LoCuSS: Calibrating Mass-Observable Scaling Relations for Cluster Cosmology with Subaru Weak Lensing Observations

Abstract

We present a joint weak-lensing/X-ray study of galaxy cluster mass-observable scaling relations, motivated by the critical importance of accurate calibration of mass proxies for future X-ray missions, including eROSITA. We use a sample of 12 clusters at z\simeq0.2 that we have observed with Subaru and XMM-Newton to construct relationships between the weak-lensing mass (M), and three X-ray observables: gas temperature (T), gas mass (Mgas), and quasi-integrated gas pressure (Yx) at overdensities of \Delta=2500, 1000, and 500 with respect to the critical density. We find that Mgas at \Delta\le1000 appears to be the most promising mass proxy of the three, because it has the lowest intrinsic scatter in mass at fixed observable: \sigma_lnM\simeq0.1, independent of cluster dynamical state. The scatter in mass at fixed T and Yx is a factor of \sim2-3 larger than at fixed Mgas, which are indicative of the structural segregation that we find in the M-T and M-Yx relationships. Undisturbed clusters are found to be \sim40% and \sim20% more massive than disturbed clusters at fixed T and Yx respectively at \sim2\sigma significance. In particular, A1914 - a well-known merging cluster - significantly increases the scatter and lowers the the normalization of the relation for disturbed clusters. We also investigated the covariance between intrinsic scatter in M-Mgas and M-T relations, finding that they are positively correlated. This contradicts the adaptive mesh refinement simulations that motivated the idea that Yx may be a low scatter mass proxy, and agrees with more recent smoothed particle hydrodynamic simulations based on the Millennium Simulation. We also propose a method to identify a robust mass proxy based on principal component analysis. The statistical precision of our results are limited by the small sample size and the presence of the extreme merging cluster in our sample. Comment: 13 pages, 6 figures : ApJ in press : proof ver

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arXiv:1007.3816v3 [astro-ph.CO] 20 Aug 2010
ApJ
Preprint typeset using LATEX style emulateapj v. 04/20/08
LOCUSS: CALIBRATING MASS-OBSERVABLE SCALING RELATIONS FOR CLUSTER COSMOLOGY WITH
SUBARU WEAK LENSING OBSERVATIONS*
N. Okabe,1,2Y.-Y. Zhang,3A. Finoguenov,4,5M. Takada,6G. P. Smith,7K. Umetsu,2T. Futamase,1
ApJ, accepted
ABSTRACT
We present a joint weak-lensing/X-ray study of galaxy cluster mass-observable scaling relations,
motivated by the critical importance of accurate calibration of mass proxies for future X-ray missions,
including eROSITA. We use a sample of 12 clusters at z ≃ 0.2 that we have observed with Subaru and
XMM-Newton to construct relationships between the weak-lensing mass (M) and three X-ray observ-
ables: gas temperature (T), gas mass (Mgas), and quasi-integrated gas pressure (YX) at overdensities
of ∆ = 2500, 1000, and 500 with respect to the critical density. We find that Mgas at ∆ ≤ 1000
appears to be the most promising mass proxy of the three because it has the lowest intrinsic scatter
in mass at fixed observable, σlnM ≃ 0.1, independent of the cluster dynamical state. The scatter in
mass at fixed T and YXis a factor of ∼ 2 − 3 larger than at fixed Mgas, which are indicative of the
structural segregation that we find in the M −T and M −YXrelationships. Undisturbed clusters are
found to be ∼ 40% and ∼ 20% more massive than disturbed clusters at fixed T and YXrespectively
at ∼ 2σ significance. In particular, A1914 – a well-known merging cluster – significantly increases the
scatter and lowers the normalization of the relation for disturbed clusters. We also investigated the
covariance between intrinsic scatter in M −Mgasand M −T relations, finding that they are positively
correlated. This contradicts the adaptive mesh refinement simulations that motivated the idea that
YXmay be a low scatter mass proxy, and agrees with more recent smoothed particle hydrodynamic
simulations based on the Millennium Simulation. We also propose a method to identify a robust mass
proxy based on principal component analysis. The statistical precision of our results is limited by the
small sample size and the presence of the extreme merging cluster in our sample. We therefore look
forward to studying a larger, more complete sample in the future.
Subject headings: Cosmology: observations – dark matter – galaxies: clusters: general – gravitational
lensing: weak – X-rays: galaxies: clusters.
1. INTRODUCTION
Galaxy clusters are the largest virialized objects in
the universe; they formed from high amplitude peaks
of the primordial density field.
cupy the high mass exponential tail of the dark mat-
ter halo mass function, which is sensitive to the matter
density and expansion history of the universe, and to
modifications of the laws of gravity. Measurements of
the evolution of the galaxy cluster mass function across
a broad range of redshifts can thus provide a powerful
Clusters therefore oc-
Electronic address: okabe@asiaa.sinica.edu.tw
*This work is based in part on data collected at the Subaru
Telescope and obtained from the SMOKA, which is operated by
the Astronomy Data Center, National Astronomical Observatory
of Japan. Based on observations made with the XMM-Newton, an
ESA science mission with instruments and contributions directly
funded by ESA member states and the USA (NASA).
1Astronomical institute, Tohoku University, Aramaki, Aoba-ku,
Sendai, 980-8578, Japan
2Academia Sinica Institute of Astronomy and Astrophysics
(ASIAA), P.O. Box 23-141, Taipei 10617, Taiwan
3Argelander-Institut f¨ ur Astronomie, Universit¨ at Bonn, Auf
dem H¨ ugel 71, 53121 Bonn, Germany
4Max-Planck-Institut f¨ ur extraterrestrische Physik, Giessen-
bachstraße, 85748 Garching, Germany
5University of Maryland, Baltimore County, 1000 Hilltop Circle,
Baltimore, MD 21250, USA
6Institute for the Physics and Mathematics of the Universe
(IPMU), The University of Tokyo
5-1-5 Kashiwa-no-Ha, Kashiwa City, Chiba 277-8568, Japan
7School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham, B15 2TT, UK
tool for constraining the cosmological parameters (e.g.,
Vikhlinin et al. 2009a, 2009b). Numerous galaxy cluster
surveys will soon begin delivering a huge amount of data
at optical, X-ray, and millimeter wavelengths, e.g. from
Subaru/Hyper-Suprime-Cam, eROSITA, SPT and ACT.
One of the main goals of these surveys is to measure the
evolution of the galaxy cluster mass function, and thus
to probe the expansion history of the universe. However,
the mass of a galaxy cluster is not directly measurable.
These surveys will therefore rely on “mass-like” observ-
ables (e.g., X-ray temperature – Evrard et al. 1996) and
scaling relations between these observables and mass, to
construct the all-important mass functions. Calibration
of mass-observable scaling relations is therefore currently
a high priority observational goal.
Traditionally,observational studies of the mass-
observable scaling relations have relied solely on X-
ray observations, typically concentrating on the mass-
temperature relation (e.g., Finoguenov et al., 2001;
Sanderson et al., 2003; Ettori et al., 2004; Arnaud et
al., 2005). X-ray-based mass measurements require hy-
drostatic equilibrium (H.E.) and spherical symmetry to
be assumed, and either measurement of the temperature
profile, or an assumption of isothermality. Inclusion of
X-ray temperature information in both axes of the mass-
temperature relation may therefore induce intrinsic cor-
relations into the measured relation. The validity of the
underlying assumptions also warrants careful testing.
Gravitational lensing offers cluster mass measurements

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2Okabe, Zhang, Finoguenov et al.
that are independent of X-ray observations, and do not
rely on assuming H.E.. Joint lensing/X-ray studies (e.g.,
Okabe & Umetsu 2008; Kawaharada et al. 2010) are
therefore a promising route for calibrating cluster mass-
observable scaling relations. Indeed, early lensing/X-ray
studies of cluster cores indicated that the scatter in clus-
ter temperature may be as large as 40% at fixed mass,
and that the scatter is dominated by disturbed, merging
clusters, in which H.E. may not hold (Smith et al. 2005).
Subsequent work has concentrated on using weak-lensing
data to extend this pioneering work beyond cluster cores
to overdensities of 500∼< ∆∼< 2500 with respect to the
critical density (Bardeau et al. 2007; Hoekstra 2007; Ped-
ersen & Dahle 2007; Zhang et al. 2007, 2008). The main
limiting factors in these weak-lensing/X-ray studies have
been the limited statistical precision and heterogeneity
of the available weak-lensing data, and also the small
samples observed to date.
On the theoretical side,
proposedthe so-called
YX≡Mgas×T as a “new robust low-scatter X-ray mass
indicator”, or, a mass-like observable.
tivated by analysis of their hydrodynamic numerical
simulations of clusters using an adaptive mesh refine-
ment (AMR) code. They found that the temperature
deviations from the M − T relation are anti-correlated
with the gas mass deviations from the M−Mgasrelation.
This anti-correlation found in their simulations acts to
suppress the scatter in the M −YXrelation, independent
of the dynamical state of the clusters. This prediction
has stimulated much observational effort within the
X-ray community that has broadly supported the idea
that YXis the optimal X-ray mass proxy (e.g., Maughan
2007; Arnaud et al., 2007; Vikhlinin et al. 2009a).
However, Stanek et al.’s (2010) smoothed particle hy-
drodynamic (SPH) Millennium Gas Simulations contra-
dict Kravtsov et al.’s simulations. Stanek et al. predict
that the temperature and gas-mass deviations are posi-
tively correlated; this result appears to be independent of
the range of gas physics (gravity only, cooling, preheat-
ing) implemented in the simulations. Juett et al. (2010)
have also recently suggested that previous X-ray-only
studies may have underestimated the scatter in mass-
observable scaling relations by a factor of ∼ 2−3. In sum-
mary, a joint lensing/X-ray observational investigation of
the relationships between mass and gas mass, tempera-
ture, and YX, is urgently needed. Such joint studies also
lend themselves well to the task of observationally testing
various corrections that have been derived from numeri-
cal simulations to account for deviations from H.E.. For
example, numerous authors have pointed out that H.E.
mass estimates may underestimate the cluster mass be-
cause of non-thermal pressure support due to turbulence
caused by bulk motion of the cluster gas (e.g., Evrard
1990; Rasia et al. 2006; Nagai et al. 2007; Piffaretti &
Valdarnini 2008; Fang et al. 2009), and Vikhlinin et al.
(2009a) applied a 17% upward correction to X-ray masses
of disturbed clusters, based on the results of simulations.
A key goal of the Local Cluster Substructure Survey
(LoCuSS9) is to calibrate cluster mass-observable scaling
relations for future cosmological experiments. LoCuSS
is a multi-wavelength survey of galaxy clusters at 0.15 <
Kravtsov et al. (2006)
quasi-integratedpressure,
This was mo-
9http://www.sr.bham.ac.uk/locuss
z < 0.3 selected from the ROSAT All-sky Survey cata-
logs (Ebeling et al. 1998, 2000; B¨ ohringer et al. 2004).
To date we have published the first lensing/Sunyaev-
Zeldovich effect comparison (Marrone et al. 2009), be-
gun our lensing/X-ray scaling relation work with a pilot
study (Zhang et al. 2008), and compared lensing-based
masses with H.E. masses on both small (Richard et al.,
2010) and large (Zhang et al. 2010) scales. This article
is a continuation of our pilot study (Zhang et al. 2008),
in which we combined weak-lensing mass measurements
from the Canada-France-Hawaii Telescope (Bardeau et
al. 2005, 2007) and from the Nordic Optical Telescope
and UH 88in (Dahle 2006) with XMM-Newton observa-
tions to calibrate the mass-observable scaling relations.
As alluded to above, Zhang et al.’s results were lim-
ited by the quality of the weak-lensing mass measure-
ments, because the underlying data were heterogeneous
in observing facilities, fields of view, and filters used.
In this article we address these issues by using our own
weak-lensing mass measurements based on uniform anal-
ysis of our Subaru/Suprime-cam observations (Okabe &
Umetsu 2008; Okabe et al., 2010).
Subaru/XMM-Newton sample remains small, at just 12
clusters. As we discuss throughout this article, sample
size therefore remains an issue, and we will address this
in a future article.
The outline of this paper is as follows. In Sec. 2 we
briefly describe the weak lensing and X-ray analysis, and
measure the dynamical state of each cluster using XMM-
Newton data. We present the main results on the mass-
observable scaling relations in Sec. 3, discuss the results
in Sec. 4, and summarize our work in Sec. 5. Throughout
this paper, we assume Ωm,0= 0.3, ΩΛ = 0.7, and h =
H0/100 kms−1Mpc−1= 0.7.
Nevertheless, our
2. SAMPLE AND DATA ANALYSIS
2.1. Sample
For the purpose of this paper, we compiled a sample
of 12 clusters – A68, A115, A209, A267, A383, A1835,
A1914, Z7160, A2261, RXJ2129.6+0005, A2390, and
A2631 – that represents the overlap between the sam-
ples for which Subaru/Suprime-Cam and XMM-Newton
data are available, and that we have previously pub-
lished (Zhang et al. 2008; Okabe & Umetsu 2008; Ok-
abe et al. 2010). The sample does not suffer, by de-
sign, any strong biases to extreme merging or extreme
cool core clusters, and therefore can be regarded, qual-
itatively, as representative of massive, X-ray luminous
clusters. However, given the small sample size, we re-
frain from attempting to quantify how these 12 might
be biased with respect to the underlying cluster popu-
lation in this article. Instead, this article presents some
early results from our Subaru/XMM-Newton program,
that benefit from the use of our Subaru data, as opposed
to the CFH12k/UH8k/NOT data that we used in Zhang
et al. (2008). We defer detailed discussion of sample def-
inition and possible biases to future articles in this series
that will address larger, more complete samples.
2.2. Weak-lensing mass measurements
The details of our weak-lensing analysis are described
in by Okabe & Umetsu (2008), and Okabe et al. (2010);
here we provide a brief outline of some important aspects

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LoCuSS: Calibrating Mass-Observable Scaling Relations for Cluster Cosmology3
of our methods.
We selected background galaxies based on their loca-
tion in the color-magnitude plane – typically (V −i′)/i′–
bluer or redder than cluster red-sequence by a minimum
color-offset (Umetsu & Broadhurst 2008; Umetsu et al.
2009; Okabe et al. 2010). As demonstrated by Okabe
et al. (2010), contamination of the background galaxy
catalogs by faint (unlensed) cluster members dilutes the
weak-lensing signal. This effect is more pronounced at
smaller clustercentric radii because the number density of
cluster galaxies rises towards the cluster centers. In the
absence of our color-selection techniques, weak-lensing
M500 and M2500 measurements can be biased low by
∼ 20%–50%.
We used the COSMOS photometric redshift catalog
(Ilbert et al. 2009) to estimate the redshift of the back-
ground galaxies. Specifically, we calculated the average
lensing weight, ?DLS/DOS? =
(see also Equation (10) in Okabe et al. 2010), of each
background galaxy catalog by selecting galaxies identi-
cal to both our catalogs, and the COSMOS catalog. DOS
and DLSare the angular diameter distances between the
observer and source (background galaxy) and lens and
source respectively.
In cosmology the three-dimensional spherical mass,
M∆, enclosed within a sphere of radius r∆ for a given
overdensity ∆ is most relevant for the cluster mass func-
tion, where r∆ is chosen such that the average density
within the sphere is equal to the critical mass density at
the cluster redshift, ρcr, times the overdensity ∆. We
estimated M∆ for each cluster by fitting the measured
radial profile of lensing distortion signals to the NFW
model prediction parameterized by the mass M∆and c∆,
where the NFW mass profile (Navarro, Frenk & White
1996, 1997) is given as ρ ∝ r−1(1 + c∆r/r∆)−2with c∆
being the concentration parameter.
Describing cluster-scale dark matter halos as spherical
objects may cause systematic errors in individual mass
measurements because clusters are predicted to be triax-
ial in the collisionless CDM model (Jing & Suto 2002).
For example, if the major axis of a triaxial halo is aligned
with or perpendicular to the line of sight, a spherical
model would overestimate or underestimate the mass, re-
spectively, and also cause systematic errors in the mea-
surement of the concentration parameter (Oguri et al.
2005; Gavazzi 2005; Corless et al. 2009). However, if
the distribution of cluster orientations is random, then
adopting spherical mass models should not introduce a
significant bias into the properties of the sample. We
therefore check that this is the case for our sample by
comparing the spherical mass measurements from Okabe
et al. (2010) that we use here with triaxial mass measure-
ments of the same clusters using the same background
galaxy catalogs from Oguri et al. (2010). On average the
spherical (M(sph)
∆
– ?M(tri)
∆∆
? = 0.98 ± 0.15,0.90 ± 0.17 and 0.83 ±
0.21, for ∆ = 500,1000 and 2500 – confirming the expec-
tation of negligible bias.
?
zddzdPWL/dzDLS/DOS
) and triaxial (M(tri)
∆
) masses agree well
/M(ave)
2.3. X-ray observables
The observations and data reduction are described in
detail by Zhang et al. (2007, 2008). In brief, the three
mass proxies considered in this article are calculated as
follows.
of the spectrally measured, radial temperature profile
limited to the radial range of (0.2 − 0.5)r500. The gas
mass Mgas(r) was obtained for each cluster by integrat-
ing a double-β model of the electron density that was
fitted to the X-ray surface brightness profile. The quasi-
integrated pressure is the product of the gas mass and the
global temperature: YX(r) = Mgas(r)×T0.2−0.5r500. Note
that Mgas(r), T0.2−0.5r500and YX(r) have been calculated
using radii obtained from the weak-lensing analysis, and
not using radii calculated from the X-ray analysis as in
Zhang et al. (2008). This definition of radii introduces
a subtle correlation with weak lensing mass – we will
explore this when estimating the intrinsic scatter in the
mass-observable scaling relations in Sec. 3.6 and the Ap-
pendix. Finally, we adopted a self-consistent definition
of the cluster centers based on the weak lensing analysis.
This caused us to change the centers of just two clusters
– A1914 and A2631 – from those used by Zhang et al.
(2010).
The global temperature is a volume average
2.4. X-ray morphology and dynamical state
Previous joint lensing/X-ray studies have identified the
dynamical state of clusters as a significant source of scat-
ter in mass-observable scaling relations (Smith et al.
2005; Pedersen & Dahle 2007; Zhang et al. 2008, 2009).
In this section we therefore classify the clusters as either
“disturbed” or “undisturbed”, based on a new method
patterned on those developed for the morphological clas-
sification of galaxies (e.g., Conselice 2003).
We calculate the asymmetry (A) and fluctuation (F)
of the X-ray surface brightness distribution in the 0.7 −
2 keV band. Asymmetry is defined as A = (?
of the flux residuals where Iijis a matrix element of the
combined MOS1+MOS2 XMM-Newton frame in the 0.7-
2.0 keV band, flat fielded, point source subtracted and
refilled assuming a Poisson distribution, and Rijare the
matrix elements obtained by rotating the above frame by
180◦. The pixel size of both frames is 4′′×4′′. The fluc-
tuation, F, measures deviations from a smooth flux dis-
tribution and is defined as F = (?
scales, which corresponds to a physical scale of 400 kpc at
z = 0.2. Such smoothing also suppresses the effect of the
complex shape of the XMM-Newton point spread func-
tion (Ghizzardi 2001). We estimate the statistical errors
of A and F assuming Poisson noise computed within a
radius of r500, excluding CCD gaps and bad pixels. We
also estimate the systematic error of A caused by uncer-
tainties in the cluster centers by recalculating A, each
time moving the cluster centers onto one of the neigh-
boring pixels within the r ≤ 4′′circle from the nominal
cluster center.
The clusters span the range A ∼ 0.07 − 0.15 and
F ∼ 0 − 0.14 (Fig. 1). Dynamically disturbed clusters
generally have an asymmetric X-ray morphology, with an
offset between optical and X-ray centers, and are there-
fore expected to have larger A and F than undisturbed
clusters. To separate the clusters into two subsamples
that represent relatively disturbed and relatively undis-
turbed systems, we subdivided the A−F plane into four
quadrants: (1) A < 1.1 and F < 0.05 – RXJ2129, A209,
ij|Iij−
Rij|)/?
ijIij, the normalized sum of the absolute value
ijIij− Bij)/?
ijIij,
where Bijis an element in a frame smoothed on 2arcmin

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4Okabe, Zhang, Finoguenov et al.
A383, A1835, and A2390, (2) A > 1.1 and F < 0.05
– A2261 and A1914, (3) A < 1.1 and F > 0.05 –
A68, A2631, A267, and Z7160, and (4) A > 1.1 and
F > 0.05 – A115. We classify the five clusters in quad-
rant (1) – low A and low F – as undisturbed clusters,
and the remaining seven as disturbed clusters. It is im-
mediately obvious that this classification matches other
possible classification schemes well. For example, four of
the five undisturbed clusters host a cool core (e.g. Smith
et al. 2003; Allen et al. 2001; Peterson et al. 2002), and
the disturbed clusters have been discussed extensively as
merging/cold-front clusters (e.g., Okabe & Umetsu 2008;
Mazzotta & Giacintucci 2008; Gutierrez & Krawczynski
2005), in which complicated temperature/entropy distri-
butions or large offsets between lensing/optical and X-
ray centroids exits (e.g., Finoguenov et al. 2005; Smith
et al. 2005; Sanderson et al. 2009a). In summary, all of
the clusters identified as disturbed in the A−F plane are
independently confirmed as disturbed by other methods
in the literature. However we stress again the relative na-
ture of the disturbed/undisturbed classification, and ac-
knowledge that the disturbed clusters in particular likely
comprise clusters in a wide variety of stages in their dy-
namical evolution. We will return to this issue later when
we assess the impact of a single extreme merging cluster
on our attempts to calibrate the mass-observable scaling
relations.
3. RESULTS
In this section, we present the main empirical results
of the slope, normalization, and intrinsic scatter in the
mass-observable scaling relations and how these depend
on the dynamical state of the clusters. We also discuss
the correlation between gas mass and temperature devi-
ations.
3.1. Scaling relations and fitting methods
If gravitational heating is the dominant mechanism re-
sponsible for the X-ray properties of galaxy clusters, the
following scaling relations are expected to hold:
ME(z) ∝ (YXE(z))3/5h1/2,
ME(z) ∝ MgasE(z)h3/2,
ME(z) ∝ T3/2h−1,
(1)
(2)
(3)
where M,Mgas, and T are the total mass, gas mass, and
temperature of a cluster, respectively, and YX= Mgas×T
is the quasi-integrated pressure. These relations, specif-
ically the exponents of M,Mgas, and T, are usually re-
ferred to as self-similar, following Kaiser (1986). Note
that the term E(z) = H(z)/H0= [Ωm,0(1+z)3+ΩΛ]1/2
accounts for the redshift evolution of the clusters in a flat
universe.
In the following subsections we therefore fit the func-
tional form Mz = M0Xγ
M E(z), M0is the normalization, Xzis the X-ray observ-
able (i.e., YX, T, or Mgas) multiplied by E(z) or not as ap-
propriate based on Equations 1-3, and γ is the logarith-
mic slope. These fits are done at three overdensities with
respect to the critical density: ∆ = 2500,1000 and 500.
The scaling relation slope and normalization measure-
ments are based on orthogonal regression performed us-
ing the Orthogonal Distance Regression package (ODR-
zto the data, where Mz =
PACK, e.g. Boggs et al. 1987) taking into account the
measurement errors. In general we ignore the subtle
correlations introduced by measuring the X-ray observ-
ables within radii defined by the weak-lensing analysis,
although we do take them into account in Sec.3.3 when
we measure the intrinsic scatter. To check for consistency
with other work, we have also refitted the relations using
the bisector modification of the BCES method (Akritas
& Bershady 1996). The difference on best-fit scaling re-
lation parameters between the two fitting methods is a
small fraction of statistical uncertainties. For example,
the difference on best-fit slopes and normalizations be-
tween the two methods is typically ∼ 30% and ∼ 6% of
the statistical error respectively. We also did the boot-
strap resampling to estimate the sample variance on the
slope parameter, and found that it is∼< 20% of the sta-
tistical errors.
3.2. Slope and normalization
We first fit the scaling relations to the full sample of 12
clusters with both slope γ and normalization M0as free
parameters. At ∆ = 500 the best-fit slopes of all three
relations agree well with the self-similar model (Table 1).
At higher overdensities, the agreement deteriorates for all
three relations, indeed the slopes of the M2500− Mgas,
M2500− YXrelations are discrepant from self-similar at
∼ 2 − 3σ at ∆ = 2500. This flattening in the scaling
relations at higher ∆ can also be seen graphically in Fig-
ures 2, 3 and 4, in which we show the M∆–YX, M∆–TX
and M∆–Mgasrelations respectively.
To constrain the normalization parameter M0, we fix
the slope parameters to the self-similar values and repeat
the fits. The measured normalizations are all consistent
with those obtained by Zhang et al. (2008) using the
same XMM-Newton data and independent weak-lensing
data. The superior quality and uniformity of our Sub-
aru data shows differences between the normalizations
for disturbed and undisturbed clusters. These differences
are most pronounced at ∆ = 500 – see Table 2 – specifi-
cally, at fixed YXundisturbed clusters are measured to be
∼ 22% more massive than disturbed clusters at ∼ 1.5σ
significance. Similarly, at fixed T undisturbed clusters
are measured to be ∼ 43% more massive than disturbed
clusters at ∼ 1.8σ significance. We confirm that our re-
sults are insensitive to whether or not the slopes are fixed
to the self-similar value.
3.3. Scatter
We also measured the intrinsic scatter, σlnM, for
the logarithm of the Y -axis, ME(z), for each mass-
observable scaling relation using the Bayesian method
described in the Appendix. Here we take into account
the correlations caused by measuring X-ray observables
within radii defined by the lensing analysis – see the
Appendix. We also confirmed that the best-fit slopes
and normalizations obtained using the ODR methods
discussed above are consistent within errors with those
obtained using the more sophisticated Bayesian method
considered here. The intrinsic scatter in all three rela-
tions is well described by a lognormal distribution.
The M−T relation exhibits the largest intrinsic scatter
(∼ 0.23−0.33; Table 3) among the three mass-observable
relations. We also observe an increase in the intrinsic

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LoCuSS: Calibrating Mass-Observable Scaling Relations for Cluster Cosmology5
scatter with increasing radius (i.e. decreasing the interior
overdensity ∆ ). The same trend is found in undisturbed
clusters, while the opposite trend is found in disturbed
clusters. However, this trend is not a physical feature
of the intracluster gas affected by gravitational heating,
because we used a fixed global temperature measurement
in the radial range of 0.2−0.5r500for all the overdensities.
The M − Mgas relation is the tightest of the three,
with an intrinsic scatter in mass of σlnM∼ 0.12−0.16 at
∆ = 500 and 1000. At ∆ = 2500 the scatter is roughly
double that at lower overdensity (Table 3), which may
be due to different core properties of individual clusters.
For example, cool-core clusters have denser, cuspier cores
than non-cool core clusters (e.g., Croston et al. 2008;
Sanderson et al. 2009b; McCarthy et al. 2008). Such
differences between cluster cores have a much smaller ef-
fect on measurements at larger radii because the core
regions make a small contribution to the total gas mass
measured out to ∆ = 1000 and 500. However, note that
the intrinsic scatter is not well constrained for M −Mgas
because the scatter is dominated by statistical errors. A
larger sample is clearly needed to improve the constraints
on the intrinsic scatter in M − Mgas, however, it is im-
portant to note that this is the only relation that appears
to have ∼ 10% intrinsic scatter.
The observed intrinsic scatter in the M −YXrelation is
intermediate between that of the M − Mgasand M − T
relations, at σlnM ∼ 0.20 − 0.25 (Table 3). This is a
factor of∼> 2 greater than that originally predicted by
Kravtsov et al. (2006) based on their AMR simulations.
3.4. The impact of an outlier
In this section we highlight the impact of one cluster,
A1914, on our results. This cluster has previously been
identified as a merging cluster with a complex X-ray mor-
phology, radio halo, and weak-lensing-based dark matter
distribution (Buote & Tsai 1996; Bacchi et al. 2003; Gov-
oni et al. 2004; Okabe & Umetsu 2008). We have also
identified it as having the most extreme X-ray/lensing
mass discrepancy among the 12 clusters considered here
(Zhang et al. 2010).
To assess the impact of such clusters on the measured
reliability of X-ray observables as mass proxies we re-
peated the calculations of normalization and scatter dis-
cussed in Sec. 3.2 & 3.3 excluding A1914 (Table 2). At
∆ = 500 the normalization of the M − YXrelations for
disturbed and undisturbed clusters are different at just
∼ 1.2σ significance when A1914 is excluded from the dis-
turbed sample, in contrast to the ∼ 1.8σ difference based
on the full sample of 12 clusters. We also find that ex-
cluding A1914 reduces the intrinsic scatter on all of the
scaling relations. In particular, the intrinsic scatter on
M − YXis reduced by ∼ 25% from σlnM∼ 0.20 to 0.15.
Jack-knife tests on samples of 11 clusters (i.e. removing
each cluster in turn) also confirm that A1914 is indeed
the most significant outlier among our sample.
These results indicate that outliers in the cluster pop-
ulation require careful treatment in the construction and
application of mass-observable scaling relations. In sum-
mary, reliable cluster selection functions are required to
gain robust constraints. This will be especially true for
future high redshift surveys because the fraction of merg-
ing clusters is expected to increase with look-back time
(Vikhlinin et al. 2009a).
3.5. Covariance of deviations
We investigate the covariance of deviations from the
best-fit M − Mgas and M − T relations following re-
cent numerical simulation studies (e.g. Kravtsov et al.
2006; Stanek et al. 2010). For a given mean scaling rela-
tion Y = f(X), the deviations of each cluster from the
mean relation are quantified as δY ≡ [Y − f(X)] and
δX ≡ [X − f−1(Y )]. We use the mean normalizations
for a full sample of 12 clusters, however we found that
the following results do not change significantly when
the best-fit normalizations of undisturbed and disturbed
clusters are used instead.
The temperature and Mgas deviations, δT/T(M∆)
and δMgas/Mgas(M∆), appear to be positively corre-
lated (Fig. 6). We test this quantitatively using Spear-
man’s rank correlation coefficient test, obtaining rs =
0.531 ± 0.009. The probability of obtaining a value of
rs greater than or equal to the measured value is low:
P = 0.075±0.006. This test therefore indicates that the
positive correlation is significant. However, the appar-
ent positive correlation between the temperature and gas
mass deviations does not show the correlation between
intrinsic scatter, but between total scatter which is a
convolution of measurement errors and intrinsic scatter,
because we here did not take into account for measure-
ment uncertainties. When dealing with observational
constraints on scaling relations, it therefore essential to
include both the covariance of intrinsic scatter, and the
measurement errors with which the scatter is convolved
in robust calculations.
3.6. Covariance of intrinsic scatter
We simultaneously fit M − T and M − Mgasrelations
and measure the covariance of the intrinsic scatter us-
ing a multi-dimensional fitting method described in the
Appendix. This method considers not only the matrix
of the observational errors for individual clusters, Σobs,i,
but also the covariance matrix of the intrinsic scatter,
Σint. The covariance of the intrinsic scatter is given by
Σint=
?σ2
tσtg
σtgσ2
g
?
(4)
where σ2
perature and gas mass, respectively, and σtg= rσtσgis a
covariance with a coefficient r. Here, we do not need to
take into account for intrinsic scatter on mass, because
the gas properties, under a cluster mass given by the cos-
mology, only physically have intrinsic scatter due to the
gas evolution. When we estimate cluster masses from
X-ray observables via scaling relations, there is intrinsic
scatter on mass due to the propagation from gas intrin-
sic scatter. As shown in the Appendix, the observational
error matrix for individual clusters is given by
tand σ2
gare variances for the logarithm of tem-
Σobs=
?
4
9e2
m+ e2
∂ lnM)e2
t
2
3(1 −∂ lnMgas
m(1 −∂ lnMgas
∂ lnM)e2
∂ lnM)2e2
m
2
3(1 −∂ lnMgas
m+ e2
g
?
where em,etand egare observational errors for the log-
arithm of the mass, temperature and gas mass, respec-
tively. The coefficients of emare the slopes of the mass