# SPIDERS: An Overview of The Largest Catalogue of Spectroscopically Confirmed
X-ray Galaxy Clusters

C. C. Kirkpatrick,1,2 N. Clerc,3 A. Finoguenov,1 S. Damsted,1 J. Ider
Chitham,4 A. E. Kukkola,1 A. Gueguen,4 K. Furnell,5 E. Rykoff,6,7 J.
Comparat,4 A. Saro,8,9,10 R. Capasso,11 N. Padilla,12 G. Erfanianfar,4 G. A.
Mamon,13 C. Collins,5 A. Merloni,4 J. R. Brownstein,14 and D. P.
Schneider15,16
1Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2,
FI-00014 Helsinki, Finland
2Helsinki Institute of Physics, Gustaf Hällströmin katu 2, FI-00014 Helsinki,
Finland
3IRAP, Université de Toulouse, CNRS, UPS, CNES, Toulouse, France
4Max-Planck-Institut für extraterrestrische Physik, Giessenbachstrae, 85748
Garching, Germany
5Astrophyics Research Institute, Liverpool John Moores University, IC2,
Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK
6Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450,
Stanford University, Stanford, CA 94305, USA
7SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
8Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11,
I-34131 Trieste, Italy
9IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014
Trieste, Italy
10INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143
Trieste, Italy
11The Oskar Klein Centre, Department of Physics, Stockholm University,
Albanova University Center, SE 106 91 Stockholm, Sweden
12Instituto de Astrofísica, Pontificia Universidad Católica de Chile, Av.
Vicuna Mackenna 4860, 782-0436 Macul, Santiago, Chile
13Institut d’Astrophysique de Paris, (UMR 7095: CNRS $\And$ UPMC, Sorbonne
Université), F-75014 Paris, France
14Department of Physics and Astronomy, University of Utah, 115 S. 1400 E.,
Salt Lake City, UT 84112, USA
15Department of Astronomy and Astrophysics, The Pennsylvania State University,
University Park, PA 16802
16Institute for Gravitation and the Cosmos, The Pennsylvania State University,
University Park, PA 16802 E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

SPIDERS is the spectroscopic follow-up effort of the Sloan Digital Sky Survey
IV (SDSS-IV) project for the identification of X-ray selected galaxy clusters.
We present our catalogue of 2740 visually inspected galaxy clusters as a part
of the SDSS Data Release 16 (DR16). Here we detail the target selection, our
methods for validation of the candidate clusters, performance of the survey,
the construction of the final sample, and a full description of what is found
in the catalogue. Of the sample, the median number of members per cluster is
approximately 10, with 818 having 15 or greater. We find that we are capable
of validating over 99% of clusters when 5 redshifts are obtained below $z<0.3$
and when 9 redshifts are obtained above $z>0.3$. We discuss the improvements
of this catalogue’s identification of cluster using 33,340 redshifts, with
$\Delta z_{\rm{phot}}/\Delta z_{\rm{spec}}\sim 100$, over other photometric
and spectroscopic surveys, as well as present an update to previous ($\sigma-
L_{X}$) and ($\sigma-\lambda$) relations. Finally, we present our cosmological
constraints derived using the velocity dispersion function.

###### keywords:

catalogues – galaxies: clusters: general – cosmology: observations

††pubyear: 2019††pagerange: SPIDERS: An Overview of The Largest Catalogue of
Spectroscopically Confirmed X-ray Galaxy Clusters–SPIDERS: An Overview of The
Largest Catalogue of Spectroscopically Confirmed X-ray Galaxy Clusters

##  1 Introduction

Galaxy clusters are the most massive bound objects in the Universe. The more
massive, $10^{14}-10^{15}$ M⊙, the more rare. As a trace of the underlying
cosmic web, they can be used as an independent probe of the expansion rate of
the Universe and growth of structure. Well measured populations can place
constraints on cosmological models based on their predictions (e.g. Böhringer
et al., 2004; Vikhlinin et al., 2009; Mantz et al., 2010; Weinberg et al.,
2013). Clusters are X-ray luminous sources due to the large amount of hot gas
trapped in their gravitational potential wells (Jones & Forman, 1999). The
extended nature of the sources make them distinct from most other extra-
galactic sources, allowing more ease in performing surveys of high volume and
completeness. Additionally, X-ray properties of clusters scale with the mass
of the system (Kaiser, 1986), making these types of surveys even more
important for cosmological studies.

The evolution of the halo mass function is an important cosmological tool.
Clusters are best detected from the X-rays emitted by their hot diffuse gas.
The first major X-ray cluster survey came with the ROSAT All Sky Survey
(Ebeling et al., 2000; Ikebe et al., 2002; Reiprich & Böhringer, 2002;
Schuecker et al., 2003), mapping the brightest galaxy clusters outside of the
Galactic Plane. Subsequent surveys (Rosati et al., 1998; Romer et al., 2000;
Burke et al., 2003; Burenin et al., 2007) and other generations of X-ray
telescopes have contributed to cluster cosmology (Pacaud et al., 2006;
Vikhlinin et al., 2009; Mantz et al., 2010; Finoguenov et al., 2010; Clerc et
al., 2014; Pierre et al., 2016), though not on an all-sky scale. The next
advance in X-ray astronomy is the newly launched extended ROentgen Survey with
an Imaging Telescope Array (eROSITA). Performing a new all-sky survey in the
soft X-ray band, the improved resolution and sensitivity will lead to
$\sim$100 000 galaxy cluster detections ($\sim 3\times 10^{-14}$ erg s-1 cm-2
flux limit at 0.5-2 keV) out to a redshift of $\sim$1 (Merloni et al., 2012;
Borm et al., 2014).

Though X-rays are most efficient for the purpose of cluster detection and mass
proxy, the understanding of their redshift to high accuracy is lacking. This
highlights the importance of optical follow-ups. The Sloan Digital Sky Survey
(SDSS) has a played a critical role in optical confirmation of galaxy
clusters. Measuring the red-sequence of galaxy cluster member galaxies using
multi-waveband optical imaging (Gladders & Yee, 2000; Rykoff et al., 2014)
provides a useful tool for identifying the optical counterparts of clusters
and characterising their redshifts. For the ultimate purpose of constraining
cosmological models, spectroscopic redshifts are needed. The extra precision
on redshift allows for the ability to disentangle projection effects of line-
of-sight alignments, infalling substructure, and other correlated structure
from the underlying dark matter halo concentration. The next important step is
the difficult task of obtaining galaxy spectra, which requires deep imaging
for targeting and longer spectroscopic integration times.

The SPectroscopic IDentification of eROSITA Sources (SPIDERS) cluster program
within SDSS-IV (Blanton et al., 2017) was designed to carry out the necessary
follow-up for the next generation of all-sky X-ray surveys. The survey itself
fully relies on the use of the Baryon Oscillation Spectroscopic Survey (BOSS)
spectrograph (Smee et al., 2013) mounted on the SDSS-2.5m telescope at Apache
Point Observatory (Gunn et al., 2006). Our observing strategy is shared with
the cosmology project eBOSS (extended Baryon OScillation Survey, Dawson et al.
(2016)) in order to maximize the ability to obtain large amounts of
spectroscopic redshifts. SPIDERS targets are few in comparison to eBOSS, but
are given the highest priority when assigning fibers for observing.

An initial study of the first 300 deg2 of the survey were provided by Clerc et
al. (2016). A small sample of galaxy clusters along with the full target list
were published in the form of a Value Added
Catalogue111https://www.sdss.org/dr14/data_access/value-added-
catalogs/?vac_id=the-spiders-clusters-demonstration-sample-catalogue (VAC).
This study demonstrated the survey’s feasibility and usefulness of the data
set. Additional studies have been preformed using an updated VAC
catalogue222https://www.sdss.org/dr14/data_access/value-added-
catalogs/?vac_id=spiders-x-ray-galaxy-cluster-catalogue-for-dr14 as a part of
Data Release 14 (DR14). This has been used for topics including characterising
scaling relations with cluster richness between both mass and X-ray luminosity
(Capasso et al., 2019, 2020) and properties of brightest cluster galaxies
(Furnell et al., 2018; Erfanianfar et al., 2019).

The COnstraining Dark Energy with X-ray (CODEX) clusters survey (Finoguenov et
al., 2020) describes the X-ray source catalogue that forms the basis for the
SPIDERS follow-up program. They present the advanced wavelet filtering
techniques used on X-ray images and provides the modeling of the sample
selection. A companion paper to this work describes in detail the final status
of the survey including: overall characteristics, final targeting strategies,
achieved completeness and spectral quality. Cosmological applications of the
galaxy cluster sample are explored.

This paper describes the entire process for the construction of the final
SPIDERS galaxy cluster catalogue as part of DR16 (Ahumada et al., 2019). It
encompasses all data obtained throughout the entire SDSS-IV program, including
the original pilot sample of Clerc et al. (2016). Every aspect of the process
is detailed from target selection, manual inspection of the spectroscopic
members and analysis of the full data set. We present an update to the first
results from the SPIDERS cluster program.

The outline of this paper is as follows. In Section 2 we describe the SPIDERS
sample and how targets were selected for the survey. In Section 3 we describe
our method of visual inspection for validating the spectroscopic redshift of
the cluster. In Section 4 we present the final results of the validation
effort. Section 5 details the construction of what goes into the final
catalogue. Section 6 is the full description of the value added catalogue with
highlights of its scientific importance. Finally, in Section 7 we present our
initial cosmological constraints based on our velocity dispersion
measurements.

Unless otherwise stated, the cosmological model used in this paper is a flat
$\Lambda$ Cold Dark Matter with $\Omega_{\rm{m}}=0.3$ and $H_{\rm{o}}=70$ km
s-1 Mpc-1. Magnitudes are expressed in their native SDSS (AB) system (Fukugita
et al., 1996).

##  2 The SPIDERS Cluster Sample

Cluster candidates for SPIDERS have been drawn from a subset of CODEX
(Finoguenov et al., 2020). CODEX is an X-ray selected catalogue of clusters
reaching fluxes of 10-13 ergs s-1 cm-2. CODEX uses data from the ROSAT All Sky
survey (RASS), specifically in the $\sim$10,000 square degree BOSS imaging
footprint. A comprehensive description of source detection is presented in
Finoguenov et al. (2020). A total of 10,415 CODEX entries were considered for
follow-up, of which 4,114 made it into the final area surveyed by SPIDERS.

Target galaxies for spectroscopic follow-up are identified using the red-
sequence technique. For each X-ray source detection, redMaPPer version 5.2
(Rykoff et al., 2014) was run at that position on the sky using the SDSS
imaging data of Data Release 8 (DR8). The calibration of the red-sequence is
carried out using existing SDSS spectroscopy from DR9 and only needs a small
set of training clusters for calibration. The full details of this procedure
is presented in Rykoff et al. (2014). Given a new red-sequence model, cluster
finding begins by looking for galaxy members in a 0.7 Mpc radius of the X-ray
centre. The algorithm proceeds to iterate over each guess for the redshift. An
optimized richness estimator (Rozo et al., 2009; Rykoff et al., 2012) uses
colour offset from the red-sequence, $i$-band magnitude, and projected
distance from the cluster centre to determine the probability of a galaxy
being a member, ultimately producing a richness and it’s corresponding
likelihood. Richness is defined as the sum of the probabilities, with
corrections for missing area and photometric depth (see Rykoff et al. (2014)
for full details). If more than 3 members are detected, the highest
probability members are simultaneously fit to the red-sequence model to find
the new cluster photometric redshift. This process iterates until convergence
on redshift ($|z_{i+1}-z_{i}|<0.0002$) is achieved. The candidate cluster with
the highest likelihood at the end of this procedure is selected as the optical
counterpart.

Due to the uncertainty associated with the position of RASS detections, the
entire algorithm is run again allowing for the centre to vary within 3 arcmin.
This new red-sequence provides an alternate cluster centre and the final
estimate of the cluster’s photometric redshift ($z_{\rm{\lambda}}$) and
richness ($\lambda_{\rm{OPT}}$) that are entered into the SPIDERS cluster
sample. We refer to this as the ‘optical detection’.

The reproducible process of associating initial red-sequence candidates to
spectroscopic fibres is referred to as ’targeting’. It involves all sub-
components of the eBOSS programme. It is designed to optimize efficiency
metrics defined for each survey; in the case of SPIDERS, the figure of merit
is spectroscopic completeness. The targeting algorithm uses priority flags
associated to each targets. We use the galaxy membership probability as the
main factor in determining SPIDERS clusters targeting priority. Before
ranking, galaxies that we do not want to be targeted are filtered out. All
galaxies associated with clusters of richness less than 10 are removed from
the target list. All galaxies with a probability less than 5 percent are also
removed. Members already with a spectroscopic redshift from past observations
are matched and removed. For the remaining galaxies, only members with a fiber
i-band magnitude between 17.0 and 21.2 will be considered for targeting to
maximize detection efficiency. These filtering steps led from an initial list
of 158,368 CODEX red-sequence galaxies to a pool of 44,367 galaxies submitted
for follow-up. This defines the full SPIDERS member candidate sample.

The final target ranking is determined, in general, as a function of cluster
richness and member probability. Cluster richness is divided into bins of:
$\lambda_{\rm{OPT}}>40$, $30<\lambda_{\rm{OPT}}<40$,
$20<\lambda_{\rm{OPT}}<30$, $10<\lambda_{\rm{OPT}}<20$. Each cluster’s red-
sequence is ranked from highest to lowest probability. The three top ranked
members of all clusters are assigned the same priority flag of 1, 2, and 3,
regardless of richness. From rank 4 and up, priority is given to galaxies in
higher richness bins. After this ranking, a subsequent filtering occurs due to
physical conditions during observing: sky fiber density is restricted to
$\sim$10/deg2 and fiber collisions at a radius of 62” within an SDSS plate.
Finally, 30,236 galaxies were selected for follow-up within the DR16 sky area.
The full details and visualisation of this scheme are found in Clerc et al.
(2016) and more recent updates reported in Clerc et al. (2020).

##  3 Spectroscopic Validation

Trained astronomers are required to verify the existence of a cluster in order
to disentangle their complex nature (Guzzo et al., 2009; Adami et al., 2011).
An automatic membership assignment algorithm was developed to ease the
validation efforts of such a large sample (Clerc et al., 2016). This was
implemented as the first step before any visual inspection takes place. Our
approach is broad in order to account for varying mass throughout the sample.
Each cluster with red-sequence members associated with a spectroscopic
redshift are considered individually. An initial redshift is computed using
the bi-weight average (see, Beers et al., 1990) of all observed members,
$N_{\rm{zspec,0}}$. All members are rejected with velocities offset by greater
then 5000 km s-1. A $3\sigma$ clipping procedure is run on the remaining
members, $N_{\rm{zspec,1}}$, to converge on a final preliminary redshift and
velocity dispersion. The procedure starts with recomputing the bi-weight
average, $z$, of the remaining members. The proper velocity for each galaxy is
estimated by $v_{\rm{prop}}=c(z_{i}-z)/(1+z)$ (Danese et al., 1980), where
$z_{i}$ is the redshift of the individual galaxy. Velocity dispersion is then
calculated depending on the number of spectroscopic members. For cases with
$N_{\rm{zspec,1}}\geq 15$, velocity dispersion is calculated as the square
root of the bi-weight variance of the member galaxies’ proper velocities.
Otherwise, the gapper method is used for cases with a lower spectroscopic
sampling rate, as it is known to give more robust results (Beers et al.,
1990). The entire procedure is allowed to iterate until convergence or is
stopped after 10 steps.

Figure 1 highlights two cases, one where the automatic membership assignment
works successfully and one where it fails. The upper panel of this figure is
the typical example where no interference is required during visual
inspection. Little time is needed in a case like this when the cluster is well
sampled and no signs of outliers are present. The lower panel shows where the
automatic assignment has trouble when line-of-sight contamination is strong. A
combination of studying the phase space diagrams, 2D projected maps, velocity
histograms, optical images, and X-ray detection information is needed to
understand and recalculate membership. The details of what is available during
visual inspection is described below.

Figure 1: Automatic analysis of redshift (velocities) for two different
clusters. Plotted are the difference between spectroscopic and cluster
photometric redshift versus RedMaPPer membership probability. Points
highlighted in purple were selected as members by the automatic algorithm and
the blue line is the cluster redshift determined by that selection. The dash-
dot and dotted lines are the photometric redshift and it’s associated error.
Upper panel: This example shows a cluster where the automatic assignment has
no problem identifying based solely on the $3\sigma$ clipping procedure. Lower
panel: Another example where automatic membership assignment fails due to
confusion with possible nearby structure. The grey points are galaxies not
selected as members by the automatic algorithm. These figures are
representative of diagnostic plots used during inspection. Figure 2: Screening
Web Interface. Various phase-space plots are used to help inspectors make
their decision whether or not to validate a cluster. /bf Red-sequence galaxies
with a spectroscopic redshift are represented in the uppermost panels, as a
function of their redshift offset to the cluster photometric redshift (on the
left) of their position on sky (on the right). Members are colour coded by
negative (blue) or positive (red) proper velocity in the middle-left and
middle-right panel, showing respectively the velocity-distance distribution
and sky distribution of current selected members. The lower left histogram
shows the redshift distribution of all red-sequence galaxies. The lower right
histogram stands for current selected members only. Tools for manually
assigning members for recalculating redshift and velocity dispersion are also
available from the command panels on the left-hand side of the interface.

Figure 2 shows a typical snapshot of the screening web interface used by the
visual inspectors. The six panels within the interface are as follow: the top-
left panel is the difference between spectroscopic redshift and photometric
redshift plotted against the natural log of the redMaPPer membership
probability for all observed red-sequence members. Red-sequence galaxies
selected as cluster members initially by the algorithm (or modified later by
the user) are highlighted in purple, otherwise they would be shown as grey
points. The top-right panel is the 2D projected sky positions of all observed
red-sequence members, the middle-left panel is the velocity offset to the
computed cluster redshift plotted against the projected distance from the
cluster center in Mpc for only the selected cluster members, the middle-right
panel is the 2D projected sky positions of only the selected members with the
physical distance from the cluster center highlighted in 0.5 Mpc intervals,
the bottom-left panel is the distribution of red-sequence member redshifts,
and the bottom-right panel is the velocity offset distribution of only the
selected cluster members.

Catalogue information is available to the inspectors in the upper tabs of the
interface. They contain, most importantly for inspection, the photometric
redshift, the richness estimate, and how many red-sequence members have
spectroscopy. The tabs on the left side allow the inspectors to access more
information about the cluster and utilize tools to aid in the manual selection
of members in cases where the automatic algorithm fails. During validation,
inspectors have access to all the individual spectra, direct links to the
cluster location in legacy sky viewer (Dey et al., 2019), and visual displays
of all SDSS related spectroscopy in the field of view together with all red-
sequence members. Different tools allow the inspector to manually select or
de-select single or groups of members. Redshift and velocity dispersion is
always recalculated using the same routines used by the automatic algorithm.

During validation, no requirements were given to the inspectors in order to
ensure independence between visual inspections. Loose guidelines were
discussed amongst inspectors, mainly concerning available literature on
previous work. Studies of nearby systems (Rines et al., 2013; Munari et al.,
2014) and scaling relations based on richness (Simet et al., 2017; Murata et
al., 2018) where important for understanding the make-up of galaxy clusters.
The former benefit from galaxy phase-space diagrams sampled by factors several
tens of hundreds denser than typical SPIDERS clusters; these clusters were
considered are high signal-to-noise analogs. The latter provide a support for
excluding configurations incompatible with the expected scaling of cluster
observables with total mass (e.g. low-richness clusters should have small
velocity dispersions and conversely).

To further illustrate the validation procedure, we will highlight a few
examples of common scenarios encountered during the process. A minimum of
three members are required to converge on a final redshift solution. These
members should lie close to each other in the velocity-distance diagram.
Explicit formulations of exact threshold criteria was avoided and everything
was left to inspectors as much as possible; such criteria emerged in the later
stage of reconciliation among inspectors (as described below). In Figure 3,
the upper panels shows a case where three members are sufficient to validate a
redshift. In the velocity-distance phase space plot, all three members are at
similar velocities, and are also clustered in 1D distance and in 2D sky
projection. Coinciding with an X-ray detection as well gives us confidence
this is more than a chance pairing. In the lower panels is a case where three
observations is not sufficient to validate a redshift. One galaxy is offset in
velocity from the other two members by more than 6000 km s-1, while these same
two members are separated on the sky by over a megaparsec. More observations
would be needed to determine which of these three, if any, are a part of a
cluster at this location.

Figure 3: Illustration of two clusters with only 3 candidate member galaxies.
A blue-to-red colour gradient is used to highlight the negative-to-positive
velocity offset from the centre. Upper panels: Velocity offset and distance
phase space plot with the 2D sky projecting from the screening web tool. This
example illustrates the case when the minimum number of red-sequence members
are able to validate the cluster redshift. Lower panels: Same plots as above,
but for a case where validation cannot be confirmed, as not even two of
galaxies appear to likely be associated with one another. Solid and dashed
lines are the same as shown in Figure 1.

An example of a cluster with 10 members, the median value after validation, is
illustrated in the upper panel of Figure 4. Universally these are agreed upon
that they are in fact clusters, but on an inspector by inspector basis, there
may not be agreement on the exact membership make up. These differences are
mediated in the reconciliation round of validation. For comparison, the bottom
panel of Figure 4 shows the more rare instance of a well sampled system where
it is easier to find a consensus, since agreement on clusters status and
membership list are reached by all.

The most time consuming aspect of validation belongs to the targets where more
than one “component” is apparent along the line-of-sight. This affects $\sim
8\%$ of the total validated sample. We define a component as a grouping of 3
or more galaxies associated with an X-ray source. There is no limit on how
many component can be classified. Upon the conclusion of the validation
process, inspectors agreed on three main scenarios that were encountered: the
redMaPPer redshift ($z_{\rm{\lambda}}$) corresponds to the main component,
corresponds to a secondary component, or does not correspond to any component.
The first is the most common case we find representing approximately 50% of
all multi-component systems. Multiple components can be apparent in the
foreground or background, but the majority of observed galaxies are consistent
with $z_{\rm{\lambda}}$, suggesting the performance of redMaPPer is mostly not
affected by the presence of smaller systems, though the estimated richness has
been overestimated by a small factor. The rarest case ($\sim 10\%$) is the
second example. A small grouping of galaxies is consistent with
$z_{\rm{\lambda}}$, but a larger grouping is found in the foreground or
background, outside the photometric errors. Small components are defined to
have less than half the number of members compared to the main component. We
find the median red-sequence width before membership cleaning, $\Delta z$, to
be a factor of 2 larger than the errors on $z_{\rm{\lambda}}$ for SPIDERS
clusters. This could be the possible explanation for why this offset of the
main component occurs. The last case describes the rest of the systems, for
which neither of the component’s redshift are consistent with
$z_{\rm{\lambda}}$, but the richness estimate is still accurate, or multiple
components are confusing redMaPPer entirely. Figure 5 presents an example of
two equal components, split in between by $z_{\rm{\lambda}}$. We find that
when two, likely equal mass clusters are in near proximity to each other,
redMaPPer interprets this as one cluster located between the real clusters
with a richness overestimated by at least a factor of two. We discuss in more
detail how we define and report the main component of clusters below in
section 5.

Figure 4: Illustration of clusters with intermediate and high number of
candidate member galaxies. The points are highlighted using the same colour
scheme of Figure 3. Upper panel: Velocity offset and distance phase space plot
for a cluster with the number of members equal to the median. Lower panel: The
same phase space plot for a rare, well sampled system. Solid and dashed lines
are the same as shown in Figure 1.

The final stage of validation is the reconciliation between the decisions from
all inspectors. Individual evaluations are collected in the form of digital
pre-formatted files at the end of an evaluation campaign. A moderator checks
the usability of the submitted results and for each cluster candidate compares
the output of all inspectors. This consists of a two-step process. The first
step involves addressing the agreement between evaluations. It requires that a
majority emerges among inspectors concerning the validation status of the
cluster (either validated or unvalidated) and that redshift values agree
within their 95% uncertainty ranges. An automated algorithm attempts to find
disjunct groups of redshift values close to each other. If a group emerges
gathering a majority of inspections (or if all evaluations belong to one
single group), the redshifts are said to agree with each other. In case of
multiple component splits, the moderator is prompted for an explicit
agreement. All candidates that do not reach an agreement are put back to the
pool of inspections in order to collect more votes. The second step involves
building a conciliated catalogue based on the agreed upon evaluations. If a
majority of inspectors validated a candidate, the corresponding membership
flags are averaged (column SCREEN_NMEMBERS_W) and so are the cluster systemic
redshift, velocity dispersion and uncertainties. A redshift spread is computed
as the standard deviation of the inspectors’ cluster redshift values,
reflecting the amplitude of the agreement between inspectors. It is then added
in quadrature to the cluster redshift uncertainty. In cases of multiple
components, a manual association between components is made and averages are
performed sequentially for each component. All unvalidated evaluations are
discarded. Candidates collecting a majority of unvalidated statuses are
flagged as such.

Figure 5: Redshift versus distance from cluster centre. The solid line is the photometric redshift from redMaPPer and the dashed lines are the associated uncertainties on that estimation. Two components are apparently at a lower and higher redshift compared to the photometric estimate are confusing the red-sequence finder algorithm. They are likely similar in mass with a detection of 9 members in the lower redshift component and 13 members in the higher redshift component. This results in an overestimation of richness by approximately a factor of two. | Inspection |  |  | Validated | Unique |
---|---|---|---|---|---|---
Run | round | Selection | Candidates | components | components | Inspectors
2016-07-04 | Round-1 |  $\mathrm{\lambda}_{\text{OPT}}$ $>$ 30 | 573 | 520 | 478 | 8
2016-07-04 | Round-2 | 20 $<$ $\mathrm{\lambda}_{\text{OPT}}$ $<$ 30 | 344 | 246 | 230 | 8
2016-07-04 | Round-3 | 10 $<$ $\mathrm{\lambda}_{\text{OPT}}$ $<$ 20 | 618 | 396 | 344 | 2
2018-04-27 | Round-1 |  $\mathrm{\lambda}_{\text{OPT}}$ $>$ 30 | 434 | 379 | 365 | 6
2018-04-27 | Round-2 | 20 $<$ $\mathrm{\lambda}_{\text{OPT}}$ $<$ 30 | 216 | 166 | 145 | 4
2018-04-27 | Round-3 | 10 $<$ $\mathrm{\lambda}_{\text{OPT}}$ $<$ 20 | 432 | 281 | 264 | 5
2018-04-27 | Round-4 |  $\mathrm{\lambda}_{\text{OPT}}$ $>$ 10, incomplete (SEQUELS) | 193 | 175 | 159 | 2
2018-12-04 | Round-1 |  $\mathrm{\lambda}_{\text{OPT}}$ $>$ 10, w/ revised redshift | 383 | 299 | 276 | 4
2019-03-22 | Round-1 |  $\mathrm{\lambda}_{\text{OPT}}$ $>$ 10 | 818 | 616 | 571 | 3
2019-03-22 | Round-2 |  $\mathrm{\lambda}_{\text{OPT}}$ $>$ 10, remaining partially observed | 140 | 101 | 87 | 3
2019-03-22 | Round-3 | re-inspection required | 14 | 14 | 14 | 2
Table 1: Statistics for each round during the almost three year validation
effort. Subsequent runs are for the inspection of completed observations of
candidates not covered in earlier runs, unless otherwise noted in the text.

##  4 Validation results

Over the course of the SDSS-IV survey, periodic validation rounds took place
as new data became available. Upon assembly of a new “run”, a call for
inspectors was announced to the entire collaboration. A minimum number of
groups consisting of 50 targets each were assigned to insure that at least 2
votes are recorded in every case, with no restriction on the maximum number of
groups an inspector wants to validate. Below is the summary for each sample
making up the final catalogue. The name of the run signifies the cut off date
for which data was included. General details for each run are found in Table
1.

###  4.1 Run 2016-07-04

Just for the purpose of breaking up the work load into more equal parts, each
round was defined by a richness cut. Round one consisted of all observed
candidates, marked as complete, with optical richness
($\mathrm{\lambda}_{\text{OPT}}$) greater than 30. Candidates marked as
complete are systems with as many observations as their allocated amount of
spectroscopic fibres. Except for marginal changes due to reprocessing of the
data, their observational status is therefore definitive in the context of the
project. A total of 573 candidates were inspected in this round. This round
makes up the entirety of the DR14 catalogue release (Abolfathi et al., 2018).
For the second round, only completed candidates with the cut 20 $<$
$\mathrm{\lambda}_{\text{OPT}}$ $<$ 30 were considered. This amounted to 344
new inspections. For the final round, the remaining 618 candidates with 10 $<$
$\mathrm{\lambda}_{\text{OPT}}$ $<$ 20 were inspected. The two available
inspectors for this round disagreed on a total of 21 candidates. Due to there
being no way to break the disagreements, two additional inspectors were added
during the reconciliation round to find a consensus for these candidates. A
final total of 1052 validated components were found. The full breakdown for
each round is presented in Table 1.

###  4.2 Run 2018-04-27

This run consists of the next 22 months of observations, all were new to the
inspectors. The first three rounds were broken up into the same richness bins
as the previous run. Round 1 consisted of inspections of 434 candidates, round
2 consisted of inspections of 216 candidates, and round 3 consisted of
inspections of 432 candidates. The 4th round was initiated as a special round
to catch up on missing candidates. It was determined that candidates labeled
incomplete in the original SEQUELS pilot area (Clerc et al., 2016) no longer
had the potential to be re-observed due to changing survey strategy and
targeting. Every candidate in this area was included for inspection, totaling
193. Only two inspectors were available for this round again. A third
inspector voted to break the disagreement for the 4 candidates with no
consensus. A final total of 1001 validated components were found. Table 1
contains more details for each round within this run.

###  4.3 Run 2018-12-04

A third run was initiated in order to reassess the effect of major changes to
the spectra processing pipeline (Ahumada et al., 2019). Included in this run
are all completed candidates observed over the previous 7 months and all
candidates from the previous two runs where 1 or more galaxies show a
different redshift, redshift error, or warning flag. Round 1 consisted of a
total of 383 candidates. Among them, 200 were new inspections and 183 were
reanalyses of previous systems. Among clusters validated in the previous runs,
52 were validated again and one was unvalidated thanks to one galaxy redshift
change in its red-sequence and a more careful examination. Among the clusters
unvalidated in previous run, 88 became validated. A final total of 299
validated components were found. See Table 1 for more details.

###  4.4 Run 2019-03-22

The final run took place after the completion of the SPIDERS survey. Round 1
consisted of the remaining 818 candidates with completed observations. Round 2
consisted of the remaining 140 candidates that had partially completed
observations. About 720 were new inspections and little less than 250 were
reanalyses of systems inspected in earlier runs. Among the previously
validated systems, 98 were validated again, none were unvalidated. Among the
previously unvalidated systems, 72 became validated. A final total of 717
validated components were found. An additional third round was needed during
the construction of the final catalogue for re-inspecting multi-component
clusters that did not fall into any of our classification categories. The
details of this can be found in section 5. Additional details of the rounds
are presented in Table 1.

###  4.5 Inspection statistics

The mean number of inspectors per cluster candidate in this sample was
approximately 3. Over the full sample of 2740 validated clusters, there are
only 71 instances where a full consensus was not reached between inspectors.
For spectroscopic redshift determination, there are 622 instances where there
was not exact agreement. The mean spread for these instances is 0.00049 (147
km s-1), with the maximum disagreement being 0.0055 (1 650 km s-1).

Comparing to the automatic membership assignment, there are 955 instances
where the redshift was changed after inspection. Within this sample, there are
228 cases where inspectors additionally broke the candidates up into 2, 3, or
4 separate components.

The chances for a candidate to be validated were highly dependent on redshift,
richness, and number of spectra obtained. Figure 6 shows the optical richness
versus photometric redshift for the all validated and unvalidated clusters
from the entire program. There is very little dependence below a redshift of
0.3. Out to a redshift of 0.4 though, most clusters below richness 20 are
unvalidated. This trend increases until a redshift of 0.6 where almost all
clusters are unvalidated except the most extreme objects
($\mathrm{\lambda}_{\text{OPT}}$ $>$ 100). This follows approximately
$15\times\zeta(z_{\rm{\lambda}})$, where $\zeta=e^{5.5(z-0.35)}-0.12$ (at
$z>0.37$) is the redshift-dependent scaling adopted by redMaPPer to account
for galaxies brighter than the $0.2L^{*}$ limit, but are fainter than can be
detected in SDSS (Rykoff et al., 2014, their Fig. 19). This empirical
delineation is materialized with the red dashed line in Fig. 6; it reflects
that about fifteen $L>0.2L^{*}$ galaxies must appear in SDSS images in order
for a cluster to be validated. The red line has been obtained by setting a
requirement on targeting at least 8 of the cluster member candidates, as with
an average success rate of measuring the redshift in accordance with cluster
redshift of 0.7, which leads to at least 5 confirmed member galaxies.

Figure 7 illustrates the targeting efficiency. We present the ratio of
observed galaxies to all red-sequence member candidates versus the cluster
photometric redshift, where a minimum of 9 spectra were available per cluster.
Below a redshift of 0.2, we are able to reach a maximum 1:1 ratio. This
maximum ratio drops off to 1:2 out to redshift 0.3, and further to 1:3 out to
redshift 0.4. Beyond that, no trend is discernible. The red points represent
all clusters that were unvalidated with a minimum of 9 spectra obtained. We
achieved a $>$99% validation rate across the entire sample at this level.
Considering only clusters at $z<0.3$, the validation rate improves to 99% when
at least 5 spectra are obtained.

Figure 6: The validation results of all inspection rounds are plotted showing
richness versus photometric redshift. Validated clusters are shown in black,
unvalidated candidates are shown in yellow. Only the highest richness (mass)
clusters are validated with increasing redshift. The dashed red line roughly
indicates the transition between mostly validated and mostly unvalidated,
where $\zeta(z_{\rm{\lambda}})$ is the redshift dependent photometric depth
correction factor for richness estimates. Figure 7: The ratio of observed
members (i.e., spectroscopic redshift was obtained) to total red-sequence
members for each cluster candidate (where $N_{\rm{observed}}\geq 9$) is
plotted against the photometric redshift. Validated points are shown in grey
and unvalidated points are shown in red. Over this entire sub-sample, less
than 1% of candidates are unvalidated. At a redshift of 0.4 (black dashed
line), the maximum observing efficiency appears to flatten out at an
approximate 1:3 ratio.

##  5 Catalogue Construction

Basic X-ray properties for every cluster that has been validated have been
calculated. We report the fluxes corrected for Galactic absorption. In
performing this calculation, we assume a constant spectral shape of the
source, and perform a correction for this assumption as a part of the
K-correction, which accounts for the effect of the source spectral shape
(defined by temperature of the emission and the redshift). We use the XXL
$L-T$ (Giles et al., 2016) relation and $M-T$ (Lieu et al., 2016) relation and
a flat $\Omega_{m}=0.3$ $\Lambda$CDM cosmology in the calculations of $L_{X}$,
as those agree with the CODEX weak lensing work of Kettula et al. (2015).

The fluxes are measured within the apertures defined to contain a significant
flux measurement, with respect to the background, but constrained to be within
$12^{\prime}$. As these do not necessarily cover the whole cluster, we
extrapolate the measurement using the expected surface brightness profiles of
clusters. The procedure is described in Finoguenov et al. (2007) and has to be
applied iteratively, allowing the cluster size to change with iterations,
based on the obtained $L_{X}$. For nearby clusters with large corrections
factors (greater than 2), we re-extracted fluxes using much larger apertures,
extending to $48^{\prime}$, confirming the results.

Further filtering must be carried out before merging the separate validations
runs into the final catalogue. The first step is to classify all clusters that
have been validated with multiple components in the line of sight. Inspectors
are tasked with ordering the components from most to least significant. The
most significant component is considered the main gravitating halo along the
line of sight, and this is defined as the component with the most
spectroscopic members associated with it. For instances where membership is
equal between components, the component that is located at a lower redshift is
considered the most significant. Inspectors may also choose to “re-merge”
components if they believe the cluster is experiencing a near 1:1 merger, as
there is no clear indication of which part is the most significant. This
results mainly when components are within 4000 km/s of each other. Round 3 of
run 2019-03-22 was the re-inspection of clusters deemed to be likely
experiencing this type of merger. For these cases, the redshift of gravitating
halo is accurate, but the velocity dispersion reported is likely unreliable as
a mass proxy. Once the classification is complete, agreement is required by
all inspectors before a cluster is entered into the final catalogue. Where
disagreements exist still, further discussion takes place until a unanimous
decision is reached. Only the main component is entered into the catalogue. A
flag in the final catalogue, $\rm{NCOMPONENT}$, indicates the number of
visually identified components, or $-1$ for clusters identified as mergers.

The next step is to concatenate the separate catalogues while simultaneously
removing duplicate clusters. Due to some clusters being inspected multiple
times throughout the validation process, we took care to choose the newest
evaluation as the one to enter our catalogue. Following this, duplicates with
unique IDs are also filtered out. As the CODEX catalogue is compiled from two
separate X-ray detection and RedMaPPer runs, a few rare instances occur where
candidates are entered twice into the target list under different IDs. These
are determined by matching clusters based on RA, Dec, and spectroscopic
redshift. The few cases are checked visually as to whether they are sharing
the same member galaxies. These are removed, keeping the ID from the second
RedMapper run (IDs starting with 2).

Next, we attempted to remove all clusters still remaining that were validated
under unique IDs, but are still likely to be the same structure being counted
twice. Clusters are matched with any other cluster in the sample that is
within 5 times R200 and $0.03(1+z)$ in redshift ($<9000$ km s-1) of itself. In
Figure 8, we plot the absolute velocity offset against the projected sky
separation for each pair of clusters. It is likely that some these clusters
are in some phase of merging. Most pairs are found to be separated within 10
to 100 arcminutes, though a small number are within a few arcminutes of each
other. With centres being based on RASS detections, it is not possible to
distinguish the difference between X-ray sources with less than 2 arc minute
separation. Upon visually inspecting pairs within 5 arcminutes of each other,
we discover most pairs have inter-mixed galaxy members. In these cases, we
enter the pair into the final catalogue as a single object.

For the final step, we make one last check for mistakenly unvalidated
candidates. Candidates that appear to be outliers from the rest of the sample
(see the red points in Figure 7) are visually inspected one last time. We
collaboratively agree that the data available is not sufficient enough to
determine a cluster redshift. This finalises the catalogue.

Figure 8: Absolute velocity offset versus projected sky separation between
cluster pairs before cleaning the sample. The pairs to the left of the
vertical dashed line were visually inspected individually for common galaxy
members. Some clusters are represented multiple times due to proximity of more
than one other cluster.

##  6 The Catalogue

The full SPIDERS cluster catalogue of optical properties, redshifts, dynamical
properties, and X-ray properties can be found
online333https://www.sdss.org/dr16/data_access/value-added-
catalogs/?vac_id=spiders-x-ray-galaxy-cluster-catalogue-for-dr16. In total,
our catalogue contains 2740 visually inspected galaxy clusters. Contained
within these clusters are 33,340 individual galaxy members. Along with this
cluster catalogue, we have released the database of redshifts used for
identification (Ahumada et al., 2019), the targeting scheme (Clerc et al.,
2016), and the full target
catalogue444https://www.sdss.org/dr16/data_access/value-added-
catalogs/?vac_id=spiders-target-selection-catalogues.

Column | Unit | Description | Example
---|---|---|---
CLUS_ID (1) |  | SPIDERS/CODEX identification number | $1\\_6939$
NCOMPONENT (2) |  | Number of validated components, or merger flag | 1
CODEX (3) |  | The CODEX cluster candidate unique identifier | 26240
RA (4) | deg | CODEX X-ray detection right ascension (J2000) | 233.106
DEC (5) | deg | CODEX X-ray detection declination (J2000) | 39.003
RA_OPT (6) | deg | CODEX optical detection right ascension (J2000) | 233.083
DEC_OPT (7) | deg | CODEX optical detection declination (J2000) | 39.024
LAMBDA_CHISQ_OPT (8) |  | Richness $\mathrm{\lambda}_{\text{OPT}}$ of the CODEX optical detection | 38.8
Z_LAMBDA (9) |  | Photometric redshift (zλ) of the CODEX optical detection | 0.27
Z_LAMBDA_ERR (10) |  | Uncertainty on zλ | 0.01
NMEM (11) |  | Number of objects in the CODEX red-sequence | 50
NOKZ (12) |  | Number of red-sequence members with a spectroscopic redshift | 14
SCREEN_CLUZSPEC (13) |  | Galaxy cluster redshift, assigned after visual inspection | 0.2863
SCREEN_CLUZSPEC_ERR (14) |  | Bootstrap uncertainty on SCREEN_CLUZSPEC | 0.0012
SCREEN_CLUZSPEC_SPREAD (15) |  | Dispersion in the inspection cluster redshifts | 0
SCREEN_CLUZSPEC_GAP (16) | km s-1 | Gapper estimate of the cluster velocity dispersion | 790.0
SCREEN_CLUVDISP_BWT (17) | km s-1 | Square root of the bi-weight variance velocity dispersion | 805.5
SCREEN_CLUVDISP_BEST (18) | km s-1 | Value of the “best velocity dispersion” | 790.0
SCREEN_NMEMBERS_W (19) |  | Weighted number of red-sequence members identified as members | 11
STATUS (20) |  | Validation status of the cluster assigned by the visual inspector | validated
NINSPECTORS (21) |  | Number of individual inspections for this system | 3
NVALID (22) |  | Number of inspectors validating this system as a galaxy cluster | 3
LX0124 (23) | ergs s-1 | Luminosity in the (0.1-2.4) keV band of the cluster, aperture $R_{\text{500c}}$ | $1.5\times 10^{44}$
ELX (24) | ergs s-1 | Uncertainty on LX0124 | $0.5\times 10^{44}$
R200C_DEG (25) | deg | Apparent $R_{\text{200c}}$ radius of the galaxy cluster | 0.088
FLUX052 (26) | ergs s-1 cm-2 | Galaxy cluster X-ray flux in the 0.5-2.0 keV band | $3.9\times 10^{-13}$
EFLUX052 (27) | ergs s-1 cm-2 | Uncertainty on FLUX052 | $1.2\times 10^{-13}$
MCXC (28) |  | Identifier in the MCXC catalogue (Piffaretti et al., 2011), if present | n/a
ANAME (29) |  | Alternative name in (Piffaretti et al., 2011), if present | n/a
Table 2: Descriptions of the columns for the validated SPIDERS/CODEX galaxy
cluster catalogue. The full VAC, and all others for DR16, are available online
at: https://www.sdss.org/dr16/data_access/value-added-catalogs/

The names of the included columns are presented in Table 2 with units, a short
description, and an example of the data formatting. The columns of the
catalogue provide: (1) the SPIDERS name, (2) the total number of individually
identified cluster components along the line-of-sight, or a value of $-1$ is
given if the multiple components are undergoing a major merger, agreed upon by
at least two inspectors, (3) the CODEX target candidate name, (4) and (5) the
right ascension and declination for the epoch J2000 in degrees of the RASS
X-ray detection, (6) and (7) the right ascension and declination for the epoch
J2000 in degrees of the optical centre determined by redMaPPer, (8) the
redMaPPer richness based on aperture at the optical centre, (9) and (10) the
photometric redshift of the red-sequence and the associated error estimation,
(11) the total number of member candidates in the CODEX targeting catalogue,
(12) the total number of member candidates with a spectroscopic redshift, (13)
the assigned redshift for the galaxy cluster after visual inspection, (14) the
bootstrap uncertainty on the assigned galaxy cluster redshift, (15) the
dispersion in redshift between inspectors as described in Section 3, (16) the
velocity dispersion calculated using the gapper estimate (GAP) in units of km
s-1, (17) the velocity dispersion using the square root of the bi-weight
variance (BWT) in units of km s-1, (18) the best velocity dispersion in units
of km s-1 based on Beers et al. (1990) where we report GAP if less than 15
members are used in the calculation and BWT for well sampled systems of 15 or
more members, (19) the sum total of weighted membership flags of red-sequence
galaxies determined after inspection, (20) the validation status assigned
after inspection reconciliation, (21) the number of individual inspections for
the given validated galaxy cluster, (22) the number of inspectors to give a
status of validated, (23) and (24) the calculated X-ray luminosity in units of
ergs s-1 for the 0.1-2.4 keV band using an aperture radius of
$R_{\text{500c}}$ and the associated uncertainties as described in Section 5,
(25) the $R_{\text{200c}}$ radius of the galaxy cluster in units of degrees,
(26) and (27) the measured X-ray flux in units of ergs s-1 cm-2 for the
0.5-2.0 keV band and associated uncertainties as described in Section 5, (28)
and (29) the matched identifier in the MCXC catalogue or an alternate name
given by Piffaretti et al. (2011) in cases where the galaxy cluster appears in
both catalogues.

The optical RA and DEC coordinates given in columns (6) and (7) are determined
by running redMaPPer a second time with a relaxed constraint on the centre.
The uncertainties on the initial X-ray position of RASS sources is
approximately 3 arcmin, which the red-sequence finder is allowed to vary
within in order to optimize the centre. The richness given in column (8)
refers to the redMaPPer richness estimator value determined, prior to SPIDERS
follow-up, from red-sequence members found using the optical centre. It is
equal to the sum of the membership probabilities over the field of view, the
details of which are described in Rykoff et al. (2014). There are additional
factors taking into account the unobserved, masked areas within the search
aperture and the photometric depth correction, modeled as an exponential
increase towards high-$z$ (Finoguenov et al., 2020). The radii
$R_{\text{500c}}$ and $R_{\text{200c}}$ referenced in columns (23), (24), and
(25) are determined using the $L_{\text{X}}$-$M$ scaling relations of
Leauthaud et al. (2010).

Here we present a few examples of the improvements this catalogue achieves due
to large scale spectroscopic follow-up. Figure 9 highlights the improved
accuracy on redshifts. We plot the ratio of photometric to spectroscopic
redshift errors against the spectroscopic redshift. The typical statistical
uncertainty on cluster redshifts is $\Delta z/(1+z)=6\times 10^{-4}$ (Clerc et
al. accepted). We demonstrate that this translates into an improvement of a
factor of $\sim 10$ compared to the photometric estimation. Redshifts below
0.1 are improved even greater due to the known short comings of red-sequence
finders on nearby objects (Rykoff et al., 2014). Overall, this is achieved due
to a redshift improvement by a factor of $\sim 100$ for individual galaxies,
demonstrated by density shaded points.

Figure 9: Ratio of photometric to spectroscopic redshift errors plotted
against the spectroscopic redshift. Clusters are represented by red circles.
The individual member galaxies are represented as squares highlighted by point
density.

When comparing our redshift determinations to the 220 clusters in common with
the MCXC catalogue (Piffaretti et al., 2011), we find good agreement in
general with only a handful of outliers. Figure 10 illustrates the
distribution of the velocity offset between SPIDERS redshifts and MCXC
redshifts, normalized by $\sigma_{\rm{BWT}}$. Only 5 systems in the sample are
offset by more than 3$\sigma$, a minimum requirement that suggests these
matches would not be considered one object during visual inspection. On re-
inspection of these outliers, all but one case are of systems that are highly
sampled (the other system only having 5 redshifts available), and only one
system has evidence for multiple components along the line-of-sight. Inset
into Figure 10, we have plotted an example of a cluster match with an
approximate 5$\sigma$ offset. The red cross indicates the MCXC centre (X-ray
peak) of the cluster, where the blue crosses are the individual member
galaxies, 23 total, used in the calculation of the cluster redshift. No
projected offset is apparent, indicating both catalogs are referring to the
same extended X-ray source. Without knowing the exact method for determining
the individual redshifts in MCXC, we cannot explain the discrepancy in
velocity offset, though we can conclude our measurement is robust. A detailed
comparison of our redMaPPer sample with Planck clusters (Rozo & Rykoff, 2014;
Rozo et al., 2015) has also been carried out. These results are encouraging
and a thorough comparison of the spectroscopic properties of all available
overlapping surveys is required, but is beyond the scope of this current work.

Figure 10: Distribution of velocity offset between matched SPIDERS and MCXC
cluster, normalised by $\sigma_{\rm{BWT}}$. Offset is determined by the same
calculation of proper velocity used during visual inspection. Inset: Projected
sky coordinates of SPIDERS member galaxies (blue crosses) and the X-ray peak
centre from the MCXC catalogue (red cross). The example matched cluster
(1_2788/J1025.0+4750) has a discrepancy in proper velocity of approximately
5$\sigma$.

In Figure 11, we present the updated $\sigma-L_{\rm{X}}$ relation as presented
in Clerc et al. (2016). This improves the previous DR14 release by increasing
the highly sampled systems from 39 clusters to 755 clusters. We define a
highly sampled system as having 15 or more spectroscopic members used in the
calculation of velocity dispersion as well as only one detected component
along the line of sight to insure no external force is affecting the
measurement. We also make a cut in redshift at $z<0.4$, as the sample above
that redshift (32 clusters only) is highly incomplete. The raw values used in
the $\sigma-L_{\rm{X}}$ plane are the reported values in column (17) of the
catalogue. The bias corrected values are calculated using an updated approach
from the previous release (Clerc et al., 2016). Details of this are described
in section 7. Associated errors are calculated using the method of re-sampling
observations of clusters with a well determined velocity dispersion using a
high number of spectroscopic observations (Ruel et al., 2014). We again
compute the best-fitting power law to the bias corrected data points using the
BCES method (Akritas & Bershady, 1996), with the $y$-axis as the dependent
variable, to illustrate our samples trend. We fit constants A and B defined
as,

$\log\left(\frac{\sigma_{\rm{BWT}}}{700~{}\rm{km~{}s^{-1}}}\right)=A+B\log\left(\frac{L_{\rm{X}}}{10^{44}\rm{ergs~{}s^{-1}}}\right),$
(1)

where we find the best fits to be $(-5.91\pm 1.47)\times 10^{-2}$ and
$(1.29\pm 0.35)\times 10^{-1}$, respectively. We find a standard deviation of
0.13 along the vertical axis of the fit. Over-plotted is the comparison to the
$L_{\rm{X}}-M_{\rm{200c}}-z$ relation of Capasso et al. (2020). Both samples
are drawn from the same catalogue, though they use an earlier release with no
redshift cut, for a total of 344 systems. Also used are an updated measurement
of X-ray luminosity Finoguenov et al. (2020), where flux is extracted through
larger apertures for nearby clusters instead of the smaller fixed aperture of
our catalogue. We evaluate the relation at fixed mass and redshift,
$M_{\rm{200c}}=3\times 10^{14}$ and $z=0.16$, the approximate mean of both
samples. Following Carlberg et al. (1997), we scale with velocity dispersion
as,

$\sigma=\frac{10R_{\rm{200}}H(z)}{\sqrt{3}},$ (2)
$R_{\rm{200}}=\left(\frac{3M_{\rm{200c}}}{4\pi
200\rho_{\rm{cr}}}\left(\frac{H(z)}{100}\right)^{-2}\right)^{1/3}.$ (3)

With twice the number of systems with a higher quality selection, it is
encouraging that out simple analysis shows good agreement. There is also no
effect on the fit whether using X-ray luminosity from our catalogue, or the
updated values. A proper derivation utilising a fully consistent statistical
treatment is the important next step.

Figure 11: $\sigma-L_{\rm{X}}$ relation for SPIDERS clusters. The plotted
points are clusters with 15 or more validated members for $z<0.4$. The
triangles represent the raw bi-weight variance calculations. The crosses
represent the bias corrected values together with their uncertainty. The solid
and dashed red lines show the BCES fit to the bias corrected values and the
1$\sigma$ uncertainty range. The solid blue line represents the translated
$L_{\rm{X}}-M_{\rm{200c}}$ scaling relation from Capasso et al. (2020). Figure
12: $\sigma$-$\lambda$ relation for SPIDERS clusters. This sample is drawn the
same cluster sample used in Figure 11. Raw and bias corrected bi-weight
variances are represented by triangles and crosses, respectively. The solid
and dashed red lines shows the BCES fit and $1\sigma$ uncertainty for the bias
corrected values and updated richness calculations. The solid blue and green
lines represent the translated predictions from Kiiveri et al. (2020) and
Mulroy et al. (2019).

We also obtain a fit for the $\sigma-\lambda$ relation, presented in Figure
12. Starting with the same sample, we find all clusters overlap with the DESI
Legacy imaging surveys (Dey et al., 2019), for a total of 611 clusters. With
deeper imaging data, a more accurate calibration of richness is achieved.
RedMaPPer is re-run at the position of the optical centre for all CODEX
clusters, allowing for a new centre to be found (Chitham et al., 2020). This
improvement reduces the up-scatter of low mass systems that occurs due to
shallow imaging, especially at high redshift. At the mean redshift of our
sample, this improvement scales as $\lambda_{\rm{Legacy}}\sim
0.89\cdot\lambda_{\rm{SDSS}}$.

We aim to compute the best-fitting power law to these new data using the same
BCES method as before to illustrate the sample trend, compare to predictions
from the literature and look for evidence of redshift evolution. We fit
constants A and B defined as,

$\log\left(\frac{\sigma_{\rm{BWT}}}{700~{}\rm{km~{}s^{-1}}}\right)=A+B\log\left(\frac{\lambda_{\text{Legacy}}}{35}\right),$
(4)

where we find the best fits to be ($-5.36\pm 0.45)\times 10^{-2}$ and
($3.23\pm 0.57)\times 10^{-1}$, respectively. We find an overall standard
deviation of 0.11 for the fit. Dividing the sample into low (< 35) and high (>
35) richness, the standard deviation along the vertical axis is 0.11 and 0.12,
respectively, indicating it is relatively constant along the fit. For
comparison, we translate results based on studies of mass and richness for
different cluster studies. Kiiveri et al. (2020) studied a small sample of
high-$z$, high-$\lambda$ clusters using deep CFHT multi-band imaging. They use
a hierarchical Bayesian model to determine their
$\lambda_{\rm{CFHT}}-M_{\rm{WL}}$ relation. We have compared the legacy
richness measurements to the their own CFHT measurements, finding the richness
scaling as $\lambda_{\rm{Legacy}}\sim 0.94\cdot\lambda_{\rm{CFHT}}$. Using
this scaling, their fit compares favorably with our low-z sample. The good
agreement, even with lower mass systems, is likely due to using the slope of
Bleem et al. (2020) as a prior, a large SZ detected cluster sample that
extends to lower masses.

A similar study by Mulroy et al. (2019) is composed of low-z, high-$\lambda$
clusters based on Subaru/Suprime-Cam observations. Taking the already
converted scaling relation ($M_{500c}$ to $M_{200c}$) from Kiiveri et al.
(2020), the prediction has a steeper slope, but agreement in the richness
range they fit for ($\lambda>50$) is good. Extrapolating to lower richness
though, agreement is poor in general. Our sample demonstrates the importance
of low mass clusters when studying scaling relations.

Carrying out the same analysis on the high redshift sample produced a
completely unconstrained fit. With less than 30 examples of clusters at
$z>0.4$, all with high richness as well, much more low richness data is
required.

##  7 Cosmology constraints

Because velocity dispersion is a proxy for cluster mass, the evolution of the
velocity dispersion function is a useful cosmological tool. The primary
cosmological analysis for the CODEX component of the SPIDERS DR16 cluster
sample is carried out using optical richness as a mass proxy (Chitham et al.,
2020). Constraints can be derived in an analogous way using velocity
dispersion as a dynamical mass proxy. To accomplish this, a modification to
the PDF which describes the relation between the observed and true quantities
is necessary i.e. replacing $P(\ln\tilde{\lambda}\,|\,\ln\lambda(\mu,z)$ with
$P(\ln\tilde{\sigma}\,|\,\ln\sigma(\mu,z))$ in the likelihood function
presented in (Chitham et al., 2020), where
$P(\ln\tilde{\sigma}\,|\,\ln\sigma(\mu,z))$ is a normal distribution centred
on the value of the natural logarithm of the true velocity dispersion
$\ln\sigma$ (Carlberg et al., 1997) added to a mean bias value, with a scatter
of both statistical and intrinsic origin. Here we define $\mu=\ln M_{\rm
200c}$. Based on numerical simulations, Ferragamo et al. (2020) found the mean
bias value to depend on the radial aperture (through a multiplicative factor
$B_{\rm ape}$) and number of galaxies $N_{\rm gal}$ entering the measurement
and on the fraction of interlopers. Interlopers are galaxies entering the
computation of velocity dispersion, but they are located outside of the
virially bound cluster region. Following results of Saro et al. (2013), the
fraction of interlopers is estimated as a function of cluster redshift and
radial aperture. This fraction sets the value of the multiplicative bias
factor $B_{\rm int}$ on the velocity dispersion measurement (Ferragamo et al.,
2020). An additional 0.98 bias factor is applied, considering that only the
most massive galaxies contribute to the measurement (Ferragamo et al., 2020).
We found this model is consistent with that derived from SPIDERS-like
resampling of bright nearby clusters (Zhang et al., 2017). Combining these
four sources of bias and following our notations, we write:

$\langle\tilde{\sigma}\rangle=\frac{0.98\sigma}{0.9775+\left(\frac{0.72}{N_{\rm
gal}-1}\right)^{1.28}}B_{\rm ap}\left(\frac{R_{\rm ap}}{R_{\rm
200c}(\mu,{z})}\right)B_{\rm int}\left(\frac{R_{\rm ap}}{R_{\rm
200c}(\mu,{z})},z\right).$ (5)

As for estimating the scatter, we considered the model of Saro et al. (2013)
(their Eq. 7) which links $N_{\rm gal}$ to the scatter on
$\ln(\tilde{\sigma}/\sigma)$. This is added in quadrature to the extra
contribution from interlopers shown in their Fig. 10, that is a function
$S_{int}$ of both cluster mass and radial aperture. Therefore we write:

$Var(\ln\tilde{\sigma})=\left(-0.037+\frac{1.047}{\sqrt{N_{\rm
gal}}}\right)^{2}+\left(S_{\rm int}\left(\frac{R_{\rm ap}}{R_{\rm
200c}(\mu,z)},M_{\rm vir}(\mu,z)\right)\right)^{2},$ (6)

and for simplicity, we define
$\delta=\ln\langle\tilde{\sigma}\rangle-\ln\sigma$ and the scatter on
$\ln\tilde{\sigma}$ as $\Delta=\sqrt{Var(\ln\tilde{\sigma})}$.

Using now the assumption that $\ln\tilde{\sigma}$ is normally distributed
around $\ln\sigma$ with a bias term (equivalently, $\tilde{\sigma}$
distribution is log-normal), we write the following expression for the
additive bias $b$ on $\ln\tilde{\sigma}$:

$b(R_{\rm ap},M_{\rm vir}(\mu,z),N_{\rm
gal})=\langle\ln\tilde{\sigma}\rangle-\ln\sigma=\delta-\frac{\Delta^{2}}{2}.$
(7)

The scatter term $\Delta$ due to measurement uncertainties is added in
quadrature to an intrinsic scatter to form the resulting scatter on
$\ln\tilde{\sigma}$:

$f=\sqrt{\Delta^{2}+\Sigma_{0}^{2}}.$ (8)

Presented in section 6, velocity dispersion can be linked to the total mass by
equation 2. This relation describes the mean value of $\sigma$. We consider
the scatter term as a part of $\tilde{\sigma}$. The discussion above can be
summarized as a probability to observe a velocity dispersion, for a given true
value of it and a redshift (which defines the quality of the data). We denote
this as $P(\ln\tilde{\sigma}|\ln\sigma(\mu,z),z)$. The relation is controlled
by a single number, $N_{\rm gal}$, that we tabulated based on the data and
inverted the relation to read $N_{\rm gal}(\sigma(\mu,z),z)$, a function
called by the code, which subsequently defines the bias $b$ and the scatter
$f$ of the observed values,

$P(\ln\tilde{\sigma}|\ln\sigma(\mu,z),z)=\mathcal{N}(\ln\tilde{\sigma},\\\
\ln\sigma+b(R_{\rm ap},M_{\rm vir}(\mu,z),N_{\rm gal}),\\\ f(R_{\rm ap},M_{\rm
vir}(\mu,z),N_{\rm gal})).$ (9)

The error term is done by sampling the data and determining the typical number
of members entering a velocity dispersion calculation at a given dispersion
and redshift bin $f(\sigma,z)$. The way to estimate the error using an average
number of members is given above.

A 10% sensitivity cut to the sample, proposed by Finoguenov et al. (2020) to
obtain clean X-ray identification, corresponds to

$P^{\mathrm{RASS}}(I|\tilde{\sigma},z)=\theta(\tilde{\sigma}-375(z/0.15)^{0.38}),$
(10)

where $\theta$ denotes a step-function. To avoid incompleteness of SDSS, we
introduce an additional cut:

$P^{\mathrm{SDSS}}(I>0.9|\tilde{\sigma},z)=\theta(\tilde{\sigma}-(372+2.76e^{3.3z^{2}})).$
(11)

This equation describes the 90% completeness of SDSS. Since the cuts are
identical between the data and the model, to ensure the cut is applied
consistently to both data and the model, we include it in a definition of
velocity dispersion bins, $\Delta\ln\tilde{\sigma}_{i}$, to stay above the
completeness limits. The CODEX survey function $P_{X}(I|\mu,z,\nu=0)$
(Finoguenov et al., 2020) is included, where $\nu=0$ notation means that we
use the part of the calculation, which corresponds to a zero value of
covariance between the velocity dispersion and X-ray luminosity:

$\langle N(\Delta\ln\tilde{\sigma}_{i},\Delta z_{j})\rangle=\int_{\Delta
z_{j}}\mathrm{d}z\frac{\mathrm{d}V}{\mathrm{d}z}\int_{\Delta\ln\tilde{\sigma}_{i}}\mathrm{d}\ln\tilde{\sigma}\\\
\int\mathrm{d}\mu
P(\ln\tilde{\sigma}|\ln\sigma(\mu,z),z)P_{X}(I|\mu,z,\nu=0)\frac{\mathrm{d}n(\mu,z)}{\mathrm{d}\mu}.$
(12)

Figure 13:  The abundance of SPIDERS clusters as a function in bins ($\Delta$)
of observed redshift ($\Delta{\tilde{z}}_{j}$) and velocity dispersion
($\Delta\tilde{\sigma}_{i}$) where $\tilde{z}\in\left[0.1,0.3\right)$ and
$\tilde{\sigma}\in\left[497,2230\right).$ Systems are selected according to
Equation 10 and Equation 11. These cuts on the observable space is accounted
for in the cosmological likelihood function. Steps represent the observed data
and shaded regions trace the expectation value of the model, centred on the
median. The upper and lower limit correspond to the 15% and 85% confidence
intervals of the MCMC chains.

The binned data and best fit model are shown in Figure 13 for three
logarithmically distributed bins of velocity dispersion over a redshift range
of $[0.1,0.3)$ with a bin-width of 0.05. The low velocity dispersion bin is
dominated by selection effects, yet it provides the strongest constraints on
the $z\sim 0.15$ cluster abundance. The two high velocity dispersion bins show
the effect of increasing volume, which balances out the evolution of the
cluster mass function. The evolution of the highest bin shall be stronger and
so, the resulting trend is flatter.

The corresponding constraints are $\sigma_{8}=0.74^{+0.03}_{-0.02}$,
$\Omega_{m_{0}}=0.33^{+0.02}_{-0.02}$ and the velocity dispersion of the
intrinsic scatter $\Sigma_{0}=0.24^{+0.02}_{-0.02}$ derived from the Markov
Chain Monte Carlo (MCMC) process are shown in Figure 14. These results depend
on a large number of fixed parameters which reduce the flexibility of the
likelihood, causing a significant underestimation of the apparent
uncertainties. However, our solution still overlaps with other cluster surveys
(Vikhlinin et al., 2009; Böhringer et al., 2014; Bocquet et al., 2019;
Finoguenov et al., 2020). Cosmology using a non-baryonic tracer of mass is
similar to that achieved through calibration of baryonic tracers. Since the
need for cluster calibration is removed, the tension to Planck18 cosmology can
no longer be explained through deficiencies of mass calibration. In the
calibration of $\sigma$, we rely on numerical simulations (Carlberg et al.,
1997) and not on the empirical calibrations. Since it is these empirical
calibrations that have been questioned during the Planck CMB vs. Planck
cluster cosmology comparison (Planck Collaboration et al., 2016), this
argument cannot be used to explain the disagreement of our results with
Planck18 cosmology and other explanations shall be sought. We have estimated,
for the first time the intrinsic scatter of $\sigma$. Compared to simulations,
our estimate falls between the higher prediction of Saro et al. (2013) and the
lower prediction of Munari et al. (2013). The low intrinsic scatter is also
comparable to other cluster mass proxies such as X-ray luminosity, X-ray
temperature, optical richness and SZ thermal energy (Capasso et al., 2019,
2020; Mulroy et al., 2019; Bleem et al., 2020; Kiiveri et al., 2020)

Figure 14:  Constraints on cosmological parameters and the intrinsic scatter
of the velocity dispersion constraints for SPIDERS DR16 as. Contours depict
the 68% and 95% confidence levels where posterior distributions are obtained
using the likelihood function described in section 7.

##  8 Summary

We present the largest catalogue of visually confirmed, spectroscopic galaxy
clusters. Herein we describe the entire validation process. The SPIDERS
program successfully followed-up 2740 X-ray sources identified as galaxy
clusters detected in RASS. A total of 33,340 redshifts were used with an
improved $\Delta z_{\rm{spec}}/(1+z)=6\times 10^{-4}$ compared to $\Delta
z_{\rm{phot}}/(1+z)=5\times 10^{-2}$. The overall efficiency of successful
validation reached 99% when $\geq 9$ red-sequence members were observed. This
improves to $\geq 5$ red-sequence members observed at redshifts $z<0.3$. The
number of confirmed members per cluster ranges from 3 to 75, with a median of
10 members.

We fit a scaling relation as $\log\sigma\propto A+B\log L_{\rm{X}}$ (equation
1) using the BCES method. The resulting normalization and slope are:

$\begin{split}A=(-5.91\pm 1.47)\times 10^{-2}\\\ B=(1.29\pm 0.35)\times
10^{-1}.\end{split}$ (13)

Our relation is in excellent agreement with a previous (Capasso et al., 2020)
study of dynamical masses using a subset of our sample.

We fit as well a scaling relation as $\log\sigma\propto
A+B\log\lambda_{\rm{Legacy}}$ (equation 3) using the BCES method as well. The
resulting normalization and slope are:

$\begin{split}A=(-5.36\pm 0.45)\times 10^{-2}\\\ B=(3.23\pm 0.57)\times
10^{-1}.\end{split}$ (14)

Comparing to weak lensing (WL) samples (Mulroy et al., 2019; Kiiveri et al.,
2020), our relation compares well in the high richness range. Extrapolating to
low richness the two WL samples diverge from agreement. Our sample highlights
the importance the large richness range is for constraining scaling relation
models.

On a first attempt to constrain cosmology with this sample, velocity
dispersion is used as a mass proxy. The constraints we obtain for as follow:

$\begin{split}\sigma_{8}=0.74^{+0.03}_{-0.02}\\\
\Omega_{m_{0}}=0.33^{+0.02}_{-0.02}\\\
\Sigma_{0}=0.24^{+0.02}_{-0.02}.\end{split}$ (15)

Though the errors are underestimated, our solution overlaps with other galaxy
cluster surveys. We find the values of scatter being comparably low to other
cluster mass tracers, which makes $\sigma$ a viable cosmology tool and
provides good prospects for the eROSITA cluster follow-up program on 4MOST
(Finoguenov et al., 2019).

## Data Availability

The data underlying this article are available in the SDSS website at
https://www.sdss.org.

## Acknowledgements

CK, NC, AF and JIC acknowledge the financial support from the visitor and
mobility program of the Finnish Centre for Astronomy with ESO (FINCA), funded
by the Academy of Finland grant nr 306531.

CK and NC acknowledge this research was supported by the DFG cluster of
excellence ‘Origin and Structure of the Universe’ (www.universe-cluster.de).

NC acknowledges travel support from the Institut Francais de Finlande, the
Embassy of France in Finland, the French Ministry of Higher Education,
Research and Innovation, the Finnish Society of Sciences and Letters and the
Finnish Academy of Science and Letters.

AS is supported by the ERC-StG ‘ClustersXCosmo’ grant agreement 716762, and by
the FARE-MIUR grant ‘ClustersXEuclid’ R165SBKTMA.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
web site is www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the Brazilian
Participation Group, the Carnegie Institution for Science, Carnegie Mellon
University, the Chilean Participation Group, the French Participation Group,
Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de
Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, the Korean
Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut
für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-
Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatário Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.

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