LoCuSS: THE SUNYAEV–ZEL'DOVICH EFFECT AND WEAK-LENSING MASS SCALING RELATION

The clusters considered in this work are drawn from the LoCuSS "high-L_X" sample discussed by Ok10 (G. P. Smith, in preparation), which was selected from the ROSAT All-Sky Survey to have 0.15 < z < 0.3, −20° ⩽ δ ⩽ +60°, and n_H < 7 × 10²⁰cm⁻². This sample is subject to a luminosity limit of L_X/E(z)^2.7 > 4.1 × 10⁴⁴ erg, where E(z) describes the redshift evolution of the Hubble parameter (E(z)² ≡ H(z)²/H²₀ = Ω_M(1 + z)³ + Ω_Λ for a flat cosmology). Specifically, we consider 21 clusters from the high-L_X sample (Table 1) for which high-quality V- and i'-band Subaru/Suprime-Cam data are available, and in which a central mass concentration is identifiable in the shear maps (see Table 5 of Ok10). The sample spans a factor of four in X-ray luminosity and includes clusters with a broad range of X-ray morphologies, including cool-core (hereafter CC) clusters (e.g., A 383, A 1835: Smith et al. 2001; Schmidt et al. 2001), cold front clusters (e.g., RX J1720.1+2638: Mazzotta et al. 2001), and clusters known to be undergoing mergers (e.g., A 209: Giovannini et al. 2009 and references therein).

The acquisition and processing of the Subaru/Suprime-Cam²³ data upon which the weak-lensing mass measurements are based are described in detail by Ok10. Briefly, the data were acquired in the V and i' bands in excellent atmospheric conditions FWHM ≃ 0.7'', consistent with the stringent requirements for precise measurement of galaxy shapes. Weakly lensed background galaxies were selected to have a minimum color offset from the cluster red sequence in the color–magnitude plane following Umetsu & Broadhurst (2008) and Umetsu et al. (2009). Background galaxy shapes were measured using KSB (Kaiser et al. 1995), achieving a residual mean ellipticity of point sources after removal of the point spread function of ∼10⁻⁴, and a typical number density of galaxies for use in the weak-lensing analysis of ∼10–30 arcmin⁻².

The sample of 21 clusters was observed with the SZA, an eight-element radio interferometer optimized for measurements of the SZ effect. The SZA is a subset of the 23 element Combined Array for Research in Millimeter-wave Astronomy (CARMA). Data were obtained in an 8 GHz passband centered at 31 GHz between 2006 and 2009 (Table 1). Over the period of these observations, the array was sited at two different locations and occupied three different configurations, as described in Culverhouse et al. (2010). For most of this period, the SZA was configured with six elements in a compact array sensitive to arcminute-scale SZ signals, and two outrigger elements providing discrimination for compact radio source emission. The compact array and outrigger baselines provide baseline lengths, in units of number of wavelengths λ, of 0.35−1.3 kλ and 2−7.5 kλ, respectively. Two of the clusters (A209 and ZwCl 1454.8+2233) were also briefly observed with an eight-compact-element array configuration, with greater SZ sensitivity but little intrinsic power to distinguish SZ signal from contamination. Integration times were tailored to the magnitude of the SZ signal in each cluster and, secondarily, to the degree of contamination from radio sources. Clusters with complex radio source environments in 1.4 GHz survey images were given more integration time. The total on-source unflagged integration times range from 5 to 69 hr, and the corresponding noise levels achieved by the short (SZ-sensitive) baselines were 0.14–0.45 mJy (12–40 μK), with synthesized beams of ∼2' FWHM. Absolute calibration of the data is determined through periodic measurements of Mars, calibrated against the Rudy et al. (1987) model for this source. A systematic uncertainty of 10% applies to the SZ measurements due to the uncertainty in the absolute calibration (Sharp et al. 2010), this is not included in the reported errors.

Exclusion of emissive sources from the SZA data is most simply achieved when the sources are spatially compact. For such sources the emission measured on the longer outrigger baselines (smaller angular scales) can be assumed to be representative of the emission measured on the short, SZ-sensitive baselines (larger angular scales) and thereby separated from the SZ signal. Sources that are not point-like on the scale of 20'' leave residual emission in the short baseline maps that fills in the cluster SZ signal. Three clusters in the sample suffer from extended radio source contamination.

• 1.  
  A115 hosts the bright radio source 3C 28, which is >100 mJy at 5 GHz and has lobes extending more than an arcminute across the cluster (Giovannini et al. 1987). While there is some discrimination on the short baselines between the extended central source and the more extended SZ decrement, reliable separation of these two resolved structures is not possible in these data.
• 2.  
  RX J1720.1+2638 contains a bright source near the cluster center that is resolved on scales of 30'' at 1.4 GHz in the Very Large Array Faint Images of the Radio Sky at Twenty cm (FIRST) survey (Becker et al. 1995). The SZA data suggest that this source is also extended at 31 GHz, where the SZ decrement appears to be split through its center by an emissive source after a point-like source model is determined from the long baselines and removed.
• 3.  
  A291 hosts three 31 GHz sources within an arcminute of the pointing center. One of these is revealed to be a double-lobed radio source in FIRST, and residual emission at this position appears to fill in the SZ decrement, leading to an offset of >20'' between the SZ centroid and brightest cluster galaxy (BCG) position.

Observations of these objects in the 90 GHz SZA band would likely provide greater spectral discrimination between the radio source emission and SZ signal; this possibility will be examined through future observations.

In subsequent sections, we analyze the sample of 18 clusters that remains after excluding the three objects described above. As in Ok10, we examine whether the 18 cluster subsample is representative of the parent sample by constructing randomized (with replacement) groups of the same size from the full sample. We find that the mean value of L_X/E(z)^2.7 for our sample is very similar to the parent population, falling in the 47th percentile of random groups. The spread in L_X/E(z)^2.7 within the sample is also very representative, the standard deviation in this quantity for our sample is again at the 47th percentile of the random groups.

The procedures for determining galaxy cluster masses and related radii are described in full by Ok10; we summarize salient details here. We adopt Ok10's "red+blue" galaxy sample—i.e., we use only those galaxies that are redder or bluer than the cluster red sequence. Exclusion of unlensed galaxies in this manner is essential because cluster/foreground galaxies can bias M₂₅₀₀ and M₅₀₀ low by ∼50%–20%. The mean redshift of the color-selected background galaxies is estimated by matching their colors and magnitudes to the COSMOS photometric redshift catalog (Ilbert et al. 2009). The tangential shear signal is then measured in logarithmically spaced bins, the measurement in each bin being the error-weighted mean tangential shear of the galaxies in that bin. The bins are centered on the BCG of each cluster.

The spherical mass (or spherical overdensity mass) M_Δ is defined as the mass enclosed within a radius r_Δ for which the average interior density is Δ times the critical density of the universe (ρ_c ≡ 3H(z)²/8πG). M_Δ is estimated by fitting the measured shear profile to the NFW model prediction parameterized by M_Δ and concentration c_Δ, where the NFW mass profile (Navarro et al. 1996) is given as ρ ∝ r⁻¹(1 + c_Δr/r_Δ)⁻². Both M_Δ and c_Δ are allowed to be free in the fit—no mass–concentration relation is assumed, as is often done in other works (e.g., Hoekstra 2007)—and the M_Δ values for each cluster are determined by marginalizing over the posterior distribution of concentration values. Spherical weak-lensing masses at two of the three overdensities used in this work, Δ = 2500 and 500, are listed in Table 8 of Ok10. Values for Δ = 1000 are calculated from the mass profiles fitted in that work. Overdensity radii, which are calculable from the masses (M_Δ = 4πr³_ΔΔρ_c/3), are given in Table 2.

For scaling relation analyses, the SZ signal is typically quantified using the Compton y-parameter integrated within a region centered on a cluster:

$$\begin{eqnarray} Y &=& \int y d\Omega = \frac{1}{D_\mathrm{A}^2} \frac{k_B \sigma _T}{m_\mathrm{e} c^2} \int dl \int n_\mathrm{e} T_\mathrm{e} dA \nonumber \\ &=& \frac{1}{D_\mathrm{A}^2} \frac{\sigma _T}{m_\mathrm{e} c^2} \int dl \int P(r) dA. \end{eqnarray} \tag{ 1 }$$

The quantity YD²_A is called the intrinsic y-parameter, as it removes the distance dependence of the right side of Equation (1).

It is customary to integrate Compton-y over the solid angle subtended by the cluster, as the average change in the CMB temperature within an aperture ($\overline{\Delta T}\propto Y T_\mathrm{CMB}$) is expected to be a robust observable. One standard method of accomplishing this is to compare the observed sky signal with a parameterized radial SZ profile, projected to two dimensions and filtered in the same way as the sky data. However, both in interferometric data such as ours and in data from single-aperture telescopes, the large-scale SZ signal is attenuated by spatial filtering and/or confused with the anisotropy of the primary CMB (e.g., Plagge et al. 2010). The poor constraint on the large-scale behavior of the profile is coupled into the derived Y as additional uncertainty from the unknown large l (line-of-sight distance) behavior of the profile.

This effect can be partially mitigated by adopting as an observable the integral of the model profile over a spherical volume,

$$\begin{equation} Y_\mathrm{sph} = \frac{1}{D_\mathrm{A}^2} \frac{\sigma _T}{m_\mathrm{e} c^2} \int P(r) dV. \end{equation} \tag{ 2 }$$

Y_sph is relatively insensitive to the unconstrained modes in the data and can be calculated without resorting to arbitrary outer radius cutoffs for the line-of-sight integration (as noted by Arnaud et al. 2010). Furthermore, such integration is standard in the analysis of X-ray measurements of clusters (e.g., Zhang et al. 2008; Vikhlinin et al. 2009a). We therefore perform our scaling relation analysis using the spherically integrated intrinsic y-parameter, Y_sphD²_A.

Since interferometers do not measure the total sky signal, but instead sample the Fourier transform of the sky signal only at the spatial frequencies determined by their uv coverage, we must restore missing spatial information in order to determine our observable. We accomplish this by adopting a model profile to fit to our visibility data; specifically, the generalized NFW pressure profile proposed by Nagai et al. (2007),

$$\begin{equation} P(r) = \frac{P_0}{ x^{\gamma } \left(1 + x^{\alpha } \right)^{(\beta -\gamma)/\alpha }}, \end{equation} \tag{ 3 }$$

where x ≡ r/r_s. The model contains five parameters: three exponents (α, β, γ) which determine the shape of the profile, a normalization P₀, and a scale radius r_s (alternatively, a concentration parameter).

Our data are not capable of placing useful constraints on the three shape parameters due to degeneracies between them, the moderate signal to noise, and the limited range of angular scales probed by the interferometer. We fix the values of the shape parameters to the Arnaud et al. (2010) average values, [α, β, γ] = [1.05, 5.49, 0.31], which were determined from simulations and scaled X-ray pressure profiles of galaxy clusters in the low-redshift (z < 0.2) REXCESS sample. We consider other observationally motivated values of α, β, and γ (derived from morphological subsamples in Arnaud et al. 2010, see also Section 5.4) to determine the range of systematic error that may be introduced by the fixed profile shape. The systematic uncertainty in Y is found to range from 1% to 9% between r₂₅₀₀ and r₅₀₀ in our data.

We estimate the remaining pressure model parameters (P₀, r_s, and the centroid position) and their uncertainties using a Markov Chain Monte Carlo method (MCMC; Bonamente et al. 2006). At each step of the MCMC, we integrate the spherical profile along the line of sight, filter it with the primary beam pattern of the SZA antennas (approximately a Gaussian with FWHM 110), and Fourier transform for comparison with the measured Fourier components (visibilities). The positions and fluxes of detected radio sources are included in the MCMC as additional free parameters and are marginalized over to determine the pressure model parameters.

The centroid of the SZ signal is a free parameter in our model to allow for the possibility of real offsets between the center of the ICM and the BCG. The typical uncertainty in the best-fit SZ centroid is 0.03r₅₀₀, corresponding to ∼10''. For all 18 clusters, the best-fit SZ centroid is within 0.15r₅₀₀ of the BCG position (the assumed center of shear), with the SZ centroid in 75% of our sample lying within 0.04r₅₀₀ of the respective BCGs. Larger offsets were typically found in clusters that lack a cool core, or show signs of merger activity. For example, the most significant offset is seen in A 2390 (r_SZ-WL = 0.15 ± 0.04r₅₀₀), which shows both a cool core and merger activity (Allen et al. 2001). After accounting for the measurement uncertainty, the distribution of offsets is reasonably consistent with recent X-ray/optical studies (Sanderson et al. 2009; Haarsma et al. 2010).

To calculate Y_sph for each cluster, we integrate the respective ICM model (Equations (2) and (3)) over the spherical volume defined by the overdensity radii obtained from the weak-lensing models (Section 3.1; Table 2). These radii are not constrained from the SZ data alone, so a completely independent determination of Y_sph at the overdensities of interest is not possible. The correlation between M_WL and Y_sph introduced by this procedure is discussed in Section 5.2. Note that the shared value of r_Δ is the only coupling between the measurements of Y_sph and the lensing analysis—there is no joint fitting of the mass and pressure profiles. The final Y_sph measurements are listed in Table 2. Because of parameter correlations, chiefly between P₀ and r_s, we do not report best-fit values of the individual profile parameters for each cluster. However, as shown in Section 4.4 of Mroczkowski et al. (2009), the value of Y_sph determined from the P₀–r_s pairs in the Markov Chain is better constrained than either of these two parameters alone.

One of our aims is to check whether the shape, normalization, and intrinsic scatter of the mass–Y_sph relation depends on cluster properties. Of particular interest is whether clusters might exhibit different behavior based on their dynamical state, which we often classify simply as merging or non-merging; however, dynamical state is difficult to estimate from SZ and weak-lensing measurements alone. Here, we employ a simple and broadly accepted indicator of dynamical state: the centroid shift parameter <w >, defined as the standard deviation of the projected separation between the X-ray peak and the centroid of emission calculated in circular apertures centered on the X-ray peak.

We measure <w > for each cluster in our sample using archival Chandra data, following the method of Mohr et al. (1995) as implemented by Maughan et al. (2008). The analysis is performed on exposure-corrected images with sources masked. The core (<30 kpc) is excised from the calculation of the centroid, and the radii span 0.05r₅₀₀–r₅₀₀ in steps of 5% of r₅₀₀. The core is included in the calculation of the X-ray peak. The error on <w > is derived from the standard deviation observed in cluster images resimulated with Poisson noise, following Böhringer et al. (2010). Previous observational studies of the <w > distribution (e.g., Maughan et al. 2008; Pratt et al. 2009; Böhringer et al. 2010) have often divided clusters into "disturbed" and "undisturbed" subsamples at < w > = 10⁻²r₅₀₀. We adopt this value to aid comparison of our results with the literature. Six of the eighteen clusters are classified as disturbed ( < w> > 10⁻²r₅₀₀), and the remaining 12 as undisturbed ( < w > < 10⁻²r₅₀₀; Table 2).²⁴ Note that although much of our discussion makes use of this binary morphological classification system, equating the disturbed systems with mergers and undisturbed systems with non-mergers, the true relationship between centroid shift and dynamical state is likely to be more complex.

Without significant influence from non-gravitational processes, the mass, temperature, size, and other properties of galaxy clusters are expected to follow self-similar scaling relations (Kaiser 1986). These power-law relationships follow directly from the assumptions of HSE and an isothermal distribution of baryons and dark matter (e.g., Bryan & Norman 1998), and form a useful reference point for measurements of scaling relations.

The simplest self-similar model, as derived in the above-cited papers, predicts that cluster mass (M) and temperature (T) are related as

$$\begin{equation} T\propto M^{2/3} E(z)^{2/3} \Delta ^{1/3}. \end{equation} \tag{ 4 }$$

To derive the expected scaling between mass and the SZ signal, we note that the double integral on the right-hand side of Equation (1) can be written as M_gasT_e for an isothermal ICM, or equivalently as f_gasMT_e for f_gas ≡ M_gas/M. Combining this with Equation (4), we find the M_WL–Y_sph scaling relation:

$$\begin{equation} M \propto f_{{\rm gas}}^{-3/5} \Delta ^{-1/5} \left[YD_\mathrm{A}^2 E(z)^{-2/3}\right]^{3/5}. \end{equation} \tag{ 5 }$$

This equation defines the self-similar reference model for our scaling analysis.

We analyze the scaling between mass and Y_sph at three overdensity radii (Δ = 500, 1000, and 2500) determined from the weak-lensing measurements. We fit the data with a power law of the form

$$\begin{equation} \frac{M(r_\Delta)}{10^{14} M_\odot }= 10^A\left(\frac{Y_\mathrm{sph}D_\mathrm{A}^2E(z)^{-2/3}}{10^{-5}\,\mathrm{Mpc}^2}\right)^B. \end{equation} \tag{ 6 }$$

The regression is performed in linearized coordinates by using the base-10 logarithm of the data points.

Our data have significant variations in their measurement precision, and simulations suggest that the M_WL–Y_sph correlation should have intrinsic scatter, placing significant demands on our fitting technique. The problem of linear regression with uncertainty in both axes and intrinsic scatter is complex, and several methods of parameter estimation have been formulated for this circumstance (e.g., Akritas & Bershady 1996; Tremaine et al. 2002; Weiner et al. 2006). Methods that ignore the intrinsic scatter, particularly when accompanied by heterogeneous measurement errors, lead to biased regression parameters (Kelly 2007).

We perform our linear regression using a three-parameter model: normalization A, slope B, and intrinsic scatter σ_M|Y. Parameter determinations are achieved through the Bayesian regression method presented in Kelly (2007). In that work Kelly demonstrates that this method performs well in regimes where parameter estimates from other methods (OLS, BCES, and FITEXY) are significantly biased. The regression is performed with the publicly available IDL code written by Kelly.

We fit the scaling relation (Equation (6)) to the full sample of clusters with the slope B as a free parameter, referring to the result as the "free slope" (FS) fit (Table 3). We also report results with the slope fixed to the self-similar value of B = 0.6, which we refer to as the "self-similar" (SS) fit. The posterior distributions of the normalization and scatter in the SS fit are obtained by taking links with the appropriate slope from the Markov Chains generated by the fitting routine.

At all three overdensity radii, the FS mass–Y_sph relations are slightly shallower than the self-similar prediction of B = 0.6 (Figure 1), although the differences from self-similarity are not statistically significant; the relation at Δ = 500 is the most discrepant at 1.4σ significance. A jackknife test of the r₅₀₀ data, dropping one cluster at a time and refitting the 17 cluster scaling relations, shows that the cluster with the lowest Y_sph,500 (A383) affects the slope far more than any other. Without this cluster the r₅₀₀ scaling relation would have parameters A = 0.259^+0.131_−0.138, B = 0.56^+0.16_−0.15, and σ_M|Y = 20⁺¹⁰₋₈%; in this case B is in no tension with the self-similar prediction. We have no strong reason to exclude this cluster, and the removal of outliers may bias the average scaling relation parameters, so no outlier clusters are excluded from the scaling relations presented in this paper.

The scaling relations reported in Table 3 are subject to some bias because of the L_X selection threshold of the parent LoCuSS sample. If there is a correlation between the scatter in the two ICM observables, L_X and Y_sph (or M_WL), selection on the former will affect the distribution of the latter. Such correlation is not yet well-characterized observationally, however. The possibility of systematic effects in the scatter (Section 5.2), the previously noted influence of outliers in this small sample, and the possible bias from the X-ray selection demand caution when employing these scaling relations for other analyses.

The intrinsic scatter, σ_M|Y, is found to be ∼20% at all overdensity radii, which is larger than the ∼10% predicted in numerical simulations (e.g., Nagai 2006). This holds for the SS fits as well, confirming that the measured scatter is not strongly dependent on the slope parameter.

The low scatter reported for simulated Y–M scaling relations does not capture additional complications introduced by the measurement techniques. One potentially important source of scatter in our data is the contribution to Y_sph from lower mass haloes along the line of sight. This was examined by Holder et al. (2007) and found to be insignificant for our mass and redshift range. The scatter between the weak-lensing-derived mass and true halo mass may also be important. In measurements of the type used here it is expected to be ∼20% (e.g., Becker & Kravtsov 2011), which is in good agreement with the scatter we observe.

The M_WL–Y_sph scaling relations presented above are the first to combine SZ measurements with weak gravitational lensing. Previous observational studies have assumed HSE to derive mass measurements, and consequently have different sets of systematics and biases. The assumption of HSE introduces an intrinsic correlation between mass and Y_sph, while the intrinsic correlation between the lensing-based mass and Y_sph in our study is expected to be lower (although not necessarily zero, as discussed in Section 5.3). Moreover, systematic biases between HSE-based masses and lensing-based masses have been measured (Mahdavi et al. 2008) or limited at the 10% level (Zhang et al. 2008, 2010) in cluster observations, and found between HSE-based masses and true masses in simulated clusters (Rasia et al. 2006; Lau et al. 2009; Burns et al. 2010; Cavaliere et al. 2011). These simulations suggest that hydrostatic masses are biased low by ∼10%–20% at r₅₀₀, with smaller biases at higher overdensities. In order to explore these effects, we compare our results with previous observational and theoretical studies in Sections 5.1.1 and 5.1.2, respectively.

5.1.1. Previous Observational Results

5.1.2. Previous Simulations

Accurate calibration of numerical simulations is a key step toward cosmological application of SZ surveys. Many authors have therefore developed simulations of increasing sophistication, five recent examples of which are overplotted on our data in Figure 2. In all cases we show the predicted scaling relation between mass and Y_sph, using fits provided by the authors for those simulation studies that did not publish a mass–Y_sph relation. The simulated relations are very close to self-similar (B = 0.6; Table 4), and slightly underpredict Y_sph at fixed mass relative to our best-fit self-similar model (red line in Figure 2). The fractional offset in Y_sph of the simulated relations from the observed relation is ∼10%–35% at M₅₀₀ = 5 × 10¹⁴ M_☉, the mean mass of the observed sample. We also rank the simulations using a simple χ² comparison of their predicted scaling relations with the observational data, assuming an intrinsic scatter of 20% in mass at fixed Y_sph (Table 4). We find that the Nagai (2006; blue solid line in Figure 2) simulation is the closest match to the observations, and the Battaglia et al. (2010) simulation is the poorest match, though not even ∼2σ deviant from our self-similar relation. Note that this ranking does not depend on the adopted intrinsic scatter, although the absolute value of χ² does.

Zoom In Zoom Out Reset image size

Table 4. Simulated Mass–Y_sph Scaling Relations

Refa	Ab	Bb	Notes	χ2c
(1)	0.27	0.60	M > 1014 M☉, z = 0, 0.6	15.8
(2)	0.35	0.55		18.9
(3)	0.32	0.58	M > 1.5 × 1014 M☉, 0.1 ⩽ z ⩽ 0.2	17.8
(4)	0.36	0.54	Preheating model	19.1
(5)	0.37	0.58	AGN feedback model	26.0

Notes. ^a(1) Nagai 2006; (2) Shaw et al. 2008; (3) Sehgal et al. 2010; (4) Stanek et al. 2010; (5) Battaglia et al. 2010. ^bScaling relations parameterized by A and B as in Equation (6). ^cχ² between model and the 18 data points (17 dof) for an assumed intrinsic scatter of 20%.

Download table as:  ASCIITypeset image

In a hybrid observational and simulated calibration of the M_WL–Y_sph relation, Arnaud et al. (2010) constructed an "observed" scaling relation between Y_sph and M₅₀₀ without SZ measurements. They use the universal pressure profile described in Section 3.2, which they calibrate against X-ray pressure profiles and simulated clusters, inside and outside of r₅₀₀, respectively. Their best-fit relation (A = 0.34 and B = 0.56) also appears in Figure 2. It matches well with the simulations of Shaw et al. (2008) and Sehgal et al. (2010), similarly underpredicting Y at fixed M. The mass scale for this scaling relation comes from their M–Y_X scaling relation, which is calibrated against hydrostatic masses, so the line might instead have been expected to overpredict Y due to the hydrostatic bias. At the measurement precision and level of scatter in this work we do not detect such a bias, even though it is expected to be more important at r₅₀₀ than at r₂₅₀₀ where we are able to compare with the results of Bonamente et al. (2008).

The distribution of fractional deviations in mass (ΔM_Δ/M_Δ) of clusters from the mean M_Δ–Y_sph relations is asymmetric, particularly at Δ = 500 (Figure 3), where it peaks to the low-mass side of the relations (negative deviations) and has an extended tail to the high-mass side (positive deviations). Interestingly, the clusters classified as "disturbed" on the basis of the centroid shift measurements discussed in Section 3.3 all have ΔM₅₀₀/M₅₀₀ < 0, and the tail of the distribution at ΔM₅₀₀/M₅₀₀ > 0.3 comprises solely undisturbed clusters (compare the blue and red in Figure 3). This suggests that the large intrinsic scatter in our mass–Y_sph relations may be related to cluster morphology.

Zoom In Zoom Out Reset image size

We detect a significant difference in the normalization of the self-similar mass–Y_sph relations between disturbed and undisturbed clusters at all three overdensity radii. At fixed Y_sph, the mass of undisturbed clusters exceeds that of disturbed clusters by 13% ± 6%, 28% ± 5%, and 41% ± 6% at Δ = 2500, 1000, and 500, respectively (Table 3 and Figure 4). The probability of observing such an offset randomly is low even in scattered data like these; drawing random samples (with replacement) of 6 and 12 clusters from our data and repeating our fits, we find that only 1% of random subsamples are more significantly offset than the real data at r₅₀₀.

Zoom In Zoom Out Reset image size

If confirmed as a real physical effect, the morphological segregation of clusters in the mass–Y_sph plane would have major implications for SZ surveys, which expect to produce nearly mass-limited cluster samples minimally biased with respect to dynamical state. However, morphological segregation has not been seen in mass–Y_sph relations predicted from numerical simulations. Indeed, it has generally been found that the SZ signal is a robust mass proxy (merging and non-merging clusters are co-located in the mass–Y_sph plane) even during periods of mass accretion (e.g., Motl et al. 2005; Poole et al. 2007).

Simulations of merging clusters do find that cluster mergers produce transitory boosts in the SZ signal, although when integrated over the entire cluster these boosts are small and short-lived (Poole et al. 2007; Wik et al. 2008). For example, Poole et al. showed that for most merger mass ratios and relative velocities, the observed Y is usually below the final "post-merger" value during the merger. This is a natural consequence of the finite time required to heat the ICM to the higher equilibrated temperature of the post-merger halo. The observational signature of this effect is that at fixed post-merger halo mass, merging (disturbed) clusters should have Y suppressed relative to non-merging (undisturbed) clusters. These theoretical predictions are in the opposite sense to the morphological segregation detected in our observed mass–Y_sph relations.

The ideal scaling relation measurement compares two observables that have been derived independently. X-ray scaling relations have typically been constructed from HSE-based masses derived from the same X-ray data that appear in the observable axis. HSE-based scaling relations therefore inevitably suffer varying degrees of correlation that depend on the details of the measurement methods and the observable against which mass is plotted (e.g., Mantz et al. 2010a). This correlation acts to suppress the scatter inferred for HSE-based scaling relations.

The M_WL and Y_sph measurements presented in Section 3 are also not completely independent, as the outer integration boundaries for the SZ profile are set by the lensing-derived overdensity radii. This introduces a correlation between observables. The mass error (δM ≡ M_WL − M_true) for an individual cluster corresponds to an error in the overdensity radius (δr),

$$\begin{equation} \delta M = 4\pi r^2 \rho (r) \delta r, \end{equation} \tag{ 7 }$$

and this error is transferred to the calculation of Y_sph. Equation (7) can be converted to an equation involving fractional errors in M and r by dividing by $M(r)=4\pi r^3 \bar{\rho }$/3,

$$\begin{equation} \delta ({\rm log}(M)) = 3\frac{\rho (r)}{\bar{\rho }}\delta ({\rm log}(r)). \end{equation} \tag{ 8 }$$

The perturbation in the calculated Y_sph introduced by δr follows the same form as Equation (8), with the pressure substituted for ρ and Y_sph for M.

From the equations for the logarithmic radial slopes of M and Y_sph we can infer the response of these two quantities to the errors in r_Δ. The direction of motion in the M_WL–Y_sph plane for an error δr, expressed as the slope δlog(M)/δlog(Y), is just the ratio of the logarithmic slopes in density (δlog(M)/δlog(r)) and pressure at the radius of interest. This ratio varies from cluster to cluster and across overdensities; in the present sample, it is (on average) 2.3 at r₂₅₀₀, but 1.7 at r₅₀₀. The latter is very close to the slope of the self-similar scaling relation, 1/B = 5/3. On average then, the scatter between M_WL and M_true leads to errors in the derived Y_sph that move the clusters along the scaling relation at r₅₀₀.

It was noted in Section 4.3 that the scatter between the mass derived from the WL shear profile fitting technique and the true cluster mass is expected to be ∼20% (Becker & Kravtsov 2011), which matches the values in Table 3. That the observed and predicted scatter agree so well may be coincidence; simulations suggest that the astrophysics of the ICM can contribute another 10%–15% scatter between Y and true mass, and systematic effects like those discussed in subsequent sections can be expected to further increase the observed scatter. The above analysis suggests that σ_M|Y could be larger if the Y_sph and M_WL measurements were decoupled. The clusters in this sample all have Chandra data, which provides us with an independent measurement of r₅₀₀ for the calculation of Y_sph. Using the X-ray values of r₅₀₀ from Sanderson et al. (2009) as integration radii for both the NFW-halo fits to the weak-lensing data and the SZ profiles, we again find the 20% scatter in the scaling relation. When these radii are used for the SZ data alone, the scatter increases to 26%. The coupling between observables through the integration radius therefore does appear to lessen the observed scatter, and the scatter of fully independent measurements may be more representative of the true intrinsic scatter.

Reliable measurements of Y_sph depend on the appropriate choice of pressure profile shape in the ICM models. Systematic errors in the choice of pressure profile may propagate to systematic errors in Y_sph measurements, and thus contribute to the observed scatter and segregation. Our Y_sph measurements employ the Arnaud et al. (2010) average pressure profile. However, these authors noted that CC clusters have more concentrated pressure profiles than disturbed clusters.²⁵

As a test of the importance of the profile choice on our SZ measurements, we recalculate our Y_sph values using the morphology-specific pressure profiles presented in Arnaud et al. (2010). For clusters we classify as "disturbed" and "undisturbed" we use the Arnaud et al. "morphologically disturbed" and "cool-core" profiles, respectively. The change in profile leads to small systematic changes, the disturbed clusters decrease in Y_sph by 1% and 3% at r₂₅₀₀ and r₅₀₀, respectively, and undisturbed clusters increase by 1% and 5% at these radii. These changes can, at most, reduce the morphological segregation in the M₅₀₀–Y_sph relation from 41% to 31%.

We expect this decrease in morphological segregation to be an upper limit on the effect of pressure profile choice on our results. Although we use the Arnaud et al. (2010) pressure profile for all of our undisturbed clusters, not all of them host a cool core (Figure 5). In the original measurement of the CC profile, Arnaud et al. excluded undisturbed clusters that lacked cool cores from their subsample, and these excluded clusters had profiles closest to the one we adopted in Section 3.2. We therefore conclude that although up to one-fourth of the morphological segregation can be attributed to systematic errors in Y_sph due to incorrect pressure profile choice in the ICM models, the majority of the segregation remains to be explained.

Zoom In Zoom Out Reset image size

Reliable measurements of M_WL also depend on appropriate model choice. The weak-lensing masses used in this article are based on the initial Ok10 analysis of data from our Subaru observing program, in which the mass distribution of each cluster was described by a model containing a single spherical NFW halo (Section 3.1). We therefore explore whether this choice may contribute to the morphological segregation in the M_WL–Y_sph plane.

Additional halos that might be justified by the data include sub-halos within each cluster, and large-scale structure projected along the same line of sight. We attempt the lowest order correction to our lensing analysis by adding a second dark matter halo to each mass model at the position of the most significant sub-peak in each of the density maps presented in Appendix 1 of Ok10. These two-halo models were fitted to the two-dimensional weak-shear field using LENSTOOL²⁶ (Jullo et al. 2007). Comparing these models to the original one-halo models, we found that the Bayesian evidence ratio significantly favored a two-halo model for all but three clusters. We re-calculated M_WL at Δ = 500 for each cluster using the more probable of the two models, excluding the sub-halo if it is known to lie outside the cluster (e.g., RXJ 0153.2+0102 adjacent to A 267; see Figure 21 in Ok10) and assuming that the projected separation of the two halos is the true three-dimensional separation of the halos in all other cases. In the resulting M₅₀₀–Y_sph scaling relation we find that neither the scatter nor segregation are reduced within the uncertainties.

The one-dimensional radial profiles used to model the cluster weak-lensing signal may also be a source of scatter. Oguri et al. (2010) fitted elliptical weak-lensing mass models to Ok10's data, thus relaxing the assumption of azimuthal symmetry in the single NFW halo of the latter's models. We find that the scatter and segregation in the M_WL–Y_sph relation are all unchanged within the uncertainties if Oguri et al.'s masses are substituted for Ok10's masses. Both of these tests suggest that the description of the projected mass distribution is not the dominant source of scatter.

The line-of-sight structure of the cluster dark matter is predicted to introduce systematic errors to lensing masses derived assuming a spherically symmetric profile (e.g., Corless & King 2007; Meneghetti et al. 2010). An observational indicator of the line-of-sight elongation of the mass distribution would therefore be a powerful tool. First, we consider the X-ray centroid shift measurements upon which the disturbed/undisturbed classification was made in Section 3.3. We plot centroid shift versus fractional deviation in mass from the best-fit SS scaling relation in Figure 6, finding no reliable trend between the mass deviation and <w >. Though the large-<w > clusters (disturbed) lie exclusively on the low-mass side of the relation, as noted in Section 5.2, the small-<w > (undisturbed) clusters span a wide range in deviation (ΔM₅₀₀/M₅₀₀ ∼ 0–0.75) at all centroid shift values. The largest deviations suggest that such clusters may be prolate spheroids viewed along an axis close to their major axis (Corless & King 2007), while the smallest deviations suggest that the dark matter in such clusters is relatively undisturbed and spherical. This implies that the interpretation of small centroid shifts is degenerate between a relatively undisturbed ICM and the disturbed ICM of a prolate cluster whose major axis is closely aligned with the line of sight. A "cleaner" indicator of the orientation of cluster dark matter halos is therefore required.

Zoom In Zoom Out Reset image size

BCGs are prolate stellar systems with their major axis closely aligned with the major axis of the cluster dark matter halo and ICM (e.g., Hashimoto et al. 2008; Fasano et al. 2010). The major axis of a prolate BCG that is measured to be strongly elliptical in projection on the sky likely lies close to the plane of the sky, and thus so too does the major axis of the cluster dark matter halo. An almost circular BCG, on the other hand, indicates that the major axis is close to the line of sight through the cluster. BCG ellipticity measurements are available from surface photometry of Hubble Space Telescope (HST) observations of 15 of the 18 clusters in our sample (Smith et al. 2010). We analyzed two of the remaining clusters (RX J0142.0+2131 and A 697) in the same manner (the last cluster, ZwCl 1459.4+4240, has not been observed with HST). We find that the clusters with the roundest BCGs suffer the strongest positive deviations from the M_WL–Y_sph relation, and that clusters with the most elliptical BCGs suffer negative deviations (right panel of Figure 6). Most significantly, clusters with _BCG < 0.15 have exclusively large deviations (ΔM₅₀₀/M₅₀₀ ≳ 0.5). The mean deviation of these "round" BCGs ( < 0.15) is 〈ΔM₅₀₀/M₅₀₀〉 = 0.76 ± 0.17 and of "elliptical" BCGs ( > 0.15) is 〈ΔM₅₀₀/M₅₀₀〉 = −0.10 ±  0.17, where the error bars are the 1σ standard deviations on the means.

The correlation between ΔM₅₀₀/M₅₀₀ and _BCG is strikingly similar to the trend between M_{3D, WL}/M_{3D, true} and viewing angle in the numerically simulated clusters presented in Meneghetti et al. (2010). Indeed, visual inspection of Meneghetti et al.'s Figure 17 suggests that we are viewing clusters with "round" BCGs within ∼20°–30° of the major axis of the cluster inertia ellipsoid, and with "elliptical" BCGs at larger viewing angles. An observed BCG ellipticity of _BCG = 0.15 (corresponding to an observed axis ratio of q = b/a = 0.85) can be converted to a viewing angle with respect to the major axis by assuming an intrinsic BCG axis ratio (β). We adopt β = 0.67, following the measurements of Fasano et al. (2010). The viewing angle is then estimated to be

$$\begin{equation} \psi =\arccos \left(\!\sqrt{\frac{1-(\beta /q)^2}{1-\beta ^2}}\,\,\right)\simeq 34^\circ \end{equation} \tag{ 9 }$$

which is consistent with the visual comparison of our Figure 6 and Figure 17 of Meneghetti et al. (2010).

Additional anecdotal support for this picture comes from detailed measurements of the three-dimensional structure of A383, which is the cluster with the second largest value of ΔM₅₀₀/M₅₀₀ in our sample and the outlier with the greatest effect on our scaling relation fits (the lowest-Y_sph cluster in Figure 1). Newman et al. (2011) combined strong- and weak-lensing data, stellar kinematics in the BCG, and X-ray data to infer the three-dimensional shape of the cluster dark matter halo. They found that the cluster is very elongated along the line of sight, with an axis ratio of 2:1 between the line of sight and plane of sky. This agrees well with our inference that halo orientation is significantly affecting our weak-lensing masses.

Our data provide observational support for previous numerical studies that have highlighted projection effects as an important source of systematic uncertainties in weak-lensing cluster mass measurements. These uncertainties arise from both the underlying shape of the main cluster halo and sub-halos that reside within the cluster due to a merger and/or in the surrounding large-scale structure. Future progress on the use of weak lensing to calibrate mass-observable scaling relations will therefore benefit from intrinsically asymmetric lens models that include halo triaxiality and/or multiple halos. Direct measurements of the ratio of the line-of-sight depth to the plane-of-sky size of the cluster through joint X-ray and SZ measurements of the ICM may aid in constructing a more three-dimensional halo model. These efforts should lead to a reduction in the scatter and segregation in the M_WL–Y_sph relation.

While the mass scatter observed here is plausibly consistent with that predicted by lensing simulations, correlated scatter between mass and Y_sph due to halo orientation and nearby structure will affect the scatter measured in scaling analyses like ours. Both the SZ and lensing signals can be expected to suffer projection effects operating in the same direction, which will reduce the scatter observed between Y_sph and M_WL from that predicted for M_WL–M_true in lensing simulations. The SZ projection effects differ from the lensing projection in important ways, which prevents perfect cancellation of the scatter. First, the ICM (in equilibrium) will follow isopotential contours rather than isodensity contours (Buote & Canizares 1992), which makes it much rounder than the dark matter. Lee & Suto (2003) show that the expected ICM eccentricity is typically 0.6 times that of the dark matter, corresponding to an ICM axis ratio of 1.2:1 for a matter axis ratio of 2:1. Lau et al. (2011) show that for simulated clusters the ICM is typically more spherical than either the dark matter or the potential at r₅₀₀. Second, the SZ signal also depends on the ICM temperature, so that nearby but as-of-yet unassimilated halos will have a lower SZ signal per unit mass than the same material once it has passed through the virial shock of the main halo. For this reason, projection of adjacent halos is a weaker effect for the SZ signal. Determinations of the scatter between Y_sph and mass, as well as attempts at normalizing the Y_sph–mass scaling relation using weak lensing, will need to constrain the covariance of the two observables (e.g., Allen et al. 2011).