Galaxy Kinematics and Masses of Clusters to z=1.3

Galaxy Kinematics and Mass Calibration in Massive SZE Selected Galaxy Clusters to z=1.3

R. Capasso, A. Saro, J. J. Mohr, A. Biviano, V. Strazzullo, S. Grandis, D. E. Applegate, M. B. Bayliss, B. A. Benson, L. E. Bleem, S. Bocquet, M. Brodwin, E. Bulbul, J. E. Carlstrom, I. Chiu, J. P. Dietrich, N. Gupta, T. de Haan, J. Hlavacek-Larrondo, M. Klein, A. von der Linden, M. McDonald, D. Rapetti, C. L. Reichardt, K. Sharon, B. Stalder, S. A. Stanford, A. A. Stark, C. Stern, A. Zenteno
Affilitations are listed at the end of the paperJuly 19, 2019
Affilitations are listed at the end of the paperJuly 19, 2019

The galaxy phase-space distribution in galaxy clusters provides insights into the formation and evolution of cluster galaxies, and it can also be used to measure cluster mass profiles. We present a dynamical study based on 3000 passive, non-emission line cluster galaxies drawn from 110 galaxy clusters. The galaxy clusters were selected using the Sunyaev-Zel’dovich effect (SZE) in the 2500 deg SPT-SZ survey and cover the redshift range . We model the clusters using the Jeans equation, while adopting NFW mass profiles and a broad range of velocity dispersion anisotropy profiles. The data prefer velocity dispersion anisotropy profiles that are approximately isotropic near the center and increasingly radial toward the cluster virial radius, and this is true for all redshifts and masses we study. The pseudo-phase-space density profile of the passive galaxies is consistent with expectations for dark matter particles and subhalos from cosmological -body simulations. The dynamical mass constraints are in good agreement with external mass estimates of the SPT cluster sample from either weak lensing, velocity dispersions, or X-ray measurements. However, the dynamical masses are lower (at the 2.2 level) when compared to the mass calibration favored when fitting the SPT cluster data to a LCDM model with external cosmological priors, including CMB anisotropy data from Planck. The tension grows with redshift, where in the highest redshift bin the ratio of dynamical to SPT+Planck masses is (statistical and systematic), corresponding to 2.6 tension.

1 Introduction

In the current paradigm of structure formation, halos form through the gravitational collapse of overdense regions that are seeded by processes in the early universe. The formation of cold dark matter (CDM) dominated halos proceeds through a sequence of mergers and the accretion of surrounding material, leading to the formation of the galaxy groups and clusters we observe. Baryonic processes associated with the intracluster medium (ICM) and the galaxies also play a role, making galaxy clusters important laboratories for investigations of structure formation and galaxy evolution as well as useful cosmological probes.

Studies of structure formation using cosmological -body simulations have been used to demonstrate that halos formed from collisionless CDM have, on average, a universal mass density profile (Navarro et al., 1996, 1997, hereinafter NFW). This profile is characterized by two parameters: the virial radius 111 defines the sphere within which the cluster overdensity with respect to the critical density at the cluster redshift is . Throughout this paper, we consider and refer to simply as the virial radius., and the scale radius , which is the radius at which the logarithmic slope of the density profile is . Numerous observational studies have found the mass distributions of clusters to be well described by this model (e.g., Carlberg et al., 1997; van der Marel et al., 2000; Biviano & Girardi, 2003; Katgert et al., 2004; Umetsu et al., 2014).

Another interesting feature is the finding in -body simulations that the quantity , where is the mass density and the velocity dispersion, has a power-law form. This quantity is known as pseudo-phase-space density (PPSD) profile, , and its power-law form resembles that of the self-similar solution for halo collapse by Bertschinger (1985) and is thought of as a dynamical equivalent of the NFW mass density profile (Taylor & Navarro, 2001). Others (Austin et al., 2005; Barnes et al., 2006) have suggested that the PPSD profile results from dynamical collapse processes, and should therefore be a robust feature of approximately virialized halos that have undergone violent relaxation (Lynden-Bell, 1967).

The galaxy population is more difficult to study in simulations, because of the over-merging problem, i.e. the premature destruction of dark matter halos in the dense clusters environments in dissipationless -body simulations (e.g. Moore et al., 1996), and the additional baryonic physics that must be included. However, from the observational side the properties of the galaxy population and trends with mass and redshift can be readily measured and interpreted as long as: (1) selection effects are understood and (2) precise cluster mass measurements are available to ensure that the same portion of the virial region is being studied in clusters of all masses and redshifts. By comparing the galaxy properties to the expectations for collisionless particles studied through -body simulations, one can characterize the impact of possible additional interactions beyond gravity that are playing a role in the formation of the galaxy population.

As an example, the radial distribution of galaxies in clusters is well fit by an NFW model when clusters are stacked in the space of (e.g. Lin et al., 2004; Muzzin et al., 2007; Zenteno et al., 2016). In cluster samples extending to redshift , it is clear that the concentration , defined as the ratio between and , varies dramatically from cluster to cluster, and that when stacked, the varies systematically with the prevalence of star formation (Hennig et al., 2017, hereinafter H17). The red, passively evolving galaxies have concentrations similar to those expected for the dark matter on these halo mass scales, while the star forming, and presumably infalling blue galaxies are far less concentrated. The number of luminous cluster galaxies (magnitudes ) within the virial region scales with cluster mass as where (Lin et al., 2004), and this property appears to be unchanged since redshift (H17). The departure from in this relation is puzzling, given that massive clusters accrete lower mass clusters and groups (Lin & Mohr, 2004) and this is presumably evidence for galaxy destruction processes that are more efficient in the most massive halos or for a mass accretion history that varies with mass on cluster scales (see discussion in Chiu et al., 2016b). There is evidence for an increase in the fraction of cluster galaxies that are dominated by passively evolving stellar populations since (H17), which provides constraints on the timescales over which quenching of star formation occurs as galaxies fall into clusters (McGee et al., 2009).

Understanding the dynamics of galaxy accretion into clusters from either lower mass clusters and groups or even individual systems from within the surrounding, lower density infall region can in principle shed additional light on galaxy evolution. A simulation based study argues that satellite orbits should become marginally more radial at higher redshifts, especially for systems with a higher host halo mass (Wetzel, 2011). Probes of redshift trends in the orbital characteristics of cluster galaxies have already been carried out (Biviano & Poggianti, 2009), providing some indication that passive galaxies have systematically different orbits at low and high redshift. In other studies of high redshift, relatively low mass systems, evidence has emerged that recently quenched galaxies have a preferred phase space distribution that is different from that of passive galaxies (Muzzin et al., 2014; Noble et al., 2016).

In this paper, we attempt to build upon these studies by focusing on a dynamical analysis of galaxies within a large ensemble of Sunyaev-Zel’dovich effect (SZE) selected galaxy clusters extending to redshift . In contrast to these previous dynamical studies, our cluster sample has a well understood selection that does not depend on the galaxy properties, and the sample extends over a broad redshift range, allowing cleaner probes for redshift trends. Moreover, each cluster has an SZE based mass estimate with 20 percent uncertainty (Bocquet et al., 2015), enabling us to estimate virial radii with  percent uncertainties and thereby ensuring that we are examining comparable regions of the cluster at all masses and redshifts.

Our goals are to study (1) whether there is evidence that the orbital characteristics of the passive galaxies are changing with redshift or mass in the cluster ensemble, (2) whether there is evidence within the galaxy dynamics for dynamical equilibrium and self-similarity with mass and redshift, and (3) whether the cluster mass constraints from our analysis are consistent with masses obtained through independent calibration in previously published SPT analyses.

Combining spectroscopic observations obtained at Gemini South, the VLT and the Magellan telescopes in a sample of 110 SPT-detected galaxy clusters, we construct a large sample of 3000 passive cluster members, spanning the wide redshift range of . With this dataset we carry out a Jeans analysis (e.g. Binney & Tremaine, 1987) that adopts a framework of spherical symmetry and allows for a range of different velocity dispersion anisotropy profiles. Specifically, we use the Modeling Anisotropy and Mass Profiles of Observed Spherical Systems code (Mamon et al., 2013, hereafter MAMPOSSt) to explore the range of models consistent with the data, and then we use the results to characterize the velocity dispersion anisotropy profile, to test for evidence of virialization with the pseudo-phase-space density (PPSD) profile and to probe for trends with cluster mass or redshift in both. Exploring a broad range of possible velocity dispersion anisotropy profiles then allows us to extract robust constraints on the cluster virial masses as well. Throughout this paper, we address a number of limitations that have to be taken into account, such as the degeneracy between the mass and the velocity anisotropy profiles (see Section 2), the assumptions of spherical symmetry and dynamical equilibrium, and the presence of foreground/background interloper galaxies projected onto the cluster virial region. Mamon et al. (2013) have tested the accuracy of MAMPOSSt by analysing a sample of clusters extracted from numerical simulations, recovering estimates with mean bias at % and rms scatter of 6% for kinematic samples with 500 tracers.

The paper is organized as follows: In Section 2 we give an overview of the theoretical framework. In Section 3 we summarize the dataset used for our analysis. The results are presented in Section 4, where we discuss the outcome of our analysis of the velocity dispersion anisotropy profile, the PPSD profiles, the virial mass comparisons, and the impact of disturbed clusters on our analysis. We present our conclusions in Section 5. Throughout this paper we adopt a flat CDM cosmology with the Hubble constant , and assume the matter density parameter . The virial quantities are computed at radius . All quoted uncertainties are equivalent to Gaussian confidence regions, unless otherwise stated.

2 Theoretical Framework

The dynamical analysis implemented in this paper is based on the application of the Jeans equation to spherical systems (Binney & Tremaine, 1987). In spherical coordinates, the Jeans equation can be written as


where is the number density profile of the tracer galaxy population, is the gravitational potential, and are the components of the velocity dispersion along the three spherical coordinates . It is convenient to write this equation as


where is the enclosed mass within radius , is Newton’s constant, is the velocity dispersion anisotropy that is generically a function of radius, and .

In principle, it is therefore possible to use Eq. 2 to estimate the mass distribution of the system. However, as external observers we can measure only projected quantities, including the surface density profile of the tracer population and the line-of-sight (LOS) velocity distribution or simply its dispersion . These two observed functions are not sufficient to derive a unique mass model— at least within the context of the typical observational uncertainties where full knowledge of the line of sight velocity distribution is lacking (Merritt, 1987). This degeneracy between the mass and the velocity anisotropy profiles can be addressed in several ways.

2.1 Dynamical analysis with MAMPOSSt

A first method consists of assuming that, at a given projected radius, the LOS velocity distribution can be described by a Gaussian (Strigari et al., 2008; Wolf et al., 2010). However, because the true distribution may deviate from Gaussianity— including in the case of anisotropic systems (Merritt, 1987)— the constraints from this approach on the anisotropy are weak (Walker et al., 2009). A step forward, therefore, consists of analysing the kurtosis of the LOS velocity distribution. This was found to be a powerful tool to break the mass-anisotropy degeneracy (Merritt, 1987; Gerhard, 1993; van der Marel & Franx, 1993; Zabludoff et al., 1993; Łokas, 2002; Łokas & Mamon, 2003). Much effort has been put into further constraining the anisotropy taking into account more of the information contained in the projected phase-space velocity distribution as a function of projected radius, considering, for example, the full set of even moments (Kronawitter et al., 2000; Wojtak et al., 2008, 2009).

In this work, we fit the whole projected phase-space velocity distribution using the MAMPOSSt code (Mamon et al., 2013). We use this code to determine the mass and anisotropy profiles of a cluster in parametrized form by performing a likelihood exploration to the distribution of the cluster galaxies in projected phase-space, constraining the parameters describing these two profiles. This method is based on the assumption of spherical symmetry, and adopts a Gaussian as the shape of the distribution of the 3D velocities without demanding Gaussian LOS velocity distributions. We emphasize that it does not assume that light traces mass, allowing the scale radius of the total mass distribution to differ from that of the galaxy distribution. For more details on the code, we refer the reader to Mamon et al. (2013).

MAMPOSSt requires parametrized models for the number density profile, the mass profile and the velocity anisotropy profile, without any limitation on the choice of these models. We will address the issue of the number density profile in Section 3.3.3.

2.2 Mass and anisotropy profiles

For the mass profile in our analyses, we consider 5 models, namely the Navarro, Frenk and White profile (NFW; Navarro et al., 1996), the Einasto profile (Einasto, 1965), the Burkert profile (Burkert, 1995), the Hernquist profile (Hernquist, 1990), and the Softened Isothermal Sphere (SIS; Geller et al., 1999). All these models have been applied to galaxy clusters in previous works (e.g. Mohr et al., 1996; Rines et al., 2003; Katgert et al., 2004; Rines & Diaferio, 2006; Biviano et al., 2006). As we show later, our data cannot distinguish among these different mass profiles, and so we adopt the NFW model and its associated scale radius in the discussion below.

For the velocity anisotropy profile, we consider the following five models that have been used in previous MAMPOSSt analyses:

  1. a radially constant anisotropy model,

  2. the Tiret anisotropy profile (Tiret et al., 2007),


    which is isotropic at the center, characterized by the anisotropy value at large radii, and with the anisotropy radius fixed to the of the NFW mass profile;

  3. a model with anisotropy of opposite sign at the center and at large radii,


    where again the anisotropy radius has been fixed to be identical to ;

  4. the Mamon & Łokas (2005) profile,

  5. the Osipkov-Merritt anisotropy profile (Osipkov, 1979; Merritt, 1985),


Summarizing, we run MAMPOSSt with 3 free parameters: the virial radius , the scale radius of the mass distribution, and a velocity anisotropy parameter (e.g., , or , depending on the velocity anisotropy profile model). The maximum likelihood solutions are obtained using the NEWUOA software (Powell, 2006) and are shown in Section 4.

SPT-CL J0000-5748 27 24 0.702 6.44 1.37
SPT-CL J0013-4906 37 36 0.408 10.39 1.81
SPT-CL J0014-4952 38 27 0.752 7.67 1.42
SPT-CL J0033-6326 28 14 0.599 6.77 1.45
SPT-CL J0037-5047 37 17 1.026 5.34 1.13
SPT-CL J0040-4407 33 33 0.350 13.95 2.04
SPT-CL J0102-4603 35 16 0.841 6.46 1.30
SPT-CL J0102-4915 81 80 0.870 17.84 1.80
SPT-CL J0106-5943 35 26 0.348 9.04 1.76
SPT-CL J0118-5156 8 7 0.705 5.40 1.29
SPT-CL J0123-4821 26 18 0.655 6.43 1.40
SPT-CL J0142-5032 28 23 0.679 8.17 1.50
SPT-CL J0200-4852 45 34 0.499 7.26 1.55
SPT-CL J0205-5829 15 8 1.322 6.07 1.06
SPT-CL J0205-6432 19 12 0.744 4.90 1.23
SPT-CL J0212-4657 26 20 0.654 8.06 1.51
SPT-CL J0232-5257 77 61 0.556 7.61 1.54
SPT-CL J0233-5819 9 9 0.664 5.68 1.33
SPT-CL J0234-5831 24 21 0.415 11.56 1.87
SPT-CL J0235-5121 96 82 0.278 9.41 1.84
SPT-CL J0236-4938 66 63 0.334 6.39 1.58
SPT-CL J0240-5946 19 17 0.400 8.06 1.67
SPT-CL J0243-4833 39 37 0.498 10.68 1.76
SPT-CL J0243-5930 32 25 0.634 6.53 1.42
SPT-CL J0252-4824 27 22 0.421 6.92 1.57
SPT-CL J0254-5857 37 32 0.438 11.18 1.83
SPT-CL J0257-5732 15 14 0.434 4.91 1.39
SPT-CL J0304-4401 45 35 0.458 11.62 1.84
SPT-CL J0304-4921 79 72 0.392 11.02 1.85
SPT-CL J0307-6225 26 17 0.580 7.21 1.49
SPT-CL J0310-4647 33 28 0.707 6.15 1.35
SPT-CL J0317-5935 25 18 0.469 5.97 1.47
SPT-CL J0324-6236 19 9 0.750 7.28 1.40
SPT-CL J0330-5228 80 71 0.442 9.77 1.75
SPT-CL J0334-4659 29 25 0.486 8.23 1.62
SPT-CL J0346-5439 85 79 0.530 8.11 1.59
SPT-CL J0348-4515 31 24 0.359 8.93 1.75
SPT-CL J0352-5647 22 16 0.649 6.07 1.37
SPT-CL J0356-5337 26 5 1.036 4.98 1.10
SPT-CL J0403-5719 31 24 0.467 5.62 1.44
SPT-CL J0406-4805 28 26 0.736 6.46 1.35
SPT-CL J0411-4819 45 42 0.424 11.70 1.87
SPT-CL J0417-4748 40 30 0.579 10.49 1.69
SPT-CL J0426-5455 15 11 0.642 7.28 1.46
SPT-CL J0433-5630 24 18 0.692 4.84 1.25
SPT-CL J0438-5419 92 87 0.422 14.44 2.00
SPT-CL J0449-4901 20 16 0.792 7.13 1.37
SPT-CL J0456-5116 31 20 0.562 7.27 1.51
SPT-CL J0509-5342 93 88 0.461 7.34 1.58
SPT-CL J0511-5154 18 14 0.645 5.90 1.36
SPT-CL J0516-5430 51 47 0.295 10.22 1.88
SPT-CL J0521-5104 21 21 0.675 5.92 1.35
SPT-CL J0528-5300 75 63 0.768 5.24 1.25
SPT-CL J0533-5005 4 4 0.881 5.33 1.20
SPT-CL J0534-5937 3 3 0.576 4.24 1.25
SPT-CL J0540-5744 24 17 0.760 5.38 1.26
SPT-CL J0542-4100 36 29 0.640 7.27 1.46
SPT-CL J0546-5345 54 49 1.066 7.01 1.22
SPT-CL J0549-6205 31 26 0.375 16.45 2.13
SPT-CL J0551-5709 39 30 0.423 7.30 1.60
Table 1: Cluster spectroscopic dataset: We present the cluster name, total number of galaxies in the spectroscopic sample , members used in our analysis after interloper rejection , cluster redshift , and cluster mass  and corresponding virial radius from Bocquet et al. (2015).
SPT-CL J0555-6406 34 27 0.345 10.82 1.88
SPT-CL J0559-5249 72 67 0.609 8.04 1.53
SPT-CL J0655-5234 33 30 0.472 7.74 1.60
SPT-CL J2017-6258 38 35 0.535 5.73 1.41
SPT-CL J2020-6314 27 18 0.537 4.91 1.34
SPT-CL J2022-6323 34 29 0.383 6.63 1.57
SPT-CL J2026-4513 15 11 0.688 5.05 1.27
SPT-CL J2030-5638 42 36 0.394 5.67 1.48
SPT-CL J2032-5627 39 32 0.284 8.16 1.75
SPT-CL J2035-5251 42 29 0.529 9.05 1.65
SPT-CL J2040-5725 7 4 0.930 5.00 1.15
SPT-CL J2043-5035 33 21 0.723 6.40 1.36
SPT-CL J2056-5459 13 11 0.718 5.06 1.26
SPT-CL J2058-5608 10 6 0.607 4.40 1.25
SPT-CL J2100-4548 37 18 0.712 4.68 1.23
SPT-CL J2106-5844 16 16 1.132 10.74 1.37
SPT-CL J2115-4659 31 28 0.299 6.03 1.57
SPT-CL J2118-5055 55 33 0.624 5.57 1.35
SPT-CL J2124-6124 23 21 0.435 7.56 1.61
SPT-CL J2130-6458 41 40 0.316 7.69 1.69
SPT-CL J2135-5726 31 30 0.427 9.15 1.72
SPT-CL J2136-4704 19 19 0.424 6.50 1.53
SPT-CL J2136-6307 6 6 0.926 5.02 1.15
SPT-CL J2138-6008 39 32 0.319 10.37 1.87
SPT-CL J2140-5727 17 11 0.404 5.49 1.46
SPT-CL J2145-5644 43 35 0.480 9.99 1.73
SPT-CL J2146-4633 15 7 0.933 7.26 1.30
SPT-CL J2146-4846 26 25 0.623 6.00 1.38
SPT-CL J2146-5736 34 23 0.602 5.77 1.38
SPT-CL J2148-6116 24 24 0.571 6.45 1.45
SPT-CL J2155-6048 23 19 0.539 5.23 1.36
SPT-CL J2159-6244 38 36 0.391 6.59 1.56
SPT-CL J2222-4834 29 25 0.652 7.56 1.48
SPT-CL J2232-5959 34 26 0.595 7.82 1.53
SPT-CL J2233-5339 33 28 0.440 7.84 1.62
SPT-CL J2248-4431 15 14 0.351 22.11 2.37
SPT-CL J2300-5331 25 21 0.262 6.63 1.64
SPT-CL J2301-5546 12 8 0.748 4.21 1.17
SPT-CL J2306-6505 46 42 0.530 8.46 1.61
SPT-CL J2325-4111 33 27 0.357 10.61 1.86
SPT-CL J2331-5051 119 108 0.576 7.85 1.54
SPT-CL J2332-5358 47 45 0.402 7.74 1.64
SPT-CL J2335-4544 37 33 0.547 9.04 1.63
SPT-CL J2337-5942 28 19 0.775 10.88 1.59
SPT-CL J2341-5119 18 13 1.003 7.65 1.29
SPT-CL J2342-5411 12 7 1.075 5.67 1.14
SPT-CL J2344-4243 33 25 0.595 15.44 1.91
SPT-CL J2351-5452 42 30 0.384 6.18 1.53
SPT-CL J2355-5055 36 33 0.320 6.59 1.61
SPT-CL J2359-5009 44 37 0.775 5.17 1.24
Table 1: Cluster spectroscopic dataset - Continued.

3 Cluster Data

In this section we present the cluster sample, spectroscopic data, and the method for constructing composite clusters.

3.1 Cluster sample

The cluster sample analysed in this study consists of galaxy clusters detected with the South Pole Telescope (SPT), a 10-meter telescope located within 1 km of the geographical South Pole, observing in three mm-wave bands centered at 95, 150 and 220 GHz (see Carlstrom et al., 2011). The SPT-SZ survey, imaging 2500 of the southern sky, has produced data that are used to select galaxy clusters via their thermal SZE signature in the 95 and 150 GHz maps. This SZE signature arises through the inverse Compton scattering of the Cosmic Microwave Background (CMB) photons and the hot intracluster medium (ICM; Sunyaev & Zel’dovich, 1972). All of the clusters studied in this work have a high detection significance () and appear in published catalogs (Williamson et al., 2011; Reichardt et al., 2013; Song et al., 2012; Bleem et al., 2015). We refer the reader to Schaffer et al. (2011) for details on the survey strategy and data processing.

A major advantage of selecting clusters with the SZE rather than other cluster observables lies in the fact that the surface brightness of the SZE signature is independent of the cluster redshift, which together with the expected temperature and density evolution of the cluster ICM at fixed virial mass and the changing solid angle of clusters with redshift lead an SZE signal to noise selected sample to be approximately mass-limited (Haiman et al., 2001; Holder et al., 2001).

Because we plan to carry out a dynamical analysis, the cluster sample includes only those SPT systems with spectroscopic follow-up. Full information on the cluster sample is provided in Table 1. From left to right the columns correspond to the SPT cluster designation, the total number of redshifts available, the number of member redshifts, the cluster redshift, the SZE based mass and the corresponding virial radius (described in Section 3.3.1).

3.2 Spectroscopic sample

We use spectroscopic follow-up including data taken 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 (LDSS3; Allington-Smith et al., 1994) on Magellan Clay.

We combine the datasets presented in Ruel et al. (2014) and in Bayliss et al. (2016) with the one described in Sifón et al. (2013), obtaining a sample of 4695 redshifts. In what follows, we split the spectroscopic galaxy sample according to the spectral features into two subsamples of those with emission lines (EL) and those without emission lines (nEL), adopting the determinations presented in the aforementioned analyses.

Due to the observational strategy of these surveys, which targeted red-sequence (RS) galaxies, the number of nEL galaxies (3834) greatly exceeds the numer of EL galaxies. Because EL and nEL galaxies are characterized by different spatial and kinematic properties (see, e.g., Mohr et al., 1996; Biviano et al., 1997; Dressler et al., 1999; Biviano et al., 2002, 2016; Bayliss et al., 2017; Hennig et al., 2017), in the analysis that follows we focus only on the passive nEL population, for which we have sufficient statistics to properly carry out the dynamical analysis. In Fig. 1 we show the normalized distribution of galaxies and clusters as a function of redshift.

Figure 1: Redshift distribution of member galaxies (red) and clusters (orange) in our sample, normalized to the total number of objects in each case. The two distributions are similar, but a clear trend to have fewer galaxies per cluster at high redshift is visible.
Bin Redshift
1 0.26-0.38 0.33 18 712 593 9.23
2 0.39-0.44 0.42 19 744 644 9.50
3 0.46-0.56 0.51 17 758 615 7.98
4 0.56-0.71 0.62 30 904 675 7.31
5 0.71-1.32 0.86 26 716 459 7.60
- 0.26-1.32 0.55 110 3834 2966 8.28
Table 2: Characteristics of the composite clusters in redshift: columns show the bin number, redshift range, mean redshift, number of included clusters , number of total galaxies , number of member galaxies and mean SPT based mass .

3.3 Construction of composite clusters

To enable a precise determination of the cluster masses and the velocity anisotropy profiles, we cannot rely on the few spectroscopic members in the individual clusters of our sample. For this reason, we either create composite clusters with much more dynamical information or fit to a common model across the cluster ensemble by combining the likelihoods associated with each individual cluster. The composite cluster approach has been adopted in previous analyses (Carlberg et al., 1997; van der Marel et al., 2000; Katgert et al., 2004; Biviano & Poggianti, 2009), and it is supported by cosmological simulations that predict cosmological halos can be characterized by a universal mass density profile with a concentration that depends mildly on the halo mass (Navarro et al., 1997). We return to the method using a combination of individual cluster likelihoods in Section 4.3.2.

We create composite clusters by combining cluster subsamples within different mass and redshift ranges in such a way that each subsample includes an adequate number of members. Considering that tests done using MAMPOSSt on cosmological simulations indicate that with 500 tracers the code is able to recover the mass and anisotropy parameters with a suitably small uncertainty (Mamon et al., 2013), we create similarly sized subsamples. With this approach we can construct 5 composite clusters selected either as redshift or mass subsamples.

Table 2 lists the characteristics of the composite clusters created within redshift bins. From left to right the columns correspond to the bin number, the redshift range of the clusters combined to create the composite system in that bin, the mean redshift of these clusters, the number of clusters, the total number of galaxy redshifts, the number of cluster members with redshifts, and the average SPT based mass of the individual clusters in the sample. In the following subsections we describe in detail how these composite clusters are created and how interlopers are removed from the spectroscopic samples.

3.3.1 Rescaling observables with SZE based mass estimates

To create composite clusters we choose the Brightest Cluster Galaxy (BCG) as the cluster center. We determine these positions using results from previous and ongoing work (Song et al., 2012; Sifón et al., 2013, Stalder et al., in prep), allowing us to calculate a projected separation for each galaxy from the cluster center. We extract the rest frame LOS velocity from the galaxy redshift and equivalent velocity as , where is the cluster redshift.

Because our clusters span a wide range of mass and redshift, we must scale the galaxy projected cluster distances and LOS velocities to the values of a fiducial cluster mass using an estimate of the individual cluster virial radius and velocity. For convenience, in the composite cluster we use the mean mass and the mean redshift of the ensemble of clusters in that bin to calculate the associated and that we adopt as fiducial values for the bin. Thus, the rescaled observables for each galaxy within a cluster in a particular bin are and , where and are the virial radius and circular velocity, respectively, of cluster . The circular velocity is defined using the virial condition .

To perform the normalization, we need precise estimates of the cluster mass. Given that the SPT sample is SZE selected, we adopt the SZE observable , which is the detection signal to noise and is correlated to the underlying cluster virial mass (Andersson et al., 2011; Benson et al., 2013), as the source of our cluster mass estimates. SPT masses derived from have a statistical uncertainty that depends on the intrinsic scatter in the –mass relation and on the observational uncertainty in the signal to noise of the detection; together, these lead characteristically to 20 percent statistical uncertainty in the cluster virial mass. In addition, there are systematic uncertainties remaining from the calibration of the mass–observable relation that are currently at the 15 percent level. As discussed further below, the systematic mass scale uncertainties at this level have no impact on our analysis, but the cluster to cluster statistical mass uncertainty of 20 percent introduces a corresponding uncertainty in and of 7 percent.

The mass–observable relation for the SPT sample can be calibrated in different ways, and in the current analysis we use two different approaches. The first approach uses direct cluster mass measurements from weak lensing, velocity dispersions, X-ray measurements to calibrate the SZE-mass observable relation. The second approach fits the SPT cluster distribution in signal to noise and redshift and the mass-observable relation to a flat CDM model with external cosmological priors from other datasets. These constraints from, e.g., CMB anisotropy, baryon acoustic oscillations, etc., effectively constrain the mass-observable relation so that the cluster mass function implied by SPT cluster distribution in and is consistent with that expected given the external cosmological priors. As an example (see Bocquet et al., 2015), the inclusion of external cosmological constraints from Planck CMB in the SPT mass–observable calibration leads to a shift of cluster masses to 25 percent higher values in comparison to the case where the mass calibration is undertaken with direct cluster mass measurements from velocity dispersions.

Given the need to have an initial mass estimate to enable the stacking of the clusters for a dynamical analysis, we carry out the analysis with initial SPT based masses calibrated in two different ways. The first uses the SPT cluster counts together with direct cluster mass information from velocity dispersions. Here we refer to these masses as . The second uses the SPT cluster counts together with external cosmological priors from Planck CMB anisotropy and distance measurements (BAO, Supernovae), specifically we use mass-cabliration inferred from the SPTCL +Planck +WP+BAO+SNIa data set in Bocquet et al. (2015). We refer to these masses as . In both cases the SPT mass–observable relation is calibrated using (Bocquet et al., 2015), and so we transform from by assuming an NFW model with a concentration sampled from cosmological -body simulations (Duffy et al., 2008). The and are then easily obtained, depending only on the cluster redshift and the adopted cosmology.

Because we are carrying out a Jeans analysis, which is based on the assumption of dynamical equilibrium, we restrict our analysis to the cluster virial region (). Moreover, we exclude the very central cluster region ( 50 kpc). In fact, the composite clusters are centered on the BCG, which we exclude from the dynamical analysis. We note also that in the composite clusters the characteristic asphericity of individual clusters is averaged down, leading to a combined system that is approximately spherical in agreement with the dynamical model we are employing.

Figure 2: The projected phase-space diagram in the rescaled coordinates for the composite cluster built within the lowest redshift bin (). Green lines represent the radially-dependent 2.7 cut used to reject interlopers (indicated by empty dots).

3.3.2 Interloper rejection

One benefit of constructing composite clusters is that we can more easily identify and reject some interloper galaxies, i.e., galaxies that are projected inside the cluster virial region, but do not actually lie inside it. We do so by using the “Clean” method (Mamon et al., 2013), which is based on the identification of the cluster members on the basis of their projected phase-space location (). The LOS velocity dispersion of the composite cluster is used to estimate the cluster mass using a scaling relation calibrated using numerical simulations (e.g., Saro et al., 2013), and an NFW mass profile with concentration sampled from the theoretical mass-concentration relation of Macciò et al. (2008). Thereafter, assuming the MŁ velocity anisotropy profile model, and given the of the cluster, an LOS velocity dispersion profile is calculated and used to iteratively reject galaxies with at any clustercentric distance (see Mamon et al., 2010, 2013). As an example, we present in Fig. 2 the location of galaxies in projected phase-space with the identification of cluster member galaxies for the low redshift composite cluster, constructed using those clusters with redshifts lying in the range .

We note that even after cleaning the dynamical sample, it is still contaminated to some degree by interlopers. One reason for this is that galaxies near the cluster turn-around radius (i.e., well outside the virial radius) will have small LOS velocities. Therefore, if those galaxies are projected onto the cluster virial region they simply cannot be separated from the galaxies that actually lie within the cluster virial radius. Analysis of cosmological -body simulations shows that when passive galaxies are selected this contamination is characteristically  percent (Saro et al., 2013) for SPT mass scale clusters and represents a systematic, whose effects on the dynamical analysis requires further exploration.

3.3.3 Galaxy number density profiles

We cannot simply adopt the spectroscopic sample to measure the number density profiles of the galaxy population, because the structure of the spectrographs and our observing strategy of visiting each cluster typically with only two masks will generally result in a radially dependent incompleteness. This incompleteness, if not accounted for, would affect the determination of the cluster projected number density profile , related to the 3D number density profile through the Abel inversion (Binney & Tremaine, 1987). As only the logarithmic derivative of enters the Jeans equation (see equation (2)), the absolute normalization of the galaxy number density profile has no impact on our analysis. However, a radially dependent incompleteness in the velocity sample would impact our analysis.

Thus, for our analysis we rely on a study of the galaxy populations in 74 SZE selected clusters from the SPT survey that were imaged as part of the Dark Energy Survey (DES) Science Verification phase (H17). The number density profile of the red sequence population of galaxies is found to be well fit by an NFW model out to radii of , with a concentration of cluster galaxies . The non-red sequence population is much less centrally concentrated, but in our analysis we are focused only on the nEL galaxy populations. No statistically significant redshift or mass trends in the galaxy concentration were identified. Therefore, we adopt the number density profile described by an NFW profile with the above-mentioned value of and a scale radius . Implicit in this approach is the assumption that the dynamical properties of our spectroscopic sample are consistent with the dynamical properties of the red sequence galaxy population used to measure the radial profiles.

In our analysis, we test this assumption of a fixed by including in the Likelihood an additional parameter , given by the ratio , where is given by H17


where, for red sequence galaxies, . We choose mass and redshift pivot points and , respectively the median mass and redshift of our sample. With this additional parameter, we adopt the Markov Chain Monte Carlo (MCMC) method for sampling from the probability distribution, utilizing the emcee code (Foreman-Mackey et al., 2013). Our analysis provides values of the scale radius and virial radius consistent within of the measured values reported in H17.

Aniso- Bayesian
model [Mpc] [Mpc] tropy Weight
; Mpc
C 0.39
OM 1.00
O 0.15
T 0.28
; Mpc
C 0.09
OM 0.25
O 0.12
T 0.18
; Mpc
C 0.02
OM 0.13
O 1.00
T 0.21
; Mpc
C 0.27
OM 1.00
O 0.09
T 0.22
; Mpc
C 0.12
OM 0.32
O 0.33
T 0.36
Table 3: Parameters constraints from the MAMPOSSt analysis of the composite clusters defined in Table 2. Columns represent the velocity anisotropy model , the virial radius , the scale radius , the anisotropy parameter and the Bayes factor from equation 11.
Bin Redshift
range [Mpc] [Mpc]
1 0.26-0.38
2 0.39-0.44
3 0.46-0.56
4 0.56-0.71
5 0.71-1.32
- 0.26-1.32
Table 4: Parameters constraints for the composite cluster mass profiles. Columns represent the redshift range, virial radius , scale radius , dynamical mass  and concentration. These results are obtained using Bayesian model averaging of the different anisotropy models.

4 Results

In this section we present the results of our dynamical analysis. As a first step, we run MAMPOSSt on all mass and anisotropy models described in Section  2.1, with 3 free parameters: the virial radius , the scale radius of the mass distribution (, in the case of NFW) and the anisotropy parameter , or . The results of this analysis indicate that we cannot reject any of the mass models we have considered. All of them are in fact consistent within confidence. Taking guidance from both theoretical expectation and observational results, we select the NFW model as a good description of the mass profile. As mentioned in Section 1, in fact, cosmological simulations produce DM halos with mass profiles well described by an NFW profile. This result is in good agreement with a variety of observational analyses using both dynamics and weak lensing (Carlberg et al., 1997; van der Marel et al., 2000; Biviano & Girardi, 2003; Katgert et al., 2004; Umetsu et al., 2011), even though some results have preferred different models (Merritt et al., 2006; Navarro et al., 2010; Dutton & Macciò, 2014; van der Burg et al., 2015; Sereno & Ettori, 2017).

On the other hand, the literature does not provide us with strong predictions for the radial form of the velocity anisotropy or profile. Therefore, we employ all the profiles described in Section 2.1. To discriminate among these five models, we calculate the ratio of the marginalized likelihoods of the models, i.e. the so-called Bayes factor (see Hoeting et al., 1999, and references therein). Considering models , …, , the posterior probability for the model given data is:




being the marginalized likelihood of the model . The vector of parameters for the model is represented by (in our case, ), while is the prior density of under model , is the likelihood, and is the prior probability that is the “true” model, which – in the absence of strong theoretical guidance – we assume to be equal for all models.

We measure the Bayes factor defined as the ratio of the posterior probabilities, namely


where indicates the model with highest marginalized likelihood. In Table 3 we list the results of the MAMPOSSt analysis, with the anisotropy parameter being for the C model, for the T and O models and for the MŁ and OM models. The errors on each of the parameters listed in the table are obtained by a marginalization procedure, i.e. by integrating the probabilities provided by MAMPOSSt over the remaining two free parameters. In addition to the anisotropy parameters, Table 3 contains the dynamical constraints on the composite cluster virial radius  and the Bayesian weight described above.

According to the Jeffreys scale (Jeffreys, 1998), is considered decisively rejectable if . We cannot strongly reject any of the models, and therefore, we perform a Bayesian Model Averaging, weighting every model by its Bayes factor and combining statistics from the different models. The results of this analysis are shown in Table 4 where we present the virial radius , the NFW scale radius , the virial mass , and the concentration . The 1 parameter uncertainties are computed through a marginalization procedure, as before.

As mentioned in Section 3.3.1, when creating the composite clusters we perform a rescaling of the observables using two different initial mass estimates,  and  . The dynamical mass constraints do not differ significantly when applying a different normalization, demonstrating the stability of our analysis to underlying systematic mass uncertainties in the initial mass estimates used to combine the clusters. We return to this point in Section 4.3.1 below, where we show the dynamical masses obtained in the two cases in Table 5. Moreover, the masses derived through the full dynamical analysis are in good agreement with the mean SPT based masses  listed in Table 2. Regarding the precision of the constraints, it is clear that a composite cluster with 600 cluster members allows one to determine the dynamical mass with what is effectively a % uncertainty (8% for the full dataset, using 33000 tracers). In contrast, the NFW scale radius and the corresponding concentration are only weakly constrained.

4.1 Velocity dispersion anisotropy profiles

Figure 3: Velocity anisotropy profile for each redshift bin. The red dashed line represents the profile obtained implementing a Bayesian model averaging, with the pink shaded region indicating the 1 confidence region around this solution. There is no clear evidence of a redshift trend. The blue line in the lower-right panel (full sample) shows the result obtained when adopting the best fit NFW model and the Jeans equation inversion to solve for the anisotropy profile. This result is in good agreement with the model averaging result. Our analysis shows that passive galaxies preferentially move on nearly isotropic orbits close to the cluster center, and on increasingly radial orbits as one moves to the virial radius.

Fig. 3 contains the measured anisotropy profiles and their 1 confidence regions for the five composite clusters in different redshift ranges together with the results from the full sample (lower, right-most panel). The dashed red line in each panel corresponds to the Bayesian model averaging result from the analysis of the composite cluster with the five different models of the velocity dispersion anisotropy profile. As noted in the previous section, the mass profile model in all cases is an NFW with concentration and mass free to vary. The shaded region encompasses the 1 allowed region for the Bayesian model averaged anisotropy profile.

Our analysis indicates that the orbits of passive, red galaxies are nearly isotropic close to the cluster center, and become increasingly radial going towards larger radii, reaching a radial anisotropy at . There is no clear evidence for a redshift trend in the anisotropy profile of the passive galaxy population out to , and we find no evidence for trends with mass, either. For this reason we analyse also the full sample, providing our best available constraints. The orbital anisotropy varies from values consistent with zero in the cluster core to a value at the virial radius. For reference, anisotropy values of 0.4 correspond to tangential components of the velocity dispersion ellipsoid having amplitudes that are only 60 percent as large as the radial component. For the full sample we show (blue dashed line) also the anisotropy profile recovered using the best fit NFW parameters and using the Jeans equation to solve for the velocity dispersion anisotropy profile (Binney & Mamon, 1982; Solanes & Salvador-Sole, 1990; Biviano et al., 2013). This result is in good agreement with the solution recovered using the Bayesian model averaging over the five adopted anisotropy profiles.

The behavior of the anisotropy profile is consistent with the theoretical model discussed in Lapi & Cavaliere (2009), according to which the growth of structure proceeds in two phases: an early, fast accretion phase during which the cluster undergoes major merging events, and a second slower accretion phase involving minor mergers and smooth accretion (see, e.g., White, 1986; Zhao et al., 2003; Diemand et al., 2007). Lynden-Bell (1967) discusses how a dynamical system rapidly relaxes from a chaotic initial state to a quasi-equilibrium. The first stage of fast accretion provokes rapid changes in the cluster gravitational potential, inducing the collisionless components of the cluster to undergo violent relaxation, resulting in orbits that are more isotropic. Galaxies accreted by the cluster during the fast accretion phase would then be expected to exhibit approximately isotropic orbits, while galaxies accreted during the second phase would maintain their preferentially radial orbitals over longer timescales. Given that as the cluster accretes its mass and virial radius also grow, a two phase scenario like this would tend to lead to anisotropy profiles that are isotropic in the core and become more radial at larger radius.

However, one might also expect to see a time or redshift variation of the anisotropy profile, with typical galaxy orbits in high redshift clusters showing less of a tendency for radial orbits near the virial radius. The fact that our analysis shows no strong redshift trend in the anisotropy profiles is an indication that, in massive galaxy clusters, the passive galaxy population orbits are not changing significantly with cosmic time since . This suggests that the merging and relaxation processes responsible for the anisotropy profiles are underway at all cosmic epochs probed here. Other indicators of cluster merging have shown similar results; namely, the fraction of systems with disturbed X-ray morphologies (typically measured using centroid variations or ellipticities; Mohr et al., 1993) does not change significantly with redshift in samples of homogeneously SZE selected cluster samples (Nurgaliev et al., 2017; McDonald et al., 2017). This may suggest that the fast accretion phase of such a two phase model could be very short and happening primarily at redshifts above those probed by our sample. Clearly, a detailed examination of cosmological -body simulations with sufficient volume to contain rare, massive clusters and sufficient mass resolution to ensure the survival of galaxy scale subhalos after accretion into the cluster is warranted. Moreover, a dynamical analysis like the one we have carried out that focuses on systems at redshift would enable a more sensitive probe for time variation in the growth of structure.

Previous studies of the velocity dispersion anisotropy conducted on passive cluster members at intermediate to high redshifts present hints of an anisotropy profile that is nearly isotropic close to the cluster center, and increasingly radial at larger radii. Biviano & Poggianti (2009), stacking 19 clusters between redshift with a mean mass of , suggest radially anisotropic orbits. Biviano et al. (2016) reach the same conclusion when analysing a stacked sample of 10 clusters at . Studying a single cluster, Annunziatella et al. (2016) show that the same trend is found for galaxies characterized by a stellar mass , while lower-mass galaxies move on more tangential orbits, avoiding small pericenters, presumably because those that cross the cluster center are more likelly to have been tidally destroyed. These results are consistent with ours. Our result, obtained through the analysis of a large sample of passive galaxies within a homogeneously selected sample of massive clusters over a wide redshift range and with low scatter mass estimates, allows us to cleanly probe for redshift and mass trends in the velocity dispersion anisotropy profile.

Some published analyses carried out at lower redshifts () than our sample show similar results. Wojtak & Łokas (2010) analysed a sample of 41 nearby relaxed clusters, finding that galaxy orbits are isotropic at the cluster centers and more radial at the cluster virial radius. A similar result is obtained by Lemze et al. (2009) and Aguerri et al. (2017). However, other analyses show that the orbits of passive galaxies at these redshifts are more isotropic at all radii (Biviano & Katgert, 2004; Katgert et al., 2004; Biviano & Poggianti, 2009; Munari et al., 2014), hinting at a possible change in galaxy orbits over time due to processes such as violent relaxation, dynamical friction, and radial orbital instability (Bellovary et al., 2008). At present, results from numerical simulations predict a range of behavior (Wetzel, 2011; Iannuzzi & Dolag, 2012; Munari et al., 2013), so further study is definitely needed. Extending our own observational analysis towards lower redshifts could also help clarify this picture.

4.2 Pseudo phase-space density profile

Figure 4: The ratios of the passive galaxy pseudo-phase space density profiles to that of dark matter particles in simulations (Dehnen & McLaughlin, 2005) are shown in green for the total PPSD and red for the radial PPSD for five redshift bins and the full sample. Shading corresponds to the 1 confidence region. The profiles are normalized to the full sample in each case. The solid black line marks the expected amplitude in each redshift bin under the assumption of cluster self-similarity. Both passive galaxy PPSD profiles are in reasonably good agreement with -body simulations, suggesting that the kinematics and radial distributions of the passive galaxies do not differ significantly from those of the underlying dark matter.

The determination of the anisotropy profile allows us to investigate the behavior of the Pseudo Phase-Space Density profile introduced in Section 1. According to numerical simulations of virialized halos (Taylor & Navarro, 2001; Dehnen & McLaughlin, 2005; Lapi & Cavaliere, 2009), there is a scaling between the density and the velocity dispersion, best appreciated by considering the quantity . The profile of this quantity is found to follow a universal power-law of fixed slope, . Remarkably, this is the same power-law predicted by the similarity solution of Bertschinger (1985) for secondary infall and accretion onto an initially overdense perturbation in an Einstein-de Sitter universe. In that work the authors found that the relaxation process is self-similar, meaning that each new shell falling in is virialized and adds a constant contribution to the resulting power-law density profile.

Because the velocity dispersion profile is constrained by the galaxies, we derive the PPSD profile using the number density profile of galaxies instead of the mass density profile. For each stacked cluster, we fix the Halo Occupation Number of red galaxies to that found by H17 to find the central density of the NFW profile, such that


We present the results obtained investigating both the total PPSD profile and the radial PPSD profile , where is obtained as described above and is recovered using the following equation (van der Marel, 1994; Mamon & Łokas, 2005; Mamon et al., 2013)


evaluated over an adequate grid of , and is given by


Figure 4 shows the ratio between the derived galaxy PPSD profiles and the fixed-slope best-fit relations and , where the slopes are those obtained by Dehnen & McLaughlin (2005) by studying DM particles in numerically simulated halos, while the normalizations are fitted to the data of the full sample. A similar value for the slope of is found by Faltenbacher et al. (2007), a work based on numerical simulations and focused on the gas and DM entropy profiles in galaxy clusters. The similarity between radial profiles of the passive galaxy PPSD and simulated PPSD profiles is an indication that the passive galaxy radial distribution and kinematics are similar to those of the dark matter particles in those simulations.

We emphasize here that it is not possible for us to predict the PPSD profile for the underlying dark matter in our sample, because in general the blue or EL galaxy population exhibits different kinematical properties than those of the passive, red or nEL galaxies. One might expect the full dark matter kinematics to exhibit a mix of the properties of the different galaxy populations. We note that because the measured concentration of the passive population (H17) is similar to the derived concentration of the mass profiles from our analysis, that if we replace the galaxy density profile with our derived mass density profile, the PPSD profiles have a very similar radial behavior.

In Fig. 4 we also show (solid black lines) the expected amplitude if there is self-similar behaviour of passive galaxy profiles and kinematics. That is, we have , where is the Hubble parameter. The passive galaxy PPSD amplitudes are in reasonably good agreement with the amplitudes expected under self-similarity, providing some additional evidence that this population is approximately self-similar.

In the previous sub-section, we mentioned that the anisotropy profiles could be the result of violent relaxation. This process, driven by gravity alone, would also tend to create a scale-invariant phase-space density. Relaxation into dynamical equilibrium would then lead the slope of the PPSD profile to approach a critical value, resulting in the particular form of the density profile for DM particles within simulated halos. An anisotropy profile isotropic in the inner regions and increasingly radial at larger radii gives the pseudo phase space density profile slope observed in numerical simulations. The agreement then suggests that the passive galaxies we analyse here have reached a similar level of dynamical equilibrium as the dark matter particles in the simulations, and that this is true for all redshifts up to .

It is difficult to compare our results directly to previous studies, because most of those studies have tended to focus on and using the inferred total matter density from their dynamical analyses rather than the galaxy density profile. Such an approach produces a mixed PPSD profile that contains both total matter and galaxy properties, because the dispersion profiles used are necessarily those of the galaxies rather than the dark matter. This complicates the interpretation, because a mismatch between the observed and simulated PPSD profiles could be because the total matter density profile doesn’t match the simulated dark matter profile or it could mean that the velocity dispersion profile of the dark matter and a particular galaxy population do not match. In general, given that different galaxy populations tend to exhibit different kinematic properties (i.e. radial distributions and velocity dispersion and anisotropy profiles), one would not expect the mixed PPSD profiles defined in this way to agree for the different populations.

Indeed, Biviano et al. (2013) measured using the anisotropy profile of the galaxies and the NFW mass density profile inferred from their analysis, separately for star forming (SF) and passive galaxies in a single cluster. They found that their observed PPSD profile agreed with simulations only for the passive galaxy population. They argue that passive members have undergone violent relaxation and have reached dynamical equilibrium, while SF members have not yet reached equilibrium. But another possible interpretation would be that the Jeans analyses of both populations are equally accurate, but that the passive population exhibits a velocity dispersion and anisotropy profile more similar to that of the simulated dark matter. A more recent study came to similar conclusions (Munari et al., 2015).

On the other hand, a recent study of the nearby cluster Abell 85 (Aguerri et al., 2017) shows that follows the theoretical power-law form independent of the galaxy colour or luminosity, concluding that all the different families of galaxies under study reached a virialized state. Given the discussion above, this agreement in the mixed PPSD profile derived from different galaxy populations also indicates that the different populations must have similar kinematics (i.e. velocity dispersion and anisotropy profiles). In their study, they emphasize that the anisotropy profiles of the blue and red galaxies are different, which would make the agreement in surprising. However, Figure 3 in their paper suggests that the anisotropy profiles have similar character (isotropic in the center, more radial at larger radius) and present evidence for inconsistency that is weak ().

Table 5: Comparisons of dynamical masses from composite clusters calculated using different initial masses. From left to right the columns contain the redshift range of the cluster sample, the derived dynamical mass given an initial SPT plus velocity dispersion mass, the derived dynamical mass given an SPT + Planck initial mass, and the ratios of these dynamical to initial masses in each case. Finally, we report the constraints on and as described in Section 4.3.2.

4.3 Dynamical mass constraints

We use the dynamical information to study the masses of these clusters in two different ways. In Section 4.3.1, we analyse the composite clusters and the consistency of the dynamical masses when using different initial masses to scale the galaxy observables. In Section 4.3.2, we examine the differences between the dynamical masses and the SPT+Planck masses using a different approach where the scaling values for each cluster and are altered self-consistently in each iteration of the Markov chain to reflect the SPT+Planck masses scaled by a free parameter , defined in equation (15).

4.3.1 Mass constraints on the composite clusters

Figure 5: Marginalized distribution of the dynamical masses. Each panel corresponds to a different redshift range, and the final panel shows the results of the analysis of the full sample. In green we highlight the 1 confidence region. The red line represents the mean SPT+ mass for the clusters in the bin, weighted by the number of member galaxies in the individual clusters. There is a good agreement between the dynamical masses and the originally inferred SPT masses in all cases (see Table 5).

Constructing the composite clusters (described fully in Section 3.3) requires a scaling of our galaxy observables, and , by estimates of the virial radius and velocity and respectively, and therefore requires an initial mass estimate. As described above, this is a potential problem, because there is currently a 25 percent shift between the SPT cluster masses derived using cluster counts and velocity dispersion information, and the masses derived using the cluster counts and external cosmological priors from Planck (Bocquet et al., 2015; de Haan et al., 2016). Thus, we examine the dynamical mass constraints in each case: (1) those derived using initial masses derived from the cluster counts and velocity dispersion measurements and (2) those derived using initial masses from the cluster counts and external cosmological constraints from Planck .

In Fig. 5 we display the marginalized distribution of the dynamical masses obtained with our analysis where the masses were adopted for the initial scaling. The green regions mark the 1 confidence regions, and the red lines represent the mean initial masses derived from the cluster counts and velocity dispersion measurements , where the masses from each cluster are weighted by the number of galaxy velocities available for that cluster. There is good agreement between the dynamical masses and the initial masses in all redshift bins and also for the full sample (lower, right-most panel). The second column of Table 5 contains the measurement results and uncertainties for each subset. Characteristic dynamical mass uncertainties are at the 15 percent level for individual subsamples and at the 8 percent level for the full sample. These are impressive mass constraints given that they are marginalized over the velocity dispersion anisotropy profile uncertainties.

To test the stability of the recovered dynamical masses to the initial input masses used for scaling, we perform the same analysis using the SPT+Planck cluster masses. The third column of Table 5 shows these results. These dynamical masses with the different initial masses are quite close to the values derived with the other set of initial masses. This shows that there is no strong dependence of the dynamical mass on the initial mass. This is because any change in the masses used for rescaling the cluster observables during stacking will impact, on average, the individual cluster masses and the final mean mass in the bin in a similar manner. The overall scale of the dynamical data in projected radius and LOS velocity remains approximately invariant.

As columns four and five of Table 5 make clear, the dynamical masses, while being in good agreement with the cluster counts plus velocity dispersion masses , exhibit some tension with the cluster counts plus external cosmological constraint masses . While the three lowest redshift bins show no significant tension, the upper two redshift bins show masses that are only 73 percent and 55 percent as large as the SPT+Planck masses (offsets that are statistically significant at the 2.5 and 6.5 levels, respectively). The full sample has a dynamical mass that is only 80 percent of the SPT+Planck masses, a difference that is significant at the 3.8 level (statistical only). The direction and scale of this mass shift is similar to that highlighted already by Bocquet et al. (2015). However, with our analysis we are able to show that this tension seems to grow with redshift.

Figure 6: Posterior distribution of (solid lines) arising from a dynamical analysis of each cluster subsample. The green region shows the 1 region, while the vertical dotted black line marks the value . These distributions show that in the two high redshift bins and for the full sample there is tension between the dynamical masses and the SPT+Planck calibrated cluster masses. For the full sample the tension is 2.2 significant when including estimates of the systematic uncertainties. In the highest redshift bin, the tension is 2.6. The dashed lines show the estimated posterior for , where represents the de Haan et al. (2016) masses calibrated using the SPT mass function and measurements for many of the systems. In contrast to the SPT+Planck masses, these masses are in good agreement ( offset) with the dynamical masses.

4.3.2 Comparison with SZE based masses

To examine this tension more carefully, we use the dynamical analysis to examine the masses of these clusters and compare them to the masses derived separately from the SPT cluster counts in combination with external cosmological constraints from the Planck CMB anisotropy. We combine the likelihoods for each cluster and explore constraints on an overall mass scaling parameter , that is defined as


We do this by running MAMPOSSt for each individual cluster in our sample. We calculate the posterior distribution of by using a multimodal nested sampling algorithm, namely MultiNest (Feroz & Hobson, 2008; Feroz et al., 2009, 2013), which provides us with the evidence for each model, and allows us to perform a Bayesian model averaging over different subsets of clusters.

Figure 6 contains a plot of the posterior distributions of from our analysis within each redshift bin and for the full sample. Results are largely consistent with the results from the composite clusters. Column six of Table 5 contains the best fit values and associated uncertainties. The preferred value for the full sample is . The constraint is followed by a statistical uncertainty and then a systematic uncertainty. The current estimates of the systematic error on MAMPOSSt derived dynamical masses are , where this number comes from an analysis of clusters extracted from numerical simulations (see Mamon et al., 2013). In their study, they find that for the clusters defined to lie within a sphere of that the estimate of the virial radius is biased at % (see Table 2; Mamon et al., 2013). Thus, as a prior on the virial mass bias, we adopt a Gaussian with % centered at no bias. If one combines the statistical and systematic uncertainty in quadrature, the implication would be a tension at the level. As mentioned already, this tendency for the dynamical masses to be lower than those masses derived from the cluster counts in combination with external cosmological constraints is consistent with the tendencies seen previously using simply dispersions and the cluster mass function in Bocquet et al. (2015). More recent weak lensing analyses also support the lower mass scale of SPT clusters (Dietrich et al., 2017; Stern et al., in prep.).

To emphasize, Figure 6 also shows (dotted line) the distribution of , where is the geometrical mean of the mass ratio in each redshift bin, and are masses calibrated from a cosmological analysis carried out in de Haan et al. (2016) using the X-ray mass proxy and the abundance of clusters as a function of redshift, and the SZE mass proxy , without the inclusion of the external Planck cosmological constraints. These results indicate that the dynamical masses are in good agreement with the masses at all redshifts and for the full sample. The distribution for the full sample prefers a value of , consistent with no tension.

To summarize, our dynamical mass measurements, which are derived using only dynamical information and no information from the mass function or cluster counts, are in good agreement with masses derived using information from the the cluster counts together with additional information from either velocity dispersions or from X-ray measurements that have been externally calibrated. However, our mass measurements exhibit moderate tension with those masses obtained similarly but when also adopting external cosmological priors from Planck CMB anisotropy. Progress in testing these two mass scales would require better control of the systematic uncertainties in the dynamical masses (Mamon et al., 2013). The agreement between the dynamical and the Planck based masses is best at low redshift, with the dynamical masses preferring ever smaller with increasing redshift. In the highest redshift bin () we measure , tension at the 2.6 level.

Table 6: Sensitivity of dynamical mass measurements to the dynamical state of clusters. We compare the masses for those clusters exhibiting large X-ray surface brightness asymmetries (unrelaxed) and those exhibiting small asymmetries (relaxed). Columns list the subsample, the mean redshift, concentration, dynamical mass given initial scaling using SPT dispersion based masses, and ratio of the dynamical mass to the SZE dispersion based mean mass for the subsample.

4.4 Impact of disturbed clusters

One assumption in applying the Jeans equation to analyze our sample is that the galaxies we are analyzing are in approximate dynamical equilibrium. Thus, it would seem important to remove the obviously disturbed clusters— those undergoing or having undergone recent, major mergers. As discussed in Mohr et al. (1993), the asymmetry and isophotal ellipticity of the X-ray surface brightness distribution provide information about the merger state of galaxy clusters, which is generally superior to the constraints possible from the galaxy distribution (Geller & Beers, 1982) due to higher signal to noise. However, neither of these measures are sensitive to mergers along the line of sight, where the galaxy velocity distribution typically provides more information (Dressler & Shectman, 1988).

Merger signatures from X-ray cluster surface brightness distributions were extracted uniformly from a large X-ray flux limited sample of nearby clusters, indicating that over half of them exhibit statistically significant centroid variations and that the bulk of them are elliptical (Mohr et al., 1995). That study provided clear evidence that even massive clusters at low redshift are still undergoing continued accretion of subclusters. This observation is in agreement with expectations from structure formation within our standard CDM model. We can use a similar approach to identify major mergers (not along the line of sight) in our cluster sample. Nurgaliev et al. (2017) quantified the X-ray morphology of a subsample of 90 SZE selected galaxy clusters using a measure of the photon asymmetry (Nurgaliev et al., 2013) closely related to the centroid variation (Mohr et al., 1993, 1995) and power ratios (Buote & Tsai, 1995, 1996) introduced before.

To test the dependence of our results on the dynamical state of the clusters, we adopt these values as a measure of departures from equilibrium and separately analyse relaxed and un-relaxed clusters. We take here relaxed clusters to be those with . Out of our 110 cluster sample, 68 (3279 spectra) have measured and 39 of these (1929 galaxies) are classified as relaxed. In Table 6 we show the results for the mass and concentration having performed the analysis on the stacked relaxed/non-relaxed populations. We find that the  and the dynamical masses are in agreement at level for the relaxed sample and at the level for the un-relaxed sample (statistical only). The recovered anisotropy profiles are consistent with both isotropy and mild radial anisotropy, allowing no clear differentiation of the two samples. Thus, we conclude that there is no driver for rejecting some of the clusters in our sample due to large asymmetries in their ICM distributions.

5 Conclusions

We present a dynamical analysis of 110 SZE selected galaxy clusters from the SPT-SZ survey with redshifts between 0.2 to 1.3, that have an associated spectroscopic sample of more than 3000 passive galaxies. We examine subsets of this cluster sample in redshift and mass, each comprising 600 cluster members. These subsets are either combined to form composite clusters or are analysed individually using a Jeans equation based code called MAMPOSSt (Mamon et al., 2013) that allows one to adopt different parametric models for the mass profile, galaxy profile and velocity dispersion anisotropy profile. In our analysis we adopt an NFW mass profile, and use the measured concentration of the red sequence galaxy population from a complete subsample of the SPT SZE selected cluster sample (H17), and employ five different velocity dispersion anisotropy profiles (see Section 2.2). We perform Bayesian model averaging to combine results from the different dispersion anisotropy models, because none of the five models are excluded by the data.

The velocity dispersion anisotropy profiles show the same radial features at all redshifts: orbits are isotropic near the center and increasingly radial at larger radii. We also find no variations with cluster mass. The radial variation is broadly consistent with that seen in a recent analysis of near-infrared selected clusters at (Biviano et al., 2016) and also studies at low redshift (; Lemze et al., 2009; Wojtak & Łokas, 2010). These trends of anisotropy with radius resemble those of DM particles in halos extracted from cosmological numerical simulations (Mamon & Łokas, 2005; Mamon et al., 2010, 2013, and references therein). The absence of a redshift trend is inconsistent with the results presented in Biviano & Poggianti (2009), where they report that passive galaxy orbits are becoming more isotropic over time. The absence of a redshift trend in the velocity anisotropy profiles suggests that the process of infall and relaxation for the passive galaxy population is occurring similarly at all redshifts since at least .

We measure the PPSD profiles and , using quantities derived from cluster galaxies. We find good agreement with theoretical predictions from -body simulations of DM particles (Dehnen & McLaughlin, 2005). We examine whether the amplitude of the profile scales as expected with redshift and mass under the assumption of self-similarity, finding that they do. To the extent that the PPSD profile provides constraints on the equilibrium nature of the galaxy dynamics, the good agreement with simulations suggests that galaxies behave approximately as collisionless particles and are as relaxed as the DM particles in halos forming within cosmological structure formation simulations. Moreover, the lack of evidence for redshift trends in the power law index of the PPSD profiles suggests again that the passive galaxy population in clusters is dynamically similar at all redshifts and mass ranges probed in our study.

We carry out a consistency check between our dynamical masses , which are marginalized over uncertainties in the velocity dispersion anisotropy profiles, with masses calibrated using the SPT cluster counts with and without strong external cosmological priors. We find that our masses are smaller than those derived with strong external cosmological priors  by , corresponding to a 2.2 tension when systematic uncertainties in our dynamical masses are included. Moreover, our analysis shows that the agreement is best at low redshift and lower values of are preferred at higher redshift. In the highest redshift bin the best fit mass ratio is , which corresponds to tension at the 2.6 level. In addition, we find good agreement between our dynamical masses and those masses extracted from the SPT cluster counts and either a large number of velocity dispersions (Bocquet et al., 2015) or a large number of externally calibrated X-ray mass constraints (de Haan et al., 2016) when the cosmological parameters are allowed to vary. Our mass constraints are also consistent with those from related studies of SPT selected clusters, using both the weak lensing magnification (Chiu et al., 2016a) and the tangential shear (Schrabback et al., 2016; Dietrich et al., 2017; Stern et al., in prep.).

Using Chandra X-ray data, we examine the impact of the dynamical state of the clusters on our dynamical analysis by separately analysing relaxed and un-relaxed clusters. We find dynamical masses to be in good agreement with our combined sample for both the relaxed and unrelaxed clusters. To within the uncertainties there are no differences in orbital anisotropy in the two subsamples.

As a next step, our analysis can be extended to cluster samples that include many low mass systems. One such sample that is being analysed presently has been defined in the project known as SPIDERS (SPectroscopic IDentification of eROSITA Sources, Clerc et al., 2016), an optical spectroscopic survey of X-ray-selected galaxy clusters discovered in ROSAT and XMM-Newton imaging. Another sample is being built up through spectroscopic observations of optically selected clusters within the Dark Energy Survey. Longer term, we expect deep spectroscopic followup of SZE and X-ray selected clusters to provide ever larger galaxy samples that include both emission line and passive galaxies. These samples will allow cluster masses to be constrained in a redshift regime where weak lensing is challenging, while also enabling studies of the kinematic relationship between cluster emission line and passive galaxies out to redshifts well beyond 1.


We acknowledge the support by the DFG Cluster of Excellence “Origin and Structure of the Universe”, the Transregio program TR33 “The Dark Universe” and the Ludwig-Maximilians University. The South Pole Telescope is supported by the National Science Foundation through grant PLR-1248097. Partial support is also provided by the NSF Physics Frontier Center grant PHY-1125897 to the Kavli Institute of Cosmological Physics at the University of Chicago, the Kavli Foundation and the Gordon and Betty Moore Foundation grant GBMF 947. The Melbourne group acknowledges support from the Australian Research Council’s Discovery Projects funding scheme (DP150103208). DR is supported by a NASA Postdoctoral Program Senior Fellowship at NASA’s Ames Research Center, administered by the Universities Space Research Association under contract with NASA. Work at Argonne National Laboratory was supported under U.S. Department of Energy contract DE-AC02-06CH11357. AB acknowledges the hospitality of the LMU, and partial financial support from PRIN-INAF 2014 ”Glittering kaleidoscopes in the sky: the multifaceted nature and role of Galaxy Clusters”, P.I.: Mario Nonino. BB has been supported by the Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics.


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Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany
Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany
Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 85748 Garching, Germany
INAF-Osservatorio Astronomico di Trieste via G.B. Tiepolo 11, 34143 Trieste, Italy
Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
Fermi National Accelerator Laboratory, Batavia, IL 60510-0500, USA
Department of Physics, University of Chicago, Chicago, IL, USA 60637
Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL, USA 60439
Department of Physics and Astronomy, University of Missouri, 5110 Rockhill Road, Kansas City, MO 64110
Enrico Fermi Institute, University of Chicago, Chicago, IL, USA 60637
Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617, Taiwan
Department of Physics, University of California, Berkeley, CA, USA 94720
Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada
Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA 94305-4085, USA
Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305
Department of Physics, Université de Montréal, Montrea, Quebec H3T 1J4, Canada
Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA
Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen Juliane Maries Vej 30, 2100 Copenhagen, Denmark
Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO, 80309
NASA Ames Research Center, Moffett Field, CA 94035, USA
School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
Department of Astronomy, University of Michigan, 1085 S. University Ave, Ann Arbor, MI 48109, USA
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, USA 02138
Department of Physics, University of California, Davis, CA, USA 95616
Institute of Geophysics and Planetary Physics, Lawrence Livermore National Laboratory, Livermore, CA 94551
Cerro Tololo Inter-American Observatory, Casilla 603, La Serena, Chile