LoCuSS: THE MASS DENSITY PROFILE OF MASSIVE GALAXY CLUSTERS AT z = 0.2^*,†

Gravitational lensing is a powerful probe of the matter distribution in galaxy clusters, because the observed signal is sensitive to the total matter distribution and insensitive to the physical processes at play within clusters. Many studies have therefore employed gravitational lensing to probe the mass and internal structure of galaxy clusters (Kneib & Natarajan 2011 and references therein). Prominent among these studies are those that aim to measure the dependence of cluster density on cluster-centric radius, i.e., the "density profile" of clusters (e.g., Miralda-Escude 1995; Smith et al. 2001; Gavazzi et al. 2003; Kneib et al. 2003; Dahle et al. 2003; Sand et al. 2004; Broadhurst et al. 2005; Limousin et al. 2007; Johnston et al. 2007; Oguri et al. 2009, 2012; Okabe et al. 2010; Umetsu et al. 2011; Newman et al. 2013). A major motivation is to test key predictions from the cold dark matter theory of structure formation: (1) the density profile of the dark matter halos posited to host galaxies and cluster of galaxies is predicted to be universal and follows a simple two-parameter model (Navarro et al. 1997) and (2) massive galaxy cluster-scale dark matter halos have concentrations⁵ of c₂₀₀ ≡ r₂₀₀/r_s ≃ 3–4 (e.g., Bullock et al. 2001; Dolag et al. 2004; Neto et al. 2007; Duffy et al. 2008; Zhao et al. 2009), and are thus "less concentrated" than less massive halos.

In order to probe the density profile across a large dynamic range, lensing studies have typically combined weak- and strong-lensing signals, and have thus been limited to small samples of strong-lensing-selected clusters. These studies typically find that strong-lensing clusters have high central concentrations in projection (e.g., Gavazzi et al. 2003; Kneib et al. 2003; Broadhurst et al. 2008; Umetsu et al. 2011; Oguri et al. 2012). Moreover, joint lensing and dynamical studies find that the density profile of the dark matter component may be shallower than predicted from cold dark matter simulations (Newman et al. 2013). Interpretation of these apparent tensions between observations and theory is complicated by possible selection biases, small sample size, lensing-projection bias caused by halo triaxiality, and the absence of baryons from the simulations upon which the predictions are based.

We adopt a complementary approach that aims to make progress on overcoming issues relating to sample size and selection, and lensing projection biases. Building on earlier stacked lensing studies (Dahle et al. 2003; Johnston et al. 2007; Okabe et al. 2010, hereafter Ok10; Umetsu et al. 2011; Oguri et al. 2012), we measure the mean density profile of massive clusters by stacking the weak-lensing signal from a sample of 50 approximately mass-selected clusters. Our sample comprises all clusters from the ROSAT All Sky Survey catalogs (Ebeling et al. 1998, 2000; Böhringer et al. 2004) that satisfy L_X[0.1–2.4 keV]/E(z)^2.7 ⩾ 4.2 × 10⁴⁴ erg s⁻¹, 0.15 ⩽ z ⩽ 0.30, n_H < 7 × 10²⁰ cm⁻², and −25° < δ < +65°, where E(z) ≡ H(z)/H₀ is the normalized Hubble expansion rate, and selecting on L_X/E(z)^2.7 mimics a mass selection (Popesso et al. 2005). We stress that our results are based on the full sample of 50 clusters; subsamples of clusters will be discussed in future articles.

In Section 2 we describe our data and analysis; in Section 3 we explain our results and compare with numerical simulations; and in Section 4 we summarize our conclusions. We use the concordance ΛCDM model of Ω_{M, 0} = 0.27, Ω_Λ = 0.73, and H₀ = 100 h km s⁻¹ Mpc⁻¹ (Komatsu et al. 2011). In this cosmology, the virial overdensity at the mean redshift of our cluster sample, 〈z〉 = 0.23, is Δ_vir = 113.77. All error bars are 68% confidence intervals unless otherwise stated.

We observed all 50 clusters with Suprime-Cam (Miyazaki et al. 2002) on the Subaru Telescope, as part of the Local Cluster Substructure Survey (LoCuSS⁶)—46 clusters through the V/i'-band filters, two each through the V/I_C- and g/i'-band filters. Hereafter, we refer to the bluer filter as V and the redder filter as i'. The FWHM of point sources in the V/i' bands is 06 ≲ FWHM ≲ 09 and 05 ≲ FWHM ≲ 07, respectively. Photometric calibration to ⩽10% precision in both filters was achieved via observations of Landolt standard stars, and double checked against Sloan Digital Sky Survey/DR8 stellar photometry (Eisenstein et al. 2011).

We measure the shape of faint galaxies using a modified version of Ok10's pipeline, based on the imcat⁷ (Kaiser et al. 1995, hereafter KSB). The main modification is to calibrate the KSB isotropic correction factor for individual objects using galaxies detected with high significance ν > 30 (Umetsu et al. 2010). This minimizes the inherent shear calibration bias in KSB+ methods in the presence of measurement errors (Okura & Futamase 2012).

We define a sample of background galaxies based on color with respect to the red sequence of early-type galaxies in each cluster. In principle, selecting red galaxies (ΔC ≡ (V − i') − (V  −  i')_ES0 > 0) yields a clean sample of background galaxies. In reality, a positive color cut is required to eliminate contamination by faint red cluster galaxies due to statistical errors and possible intrinsic scatter in galaxy colors (e.g., Broadhurst et al. 2005). In contrast, interpretation of "blue galaxy" (ΔC < 0) samples is complicated by star formation. For completeness, we include blue galaxies in this section; however, our results in Section 3 are based only on the red galaxy sample.

The mean tangential distortion strength averaged over (1) all galaxies satisfying each color cut, (2) cluster-centric radii of 0.1 h⁻¹ Mpc < r < 2.8 h⁻¹ Mpc, and (3) all 50 clusters, G₊ ≡ 〈〈〈g₊〉〉〉, increases monotonically with ΔC for red galaxies (Figure 1). We interpret the steep slope at 0 ≲ ΔC ≲ 0.3 as arising from contamination by cluster members. Indeed, the mean cluster-centric radius of red galaxies is an increasing function of ΔC at small ΔC (Figure 1). At ΔC ≳ 0.4, we interpret the shallow slope of G₊ as arising from a slowly increasing redshift of the faint red background population as ΔC increases. We therefore model the data with a Gaussian of width σ centered at ΔC = 0 (to represent the cluster population), and the mean lensing kernel, D(ΔC) ≡ 〈D_ls/D_s〉, for galaxies in the COSMOS photometric redshift catalog (Ilbert et al. 2009) that matches each color cut. The model for the color dependence of G₊ is therefore G₊(ΔC) = A D(ΔC) (1 − Bf(ΔC)), where A converts D into shear in a simple manner, B is the normalization of the Gaussian contaminant function at ΔC = 0, and $f(\Delta C>0)=[1-{\rm erf}(\Delta C/\sqrt{2}\sigma)]/2$. This model has three free parameters: A, B, and σ, and allows us to estimate explicitly the fraction of contaminant galaxies, f, as a function of ΔC.

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The best-fit model describes the red galaxies well (Figure 1, upper panel). We conservatively adopt a limit of 1% on contaminating fraction, which translates into a red color cut of ΔC > 0.475. We select galaxies redder than this cut for the results presented in Section 3; the mean number density of these galaxies is 5.3 ± 1.9 arcmin⁻² per cluster, where the uncertainty is the standard deviation among the 50 clusters. We therefore achieve a total stacked number density of red galaxies of 266.3 arcmin⁻².

For completeness, we applied the same methods to blue galaxies, describing the contaminating fraction as $f(\Delta C<0)=[1+{\rm erf}(\Delta C/\sqrt{2}\sigma)]/2$. The model does not describe the blue galaxies well, and we do not use them in Section 3.

Our results are based on stacking the red background galaxy sample, defined by ΔC > 0.475 (Section 2), for all 50 clusters in the sample.

3.1. Stacking and Modeling the Weak Shear Signal

We detect each individual cluster at a typical peak signal-to-noise ratio (S/N) of ≃ 4 in two-dimensional Kaiser & Squires (1993) mass reconstructions. We also stack the shear catalogs in physical length units centered on the respective brightest cluster galaxies (BCGs) and reconstruct the average cluster mass distribution for the full sample, with a peak S/N of 28 (Figure 2). Motivated by the symmetrical average mass map, we constructed the stacked tangential shear profile for the full sample (Figure 3) following the procedure of Umetsu et al. (2011). In brief, we center the catalogs on the respective BCGs, and stack in physical length units across the radial range 100 h⁻¹ kpc < r < 2.8 h⁻¹ Mpc, in 14 log-spaced bins. We detect the signal at S/N = 32.7, using the full covariance matrix to take into account projected uncorrelated large-scale structure and intrinsic ellipticity noise (e.g., Hoekstra 2003; Hoekstra et al. 2011; Oguri & Takada 2011; Umetsu et al. 2011; Oguri et al. 2012), computing the cosmic-shear contribution using the non-linear matter power spectrum (Smith et al. 2003) for the Wilkinson Microwave Anisotropy Probe 7 cosmology and the shape noise from the diagonal matrix. The 45°-rotated distortion component is consistent with a null signal, confirming that residual systematic errors are at least an order of magnitude smaller than the measured lensing signal.

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The stacked shear profile (Figure 3) is well described by the so-called Navarro–Frenk–White (NFW) profile: ρ∝x⁻¹(1 + x)⁻², where x ≡ r/r_s, and dlog ρ/dlog r = −2 at r = r_s (Navarro et al. 1997). We express our model fits in terms of the virial mass M_vir ≡ (4π/3)ρ_crΔ_virr_vir³, and the concentration parameter c_vir ≡ r_vir/r_s, where Δ_vir is the virial overdensity and ρ_cr is the critical density. We measure both parameters to sub-10% statistical precision (Table 1), obtaining a best-fit concentration parameter of c_vir = 5.4 ± 0.5. Indeed, the statistical errors on concentration are comparable with the differences between the predictions from different numerical simulations (Figure 3). Moreover, the observed concentration parameter exceeds the predicted concentration from numerical simulations (Duffy et al. 2008; Zhao et al. 2009; Bhattacharya et al. 2013; De Boni et al. 2012).

Table 1. Density Profile Models

Model	Shape Parametera	Mvir	cvirb	M200	c200b	$\chi ^2_{\rm min}/{\rm dof}$
		(1014 M☉ h−1)		(1014 M☉ h−1)
NFW	γ = 1	$7.19_{-0.50}^{+0.53}$	$5.41_{-0.45}^{+0.49}$	$5.98_{-0.38}^{+0.40}$	$4.22_{-0.36}^{+0.40}$	7.2/12
gNFW	$\gamma =1.27_{-0.37}^{+0.24}$	$7.50_{-0.65}^{+0.74}$	$4.88_{-0.86}^{+0.86}$	$6.15_{-0.44}^{+0.48}$	$3.79_{-0.69}^{+0.69}$	6.6/11
Einasto	$\alpha =0.188_{-0.058}^{+0.062}$	$7.49_{-0.73}^{+0.86}$	$4.92_{-0.80}^{+0.57}$	$6.15_{-0.45}^{+0.50}$	$3.82_{-0.66}^{+0.48}$	6.6/11

Notes. ^aParameter describing the shape of the mass density profile on small scales. ^bNFW-like concentration parameter defined by $c_\Delta ^{\rm NFW}=r_{\Delta }/r_{\rm s}$, $c_{\rm -2}^{\rm gNFW}=(r_{\Delta }/r_{\rm s})/(2-\gamma)$, and $c_{\rm \Delta }^{\rm Einasto}=r_{\Delta }/r_{-2}$.

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The NFW model fit described above does not motivate fitting more flexible models to our data (Table 1). Nevertheless, for completeness, we fit the generalized NFW (gNFW) and Einasto (1965) profiles. The former adds a free parameter γ to the NFW profile: ρ∝x^−γ(1 + x)^{−3 + γ}; the latter describes the shape of the profile slope, thus dlog ρ/dlog r = −2(r/r₋₂)^α. The best-fit gNFW profile is consistent with NFW, with $\gamma =1.27^{+0.24}_{-0.37}$. The best-fit Einasto profile has α = 0.19 ± 0.06, consistent with numerical simulations, e.g., 〈α〉 = 0.175 ± 0.046 (Gao et al. 2012), and 〈α〉 = 0.183 (Navarro et al. 2004). We also measure the inner slope of the best-fit density profile models directly, obtaining β(r = 0.01r₂₀₀) = −dlog ρ/dlog r = 1.1 for the gNFW and Einasto models, in good agreement with 〈β〉 ≃ 1.1 (Navarro et al. 2004; Gao et al. 2012).

We also examine the possible impact of adiabatic contraction on the total measured density profile (e.g., Gnedin et al. 2004) by introducing a central point mass into the model. We obtain an upper limit on the point mass of M_point ≲ 12 × 10¹² h⁻¹ M_☉, which is degenerate with the structural parameters of the smooth component in all models (NFW, gNFW, and Einasto). The best-fit mass and concentration parameters do not change significantly from those listed in Table 1. The excellent fit of the NFW model—which is based on numerical dark matter only simulations—to our weak-lensing data and the results of adding baryons to the model (albeit in a simplified form) suggest the dark matter may not suffer adiabatic contraction by baryons in the cluster core. We will return to this topic in a future article that combines strong- and weak-lensing constraints.

3.2. Systematic Errors

3.3. Comparison with Okabe et al. (2010)

We fit an NFW model to Ok10's stacked red+blue catalog and our own stacked red galaxy catalog for the 21 clusters in common between the two studies, finding that our mean masses and concentrations are ∼14%–20% and ∼15%–17% greater than theirs (Table 2). The main differences between Ok10 and our analysis relate to color selection of background galaxies and their shape measurement methods (Section 2). We attribute the differences between our respective mass measurements mainly to a combination of (1) contamination of Ok10's blue galaxy sample at large cluster-centric radii and (2) systematics in Ok10's shape measurement methods. We attribute the differences between the respective concentration measurements mainly to contamination of Ok10's red galaxy catalog—their less conservative red color cut (〈ΔC〉 = 0.33) leads to an overall ∼5% contamination by galaxies that preferentially lie at small cluster-centric radii (see right panel of Figure 1). We note that the results in this section are consistent with those of the Planck Collaboration et al. (2013) and Applegate et al. (2012).

Table 2. Comparison with Okabe et al. (2010)

Parametersa	Overdensity, Δ
	Δvir	200	500
$M_{\Delta }^{2013}/M_{\Delta }^{2010}$	1.14 ± 0.16	1.16 ± 0.14	1.20 ± 0.12
$c_{\Delta }^{2013}/c_{\Delta }^{2010}$	1.15 ± 0.19	1.16 ± 0.19	1.17 ± 0.22

Note. ^aRatio of the stacked mass and concentration obtained from our methods and those of Ok10, for the 21 clusters in common between the two studies.

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We have used sensitive high-resolution observations with Subaru to measure the average density profile of an approximately mass-selected sample of 50 galaxy clusters at 0.15 < z < 0.3. Careful treatment of systematic errors indicates that they are all smaller than the statistical errors. In particular, we achieve just 1% contamination of the background galaxy sample by foreground and cluster galaxies; tests on simulated data indicate that our shape measurement multiplicative systematic error is m ≲ 0.03; and errors from choice of binning scheme are just a few percent. When the signal from all 50 clusters is combined together, we achieve a number density of background galaxies of 266.3 arcmin⁻².

The shape of the stacked density profile is consistent with numerical simulations across the radial range 100 h⁻¹ kpc–2.8 h⁻¹ Mpc. Specifically, we find no statistical evidence for departures from the NFW profile. We constrain the mean mass and concentration of the clusters to sub-10% precision, obtaining $c_{\rm vir}=5.41^{+0.49}_{-0.45}$. This level of precision is comparable with the differences between the concentrations predicted by different numerical simulations, and therefore opens the possibility of discriminating between different simulations using observational data in the near future.

Our results emphasize the power of stacked weak lensing for constraining the average mass and shape of galaxy clusters. Surveys including Hyper-Suprime-Cam on Subaru, the Dark Energy Survey, and KIDS all hold much promise for stacked weak-lensing studies of less massive clusters, including those at higher redshifts. However, significant advances on the precision that we have achieved here on massive low-redshift clusters await future facilities such as LSST and Euclid to provide the required number density of background galaxies on these rare and massive low-redshift clusters.

We thank Ian McCarthy, Yannick Bahé, and Joop Schaye for sharing their weak shear catalogs from the Cosmo-OWLS simulation in advance of publication. We also thank our LoCuSS colleagues, especially Dan Marrone, Gus Evrard, Pasquale Mazzotta, Arif Babul, and Alexis Finoguenov for many helpful discussions and comments. We acknowledge the Subaru Support Astronomers, plus Paul May, Chris Haines, and Mathilde Jauzac, for assistance with the Subaru observations. We are grateful to N. Kaiser and M. Oguri for making their imcat and glafic packages public. This work is supported in part by Grant-in-Aid for Scientific Research on Priority Area No. 467 "Probing the Dark Energy through an Extremely Wide & Deep Survey with Subaru Telescope," by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and by the FIRST program "Subaru Measurements of Images and Redshifts (SuMIRe)." G.P.S. acknowledges support from the Royal Society. K.U. acknowledges partial support from the National Science Council of Taiwan (grant NSC100-2112-M-001-008-MY3) and from the Academia Sinica Career Development Award.