Issue 
A&A
Volume 564, April 2014



Article Number  A129  
Number of page(s)  15  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201322870  
Published online  17 April 2014 
The 400d Galaxy Cluster Survey weak lensing programme
III. Evidence for consistent WL and Xray masses at z ≈ 0.5^{⋆}
^{1}
Department of PhysicsDurham University,
South Road,
Durham
DH1 3LE,
UK
email:
holger.israel@durham.ac.uk
^{2}
ArgelanderInstitut für Astronomie, Auf dem Hügel 71, 53121
Bonn,
Germany
^{3}
Department of Astronomy, University of Virginia,
530 McCormick Road,
Charlottesville
VA
22904,
USA
^{4}
HarvardSmithsonian Center for Astrophysics,
60 Gard0en Street, Cambridge
MA
02138,
USA
Received:
17
October
2013
Accepted:
11
February
2014
Context. Scaling properties of galaxy cluster observables with cluster mass provide central insights into the processes shaping clusters. Calibrating proxies for cluster mass that are relatively cheap to observe will moreover be crucial to harvest the cosmological information available from the number and growth of clusters with upcoming surveys like eROSITA and Euclid. The recent Planck results led to suggestions that Xray masses might be biased low by ~40%, more than previously considered.
Aims. We aim to extend knowledge of the weak lensing – Xray mass scaling towards lower masses (as low as 1 × 10^{14}M_{⊙}) in a sample representative of the z ~ 0.4–0.5 population. Thus, we extend the direct calibration of cluster mass estimates to higher redshifts.
Methods. We investigate the scaling behaviour of MMT/Megacam weak lensing (WL) masses for 8 clusters at 0.39 ≤ z ≤ 0.80 as part of the 400d WL programme with hydrostatic Chandra Xray masses as well as those based on the proxies, e.g. Y_{X} = T_{X}M_{gas}.
Results. Overall, we find good agreement between WL and Xray masses, with different mass bias estimators all consistent with zero. When subdividing the sample into a lowmass and a highmass subsample, we find the highmass subsample to show no significant mass bias while for the lowmass subsample, there is a bias towards overestimated Xray masses at the ~2σ level for some mass proxies. The overall scatter in the massmass scaling relations is surprisingly low. Investigating possible causes, we find that neither the greater range in WL than in Xray masses nor the small scatter can be traced back to the parameter settings in the WL analysis.
Conclusions. We do not find evidence for a strong (~40%) underestimate in the Xray masses, as suggested to reconcile recent Planck cluster counts and cosmological constraints. For highmass clusters, our measurements are consistent with other studies in the literature. The mass dependent bias, significant at ~2σ, may hint at a physically different cluster population (less relaxed clusters with more substructure and mergers); or it may be due to small number statistics. Further studies of lowmass highz lensing clusters will elucidate their mass scaling behaviour.
Key words: galaxies: clusters: general / cosmology: observations / gravitational lensing: weak / Xrays: galaxies: clusters
Appendix A is available in electronic form at http://www.aanda.org
© ESO, 2014
1. Introduction
Galaxy cluster masses hold a crucial role in cosmology. In the paradigm of hierarchical structure formation from tiny fluctuations in the highly homogeneous early cosmos after inflation, clusters emerge via the continuous matter accretion onto local minima of the gravitational potential. Depending sensitively on cosmological parameters, the cluster mass function, i.e. their abundance as function of mass and redshift z, provides observational constraints to cosmology (e.g., Vikhlinin et al. 2009b; Allen et al. 2011; Planck Collaboration XX 2014).
Observers use several avenues to determine cluster masses: properties of the Xrayemitting intracluster medium (ICM), its imprint on the cosmic microwave background via the SunyaevZel’dovich (SZ) effect in the submm regime, galaxy richness estimates and dynamical masses via optical imaging and spectroscopy, and gravitational lensing. Across all wavelengths, cluster cosmology surveys are under preparation, aiming at a complete cluster census out to ever higher redshifts, e.g. eROSITA (Predehl et al. 2010; Merloni et al. 2012; Pillepich et al. 2012) and Athena (Nandra et al. 2013; Pointecouteau et al. 2013) in Xrays, Euclid (Laureijs et al. 2011; Amendola et al. 2013), DES and LSST in the optical/nearinfrared, CCAT (Woody et al. 2012) and SKA at submm and radio frequencies.
Careful Xray studies of clusters at low and intermediate redshifts yield highly precise cluster masses, but assume hydrostatic equilibrium, and in most cases spherical symmetry (e.g. Croston et al. 2008; Ettori et al. 2013). Observational evidence and numerical modelling challenge these assumptions for all but the most relaxed systems (e.g. Mahdavi et al. 2008; Rasia et al. 2012; Limousin et al. 2013; Newman et al. 2013). While simulations find Xray masses to only slightly underestimate the true mass of clusters that exhibit no indications of recent mergers and can be considered virialised, nonthermal pressure support can lead to a >20% bias in unrelaxed clusters (Laganá et al. 2010; Rasia et al. 2012). Shi & Komatsu (2014) modelled the pressure due to ICM turbulence analytically and found a ~10% underestimate of cluster masses compared to the hydrostatic case.
Weak lensing (WL), in contrast, is subject to larger stochastic uncertainties, but can in principle yield unbiased masses, as no equilibrium assumptions are required. Details of the mass modelling however can introduce biases, in particular concerning projection effects, the source redshift distribution and the departures from an axisymmetric mass profile (Corless & King 2009; Becker & Kravtsov 2011; Bahé et al. 2012; Hoekstra et al. 2013). For individual clusters, stochastic uncertainties dominate the budget; however, larger cluster samples benefit from improved corrections for lensing systematics, driven by cosmic shear projects (e.g. Massey et al. 2013).
Most of the leverage on cosmology and structure formation from future cluster surveys will be due to clusters at higher z than have been previously investigated. Hence, the average cluster masses and signal/noise ratios for all observables are going to be smaller. Even and especially for the deepest surveys, most objects will lie close to the detection limit. Thus the scaling of inexpensive proxies (e.g. Xray luminosity L_{X}) with total mass needs to be calibrated against representative cluster samples at low and high z. Weak lensing and SZ mass estimates are both good candidates as they exhibit independent systematics from Xrays and a weaker zdependence in their signal/noise ratios.
Theoretically, cluster scaling relations arise from their description as selfsimilar objects forming through gravitational collapse (Kaiser 1986), and deviations from simple scaling laws provide crucial insights into cluster physics. For the current state of scaling relation science, we point to the recent review by Giodini et al. (2013). As we are interested in the cluster population to be seen by upcoming surveys, we focus here on results obtained at high redshifts.
Selfsimilar modelling includes evolution of the scaling relation normalisations with the Hubble expansion, which is routinely measured (e.g. Reichert et al. 2011; Ettori 2013). Evolution effects beyond selfsimilarity, e.g. due to declining AGN feedback at low z, have been claimed and discussed (e.g. Pacaud et al. 2007; Short et al. 2010; Stanek et al. 2010; Maughan et al. 2012), but current observations are insufficient to constrain possible evolution in slopes (Giodini et al. 2013). Evidence for different scaling behaviour in groups and lowmass clusters was found by, e.g., Eckmiller et al. (2011), Stott et al. (2012), Bharadwaj et al. (2014).
Reichert et al. (2011) and Maughan et al. (2012) investigated Xray scaling relationships including clusters at z> 1, and both stressed the increasing influence of selection effects at higher z. Larger weak lensing samples of distant clusters are just in the process of being compiled (Jee et al. 2011; Foëx et al. 2012; Hoekstra et al. 2011a, 2012; Israel et al. 2012; von der Linden et al. 2014a; Postman et al. 2012). Thus most WL scaling studies are currently limited to z ≲ 0.6, and also include nearby clusters (e.g. Hoekstra et al. 2012; Mahdavi et al. 2013, M13). The latter authors find projected WL masses follow the expected correlation with the SZ signal Y_{SZ}, corroborating similar results for more local clusters by Marrone et al. (2009, 2012). Miyatake et al. (2013) performed a detailed WL analysis of a z = 0.81 cluster discovered in the SZ using the Atacama Cosmology Telescope, and compared the resulting lensing mass against the Reese et al. (2012)Y_{SZ}–M scaling relation, in what they describe as a first step towards a highz SZWL scaling study.
By compiling Hubble Space Telescope data for 27 massive clusters at 0.83 <z< 1.46, Jee et al. (2011) not only derive the relation between WL masses M^{wl} and ICM temperature T_{X}, but also notice a good correspondence between WL and hydrostatic Xray masses M^{hyd}. As they focus on directly testing cosmology with the most massive clusters, these authors however stop short of deriving the WLXray scaling. Also using HST observations, Hoekstra et al. (2011a) investigated the WL mass scaling of the optical cluster richness (i.e. galaxy counts) and L_{X} of 25 moderateL_{X} clusters at 0.3 <z< 0.6, thus initiating the study of WL scaling relations off the top of the mass function.
Comparisons between weak lensing and Xray masses for larger cluster samples were pioneered by Mahdavi et al. (2008) and Zhang et al. (2008), collecting evidence for the ratio of weak lensing to Xray masses M^{wl}/M^{hyd}> 1, indicating nonthermal pressure. Zhang et al. (2010), analysing 12 clusters from the Local Cluster Substructure Survey (LoCuSS), find this ratio to depend on the radius. Likewise, a difference between relaxed and unrelaxed clusters is found (Zhang et al. 2010; Mahdavi et al. 2013). Rasia et al. (2012) show that the gap between Xray and lensing masses is more pronounced in simulations than in observations, pointing to either an underestimate of the true mass also by WL masses (cf. Bahé et al. 2012) or to simulations overestimating the Xray mass bias.
The current disagreement between the cosmological constraints derived from Planck primary cosmic microwave background (CMB) data with Wilkinson Microwave Anisotropy Probe data, supernova data, and cluster data (Planck Collaboration XX 2014) may well be alleviated by, e.g. sliding up a bit along the Planck degeneracy curve between the Hubble factor H_{0} and the matter density parameter Ω_{m}. Nevertheless, as stronger cluster mass biases than currently favoured (~40%) have also been invoked as a possible explanation, it is very important to test the cluster mass calibration with independent methods out to high z, as we do in this work.
This article aims to test the agreement of the weak lensing and Xray masses measured by Israel et al. (2012) for 8 relatively lowmass clusters at z ≳ 0.4 with scaling relations from the recent literature. The 400d Xray sample from which our clusters are drawn has been constructed to contain typical objects at intermediate redshifts, similar in mass and redshift to upcoming surveys. Hence, it does not include extremely massive lowz clusters. We describe the observations and WL and Xray mass measurements for the 8 clusters in Sect. 2, before presenting the central scaling relations in Sect. 3. Possible explanations for the steep slopes our scaling relations exhibit are discussed in Sect. 4, and we compare to literature results in Sect. 5, leading to the conclusions in Sect. 6. Throughout this article, $\mathit{E}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathrm{=}\mathit{H}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathit{/}{\mathit{H}}_{\mathrm{0}}\mathrm{=}\sqrt{{\mathrm{\Omega}}_{\mathrm{m}}\mathrm{(}\mathit{z}\mathrm{+}\mathrm{1}{\mathrm{)}}^{\mathrm{3}}\mathrm{+}{\mathrm{\Omega}}_{\mathrm{\Lambda}}}$ denotes the selfsimilar evolution factor (Hubble factor H(z) normalised to its presentday value of H_{0} = 72 km s^{1} Mpc^{1}), computed for a flat universe with matter and dark energy densities of Ω_{m} = 0.3 and Ω_{Λ} = 0.7 in units of the critical density.
2. Observations and data analysis
2.1. The 400d weak lensing survey
This article builds on the weak lensing analysis for 8 clusters of galaxies (Israel et al. 2010, 2012, Paper I and Paper II hereafter) selected from the 400d Xray selected sample of clusters (Burenin et al. 2007; Vikhlinin et al. 2009a, V09a). From the ~400 deg^{2} of all suitable ROSAT PSPC observations, Burenin et al. (2007) compiled a catalogue of serendipitously detected clusters, i.e. discarding the intentional targets of the ROSAT pointings. For a uniquely complete subsample of 36 Xray luminous (L_{X} ≳ 10^{44} erg / s) highredshift (0.35 ≤ z ≤ 0.89) sources, V09a obtained deep Chandra data, weighing the clusters using three different mass proxies (Sect. 3.2). Starting from the cluster mass function computed by V09a, Vikhlinin et al. (2009b) went on to constrain cosmological parameters. For brevity, we will refer to the V09a highz sample as the 400d sample. The 400d weak lensing survey follows up these clusters in weak lensing, determining independent WL masses with the ultimate goals of deriving the lensingbased mass function for the complete sample and to perform detailed consistency checks. Currently, we have determined WL masses for 8 clusters observed in four dedicated MMT/Megacam runs (see Papers I and II). Thus, our scaling relation studies are largely limited to this subset of clusters, covering the sky between α_{J2000} = 13^{h}30^{m}–24^{h} with δ_{J2000}> 10° and α_{J2000} = 0^{h}–08^{h}30^{m} with δ_{J2000}> 0°.
2.2. Weak lensing analysis
Measured properties of the 400d MMT cluster sample.
We present only a brief description of the WL analysis in this paper; for more details see Paper II. Basic data reduction is performed using the THELI pipeline for multichip cameras (Erben et al. 2005; Schirmer 2013), adapted to MMT/Megacam. We employ the photometric calibration by Hildebrandt et al. (2006). Following Dietrich et al. (2007), regions of the THELI coadded images not suitable for WL shear measurements are masked. Shear is measured using an implementation of the “KSB+” algorithm (Kaiser et al. 1995; Erben et al. 2001), the “TS” pipeline (Heymans et al. 2006; Schrabback et al. 2007; Hartlap et al. 2009). Catalogues of lensed background galaxies are selected based on the available colour information. For clusters covered in three filters, we include galaxies based on their position in colour–colour–magnitude space (Paper II; see Klein et al., in prep., for a generalisation). For clusters covered only in one passband, we apply a magnitude cut. Where available, colour information also enables us to quantify and correct for the dilution by residual cluster members (Hoekstra 2007) in the shear catalogues. The mass normalisation of the WL signal is set by the mean lensing depth ⟨ β ⟩, defined as β = D_{ds}/D_{s}, the ratio of angular diameter distances between the deflector and the source, and between the observer and the source. The Ilbert et al. (2006) CFHTLS Deep fields photometric redshift catalogue serves as a proxy for estimating ⟨ β ⟩ and for calibrating the background selection.
The tangential ellipticity profiles given the ROSAT cluster centres are modelled by fitting the reduced shear profile (Bartelmann 1996; Wright & Brainerd 2000) corresponding to the Navarro et al. (1996, 1997, NFW) density profile between 0.2 Mpc and 5.0 Mpc projected radius. Input ellipticities are scaled according to the Hartlap et al. (2009) calibration factor and, where applicable, with the correction for dilution by cluster members. We consider the intrinsic source ellipticity measured from the data, accounting for its dependence on the shear (Schneider et al. 2000).
Lensing masses are inferred by evaluating a χ^{2} merit function on a grid in radius r_{200} and concentration c_{200}. The latter is poorly constrained in the direct fit, so we marginalise over it assuming an empirical massconcentration relation. In addition to the direct fit approach, in Israel et al. (2012), we report masses using two different massconcentration relations: Bullock et al. (2001, B01), and Bhattacharya et al. (2013, B13)^{1}. Finding the masses using B01 or B13 makes them less susceptible to variations in the model in Paper II, we explore their effect further in Sect. 4.1.
2.3. Choice of the overdensity contrast
Cluster scaling relations are usually given for the mass contained within a radius r_{500}, corresponding to an overdensity Δ = 500 compared to the critical density ρ_{c} of the Universe at the cluster redshift. This Δ is chosen because the bestconstrained Xray masses are found close to r_{500}, determined by the particle backgrounds of Chandra and XMMNewton (cf. Okabe et al. 2010a). Currently, only Suzaku allows direct constraints upon Xray masses at r_{200} (see Reiprich et al. 2013, and references therein). In order to compare to the results from the Vikhlinin et al. (2009a)Chandra analysis, we compute our Δ = 500 WL masses from our Δ = 200 masses, assuming the fitted NFW profiles given by (r_{200},c_{200}) to be correct. Independent of Δ > 1, the cumulative mass of a NFW halo, described by r_{Δ} and c_{Δ}, out to a test radius r is given by: $\begin{array}{ccc}{\mathit{M}}_{\mathrm{NFW}}\mathrm{\left(}\mathit{r}\mathrm{\right)}& \mathrm{=}& \mathrm{\Delta}{\mathit{\rho}}_{\mathrm{c}}\frac{\mathrm{4}\mathit{\pi}}{\mathrm{3}}{\mathit{r}}_{\mathrm{\Delta}}^{\mathrm{3}}\mathrm{\times}\frac{\mathrm{ln}\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{c}}_{\mathrm{\Delta}}\mathit{r}\mathit{/}{\mathit{r}}_{\mathrm{\Delta}}\mathrm{)}\mathrm{}\frac{{\mathit{c}}_{\mathrm{\Delta}}\mathit{r}\mathit{/}{\mathit{r}}_{\mathrm{\Delta}}}{\mathrm{1}\hspace{0.17em}\mathrm{+}\hspace{0.17em}{\mathit{c}}_{\mathrm{\Delta}}\mathit{r}\mathit{/}{\mathit{r}}_{\mathrm{\Delta}}}}{\mathrm{ln}\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{c}}_{\mathrm{\Delta}}\mathrm{)}\mathrm{}{\mathit{c}}_{\mathrm{\Delta}}\mathit{/}\mathrm{(}\mathrm{1}\mathrm{+}{\mathit{c}}_{\mathrm{\Delta}}\mathrm{)}}\\ & \mathrm{=}& {\mathit{M}}_{\mathrm{\Delta}}\mathrm{\left(}{\mathit{r}}_{\mathrm{\Delta}}\mathrm{\right)}\mathrm{\times}\mathrm{\Xi}\mathrm{\left(}\mathit{r}\mathrm{;}{\mathit{r}}_{\mathrm{\Delta}}\mathit{,}{\mathit{c}}_{\mathrm{\Delta}}\mathrm{\right)}\mathit{,}\end{array}$separating into the mass M_{Δ} and a function we call Ξ(r;r_{Δ},c_{Δ}). Equating Eq. (1) with r = r_{500} for Δ = 200 and Δ′ = 500, we arrive at this implicit equation for r_{500}, which we solve numerically: $\begin{array}{ccc}{\mathit{r}}_{\mathrm{500}}\mathrm{=}{\mathit{r}}_{\mathrm{200}}{\mathrm{(}\frac{\mathrm{2}}{\mathrm{5}}\hspace{0.17em}\mathrm{\Xi}\mathrm{(}{\mathit{r}}_{\mathrm{500}}\mathit{,}{\mathit{r}}_{\mathrm{200}}\mathit{,}{\mathit{c}}_{\mathrm{200}}{\mathrm{)}}^{\mathrm{)}}}^{\mathrm{1}\mathit{/}\mathrm{3}}\mathit{.}& & \end{array}$(3)
2.4. Xray analysis
Under the strong assumptions that the ICM is in hydrostatic equilibrium and follows a spherically symmetric mass distribution, the cluster mass within a radius r can be calculated as (see e.g. Sarazin 1988): $\begin{array}{ccc}{\mathit{M}}^{\mathrm{hyd}}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathrm{=}\frac{\mathrm{}{\mathit{k}}_{\mathrm{B}}{\mathit{T}}_{\mathrm{X}}\mathrm{\left(}\mathit{r}\mathrm{\right)}\hspace{0.17em}\mathit{r}}{\mathit{\mu}{\mathit{m}}_{\mathrm{p}}\mathit{G}}\left(\frac{\mathrm{d}\mathrm{ln}{\mathit{\rho}}_{\mathrm{g}}\mathrm{\left(}\mathit{r}\mathrm{\right)}}{\mathrm{d}\mathrm{ln}\mathit{r}}\mathrm{+}\frac{\mathrm{d}\mathrm{ln}{\mathit{T}}_{\mathrm{X}}\mathrm{\left(}\mathit{r}\mathrm{\right)}}{\mathrm{d}\mathrm{ln}\mathit{r}}\right)& & \end{array}$(4)from the ICM density and temperature profiles ρ_{g}(r) and T_{X}(r), where G is the gravitational constant, m_{p} is the proton mass, and μ = 0.5954 the mean molecular mass of the ICM. The ICM density is modelled by fitting the observed Chandra surface brightness profile, assuming a primordial He abundance and an ICM metallicity of 0.2 Z_{⊙}, such that ρ_{g}(r) = 1.274 m_{p}n(r). We use a Vikhlinin et al. (2006) particle density profile with $\mathit{n}\mathrm{\left(}\mathit{r}\mathrm{\right)}\mathrm{=}\sqrt{{\mathit{n}}_{\mathrm{p}}\mathrm{\left(}\mathit{r}\mathrm{\right)}\hspace{0.17em}{\mathit{n}}_{\mathrm{e}}\mathrm{\left(}\mathit{r}\mathrm{\right)}}$. Extending the widelyused βprofile (Cavaliere & FuscoFemiano 1978), it allows for prominent cluster cores as well as steeper surface brightness profiles in the cluster outskirts to be modelled by additional terms. Regarding the systematic differences Rozo et al. (2014b) find between Xray mass algorithms, we point out that we employ the V09a profiles directly rather than rederiving them from their parameters.
The relatively low signal/noise in the Chandra data renders the determination of individual temperature profiles difficult. Rather, we fit a global T_{X} (Table 1; V09a) and assume the empirical average temperature profile T_{X}(r) = T_{X}^{(}1.19 − 0.84r/r_{200}^{)}Reiprich et al. (2013) derive from compiling all available Suzaku temperature profiles (barring only the two most exceptional clusters). For r_{200}, we use the WL results from Paper II^{2}.
Equation (4) provides us with a cumulative mass profile. We evaluate this profile at some r_{test}, e.g. from WL, and propagate the uncertainty in r_{test}, together with the uncertainty in T_{X}.
Hydrostatic equilibrium and sphericity are known to be problematic assumptions for many clusters. Nonetheless, hydrostatic masses are commonly used in the literature in comparisons to WL masses. Our goal is to study if and how biases due to deviations from the abovementioned assumptions show up.
2.5. Mass estimates
Table 1 comprises the key results on radii r_{500} and the corresponding mass estimates. By , we denote a mass measured from data on proxy within a radius defined by proxy . We use five mass estimates: . The first two are the weak lensing (wl) and hydrostatic Xray masses (hyd), as introduced in Sects. 2.2 and 2.4. Having analysed deep Chandra observations they acquired, Vikhlinin et al. (2009a) present three further mass estimates for all 36 clusters in the complete sample. Based on the proxies T_{X}, the ICM mass M_{gas}, and Y_{X} = T_{X}M_{gas}, mass estimates M^{T}, M^{G}, and M^{Y} are quoted in Table 2 of V09a. We point out that V09a obtain these estimates by calibrating the mass scaling relations for respective proxy on local clusters (see their Table 3). V09a further provide a detailed account of the relevant systematic sources of uncertainty.
The radii ${\mathit{r}}_{\mathrm{500}}^{\mathrm{\mathcal{P}}}\mathrm{=}{{}^{\mathrm{(}}\mathrm{3}{\mathit{M}}_{\mathrm{500}}^{\mathrm{\mathcal{P}}}\mathit{/}\mathrm{(}\mathrm{2000}\mathit{\pi}{\mathit{\rho}}_{\mathrm{c}}{\mathrm{\left)}}^{\mathrm{\right)}}}^{\mathrm{1}\mathit{/}\mathrm{3}}$ listed in Table 1 are obtained from ${\mathit{M}}_{\mathrm{500}}^{\mathrm{\mathcal{P}}}\mathit{,}\hspace{0.17em}\mathrm{\mathcal{P}}\mathrm{\in}\mathrm{\left\{}\mathrm{Y}\mathit{,}\mathrm{T}\mathit{,}\mathrm{G}\mathrm{\right\}}$. Using Eqs. (1) and (4), we then derive the WL and hydrostatic masses, respectively, within these radii. We emphasise that all WL mass uncertainties quoted in Table 1 are purely statistical and do not include any of the systematics discussed in Paper II.
2.6. Fitting algorithm for scaling relations
The problem of selecting the best linear representation y = A + Bx for a sample of (astronomical) observations of two quantities { x_{i} } and { y_{i} } can be surprisingly complex. A plethora of algorithms and literature cope with the different assumptions about measurement uncertainties one can or has to make (e.g. Press et al. 1992; Akritas & Bershady 1996; Tremaine et al. 2002; Kelly 2007; Hogg et al. 2010; Williams et al. 2010; Andreon & Hurn 2012; Feigelson & Babu 2012). The challenges observational astronomers have to tackle when trying to reconcile the prerequisites of statistical estimators with the realities of astrophysical data are manifold, including heteroscedastic uncertainties (i.e. depending nontrivially on the data themselves), intrinsic scatter, poor knowledge of systematics, poor sample statistics, “outlier” points, and nonGaussian probability distributions. Tailored to the problem of galaxy cluster scaling relations, Maughan (2014) proposed a “selfconsistent” modelling approach based on the fundamental observables. A full account of these different effects exceeds the scope of this article. We choose the relatively simple fitexy algorithm (Press et al. 1992), minimising the estimator $\begin{array}{ccc}{\mathit{\chi}}_{\mathrm{P}\mathrm{92}}^{\mathrm{2}}\mathrm{=}\sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathit{N}}\frac{{\left({\mathit{y}}_{\mathit{i}}\mathrm{}\mathit{A}\mathrm{}\mathit{B}{\mathit{x}}_{\mathit{i}}\right)}^{\mathrm{2}}}{{\mathit{\sigma}}_{\mathit{y,i}}^{\mathrm{2}}\mathrm{+}{\mathit{B}}^{\mathrm{2}}{\mathit{\sigma}}_{\mathit{x,i}}^{\mathrm{2}}}\mathit{,}& & \end{array}$(5)which allows the uncertainties σ_{x,i} and σ_{y,i} to vary for different data points x_{i} and y_{i}, but assumes them to be drawn from a Gaussian distribution. To accommodate intrinsic scatter, ${\mathit{\sigma}}_{\mathit{y,i}}^{\mathrm{2}}$ in Eq. (5) can be replaced by ${\mathit{\sigma}}_{\mathit{i}}^{\mathrm{2}}\mathrm{=}{\mathit{\sigma}}_{\mathit{y,i}}^{\mathrm{2}}\mathrm{+}{\mathit{\sigma}}_{\mathrm{int}}^{\mathrm{2}}$ (e.g. Weiner et al. 2006; Andreon & Hurn 2012). We test for intrinsic scatter using mpfitexy (Markwardt 2009; Williams et al. 2010), but in most cases, due to the small χ^{2} values, find the respective parameter not invoked. Thus we decide against this additional complexity. A strength of Eq. (5) is its invariance under changing x and y (e.g. Tremaine et al. 2002); i.e., we do not assume either to be “the independent variable”.
Fig. 1 Scaling of weak lensing masses ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$ with hydrostatic masses ${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$. The upper (lower) panel is for c_{fit} (c_{B13}). Both show best fits for three cases: the default (filled, thick ring, dotted ring symbols; thick dashed line), regular shear profile clusters only (filled and thick ring symbols; dashdotted line; Sect. 4.1), and without correction for dilution by cluster members (filled, thin ring, dotted ring symbols; long dashed line; Sect. 4.3). The dotted line shows equality of the two masses, ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{=}{\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$. Shaded regions indicate the uncertainty range of the default bestfit. Some error bars were omitted for sake of clarity. 
Rather than propagating the (unknown) distribution functions in the mass uncertainties^{3}, we approximate 1σ Gaussian uncertainties in decadic logspace, applying the symmetrisation: $\begin{array}{ccc}{\mathit{\sigma}}_{\mathrm{\left(}\mathrm{log}{\mathit{\xi}}_{\mathit{i}}\mathrm{\right)}}\mathrm{=}\mathrm{log}\mathrm{\left(}\mathrm{e}\mathrm{\right)}\mathrm{(}{\mathit{\xi}}_{\mathit{i}}^{\mathrm{+}}\mathrm{}{\mathit{\xi}}_{\mathit{i}}^{\mathrm{}}\mathrm{)}\mathit{/}\mathrm{\left(}\mathrm{2}{\mathit{\xi}}_{\mathit{i}}\mathrm{\right)}\mathrm{=}\mathrm{log}\mathrm{\left(}\mathrm{e}\mathrm{\right)}\mathrm{(}{\mathit{\sigma}}_{\mathit{\xi ,i}}^{\mathrm{+}}\mathrm{+}{\mathit{\sigma}}_{\mathit{\xi ,i}}^{\mathrm{}}\mathrm{)}\mathit{/}\mathrm{\left(}\mathrm{2}{\mathit{\xi}}_{\mathit{i}}\mathrm{\right)}\mathit{,}& & \end{array}$(6)where ${\mathit{\xi}}_{\mathit{i}}^{\mathrm{+}}\mathrm{=}{\mathit{\xi}}_{\mathit{i}}\mathrm{+}{\mathit{\sigma}}_{\mathit{\xi ,i}}^{\mathrm{+}}$ and ${\mathit{\xi}}_{\mathit{i}}^{\mathrm{}}\mathrm{=}{\mathit{\xi}}_{\mathit{i}}\mathrm{}{\mathit{\sigma}}_{\mathit{\xi ,i}}^{\mathrm{}}$ are the upper and lower limits of the 1σ interval (in linear space) for the datum ξ_{i}, given the uncertainties ${\mathit{\sigma}}_{\mathit{\xi ,i}}^{\mathrm{\pm}}$. All our calculations and plots use { x_{i} } : = { log ξ_{i} } and { σ_{x,i} } : = { σ_{(log ξi)} }, with log ≡ log _{10}.
3. Results
3.1. Weak lensing and hydrostatic masses
Measurements of the Xray – WL mass bias.
Fig. 2 Ratios between Xray and WL masses as a function of WL mass. Panel A) shows log (M^{hyd}/M^{wl}) within ${\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}$, panel B) shows log (M^{Y}/M^{wl}) within ${\mathit{r}}_{\mathrm{500}}^{\mathrm{Y}}$. WL masses assume the B13 c–M relation. We show three tests for a mass bias: the overall average logarithmic bias b = ⟨ log M^{X} − log M^{wl} ⟩ is denoted by a longdashed line, and its standard error by a dark grey shading. Shortdashed lines and light grey shading denote the same quantity, but obtained from averaging over Monte Carlo realisations including the jackknife test. We also show this b_{MC} for the lowM^{wl} and highM^{wl} clusters separately, with the 1σ uncertainties presented as boxes, for sake of clarity. As a visual aid, a dotdashed line depicts the Monte Carlo/jackknife bestfit of log (M^{X}/M^{wl}) as a function of M^{wl}. In addition, panel A) also contains this bestfit line (tripledotdashed) for the case without correction for cluster member dilution; the corresponding data points follow the Fig. 1 scheme. Indicated by uncertainty bars, panel B) also presents three highz clusters from High et al. (2012). 
The first and single most important observation is that hydrostatic masses ${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$, i.e. evaluated at r_{500} as found from weak lensing, and weak lensing masses ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$ roughly agree with each other (Table 1). Our second key observation is the very tight scaling behaviour between ${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ and ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$, as Fig. 1 shows. In all cases presented in Fig. 1, and most of the ones we tested, all data points are consistent with the bestfit relation. Consequently, the fits return small values of ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}\mathit{<}\mathrm{1}$ (see Table 2). Bearing in mind that we only use stochastic uncertainties, this points to some intrinsic correlation of the WL and hydrostatic masses. We will discuss this point in Sect. 4.2.
Finally, we find the slope of the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$ relation (dashed lines in Fig. 1) to be steeper than unity (dotted line): using the “default model”, i.e. the analysis described in Sect. 2, a fitexy fit yields 1.71 ± 0.64 for the “c_{fit}” case (concentration parameters from the shear profile fits, cf. Paper II; upper panel of Fig. 1), and 1.46 ± 0.57, if the B13 massconcentration relation is applied (“c_{B13}”; lower panel). The different slopes in the c_{fit} and c_{B13} cases are mainly due to the two clusters, CL 1641+4001 and CL 1701+6414, in which the weak lensing analysis revealed shallow tangential shear profiles due to extended surface mass plateaus (cf. Figs. 3 and 5 of Paper II). This will be the starting point for further analysis and interpretation in Sect. 4.1.
Although the c_{B13} slope is consistent with the expected 1:1 relation, such a ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ relation would translate to extreme biases between Xray and WL masses if extrapolated to higher and lower masses. Especially for masses of a few 10^{15}M_{⊙}, ample observations disagree with the extrapolated M^{wl}> 2M^{hyd}. We do not claim our data to have such predicting power outside its mass range. Rather, we focus on what can be learnt about the Xray/WL mass bias in our 0.4 ~ z ~ 0.5 mass range, which we, for the first time, study in the mass range down to ~1 × 10^{14}M_{⊙}.
We are using three methods to test for the biases between Xray and WL masses. First, we compute the logarithmic bias b = ⟨ log ξ − log η ⟩, which we define as the average logarithmic difference between two general quantities ξ and η. Its interpretation is that 10^{b}η is the average value corresponding to ξ. The uncertainty in b is given by the standard error of (log ξ − log η). Hence, our measurement of b = − 0.02 ± 0.04 for c_{B13} corresponds to a vanishing fractional bias of ⟨ M^{hyd} ⟩ ≈ (0.97 ± 0.09) ⟨ M^{wl} ⟩.
Given the small sample size, large uncertainties, and the tight scaling relations in Fig. 1 pointing to some correlation between the WL and Xray masses, we base our further tests on a Monte Carlo (MC) analysis including the jackknife test. For 10^{5} realisations, we chose with random δξ_{i,k} drawn from zeromean distributions assembled from two Gaussian halves with variances ${\mathit{\sigma}}_{\mathit{\xi ,i}}^{\mathrm{}}$ for the negative and ${\mathit{\sigma}}_{\mathit{\xi ,i}}^{\mathrm{+}}$ for the positive half^{4}. This provides a simple way of accommodating asymmetric uncertainties (cf. Paper II and Table 1). Then we take the logarithm and again symmetrise the errors. We repeat for . On top, for each realisation , we discard one cluster after another, yielding a total of 8 × 10^{5} samples.
Based on those MC/jackknife realisations, we compute our second bias estimator . In order to achieve the best possible robustness against large uncertainties and small cluster numbers, we quote the ensemble median and dispersion. We find ${\mathit{b}}_{\mathrm{MC}}\mathrm{=}\mathrm{0.0}{\mathrm{0}}_{0.13}^{\mathrm{+}\mathrm{0.14}}$ for c_{B13}, in good agreement with b = − 0.02 ± 0.04, i.e. a median WL/Xray mass ratio of 1.
Fitting log (M^{X}/M^{wl}) as a function of M^{wl} and averaging over the MC/jackknife samples, we obtain our third bias estimator, an intercept A at the pivot mass of $\mathrm{log}{}^{\mathrm{(}}\mathit{M}_{\mathrm{piv}}\mathit{/}\mathrm{M}{{}_{\mathrm{\odot}}}^{\mathrm{)}}\mathrm{=}\mathrm{14.5}$. We find $\mathit{A}\mathrm{=}\mathrm{}\mathrm{0.0}{\mathrm{2}}_{0.08}^{\mathrm{+}\mathrm{0.07}}$ for c_{B13}, again consistent with vanishing bias.
3.2. Lensing masses and Xray masses from proxies
Figures 2 and A.2, as well as Table 2 present the three different Xray/WL mass bias estimates for four Xray mass observables: ${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ using c_{B13} from Sect. 3.1 in Panel A of Fig. 2, ${\mathit{M}}_{\mathrm{500}}^{\mathrm{Y}}$ in Panel B of Fig. 2, ${\mathit{M}}_{\mathrm{500}}^{\mathrm{T}}$ in Panel A of Fig. A.2, ${\mathit{M}}_{\mathrm{500}}^{\mathrm{G}}$ in Panel B of Fig. A.2. The last three are the proxybased Vikhlinin et al. (2009a) Xray mass estimators defined in Sect. 2.5. While Panel A uses ${\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}$, the other three panel use the respective ${\mathit{r}}_{\mathrm{500}}^{\mathrm{Y}\mathit{,}\mathrm{T}\mathit{,}\mathrm{G}}$.
Longdashed lines and darkgrey boxes in Fig. 2 display b and its error. Shortdashed lines and lightgrey boxes denote b_{MC}. The intercept A is located at the intersection of the dotdashed fit and dotted zero lines. We also show b_{MC} for the lowmass and highmass clusters separately, splitting at M^{wl}E(z) = M_{piv}; the respective 1σ ranges are shown as outline boxes.
We observe remarkable agreement between the Xray/WL mass ratios and bias fits from all four Xray observables (which are not fully independent). For each of them, all three bias estimates agree with each other, and all are consistent with no Xray/WL mass bias. We find no evidence for Xray masses being biased low by ~40% in our cluster sample, as it has been suggested to explain the Planck CMB – SZ cluster counts discrepancy. While b_{MC} ≥ 0.2 (≳35% mass bias) lies within the possible range of the highM^{wl} bin, in particular using the gas mass M^{G}, the overall cluster sample does not support this hypothesis. We point out that b_{MC} was designed to be both robust against possible effects of large uncertainties and the small number of clusters in this first batch of 400d WL clusters. The larger uncertainties in b_{MC} compared to b are directly caused by the jackknife test and the account for possible fit instability in the MC method.
The slopes B quantifying the M^{wl} dependence of the Xray/WL mass ratio are significantly negative in all of our measurements. This directly corresponds to the steep slope in the massmass scaling (Fig. 1). Predicting cluster masses for very massive clusters (or lowmass groups) from B = − 0.75 for M^{Y} would yield M^{X} = M^{wl}/ 2 at 10^{15} M_{⊙}/E(z) (and M^{X} = 2.8 M^{wl} at 10^{14}M_{⊙}/E(z)). Such ratios are at odds with existing measurements. Therefore, we refrain from extrapolating cluster masses, but interpret the slopes B as indicative of a possibly massdependent Xray/WL mass ratio. This evidence is more prudently presented as the ~2σ discrepancy between the lowM^{wl} and highM^{wl} mass bins for all three V09a Xray observables.
4. A massdependent bias?
In this section, we analyse two unexpected outcomes of our study in greater detail: the clear correlation between the M^{X}/M^{wl} measurement of the individual clusters and their lensing masses, and the unusually small scatter in our scaling relations. Results for ancillary scaling relations that we present in Appendix A underpin the findings of Fig. 1 and Table 2. To begin with, we emphasise that the massdependent bias is not caused by the conflation of a large z range; all but one of our clusters inhabit the range 0.39 ≤ z ≤ 0.53 across which E(z) varies by <10%, and we accounted for this variation. As Fig. A.2 shows, this also leaves us with little constraining power with regard to a zdependent bias, at least until the 400d WL survey becomes more complete.
4.1. Role of c_{200} and departures from NFW profile
Figure 1 shows that the M^{wl}–M^{X} scaling relation sensitively depends on the choice for the cluster concentration parameter c_{200}. This translates into more positive bias estimates for c_{fit} as compared to c_{B13} (Table 2). The difference is caused by the two flatprofile clusters for which NFW fits yield low masses but do not capture all the largescale mass distribution, in particular if c_{200} is determined directly from the data, rather than assuming a massconcentration relation (cf. the discussion of the concentration parameter in Paper II and Foëx et al. 2012). This induces a bias towards low masses in the r_{200} → r_{500} conversion. If these two “irregular” clusters (dotted ring symbols in Fig. 1) are excluded, the “regular clusters only” M^{wl}–M^{X} scaling relations (dashdotted lines) differ for the c_{fit}, but not for the c_{B13} case. Moreover, their mass ratios are consistent with the other highM^{wl} clusters for c_{B13}. We thus confirm that assuming a massconcentration relation and marginalising over c_{200} is advantageous for scaling relations. We note that Comerford et al. (2010) observed a correlation between the scatters in the massconcentration and masstemperature relations and advocated the inclusion of unrelaxed clusters in scaling relation studies.
Because NFW profile fits do not capture the complete projected mass morphologies of irregular clusters, the assumption of that profile for ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$ (Eq. 3), and ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{Y}}\mathrm{\right)}$, etc. (Sect. 3.2) could introduce a further bias. Aperturebased lensing masses, e.g. the ζ_{c} statistics (Clowe et al. 1998) employed by Hoekstra et al. (2012) provide an alternative. However, Okabe et al. (2013) demonstrated by the stacking of 50 clusters from the Local Cluster Substructure Survey (0.15 <z< 0.3), that the average weak lensing profile does follow NFW to a high degree, at least at low redshift. Furthermore, the Planck Collaboration (2013) finds a trend of M^{wl}/M^{hyd} with the ratio of concentration parameters measured from weak lensing and Xrays.
4.2. Correlation between mass estimates
In Table 2, we quote ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}$ for the M^{wl}–M^{X} scaling relations, using the same MC/jackknife samples as for the bias tests. For the ones involving hydrostatic masses, we measure $\mathrm{0.5}\mathit{<}{\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}\mathit{<}\mathrm{0.6}$. We evaluated ${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ at ${\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}$ in order to measure both estimates within the same physical radius, in an “apples with apples” comparison. But using the lensingderived radius also introduces an unknown amount of correlation, a possible (partial) cause of the measured low ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}$ values.
We test for the impact of the correlation by measuring both M^{wl} and M^{hyd} within a fixed physical radius for all clusters, and choose r_{fix} = 800 kpc as a rough sample average of r_{500}. Surprisingly, we find an only slightly higher ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}$, still <1 (see Table 2). As before, the bias estimators are consistent with zero. Fixed radii of 600 kpc and 1000 kpc give similar results (Table 2 and Fig. A.2). with a tentative trend of increasing ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}$ with smaller radii. Interestingly, a low ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}$ is also found for the M^{wl}–M^{T} and M^{T}–M^{hyd} relations (Table A.2). The latter is expected, because M^{hyd} are derived from the same T_{X} and depend sensitively on them. This all suggests that the small scatter is not driven by using ${\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}$, but by some other intrinsic factor.
If the uncertainties in M^{wl} were overestimated significantly, this would obviously explain the low ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}$. However, we do not even include systematic effects here. Moreover, the quoted M^{wl} uncertainties directly come from the NFW modelling of Paper II and reflect the Δχ^{2} from their Eq. (3), given the shear catalogue. The errors are dominated by the intrinsic source ellipticity σ_{ε}, for which we, after shear calibration, find values of ~0.3, consistent with other groundbased WL experiments. Therefore, despite the allure of our WL errors being overestimated, we do not find evidence for this hypothesis in our shear catalogues. Furthermore, the quoted M^{wl} uncertainties are consistent with the aperture mass detection significances we reported in Paper II.
4.3. Dilution by cluster members and foregrounds
Comparing the complete set of massmass scaling relations our data offer (Table A.2), we trace the massdependence of the bias seen in Fig. 2 back to the different ranges spanned by the estimates for r_{500}. While the ratio of minimum to maximum is ≈0.75 for ${\mathit{r}}_{\mathrm{500}}^{\mathrm{Y}}$, ${\mathit{r}}_{\mathrm{500}}^{\mathrm{T}}$, and ${\mathit{r}}_{\mathrm{500}}^{\mathrm{hyd}}$, the same ratio is 0.57 for ${\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}$, using the B13 M–c relation. In the following, we discuss the potential influence of several sources of uncertainty in the WL masses, showing that the dispersion between lowest and highest ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$ is likely an inherent feature rather than a modelling artefact.
In Paper II, we discussed the great effort we took in constructing the bestpossible homogeneous analysis from the quite heterogeneous MMT imaging data. Unfortunately, we happen to find higher M^{wl} for all clusters with imaging in three bands than for the clusters with imaging in one band. We emphasise there are no trends with limiting magnitude, seeing, or density n_{KSB} of galaxies with measurable shape (cf. Tables 1 and 2 in Paper II).
In the cases where threeband imaging is available, our WL model includes a correction for the diluting effect residual cluster member galaxies impose on the shear catalogue. For the other clusters, no such dilution correction could be applied. A rough estimate of the fraction of unlensed galaxies remaining after background selection suggests that the contamination in singleband catalogues is ~30–50% higher than with the more sophisticated galaxycolour based method. Therefore, we recalculate the scaling relations, switching off the dilution correction (long dashed line and thin ring symbols in Fig. 1). This lowers the r_{500} values by ~6% and the masses by 10–15%. For both the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ and ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{Y}}$ relations, we only observe a slightly smaller difference between b_{MC} for the high and lowM^{wl} bins (Tables 2 and A.2), not significant given the uncertainty margins.
Fig. 3 Comparisons with literature data. Left panel: black symbols show z> 0.35 clusters from Mahdavi et al. (2013), whose bestfit using Eq. (5) is shown by the dotdashed line. The cluster CL 1524+0957 is indicated by a diamond symbol. Coloured symbols and the dashed line show the “default” ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ relation for c_{B13} as in the lower panel of Fig. 1. Middle panel: the same, but comparing to Foëx et al. (2012) (black symbols and dotdashed line for best fit). Xray masses are measured within ${\mathit{r}}_{\mathrm{500}}^{\mathrm{hyd}}$. CL 1003+3253 and CL 1120+4318 are emphasised by special symbols. Right panel: scaling of lensing masses ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$ with the Y_{X} proxy. Black symbols show the z> 0.35 clusters from M13, to which the thick, dashdotted line is the best fit. Shaded regions indicate the uncertainties to this fit. The thin, dashdotted line gives the best fit M13 quote for their complete sample, while the thin solid and longdashed lines mark the M_{500}–Y_{X} relations by V09a and Arnaud et al. (2010), respectively, for z = 0.40. 
The dilution of the shear signal by an increased number of galaxies not bearing a shear signal can also be expressed as a overestimation of the mean lensing depth ⟨ β ⟩. We model a possible lensing depth bias by simultaneously adding the uncertainty σ( ⟨ β ⟩) for the threeband clusters and subtracting it for the singleband ones, maximising the leveraging effect on the masses. Similar to the previous experiment^{5} we still observe a massdependent bias, with little change to the default model.
We further tested alternative choices of cluster centre and fitting range, but do not observe significant changes to the mass dependentbias or to ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}$ (see Appendix A.3). Although ⟨ β ⟩ is calculated for all clusters from the same catalogues, related systematics would affect the mass normalisation, but not the relative stochastic uncertainties, which determine ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}$. As we consider a drastic overestimation of the purely statistic uncertainties in the WL modelling being unlikely (Sect. 4.2), the cause of the low ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}$ values remains elusive. If a WL analysis effect is responsible for one or both anomalies, it has to be of a more subtle nature than the choices investigated here.
4.4. A statistical fluke?
We summarise that the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ scaling relations we observe show an unusual lack of scatter and that we find a ~2σ difference between the Xray/WL mass ratios of our high and lowM^{wl} clusters. The latter effect can be traced back to the considerable span in cluster lensing signal, which is only partly due to different background selection procedures and the dilution correction that was only applied for clusters imaged in three bands.
The question then arises if the massdependent bias is caused by an unlucky selection of the 8 MMT clusters from V09’s wider sample of 36. The 8 clusters were chosen to be observed first merely because of convenient telescope scheduling, and appear typical of the larger sample in terms of redshift and Xray observables. The MMT clusters trace well the mass range and dispersion spanned by all 36 clusters in their ${\mathit{M}}_{\mathrm{500}}^{\mathrm{Y}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{T}}$ relation. We observe the expected vanishing slopes for log (M^{T}/M^{Y}) as a function of M^{Y}, both for the 8 MMT clusters and for the complete sample of 36 (Table A.2).
In Table 2, we observe significant scatter (${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}\mathrm{=}\mathrm{2.11}$) in the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{G}}$ relation, while Okabe et al. (2010b) and M 13 reported particularly low scatter in M^{G}, comparing to WL masses. This large intrinsic scatter seems to be a feature of the overall 400d sample: plotting M^{G} versus the two other V09 Xray masses of all 36 clusters, we also find ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}\mathit{>}\mathrm{2}$ (Table A.2), as well as significant nonzero logarithmic biases. While tracing the cause of this observation is beyond the scope of this article, it deserves further study. Because two of the clusters with highest $\left{\mathit{M}}_{\mathrm{500}}^{\mathrm{Y}\mathit{,}\mathrm{T}}\mathrm{}{\mathit{M}}_{\mathrm{500}}^{\mathrm{G}}\right$ are covered by our MMT subsample, we observe a more massdependent M^{G}/M^{Y,T} ratio than for all 36. Overall, however, the MMT subsample is not a very biased selection.
4.5. Physical causes
An alternative and likely explanation for the massdependent bias we observe could be a high rate of unrelaxed clusters, especially for our least massive objects. If the departure from hydrostatic equilibrium were stronger among the lowmass clusters than for the massive ones, this would manifest in mass ratios similar to our results. Simulations show the offset from hydrostatic equilibrium to be massdependent (Rasia et al. 2012), despite currently being focused on the highmass regime. Variability in the nonthermal pressure support with mass (Laganá et al. 2013) may be exacerbated by small number statistics. At high z, the expected fraction of merging clusters, especially of major mergers, increases. Unrelaxed cluster states are known to affect Xray observables and, via the NFW fitting, also lensing mass estimates. Indeed, the two most deviant systems in Fig. 2 are CL 1416+4446 and the flatprofile “shear plateau” cluster CL 1641+4001. Although the first shows an inconspicuous shear profile, we suspect it to be part of a possibly interacting supercluster, based on the presence of two nearby structures at the same redshift, detected in Xray as well as in our lensing maps (Paper II). Both these clusters are classified as nonmergers in the recent Nurgaliev et al. (2013) study, introducing a new substructure estimator based on Xray morphology. However, WL and Xray methods are sensitive to substructure on different radial and mass scales, such that this explanation cannot be ruled out. We summarise that the greater dynamical range in WL than in Xray masses might be linked to different sensitivities of the respective methods to substructure and mergers in the lowmass, highz cluster population we are probing, but which is currently still underexplored.
5. Comparison with previous works
5.1. The M$\begin{array}{c}\mathrm{wl}\\ \mathrm{500}\end{array}$–M$\begin{array}{c}\mathrm{hyd}\\ \mathrm{500}\end{array}$ relation
Comparison with Mahdavi et al. (2013) results:
recently, M13 published scaling relations observed between the weak lensing and Xray masses for a sample of 50 massive clusters, partly based on the brightest clusters from the Einstein Observatory Extended Medium Sensitivity Survey (Gioia et al. 1990). Weak lensing masses for the M13 sample have been obtained from CFHT/Megacam imaging (Hoekstra et al. 2012), while the Xray analysis combines XMMNewton and Chandra data. While the median redshift is z = 0.23, the distribution extends to z = 0.55, including 12 clusters at z> 0.35. Owing to their selection, these 12 clusters lie above the 400d flux and luminosity cuts, making them directly comparable to our sample.
The left panel in Fig. 3 superimposes the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$ and ${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ of the M13 highz clusters on our results. The two samples overlap at the massive (≳5 × 10^{14}M_{⊙}) end, but the 400d objects probe down to 1 × 10^{14}M_{⊙} for the first time at this z and for these scaling relations. The slopes of the scaling relations are consistent: using Eq. (5), we measure B_{M − M} = 1.13 ± 0.20 for the 12 M13 objects. A joint fit with the 400d clusters (B_{M − M} = 1.46 ± 0.57) yields B_{M − M} = 1.15 ± 0.14 and a low ${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}\mathrm{=}\mathrm{0.54}$, driven by our data. We note that the logarithmic bias of b = 0.10 ± 0.05 for the M13 highz clusters corresponds to a (20 ± 10) % mass bias, consistent with both the upper range of the 400d results and expectation from the literature (e.g. Laganá et al. 2010; Rasia et al. 2012).
Calculating the Hogg et al. (2010, H10) likelihood which Mahdavi et al. (2013) use, we find ${\mathit{B}}_{\mathrm{H}\mathrm{10}}\mathrm{=}\mathrm{1.1}{\mathrm{8}}_{0.20}^{\mathrm{+}\mathrm{0.22}}$ and intrinsic scatter σ_{int} consistent with zero, confirming our above results. If we, however, repeating our fits from Fig. 1 with the H10 likelihood, we obtain discrepant results which highlight the differences between the various regression algorithms (see Sect. 2.6)^{6}.
CL 1524+0957 at z = 0.52 is the only cluster the 400d and M13 samples share. Denoted by a black diamond in Fig. 3, its masses from the M13 lensing and hydrostatic analyses blend in with the MMT 400d clusters. If it were included in the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ relation, it would not significantly alter the best fit, but we caution that different analysis methods have been employed, e.g. M13 reporting aperture lensing masses based on the ζ_{c} statistics.
Comparison with Jee et al. (2011) results:
Jee et al. (2011, J11) studied 14 very massive and distant clusters (0.83 <z< 1.46) and found their WL and hydrostatic masses ${\mathit{M}}_{\mathrm{200}}^{\mathrm{wl}}$ and ${\mathit{M}}_{\mathrm{200}}^{\mathrm{hyd}}$ to agree well, similar to our results. However, they caution that their ${\mathit{M}}_{\mathrm{200}}^{\mathrm{hyd}}$ were obtained by extrapolating a singular isothermal sphere profile. Because we doubt that the Chandrabased Vikhlinin et al. (2006) model reliably describes the ICM out to such large radii, we refrain from deriving ${\mathit{M}}_{\mathrm{200}}^{\mathrm{hyd}}$. Nevertheless, we notice that our the J11 samples not only shows similar ${\mathit{M}}_{\mathrm{200}}^{\mathrm{wl}}$ than our most massive clusters, but also contains the only two 400d clusters exceeding the redshift of CL 0230+1836, CL 0152−1357 at z = 0.83 and CL 1226+3332 at z = 0.89. Their planned reanalysis will improve the leverage of our samples at the highz end.
Comparison with Foëx et al. (2012) results:
in the middle panel of Fig. 3, we compare our results to 11 clusters from the EXCPRES XMMNewton sample, analysed by Foëx et al. (2012, F12) and located at a similar redshift range (0.41 ≤ z ≤ 0.61) as the bulk of our sample. Selected to be representative of the cluster population at z ≈ 0.5, these objects have been studied with XMMNewton in Xrays and CFHT/Megacam in the optical. Foëx et al. (2012) explicitly quote hydrostatic and lensing masses within their respective radii; thus we also show the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{hyd}}\mathrm{\right)}$. Again, the more massive of our clusters resemble the F12 sources, with the 400d MMT sample extending towards lower masses. Indeed, F12 study two clusters which are part of our sample: these, CL 1002+3253 at z = 0.42 and CL 1120+4318 at z = 0.60 mark their lowest lensing mass objects. At similar ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$ on either sides of the bestfit 400d scaling relation, their inclusion with the quoted masses would have no immediate effect on its slope, but slightly increase its scatter.
Bearing in mind that Fig. 3 (middle panel) compares quantities measured at different radii, we notice that the significantly flat best fit regression line to the F12 cluster masses, showing a larger dispersion in hydrostatic than in WL masses, as opposed to the 400d MMT clusters. The comparisons in Fig. 3 underscore that while being broadly compatible with each other, different studies are shaped by the fine details of their sample selection and analysis methods. We will conduct a more detailed comparison between our results and the ones of Foëx et al. (2012) and Mahdavi et al. (2013) once we reanalysed the CFHT/Megacam of the three overlapping clusters, having already shown the MMT and CFHT Megacams to produce consistent lensing catalogues (Paper II).
5.2. The ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{Y}}$ relation
The right panel of Fig. 3 investigates the scaling behaviour of ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$ with Y_{X} by comparing the 400d MMT clusters to the z> 0.35 clusters from M13^{7}. The difference between the two samples is more pronounced than in the left panel, with only the lowmass end of the M13 sample, including CL 1524+0957, overlapping with our clusters. None of the 400d MMT clusters deviates significantly from the M_{500}–Y_{X} relation applied by V09a in the derivation of the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{Y}}$ masses we used. The V09a M_{500}–Y_{X} relation based on Chandra data for lowz clusters (Vikhlinin et al. 2006) is in close agreement to the M13 result for their complete sample, as well as the widely used Arnaud et al. (2010)M_{500}–Y_{X} relation. For the latter as well as V09a we show the version with a slope fixed to the selfsimilar expectation of B = 3 / 5. The best fit to the M13 z> 0.35 essentially yields the same slope as the complete sample (B = 0.55 ± 0.09 compared to B_{H10} = 0.56 ± 0.07, calculated with the H10 method). The higher normalisation for the highz subsample can be likely explained as Malmquist bias due to the effective higher mass limit in the M13 sample selection. The incompatibility of the least massive MMT clusters with this fit highlights that we sample lower mass clusters, which, at the same redshift, are likely to have different physical properties.
The Y_{X} proxy is the Xray equivalent to the integrated pressure signal Y_{SZ} seen by SZ observatories. Observations confirm a close Y_{X}–Y_{SZ} correlation, with measured departures from the 1:1 slope considered inconclusive (Andersson et al. 2011; Rozo et al. 2012). Performing a cursory comparison with SZ observations, we included in Fig. 2A data for three z> 0.35 clusters from High et al. (2012) (dashed uncertainty bars), taken from their Fig. 6. The abscissa values for the High et al. (2012) clusters (SPTCL J2022−6323, SPTCL J2030−5638, and SPTCL J2135−5726) show masses based on Y_{SZ}, derived from South Pole Telescope SZ observations (Reichardt et al. 2013). The M^{wl} are derived from observations with the same Megacam instrument we used for the 400d clusters, but after its transfer to the Magellan Clay telescope at Las Campanas Observatory, Chile. In good agreement with zero bias, the High et al. (2012) clusters are consistent with some of the lower mass 400d clusters. This result suggests that the Y_{X}–Y_{SZ} equivalence might hold once larger samples at high z and low masses will become available.
6. Summary and conclusions
In this article, we analysed the scaling relation between WL and Xray masses for 8 galaxy clusters drawn from the 400d sample of Xrayluminous 0.35 ≤ z ≤ 0.89 clusters. WL masses were measured from the Israel et al. (2012) MMT/Megacam data, and Xray masses were based on the V09a Chandra analysis. We summarise our main results as follows:

1.
We probe the WL–Xray mass scaling relation,in an unexplored region of the parameter spacefor the first time: the z ~ 0.4–0.5 redshift range, down to1 × 10^{14}M_{⊙}.

2.
Using several Xray mass estimates, we find the WL and Xray masses to be consistent with each other. Most of our clusters are compatible with the M^{X} = M^{wl} line.

3.
Assuming the M^{wl} not to be significantly biased, we do not find evidence for a systematic underestimation of the Xray masses by ~40%, as suggested as a possible solution to the discrepancy between the Planck CMB constraints on Ω_{m} and σ_{8} (the normalisation of the matter power spectrum) and the Planck SZ cluster counts (Planck Collaboration XX 2014). While our results favour a small WLXray mass bias, they are consistent with both vanishing bias and the ~20% favoured by studies of nonthermal pressure support.

4.
For the massmass scaling relations involving M^{wl}, we observe a surprisingly low scatter $\mathrm{0.5}\mathit{<}{\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}\mathit{<}\mathrm{0.6}$, although we use only stochastic uncertainties and allow for correlated errors via a Monte Carlo method. Because the errors in M^{wl} are largely determined by the intrinsic WL shape noise σ_{ε}, we however deem a drastic overestimation unlikely (Sect. 4.2). For the scaling relations involving M_{G}, however, we observe a large scatter, contrary to Okabe et al. (2010b) and M13.

5.
Looking in detail, there are intriguing indications for a massdependence of the WLXray mass ratios of our relatively lowmass z ~ 0.4–0.5 clusters. We observe a mass bias in the low–M^{wl} mass bin at the ~2σ level when splitting the sample at log (M_{piv}/M_{⊙}) = 14.5 This holds for the masses V09a report based on the Y_{X}, T_{X}, and M_{G} proxies.
The discrepant temperatures Chandra and XMMNewton measure in clusters (Nevalainen et al. 2010; Schellenberger et al. 2012) could provide a possible avenue to reconcile Planck cluster properties with the Planck cosmology.
We thoroughly investigate possible causes for the massdependent bias and tight scaling relations. First (Sect. 4.1), we confirm that by using a massconcentration relation instead of directly fitting c_{200} from WL, we already significantly reduced the bias due to conversion from r_{200} to r_{500}. We emphasise that, on average, the NFW shear profile represents a suitable fit for the cluster population (cf. Okabe et al. 2013). Measuring M^{hyd} within ${\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}$ induces correlation between the data points in Fig. 1. Removing this correlation by plotting both masses within a fixed physical radius, we still find small scatter (Sect. 4.2).
We notice that the mass range occupied by the M^{wl} exceeds the Xray mass ranges. Partially, this higher WL mass range can be explained by the correction for dilution by member galaxies, which could be applied only where colour information was available (Paper II). Coincidentally, this is the case for the more massive half of the MMT sample in terms of M^{wl}, thus boosting the range of measured WL masses (Sect. 4.3). This result underscores the importance of correcting for the unavoidable inhomogeneities in WL data due to the demanding nature of WL observations (cf. Applegate et al. 2014). We find no further indications for biases via the WL analysis. Furthermore the tight scaling precludes strong redshift effects, and we find that our small MMT subsample is largely representative of the complete sample of 36 clusters, judging from the M^{Y}–M^{T} relation (Sect. 4.4). For the M^{Y}–M^{G} and M^{T}–M^{G} relations, significant scatter (${\mathit{\chi}}_{\mathrm{red}}^{\mathrm{2}}\mathit{>}\mathrm{2}$) is present in the larger sample. The former relation also shows indications for a significant bias of M^{Y} ≈ 1.15M^{G}.
Weak lensing and hydrostatic masses for the 400d MMT clusters are in good agreement with the z> 0.35 part of the Mahdavi et al. (2013) sample and the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ relation derived from it (Sect. 5.1). The M13 and Foëx et al. (2012) samples include three 400d clusters with CFHT WL masses. These clusters neither point to significantly higher scatter nor to a less massdependent bias (Fig. 3). We are planning a reanalysis of the CFHT data, having demonstrated in Paper II that lensing catalogues from MMT and CFHT are nicely compatible. Such reanalysis is going to be helpful to identify more subtle WL analysis effects potentially responsible for the steep slopes and tight correlation of WL and Xray masses.
An alternative explanation are intrinsic differences in the lowmass cluster population. That the 400d MMT sample probes to slightly lower masses (1 × 10^{14}M_{⊙}) than M13 or F12 becomes especially obvious from the M^{wl}–Y^{X} relation (Fig. 3, Sect. 5.2). Because the 400d sample is more representative of the z ~ 0.4–0.5 cluster population, it is likely to contain more significant mergers relative to the cluster mass, skewing mass estimates (Sect. 4.5). Hence, the 400d survey might be the first to see the onset of a mass regime in which cluster physics and substructure lead the WLXray scaling to deviate from what is known at higher masses. Remarkably, Giles et al. (2014) are finding a different steep slopes in their lowmass WLXray scaling analysis. Detailed investigations of how their environment shapes clusters like CL 1416+4446 might be necessary to improve our understanding of the cluster population to be seen by future cosmology surveys. Analysis systematics might also behave differently at lower masses. A turn for WL cluster science towards lower mass objects, e.g. through the completion of the 400d WL sample, will help addressing the question of evolution in lensing mass scaling relations.
Note added in proof. After this paper was accepted, another paper (von der Linden et al. 2014b) appeared as submitted, pointing at the combined effects of hydrostatic mass bias and calibration systematics. The cross calibration effect on cosmological parameter constraints is currently being tested directly by Schellenberger et al. (in prep.). Rozo et al. (2014a,b) review and crosscalibrate the various scaling relations involved. Alternatively, Burenin (2013) suggested additional massive neutrino species.
Online material
Appendix A: Further scaling relations and tests
Fig. A.1 Lensing mass – Xray luminosity relation. The M–L_{X} relation is shown, for both ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$ (filled circles) and ${\mathit{M}}_{\mathrm{500}}^{\mathrm{Y}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{Y}}\mathrm{\right)}$ (small triangles). Open triangles represent the sample clusters for which MMT lensing masses are not available. The V09a M–L_{X} relation at z = 0.40 (z = 0.80) is denoted by a longdashed black (shortdashed red) line. Shaded (hatched) areas show the respective 1σ intrinsic scatter ranges. 
Appendix A.1: The L_{X}–M relation
To better assess the consistency of our weak lensing masses with the Vikhlinin et al. (2009a) results, we compare them to the L_{X}–M_{Y}relation derived by V09a using the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{Y}}$ masses of their lowz cluster sample. Figure A.1 inverts this relation by showing the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$ masses as a function of the 0.5–2.0 keVChandra luminosities measured by V09a. Statistical uncertainties in the Chandra fluxes and, hence, luminosities are negligible for our purposes. We calculate the expected 68 % confidence ranges in mass for a given luminosity by inverting the scatter in L_{X} at a fixed M^{Y} as given in Eq. (22) of V09a. For two fiducial redshifts, z = 0.40 and z = 0.80, spanning the unevenly populated redshift range of the eight clusters, the M–L_{X} relations and their expected scatter are shown in Fig. A.1. Small filled triangles in Fig. A.1 show the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{Y}}$ masses from which V09a derived the L_{X}–M relation. Our 8 MMT clusters are nicely tracing the distribution of the overall sample of 36 clusters (open triangles).
As an important step in the calculation of the mass function, these authors show that their procedure is able to correct for the Malmquist bias even in the presence of evolution in the L_{X}–M
relation, which they include in the model. We emphasise that the Malmquist bias correction – which is not included here – applied by V09a moves the clusters upwards in Fig. A.1, such that the sample agrees with the bestfit from the lowz sample, as Fig. 12 in V09a demonstrates.
As already seen in Fig. 2, the M^{wl} (large symbols in Fig. A.1) and M^{Y} agree well. Thus we can conclude that the WL masses are consistent with the expectations from their L_{X}. Finally, we remark that the higher Xray luminosities for the some of the same clusters reported by Maughan et al. (2012) in their study of the L_{X}–T_{X} relation are not in disagreement with V09a, as Maughan et al. (2012) used bolometric luminosities.
Appendix A.2: Redshift scaling and crossscaling of Xray masses
Here we show further results mentioned in the main body of the article. Figure A.2 shows two examples of the Xray/WL mass ratio as a function of redshift. Owing to the inhomegenous redshift coverage of our clusters, we cannot constrain a redshift evolution. All of our bias estimates are consistent with zero bias.
Table A.2 shows the fit results and bias estimates for various tests we performed modifying our default model, as well as for ancillary scaling relations. In particular, we probe the scaling behaviour of hydrostatic masses against the V09a estimates, for which we find a M^{Y}/M^{hyd} tentatively biased high by ~15%, while M^{T} and M^{G} do not show similar biases.
Appendix A.3: Choice of centre and fitting range
Weak lensing masses obtained from profile fitting have been shown to be sensitive to the choice of the fitting range (Becker & Kravtsov 2011; Hoekstra et al. 2011b; Oguri & Hamana 2011). Taking these results into account, we fitted the WL masses within a fixed physical mass range. Varying the fitting range by using r_{min} = 0 instead of 0.2 Mpc in one and r_{max} = 4.0 Mpc instead of 5.0 Mpc in another test, we find no evidence for a crucial influence on our results.
Both simulations and observations establish (e.g. Dietrich et al. 2012; George et al. 2012) that WL masses using lensing cluster centres are biased high due to random noise with respect to those based on independently obtained cluster centres, e.g. the ROSAT centres we employ. The fact that the ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ relation gives slightly milder difference between b_{MC} for the high and lowM^{wl} bins when the peak of the Sstatistics is assumed as the cluster centre (Table A.2) can be explained by the larger relative M^{wl} “boost” for clusters with larger offset between Xray and lensing peaks. This affects the flatprofile clusters (Sect. 4.1) in particular, translating into a greater effect for the c_{fit} case than for c_{B13}based masses. We find that WL cluster centres only slightly alleviate the observed massdependence.
Fig. A.2 Continuation of Fig. 2. Panel A) shows log (M^{T}/M^{wl}) within ${\mathit{r}}_{\mathrm{500}}^{\mathrm{T}}$, panel B) shows log (M^{G}/M^{wl}) within ${\mathit{r}}_{\mathrm{500}}^{\mathrm{G}}$. Like panel A) of Fig. 2, panel C) presents log (M^{hyd}/M^{wl}), but showing both WL masses measured at a fixed physical radius r_{fix}. Filled dots and dotdashed lines correspond to r_{fix} = 800 kpc, while triangles and tripledotdashed lines denote r_{fix} = 600 kpc. Uncertainties for the 600 kpc case were omitted for clarity. Panel D) shows log (M^{hyd}/M^{wl}) from Fig. 2 as a function of redshift. Thin solid lines indicating the 1σ uncertainty range of the bestfit Monte Carlo/jackknife regression line (dotdashed). 
In fact, regression lines not only depend on the likelihood or definition of the best fit, but also on the algorithm used to find its extremum, and, if applicable, how uncertainties are transferred from the linear to the logarithmic domain. Thus, our H10 slopes agree with the ones the webtool provided by M13 yield, but produce different uncertainties.
Acknowledgments
The authors express their thanks to M. Arnaud and EXCPRES collaboration (private communication) for providing the hydrostatic masses of the Foëx et al. (2012) clusters. We further thank A. Mahdavi for providing the masses of the Mahdavi et al. (2013) clusters via their helpful online interface. H.I. likes to thank M. Klein, J. Stott, and Y.Y. Zhang, and the audiences of his presentations for useful comments. The authors thank the anonymous referee for their constructive suggestions. H.I. acknowledges support for this work has come from the Deutsche Forschungsgemeinschaft (DFG) through Transregional Collaborative Research Centre TR 33 as well as through the Schwerpunkt Program 1177 and through European Research Council grant MIRGCT208994. T.H.R. acknowledges support by the DFG through Heisenberg grant RE 1462/5 and grant RE 1462/6. T.E. is supported by the DFG through project ER 327/31 and by the Transregional Collaborative Research Centre TR 33 “The Dark Universe”. R.M. is supported by a Royal Society University Research Fellowship. We acknowledge the grant of MMT observation time (program 2007B0046) through NOAO public access. MMT time was also provided through support from the F. H. Levinson Fund of the Silicon Valley Community Foundation.
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All Tables
All Figures
Fig. 1 Scaling of weak lensing masses ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$ with hydrostatic masses ${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$. The upper (lower) panel is for c_{fit} (c_{B13}). Both show best fits for three cases: the default (filled, thick ring, dotted ring symbols; thick dashed line), regular shear profile clusters only (filled and thick ring symbols; dashdotted line; Sect. 4.1), and without correction for dilution by cluster members (filled, thin ring, dotted ring symbols; long dashed line; Sect. 4.3). The dotted line shows equality of the two masses, ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{=}{\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$. Shaded regions indicate the uncertainty range of the default bestfit. Some error bars were omitted for sake of clarity. 

In the text 
Fig. 2 Ratios between Xray and WL masses as a function of WL mass. Panel A) shows log (M^{hyd}/M^{wl}) within ${\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}$, panel B) shows log (M^{Y}/M^{wl}) within ${\mathit{r}}_{\mathrm{500}}^{\mathrm{Y}}$. WL masses assume the B13 c–M relation. We show three tests for a mass bias: the overall average logarithmic bias b = ⟨ log M^{X} − log M^{wl} ⟩ is denoted by a longdashed line, and its standard error by a dark grey shading. Shortdashed lines and light grey shading denote the same quantity, but obtained from averaging over Monte Carlo realisations including the jackknife test. We also show this b_{MC} for the lowM^{wl} and highM^{wl} clusters separately, with the 1σ uncertainties presented as boxes, for sake of clarity. As a visual aid, a dotdashed line depicts the Monte Carlo/jackknife bestfit of log (M^{X}/M^{wl}) as a function of M^{wl}. In addition, panel A) also contains this bestfit line (tripledotdashed) for the case without correction for cluster member dilution; the corresponding data points follow the Fig. 1 scheme. Indicated by uncertainty bars, panel B) also presents three highz clusters from High et al. (2012). 

In the text 
Fig. 3 Comparisons with literature data. Left panel: black symbols show z> 0.35 clusters from Mahdavi et al. (2013), whose bestfit using Eq. (5) is shown by the dotdashed line. The cluster CL 1524+0957 is indicated by a diamond symbol. Coloured symbols and the dashed line show the “default” ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$–${\mathit{M}}_{\mathrm{500}}^{\mathrm{hyd}}$ relation for c_{B13} as in the lower panel of Fig. 1. Middle panel: the same, but comparing to Foëx et al. (2012) (black symbols and dotdashed line for best fit). Xray masses are measured within ${\mathit{r}}_{\mathrm{500}}^{\mathrm{hyd}}$. CL 1003+3253 and CL 1120+4318 are emphasised by special symbols. Right panel: scaling of lensing masses ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}$ with the Y_{X} proxy. Black symbols show the z> 0.35 clusters from M13, to which the thick, dashdotted line is the best fit. Shaded regions indicate the uncertainties to this fit. The thin, dashdotted line gives the best fit M13 quote for their complete sample, while the thin solid and longdashed lines mark the M_{500}–Y_{X} relations by V09a and Arnaud et al. (2010), respectively, for z = 0.40. 

In the text 
Fig. A.1 Lensing mass – Xray luminosity relation. The M–L_{X} relation is shown, for both ${\mathit{M}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{wl}}\mathrm{\right)}$ (filled circles) and ${\mathit{M}}_{\mathrm{500}}^{\mathrm{Y}}\mathrm{\left(}{\mathit{r}}_{\mathrm{500}}^{\mathrm{Y}}\mathrm{\right)}$ (small triangles). Open triangles represent the sample clusters for which MMT lensing masses are not available. The V09a M–L_{X} relation at z = 0.40 (z = 0.80) is denoted by a longdashed black (shortdashed red) line. Shaded (hatched) areas show the respective 1σ intrinsic scatter ranges. 

In the text 
Fig. A.2 Continuation of Fig. 2. Panel A) shows log (M^{T}/M^{wl}) within ${\mathit{r}}_{\mathrm{500}}^{\mathrm{T}}$, panel B) shows log (M^{G}/M^{wl}) within ${\mathit{r}}_{\mathrm{500}}^{\mathrm{G}}$. Like panel A) of Fig. 2, panel C) presents log (M^{hyd}/M^{wl}), but showing both WL masses measured at a fixed physical radius r_{fix}. Filled dots and dotdashed lines correspond to r_{fix} = 800 kpc, while triangles and tripledotdashed lines denote r_{fix} = 600 kpc. Uncertainties for the 600 kpc case were omitted for clarity. Panel D) shows log (M^{hyd}/M^{wl}) from Fig. 2 as a function of redshift. Thin solid lines indicating the 1σ uncertainty range of the bestfit Monte Carlo/jackknife regression line (dotdashed). 

In the text 
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