OPTICAL SPECTROSCOPY AND VELOCITY DISPERSIONS OF GALAXY CLUSTERS FROM THE SPT-SZ SURVEY

Most of the galaxy clusters for which we report spectroscopic observations were published as SPT cluster detections (and new discoveries) in Vanderlinde et al. (2010), Williamson et al. (2011), and Reichardt et al. (2013); we refer the reader to those publications for details of the SPT observations. In Table 1, we give the SPT identification (ID) of the clusters and their essential SZ properties. This includes the right ascension and declination of the SZ center, the cluster redshift, and the SPT detection significance ξ. We also report the SPT cluster mass estimate, M_{500c, SPT}, as reported in Reichardt et al. (2013), for those clusters at redshift z ⩾ 0.3, the redshift threshold used in the SPT cosmological analysis. As described in Reichardt et al. (2013), the SPT mass estimate is measured from the SPT SZ significance and X-ray measurements, where available, while accounting for the SPT selection, and marginalizing over all uncertainties in cosmology and the cluster observable scaling relations. The last columns indicate the source of the spectroscopy, our own measurements for 61 clusters, and a literature reference for 19 of them. Five clusters have data from both sources.

Table 1. SPT Properties and Source of Spectroscopic Data

ID and Coordinates						Source of Spectroscopy
SPT ID	R.A.	Decl.	z	ξ	M500c, SPT	This Work	Literature
	(J2000 deg)	(J2000 deg)			($10^{14}\ h_{70}^{-1}\ M_\odot$)
SPT-CL J0000-5748	0.2496	−57.8066	0.702	5.48	4.29 ± 0.71	✓
SPT-CL J0014-4952*	3.6969	−49.8772	0.752	8.87	5.14 ± 0.86	✓
SPT-CL J0037-5047*	9.4441	−50.7971	1.026	6.93	3.64 ± 0.79	✓
SPT-CL J0040-4407	10.2048	−44.1329	0.350	19.34	10.18 ± 1.32	✓
SPT-CL J0102-4915	15.7294	−49.2611	0.870	39.91	15.69 ± 1.89		1
SPT-CL J0118-5156*	19.5990	−51.9434	0.705	5.97	3.39 ± 0.82	✓
SPT-CL J0205-5829	31.4437	−58.4856	1.322	10.54	4.79 ± 1.00	✓
SPT-CL J0205-6432	31.2786	−64.5461	0.744	6.02	3.29 ± 0.79	✓
SPT-CL J0232-5257**	38.1876	−52.9578	0.556	8.65	5.04 ± 0.89		1
SPT-CL J0233-5819	38.2561	−58.3269	0.663	6.64	3.71 ± 0.86	✓
SPT-CL J0234-5831	38.6790	−58.5217	0.415	14.65	7.64 ± 1.50	✓
SPT-CL J0235-5121**	38.9468	−51.3516	0.278	9.78	⋅⋅⋅		1
SPT-CL J0236-4938**	39.2477	−49.6356	0.334	5.80	3.39 ± 0.89		1
SPT-CL J0240-5946	40.1620	−59.7703	0.400	9.04	5.29 ± 1.07	✓
SPT-CL J0245-5302	41.3780	−53.0360	0.300	⋅⋅⋅	⋅⋅⋅	✓
SPT-CL J0254-5857	43.5729	−58.9526	0.437	14.42	7.46 ± 1.46	✓
SPT-CL J0257-5732	44.3516	−57.5423	0.434	5.40	3.14 ± 0.86	✓
SPT-CL J0304-4921**	46.0619	−49.3612	0.392	12.75	7.32 ± 1.04		1
SPT-CL J0317-5935	49.3208	−59.5856	0.469	5.91	3.46 ± 0.89	✓
SPT-CL J0328-5541	52.1663	−55.6975	0.084	7.08	⋅⋅⋅		3
SPT-CL J0330-5228**	52.7287	−52.4698	0.442	11.57	6.36 ± 1.00		1
SPT-CL J0346-5439**	56.7247	−54.6505	0.530	9.25	5.07 ± 0.93		1
SPT-CL J0431-6126	67.8393	−61.4438	0.059	6.40	⋅⋅⋅		2
SPT-CL J0433-5630	68.2522	−56.5038	0.692	5.35	2.89 ± 0.82	✓
SPT-CL J0438-5419	69.5749	−54.3212	0.422	22.88	10.82 ± 1.39	✓	1
SPT-CL J0449-4901*	72.2742	−49.0246	0.790	8.91	4.57 ± 0.86	✓
SPT-CL J0509-5342	77.3360	−53.7045	0.462	6.61	5.36 ± 0.71	✓	1
SPT-CL J0511-5154	77.9202	−51.9044	0.645	5.63	3.61 ± 0.96	✓
SPT-CL J0516-5430	79.1480	−54.5062	0.294	9.42	⋅⋅⋅	✓
SPT-CL J0521-5104	80.2983	−51.0812	0.675	5.45	3.46 ± 0.96		1
SPT-CL J0528-5300	82.0173	−53.0001	0.769	5.45	3.18 ± 0.61	✓	1
SPT-CL J0533-5005	83.3984	−50.0918	0.881	5.59	2.68 ± 0.61	✓
SPT-CL J0534-5937	83.6018	−59.6289	0.576	4.57	2.71 ± 1.00	✓
SPT-CL J0546-5345	86.6541	−53.7615	1.066	7.69	5.25 ± 0.75	✓	1
SPT-CL J0551-5709	87.9016	−57.1565	0.424	6.13	3.75 ± 0.54	✓
SPT-CL J0559-5249	89.9245	−52.8265	0.609	9.28	6.79 ± 0.86	✓	1
SPT-CL J0658-5556	104.6317	−55.9465	0.296	39.05	⋅⋅⋅		4
SPT-CL J2012-5649	303.1132	−56.8308	0.055	5.99	⋅⋅⋅		2
SPT-CL J2022-6323	305.5235	−63.3973	0.383	6.58	3.82 ± 0.89	✓
SPT-CL J2032-5627	308.0800	−56.4557	0.284	8.14	⋅⋅⋅	✓
SPT-CL J2040-4451	310.2468	−44.8599	1.478	6.28	3.21 ± 0.79	✓
SPT-CL J2040-5725	310.0631	−57.4287	0.930	6.38	3.25 ± 0.75	✓
SPT-CL J2043-5035	310.8285	−50.5929	0.723	7.81	4.71 ± 1.00	✓
SPT-CL J2056-5459	314.2199	−54.9892	0.718	6.05	3.68 ± 0.89	✓
SPT-CL J2058-5608	314.5893	−56.1454	0.606	5.02	2.64 ± 0.79	✓
SPT-CL J2100-4548	315.0936	−45.8057	0.712	4.84	2.71 ± 0.93	✓
SPT-CL J2104-5224	316.2283	−52.4044	0.799	5.32	3.04 ± 0.89	✓
SPT-CL J2106-5844	316.5210	−58.7448	1.131	22.08	8.36 ± 1.71	✓
SPT-CL J2118-5055	319.7291	−50.9329	0.625	5.62	3.43 ± 0.93	✓
SPT-CL J2124-6124	321.1488	−61.4141	0.435	8.21	4.68 ± 0.96	✓
SPT-CL J2130-6458	322.7285	−64.9764	0.316	7.57	4.46 ± 0.96	✓
SPT-CL J2135-5726	323.9158	−57.4415	0.427	10.43	5.68 ± 1.11	✓
SPT-CL J2136-4704	324.1175	−47.0803	0.425	6.17	4.04 ± 0.96	✓
SPT-CL J2136-6307	324.2334	−63.1233	0.926	6.25	3.18 ± 0.75	✓
SPT-CL J2138-6007	324.5060	−60.1324	0.319	12.64	6.75 ± 1.32	✓
SPT-CL J2145-5644	326.4694	−56.7477	0.480	12.30	6.39 ± 1.25	✓
SPT-CL J2146-4633	326.6473	−46.5505	0.932	9.59	5.36 ± 1.07	✓
SPT-CL J2146-4846	326.5346	−48.7774	0.623	5.88	3.64 ± 0.93	✓
SPT-CL J2148-6116	327.1798	−61.2791	0.571	7.27	4.04 ± 0.89	✓
SPT-CL J2155-6048	328.9851	−60.8072	0.539	5.24	2.82 ± 0.82	✓
SPT-CL J2201-5956	330.4727	−59.9473	0.098	13.99	⋅⋅⋅		5
SPT-CL J2248-4431	342.1907	−44.5269	0.351	42.36	17.97 ± 2.18	✓
SPT-CL J2300-5331	345.1765	−53.5170	0.262	5.29	⋅⋅⋅	✓
SPT-CL J2301-5546	345.4688	−55.7758	0.748	5.19	3.11 ± 0.96	✓
SPT-CL J2325-4111	351.3043	−41.1959	0.358	12.50	7.29 ± 1.07	✓
SPT-CL J2331-5051	352.9584	−50.8641	0.575	8.04	5.14 ± 0.71	✓
SPT-CL J2332-5358	353.1040	−53.9733	0.402	7.30	6.50 ± 0.79	✓
SPT-CL J2337-5942	354.3544	−59.7052	0.776	14.94	8.14 ± 1.14	✓
SPT-CL J2341-5119	355.2994	−51.3328	1.002	9.65	5.61 ± 0.82	✓
SPT-CL J2342-5411	355.6903	−54.1887	1.075	6.18	3.00 ± 0.50	✓
SPT-CL J2344-4243	356.1847	−42.7209	0.595	27.44	12.50 ± 1.57	✓
SPT-CL J2347-5158*	356.9423	−51.9766	0.869	⋅⋅⋅	⋅⋅⋅	✓
SPT-CL J2351-5452	357.8877	−54.8753	0.384	4.89	3.18 ± 1.04		6
SPT-CL J2355-5056	358.9551	−50.9367	0.320	5.89	4.07 ± 0.57	✓
SPT-CL J2359-5009	359.9208	−50.1600	0.775	6.35	3.54 ± 0.54	✓

Notes. SPT ID of each cluster, right ascension and declination of its SZ center, and redshift z (from Tables 4 and 5, for reference). Also given are the SPT significance ξ and the SZ-based SPT mass, marginalized over cosmological parameters as in Reichardt et al. (2013), for those clusters at z ⩾ 0.3, except for two, as described in Section 2.1. Clusters marked with ** are reported here as SPT detections for the first time, and those with * are new discoveries. References. (1) Sifón et al. 2013; (2) Girardi et al. 1996; (3) Struble & Rood (1999); (4) Barrena et al. 2002; (5) Katgert et al. 1998; (6) Buckley-Geer et al. 2011.

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There are 11 clusters that do not appear in prior SPT publications, and are presented here as SPT detections for the first time. Five of them are new discoveries (identified with * in Table 1), and the other six were previously published as ACT detections (Marriage et al. 2011, identified with ** in Table 1). These SPT detections will be reported in an upcoming cluster catalog from the full 2500 deg² SPT-SZ survey.

One cluster, SPT-CL J0245-5302, is detected by SPT at high significance; however, because of its proximity to a bright point source (<8 arcmin away), it is not included in the official catalog. SPT-CL J2347-5158 had a higher SPT significance in early maps of the survey, but has ξ < 4.0 in the 2500 deg² survey. The SPT significance and mass are not given for these two clusters.

The spectroscopic observations presented in this work are the first of our ongoing follow-up program. The data were taken from 2008 to 2012 using the Gemini Multi Object Spectrograph (GMOS; Hook et al. 2004) on Gemini South, the Focal Reducer and low dispersion Spectrograph (FORS2; Appenzeller et al. 1998) on VLT Antu, the Inamori Magellan Areal Camera and Spectrograph (IMACS; Dressler et al. 2006) on Magellan Baade, and the Low Dispersion Survey Spectrograph (LDSS3³⁹; Allington-Smith et al. 1994) on Magellan Clay.

In order to place a large number of slitlets in the central region of the cluster, most of the IMACS observations were conducted with the Gladders Image-Slicing Multi-slit Option (GISMO⁴⁰). GISMO optically remaps the central region of the IMACS field of view (roughly 35 × 32) to sixteen evenly spaced regions of the focal plane, allowing for a large density of slitlets in the cluster core while minimizing slit collisions on the CCD.

Details about the observations pertaining to each cluster, including the instrument, optical configuration, number of masks, total exposure time, and measured spectral resolution, are listed in Table 2.

Optical and infrared follow-up imaging observations of SPT clusters are presented alongside our group's photometric redshift methodology in High et al. (2010), Song et al. (2012) and Desai et al. (2012). Those photometric redshifts (and in a few cases, spectroscopic redshifts from the literature) were used to guide the design of the spectroscopic observations. Multislit masks were designed using the best imaging available to us, usually a combination of ground-based griz (on Blanco/MOSAIC II, Magellan/IMACS, Magellan/LDSS3, or BVRI on Swope) and Spitzer/IRAC 3.6 μm. In addition, spectroscopic observations at Gemini and VLT were preceded by at least one-band (r or i) pre-imaging for relative astrometry, and two-band (r and i) pre-imaging for red-sequence target selection in the cases where the existing imaging was not deep enough. The exposure times for this pre-imaging were chosen to reach a magnitude depth for galaxy photometry of m^⋆ + 1 at 10σ at the cluster redshift.

In designing the multislit masks, top priority for slit placement was given to bright red-sequence galaxies (the red sequence of SPT clusters is discussed in the context of photometric redshifts in High et al. 2010; Song et al. 2012), as defined by their distance to either a theoretical or an empirically fit red-sequence model. The details varied depending on the quality of the available imaging, the program, and the prioritization weighting scheme of the instrument's mask-making software. In many of the GISMO observations, blue galaxies were given higher priority than faint red galaxies because, especially at high redshift, they were expected to be more likely to yield a redshift. The results from the different red-sequence weighting schemes are very similar, and few emission lines are found, even at z ≳ 1 (Brodwin et al. 2010; Foley et al. 2011; Stalder et al. 2013, these articles also provide more details about the red-sequence nature of spectroscopic members). The case of SPT-CL J2040-4451 at z = 1.478 is different and redshifts were only obtained for emission-line galaxies (Bayliss et al. 2013). In all cases, non-red-sequence objects were used to fill out any remaining space in the mask.

The dispersers and filters, listed in Table 2, were chosen (within the uncertainty on the photo-z) to obtain low- to medium-resolution spectra covering at least the wavelengths of the main spectral features that we use to identify the galaxy redshifts: [O ii] emission, and the Ca ii H&K absorption lines and break.

The spectroscopic exposure times (also in Table 2) for GMOS and FORS2 observations were chosen to reach S/N = 5 (S/N = 3) per spectral element just below the 4000 Å break for a red galaxy of magnitude m^⋆ + 1 (m^⋆ + 0.5) at z < 1 (z > 1). Under the conditions prevailing at the telescope during classical observing, the exposure times for the Magellan observations were determined by a combination of experience, real-time quick-look reductions, and airmass limitations.

2.2.1. Data Processing

Modern multi-object spectrographs use slit masks, so that the investment in telescope time is quantized by how many masks are allocated to each cluster. The optimization problem is, therefore, to allocate the observation of m masks across n clusters so as to minimize the uncertainty on the ensemble cluster mass normalization.

We pursue a strategy for spectroscopic observations informed by the expectation (from N-body simulations; see, e.g., Kasun & Evrard 2005; White et al. 2010; Saro et al. 2013) that line-of-sight projection effects induce an unavoidable intrinsic scatter of 12% in log dispersion (ln σ) at fixed mass, implying a 35% scatter in dynamical mass (Saro et al. 2013, see Equation (15) of the present paper). As this 35% intrinsic scatter needs to be added to the dynamical mass uncertainty of any one cluster, for the purpose of mass calibration, obtaining coarser dispersions on more clusters is more informative than measuring higher-precision velocity dispersions on a few clusters. Considering the results of those simulations and the experience encapsulated in the velocity dispersion literature (e.g., Girardi et al. 1993), we have adopted a target of N_members ∼ 20–30, where N_members is the number of spectroscopic member galaxies in a cluster. This target range of N_members can be obtained by observing two masks per cluster on the spectrographs available to us.⁴³ The use of a red-sequence selection to target likely cluster members is a necessary feature of this strategy, as a small number of multislit masks only allows us to target a small fraction of the galaxies in the region of the sky around the SZ center.

In discussions throughout this paper, we often use a N_members ⩾ 15 cut. We note that this number is chosen somewhat arbitrarily for the conservative exclusion of systems with very few members. As we will see in the resampling analysis of Section 4, no special statistical transition happens at N_members = 15, and dispersions with fewer members could potentially be used for reliable mass estimates.

Recent simulations and our data suggest that this choice of few-member strategy may increase the scatter due to systematics in the measured dispersions. This is discussed in Section 4.1.

The full sample of redshifts for both member and non-member galaxies is available in electronic format. In Table 3, we present a subset composed of central galaxies, for the 50 clusters where we have the central galaxy redshift. We have visually selected the central galaxy for each cluster to be a large, bright, typically cD-type galaxy that is close to the SZ center and that appears to be central to the distribution of galaxies. For each galaxy, the table lists the SPT ID of the associated cluster, a galaxy ID, right ascension and declination, the redshift and redshift-measurement method, and notable spectral features.

Table 4 lists the cluster redshifts and velocity dispersions measured from the galaxy redshifts.

The cluster redshift z is the biweight average (Beers et al. 1990) of member galaxy redshifts (see below) with an uncertainty given by the standard error, as explained in Section 4.2. Once the cluster redshift is computed, the galaxy proper velocities v_i are obtained from their redshifts z_i by v_i = c(z_i − z)/(1 + z) (Danese et al. 1980). The velocity dispersion σ_BI is the square root of the biweight sample variance of proper velocities, the uncertainty of which we found to be well described by $0.92 \sigma _{\mathrm{BI}} / \sqrt{N_{\mathrm{members}}-1}$ when including the effect of membership selection (Section 4.2; see Section 4.1 for the formula of the biweight sample variance). We also report the dispersion σ_G determined from the gapper estimator, which is a preferred measurement, according to Beers et al. (1990), for those clusters with fewer than 15 member redshifts.

The cluster redshifts and velocity dispersions are calculated using only galaxies identified as members, where membership is established using iterative 3σ clipping on the velocities (Yahil & Vidal 1977; Mamon et al. 2010; Saro et al. 2013). The center at each iteration of 3σ clipping is the biweight average, and σ is calculated from the biweight variance, or the gapper estimator in the case where there are fewer than 15 members. We do not make a hard velocity cut; the initial estimate of σ used in the iterative clipping is determined from the galaxies located within 4000 km s⁻¹ of the center, in the rest frame.

Figure 1 shows the velocity histogram for each cluster with 15 members or more, as well as an indication of emission-line objects and our determination of member and non-member galaxies.

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Some entries in Table 4 have a star-shaped flag ⋆ in the SPT ID column, which highlights possibly less reliable dispersion measurements. These include 8 clusters that have fewer than 15 measured member redshifts,⁴⁴ as well as SPT-CL J0205-6432 with N_members = 15, for which the gapper and biweight dispersions differ by more than 1σ. Since these are not independent measurements but rather two estimates of the same quantity from the same data, we consider a 1σ discrepancy to be large and an indication that the sampling is inadequate.

3.2.1. The Stacked Cluster

To examine the ensemble phase-space galaxy selection, we produce a stacked cluster from our observations; this stacked cluster will also be useful for evaluating our confidence intervals via resampling (see Section 4.2). We generate it in a way that is independent of cluster membership determination. As the calculation of the velocity dispersion and membership selection are unavoidably intertwined, we use the SPT mass—the other uniform mass measurement that we have for all clusters—to normalize the velocities before stacking.

We make a stacked proper-velocity distribution independent of any measurement of the velocity dispersion by calculating the "equivalent dispersion" from the SPT mass. We convert the M_{500c, SPT} to M_{200c, SPT} assuming an Navarro–Frenk–White (NFW) profile and the Duffy et al. (2008) concentration, and then convert the M_{200c, SPT} to a σ_SPT (in km s⁻¹) using the Saro et al. (2013) scaling relation. This σ_SPT is listed in Table 4 for reference. We also normalize the distance to the SZ center by R_{200c, SPT}. The resulting phase-space diagram of the normalized proper velocities v_i/σ_SPT versus r_i/R_{200c, SPT} is shown in Figure 2. For reference, different velocity cuts are plotted. The black dashed line is a 3σ cut. The blue dotted line is a radially dependent 2.7σ(R) cut, where again the σ(R) is from an NFW profile; this velocity cut is found to be optimal for rejecting interlopers by Mamon et al. (2010; although when considering systems without red-sequence selection). While 3σ clipping was a natural choice of membership selection algorithm (given our sometimes small sample size for individual clusters), these different cuts demonstrate that we were generally successful at selecting member galaxies. The histogram of proper velocities is shown in the right panel, together with a Gaussian of mean zero and standard deviation of one. The agreement between the distributions is difficult to quantify due to the expected presence of non-members in the histogram. We will see in Section 5 that we measure a systematic bias in normalization.

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Table 5 contains spectroscopic redshifts and velocity dispersions from the literature for clusters detected by the SPT. Notably, 14 of these clusters are from Sifón et al. (2013), which presents spectroscopic follow-up of galaxy clusters that were detected by the ACT. Because SPT and ACT are both SZ surveys based in the southern hemisphere, there is some overlap between the galaxy clusters detected with the two telescopes. We independently obtained data for five of the clusters that appear in Sifón et al. (2013), and there is some overlap between the cluster members for which we have measured redshifts. These clusters are of some interest for evaluating our follow-up strategy, because the typical number of SPT-reported member galaxies per dispersion is 25 (for N_members ⩾ 15), while for the overlapping Sifón et al. (2013) sample it is 55. All of the overlapping cluster redshifts and dispersions are consistent between our work and Sifón et al. (2013) at the 1σ level, except for the velocity dispersion of SPT-CL J0528-5300. Its velocity histogram shows extended structure (Figure 1). The galaxies responsible for this extended structure are kept by the membership selection algorithm in this paper, but they were either not observed or not classified as cluster members by Sifón et al. (2013). It is not possible to determine from the data in hand whether this discrepancy is statistical or systematic in origin.

Table 5. Cluster Redshifts and Velocity Dispersions from the Literature

SPT/This Work				Literature
SPT ID	N	z	σBI	Lit. ID	N	z	σBI	Ref.
			(km s−1)				(km s−1)
z < 0.3
SPT-CL J0235-5121	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	ACT-CL J0235-5121	82	0.2777 ± 0.0005	1063 ± 101	1
SPT-CL J0328-5541	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	A3126	38	0.0844	1041	3
SPT-CL J0431-6126	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	A3266	132	0.0594 ± 0.0003	$1182^{+100}_{-85}$	2
SPT-CL J0658-5556	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	1E0657-56	71	0.2958 ± 0.0003	$1249^{+109}_{-100}$	4
SPT-CL J2012-5649	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	A3667	123	0.0550 ± 0.0004	$1208^{+95}_{-84}$	2
SPT-CL J2201-5956	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	A3827	22	0.0983 ± 0.0010	$1103^{+252}_{-138}$	5
z ⩾ 0.3
SPT-CL J0102-4915	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	ACT-CL J0102-4915	89	0.8701 ± 0.0009	1321 ± 106	1
SPT-CL J0232-5257	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	ACT-CL J0232-5257	64	0.5559 ± 0.0007	884 ± 110	1
SPT-CL J0236-4938	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	ACT-CL J0237-4939	65	0.3344 ± 0.0007	1280 ± 89	1
SPT-CL J0304-4921	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	ACT-CL J0304-4921	71	0.3922 ± 0.0007	1109 ± 89	1
SPT-CL J0330-5228	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	ACT-CL J0330-5227	71	0.4417 ± 0.0008	1238 ± 98	1
SPT-CL J0346-5439	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	ACT-CL J0346-5438	88	0.5297 ± 0.0007	1075 ± 74	1
SPT-CL J0438-5419	18	0.4223 ± 0.0016	1422 ± 317	ACT-CL J0438-5419	65	0.4214 ± 0.0009	1324 ± 105	1
SPT-CL J0509-5342	21	0.4616 ± 0.0007	678 ± 139	ACT-CL J0509-5341	76	0.4607 ± 0.0005	846 ± 111	1
SPT-CL J0521-5104	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	ACT-CL J0521-5104	24	0.6755 ± 0.0016	1150 ± 163	1
SPT-CL J0528-5300	21	0.7694 ± 0.0017	1318 ± 271	ACT-CL J0528-5259	55	0.7678 ± 0.0007	928 ± 111	1
SPT-CL J0546-5345	21	1.0661 ± 0.0018	1191 ± 245	ACT-CL J0546-5345	48	1.0663 ± 0.0014	1082 ± 187	1
SPT-CL J0559-5249	37	0.6092 ± 0.0010	1146 ± 176	ACT-CL J0559-5249	31	0.6091 ± 0.0014	1219 ± 118	1
SPT-CL J2351-5452	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	SCSOJ235138-545253	30	0.3838 ± 0.0008	$855^{+108}_{-96}$	6

Notes. Number of member-galaxy redshifts N (≡ N_members), cluster redshift, and velocity dispersion for clusters found in the literature that are also SPT detections. In the cases where we are presenting our own spectroscopic observations, some of the information from Table 4 is repeated in the left half of the present table for reference. References. (1) Sifón et al. 2013; (2) Girardi et al. 1996; (3) Struble & Rood (1999, this paper does not contain confidence intervals); (4) Barrena et al. 2002; (5) Katgert et al. 1998; (6) Buckley-Geer et al. 2011.

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We note that ACT-CL J0616-5227, also studied in Sifón et al. (2013), is seen in SPT maps but is excluded from the survey because of its proximity to a point source.

Estimators and confidence intervals for velocity dispersions are discussed in Beers et al. (1990), which the reader is encouraged to review. They present estimators such as the biweight, that are resistant and robust.⁴⁵ As we are exploring the properties of the few-N_members regime, we would also like our estimators to be unbiased, meaning that the mean estimate should be independent of the number of points that are sampled.

The first point that we would like to make on the subject is that the biweight dispersion (or more correctly, the associated variance) as presented in Beers et al. (1990), is biased for samples, in the same way that the population variance, $\sum _i (v_i - \bar{v})^2 / n$, is biased and the sample variance, $\sum _i (v_i - \bar{v})^2 / (n-1)$, is not.⁴⁶

We use the biweight sample variance, which does not suffer from this bias (see, e.g., Mosteller & Tukey 1977):

$$\begin{equation} \sigma _{\mathrm{BI}}^2 = N_{\mathrm{members}} \frac{\sum _{|u_i| < 1} \left(1-u_i^2 \right)^4 (v_i-\bar{v})^2 }{D(D-1)} \end{equation} \tag{ 1 }$$

where v_i are the proper velocities, $\bar{v}$ their average,

$$\begin{equation} D = \sum _{|u_i| < 1} \left(1-u_i^2 \right)\left(1-5u_i^2 \right), \end{equation} \tag{ 2 }$$

and u_i is the usual biweight weighting

$$\begin{equation} u_i = \frac{v_i-\bar{v}}{9\, \mathrm{MAD}(v_i)}, \end{equation} \tag{ 3 }$$

where MAD(v_i) is the median absolute deviation of the velocities.

We calculate cluster redshifts using the same biweight average estimator that is presented in Beers et al. (1990). Unlike the more subtle case of the variance, the biweight average is unbiased for all N_members.

A second issue related to bias at few N_members is the possibility that the presence of velocity substructure would bias the estimation of the dispersion. We tested whether that was the case by extracting smaller pseudo-observations from the 18 individual clusters for which we obtained 30 or more member velocities. We did not use the stacked cluster here, as substructure would be lost in the averaging. For each cluster, we randomly drew 1000 pseudo-observations with 10 ⩽ N_members ⩽ 25. The cluster redshift and dispersion from those smaller, random samples was computed and compared to the value that was measured with the full data set.

Figure 3 shows the results of this resampling analysis as a function of N_members; the black solid line is the average relative error 〈(σ_BI − σ_pseudo-obs)/σ_BI〉 of the sample velocity dispersion of all samples across all clusters, while the colored solid lines depict the average relative error for the individual clusters. The average relative error departs from zero at the percent level. From this, we conclude that the observation of only a small number of velocities per cluster does not introduce significant bias in the measurement of the velocity dispersion for an ensemble of clusters.

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However, we see that for some individual clusters that have many measured galaxy velocities, the distribution of velocities is such that measuring fewer members in a pseudo-observation yields, on average, a velocity dispersion that can have several to many percent difference with the one obtained with more members. This is a way in which observing few member galaxies will increase the scatter of observed velocity dispersions at fixed mass. The size of our sample does not allow us to pursue this effect thoroughly, but Figure 3 shows that this systematic increase in the scatter is of the order of 5%, relative to dispersions computed with more than 30 members. Saro et al. (2013) isolate the scatter that is not due to statistical effects and also find that the scatter due to systematics increases at few-N_members and that this effect is most significant when N_members is less than ∼30.

We now turn to the calculation of the statistical uncertainty on our measured redshifts and velocity dispersions. Beers et al. (1990) describe a number of different ways in which the confidence intervals on biweight estimators can be calculated. They conclude that the statistical jackknife and the statistical bootstrap both yield satisfactory confidence intervals. Broadly speaking, both of these methods estimate the confidence intervals by looking at the internal variability of a sample. The statistical jackknife constructs a confidence interval for an estimate from how much it varies when data points are removed. The bootstrap generates a probability distribution function for the estimate from resampling the observed values with replacement a large number of times, often 1000 or more. The confidence intervals can then be found from the percentiles of this distribution. Many publications after Beers et al. (1990) have chosen the bootstrap; different practices seen in its use, with papers quoting asymmetric confidence intervals and others symmetric ones, have prompted us to inspect our uncertainties carefully.

The reason for using the statistical bootstrap or jackknife is the absence of an analytic expression for the distribution of the errors, given that the source distribution of velocities is unknown, as is the distribution of measured biweight dispersions. We use the stacked cluster as the best model of a cluster with our selection of potential member galaxies. As explained in Section 3.2.1, the availability of SPT masses for all clusters allows us to construct this stacked cluster independently of cluster membership determination or dispersion measurements. We draw a large number of pseudo-observations with replacement from the stacked cluster, perform member selection, and calculate the cluster's redshift and velocity dispersion from each pseudo-observation. Thus, we generate a probability distribution function for those quantities.

We find that the distribution of the measured cluster redshift is close to a normal distribution whose standard deviation is well described by:

$$\begin{equation} \Delta z = \frac{1}{c} \frac{\sigma _{\mathrm{BI}}(1+z)}{\sqrt{N_{\mathrm{members}}}}. \end{equation} \tag{ 4 }$$

This is the "usual" standard error; the 1/c factor converts between velocity and redshift, and the 1 + z factor is needed because σ_BI is defined in the rest frame. At any given N_members, the average bootstrap and jackknife uncertainties also reproduce this standard error.

In the case of the velocity dispersion, the bootstrap and jackknife give confidence intervals that are too narrow. Simply put, those estimators use a sample's internal variability to infer likely properties of the population from which it was drawn. However, the variability is reduced by the membership selection, and the effect of that step is not included in the confidence interval.

The distribution of biweight sample dispersions measured in pseudo-observations after 3σ clipping membership selection is also observed to be close to a normal distribution in our resampling analysis. We set out to model the standard deviation of this distribution, which is the uncertainty that we are looking for.

If we draw observations from a normal distribution of variance σ² and calculate the velocity dispersion as the "usual" (non-biweight) sample standard deviation from n members, without a membership selection step, then the distribution of the measured standard deviation is related to a chi distribution with n − 1 degrees of freedom. Indeed, the sample standard deviation s is

$$\begin{equation} s = \sqrt{\frac{1}{n-1} \sum _{i=1}^n (v_i - \bar{v}_i)^2 }. \end{equation} \tag{ 5 }$$

This implies that

$$\begin{eqnarray} \frac{s}{\sigma } &=& \sqrt{\frac{1}{n-1} \sum _{i=1}^n \frac{(v_i - \bar{v}_i)^2}{\sigma ^2} } \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \Rightarrow \quad \sqrt{n-1}\frac{s}{\sigma } &=& \sqrt{ \sum _{i=1}^n \frac{(v_i - \bar{v}_i)^2}{\sigma ^2} } \sim \chi _{n-1} \end{eqnarray} \tag{ 7 }$$

which is the definition of a chi distribution.

The variance of the chi distribution varies very little between k = 10 and k = 100 degrees of freedom:

$$\begin{eqnarray} \mathrm{Var}\, \chi _{k} &=& k - 2\left(\frac{\Gamma ((k+1)/2)}{\Gamma (k/2)} \right)^2 \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} &=& 0.49 \quad (k=10) \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} &=& 0.50 \quad (k=100). \end{eqnarray} \tag{ 10 }$$

Therefore, taking the square root on each side to find the standard deviation Δ of the dispersion estimate s:

$$\begin{eqnarray} \Delta \left(\sqrt{n-1}\frac{s}{\sigma }\right) &=& \sqrt{0.5} = 0.7 \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} \Rightarrow\ \Delta s &=& 0.7 \sigma / \sqrt{n-1}. \end{eqnarray} \tag{ 12 }$$

Following the above, we model the uncertainty as

$$\begin{eqnarray} \Delta \sigma _{\mathrm{BI}} &=& C_{\mathrm{BI}} \sigma _{\mathrm{BI}} / \sqrt{N_{\mathrm{members}}-1} \end{eqnarray} \tag{ 13 }$$

where C_BI is a constant. We also parameterize the uncertainty on the gapper measurement in the same way, with a constant C_G.

Figure 4 shows the relative error in σ_BI measured from the resampling analysis, as a function of N_members, for 10 ⩽ N_members ⩽ 60. The solid black line shows the average error, and the blue dashed line is the asymmetric rms error.

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We find the numerical value of C_BI as the mean ratio of the rms error and $1 / \sqrt{N_{\mathrm{members}}-1}$. We find that C_BI = 0.92. The green dotted line of Figure 4 shows the uncertainty given by our model, ${\pm }0.92 / \sqrt{N_{\mathrm{members}}-1}$. Similarly for the gapper scale, we find that C_G = 0.91.

Therefore, we find that the 3σ membership selection combined with the biweight estimation of the dispersion gives an uncertainty increased by 30% compared with random sampling from a normal distribution, and also compared to the bootstrap and jackknife estimates. The larger errors are caused by non-Gaussianity in the velocity distribution and by the cluster membership selection, which can both include non-members and reject true members, generically leading to increased scatter in the measured dispersion.

We note that this effect is different than the systematic scatter shown in Figure 3, where the measured dispersion changed significantly for some individual clusters when resampling with fewer galaxy members. This latter effect likely has both physical (e.g., velocity sub-structure in the cluster) and measurement (e.g., member selection, interlopers) origins. However, both effects will be present at some level in any dispersion measurement, and the results here are important benchmarks for simulations to compare to and reproduce.

Figure 5 shows 43 cluster biweight velocity dispersions from Table 4 plotted against the masses estimated from their SPT SZ signal (combined with X-ray observations where applicable; Table 1, Section 2.1). The clusters that are included are those with N_members ⩾ 15 and z ⩾ 0.3, except for SPT-CL J0205-6432, which was flagged as having a potentially less reliable dispersion measurement in Section 3.2. We also plot, as a solid line, the predicted scaling between dispersion and mass from Saro et al. (2013)

$$\begin{equation} M_{200c, \,\mathrm{dyn}} = \left(\frac{\sigma _{\mathrm{DM}}}{A \times h_{70}(z)^C} \right)^B 10^{15}\ M_\odot \end{equation} \tag{ 14 }$$

where A = 939, B = 2.91, and C = 0.33, with negligible statistical uncertainty compared to the systematic uncertainty, whose floor is evaluated to be at 5% in dispersion (Evrard et al. 2008). σ_DM is the dispersion computed from dark-matter subhalos, which are identified as galaxies in simulations. This has a different functional form but is consistent with the Evrard et al. (2008) scaling relation.

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Our measurements appear to have a systematic offset relative to the model prediction. To quantify this offset, we compute the mean of the log mass ratio, ln (M_200c, dyn/M_200c, SPT). For each cluster i, we compute this log mass ratio, and its associated uncertainty $\sigma _{\ln M_\mathrm{ratio}, i}$. The uncertainty in the ratio is estimated from the quadrature sum of the fractional uncertainty in the SZ and dynamical mass estimates. To the latter, we add the expected intrinsic scatter in dynamical to true mass as estimated by Saro et al. (2013):

$$\begin{equation} \Delta _{\ln (M_{200c, \,\mathrm{dyn}}/M_{200c})} = 0.3 + 0.075z. \end{equation} \tag{ 15 }$$

The uncertainty on the SZ-based SPT mass already includes the effect of intrinsic scatter.

The weighted average log mass ratio is

$$\begin{equation} \langle \ln (M_{200c, \,\mathrm{dyn}}/M_{200c, \,\mathrm{SPT}}) \rangle = 0.33 \pm 0.10, \end{equation} \tag{ 16 }$$

where the weights for each data point are given by $1/\sigma _{\ln M_\mathrm{ratio}, i}^2$, and the uncertainty on the average is given by $1/\sigma ^2 = \sum _i 1/\sigma _{\ln M_\mathrm{ratio}, i}^2$. This average log ratio means that the dynamical mass is exp (0.33) = 1.39 times the SPT mass estimate.

Figure 5 shows as a dashed blue line how the N-body scaling relation is shifted if the log mass ratio is shifted by 0.33 to make the mass estimates coincide. Because the slope is 1/B = 1/2.91, the offset in log dispersion is (0.33 ± 0.10)/2.91 = 0.11 ± 0.03. In other words, the measured velocity dispersions are on average exp (0.11) = 1.12 times their expected value given the N-body simulation work and the current normalization of the SPT mass estimate. The size of this normalization offset is consistent with the expected size of systematic biases, as discussed in Section 5.3.

We quantify the level of Gaussianity in the dispersion estimates around the best-fit dispersion–mass relation by performing the Anderson–Darling test on the residuals. We find that the residuals are non-Gaussian at the 95% confidence level. If we remove the two clusters with the lowest dispersions (SPT-CL J0317-5935 and SPT-CL J2043-5035), we find per the Anderson–Darling test that the residuals are consistent with a normal distribution. This suggests that the scatter in ln σ is normal—i.e., that the dispersion distribution is log-normal—with a tail toward low dispersion, as might be suspected from the distribution of data points in Figure 5.

If the statistical uncertainty on dispersion measurements of individual clusters has been correctly estimated and is much larger than any systematic uncertainty, then the fractional scatter in ln σ at fixed mass should roughly equal the average fractional uncertainty in the individual measurements. The mean uncertainty in log dispersion at fixed mass is 0.24, including the intrinsic scatter of the scaling relation and the uncertainty on the SPT mass. Analysis of mock observations from simulated clusters indicates that the combination of intrinsic, statistical, and systematic effects would lead to a log-normal scatter of 0.26 in dispersion at fixed mass (Saro et al. 2013). Both numbers are smaller but in general agreement with the measured scatter in ln σ at fixed mass, (0.31 ± 0.03). Systematic effects can increase the scatter, as discussed in Section 5.3.

In this section, we compare the velocity dispersion measurements to X-ray observables and mass estimates, and contrast these results with predictions from simulations. We also compare our results to those when using a separate low-redshift sample of comparable-mass clusters with similar velocity dispersion and X-ray observables.

For the clusters in this work, we primarily use X-ray measurements from a Chandra X-ray Visionary Project to observe the 80 most significantly detected clusters by the SPT at z > 0.4 (PI: B. Benson). This cluster sample has been observed and analyzed in a uniform fashion to derive cluster mass-observables (B. A. Benson et al. 2014, in preparation) and cluster cooling properties (McDonald et al. 2013). In Table 6, we give the X-ray measured intracluster medium temperature, T_X, and the Y_X-derived cluster mass, $M_{500c,\mathrm{Y_X}}$, for the 28 clusters that overlap with the sample from B. A. Benson et al. (2014, in preparation).

Table 6. X-Ray and Velocity Dispersion Data

Cluster ID	z	N	σBI	TX	$M_{500c,\mathrm{Y_X}}$
			(km s−1)	(keV)	(1014 M☉)
SPT-CL
J0000-5748	0.702	26	563 ± 104	$6.75^{+3.09}_{-1.85}$	$4.11^{+1.06}_{-0.80}$
J0014-4952	0.752	29	811 ± 141	$5.91^{+1.09}_{-0.65}$	$4.97^{+0.54}_{-0.38}$
J0037-5047	1.026	18	555 ± 124	$2.85^{+1.44}_{-0.75}$	$1.22^{+0.38}_{-0.28}$
J0040-4407	0.350	36	1277 ± 199	$5.95^{+1.09}_{-0.74}$	$5.42^{+0.62}_{-0.49}$
J0234-5831	0.415	22	926 ± 186	$9.20^{+3.98}_{-2.52}$	$6.83^{+1.62}_{-1.24}$
J0438-5419	0.422	18	1422 ± 317	$11.32^{+2.07}_{-1.76}$	$10.74^{+1.19}_{-1.11}$
J0449-4901	0.790	20	1090 ± 230	$10.39^{+3.43}_{-2.53}$	$6.50^{+1.24}_{-1.08}$
J0509-5342	0.462	21	678 ± 139	$7.28^{+1.30}_{-1.38}$	$5.62^{+0.65}_{-0.72}$
J0516-5430	0.294	48	724 ± 97	$10.95^{+2.27}_{-1.73}$	$12.25^{+1.52}_{-1.31}$
J0528-5300	0.769	21	1318 ± 271	$4.85^{+1.56}_{-0.98}$	$2.68^{+0.50}_{-0.38}$
J0546-5345	1.066	21	1191 ± 245	$7.61^{+2.45}_{-1.52}$	$5.22^{+0.98}_{-0.74}$
J0551-5709	0.424	34	966 ± 155	$3.12^{+0.28}_{-0.28}$	$3.04^{+0.23}_{-0.23}$
J0559-5249	0.609	37	1146 ± 176	$6.74^{+0.76}_{-0.71}$	$5.93^{+0.45}_{-0.45}$
J2043-5035	0.723	21	524 ± 108	$5.87^{+1.03}_{-0.63}$	$4.44^{+0.50}_{-0.38}$
J2106-5844	1.131	18	1228 ± 274	$10.36^{+2.49}_{-1.79}$	$8.37^{+1.26}_{-1.07}$
J2135-5726	0.427	33	1029 ± 167	$7.78^{+4.41}_{-2.10}$	$5.15^{+1.56}_{-0.97}$
J2145-5644	0.480	37	1638 ± 251	$5.34^{+0.90}_{-0.74}$	$5.00^{+0.57}_{-0.52}$
J2146-4633	0.932	18	1817 ± 405	$5.14^{+0.87}_{-0.82}$	$3.66^{+0.41}_{-0.42}$
J2148-6116	0.571	30	966 ± 165	$8.24^{+3.14}_{-2.18}$	$5.42^{+1.14}_{-0.93}$
J2248-4431	0.351	15	1301 ± 320	$12.37^{+1.01}_{-0.77}$	$16.35^{+0.84}_{-0.70}$
J2325-4111	0.358	33	1921 ± 312	$8.84^{+2.16}_{-1.55}$	$8.39^{+1.19}_{-0.98}$
J2331-5051	0.575	78	1382 ± 145	$6.38^{+1.84}_{-1.25}$	$4.66^{+0.81}_{-0.66}$
J2332-5358a	0.402	53	1240 ± 158	$7.40^{+1.20}_{-0.70}$	$5.66^{+0.48}_{-0.48}$
J2337-5942	0.776	19	707 ± 153	$6.95^{+1.91}_{-1.31}$	$5.76^{+0.92}_{-0.74}$
J2341-5119	1.002	15	959 ± 236	$9.30^{+2.45}_{-2.02}$	$5.77^{+0.89}_{-0.83}$
J2344-4243	0.595	32	1878 ± 310	$11.72^{+2.88}_{-2.10}$	$11.64^{+1.64}_{-1.36}$
J2355-5056	0.320	37	1104 ± 169	$4.34^{+1.15}_{-0.81}$	$3.00^{+0.54}_{-0.47}$
J2359-5009	0.775	26	950 ± 175	$4.41^{+1.18}_{-0.65}$	$2.58^{+0.41}_{-0.28}$
Literature
A3571	0.039	70	$1085^{+110}_{-107}$	6.81 ± 0.10	5.90 ± 0.06
A2199	0.030	51	$860^{+134}_{-83}$	3.99 ± 0.10	2.77 ± 0.05
A496	0.033	151	$750^{+61}_{-56}$	4.12 ± 0.07	2.96 ± 0.04
A3667	0.056	123	$1208^{+95}_{-84}$	6.33 ± 0.06	7.35 ± 0.07
A754	0.054	83	$784^{+90}_{-85}$	8.73 ± 0.00	8.47 ± 0.13
A85	0.056	131	$1069^{+105}_{-92}$	6.45 ± 0.10	5.98 ± 0.07
A1795	0.062	87	$887^{+116}_{-83}$	6.14 ± 0.10	5.46 ± 0.06
A3558	0.047	206	$997^{+61}_{-51}$	4.88 ± 0.10	4.78 ± 0.07
A2256	0.058	47	$1279^{+136}_{-117}$	8.37 ± 0.24	7.84 ± 0.15
A3266	0.060	132	$1182^{+100}_{-85}$	8.63 ± 0.18	9.00 ± 0.13
A401	0.074	123	$1142^{+80}_{-70}$	7.72 ± 0.30	8.63 ± 0.24
A2052	0.035	62	$679^{+97}_{-59}$	3.03 ± 0.07	1.84 ± 0.03
Hydra-A	0.055	82	$614^{+52}_{-43}$	3.64 ± 0.06	2.83 ± 0.03
A119	0.044	80	$850^{+108}_{-92}$	5.72 ± 0.00	4.50 ± 0.03
A2063	0.034	91	$664^{+50}_{-45}$	3.57 ± 0.19	2.21 ± 0.08
A1644	0.048	92	$937^{+107}_{-77}$	4.61 ± 0.14	4.21 ± 0.09
A3158	0.058	35	$1046^{+174}_{-99}$	4.67 ± 0.07	4.13 ± 0.05
MKW3s	0.045	30	$612^{+69}_{-52}$	3.03 ± 0.05	2.09 ± 0.03
A3395	0.051	107	$934^{+123}_{-100}$	5.10 ± 0.17	6.74 ± 0.18
A399	0.071	92	$1195^{+94}_{-79}$	6.49 ± 0.17	6.18 ± 0.11
A576	0.040	48	$1006^{+138}_{-91}$	3.68 ± 0.11	2.34 ± 0.05
A2634	0.030	69	$705^{+97}_{-61}$	2.96 ± 0.09	1.74 ± 0.04
A3391	0.055	55	$990^{+254}_{-128}$	5.39 ± 0.19	4.06 ± 0.10

Notes. SPT data and data from the literature used in Figure 6. For the SPT data, the redshift, number of member-galaxy redshifts N (≡ N_members) and velocity dispersion from Table 4 are repeated for reference, and the X-ray temperature and $M_{500c,\mathrm{Y_X}}$ are from the same Chandra XVP program, except for one case that is marked. The literature clusters draw their velocity dispersion from Girardi et al. (1996) and X-ray properties from Vikhlinin et al. (2009). ^a XMM X-ray data from Andersson et al. (2011).

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We also plot our results alongside velocity dispersion and X-ray measurements of comparable-mass low-redshift clusters taken from the literature. For the X-ray measurements, we use measurements of T_X and $M_{500c,\mathrm{Y_X}}$ from the low-z sample of Vikhlinin et al. (2009), which were produced following an analysis identical to that used in B. A. Benson et al. (2014, in preparation). The velocity dispersions for many of those galaxy clusters were calculated in a uniform way in Girardi et al. (1996). These velocity dispersion measurements were made with a different galaxy selection and more cluster members, and so will carry different systematics from our own. They nonetheless provide an interesting baseline for comparison. We will see that the scatter of those data points is smaller that that of our sample. Taking intrinsic scatter and mass uncertainties into account, the measured scatter of the literature sample at fixed mass is consistent both with our analysis from Section 4.2 and with the Girardi et al. (1996) uncertainties, and therefore is due to the lower statistical uncertainty.

Figure 6 shows the velocity dispersion versus X-ray temperature and versus $M_{500c,\mathrm{Y_X}}$. The blue points are our data, and the black crosses are the data from the literature; these literature data are listed for reference in Table 6.

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The left panel of Figure 6 shows dispersion versus T_X. The empirical best-fit scaling relation from Girardi et al. (1996), where $\sigma \propto T_X^{0.61}$, is plotted as a solid line; this scaling relation is consistent with the Vikhlinin et al. (2009) temperatures used here, although it was fit using X-ray temperatures from a different source, David et al. (1993). The comparison to the temperature is especially interesting in that there is, to first order, a simple correspondence between temperature and velocity dispersion. Assuming that the galaxies and gas are both in equilibrium with the potential (see, e.g., Voit 2005), then σ² = k_BT_X/(μm_p), where m_p is the proton mass, and μ the mean molecular weight (we take μ = 0.58; see Girardi et al. 1996). This energy equipartition line is plotted as a dashed line in the left panel of Figure 6. Real clusters show a deviation from this simple model, but it offers an interesting theoretical baseline, one independent of data or simulations. This relation implies that the temperature and velocity dispersion have a similar redshift evolution, which is why the quantities in this plot are uncorrected for redshift.

The X-ray Y_X observable, while not independent from T_X, is expected to be significantly less sensitive to cluster mergers than T_X, with simulations predicting Y_X to have both a lower scatter and to be a less biased mass indicator (see, e.g., Kravtsov et al. 2006; Fabjan et al. 2011). For this reason, we also plot the velocity dispersion against $M_{500c,\mathrm{Y_X}}$ (times a redshift-evolution factor), in the right panel of Figure 6. The dot-dashed line is the scaling relation predicted from the simulation analysis of Saro et al. (2013).

Computing the average log ratio of the dynamical and Y_X-based masses gives 0.26 ± 0.12, corresponding to a bias of 0.09  ±  0.04 in log dispersion. This was computed, in the previous section, to be 0.33 ± 0.10 in the case of dynamical and SPT masses, corresponding to 0.11 ± 0.03 in log dispersion. The residuals of the dispersion–$M_{500c,\mathrm{Y_X}}$ relation have a measured scatter in dispersion of 0.31 ± 0.03, which is the same as the measurement made using the SZ-based SPT mass. The Anderson–Darling test gives similar results to the residuals of the previous section, suggesting a normal scatter in ln σ with a tail toward low dispersion.

While there is very good agreement between the scaling relations comparing the dispersion to the SPT and X-ray mass estimates, we note that the results are not independent. Nine of the clusters included in this work from Reichardt et al. (2013) quoted joint SZ and X-ray mass estimates, which we have included in our sample of SPT mass estimates. In addition, the SPT significance–mass relation used in the SZ mass estimates was in part calibrated from a sub-sample of SPT clusters with X-ray mass estimates, which have effectively calibrated the SPT cluster mass normalization. Regardless, the majority of clusters in this work have SPT mass estimates derived only from the SPT SZ measurements, which have very different noise properties from the X-ray measurements, therefore the agreement in the measured scatters is not entirely trivial.

There are two different, although related, systematics that affect the interpretation of velocity dispersion measurements: systematics that can affect the measurement of the velocity dispersion of galaxies, and a possible velocity bias between the galaxies and the underlying dark matter halo. The velocity bias cannot be empirically measured in our data. However, both effects have been quantified in recent cluster simulation studies (Saro et al. 2013; Gifford et al. 2013; Munari et al. 2013; Wu et al. 2013). In principle, the velocity bias could explain part of the offset between our measurements and the predicted relation from N-body simulations, described in Sections 5.1 and 5.2. The velocity bias has been estimated to be of the order of ∼5% by Evrard et al. (2008). More recent studies have found a spread in the velocity bias of ∼10% when comparing different tracers and algorithms for predicting the galaxy population (Gifford et al. 2013) and comparing dark-matter with hydrodynamic simulations (Munari et al. 2013; Wu et al. 2013).

The measured velocity dispersion can be biased from the true value by the galaxy selection of the measurements, in principle being affected by systematics relating to the luminosity, color, and offset from the cluster center of the galaxies. The observations will also have some amount of imperfect membership determination, due to the presence of interlopers.

Of those effects, the luminosity of the selected galaxies has the potential to create the largest bias, according to recent simulation work showing that brighter galaxies have a smaller velocity dispersion (Saro et al. 2013; Old et al. 2013; Wu et al. 2013; Gifford et al. 2013). Observing only the 25 brightest galaxies of the halo leads to the velocity dispersion being biased low by as much as ∼5–10%. These results are difficult to directly compare to our measurements, because the simulated observations use the N brightest galaxies, while real ones target a more varied population. Nonetheless, it is true that brighter galaxies are targeted in priority in our observations. As far as our data is concerned, we took the 18 clusters for which we obtained 30 or more member velocities, and compared the dispersion of the 15 brightest galaxies (among those observed spectroscopically) with our best value. The bright galaxies have a dispersion that is (5 ± 4)% lower than the measured dispersion.

The effect of the radius at which the galaxies are sampled is discussed in Sifón et al. (2013). They conclude that there are too many uncertainties to accurately correct for a potential bias. Regardless, they estimate the systematic bias compared to sampling all the way to the virial radius by using mock observations of a simulated cluster. They find an average correction of 0.91 to the velocity dispersions and 0.79 to the dynamical masses; in other words, the measured velocity dispersions are biased high by 10%. That a small aperture radius should bias the velocity dispersions high is in line with the results of Saro et al. (2013) and Gifford et al. (2013).

We performed a related test of the radial dependence of the dispersion using our best-sampled clusters. For the 18 clusters with 30 or more member velocities, we compared the dispersion of the half of the galaxies that are the most central with the half that are farther away from the center. There is no statistically significant difference between the most central galaxies and the most distant, their dispersions differing by (− 2 ± 6)%. Our data sample cluster member galaxies out to a projected radius that is typically ∼0.5 Mpc $h_{70}^{-1}$, which is generally less that the virial radius. As a result, our data are not always directly comparable to the numbers quoted from the literature.

Gifford et al. (2013) also explores the effect of measuring velocity dispersion from galaxies that are a mix of red (passive) and blue (in-falling) cluster members, including blue galaxies alongside red galaxies in the spectroscopic sample, and find that including a few blue galaxies only has a small differential effect on the measured dispersion.

In addition to causing a bias in the measurement, systematics can also increase the scatter. The resampling of Section 4.1 implies that there is an increase in the scatter at few-N_members due to the different shapes of the velocity distribution of individual clusters. Saro et al. (2013) find that the scatter due to systematics is most significant when N_members ≲ 30.

More feedback between statistical studies of much larger spectroscopic samples than the present one and simulation work will be needed to understand precisely how those effects affect the measured velocity dispersion. One could imagine using the color, magnitude, position, and number of the galaxies with a spectroscopic redshift to compute a correction factor to the dispersion, or relative weights for the proper velocities, that would eliminate the systematic bias and scatter from the sources discussed above.