11institutetext: HH Wills Physics Laboratory, University of Bristol, Tyndall
Avenue, Bristol, BS8 1TL, UK 22institutetext: European Space Agency (ESA),
European Space Astronomy Centre (ESAC), Camino Bajo del Castillo s/n, 28692
Villanueva de la Cañada, Madrid, Spain 33institutetext: Università di Torino,
Dipartimento di Fisica, via P. Giuria 1, I-10125, Torino, Italy
44institutetext: Istituto Nazionale di Fisica Nucleare (INFN), sezione di
Torino, Via P. Giuria 1, I-10125 Torino, Italy 55institutetext: Smithsonian
Astrophysical Observatory, 60 Garden St., Cambridge, MA 02138 66institutetext:
Department of Physics & Astronomy, Western Washington University, Bellingham,
WA 98225, USA

#  Chandra follow-up of the Hectospec Cluster Survey: comparison of caustic
and hydrostatic masses and constraints on the hydrostatic bias

Crispin H. A. Logan, 1122 Ben J. Maughan email<EMAIL_ADDRESS>Antonaldo Diaferio, 3344 Ryan T. Duffy 11 Margaret J. Geller 55 Kenneth Rines
66 Jubee Sohn 55

(Received 17/02/2022; accepted 06/07/2022)

###### Abstract

Context. Clusters of galaxies are powerful probes with which to study
cosmology and astrophysics. However, for many applications, an accurate
measurement of a cluster’s mass is essential. A systematic underestimate of
hydrostatic masses from X-ray observations (the so-called hydrostatic bias)
may be responsible for tension between the results of different cosmological
measurements.

Aims. We compare X-ray hydrostatic masses with masses estimated using the
caustic method (based on galaxy velocities) in order to explore the systematic
uncertainties of both methods and place new constraints on the level of
hydrostatic bias.

Methods. Hydrostatic and caustic mass profiles were determined independently
for a sample of 44 clusters based on Chandra observations of clusters from the
Hectospec Cluster Survey. This is the largest systematic comparison of its
kind. Masses were compared at a standardised radius ($R_{500}$) using a model
that includes possible bias and scatter in both mass estimates. The
systematics affecting both mass determination methods were explored in detail.

Results. The hydrostatic masses were found to be systematically higher than
caustic masses on average, and we found evidence that the caustic method
increasingly underestimates the mass when fewer galaxies are used to measure
the caustics. We limit our analysis to the 14 clusters with the best-sampled
caustics where this bias is minimised ($\geq 210$ galaxies), and find that the
average ratio of hydrostatic-to-caustic mass at $R_{500}$ is
$M_{500,X}/M_{500,C}=1.12^{+0.11}_{-0.10}$.

Conclusions. We interpret this result as a constraint on the level of
hydrostatic bias, favouring small or zero levels of hydrostatic bias (less
than $20\%$ at the $3\sigma$ level). However, we find that systematic
uncertainties associated with both mass estimation methods remain at the
$10-15\%$ level, which would permit significantly larger levels of hydrostatic
bias.

###### Key Words.:

cosmology: observations – galaxies: clusters: general – galaxies: kinematics
and dynamics – X-rays: galaxies: clusters

##  1 Introduction

Galaxy clusters are the largest gravitationally bound objects in the Universe,
and thus they are used to study the growth of cosmic structures and as
cosmological probes. Tight constraints on cosmological parameters can be
determined from measurements of the mass function of clusters, or their baryon
fraction, as a function of redshift (see e.g. Allen et al., 2011, for a
review). However, the accuracy of these constraints crucially depends on the
accuracy with which cluster masses can be measured. For example, the
exponential drop in the number density of clusters at high masses means that
their mass function is a sensitive cosmological probe, but an error of $\sim$
10% on the cluster mass can lead to a factor of two difference in the expected
space density111For this approximation we used hmfcalc (Murray et al., 2013)..
The measurement of galaxy cluster masses is challenging, as most of the
cluster mass is in the form of dark matter. Mass estimation techniques thus
probe the masses of clusters indirectly via the effect of the gravitational
potential of the cluster on its intracluster medium (ICM), its member
galaxies, or images of background galaxies.

Optical data were used to give the first mass estimates of galaxy clusters.
Zwicky (1933) measured the line-of-sight velocities of a number of cluster
galaxies in the nearby Coma cluster, and, by computing the velocity
dispersion, used the virial theorem to estimate a mass for the cluster. In the
late 1960s and 1970s, the first X-ray satellites were launched, and it was in
the 1990s that the first X-ray samples of hundreds of clusters were produced
(Ebeling et al., 1998, 2001; Böhringer et al., 2000, 2001) from which X-ray
hydrostatic masses could be calculated (e.g. Reiprich & Böhringer, 2002). The
launch of _XMM-Newton_ and _Chandra_ in 1999 has led to even more precise
X-ray masses being measured for large samples of clusters (e.g. Martino et
al., 2014). The first work using weak gravitational lensing (WL) to map the
dark matter distribution, undertaken by Tyson et al. (1990), was followed by a
number of other papers (e.g. Blandford et al., 1991; Miralda-Escude, 1991;
Kaiser, 1992), and in the last 3 decades it has become increasingly common to
use WL as a mass estimation technique (see Umetsu, 2020 for a review).

For cosmological studies, the calibration of masses is critical to understand
the biases that may be inherent to different mass estimation techniques.
Cosmology requires large samples of clusters with masses which in general must
be estimated from simple mass proxies that must be calibrated with more
direct, higher fidelity mass measurements. Thus, the accuracy of the mass
scale from those measurements is crucial, and biases in the mass scale can
manifest as biases in the derived cosmological parameters. For example, there
is currently tension between the cosmological constraints ($\Omega_{m}$ and
$\sigma_{8}$) from the _Planck_ cluster number counts (Planck Collaboration et
al., 2016a) and the _Planck_ cosmic microwave background (CMB) results (Planck
Collaboration et al., 2016b). The Planck cluster masses were calibrated using
hydrostatic masses, and the tension in the results could be due to a negative
bias (i.e. the true mass is underestimated) in the hydrostatic masses, often
referred to as the hydrostatic bias.

More recent analyses have found lower tension between the values of
$\sigma_{8}$ obtained from the Planck CMB and from cluster number counts
measured with SZE or X-ray cluster surveys (e.g. Douspis et al., 2019; Bocquet
et al., 2019; Planck Collaboration et al., 2020; Garrel et al., 2022; Salvati
et al., 2021). The improved agreement comes from updates to the CMB analysis
and the mass calibration of the clusters, with all analyses favouring a non-
negligible hydrostatic bias. However the magnitude of the hydrostatic bias is
not agreed upon, and remains a critical point for cluster studies.

Some level of hydrostatic bias is expected a-priori, due to the assumption
that the only pressure support in the ICM is the thermal pressure measured in
X-rays. If any other sources of pressure (e.g. from turbulence, bulk motions
or clumping of the ICM, or from cosmic rays) are also present, this leads to
the X-ray hydrostatic mass method underestimating the true cluster mass (Nagai
et al., 2007; Lau et al., 2009; Nelson et al., 2014).

The level of hydrostatic bias is expected to be higher in unrelaxed clusters
compared with relaxed clusters, as unrelaxed systems are expected to have more
non-thermal pressure support from bulk motions and turbulence of the cluster
gas. However, all clusters, regardless of dynamical state, will experience
some amount of non-thermal pressure support, as gas and substructures are
always infalling onto the cluster as they undergo constant growth, leading to
non-thermal pressure due to the residual motion of the ICM and the turbulence
created. Constraining turbulence in the ICM requires high resolution X-ray
spectrometry, and the only mission thus far capable of this has been Hitomi.
Before the mission failed, Hitomi observed the Perseus cluster, finding that
in the core of the Perseus cluster the pressure support from turbulence was 4%
of the thermodynamic pressure (Hitomi Collaboration et al., 2016), which on
its own would not lead to a large value of hydrostatic bias (though we note
that the Perseus cluster is a dynamically relaxed cluster and, of course, not
representative of the cluster population as a whole).

The hydrostatic bias may also be a function of radius, as the infalling gas
may lead to stronger gas motions and therefore significant non-thermal
pressure support at the outskirts of clusters. Bonamente et al. (2013) and
Fusco-Femiano & Lapi (2018) both find evidence for significant non-thermal
pressure at the cluster outskirts (for A1835 and A2142 respectively), and work
using hydrodynamical simulations suggests that the non-thermal pressure due to
the bulk gas motion increases with radius (Nagai et al., 2007; Lau et al.,
2009; Vazza et al., 2009, 2018; Battaglia et al., 2012; Rasia et al., 2012).
Specifically, Nagai et al. (2007) and Rasia et al. (2012) both find that at
low ($<$ 0.2 $R_{500}$) and high radius ($>$ $R_{500}$) the bias
increases222We follow the convention that e.g. $R_{500}$ refers to the radius
within which the mean density of the clusters is 500 times the critical
density of the Universe at the cluster’s redshift.. Nagai et al. (2007) find
the bias to increase from $<$10% at $R_{500}$ to $\sim$ 30% at twice this
radius. Studies that have used observational data agree with this general
trend (Siegel et al., 2018; Eckert et al., 2019).

To get a grasp on the biases of the cluster mass estimation methods,
comparisons between methods themselves are undertaken, and simulated data are
also used to compare the known mass from the simulation to the mass that is
recovered if observational methods are applied to synthetic observations of
the simulated cluster. Comparisons of X-ray hydrostatic masses with WL masses
can yield insights into the magnitude of the hydrostatic bias, as WL masses
are not sensitive to the state of the ICM (in contrast to X-ray hydrostatic
masses). However, WL masses are sensitive to mass along the line of sight to
the observed cluster, which can lead to scatter of $\sim$ 20 - 30% and
positive bias of up to $\sim$ 20% (Hoekstra, 2001, 2003; Hoekstra et al.,
2011; Becker & Kravtsov, 2011; Hwang et al., 2014), which needs to be
understood and taken into account in these comparisons. A further complication
is that the uncertainty on WL masses increases with decreasing cluster mass
and decreasing cluster redshift (Hoekstra, 2003), and even for a cluster with
mass 5$\times$1014 $\text{M}_{\odot}$ at a redshift of 0.3, the one sigma
uncertainty can be as large as 40%. Some comparisons between WL masses and
X-ray hydrostatic masses suggest that X-ray hydrostatic masses underestimate
the true mass by $\sim$ 20 - 30% (von der Linden et al., 2014; Donahue et al.,
2014; Sereno et al., 2015; Hoekstra et al., 2015). Work using hydrodynamical
simulations also suggest a significant hydrostatic bias of $\sim$10 - 30%
(Rasia et al., 2006, 2012; Nagai et al., 2007; Lau et al., 2009; Nelson et
al., 2014). However Gruen et al. (2014), Israel et al. (2014), Applegate et
al. (2016) and Smith et al. (2016) find no significant evidence for
hydrostatic bias when comparing X-ray hydrostatic masses to WL masses.

Comparisons between X-ray and WL masses from observations also support the
idea that disturbed clusters will have a larger hydrostatic bias than relaxed
clusters (Zhang et al., 2010; Mahdavi et al., 2013). Biffi et al. (2016) also
find a difference in hydrostatic bias between relaxed and disturbed clusters
using simulations (though find no difference between cool core and non cool
core systems).

Comparisons between X-ray and caustic masses are much rarer. Caustic masses
(Diaferio & Geller, 1997; Diaferio, 1999) are a useful alternative to WL
masses for testing the accuracy of hydrostatic masses. Like WL masses they are
not sensitive to the state of the ICM, and biases in the caustic method are,
in principle, well understood from simulations (Serra et al., 2011). One
drawback of using caustic masses for comparison is that they have a large
scatter of $\sim$ 30% (Serra et al., 2011), which comes predominantly from the
assumption of spherical symmetry (and thus the viewing angle of the cluster).
Thus, for a meaningful comparison, a large cluster sample is needed (several
tens of clusters). Another drawback of the comparison is that the caustic mass
method was conceived to measure the mass in the outer region of clusters, from
1 - 3 $R_{200}$, where the X-ray hydrostatic mass method cannot be applied
because the X-ray surface brightness is too low to return a reliable signal at
these radii. However, Serra et al. (2011) use simulations to estimate the bias
in the caustic method at lower radii, such that a meaningful comparison
between the two cluster mass estimators can still be made.

Diaferio et al. (2005) was the first work to compare X-ray and caustic masses,
for a sample of three clusters, finding that for two clusters the mass
profiles agreed well, and for the other cluster they did not, as the cluster
was clearly out of equilibrium. Much larger cluster samples (from
observational data) were used by Maughan et al. (2016), Andreon et al. (2017a)
and Foëx et al. (2017) to infer caustic masses to be $\sim$20% lower (at
$R_{500}$), $\sim$15% larger (at $R_{500}$) and $\sim$30% larger (at
$R_{200}$) than the hydrostatic masses respectively. Armitage et al.
(2019)333We note that the caustic method used in Armitage et al. (2019) is
from Gifford & Miller (2013). Their method, unlike the other X-ray caustic
comparison papers mentioned which all use the Diaferio (1999) method, heavily
relies on the assumption that the cluster has an NFW density profile; in
addition, they are interested in the mass within $R_{200}$, which is not the
target of the Diaferio (1999) caustic method. use simulation data to compare
caustic and X-ray hydrostatic masses (at $R_{200}$) and find a similar mass
ratio to Foëx et al. (2017). Ettori et al. (2019), however, find their caustic
masses to be underestimated by $\sim$ 40% (at $R_{200}$) compared to their
hydrostatic masses (both computed using observational data). These
contradictory results clearly show that the situation is complex and
unsettled.

There have also been comparisons of caustic masses with weak lensing -
Diaferio et al. (2005) do so for a sample of three clusters, and Geller et al.
(2013) do so for a sample of 19 clusters. Both works find the mass methods to
agree (independent of the dynamical state of the cluster) at around the virial
radius to within $\sim$ 30%, consistent within the errors associated with each
mass method.

In this paper, we aim to compare hydrostatic and caustic masses for the
largest sample so far used for these purposes, in order to investigate the
accuracy of the methods and provide new constraints on the hydrostatic bias
that are independent of those based on WL masses. We improve on our earlier
work (Maughan et al., 2016) by expanding the sample from 16 to 44 clusters. We
also include a more comprehensive analysis of the systematics, and our main
conclusions are based on a subset of 14 clusters for which the systematics are
minimised.

In this paper we use the following notation to represent the masses of the
clusters. $M_{X}$ and $M_{C}$ denote the masses measured with the hydrostatic
and caustic methods, respectively. When these masses are specifically measured
within $R_{\text{500}}$, we denote them as $M_{500,X}$ and $M_{500,C}$. Unless
otherwise stated, these are always measured within the $R_{\text{500}}$
determined from the hydrostatic analysis.

The structure of the paper is as follows. In §2 we discuss the sample
selection and data preparation. §3 details the data analysis and modelling. In
§4 we present initial results for the full sample, and then our main results
from the subset found to have the most reliable mass estimates. We discuss our
results in §5, including a review of the systematics affecting the mass
estimates, and the implications for the hydrostatic bias, and summarise our
findings in §6. Throughout this paper we assume a WMAP9 cosmology of
$H_{0}=69.7$ km s-1Mpc-1, $\Omega_{\Lambda}=0.718$, and $\Omega_{m}=0.282$
(Hinshaw et al., 2013). All measurement uncertainties correspond to 68%
confidence level, unless explicitly stated otherwise. When referring to base
10 logarithm we use ‘$\log$’ or ‘$\log_{10}$’, and use ‘$\ln$’ if referring to
the natural logarithm.

##  2 Cluster sample

For our analysis, we used the X-ray flux-limited subset of the Hectospec
Cluster Survey (HeCS; Rines et al., 2013), a spectroscopic survey of X-ray
selected clusters. The HeCS sample was constructed by matching clusters
selected using the ROSAT All-Sky Survey (RASS; Voges et al., 1999) with the
imaging footprint of the SDSS Data Release 6 (Adelman-McCarthy et al., 2008).
The SDSS data were used to select candidate cluster member galaxies for
spectroscopic follow-up, which was performed with MMT/Hectospec (Fabricant et
al., 2005).

The X-ray flux limit applied to the HeCS sample in order to create the flux-
limited HeCS sub-sample was $5\times 10^{-12}\mbox{$\text{\leavevmode\nobreak\
erg}$}\mbox{$\text{\leavevmode\nobreak\
s}$}^{-1}\mbox{$\text{\leavevmode\nobreak\ cm}$}^{-2}$ (in the
$0.1-2.4\mbox{$\text{\leavevmode\nobreak\ keV}$}$ band), which excluded A750,
A2187, A2396, A2631, and A2645 from the full HeCS sample. We then excluded
three clusters from the flux-limited sample that were found to be clearly
dominated by AGN upon inspection of available X-ray observations (these were
A689 and A1366 based on Chandra data, and A2055 based on XMM-Newton data).
More specifically, A1366 and A2055 have BLLacs in the brightest cluster galaxy
(BCG), and A689 has a BLLac very close to the BCG (Giles et al., 2012). This
resulted in a flux-limited sample of 50 clusters. We then obtained Chandra
follow-up observations of those clusters lacking Chandra data.

To construct the final sample, we then excluded four clusters due to flaring
in the available Chandra observations (A267, A667, A1361, and A1902), one
cluster as it was part of a double cluster (A1758), and one cluster as it was
off-chip in the _Chandra_ observation available (A646 - the observation was of
a radio galaxy in the cluster, but around 7′ from the cluster core). The final
sample then consists of 44 clusters, which are summarised in Table 1. All
clusters are in the redshift range $0.1<$ z $<0.3$. We call our sample the
CHeCS (Chandra observations of Hectospec Cluster Survey clusters) sample. We
note that the masses from Rines et al. (2013) were converted to be consistent
with the value of $H_{0}$ we use in this paper.

Cluster | Short name | RA | DEC | z | ObsID | clean time | Ngal
---|---|---|---|---|---|---|---
|  | deg | deg |  |  | (ks) |
ZwCl 0755.8+5408 | Zw1478 | 119.9190 | 53.9990 | 0.1027 | 18248 | 20 | 82
Abell 0655 | A655 | 126.3610 | 47.1320 | 0.1271 | 15159 | 8 | 315
Abell 0697 | A697 | 130.7362 | 36.3625 | 0.2812 | 4217 | 20 | 185
MS 0906.5+1110 | MS0906 | 137.2832 | 10.9925 | 0.1767 | 924 | 30 | 101
Abell 0773 | A773 | 139.4624 | 51.7248 | 0.2173 | 5006,533,3588 | 15,10,9 | 173
Abell 0795 | A795 | 141.0240 | 14.1680 | 0.1374 | 11734⋆ | 29 | 179
ZwCl 0949.6+5207 | Zw2701 | 148.1980 | 51.8910 | 0.2160 | 12903⋆,3195⋆ | 93,22 | 93
Abell 0963 | A963 | 154.2600 | 39.0484 | 0.2041 | 903⋆ | 34 | 211
Abell 0980 | A980 | 155.6275 | 50.1017 | 0.1555 | 15105 | 14 | 222
ZwCl 1021.0+0426 | Zw3146 | 155.9117 | 04.1865 | 0.2894 | 909,9371 | 43,33 | 106
Abell 0990 | A990 | 155.9120 | 49.1450 | 0.1416 | 15114 | 10 | 91
ZwCl 1023.3+1257 | Zw3179 | 156.4840 | 12.6910 | 0.1422 | 13375 | 9 | 69
Abell 1033 | A1033 | 157.9320 | 35.0580 | 0.1220 | 15614,15084 | 33,29 | 191
Abell 1068 | A1068 | 160.1870 | 39.9510 | 0.1386 | 1652⋆ | 26 | 129
Abell 1132 | A1132 | 164.6160 | 56.7820 | 0.1351 | 19770,13376 | 20,9 | 160
Abell 1201 | A1201 | 168.2287 | 13.4448 | 0.1671 | 9616 | 47 | 165
Abell 1204 | A1204 | 168.3324 | 17.5937 | 0.1706 | 2205 | 24 | 92
Abell 1235 | A1235 | 170.8040 | 19.6160 | 0.1030 | 18247 | 18 | 131
Abell 1246 | A1246 | 170.9912 | 21.4903 | 0.1921 | 11770 | 5 | 226
Abell 1302 | A1302 | 173.3070 | 66.3990 | 0.1152 | 18245 | 19 | 162
Abell 1413 | A1413 | 178.8260 | 23.4080 | 0.1412 | 5003 | 75 | 116
Abell 1423 | A1423 | 179.3420 | 33.6320 | 0.2142 | 11724,538 | 26,10 | 230
Abell 1437 | A1437 | 180.1040 | 03.3490 | 0.1333 | 15306,15188 | 10,9 | 194
Abell 1553 | A1553 | 187.6959 | 10.5606 | 0.1668 | 12254 | 9 | 171
Abell 1682 | A1682 | 196.7278 | 46.5560 | 0.2272 | 11725 | 20 | 151
Abell 1689 | A1689 | 197.8750 | \- 1.3353 | 0.1842 | 6930,7289,5004 | 76,75,20 | 210
Abell 1763 | A1763 | 203.8257 | 40.9970 | 0.2312 | 3591 | 20 | 237
Abell 1835 | A1835 | 210.2595 | 02.8801 | 0.2506 | 6880,6881,7370 | 118,36,40 | 219
Abell 1918 | A1918 | 216.3420 | 63.1830 | 0.1388 | 18249 | 21 | 80
Abell 1914 | A1914 | 216.5068 | 37.8271 | 0.1660 | 20026,18252,20023,20025,20024 | 29,27,25,22,17 | 255
Abell 1930 | A1930 | 218.1200 | 31.6330 | 0.1308 | 11733⋆ | 35 | 76
Abell 1978 | A1978 | 222.7750 | 14.6110 | 0.1459 | 18250 | 20 | 63
Abell 2009 | A2009 | 225.0850 | 21.3620 | 0.1522 | 10438 | 20 | 195
RXCJ 1504.1-0248 | RXJ1504 | 226.0321 | \- 2.8050 | 0.2168 | 17670,17197,17669,4935 | 51,30,29,12 | 120
Abell 2034 | A2034 | 227.5450 | 33.5060 | 0.1132 | 12886,12885 | 91,81 | 182
Abell 2050 | A2050 | 229.0680 | 00.0890 | 0.1191 | 18251 | 15 | 106
Abell 2069 | A2069 | 231.0410 | 29.9210 | 0.1139 | 4965 | 39 | 441
Abell 2111 | A2111 | 234.9337 | 34.4156 | 0.2291 | 11726, 544 | 21,10 | 208
Abell 2219 | A2219 | 250.0892 | 46.7058 | 0.2257 | 14356,14431,14355,14451 | 49,39,30,20 | 461
ZwCl 1717.9+5636 | Zw8197 | 259.5480 | 56.6710 | 0.1132 | 18246 | 19 | 76
Abell 2259 | A2259 | 260.0370 | 27.6702 | 0.1605 | 3245 | 10 | 165
RXCJ 1720+2638 | RXJ1720 | 260.0370 | 26.6350 | 0.1604 | 4361,1453 | 14,8 | 376
Abell 2261 | A2261 | 260.6129 | 32.1338 | 0.2242 | 5007 | 24 | 209
RXCJ 2129+0005 | RXJ2129 | 322.4186 | 00.0973 | 0.2339 | 9370,552 | 30,10 | 325

Table 1:  Summary of sample. Column 1 gives the cluster name with a shorter
alternative name given in column 2, columns 3 and 4 give the RA and DEC of the
cluster respectively, column 5 gives the spectroscopic redshift of the
cluster, column 6 gives the Chandra ObsID(s), column 7 gives the length of the
observation after lightcurve cleaning, column 8 gives the number of galaxies
classed as member galaxies of each cluster in the caustic method (Serra &
Diaferio, 2013), as defined in Rines et al. (2013) (equivalent to the number
of galaxies within the caustics). Starred ObsIDs are ACIS-S otherwise all are
ACIS-I

##  3 Analysis

###  3.1 X-ray data

The Chandra data analysis follows closely that presented in Giles et al.
(2015), which in turn is based on the analysis presented in Maughan et al.
(2008) and Maughan et al. (2012). We summarise the main steps here, and
highlight any differences in the updated analysis used in this paper. Any
peculiarities of the analysis of individual clusters are noted in Appendix A.

All of the clusters in our sample were analysed with the ciao444See
http://cxc.harvard.edu/ciao. 4.10 software package and caldb555See
http://cxc.harvard.edu/caldb. version 4.8.1 (Fruscione et al., 2006).
Projected emissivity profiles of the ICM were obtained from the surface
brightness profile, measured in circular annuli centred on the centroid of the
X-ray emission. The centroid location was measured in an X-ray image produced
in the $0.7-2\mbox{$\text{\leavevmode\nobreak\ keV}$}$ band, and was
determined by iteration using a circular region of
$150\mathrm{{}^{\prime}}\hskip-1.00006pt\mathrm{{}^{\prime}}$ and then
$50\mathrm{{}^{\prime}}\hskip-1.00006pt\mathrm{{}^{\prime}}$ initially centred
on the X-ray peak (Maughan et al., 2008). Projected temperatures profiles were
obtained by extracting spectra in circular annuli also centred on the X-ray
centroid. The spectra were fit with an absorbed APEC plasma model (Smith et
al., 2001) in XSPEC (Arnaud, 1996). For the spectral analysis, instead of
using the $\chi^{2}$ statistic as in Maughan et al. (2016), we used the
C-statistic, which assumes Poissonian statistics, and the spectra were grouped
to contain at least five counts per energy bin. This minimal binning is
required to avoid a bias in the XSPEC implementation of the C-statistic when
there are bins in the spectrum with zero or very few counts (see Willis et
al., 2005, and the discussion in Appendix B in the XSPEC manual). We exclude
the outermost temperature measurement in the profile if the measured
temperature in the final bin is greater than 15 keV, has errors $>$50%, or
including it leads to an unphysical increase in the best fit temperature
profile.

A further minor difference from our previous analysis is that the absorbing
column used in the spectral fits are fixed at the nhtot value from Willingale
et al. (2013), instead of the absorbing column from Dickey & Lockman (1990).
The updated nhtot value now contains absorption contributions from molecular
hydrogen, not previously accounted for. We note this update does not
significantly affect our results.

In order to calculate hydrostatic mass profiles, we use the models for the 3D
temperature and gas density profiles as presented in Vikhlinin et al. (2006)
and fit them to the observed projected temperature profile and surface
brightness profile respectively (this mass fitting method is sometimes
referred to as the forward-fitting mass method e.g. Ettori et al., 2013 and
more details of our implementation are given in Appendix B). In this work we
fit the 3D temperature and gas density models simultaneously, in contrast to
the X-ray mass method used in Maughan et al. (2016). This fitting is done
using a Markov Chain Monte Carlo (MCMC) method using emcee (Foreman-Mackey et
al., 2013), and the priors used on the 3D gas density and temperature models
are presented in Table 2. Uncertainties on the mass profile are obtained
straightforwardly from the MCMC chains. We also compute the mean $R_{500}$,
and mean hydrostatic mass at $R_{500}$ from these chains (presented in Table
3). Gas mass profiles can also be obtained from the 3D gas density fit, with
uncertainties obtained from the MCMC chains in the same way. This means that
the uncertainties derived on the gas mass within $R_{\text{500}}$ self-
consistently include the uncertainties on $R_{\text{500}}$.

Density profile |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
$n_{0}$ | $r_{\text{c}}$ | $r_{\text{s}}$ | $\alpha$ | $\beta$ | $\epsilon$ | $n_{02}$ | $r_{\text{c2}}$ | $\beta_{2}$
cm-3 | kpc | kpc |  |  |  | cm-3 | kpc |
($10^{-4}$, $10^{-1}$) | (5, 800) | (100, 4000) | (0, 3) | (0.3, 1.5) | (0, 5) | (0, $10^{-1}$) | (1, 70) | (0.1, 5)
Temperature profile |  |  |  |  |  |  |  |
$T_{\text{0}}$ | $r_{\text{cool}}$ | $a_{\text{cool}}$ | $T_{\text{min}}$ | $r_{\text{t}}$ | a | b | c |
keV | kpc |  | keV | kpc |  |  |  |
(0.5, 18) | (10, 500) | (0, 3) | (0.1, 6 or $T_{\text{0}}$) | (100, 500) | (-0.5, 0.5) | (0, 5) | (0, 1) |

Table 2:  Summary of priors for the 3D gas density and 3D temperature models
(see §B for equations) used in the X-ray hydrostatic mass method described in
§3.1. All are flat uniform priors (in linear space) with lower and upper
bounds in parentheses.

To investigate the influence of the clusters’ dynamical states on our results,
we split our sample into relaxed cool core (RCC) and non-relaxed cool core
(NRCC) clusters, using the same three diagnostics as in Maughan et al. (2016).
To be classed as RCC, a cluster has to have a low central cooling time ($<$
7.7 Gyr), a highly peaked central density profile (logarithmic slope $>$ 0.7
within 0.048 $R_{500}$), and a low centroid shift ($<$ 0.009). This definition
of what constitutes a RCC cluster is a conservative one: for our sample of 44
clusters, 10 are classified as RCC (see Table 3).

This work is an extension of Maughan et al. (2016) from 16 to 44 clusters. As
noted §2, we have dropped A267 and A2631 from the present analysis, leaving 14
clusters in common. For these clusters, we compared the hydrostatic masses at
$R_{500}$ between the two analyses, finding a weighted average ratio of the
new masses to those from Maughan et al. (2016) of 0.96$\pm$0.05. This
demonstrates that the improvements in our analysis, and changes to the Chandra
calibration have not significantly affected our hydrostatic mass estimates.

###  3.2 Caustic masses

In this work, we use the caustic masses from HeCS, as presented in Rines et
al., 2013. We summarise the key aspects of the analysis in the following. The
caustic mass method uses spectroscopic redshifts for a number of member
galaxies in a cluster (on average $\sim$180 for our sample, see column 7 Table
1), to determine their line-of-sight velocity (relative to the median of the
distribution of the line-of-sight velocities of the galaxies of the main group
on the binary tree Diaferio, 1999), which when combined with their projected
distances from the cluster centre, can be used to estimate a mass profile of a
cluster.

By plotting the line-of-sight velocity versus the projected distance from the
cluster centre for each of the member galaxies (referred to as a redshift
diagram), an overpopulated region of this parameter space is clearly seen (see
e.g. Rines et al., 2013), and is bounded by a ‘trumpet shape’. The boundaries
between galaxies within this trumpet shape and outside it are known as
caustics. The galaxies within this overpopulated parameter space are assumed
to be bound by the cluster’s gravitational potential, and their velocities are
assumed to be less than the escape velocity of the cluster. The amplitude,
${\cal A}(r)$, of this trumpet shape is related to the cluster’s escape
velocity and the velocity anisotropy parameter $\beta(r)$, and decreases as a
function of projected distance from the cluster centre. Diaferio & Geller
(1997) show that this caustic amplitude, ${\cal A}(r)$, can be related to the
cluster mass within a radius, $r$:

$GM(<r)=\mathcal{F}_{\beta}\int_{0}^{r}{\cal A}^{2}(r)\,dr\,.$ (1)

Equation (1) simplifies the correct equation $GM(<r)=\int_{0}^{r}{\cal
F}_{\beta}(r){\cal A}^{2}(r){\mathrm{d}}r$ by replacing the function ${\cal
F}_{\beta}(r)$ with the constant filling factor ${\cal F}_{\beta}$. The
function ${\cal F}_{\beta}(r)$ combines the cluster density profile and the
anisotropy of the velocity field, both of which vary as a function of $r$.
Beyond $\sim 0.5R_{200}$, ${\cal F}_{\beta}(r)$ is roughly constant with
radius, so assuming a constant filling factor ${\cal F}_{\beta}$ should not
lead to significant overestimation or underestimation of mass at larger radii
(Diaferio, 1999). However, within $\sim 0.5R_{200}$, approximating ${\cal
F}_{\beta}(r)$ to a constant might lead to an overestimate of the cluster mass
by $\sim 15\%$ (Serra et al., 2011).

We note that we use $\mathcal{F}_{\beta}=0.5$ for the filling factor, which is
appropriate for the algorithm of Diaferio (1999) used to calculate the caustic
masses presented in Rines et al. (2013).

The uncertainties on the caustic masses represent the statistic precision of
the caustic mass estimate due to the uncertainties in the location of the
caustics. They do not represent the accuracy with which the caustic mass
measures the true mass, which is $\approx 30\%$ (Serra et al., 2011), driven
largely by the effects of projection and the viewing angle of the cluster
(this is accounted for in our model below by an intrinsic scatter between
caustic and true masses).

The uncertainty on the caustic location increases with decreasing ratio $\rho$
between the number of galaxies within the caustics and the number of galaxies
in the redshift diagram. On average, this recipe is a robust estimate of the
statistical uncertainty (Serra et al., 2011). However, in sparsely-sampled
redshift diagrams, the caustic technique might underestimate the statistical
uncertainty, because the ratio $\rho$ might remain relatively large, despite
the low number of galaxies in the diagram. We verified that these cases have a
negligible effect on our results by repeating our analysis with a uniform
$50\%$ error on all caustic masses. In this test, the intrinsic scatter term
in our model is effectively removed and we prevent any points with
underestimated errors from driving the results.

The number of galaxies within the caustics, Ngal, can also have a systematic
effect on the mass derived using the caustic method. The simulations of Serra
et al. (2011) suggest an underestimate of the mass of $\sim 10\%$ at $R_{500}$
for $\text{Ngal}\sim 100$, with the masses being unbiased for $200\,\hbox
to0.0pt{\raise 1.72218pt\hbox{$<$}\hss}{\lower
2.58334pt\hbox{$\sim$}}\,\text{Ngal}\,\hbox to0.0pt{\raise
1.72218pt\hbox{$<$}\hss}{\lower 2.58334pt\hbox{$\sim$}}\,1000$. Any
underestimate of mass due to low values of Ngal is expected to approximately
cancel with the overestimate of mass at $R_{500}$ due to the assumption of
constant filling factor. However, as discussed below (§4.2), our results
suggest that the magnitude of the mass underestimate for low values of Ngal
may be larger than predicted, and our main conclusions are thus based on the
clusters with the largest values of Ngal.

###  3.3 Modelling the mass biases

Figure 1: Graphical model summarising the mass modelling framework that we
present in §3.3. The symbols are also described in that section; in summary,
the symbol $\mu$ represents mass, $\kappa$ is the bias term, $\delta$ is the
scatter term and $\sigma$ is the measurement error term (all in log space),
and the $C$ and $X$ subscripts denote whether the quantity refers to that from
the hydrostatic or caustic method. The $\hat{}$ refers to an observed
quantity. $\cal{N}$ refers to a normal distribution, and ‘$\sim$’ denotes ‘is
distributed as’.

In order to constrain the bias and scatter between the hydrostatic and caustic
mass measurement techniques, we use a Bayesian framework (of which a graphical
representation is shown in Figure 1). The modelling framework is the same as
in Maughan et al. (2016), but we summarise the key points here. In the
following, we use $\mu$ to denote the base 10 logarithm of mass.

A given cluster with a ‘true’ mass, $\mu$ also has a ‘true’ caustic mass,
$\mu_{C}$, and ‘true’ hydrostatic mass, $\mu_{X}$, which are related to the
true mass, $\mu$, as follows:

$\displaystyle\mu_{X}\sim\cal{N}(\mu+\rm\kappa_{X},\rm\delta_{X})$ (2)
$\displaystyle\mu_{C}\sim\cal{N}(\mu+\rm\kappa_{C},\rm\delta_{C})$ (3)

where $\kappa_{X}$ and $\kappa_{C}$ parametrise the bias between the true mass
and the ‘true’ hydrostatic and caustic mass, respectively, and $\delta_{X}$
and $\delta_{C}$ parametrise the intrinsic scatter between the true mass and
the ‘true’ hydrostatic and caustic mass, respectively. $\cal{N}$ refers to a
normal distribution, and “$\sim$” is equivalent to “is distributed as”.

These ‘true’ hydrostatic and caustic masses are related to the observed
hydrostatic and caustic masses ($\hat{\mu_{X}}$ and $\hat{\mu_{C}}$,
respectively), as follows:

$\displaystyle\hat{\mu_{X}}\sim\cal{N}(\rm\mu_{X},\rm\sigma_{X})$ (4)
$\displaystyle\hat{\mu_{C}}\sim\cal{N}(\rm\mu_{C},\rm\sigma_{C})$ (5)

where $\sigma_{X}$ and $\sigma_{C}$ represent the standard deviation of the
lognormal error for the observed hydrostatic and caustic masses, respectively.
Note that our model does not take into account selection effects that could
arise due to the X-ray flux limit. However, we show in §5.3 that this has a
negligible impact on our results.

We used the same (loose) priors as in Maughan et al. (2016) on the cluster
masses, bias and scatter terms. Specifically, for the cluster masses in log
space, $\mu$, we applied a uniform prior from 12 to 17. For the bias terms in
log space, $\kappa_{X}$ and $\kappa_{C}$, we applied normal priors with mean 0
and a standard deviation of 1. For the intrinsic scatter terms in log space,
$\delta_{X}$ and $\delta_{C}$, we applied normal priors (truncated at zero)
with mean 0.09 and a standard deviation of 2.2, corresponding to a weak prior
with a mean of $\sim$20% in normal space.

Using the model described above, we used the observational data that we have
for each cluster ($\hat{\mu_{X}}$, $\sigma_{X}$, $\hat{\mu_{C}}$,
$\sigma_{C}$) to constrain the ’true’ hydrostatic and caustic mass terms
($\mu_{X}$, $\mu_{C}$), the scatter terms ($\delta_{X}$ and $\delta_{C}$) and
bias terms ($\kappa_{X}$ and $\kappa_{C}$) for all of the clusters in our
sample. Both the scatter terms and the bias terms are degenerate, however, the
intrinsic scatter between the hydrostatic and caustic mass measurements

$\displaystyle\delta=\sqrt{\delta_{X}^{2}+\delta_{C}^{2}}$ (6)

and the mean bias between the hydrostatic and caustic mass measurements

$\displaystyle\kappa=\kappa_{X}-\kappa_{C}=\mu_{X}-\mu_{C}=\log_{10}\left(\frac{M_{X}}{M_{C}}\right)$
(7)

can be constrained by the data.

When reporting the values of $\kappa$ and $\delta$ in Tables 4 and 5, we
report the median value of $\delta$, with errors given as the difference
between the median and the 16th and 84th percentiles. For $\kappa$, we report
the mean value. As $\kappa=\log_{10}(M_{X}/M_{C})$, and $\kappa$ (a quantity
in log space) is normally distributed, the posterior of $M_{X}/M_{C}$ (a
quantity in linear space) is lognormally distributed. Therefore, in Tables 4
and 5 we summarise the posterior of $M_{X}/M_{C}$ by reporting its median and
errors in the same way that we did for $\delta$. In these tables, the $\kappa$
values are given in base 10 log space, and the $\delta$ values have been
converted to percentage scatters.

We also note that in Table 3 and in all figures showing mass errors, we report
the mean and standard deviation of the hydrostatic and caustic mass
measurements in linear space, which is the form in which masses are often
reported in other work. However, we find that the uncertainties on the mass
measurements are in fact better described using lognormal errors, and so
choose to model the biases associated with the caustic and X-ray hydrostatic
mass measurement methods in log space.

We use the probabilistic programming language stan666http://mc-stan.org to
implement the model described above, specifically using the No-U-Turn Sampler
(Hoffman & Gelman, 2011). We sampled the parameters in our model with four
chains of 5,000 steps each. The analysis can be done at any radius, and so by
repeating the analysis at increasing radii, we produced a profile of the mean
bias between the two mass measurement methods.

In our analysis, the mass determination methods are independent and use
different definitions of the cluster centre (the x-ray centroid for the
hydrostatic mass, and the peak of the galaxy distribution on the sky and the
median of the line-of-sight velocity distribution of the cluster members with
the caustic method). This means that the radial coordinates are not identical
and a given radius does not correspond to an identical aperture for both
methods. For our main analysis, we scale radial coordinates to the
$R_{\text{500}}$ estimated from the hydrostatic method. This is motivated by
the expectation that the hydrostatic mass should have lower intrinsic scatter
with the true mass. We show in §5.2 that using $R_{\text{500}}$ computed
independently for each method does not alter our conclusions.

##  4 Results

In the following, we first present the comparisons between hydrostatic and
caustic masses for the full CHeCS sample. We then demonstrate that the caustic
masses for clusters with lower values of Ngal are likely to be underestimated.
Finally we present our main results, based on the subset of clusters with
higher values of Ngal.

###  4.1 Initial results for the full sample

Cluster | z | Status | $R_{500}$ | $M_{500,X}$ | $M_{500,C}$ | Ngal | Ngal | nhtot |
---|---|---|---|---|---|---|---|---|---
|  |  | Mpc | $10^{14}\text{M}_{\odot}$ | $10^{14}\text{M}_{\odot}$ | sub-sample |  | cm-2 |
ZW1478 | 0.103 | NRCC | 0.80 $\pm$ 0.03 | 1.57 $\pm$ 0.27 | 0.81 $\pm$ 0.04 | low | 82 | 3.86 |
A0655 | 0.127 | NRCC | 1.08 $\pm$ 0.04 | 4.05 $\pm$ 0.83 | 3.87 $\pm$ 0.20 | high | 315 | 4.39 |
A0697 | 0.281 | NRCC | 1.47 $\pm$ 0.05 | 11.88 $\pm$ 1.81 | 5.90 $\pm$ 2.80 | mid | 185 | 3.28 |
MS0906 | 0.177 | NRCC | 1.04 $\pm$ 0.02 | 3.74 $\pm$ 0.33 | 1.78 $\pm$ 0.24 | low | 101 | 3.61 |
A0773 | 0.217 | NRCC | 1.32 $\pm$ 0.03 | 7.97 $\pm$ 0.88 | 9.78 $\pm$ 0.11 | mid | 173 | 1.33 |
A0795 | 0.137 | NRCC | 1.09 $\pm$ 0.04 | 4.18 $\pm$ 0.69 | 3.47 $\pm$ 0.06 | mid | 179 | 3.62 |
ZW2701 | 0.216 | RCC | 1.07 $\pm$ 0.03 | 4.22 $\pm$ 0.54 | 2.38 $\pm$ 0.73 | low | 93 | 0.767 |
A0963 | 0.204 | NRCC | 1.06 $\pm$ 0.02 | 4.03 $\pm$ 0.31 | 4.10 $\pm$ 0.04 | high | 211 | 1.30 |
A0980 | 0.155 | NRCC | 1.36 $\pm$ 0.07 | 8.33 $\pm$ 2.20 | 5.66 $\pm$ 1.80 | high | 222 | 0.829 |
ZW3146 | 0.289 | RCC | 1.29 $\pm$ 0.03 | 8.02 $\pm$ 0.95 | 3.99 $\pm$ 1.78 | low | 106 | 2.69 |
A0990 | 0.142 | NRCC | 1.29 $\pm$ 0.05 | 6.87 $\pm$ 1.28 | 2.22 $\pm$ 0.82 | low | 91 | 0.808 |
ZW3179 | 0.142 | NRCC | 1.44 $\pm$ 0.13 | 10.19 $\pm$ 4.57 | 1.52 $\pm$ 0.11 | low | 69 | 4.08 |
A1033 | 0.122 | NRCC | 1.11 $\pm$ 0.02 | 4.35 $\pm$ 0.35 | 3.18 $\pm$ 0.04 | mid | 191 | 1.73 |
A1068 | 0.139 | RCC | 1.15 $\pm$ 0.05 | 4.96 $\pm$ 1.09 | 8.29 $\pm$ 0.66 | low | 129 | 1.78 |
A1132 | 0.135 | NRCC | 1.64 $\pm$ 0.08 | 14.09 $\pm$ 2.87 | 5.12 $\pm$ 0.21 | mid | 160 | 0.63 |
A1201 | 0.167 | NRCC | 1.15 $\pm$ 0.02 | 5.03 $\pm$ 0.47 | 3.05 $\pm$ 0.06 | mid | 165 | 1.65 |
A1204 | 0.171 | RCC | 0.91 $\pm$ 0.02 | 2.52 $\pm$ 0.28 | 1.42 $\pm$ 0.18 | low | 92 | 1.36 |
A1235 | 0.103 | NRCC | 1.05 $\pm$ 0.05 | 3.66 $\pm$ 0.87 | 2.04 $\pm$ 0.21 | mid | 131 | 1.78 |
A1246 | 0.192 | NRCC | 1.12 $\pm$ 0.06 | 4.81 $\pm$ 1.26 | 5.85 $\pm$ 0.14 | high | 226 | 1.65 |
A1302 | 0.115 | NRCC | 1.05 $\pm$ 0.03 | 3.60 $\pm$ 0.46 | 2.30 $\pm$ 0.04 | mid | 162 | 0.924 |
A1413 | 0.141 | NRCC | 1.33 $\pm$ 0.02 | 7.54 $\pm$ 0.55 | 6.70 $\pm$ 0.02 | low | 116 | 1.96 |
A1423 | 0.214 | NRCC | 1.10 $\pm$ 0.03 | 4.61 $\pm$ 0.50 | 4.14 $\pm$ 0.07 | high | 230 | 1.92 |
A1437 | 0.133 | NRCC | 1.19 $\pm$ 0.02 | 5.31 $\pm$ 0.38 | 9.37 $\pm$ 1.19 | mid | 194 | 2.29 |
A1553 | 0.167 | NRCC | 1.31 $\pm$ 0.06 | 7.42 $\pm$ 1.64 | 5.72 $\pm$ 0.03 | mid | 171 | 2.29 |
A1682 | 0.227 | NRCC | 1.13 $\pm$ 0.03 | 5.05 $\pm$ 0.65 | 6.75 $\pm$ 0.04 | mid | 151 | 1.07 |
A1689 | 0.184 | RCC | 1.47 $\pm$ 0.02 | 10.56 $\pm$ 0.70 | 10.42 $\pm$ 3.26 | high | 210 | 1.97 |
A1763 | 0.231 | NRCC | 1.23 $\pm$ 0.03 | 6.47 $\pm$ 0.75 | 10.90 $\pm$ 1.24 | high | 237 | 0.835 |
A1835 | 0.251 | RCC | 1.45 $\pm$ 0.02 | 10.73 $\pm$ 0.67 | 9.57 $\pm$ 0.66 | high | 219 | 2.24 |
A1918 | 0.139 | NRCC | 1.12 $\pm$ 0.05 | 4.61 $\pm$ 1.14 | 2.55 $\pm$ 0.06 | low | 80 | 1.42 |
A1914 | 0.166 | NRCC | 1.44 $\pm$ 0.04 | 9.91 $\pm$ 1.20 | 6.08 $\pm$ 0.17 | high | 255 | 1.10 |
A1930 | 0.131 | NRCC | 1.06 $\pm$ 0.05 | 3.81 $\pm$ 0.78 | 2.35 $\pm$ 0.15 | low | 76 | 1.15 |
A1978 | 0.146 | NRCC | 0.95 $\pm$ 0.02 | 2.79 $\pm$ 0.32 | 1.19 $\pm$ 0.33 | low | 63 | 1.80 |
A2009 | 0.152 | NRCC | 1.33 $\pm$ 0.05 | 7.69 $\pm$ 1.46 | 4.29 $\pm$ 0.21 | mid | 195 | 3.89 |
RXJ1504 | 0.217 | RCC | 1.30 $\pm$ 0.01 | 7.65 $\pm$ 0.36 | 3.09 $\pm$ 2.17 | low | 120 | 8.38 |
A2034 | 0.113 | NRCC | 1.31 $\pm$ 0.02 | 7.05 $\pm$ 0.35 | 5.46 $\pm$ 0.04 | mid | 182 | 1.62 |
A2050 | 0.119 | NRCC | 1.04 $\pm$ 0.03 | 3.49 $\pm$ 0.40 | 3.91 $\pm$ 1.08 | low | 106 | 5.20 |
A2069 | 0.114 | NRCC | 1.33 $\pm$ 0.04 | 7.46 $\pm$ 1.14 | 6.10 $\pm$ 0.06 | high | 441 | 2.03 |
A2111 | 0.229 | NRCC | 1.19 $\pm$ 0.03 | 5.85 $\pm$ 0.66 | 3.71 $\pm$ 0.42 | mid | 208 | 1.99 |
A2219 | 0.226 | NRCC | 1.55 $\pm$ 0.02 | 12.86 $\pm$ 0.58 | 10.07 $\pm$ 2.61 | high | 461 | 1.86 |
ZW8197 | 0.113 | NRCC | 0.87 $\pm$ 0.03 | 2.08 $\pm$ 0.36 | 1.86 $\pm$ 0.03 | low | 76 | 2.39 |
A2259 | 0.161 | NRCC | 1.16 $\pm$ 0.05 | 5.14 $\pm$ 0.95 | 4.58 $\pm$ 0.81 | mid | 165 | 3.81 |
RXJ1720 | 0.160 | RCC | 1.24 $\pm$ 0.03 | 6.30 $\pm$ 0.68 | 5.06 $\pm$ 0.32 | high | 376 | 3.88 |
A2261 | 0.224 | RCC | 1.25 $\pm$ 0.03 | 6.83 $\pm$ 0.70 | 3.34 $\pm$ 1.20 | high | 209 | 3.62 |
RXJ2129 | 0.234 | RCC | 1.31 $\pm$ 0.05 | 8.07 $\pm$ 1.48 | 6.06 $\pm$ 1.33 | high | 325 | 4.21 |

Table 3: Summary of the caustic and hydrostatic masses as calculated within
the hydrostatic $R_{500}$ for each cluster. Column 1 is the cluster name;
column 2 is the redshift of the cluster; column 3 is the dynamical state of
the cluster; column 4 is the mean $R_{500}$; column 5 is the hydrostatic mass
at $R_{500}$; column 6 is the caustic mass at the hydrostatic $R_{500}$;
column 7 gives the Ngal sub-sample to which each cluster belongs (see §4.2);
column 8 is the number of galaxies of each cluster within the caustics (Serra
& Diaferio, 2013), as given in Rines et al. (2013); column 9 gives the nhtot
value (Willingale et al., 2013) that we used for each cluster (see §3.1). All
errors are $1\sigma$ errors.

The caustic and hydrostatic mass profiles for each cluster are shown in Figure
10777For A1835, the hydrostatic mass profile decreases unphysically at around
$R_{500}$. This is believed to be due to the hydrostatic assumption becoming
worse at large radii and is discussed in detail in Bonamente et al. (2013)..
These profiles were used to calculate the $M_{500,X}$ and $M_{500,C}$, using
the $R_{500}$ value calculated from the hydrostatic mass profile for each
cluster (unless stated otherwise, $R_{500}$ always refers to that derived from
the hydrostatic mass profile)888We note that this is the maximum radius that
can usefully be used for comparisons with X-ray data. The $R_{500}$ radius is
not optimal for caustic mass profiles, as the caustic method performs better
at larger radii, though the bias at this radius can be estimated from
simulations (Serra et al., 2011) which makes a meaningful comparison
possible.. The hydrostatic and caustic masses are reported in Table 3 and are
compared in Figure 2.

Figure 2: The hydrostatic versus the caustic masses are plotted for the full
sample of clusters (with masses calculated at the hydrostatic $R_{500}$). NRCC
clusters are red circles and RCC clusters are blue triangles. The solid black
line shows the best fitting model to the full data, with the shaded region
indicating the $1\sigma$ uncertainty. The 1:1 line is also plotted as a dashed
black line.

The temperature profiles required for the hydrostatic mass estimates were
measured out to radii close to, or beyond, $R_{500}$ for the majority of
clusters (see Figure 10). For the mass profiles at radii greater than the
extent of the temperature profile, we extrapolated the best fitting 3D
temperature profile. As a consequence, our hydrostatic mass profiles beyond
the extent of the measured temperature profiles are less robust. The median
radius out to which the temperature profiles were measured is 0.95 $R_{500}$,
and the range of radii is 0.51 - 1.62 $R_{500}$.

The use of the hydrostatic estimate of $R_{500}$ to determine both sets of
masses introduces a covariance between the hydrostatic and caustic masses that
is not included in our model. For this reason, we verified that using a fixed
aperture of radius 1 Mpc for the mass measurements made a negligible
difference to our results (as found in Maughan et al., 2016).

The $M_{X}/M_{C}$ profile of each cluster (calculated as
$\hat{\mu_{X}}-\hat{\mu_{C}}$) is shown in Figure 3. We also plot the mean
$M_{X}/M_{C}$ profile (calculated as the mean bias $\kappa$, see equation 7).
The $M_{X}/M_{C}$ profile is consistent with the bias being constant with
radius. At smaller radii the $M_{X}/M_{C}$ value decreases, but this is likely
due to the caustic masses being overestimated at small radii (see e.g. Figure
12 Serra et al., 2011).

Figure 3: The $M_{X}/M_{C}$ profiles for all 44 clusters in our sample, scaled
in radius by the hydrostatic $R_{500}$. NRCC clusters are solid red lines and
RCC clusters are blue dot dashed lines. The 1:1 line is also plotted as a thin
dashed black line. The thick dashed black line shows the average $M_{X}/M_{C}$
ratio, and the shaded region shows the 1$\sigma$ uncertainty.

The $M_{500,X}/M_{500,C}$ ratios at $R_{500}$ for the individual clusters is
shown in Figure 4, and the average ratios are summarised in Table 4. At
$R_{500}$ the caustic and hydrostatic masses generally do not agree well. For
the full sample, the hydrostatic mass on average $\sim$ 30% higher than the
caustic mass (significant at $\approx 3\sigma$). Similar results are found for
both the NRCC and RCC clusters.

Figure 4: We show the $M_{500,X}/M_{500,C}$ ratio for all 44 clusters in our
sample. NRCC clusters are red circles and RCC clusters are blue triangles. The
solid black line and shaded region show the average $M_{500,X}/M_{500,C}$
ratio at the hydrostatic $R_{500}$. 1$\sigma$ errors are shown. The clusters
are ranked by Ngal with the highest at the top, and the horizontal lines
separate the clusters into the low, mid and high Ngal subsets (see §4.2).

This $M_{500,X}/M_{500,C}$ value is consistent with the our earlier findings
(Maughan et al., 2016), but rules out $M_{500,X}/M_{500,C}=1$ at higher
significance, and remains counter-intuitive. The hydrostatic mass is expected
to be underestimated compared to the true mass (e.g. Nelson et al., 2014;
Eckert et al., 2019) while the caustic mass expected to be overestimated
compared to the true mass at $R_{500}$ (e.g. Serra et al., 2011); thus we
expect the $M_{500,X}/M_{500,C}$ ratio to be less than 1.

Aperture | Subset | $N_{C}$ | $\kappa$ | $M_{500,X}/M_{500,C}$ |  $\delta$ (%)
---|---|---|---|---|---
$R_{500}$ | All | 44 | 0.123$\pm$0.031 | $1.33^{+0.10}_{-0.09}$ | $39^{+06}_{-06}$
$R_{500}$ | RCC | 10 | 0.126$\pm$0.102 | $1.32^{+0.33}_{-0.24}$ | $51^{+25}_{-16}$
$R_{500}$ | NRCC | 34 | 0.125$\pm$0.035 | $1.33^{+0.11}_{-0.10}$ | $41^{+07}_{-06}$

Table 4: Summary of the results from our mass comparison analysis. The first
column gives the aperture within which the caustic and hydrostatic masses were
calculated. For this column, $R_{500}$ refers to the hydrostatic $R_{500}$.
The second column gives the subset for which the summary statistics are given,
and the third column $N_{C}$ gives the number of clusters in that subset.
Columns 4, 5 and 6 give the summary statistics. The mean bias is given in
column 4, the median ratio of masses is given in column 5, and the intrinsic
scatter between the caustic and hydrostatic mass is given in column 6. See
§3.3 for details on how these values were derived.

###  4.2 Dependency on Ngal

Figure 2 suggests that the high value of the $M_{500,X}/M_{500,C}$ ratio for
the full sample may be driven by clusters with lower caustic masses (with
hydrostatic masses being overestimated and/or caustic masses being
underestimated). However, simulations indicate that dynamical mass estimators
in general (including a different implementation of the caustic method to that
used here) tend to underestimate masses when the number of galaxies used is
low (Wojtak et al., 2018). This is a confounding factor since the lower mass
clusters tend to have lower Ngal.

In order to investigate the role of Ngal in our results, the sample was split
on Ngal into three approximately equal subsets of 15, 15 and 14 clusters (see
the Ngal sub-sample column in Table 3). These low, mid and high Ngal sub-
samples have $63\leq\text{Ngal}\leq 129$ (mean 93), $131\leq\text{Ngal}\leq
208$ (mean 174), and $210\leq\text{Ngal}\leq 461$ (mean 281) respectively.
Figure 5 shows the comparison between hydrostatic and caustic masses with the
different Ngal subsets indicated. This illustrates the trend that lower mass
clusters tend to have lower Ngal. It is also clear that, at a given
hydrostatic mass, the clusters with the lowest Ngal are most discrepant in
their caustic mass.

Figure 5: The hydrostatic and caustic masses measured within the hydrostatic
$R_{500}$ are plotted for the full sample. This is the same data as plotted in
Figure 2, but now showing the Ngal subsets. The high, mid and low Ngal bins
are green squares, blue triangles and red circles respectively. The 1:1 line
is also plotted as a dashed black line.

To investigate this further, the gas fraction, $f_{\text{gas}}$ (defined as
the gas mass, $m_{\text{gas}}$, within some radius divided by the total mass
within the same radius), was found to be a useful discriminator between
possible systematics in the analysis. This is because $m_{\text{gas}}$ can be
measured accurately from X-ray observations (e.g. Nagai et al., 2007), and
there are robust theoretical predictions for the range of $f_{\text{gas}}$
expected. We thus computed $f_{\text{gas}}$ at a radius of 1 Mpc (chosen to
avoid dependency of the radius on any mass estimation method) with either the
hydrostatic and caustic masses in the denominator. The results are plotted in
Figure 6. The plot also indicates the approximate range of $f_{\text{gas}}$
($\sim$ 0.1 - 0.15) expected at this radius (e.g. Eckert et al., 2016).

Figure 6: Left: $f_{\text{gas}}$ measured at 1 Mpc using caustic mass in the
denominator is plotted against Ngal. Right: The same plot, but with
$f_{\text{gas}}$ calculated with hydrostatic mass in the denominator. In both
panels, the black dashed lines indicate $f_{\text{gas}}=0.10$ and
$f_{\text{gas}}=0.15$ for reference, encompassing the approximate expected
range of $f_{\text{gas}}$.

When the caustic mass is used as the denominator in $f_{\text{gas}}$, there is
a strong indication of a trend with Ngal, with unphysically high values of
$f_{\text{gas}}$ for the clusters with lower values of Ngal. The mean and
standard error of $f_{\text{gas}}$ was 0.23$\pm$0.03, 0.15$\pm$0.01 and
0.15$\pm$0.01 for the low, mid and high Ngal sub-samples respectively.

In contrast, when the hydrostatic mass is used to compute $f_{\text{gas}}$,
the trend with Ngal disappears, and the average values of 0.12$\pm$0.01,
0.11$\pm$0.01, 0.12$\pm$0.01, for the low, mid and high Ngal sub-samples
respectively are consistent with the expected range.

These $f_{\text{gas}}$ measurements suggest that Ngal is responsible for the
high values of $M_{500,X}/M_{500,C}$ due to an underestimate of $M_{C}$.
However, due to the correlation between Ngal and mass, the possibility remains
that a trend with mass is more fundamental. During the completion of the work
presented here, additional caustic mass measurements became available in the
HeCS-Omnibus catalogue (Sohn et al., 2020) that allow the effects of mass and
Ngal to be separated. The new catalogue includes new caustic mass estimates
for clusters in our sample, using the same caustic method, but based (in most
cases) on a larger number of galaxies per cluster.

When comparing the earlier and newer caustic mass measurements, it was found
that for well-sampled clusters, increasing the sampling had little effect on
the mass. On the contrary, for more poorly sampled clusters, increasing Ngal
systematically led to an increased mass estimate. This is illustrated in Fig.
7, which shows the ratio of the caustic masses (measured at the hydrostatic
$R_{500}$) from Sohn et al. (2020) to those from both Rines et al. (2013) (as
used in our work), plotted against the difference in Ngal between the two
analyses.

The clusters were divided into subsets based on the minimum Ngal used in the
two caustic calculations. For clusters where at least 200 galaxies were used
in both analyses (green squares in Fig. 7), increasing the sampling has a weak
or null effect on the mass estimate. The green line is a straight line fit to
the data in this parameter space (i.e.
$\log(M_{\text{Sohn}}/M_{\text{Rines}})$ versus
$N_{\text{Sohn}}-N_{\text{Rines}}$) using BCES y on x regression (Akritas &
Bershady, 1996). This is a simplistic model and is not intended to fully
describe the systematic effects in the analyses, but is an adequate
description of the data presented here. The slope of the line is $0.001\pm
0.001$.

The impact of increasing Ngal for less well-sampled clusters is seen in the
blue triangles in Fig. 7. These are clusters for which Ngal is $<120$ in at
least one of the two analyses. It is clear in the figure that increasing the
sampling leads to systematically higher caustic mass estimates. The best
fitting line to this subset has a slope of $0.008\pm 0.002$. The subsets were
defined to split the sample into three equal groups, and the intermediate
subset falls between the two outer cases and are omitted from the figure for
clarity.

Figure 7: The ratio of the caustic masses (measured at the hydrostatic
$R_{500}$) from Sohn et al. (2020) to those from Rines et al. (2013) are
plotted as a function of the difference in Ngal between the analyses. The
clusters are split into subsets containing those where the minimum Ngal used
in the two analyses was at least 200 (green squares) and those where the
minimum Ngal was less then 120 (red circles). The lines show the best fits to
each subset.

We further test the role of Ngal as follows. We undersample the galaxy
catalogues of the 14 clusters in our high-Ngal sub-sample, by progressively
removing the faintest galaxies. When Ngal drops by $\sim 50\%$, falling in the
range $\sim 100-200$, the caustic mass within $R_{500}$ decreases by $\sim
15\%$, on average. This average mass underestimate is consistent with the
analysis of Serra et al (2011) who found that with $\text{Ngal}\sim 100$ the
escape velocity at $R_{500}$ is underestimated by $\sim 5-10\%$, whereas
$\text{Ngal}\gtrsim 200$ is required to return unbiased escape velocity
estimates (see also Halenka et al., 2022). The lower Ngal of the undersampled
clusters in our high-Ngal sample would move these clusters in our mid-Ngal
sample: the $\sim 15\%$ mass underestimate we find in this test is thus also
consistent with the larger ratio $M_{500,X}/M_{500,C}$ we find for our mid-
Ngal sample, as illustrated in §4.3 below (Table 5).

Based on these results, we conclude that a systematic underestimate of the
caustic mass at $R_{500}$ for low values of Ngal is responsible for the high
apparent $M_{500,X}/M_{500,C}$ ratio found at low masses, and that the high
Ngal sub-sample ($\text{Ngal}\geq 210$) is not significantly affected by this
systematic. We thus repeat our analysis on the separate Ngal subsets in the
following section, and will draw our final conclusions based on the results
for the high Ngal subset alone.

###  4.3 Final results

The analysis described in §4.1 was repeated for each of the Ngal subsets
separately, and the results are summarised in Table 5. Figure 8 shows the
profiles of $M_{X}/M_{C}$ for the different Ngal subsets. The separation
between the subsets is clear, with no radial dependency.

Aperture | Subset | $N_{C}$ | $\kappa$ | $M_{500,X}/M_{500,C}$ |  $\delta$ (%)
---|---|---|---|---|---
$\mathbf{R_{500}}$ | High | 14 | $\mathbf{0.050\pm 0.043}$ | $\mathbf{1.12^{+0.11}_{-0.10}}$ | $\mathbf{25^{+12}_{-08}}$
$R_{500}$ | Mid | 15 | 0.109$\pm$0.053 | $1.28^{+0.16}_{-0.14}$ | $41^{+12}_{-09}$
$R_{500}$ | Low | 15 | 0.230$\pm$0.082 | $1.69^{+0.33}_{-0.27}$ | $59^{+18}_{-13}$

Table 5: Summary of the results from our mass comparison analysis when we
define subsets of our clusters based on the Ngal value. The columns are the
same as in Table 4. The line in bold represents the most reliable results,
based on the high Ngal subset. Figure 8: The average $M_{X}/M_{C}$ profiles
for the high Ngal, mid Ngal and low Ngal clusters in green lines, blue lines,
and red dot-dashed lines respectively. The 1:1 line is also plotted as a thin
black dashed line. The thick dashed black line shows the average $M_{X}/M_{C}$
profile for all clusters. The profiles are scaled in radius by the hydrostatic
$R_{500}$, and the shaded regions show the 1$\sigma$ uncertainties.

As expected based on the preceding analysis, the results in Table 5 show a
trend towards lower $M_{X}$/$M_{C}$ ratios with increasing Ngal. The most
reliable results are those based on the sample of 14 clusters with the highest
values of Ngal, and these are highlighted in boldface in the table. For this
subset, the average ratio of hydrostatic to caustic masses is
$M_{500,X}/M_{500,C}=1.12^{+0.11}_{-0.10}$ with an intrinsic scatter between
the masses of $\delta=(25^{+12}_{-08})\%$.

The comparison between the hydrostatic and caustic masses for the high Ngal
subset is shown in Figure 9, with the RCC/NRCC classification of the clusters
indicated. There does not appear to be a significant trend with dynamical
state but given the reduced size of this high Ngal subset, a more detailed
analysis would not be useful.

Figure 9: We show the hydrostatic masses versus the caustic masses for the 14
clusters in our sample with the highest Ngal values, both calculated at the
hydrostatic $R_{500}$. We show RCC clusters as blue triangles and NRCC
clusters as red circles. The solid black line shows the best fitting model,
with the shaded region indicating the $1\sigma$ uncertainty. The 1:1 line is
also plotted as a dashed black line.

Recall that 14 of the 16 clusters used in our earlier analysis in Maughan et
al. (2016) are in common with the current analysis. Of these, six are in our
high Ngal subset, seven are in the mid Ngal sub-sample, and one is in the low
Ngal sub-sample. This mix of mid- and high-Ngal clusters naturally explains
why Maughan et al. (2016) found an $M_{X}$/$M_{C}$ ratio that was slightly
higher ($M_{500,X}/M_{500,C}=1.20\pm 0.12$) than our final result for the high
Ngal subset.

##  5 Discussion

In the following section we consider the possible systematics that could have
impacted our results. We explore in more detail our conclusion that the
apparent increase in $M_{500,X}/M_{500,C}$ at low masses is driven by Ngal
rather than by mass (or some other factor). Finally, we compare our results
with other related analyses, and discuss the constraints that can be derived
on the level of hydrostatic bias in massive clusters.

###  5.1 The impact of Ngal

Our investigation of the comparison between hydrostatic and caustic masses
suggested that the caustic masses were underestimated when fewer galaxies were
used. While a general result of dynamical masses being biased low when based
on smaller numbers of galaxies was shown in Wojtak et al. (2018), the caustic
method examined in that work differed from the implementation used here.

The dependence on Ngal of the caustic mass was investigated for our
implementation of the caustic method by Serra et al. (2011). They found that
the true escape velocity profile will be underestimated by $\sim$5-10% at
$R_{500}$ (Figure 20, Serra et al., 2011) for clusters with Ngal $\sim$ 100
(compared to clusters with a higher Ngal for which the true escape velocity is
not underestimated). This would lead to the caustic mass being underestimated
by $\sim$10-20% at $R_{500}$, were it not for the assumption of a constant
$\mathcal{F}_{\beta}$ (see §3.2), which leads to a $\sim$10-15% overestimate
of the true mass at $R_{500}$ for clusters with a higher Ngal (Figure 12,
Serra et al., 2011). Thus, the simulations suggest that these two effects
approximately cancel out at our comparison radius of $R_{500}$, such that a
large underestimation of the cluster mass using the caustic method for low
Ngal clusters (like that implied by our results) is not expected.

As discussed in §4.2, we consider the caustic masses for the high Ngal subset
to be unaffected by this apparent bias, and base our main conclusions on this
subset. As a test of the robustness of our results, we repeated the
measurement of $M_{500,X}/M_{500,C}$ using the HeCS-Omnibus caustic data of
Sohn et al. (2020) in place of the original HeCS caustic data of Rines et al.
(2013) that was used for our main analysis. We selected the 14 clusters in our
sample with the highest Ngal in HeCS-Omnibus ($\text{Ngal}>274$ with a median
of $413$). For these best-sampled clusters, using the HeCS-Omnibus caustic
masses gives an average hydrostatic to caustic mass ratio of
$M_{500,X}/M_{500,C}=1.10$ at the hydrostatic $R_{500}$. This agrees very well
with the $M_{500,X}/M_{500,C}=1.12$ we found for the 14 clusters in our high
Ngal subset when using the Rines et al. (2013) data (for which
$\text{Ngal}>210$ with a median of 234). This supports our conclusion that our
high Ngal subset is unaffected by systematics due to low numbers of galaxies
(i.e. $\text{Ngal}\,\hbox to0.0pt{\raise 1.72218pt\hbox{$>$}\hss}{\lower
2.58334pt\hbox{$\sim$}}\,200$ is sufficient to avoid biases in the caustic
mass at $R_{500}$). Indeed, when we apply the caustic technique to the cluster
catalogues in the high-Ngal sample where we undersampled Ngal such that
$\text{Ngal}\lesssim 200$, the caustic mass within $R_{500}$ is underestimated
by $\sim 15\%$, on average, suggesting that the increase of the ratio
$M_{500,X}/M_{500,C}$ from $\sim 1.12$ of the high-Ngal sample to $\sim 1.28$
of the mid-Ngal sample is mostly driven by the lower number of the caustic
members.

Poorer sampling of the caustics is likely to be responsible (at least in part)
for some of the outliers seen in the comparisons between the hydrostatic and
caustic masses (e.g in Figure 4). When restricted to the high Ngal subset,
there are no significant outliers in the comparison between hydrostatic and
caustic masses, given the $\sim 30\%$ intrinsic scatter between the mass
estimators (e.g. Figure 9).

###  5.2 Cluster centres and apertures

In the caustic analysis, the cluster centre is located as part of the process
(Rines et al., 2013), while the centre is fixed at the X-ray centroid for the
hydrostatic analysis. This means that the profiles are not generally centred
at the same location. The separation between the centres used for the X-ray
and caustic analyses were generally small, and showed no trend with Ngal or
with $M_{500,X}/M_{500,C}$ (the median separation was $\approx
130\mbox{$\text{\leavevmode\nobreak\ kpc}$}$ in each of the Ngal subsets). We
thus conclude that the increasing disagreement between hydrostatic and caustic
masses at low Ngal is not being driven by miscentring.

Serra et al. (2011) investigated the impact of miscentering of the caustic
analyses on the mass determination in their simulations. At $R_{500}$, the
caustic mass estimate was increased by $\approx 10\%$ when the profile was
forced to use the true centre. This suggests that our measurements of
$M_{500,X}/M_{500,C}$ could be overestimated by $\approx 0.1$ due to
miscentring of the profiles.

When the separation between the centres used for the caustic and hydrostatic
analyses is small, then the use of a common aperture size (i.e. the
hydrostatic $R_{\text{500}}$) allows the mass estimators to be compared
without adding scatter from different aperture definitions. However, given
that the centres are not identical, it is also helpful to compare the masses
when they are derived fully independently (i.e. comparing the caustic mass
within the caustic $R_{\text{500}}$ with the hydrostatic mass within the
hydrostatic $R_{\text{500}}$).

We repeated our analysis using these independent definitions of
$R_{\text{500}}$. For the full sample of 44 clusters, we found that mass ratio
increased from $M_{500,X}/M_{500,C}=1.33^{+0.10}_{-0.09}$ (using the
hydrostatic $R_{\text{500}}$) to $M_{500,X}/M_{500,C}=1.52^{+0.16}_{-0.14}$
(using independent $R_{\text{500}}$). In the case of the high Ngal sub-sample,
the original $M_{500,X}/M_{500,C}=1.12^{+0.11}_{-0.10}$ (hydrostatic
$R_{\text{500}}$) increased slightly to
$M_{500,X}/M_{500,C}=1.16^{+0.16}_{-0.13}$ (independent $R_{\text{500}}$).

These results make sense. When the masses agree well (as in the high Ngal sub-
sample) then the $R_{\text{500}}$ definitions also agree so the results do not
change. If the masses are more discrepant (as in the full sample) then using
the independent $R_{\text{500}}$ will tend to amplify the difference.

As might be expected, the scatter between the mass estimators is somewhat
larger when the independent $R_{\text{500}}$ are used. The scatter increases
from $(39\pm 6)\%$ to $(55^{+9}_{-8})\%$ for the full sample, and from
$(25^{+12}_{-08})\%$ to $(35^{+15}_{-11})\%$ for the high Ngal sub-sample.

Based on this analysis, we conclude that the use of the hydrostatic
$R_{\text{500}}$ for both mass determinations does not have a strong effect on
our results.

###  5.3 Impact of selection effects

In our work we do not account for the impact of selection effects in our
model. The sample is selected on the basis of X-ray flux, so covariance
between X-ray flux and $M_{X}$ could have an impact on the inferred
$M_{500,X}/M_{500,C}$ ratio. Specifically, if there were a strong positive
covariance between X-ray luminosity, $L_{X}$, and hydrostatic mass, $M_{X}$,
given the true mass, then X-ray selection would select clusters that were both
more luminous than average and had higher hydrostatic masses than average,
given their true mass. The size of this effect would depend both on the
scatter in the relevant relations and the degree of covariance between $M_{X}$
and $L_{X}$.

We assessed the possible impact of this selection effect by simulating cluster
populations with different degrees of covariance between $L_{X}$ and $M_{X}$,
imposing the sample selection function, and then applying our model (described
in §3.3) to the simulated population to determine the impact of the sample
selection on the recovered $M_{500,X}/M_{500,C}$ ratio.

In more detail, we sampled a large number of points from a mass function where
each point consists of a true mass, $M_{\text{true}}$, and a redshift. We use
the mass - luminosity relation from Mantz et al. (2010) to assign the true
luminosity, $L_{\text{true}}$ to each cluster. We then assume a scatter
between $L_{\text{true}}$ and $L_{X}$ of 30%, a scatter between
$M_{\text{true}}$ and $M_{X}$ of 10% and a scatter between $M_{\text{true}}$
and $M_{C}$ of 30%. We also require a correlation coefficient, $\rho_{LM}$,
between $L_{X}$ and $M_{X}$ to create the covariance matrix for the multi-
dimensional Gaussian from which we sample $L_{X}$ and $M_{X}$ from
$L_{\text{true}}$ and $M_{\text{true}}$. It is the strength of this
correlation coefficient that dictates how much a selection on X-ray flux might
bias an inference based on hydrostatic masses, and we investigate the impact
of different assumed values below. Finally, we apply our X-ray selection to
the fluxes derived from the $L_{\text{true}}$ and redshift values, and use our
model to measure the $M_{500,X}/M_{500,C}$ ratio for the resulting simulated
sample. We do not set any hydrostatic or caustic bias in the simulation, so in
the absence of selection biases, we would measure $M_{500,X}/M_{500,C}=1$.

To our knowledge, covariances between $L_{X}$ and hydrostatic mass have not
been measured in the literature, but the covariances between other ICM
properties have been investigated in a few studies (e.g. Maughan, 2014; Mantz
et al., 2016; Andreon et al., 2017b; Farahi et al., 2018). For example, for
the covariance between $L_{X}$ and $m_{\text{gas}}$, Mantz et al. (2016) found
a correlation coefficient of 0.43, while Farahi et al. (2018) found 0.76.
Meanwhile, for the covariance between $L_{X}$ and $T$, Mantz et al. (2016)
found a correlation coefficient of 0.53, while Farahi et al. (2018) found
0.49. In both cases, the values obtained by Mantz et al. (2016) are the more
relevant here since they make use of a soft-band, core-included X-ray
luminosity, which matches the selection method used for our sample (Farahi et
al., 2018, made use of bolometric luminosities with the core regions removed,
but we include their results here for comparison). We therefore conclude that
a correlation coefficient of $\rho_{LM}=0.5$ is a reasonable upper limit on
the strength of the covariance between $L_{X}$ and $M_{X}$ and use this as a
reference value for our simulations. We also performed the simulations with
with $\rho_{LM}=1$ as a maximally pessimistic case.

We find that for $\rho_{LM}=0.5$, the recovered $M_{500,X}/M_{500,C}$ value is
1.03$\pm$0.01 and for $\rho_{LM}=1$ we measured $M_{500,X}/M_{500,C}=1.07\pm
0.01$ (for a sample of 500 simulated clusters).

We also checked for a mass dependence of the selection bias by splitting the
simulated clusters into low (log($M_{X}$)¡14.5), mid
(14.5¡log($M_{X}$)¡14.75), and high (log($M_{X}$)¿14.75) mass sub-samples
(each containing 500 simulated clusters). The measured $M_{500,X}/M_{500,C}$
ratio was largest for the low-mass sub-sample, due to a larger proportion of
the clusters being close to the flux limit. For the low-mass sub-sample,
assuming $\rho_{LM}=0.5$ we found $M_{X}/M_{X}=1.06\pm 0.01$, increasing to
$1.15\pm 0.01$ for $\rho_{LM}=1$. For the high mass sub-sample (which is
similar in mass to our high Ngal subset), the measured $M_{500,X}/M_{500,C}$
ratios dropped to 1.02$\pm$0.01 and 1.03$\pm$0.01 for $\rho_{LM}=0.5$ and
$\rho_{LM}=1$ respectively.

We conclude that, while neglecting selection effects tends to bias the
$M_{500,X}/M_{500,C}$ ratio high, the impact is not significant for the mass
range of the high Ngal subset, and that selection biases alone cannot explain
the high values of $M_{500,X}/M_{500,C}$ that we find in the lower mass, lower
Ngal clusters.

###  5.4 Impact of X-ray temperature calibration

It is established that there are systematic differences in cluster
temperatures measured by _Chandra_ and _XMM-Newton_ , temperatures measured
with _Chandra_ being systematically hotter (Schellenberger et al., 2015). This
has been found to have a corresponding systematic impact on the derived
hydrostatic masses Mahdavi et al., 2013; Schellenberger et al., 2015). We
note, however, that Martino et al. (2014) found no systematic differences in
hydrostatic masses for a sample of 50 clusters with both _XMM-Newton_ and
_Chandra_ data.

Schellenberger et al. (2015) found that the hydrostatic masses measured from
Chandra data were $14\pm 2\%$ higher than those from XMM-Newton. They found no
significant trend with cluster mass, suggesting that the apparent trend of
$M_{500,X}/M_{500,C}$ with mass or Ngal that we find is not driven by this
systematic. If our hydrostatic masses were systematically overestimated by
$\approx 15\%$, this would have a direct effect of reducing our derived
$M_{500,X}/M_{500,C}$ ratios by $0.15$.

###  5.5 Summary of systematics

We have identified a number of possible systematics that could impact our
comparison between caustic and hydrostatic masses. In the case of the caustic
masses, poorer sampling of the caustics was found to result in an
underestimate of the caustic masses for $\text{Ngal}\la 200$, producing an
apparent trend of $M_{500,X}/M_{500,C}$ with mass. For the better-sampled
clusters, this systematic appears to be absent, but the assumption of constant
$\mathcal{F}_{\beta}$ is expected to lead to an overestimate of the caustic
mass by $10-15\%$ at $R_{500}$ (i.e. biasing $M_{500,X}/M_{500,C}$ low by up
to $0.15$). The miscentering of the caustic profiles leads to an estimated
underestimate of the caustic masses of $\approx 10\%$ at $R_{500}$ (biasing
$M_{500,X}/M_{500,C}$ high by $\approx 0.1$).

We find that the use of X-ray flux to select the sample combined with a
possible covariance between X-ray luminosity and hydrostatic mass could bias
the hydrostatic masses to be high. However, the effect is estimated at a few
percent for the more massive clusters comprising our high Ngal subset, and the
trend with mass is not sufficient to explain the strength of the trend of
$M_{500,X}/M_{500,C}$ with mass or Ngal. Finally, systematics on the
calibration of Chandra could lead to the hydrostatic masses being
overestimated by $\approx 15\%$ (biasing $M_{500,X}/M_{500,C}$ high by about
$0.15$).

###  5.6 Comparison with other work

Direct comparisons between hydrostatic and caustic masses for samples of
clusters are scarce in the literature. Andreon et al. (2017a) compared the
mean $f_{gas}$ for a sample of 34 clusters with caustic masses, with the mean
value from separate samples with hydrostatic masses from Chandra data. They
inferred that caustic masses were slightly higher than, but consistent with,
hydrostatic masses at $R_{500}$. This is consistent with the result we find
for the high Ngal subset. However, the clusters with caustic masses used in
Andreon et al. (2017a) cover a range in mass and Ngal that is comparable with
our low and mid Ngal subsets, and are in tension (at the $\approx 2\sigma$
level) with those subsets for which we find the caustic masses to be lower
than hydrostatic masses (although we consider our results for those subsets to
be less reliable due to the poorer sampling of the caustics).

Foëx et al. (2017) compared hydrostatic and caustic masses for a sample of 13
clusters with masses, redshifts and Ngal similar to our high Ngal subset. They
found a mean $M_{C}/M_{X}=1.32\pm 0.18$ at $R_{500}$ (we note that this is the
inverse of our mass ratio), and hence caustic masses were higher than
hydrostatic masses by $\sim 30\%$ (although the ratio dropped to unity when
some substructures were removed in the dynamical analysis). While this
contrasts with our results, the difference is not strongly significant
($\approx 2\sigma$). Furthermore, the Foëx et al. (2017) analysis was based on
hydrostatic masses from XMM-Newton; adjusting these masses to match the
Chandra calibration would reduce the tension with our result to less than
$1\sigma$.

Ettori et al. (2019) compared hydrostatic masses (from XMM-Newton) and caustic
masses at $R_{200}$ for 6 clusters, finding that the caustic masses were lower
than the hydrostatic masses by about $30\%$ on average. This is in the same
sense as the difference we find for our sample, but the clusters include some
with lower Ngal than our high Ngal subset, so a direct comparison is not
possible.

###  5.7 Implications for the hydrostatic bias

Conventionally, the hydrostatic bias is defined as $b=1-M_{X}/M$, where
$M_{X}$ is the hydrostatic mass and $M$ is the true mass. Our best estimate of
the ratio of hydrostatic to caustic masses, with statistical error is
$M_{500,X}/M_{500,C}=1.12^{+0.11}_{-0.10}$, for the high Ngal subset. Under
the strong assumption that the caustic masses are unbiased, we can write
$b=1-M_{500,X}/M_{500,C}$, in which case our results exclude a hydrostatic
bias of more than $20\%$ at the $3\sigma$ level based on statistical
uncertainty alone. However, taking the possible systematic uncertainties on
both the caustic and hydrostatic masses into account, we cannot rule out
larger values of the hydrostatic bias.

Constraints on the hydrostatic bias from direct comparisons between
hydrostatic and WL masses currently encompass a range from approximately zero
(e.g. Gruen et al., 2014; Israel et al., 2014; Applegate et al., 2016; Smith
et al., 2016) up to around $30\%$ (e.g von der Linden et al., 2014; Donahue et
al., 2014; Hoekstra et al., 2015). Meanwhile, constraints on the level of bias
have also been estimated indirectly by comparisons of WL masses with masses
derived from scaling relations of other observables that were calibrated to
hydrostatic masses (e.g. from X-ray luminosity Sereno & Ettori (2017) or the
Sunyaev-Zel’dovich effect signals Medezinski et al. (2018); Miyatake et al.
(2019)). These also tend to favour biases of $20-30\%$.

The question of hydrostatic bias has also been addressed with simulations,
which typically find biases of $10-30\%$, but with radial and mass
dependences, and sensitivity to the method used to reconstruct the hydrostatic
masses from the simulation data (e.g. Nagai et al., 2007; Rasia et al., 2012;
Pearce et al., 2020; Barnes et al., 2021).

Simulations suggest a significant mass dependence of the hydrostatic bias,
with the bias increasing for higher mass clusters (e.g Barnes et al., 2021).
However, due to the relatively poorer sampling of the caustics in our lower
mass clusters, we are not able to test these results observationally.

Overall, there is not a clear consensus on the magnitude of the hydrostatic
bias. Our results are more consistent with those estimates from the low end of
the range and add a useful new datapoint to this still-open question, since
the caustic mass determination is subject to different systematics and
assumptions to those affecting the WL masses which are primarily used to
address this question observationally.

##  6 Summary and conclusions

We compared masses of 44 galaxy clusters determined independently from X-ray
data under the assumption of hydrostatic equilibrium, and galaxy velocities
using the caustic method. Our initial results showed significant discrepancies
between the masses which appeared to increase for lower mass clusters. We
found that this was driven by the poorer sampling of the caustics for the
lower mass clusters leading to an underestimate of the mass using the caustic
method at $R_{500}$. The caustic method is not optimised for mass
determinations at these radii, but this finding suggests that simulations have
underestimated the impact of poorer sampling at smaller radii.

Restricting our analysis to the 14 clusters with best-sampled caustics (the
high Ngal subset), we find an average ratio of the hydrostatic to caustic
masses of $M_{500,X}/M_{500,C}=1.12^{+0.11}_{-0.10}$. We find no evidence for
radial dependence of this ratio or dependence on dynamical state of the
cluster, but these conclusions are limited by the reduced sample size.

We investigated possible systematics that could affect the mass estimates for
this high Ngal subset and find that the most significant are as follows. The
assumption of constant $\mathcal{F}_{\beta}$ and possible miscentering of the
caustic profiles could bias the caustic masses high and low respectively by
$10-15\%$. The uncertainties on the absolute calibration of the X-ray
temperature measurements could bias the hydrostatic masses high by about
$15\%$ compared to those measured with XMM-Newton.

We interpret our results as favouring a hydrostatic bias that is close to
zero. However, the systematics we identified would allow for significantly
larger levels of hydrostatic bias.

###### Acknowledgements.

CL, BJM and RTD acknowledge support from the UK Science and Technology
Facilities Council (STFC) grant ST/R00700/1. BJM acknowledges further support
from STFC grant ST/V000454/1. CL also acknowledges support from an ESA
Research Fellowship. AD acknowledges partial support from the Italian Ministry
of Education, University and Research (MIUR) under the Departments of
Excellence grant L.232/2016, and from the INFN grant InDark. This research has
made use of NASA’s Astrophysics Data System Bibliographic Services.

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##  Appendix A Notes on individual clusters

In this section we detail where we differed from the analysis methods
presented in §3, or any particular points of interest for each cluster in our
sample.

When extended sources other than the cluster were detected in the images, we
attempted to determine if they were physically associated with the cluster or
were unrelated projected sources. We did this by querying the NASA
Extragalactic Database999https://ned.ipac.caltech.edu/ to see if the source
has a known counterpart.

  * •

A773 - For ObsID 5006, due to flaring we only use the first 15 ks of the
observation.

  * •

MS0906 - At 9:08:58.02, 11:01:58.36 (RA, DEC = 137.24, 11.03), there is a
large region of diffuse extended emission (which corresponds to cluster A750
(Abell et al., 1989)) which we mask with a 230′′ radius circle.

  * •

ZW2701 - At 9:53:05.85, 51:49:16.39 (148.27, 51.82) there is a region of
extended emission that we mask with a 150′′ radius circle.

  * •

A963 - For ObsID 903, a temperature of 0.28 keV was used when fitting the APEC
model to the soft background residuals. We note that the standard temperature
used for this is 0.18 keV, as it is found that this temperature generally
models these residuals well (Giles et al., 2015). The higher temperature
implies that this observation includes a hotter region of the galactic
foreground emission.

  * •

ZW3146 - For ObsID 9371, due to flaring we only use the first 34 ks of the
observation.

  * •

A1423 - For ObsID 538, due to flaring we only use the first 11 ks of the
observation.

  * •

A1553 - For ObsID 12254, due to flaring we only use the first 10 ks of the
observation.

  * •

A1682 - At 13:06:59.904, 46:31:40.65 (196.75, 46.53) there is a cluster galaxy
(Morrison et al., 2003) and at 13:07:13.47, 46:29:02.31 (196.81, 46.48) there
is another small region of extended emission. Both sources are masked by
radius 40′′ circles.

  * •

A1763 - At 13:34:52.8, 40:57:21.6 (203.72, 40.96) we use a 90′′ radius circle
to mask a region of extended emission, which is likely to be associated with a
previously known X-ray source (Evans et al., 2010).

  * •

A1930 - At 14:32:42.72, 31:33:50.4 (218.18, 31.56) there is a region of
extended emission that we mask with a circle of radius 160′′. The extended
emission is the cluster RM J143242.6+313407.1 (Rozo et al., 2015) at redshift
$z=0.137$, and is likely part of the same dark matter halo as A1930 (z =
0.1308).

  * •

A2009 - For ObsID 10438, a temperature of 0.27 keV was used when fitting the
APEC model to the soft background residuals.

  * •

A2069 - There is a separate cluster at 15:24:25.846, +30:00:16.039 (231.11,
30.00) at redshift $z=0.119$ (MaxBCG J231.10029+30.00604, Koester et al.,
2007); it is likely part of the same dark matter halo as A2069. We mask it
with a rectangle of 23′ by 14′ at an inclination angle of 35∘. There is either
a group of point sources or a filament at 15:23:38.87, +29:58:30.226 (230.91,
29.98) that we mask using an ellipse. In addition, for ObsID 4965, due to
flaring we only use the first 40 ks of the observation.

  * •

A2261 - At 17:22:12.78, 32:06:36.95 (260.55, 32.11) there is a region of
diffuse extended emission that we mask with a 80′′ radius circle. This source
is associated with a galaxy cluster (GMBCG J260.55436+32.11438) at $z=0.304$
(Hao et al., 2010), so is not associated with A2261 which is at redshift
$z=0.2242$.

  * •

RXJ1720 - For ObsID 4361, due to flaring we only use the first 15 ks of the
observation.

##  Appendix B Temperature and density profiles

In this Appendix we expand on the method used to model the temperature and
density profiles used to determine the hydrostatic masses in §3.

###  B.1 Temperature Profiles

The model used for the 3D distribution of the temperature (Vikhlinin et al.,
2006) is sufficient to describe a temperature decline (if present) in the
central core region of the cluster in addition to the rest of the cluster
temperature profile outside the central cooling region of a cluster. The part
outside the central cooling region is modelled by

$t(r)=-\frac{(r/r_{t})^{-a}}{\left[1+(r/r_{t})^{b}\right]^{c/b}}\quad,$ (8)

where $r$ is the radius of the cluster and all other symbols are free
parameters: $a$ models the slope at small radii, $c$ models the slope at large
radii, and $b$ models the width of the transition between these slopes. This
transition occurs at a radius $r_{t}$. The decrease in temperature found in
the central cooling region of many clusters can be modelled as

$t_{cool}(r)=-\frac{(x+T_{min}/T_{0})}{x+1}\quad,\qquad
x=\left(\frac{r}{r_{cool}}\right)^{a_{cool}},$ (9)

where $r$ is cluster radius and all other symbols are free parameters: $T_{0}$
is the normalisation, $T_{min}$ is the minimum temperature, $r_{cool}$ models
radius of the central cooling region and $a_{cool}$ models the slope of the
cool region which extends to a radius $r_{cool}$.

The 3D temperature profile model is a product of equations 8 and 9

$T_{3D}(r)=T_{0}\,t_{cool}(r)t(r)\quad.$ (10)

###  B.2 ICM density profiles

The model used to describe the 3D distribution of the density (Vikhlinin et
al., 2006) is given by:

$n_{\text{e}}n_{\text{p}}=n_{0}^{2}\frac{(r/r_{c})^{-\alpha}}{(1+r^{2}/r_{c}^{2})^{3\beta-\alpha/2}}\frac{1}{(1+r^{\gamma}/r_{s})^{\epsilon/\gamma}}+\frac{n_{02}^{2}}{(1+r^{2}/r_{c2}^{2})^{3\beta_{2}}},$
(11)

with $r$ being the cluster radius and the nine free parameters able to model
both the central and outer parts of the density profile sufficiently. We fix
$\gamma=3$ as in Vikhlinin et al. (2006), and all the other parameters are
free. $n_{\text{e}}$ and $n_{\text{p}}$ are the electron and proton density
respectively, and $r$ is the distance from the centre of the cluster. The
first term in the equation is a modification of the $\beta$-model (Cavaliere &
Fusco-Femiano, 1978) that allows for a power-law type cusp, with slope
$\alpha$ at small radii, which is expected in the centres of clusters that are
dynamically relaxed. The second term models the change of the slope by
$\epsilon$ near the radius $r_{s}$, and the width of this transition region is
controlled by $\gamma$. The final term is another $\beta$-model component,
which increases the freedom of the overall model near the cluster centre.

###  B.3 Projection of 3D profiles

The 3D temperature and density profiles must be projected before they can be
compared with the observed profiles. For this projection we assume spherical
symmetry (see e.g. Ettori et al., 2013, for the geometry of this projection).
We note that whenever we use an abundance profile in our analysis, we use the
measured projected abundance profile (we lack the very high quality data
needed to deproject abundance profiles), and at large radii set the abundance
to 0.3 $Z_{\odot}$, in line with observed flattening of abundance profiles at
large radii (Leccardi & Molendi, 2008).

The projected temperature profile was computed using the method described in
Vikhlinin et al. (2006). Briefly, the temperature that would be measured when
fitting a single temperature thermal plasma model to the projected emission is
computed by weighting the different temperature components of the 3D model
along the line of sight by their emissivity. The emissivity is derived from
the projected abundance and 3D density model, and the weighting algorithm of
Vikhlinin (2006) was used.

The 3D density profile is projected to produce a model surface brightness
profile. The count rate from each volume element along the line of sight is
computed assuming an APEC plasma emission model with the normalisation set by
the density in the volume element, and the temperature given by the 3D
temperature profile model. This conversion from 3D density to projected
surface brightness uses the appropriate instrument response for the detector
location.

In the fitting process, the projection of the temperature and density profiles
are performed simultaneously for a given set of model parameters to produce
projected temperature and surface brightness profiles that are compared with
the data. This is an improvement over the method used in Maughan et al.
(2016), where a normalised average temperature profile was assumed when
converting from density to emissivity.

##  Appendix C Mass profile plots

Figure 10 shows the X-ray hydrostatic and caustic mass profiles for the 44
clusters in our sample.

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 10: The caustic and hydrostatic mass profiles for each cluster as black
solid and green dashed lines respectively in the upper panels of each plot.
The ratio of the hydrostatic to caustic mass, $M_{X}/M_{C}$, are shown in the
lower panels of each plot. 1$\sigma$ uncertainties are shown by the shaded
regions. The vertical black line is at the value of $R_{500}$ as calculated
from the hydrostatic mass profile; the solid red vertical line is at the outer
radius of the measured temperature profile (note that hydrostatic masses
beyond this radius are based on extrapolation).

(i)  (j)

(k)  (l)

(m)  (n)

(o)  (p)

Figure 10: _\- continued_

(q)  (r)

(s)  (t)

(u)  (v)

(w)  (x)

Figure 10: _\- continued_

(y)  (z)

(aa)  (ab)

(ac)  (ad)

(ae)  (af)

Figure 10: _\- continued_

(ag)  (ah)

(ai)  (aj)

(ak)  (al)

(am)  (an)

Figure 10: _\- continued_

(ao)  (ap)

(aq)  (ar)

Figure 10: _\- continued_