalogue, in case of missing z (solid red), i (solid green), r (dashed orange), and g (dashed
grey) bands. The blue curves represent the results from the griz and grizYEJEHE selections reported in Tables A.1 and A.2. Top panels, from left
to right: completeness, purity, foreground failure rate, and background density are shown, in the case of ground-only photometry. Middle panels:
colour selections from the combination of ground-based and Euclid photometry. The plot structure is analogous to that of top panels. Bottom
panels: difference between Cn ftot and C, for ground-only observations (left panel) and for the combination of ground-based and Euclid photometry
(right panel).
e.g. Jarvis et al. 2016; Melchior et al. 2017; Giblin et al. 2021).
To assess the impact of colour selections on m, we consid-
ered the shape catalogues of Heymans et al. (2012), based on
CFHTLS, and of Mandelbaum et al. (2018), based on the HSC
Subaru Strategic Program (HSC-SSP; Miyazaki et al. 2018;
Aihara et al. 2018). Throughout this section, we adopted a lens
redshift of zl = 0.5. By applying the griz selection calibrated in
this work, we derived a shift in the mean shear multiplicative bias
of ∆m = 7 × 10−3 in CFHTLS and of ∆m = −2 × 10−3 in HSC-
SSP. In addition, the Oguri et al. (2012) and Medezinski et al.
(2018) colour selections provide ∆m = −3 × 10−3 and ∆m =
−1 × 10−2 from CFHTLS, respectively, while from HSC-SSP
we obtained ∆m = −5 × 10−3 and ∆m = −7 × 10−3, respec-
tively. Thus, galaxy population differences due to colour cuts
provide systematic effects that are within the typical m uncer-
tainty in stage III surveys. By combining colour and photo-z
selections, we expect ∆m to become closer to zero. In Euclid-
like surveys, shear has to be calibrated within an accuracy of
2 × 10−3 (Cropper et al. 2013). As we discussed in Sect. 4.5,
the combination of grizYEJEHE and photo-z selections leads to
90% background completeness in the Euclid Wide Survey, on
average; thus, we may expect the bias on m to be subdomi-
nant with respect to the required shear accuracy. Indeed, let us
assume that 90% of galaxies, selected through the combination
of grizYEJEHE and photo-z selections, have an average m similar
to that derived from stage III surveys, namely 〈m〉 = 0.01. We
assume that the remaining 10% of galaxies have a very biased m,
namely 〈m〉 = 0.02, compared to the selected population. This
would imply a systematic error of ∆m = 10−3 in the average m
of the selected population. We will delve deeper into these vari-
ations in m by examining the first data releases of the Euclid
surveys.
4.7. Selection efficiency with missing bands
In this work, we derived colour selections based on griz
and grizYEJEHE photometry. In some cases, however, the full
ground-based griz photometry may be not available. For exam-
ple, the DES Year 3 galaxy shape catalogue was not based on g
band (Gatti et al. 2021), due to issues in the point spread func-
tion estimation (Jarvis et al. 2021). Thus, we investigated the
efficiency of griz and grizYEJEHE selections in the case of a
A139, page 11 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
0.2 0.4 0.6 0.8
zl
40
60
80
(z
l) 
[%
]
This work, griz
Medezinski+10
Medezinski+10, red only
Oguri+12
Oguri+12, restricted
Medezinski+18
Medezinski+18, red only
0.2 0.4 0.6 0.8
zl
87.5
90.0
92.5
95.0
97.5
(z
l) 
[%
]
0.2 0.4 0.6 0.8
zl
0
10
20
30
40
50
(z
l) 
[%
]
0.2 0.4 0.6 0.8
zl
5.0
7.5
10.0
12.5
15.0
n b
(z
l) 
[a
rc
m
in
2 ]
Fig. 11.Comparison of the griz selection described in Sect. 4.1 with literature colour selections. From left to right: completeness, purity, foreground
failure rate, and background density, derived from the B20 catalogue. The blue solid lines represent the griz selection derived in this work. The
red curves refer to the Medezinski et al. (2010) selection, where the solid lines are given by Eqs. (11)–(12), while the dashed lines are given by
Eq. (11). The green curves represent the results from the Oguri et al. (2012) selection, where the solid lines are given by Eqs. (13)–(16), while
the dashed lines are given by Eqs. (13)–(15). The grey curves refer to the Medezinski et al. (2018) selection, where the solid lines are given by
Eqs. (17)–(18), while the dashed lines are given by Eq. (17).
missing band, based on the B20 calibration sample described
in Sect. 2. In performing this test, we excluded the colour condi-
tions in Tables A.1 and A.2 containing the chosen missing bands.
In Fig. 10 we show that, in the case of ground-only observa-
tions, the absence of the r band implies the largest completeness
decrease, providing C(zl) < 60%. In addition, the zl range is
substantially reduced, corresponding to zl ∈ [0.2, 0.6]. Also the
absence of i and z bands implies a reduction of the maximum
zl for the ground-based selection, corresponding to zl = 0.7 and
zl = 0.6, respectively, and a completeness decrease of up to 10
and 20 percentage points, respectively. On average, a 20 percent-
age points drop in completeness is found in absence of g band
photometry. Nevertheless, in the latter case the zl range is not
reduced. We remark that the considered samples differ from case
to case, as they contain only galaxies with photometry available
in the required bands.
In Fig. 10 we show the effect of missing photometric bands
on the combination of ground-based and Euclid observations. In
this case, the lack of r band does not imply changes in C(zl) for
zl > 1. In the absence of i band, C(zl) significantly decreases for
zl & 0.7, being below 30%, while the zl range is not reduced. A
zl range reduction is obtained in the case of missing z or g bands,
as we derived zl ∈ [0.2, 1.1] and zl ∈ [0.2, 1.3], respectively.
On average, in the case of the combination of ground-based and
Euclid observations, the largest completeness decrease is caused
by the lack of the g band.
In this section, we defined colour selections with missing g,
r, i, or z band, as subsets of the colour conditions defining the
griz and grizYEJEHE selections. In order to assess the difference
between the selections defined by such subsets and those that
would be derived from the colour selection calibration described
in Sect. 3, we compute Cn ftot for each case. In Fig. 10, we show
the difference between Cn ftot and C, the latter being derived by
subsets of the colour conditions defining griz and grizYEJEHE
selections. In the ground-only case, the lack of r band provides
the largest C underestimation, as Cn ftot − C ∼15 percentage points
for zl ∈ [0.3, 0.5]. Nevertheless, in case of other missing bands,
the average Cn ftot − C is close to 0. The same holds for the com-
bination of ground-based and Euclid photometry. We conclude
that griz and grizYEJEHE selections provide robust results in
the case of a missing band, except for ground-only observations
without the r band, for which a dedicated calibration might be
needed.
5. Comparison with literature ground-based
selections
Based on the B20 sample considered in Sect. 4.1, we compared
our griz colour selection to those derived by Medezinski et al.
(2010, 2018) and Oguri et al. (2012), which are also imple-
mented in COMB-CL. As detailed below, for each of these selec-
tions, we considered two versions. One includes all the colour
conditions provided by the corresponding authors, while the
other comprises only a subsample of such conditions, providing
lower foreground contamination. COMB-CL includes both ver-
sions of each colour selection.
Medezinski et al. (2010) derived colour selections for three
massive clusters, identified through deep Subaru imaging, by
maximising their weak-lensing signal. COMB-CL provides the
selection calibrated for the A1703 cluster at redshift zl ∼ 0.26,
as this is the one based on gri photometry. This selection is
expressed as follows,[
(g − r) < 2.17 (r − i) − 0.37 ∧
(g − r) < −0.6 (r − i) + 1.85 ∧ (r − i) > 0.3 ] ∨ (11)[
(g − r) < −0.4 (r − i) + 0.47 ∧ (r − i) < 0.3 ] ∨
(r − i) < −0.06, (12)
where ∧ is the logical “and” operator. Notably, Eq. (11) defines
the red background sample, namely the galaxies redder than
cluster galaxies, while Eq. (12) defines the blue background
sample. In Fig. 11, we can see that this selection provides
C(zl) = 60%, which is 20 percentage points lower than that
provided by the griz selection calibrated in this work. In addi-
tion, from the Medezinski et al. (2010) selection we derived
P(zl) > 96%, which is slightly lower than that obtained from
the griz selection discussed in Sect. 4.1. Foreground contamina-
tion can be attenuated by considering the red sample selection
only, namely Eq. (11), as shown in Fig. 11. In this case, how-
ever, the completeness is lowered by 20 percentage points. In
Fig. 12 (upper panel), we show a comparison between the selec-
tion by Medezinski et al. (2010), namely Eqs. (11) and (12), and
our griz selection in the (r − i) − (g − r) colour-colour space,
by assuming zl = 0.26. Within this colour-colour space, we
obtained C(zl) = 55% and P(zl) = 98% from the griz selec-
tion, while through the Medezinski et al. (2010) selection we
found C(zl) = 62% and P(zl) = 95%. We remark that the full
set of colour conditions defining the griz selection yields 80%
A139, page 12 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
0.5 0.0 0.5 1.0 1.5 2.0
g r
0.5
0.0
0.5
1.0
1.5
r
i
zl = 0.26
This work, griz
= 55%, = 98%
Medezinski+10
= 62%, = 95%
0.5 0.0 0.5 1.0 1.5 2.0
g r
0.5
0.0
0.5
1.0
1.5
r
i
zl = 0.6
This work, griz
= 27%, = 98%
Oguri+12
= 60%, = 94%
Fig. 12. Comparison of colour selections in the (r − i) − (g − r) colour-
colour space. The solid grey contours indicate the 68%, 95%, and 99%
galaxy number density in the calibration sample, described in Sect. 2.
The blue shaded areas represent the regions excluded by applying the
griz selection calibrated in Sect. 4.1. Completeness and purity of the
selections are reported in the legends. For the griz selection, C andP are
computed by considering the colour conditions in Table A.1 which are
defined in the (r−i)−(g−r) space. Top panel: the red hatched area shows
the region excluded through the Medezinski et al. (2010) selection, and
zl = 0.26 is assumed. Bottom panel: the green hatched area shows the
region excluded through the Oguri et al. (2012) selection, and zl = 0.6
is assumed.
completeness for zl = 0.26, and that a calibration based on
gri bands only would yield larger completeness values in the
(r − i) − (g − r) space (see Sect. 4.7). In addition, Fig. 12 shows
that the griz selection extends the selected region defined by
Eq. (11), thus enhancing the red background sample compared
to Medezinski et al. (2010). On the other hand, the griz selec-
tion shows a more conservative definition of the blue background
sample, compared to Eq. (12).
Oguri et al. (2012) calibrated a selection based on gri pho-
tometry from the COSMOS catalogue by Ilbert et al. (2009),
providing reliable results for lenses at redshift zl . 0.7. This
selection is expressed as
(g − r) < 0.3 ∨ (13)
(r − i) > 1.3 ∨ (14)
(r − i) > (g − r) ∨ (15)
(g − r) > 1 ∧ (r − i) < 0.4 (g − r) − 0.5. (16)
The inclusion of Eq. (16) does not provide significant improve-
ment in the completeness, causing a lower selection purity
(Covone et al. 2014). In fact, Fig. 11 shows that the selection
including Eqs. (13)–(16) provides sub-percent improvements in
C(zl), compared to the selection including Eqs. (13)–(15) only,
while P(zl) and F (zl) are up to 1 percentage point lower and
higher, respectively. With respect to the griz selection calibrated
in this work, the Oguri et al. (2012) selection provides a purity
which is up to 5 percentage points lower. This explains the higher
completeness values for zl > 0.6. For lower zl, the Oguri et al.
(2012) selection provides a completeness up to 35 percentage
points lower, which is expected as the selection by Oguri et al.
(2012) was calibrated for clusters at zl ∼ 0.7. Similarly to
the comparison with the Medezinski et al. (2010) selection dis-
cussed above, in Fig. 12 (lower panel) we compare the griz and
Oguri et al. (2012) selections in the (r− i)− (g− r) colour-colour
space, assuming zl = 0.6. We obtained C(zl) = 27% and P(zl) =
98% from the griz selection, while with the Oguri et al. (2012)
selection we found C(zl) = 60% and P(zl) = 94%. With respect
to what we found by comparing griz and Medezinski et al.
(2010) selections at zl = 0.26, the decrease in completeness due
to a purity enhancement is much larger at zl = 0.6. This depends
on the overlap of the foreground and background galaxy distri-
butions in the (r − i) − (g − r) space. In addition, we remark that
the calibration process presented in Sect. 3 excludes redundant
colour conditions. This may partially explain the 27% complete-
ness found in the case of the griz selection.
Medezinski et al. (2018) calibrated a colour selection based
on HSC observations, including griz bands, following an
approach similar to Medezinski et al. (2010). This colour selec-
tion is expressed as[
(g − i) < 2.276 (r − z) − 0.152 + a(zl) ∧
(g − i) < 1
2.276
(r − z) + b(zl)
(
1 +
1
2.2762
)
− 0.152
2.2762
∧
(r − z) > 0.5 ∧
z > 21
]
∨ (17){
(r − z) < −0.0248 z + 1.604 + c(zl) ∨[
(g − i) < 1
2.276
(r − z) + d(zl)
(
1 +
1
2.2762
)
− 0.152
2.2762
∧
(g − i) < 4
]
∨
(r − z) < 0.5 ∨
z > 22
}
, (18)
where
a(zl) =
{−0.7 if zl < 0.4
−0.8 if zl ≥ 0.4, (19)
A139, page 13 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
0 1 2 3
zg
0
1000
2000
3000
4000
5000
6000
7000
N
(z g
)
zl = 0.26
This work, griz
Medezinski+10
0 1 2 3
zg
zl = 0.6
Oguri+12
0 1 2 3
zg
zl = 0.8
Medezinski+18
Fig. 13. From left to right: number of selected galaxies as a function of zg assuming zl = 0.26, zl = 0.6, and zl = 0.8. The zl values are represented
by vertical black dashed lines. The blue histograms represent the griz selection calibrated in this work. The galaxy redshift distributions derived
with the Medezinski et al. (2010, 2018) and Oguri et al. (2012), selections are represented by red, green, and grey hatched histograms, respectively.
b(zl) =
{
4.0 if zl < 0.4
1.7 if zl ≥ 0.4, (20)
c(zl) =
{−0.8 if zl < 0.4
−0.9 if zl ≥ 0.4, (21)
d(zl) =
{
0.5 if zl < 0.4
0.3 if zl ≥ 0.4. (22)
Similarly to Medezinski et al. (2010), Eqs. (17) and (18) define
the red and blue background samples, respectively. This selec-
tion provides a much larger amount of contamination compared
to that derived in this work, reachingP(zl) < 90% for zl > 0.7, as
shown in Fig. 11. However, by considering the red sample selec-
tion only, the purity improves by up to 6 percentage points. In the
latter case, compared to the griz selection detailed in Table A.1,
C(zl) is up to 50 percentage points lower for zl < 0.7. For higher
zl, the lower purity from the Medezinski et al. (2018) selection
allows for higher completeness values.
In Fig. 13 we show the redshift distributions of the galax-
ies selected through the griz, Medezinski et al. (2010, 2018)
and Oguri et al. (2012) selections, assuming different zl val-
ues. At zl = 0.26, the griz selection shows a larger num-
ber of galaxies, of the order of 103, which are correctly iden-
tified as background objects with zg < 0.6, compared to
Medezinski et al. (2010). This results in the larger complete-
ness of the griz selection shown in Fig. 11. At zl = 0.6, where
the griz and Oguri et al. (2012) selections have the same com-
pleteness (see Fig. 11), the griz selection is less complete at
zg > 1.5 and more complete at lower redshifts, compared to
Oguri et al. (2012). At zl = 0.8, where the griz selection is
remarkably purer than that by Medezinski et al. (2018), a notable
incompleteness of the griz selection is evident at any zg, com-
pared to Medezinski et al. (2018). In fact, for the case of the
griz selection, Fig. 13 shows that the number of rejected galax-
ies at high redshift increases with the number of excluded fore-
ground galaxies. This reflects the overlapping of foreground and
background galaxy distributions in the considered colour-colour
spaces.
6. Discussion and conclusions
We developed a method to derive optimal galaxy colour selec-
tions for cluster weak-lensing analyses, given any set of pho-
tometric bands. To this aim, we considered all the available
colour-colour combinations. Based on the galaxy catalogue by
B20, we calibrated selections based on ground-based griz and
Euclid YEJEHE bands, with purity higher than 97%. Specifi-
cally, we showed that the griz selection provides a complete-
ness between 30% and 84%, in the lens limiting redshift range
zl ∈ [0.2, 0.8]. The inclusion of Euclid YEJEHE bands, leading
to a grizYEJEHE selection, improves the completeness by up to
25 percentage points in this zl range, allowing for a galaxy selec-
tion up to zl = 1.5. In addition, for the first time in the literature,
we expressed such selections as a continuous function of zl. In
the following, we summarise the main results obtained from the
tests presented in Sects. 4 and 5.
– The calibrated colour selections, described in Sect. 4.1, are
stable with respect to changes in the sample limiting magni-
tudes and redshift.
– By applying the griz selection to the VMLS catalogue by
Moutard et al. (2016) and to the COSMOS20 catalogue by
Weaver et al. (2022), we derived completeness and purity
estimates that are consistent with those obtained from the
calibration sample by B20. Consequently, the calibrated
selections provide stable results by assuming alternative
photometric aperture definitions, obtained from different
ground-based telescopes.
– The application of griz and grizYEJEHE selections to the
simulated Euclid Flagship galaxy catalogue v2.1.10 (Euclid
Collaboration, in prep.) provided a purity of around 99%,
on average, which is higher than that obtained from the
B20 catalogue. The completeness from the Flagship and B20
samples is compatible in the griz selection case, while the
grizYEJEHE selection provides up to 50 percentage points
higher completeness from Flagship. We verified that this dis-
crepancy does not depend on magnitude limits. In addition,
we found no significant differences in the star forming galaxy
fraction from the Flagship and B20 samples. A calibration of
the grizYEJEHE selection based on the Euclid Deep Survey
will allow for a more thorough investigation of these com-
pleteness differences.
– Based on the Flagship sample, we combined the calibrated
colour selections with photo-z selections based on the p(zg)
shape. We showed that in this case the completeness is up to
95%.
– We found no significant systematic effects on the multiplica-
tive shear bias due to colour selections for stage III surveys.
The first Euclid data releases will provide further insights
into the influence of background selections on this bias.
– The calibrated colour selections provide robust results in
the case of a missing band from ground-based observations,
A139, page 14 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
except for those without the r band, for which a dedicated
calibration might be needed.
– Compared to the ground-based colour selections provided by
Medezinski et al. (2010, 2018) and Oguri et al. (2012), the
griz selection derived in this work is purer at high redshift
and more complete at low redshift.
One potential enhancement to the calibration presented in this
work could entail the inclusion of a magnitude dependence in
the colour cuts. This might mitigate the impact of large pho-
tometric scatter at faint magnitudes (see, e.g. Schrabback et al.
2021). In addition, enhancing the set of photometric bands in
the calibration sample, for example by including the LSST y
band, could remarkably improve the effectiveness of the colour
selections. The performance of colour selections could further
improve through dedicated calibration samples. Ongoing spec-
troscopic programmes are specifically designed to calibrate the
relationship between galaxy colours and redshifts to match the
depth of the Euclid survey (Euclid Collaboration 2022c).
Furthermore, the colour selections presented in this work
could improve the shear calibration in cluster weak-lensing
analyses. The lensing signal from galaxy clusters differs from
that of large scale structure in ways that affect both shear
and photometric measurements. The dense cluster environ-
ment causes increased blending among light sources, due to
both galaxy blends (Simet & Mandelbaum 2015; Everett et al.
2022) and the presence of diffuse intra-cluster light (Gruen et al.
2019; Kluge et al. 2020; Sampaio-Santos et al. 2021). In addi-
tion, cluster lines of sight exhibit characteristically stronger
shear, especially at small scales (McClintock et al. 2019;
Ingoglia et al. 2022). These effects can lead to distinct biases
in shear measurements compared to those obtained from
calibrations primarily designed for cosmic shear analyses.
Through the combination of colour and photo-z selections,
cluster shear calibration and mass bias can be assessed based
on dedicated, multi-band cluster image simulations (see, e.g.
Hernández-Martín et al. 2020).
Acknowledgements. The Euclid Consortium acknowledges the European
Space Agency and a number of agencies and institutes that have supported
the development of Euclid, in particular the Academy of Finland, the Agen-
zia Spaziale Italiana, the Belgian Science Policy, the Canadian Euclid Con-
sortium, the French Centre National d’Etudes Spatiales, the Deutsches Zen-
trum für Luft- und Raumfahrt, the Danish Space Research Institute, the Fun-
dação para a Ciência e a Tecnologia, the Ministerio de Ciencia e Innovación,
the National Aeronautics and Space Administration, the National Astronomi-
cal Observatory of Japan, the Netherlandse Onderzoekschool Voor Astronomie,
the Norwegian Space Agency, the Romanian Space Agency, the State Sec-
retariat 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 website (https://www.euclid-ec.org) We
thank W. Hartley for his valuable advice, which remarkably enhanced the
quality of this work. G.C. and L.M. acknowledge the support from the grant
ASI n.2018-23-HH.0. L.M. and F.M. acknowledge the financial contribution
from the grant PRIN-MUR 2022 20227RNLY3 “The concordance cosmological
model: stress-tests with galaxy clusters” supported by Next Generation EU. L.B.
acknowledges financial support from PRIN-MIUR 2017–20173ML3WW_001.
M.S. acknowledges financial contributions from contract ASI-INAF n.2017-
14-H.0, contract INAF mainstream project 1.05.01.86.10, and INAF Theory
Grant 2023: Gravitational lensing detection of matter distribution at galaxy
cluster boundaries and beyond (1.05.23.06.17). G.C. thanks the support from
INAF Theory Grant 2022: Illuminating Dark Matter using Weak Lensing by
Cluster Satellites, PI: Carlo Giocoli. This work has made use of CosmoHub
(Carretero et al. 2017; Tallada et al. 2020). CosmoHub has been developed by
the Port d’Informació Científica (PIC), maintained through a collaboration of
the Institut de Física d’Altes Energies (IFAE) and the Centro de Investigaciones
Energéticas, Medioambientales y Tecnológicas (CIEMAT) and the Institute of
Space Sciences (CSIC & IEEC), and was partially funded by the “Plan Estatal
de Investigación Científica y Técnica y de Innovación” program of the Spanish
government.
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1 Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma
Mater Studiorum Università di Bologna, Via Piero Gobetti 93/2,
40129 Bologna, Italy
e-mail: giorgio.lesci2@unibo.it
2 INAF-Osservatorio di Astrofisica e Scienza dello Spazio di
Bologna, Via Piero Gobetti 93/3, 40129 Bologna, Italy
3 INFN-Sezione di Bologna, Viale Berti Pichat 6/2, 40127 Bologna,
Italy
4 INAF-Osservatorio Astronomico di Padova, Via dell’Osservatorio
5, 35122 Padova, Italy
5 Dipartimento di Fisica e Astronomia “G. Galilei”, Università di
Padova, Via Marzolo 8, 35131 Padova, Italy
6 Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM,
91191 Gif-sur-Yvette, France
7 Department of Physics “E. Pancini”, University Federico II, Via
Cinthia 6, 80126 Napoli, Italy
8 INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello
16, 80131 Napoli, Italy
9 INFN Section of Naples, Via Cinthia 6, 80126 Napoli, Italy
10 Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irne-
rio 46, 40126 Bologna, Italy
11 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS,
Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304
Nice Cedex 4, France
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34143 Trieste, Italy
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2, 34151 Trieste, Italy
14 Université Paris-Saclay, CNRS, Institut d’Astrophysique Spatiale,
91405 Orsay, France
15 Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth PO1 3FX, UK
16 INAF-Osservatorio Astronomico di Brera, Via Brera 28, 20122
Milano, Italy
17 Dipartimento di Fisica e Astronomia, Università di Bologna, Via
Gobetti 93/2, 40129 Bologna, Italy
18 Max Planck Institute for Extraterrestrial Physics, Giessenbachstr.
1, 85748 Garching, Germany
19 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-
Maximilians-Universität München, Scheinerstrasse 1, 81679
München, Germany
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10025 Pino Torinese, TO, Italy
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16146 Genova, Italy
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Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal
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Giuria 1, 10125 Torino, Italy
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30 Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de
Can Magrans, s/n, 08193 Barcelona, Spain
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Capitá 2-4, 08034 Barcelona, Spain
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Monteporzio Catone, Italy
33 Dipartimento di Fisica e Astronomia “Augusto Righi” – Alma
Mater Studiorum Università di Bologna, Viale Berti Pichat 6/2,
40127 Bologna, Italy
34 Institute for Astronomy, University of Edinburgh, Royal Observa-
tory, Blackford Hill, Edinburgh EH9 3HJ, UK
35 Jodrell Bank Centre for Astrophysics, Department of Physics and
Astronomy, University of Manchester, Oxford Road, Manchester
M13 9PL, UK
36 European Space Agency/ESRIN, Largo Galileo Galilei 1, 00044
Frascati, Roma, Italy
37 ESAC/ESA, Camino Bajo del Castillo, s/n, Urb. Villafranca del
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38 University of Lyon, Univ. Claude Bernard Lyon 1, CNRS/IN2P3,
IP2I Lyon, UMR 5822, 69622 Villeurbanne, France
39 Institute of Physics, Laboratory of Astrophysics, Ecole Polytech-
nique Fédérale de Lausanne (EPFL), Observatoire de Sauverny,
1290 Versoix, Switzerland
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41 Departamento de Física, Faculdade de Ciências, Universidade de
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45 Department of Physics, Oxford University, Keble Road, Oxford
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47 Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB, Car-
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48 School of Physics, HH Wills Physics Laboratory, University of
Bristol, Tyndall Avenue, Bristol BS8 1TL, UK
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50 Institute of Theoretical Astrophysics, University of Oslo, PO Box
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51 Department of Physics, Lancaster University, Lancaster LA1 4YB,
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52 von Hoerner & Sulger GmbH, SchloßPlatz 8, 68723 Schwetzingen,
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53 Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyn-
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54 Cosmic Dawn Center (DAWN), Copenhagen, Denmark
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87 Department of Mathematics and Physics E. De Giorgi, University
of Salento, Via per Arnesano, CP-I93, 73100 Lecce, Italy
88 INAF-Sezione di Lecce, c/o Dipartimento Matematica e Fisica, Via
per Arnesano, 73100 Lecce, Italy
89 INFN, Sezione di Lecce, Via per Arnesano, CP-193, 73100 Lecce,
Italy
90 Institut für Theoretische Physik, University of Heidelberg,
Philosophenweg 16, 69120 Heidelberg, Germany
91 Université St Joseph; Faculty of Sciences, Beirut, Lebanon
92 Junia, EPA department, 41 Bd Vauban, 59800 Lille, France
93 SISSA, International School for Advanced Studies, Via Bonomea
265, 34136 Trieste, TS, Italy
94 INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, TS, Italy
95 ICSC – Centro Nazionale di Ricerca in High Performance Comput-
ing, Big Data e Quantum Computing, Via Magnanelli 2, Bologna,
Italy
96 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco,
28049 Madrid, Spain
97 CERCA/ISO, Department of Physics, Case Western Reserve Uni-
versity, 10900 Euclid Avenue, Cleveland, OH 44106, USA
98 Laboratoire Univers et Théorie, Observatoire de Paris, Université
PSL, Université Paris Cité, CNRS, 92190 Meudon, France
99 Dipartimento di Fisica e Scienze della Terra, Università degli Studi
di Ferrara, Via Giuseppe Saragat 1, 44122 Ferrara, Italy
100 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via
Giuseppe Saragat 1, 44122 Ferrara, Italy
101 Dipartimento di Fisica – Sezione di Astronomia, Università di Tri-
este, Via Tiepolo 11, 34131 Trieste, Italy
102 NASA Ames Research Center, Moffett Field, CA 94035, USA
103 Kavli Institute for Particle Astrophysics & Cosmology (KIPAC),
Stanford University, Stanford, CA 94305, USA
104 Bay Area Environmental Research Institute, Moffett Field, CA
94035, USA
105 Minnesota Institute for Astrophysics, University of Minnesota, 116
Church St SE, Minneapolis, MN 55455, USA
106 INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129
Bologna, Italy
107 Institute Lorentz, Leiden University, PO Box 9506, Leiden 2300
RA, The Netherlands
108 Institute for Astronomy, University of Hawaii, 2680 Woodlawn
Drive, Honolulu, HI 96822, USA
109 Department of Physics & Astronomy, University of California
Irvine, Irvine, CA 92697, USA
110 Department of Astronomy & Physics and Institute for Compu-
tational Astrophysics, Saint Mary’s University, 923 Robie Street,
Halifax, Nova Scotia B3H 3C3, Canada
111 Departamento Física Aplicada, Universidad Politécnica de Carta-
gena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain
112 Université Paris Cité, CNRS, Astroparticule et Cosmologie, 75013
Paris, France
113 Department of Computer Science, Aalto University, PO Box
15400, Espoo 00 076, Finland
114 NRC Herzberg, 5071 West Saanich Rd, Victoria, BC V9E 2E7,
Canada
115 Ruhr University Bochum, Faculty of Physics and Astronomy,
Astronomical Institute (AIRUB), German Centre for Cosmological
Lensing (GCCL), 44780 Bochum, Germany
116 Instituto de Astrofísica de Canarias (IAC), Departamento de
Astrofísica, Universidad de La Laguna (ULL), 38200 La Laguna,
Tenerife, Spain
117 Université PSL, Observatoire de Paris, Sorbonne Université,
CNRS, LERMA, 75014 Paris, France
118 Université Paris-Cité, 5 Rue Thomas Mann, 75013 Paris, France
A139, page 17 of 19
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119 Univ. Grenoble Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 53,
Avenue des Martyrs, 38000 Grenoble, France
120 Department of Physics and Astronomy, University of Turku,
Vesilinnantie 5, 20014 Turku, Finland
121 Serco for European Space Agency (ESA), Camino bajo del
Castillo, s/n, Urbanizacion Villafranca del Castillo, Villanueva de
la Cañada 28692, Madrid, Spain
122 Centre for Astrophysics & Supercomputing, Swinburne University
of Technology, Victoria 3122, Australia
123 ARC Centre of Excellence for Dark Matter Particle Physics, Mel-
bourne, Australia
124 Department of Physics and Helsinki Institute of Physics, University
of Helsinki, Gustaf Hällströmin katu 2, 00014 Helsinki, Finland
125 Oskar Klein Centre for Cosmoparticle Physics, Department of
Physics, Stockholm University, Stockholm 106 91, Sweden
126 Astrophysics Group, Blackett Laboratory, Imperial College Lon-
don, London SW7 2AZ, UK
127 Centre de Calcul de l’IN2P3/CNRS, 21 Avenue Pierre de Cou-
bertin, 69627 Villeurbanne Cedex, France
128 Dipartimento di Fisica, Sapienza Università di Roma, Piazzale
Aldo Moro 2, 00185 Roma, Italy
129 INFN-Sezione di Roma, Piazzale Aldo Moro 2 – c/o Dipartimento
di Fisica, Edificio G. Marconi, 00185 Roma, Italy
130 Centro de Astrofísica da Universidade do Porto, Rua das Estrelas,
4150-762 Porto, Portugal
131 Zentrum für Astronomie, Universität Heidelberg, Philosophenweg
12, 69120 Heidelberg, Germany
132 Dipartimento di Fisica, Università di Roma Tor Vergata, Via della
Ricerca Scientifica 1, Roma, Italy
133 INFN, Sezione di Roma 2, Via della Ricerca Scientifica 1, Roma,
Italy
134 Institute for Computational Science, University of Zurich, Win-
terthurerstrasse 190, 8057 Zurich, Switzerland
135 Mullard Space Science Laboratory, University College London,
Holmbury St Mary, Dorking, Surrey RH5 6NT, UK
136 Department of Physics and Astronomy, University of California,
Davis, CA 95616, USA
137 Department of Astrophysical Sciences, Peyton Hall, Princeton Uni-
versity, Princeton, NJ 08544, USA
138 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200
Copenhagen, Denmark
A139, page 18 of 19
Euclid Collaboration: A&A, 684, A139 (2024)
Appendix A: Colour selection parameterisation
In Tables A.1 and A.2 we report the parameterisation of griz and grizYEJEHE selections, respectively, described in Sect. 4.1.
Table A.1. Calibrated colour selection based on griz photometry. The listed colour conditions are combined through the ∨ logical operator.
Colour condition Parameters zl range
(g − r) < s (r − z) + c s = 9.39 z3l − 7.93 z2l + 0.84 zl + 1.69; c = −31.8 z3l + 35.71 z2l − 12.78 zl + 1.04 [0.2, 0.7]
(g − r) < s (i − z) + c s = 7.64 z2l − 7.49 zl + 2.75; c = −2.53 z2l + 1.75 zl − 0.32 [0.2, 0.8]
(r − i) > c c = 1.28 zl + 0.11 [0.2, 0.6]
(g − r) < s (r − i) + c s = 2.07 z2l − 1.75 zl + 1.99; c = −25.25 z3l + 28.86 z2l − 11.72 zl + 1.34 [0.2, 0.6]
(r − i) < s (r − z) + c s = 8.05 z3l − 14.37 z2l + 6.87 zl + 0.73; c = −8.42 z3l + 16.11 z2l − 9.54 zl + 0.68 [0.2, 0.8]
(g − i) > s (i − z) + c s = −9.17 zl + 3.24; c = 7.07 zl − 0.45 [0.2, 0.3]
(g − i) < s (r − z) + c s = −47.54 z4l + 84.36 z3l − 53.03 z2l + 13.64 zl + 0.48; c = 56.05 z4l − 107.76 z3l + 72.88 z2l − 21.09 zl + 1.93 [0.2, 0.8]
(g − z) < s (r − z) + c s = 1.70; c = −21.04 z4l + 16.37 z3l + 1.47 z2l − 3.46 zl + 0.59 [0.2, 0.7]
(g − r) < s (g − z) + c s = −8.94 z3l + 13.28 z2l − 6.55 zl + 1.73; c = −2.53 z2l + 2.24 zl − 0.80 [0.2, 0.7]
(g − r) < c c = 1.46 z2l − 1.43 zl + 0.40 [0.2, 0.5]
(r − i) > s (i − z) + c s = 8.45 z2l − 6.93 zl + 1.67; c = −2.53 z2l + 3.48 zl − 0.35 [0.2, 0.5]
(i − z) > c c = −1.53 z2l + 2.15 zl − 0.01 [0.2, 0.6]
(g − z) > s (r − z) + c s = −0.58 zl − 1.42; c = −10.10 z2l + 15.76 zl − 0.52 [0.2, 0.5]
(g − i) < s (r − i) + c s = 0.24 zl + 1.53; c = −0.91 zl + 0.33 [0.2, 0.6]
Notes. In the first column, the colour conditions are listed. The parameters of such conditions are shown in the second column, while in the last
column the zl ranges are listed. From top to bottom, the ith row corresponds to the ith iteration of the iterative process described in Sect. 3.
Table A.2. Calibrated colour selection based on grizYEJEHE photometry. The listed colour conditions are combined through the ∨ logical operator.
Colour condition Parameters zl range
(g − i) < s (r − HE) + c s = 30.81 z4l − 79.74 z3l + 73.28 z2l − 28.85 zl + 5.07; c = −31.2 z4l + 78.62 z3l − 71.16 z2l + 27.57 zl − 4.41 [0.2, 1.0]
(g − i) < s (z − HE) + c s = 1.41 z2l − 2.42 zl + 2.1; c = −2.42 z3l + 3.67 z2l − 1.57 zl − 0.28 [0.2, 1.3]
(g − r) < s (r − z) + c s = 0.39 zl + 1.33; c = −5.23 z2l + 2.6 zl − 0.83 [0.2, 0.7]
(g − YE) < s (z − HE) + c s = −0.01 zl + 1.65; c = 0.37 z2l − 1.45 zl + 0.08 [0.2, 1.5]
(r − i) > s (JE − HE) + c s = 1.64 zl − 1.33; c = −0.72 z2l + 1.49 zl + 0.41 [0.2, 0.6]
(g − i) < s (z − JE) + c s = −0.05 z4l − 2.44 z3l + 6.19 z2l − 4.35 zl + 2.31; c = −0.29 z2l − 0.35 zl − 0.21 [0.2, 1.3]
(g − i) < s (i − JE) + c s = −2.17 z3l + 5.01 z2l − 3.25 zl + 2.11; c = −2.79 z3l + 4.06 z2l − 2.7 zl − 0.18 [0.2, 1.1]
(g − r) < s (r − HE) + c s = −25.93 z3l + 42.07 z2l − 22.02 zl + 4.2; c = 14.96 z3l − 30.3 z2l + 18.03 zl − 3.52 [0.2, 0.7]
(g − i) > s (i − YE) + c s = −5.31 zl + 1.99; c = 7.07 zl − 0.45 [0.2, 0.3]
(g − r) < s (i − YE) + c s = −26.87 z4l + 63.55 z3l − 51.34 z2l + 17.08 zl − 0.8; c = 18.25 z4l − 46.35 z3l + 38.77 z2l − 13.53 zl + 1.31 [0.2, 1.0]
(g − r) < s (YE − HE) + c s = 2.07 z2l − 2.39 zl + 1.76; c = −10.29 z3l + 11.18 z2l − 4.19 zl + 0.33 [0.2, 0.7]
(g − r) < s (i − HE) + c s = −4.02 z3l + 8.91 z2l − 5.87 zl + 1.75; c = −2.55 z3l + 0.24 z2l + 1.44 zl − 0.61 [0.2, 0.9]
(g − z) < s (z − HE) + c s = 1.43 z4l − 5.51 z3l + 7.29 z2l − 3.79 zl + 2.23; c = 0.32 z2l − 1.55 zl − 0.17 [0.2, 1.4]
(g − YE) < s (i − HE) + c s = −2.25 z4l + 6.28 z3l − 6.63 z2l + 3.09 zl + 1.16; c = −1.86 z3l + 3.75 z2l − 3.43 zl + 0.26 [0.2, 1.1]
(g − z) < s (i − JE) + c s = −6.33 z4l + 15.05 z3l − 11.67 z2l + 3.25 zl + 1.41; c = 8.54 z4l − 23.53 z3l + 22.33 z2l − 9.11 zl + 0.74 [0.2, 1.2]
(r − z) > s (i − z) + c s = −5.60 z2l + 2.31 zl − 0.61; c = 4.69 z2l − 0.67 zl + 0.80 [0.2, 0.7]
(g − JE) < s (r − HE) + c s = 2.76 z2l − 2.91 zl + 2.14; c = −21.51 z3l + 23.09 z2l − 7.14 zl + 0.12 [0.2, 0.7]
(r − i) > s (z − JE) + c s = −1.21 z2l + 0.8 zl + 0.2; c = 1.01 zl + 0.15 [0.2, 0.5]
(g − z) < s (i − HE) + c s = −3.66 z4l + 12.87 z3l − 16.26 z2l + 7.83 zl + 0.36; c = 6.29 z4l − 21.87 z3l + 26.63 z2l − 13.19 zl + 1.43 [0.2, 1.3]
(g − i) < s (r − JE) + c s = −2.48 z3l + 6.11 z2l − 5.46 zl + 2.59; c = 11.18 z4l − 27.18 z3l + 20.47 z2l − 4.48 zl − 0.72 [0.2, 1.0]
(g − z) < s (g − JE) + c s = −2.16 z3l + 5.88 z2l − 5.06 zl + 2.12; c = 2.33 z3l − 6.82 z2l + 6.18 zl − 2.32 [0.2, 1.3]
(i − YE) > s (z − YE) + c s = −0.02 zl + 0.42; c = 2.81 z3l − 5.05 z2l + 3.36 zl + 0.08 [0.2, 0.8]
(g − r) < s (z − JE) + c s = 13.22 z4l + 31.92 z3l − 25.03 z2l + 7.6 zl + 0.42; c = 14.13 z4l − 34.81 z3l + 27.79 z2l − 9.36 zl + 0.83 [0.2, 1.1]
(g − i) < s (i − HE) + c s = −11.75 z4l + 33.51 z3l − 33.07 z2l + 12.73 zl − 0.33; c = 10.74 z4l − 35.01 z3l + 37.81 z2l − 16.31 zl + 1.62 [0.2, 1.1]
(g − JE) < s (i − HE) + c s = −0.03 zl + 1.7; c = 21.04 z4l − 48.34 z3l + 38.51 z2l − 13.25 zl + 1.21 [0.2, 0.9]
(r − i) < s (z − JE) + c s = −0.66 z3l + 1.27 z2l − 0.3 zl + 1.37; c = 0.58 z2l − 1.84 zl − 0.38 [0.2, 1.3]
(i − YE) < s (i − HE) + c s = 2.90 zl + 0.49; c = −6.06 zl + 0.66 [0.2, 0.3]
(r − z) > s (YE − JE) + c s = 0.58 zl + 0.29; c = 1.01 zl + 0.45 [0.2, 0.6]
(g − r) < s (z − HE) + c s = 1.1 z3l + 0.69 z2l − 1.14 zl + 1.09; c = −8.67 z3l + 9.14 z2l − 3.35 zl + 0.24 [0.2, 0.9]
(g − HE) < s (r − HE) + c s = 0.78 z2l − 1.03 zl + 1.89; c = −13.09 z3l + 15.15 z2l − 5.42 zl + 0.22 [0.2, 0.7]
(g − YE) < s (r − JE) + c s = 12.07 z3l − 15.52 z2l + 5.71 zl + 1.08; c = −33.67 z3l + 42.93 z2l − 17.52 zl + 1.61 [0.2, 0.7]
Notes. The table structure is analogous to that of Table A.1.
A139, page 19 of 19