Free Access
Volume 564, April 2014
Article Number A129
Number of page(s) 15
Section Cosmology (including clusters of galaxies)
Published online 17 April 2014

Online material

Appendix A: Further scaling relations and tests

thumbnail Fig. A.1

Lensing mass – X-ray luminosity relation. The MLX relation is shown, for both (filled circles) and (small triangles). Open triangles represent the sample clusters for which MMT lensing masses are not available. The V09a MLX relation at z = 0.40 (z = 0.80) is denoted by a long-dashed black (short-dashed red) line. Shaded (hatched) areas show the respective 1σ intrinsic scatter ranges.

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Appendix A.1: The LX–M relation

To better assess the consistency of our weak lensing masses with the Vikhlinin et al. (2009a) results, we compare them to the LXMY-relation derived by V09a using the masses of their low-z cluster sample. Figure A.1 inverts this relation by showing the masses as a function of the 0.52.0 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 LX at a fixed MY 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 MLX relations and their expected scatter are shown in Fig. A.1. Small filled triangles in Fig. A.1 show the masses from which V09a derived the LXM 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 LXM

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 best-fit from the low-z sample, as Fig. 12 in V09a demonstrates.

As already seen in Fig. 2, the Mwl (large symbols in Fig. A.1) and MY agree well. Thus we can conclude that the WL masses are consistent with the expectations from their LX. Finally, we remark that the higher X-ray luminosities for the some of the same clusters reported by Maughan et al. (2012) in their study of the LXTX relation are not in disagreement with V09a, as Maughan et al. (2012) used bolometric luminosities.

Appendix A.2: Redshift scaling and cross-scaling of X-ray masses

Here we show further results mentioned in the main body of the article. Figure A.2 shows two examples of the X-ray/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 MY/Mhyd tentatively biased high by ~15%, while MT and MG 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 rmin = 0 instead of 0.2 Mpc in one and rmax = 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 relation gives slightly milder difference between bMC for the high- and low-Mwl bins when the peak of the S-statistics is assumed as the cluster centre (Table A.2) can be explained by the larger relative Mwl “boost” for clusters with larger offset between X-ray and lensing peaks. This affects the flat-profile clusters (Sect. 4.1) in particular, translating into a greater effect for the cfit case than for cB13-based masses. We find that WL cluster centres only slightly alleviate the observed mass-dependence.

thumbnail Fig. A.2

Continuation of Fig. 2. Panel A) shows log (MT/Mwl) within , panel B) shows log (MG/Mwl) within . Like panel A) of Fig. 2, panel C) presents log (Mhyd/Mwl), but showing both WL masses measured at a fixed physical radius rfix. Filled dots and dot-dashed lines correspond to rfix = 800 kpc, while triangles and triple-dot-dashed lines denote rfix = 600 kpc. Uncertainties for the 600 kpc case were omitted for clarity. Panel D) shows log (Mhyd/Mwl) from Fig. 2 as a function of redshift. Thin solid lines indicating the 1σ uncertainty range of the best-fit Monte Carlo/jackknife regression line (dot-dashed).

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Table A.1

Continuation of Table 1.

Table A.2

Continuation of Table 2.

© ESO, 2014

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