RHAPSODY: I. HALO PROPERTIES AND FORMATION HISTORY FROM A STATISTICAL SAMPLE OF RE-SIMULATED CLUSTER-SIZE HALOS

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1 Draft version July 24, 2012 Preprint typeset using L A TEX style emulateapj v. 12/16/11 RHAPSODY: I. HALO PROPERTIES AND FORMATION HISTORY FROM A STATISTICAL SAMPLE OF RE-SIMULATED CLUSTER-SIZE HALOS Hao-Yi Wu, 1,2 Oliver Hahn, 1 Risa H. Wechsler, 1 Yao-Yuan Mao, 1 Peter S. Behroozi 1 1 Kavli Institute for Particle Astrophysics and Cosmology; Physics Department, Stanford University, Stanford, CA, SLAC National Accelerator Laboratory, Menlo Park, CA, Physics Department, University of Michigan, Ann Arbor, MI 48109; hywu@umich.edu Draft version July 24, 2012 ABSTRACT We present the first results of the Rhapsody cluster re-simulation project: a sample of 96 zoomin simulations of dark matter halos of ±5 h 1 M, selected from a 1 h 3 Gpc 3 volume. This simulation suite is the first to resolve this many massive halos with particles per halo. The combination of high resolution and large sample size allows us to probe the distribution of and correlation between key halo properties in a statistical way for these very massive systems. In this first paper, we discuss results on the properties of the main halos and how they are affected by halo formation history. We are able to track the most massive progenitors of these halos up to z = 12 (with 200 snapshots) and for almost 5 decades in mass. With this wide span in time scale and high time resolution, we re-examine the mass accretion history of halos and compare various formation time proxies. We present a new 1-parameter description of the formation history that can provide a good fit to halos in this mass range, and assess how formation time impacts the density profiles, mass accretion histories, halo shapes, and the velocity structure of our sample. We explored various fitting function of halo density profiles and found that they deviate from Navarro Frenk White profile systematically at 10% level. The properties of subhalos in our sample will be presented in a second paper, where we will explore, in addition to halo formation history, how stripping is affecting the subhalo properties. Keywords: cosmology: theory dark matter galaxies: clusters: general galaxies: halos methods: N-body simulations 1. INTRODUCTION Galaxy clusters are powerful probes of cosmological parameters and have played a key role in the development of the current ΛCDM paradigm (see e.g., Allen et al for a review). For example, the spatial distribution and abundance of galaxy clusters reflect the growth rate of large-scale structure and the expansion rate of the universe, providing constraints on dark matter and dark energy (e.g., Vikhlinin et al. 2009; Mantz et al. 2010b; Rozo et al. 2010), neutrino mass (e.g., Mantz et al. 2010a; Reid et al. 2010), and the validity of general relativity on cosmic scales (e.g., Rapetti et al. 2010, 2012). In the near future, the massive influx of multi-wavelength surveys (e.g., SPT 1, ACT 2, Planck 3, erosita 4, PanSTARRS 5, DES 6, Euclid 7, LSST 8 ) will greatly enhance the sample size of galaxy clusters and reduce the statistical uncertainties in cluster cosmology. However, the constraining power of galaxy clusters will depend on how well various systematic uncertainties can be controlled, including the normalization, slope, and scatter of scaling relations between observable properties and mass (e.g., Rozo et al. 1 The South Pole Telescope; 2 Atacama Cosmology Telescope; Extended ROentgen Survey with an Imaging Telescope Array; 5 The Panoramic Survey Telescope & Rapid Response System; 6 The Dark Energy Survey; The Large Synoptic Survey Telescope; ); the robustness of cluster identification and centering (e.g., Rykoff et al. 2012); the effect of viewing angle and projection (e.g., White et al. 2010). One essential way to understand systematic uncertainties involved in galaxy cluster surveys is through N-body simulations, which have been applied to study galaxy clusters for more than a decade (e.g., Tormen et al. 1997; Moore et al. 1998; Ghigna et al. 1998). To calibrate the systematic uncertainties in the era of large-sky survey and precision cosmology, it is desirable to have controlled simulation samples that can help us understand the statistical distribution of the properties of galaxy clusters and the correlation between observables, as well as their detailed structures and evolution. Since massive galaxy clusters are rare, cosmological simulations need to cover a large volume to include a fair number of these systems (e.g., the MultiDark simulation [Prada et al. 2011] and the recent Millennium XXL simulation [Angulo et al. 2012]). However, given limited computational resources, the detailed substructures of halos are not well-resolved in these simulations. Instead of using a cosmological volume with a single resolution, one can focus on particular systems and re-simulate them with higher resolution. This so-called zoom-in technique provides a powerful way to study individual cluster systems in detail in a cosmological context (e.g., Tormen et al. 1997; Moore et al. 1999; Navarro et al. 2004; Gao et al. 2005; Reed et al. 2005). However, so far most zoom-in simulations have focused only on a small number of systems (e.g., the current high-resolution Phoenix simulation [Gao et al. 2012]) or galactic halos (e.g., the Via Lactea II simu-

2 2 WU ET AL. lation [Diemand et al. 2008] and the Aquarius simulations [Springel et al. 2008]). Therefore, few statements have been made about the statistical properties of wellresolved substructures in the mass regime of galaxy clusters. In this work, we perform re-simulations of a large number of cluster-forming regions in a cosmological volume (side length 1 h 1 Gpc) to create a high-resolution statistical cluster sample, Rhapsody, which stands for Resimulated HAlo Population for Statistical Observable mass Distribution study. The current sample includes 96 halos of mass ±5 h 1 M with mass resolution h 1 M. One of the main goals of Rhapsody is to create a sample of cluster-size halos that enables us to make statistical statements about the halo population that is relevant for current and imminent cluster surveys. In Figure 1, we compare the halo sample of several N-body simulations in the literature (Millennium [Springel 2005]; Millenium XXL [Angulo et al. 2012]; Bolshoi [Klypin et al. 2011]; MultiDark [Prada et al. 2011]; Consuelo and Carmen [from LasDamas; McBride et al. in preperation]; Phoenix [Gao et al. 2012]; Aquarius [Springel et al. 2008]) to our Rhapsody 8K (main sample) and Rhapsody 4K (a factor of 8 lower in mass resolution). The cosmological volumes are presented by curves, which indicate the halo population inside each volume, while simulations of individual halos are presented by symbols. As can be seen, Rhapsody is in a unique regime in terms of the number of halos simulated with high particle number. Our repeated implementation of the re-simulation method makes the simulation suite statistically interesting and computationally feasible. The Rhapsody sample is highly relevant to several current observational programs. For example, the Cluster Lensing And Supernova survey with Hubble Multi- Cycle Treasury Program (CLASH; Postman et al. 2011) focuses on 25 massive clusters and aims to establish unbiased measurements of cluster mass concentration relation of these clusters. In addition, various X-ray programs have been efficiently identifying massive clusters; for example, the ROSAT Brightest Cluster Sample (Ebeling et al. 2000), ROSAT-ESO Flux-Limited X- ray sample (Böhringer et al. 2004), and the MAssive Cluster Survey (Ebeling et al. 2010). These samples have achieved high completeness and provided relatively unbiased selection. Relatively recently, massive galaxy clusters have also been detected through the Sunyaev Zel dovich (SZ) effect by ACT (Marriage et al. 2011), SPT (Williamson et al. 2011), and Planck (Planck Collaboration et al. 2011), which have ushered in an era of high-purity detection of high-redshift galaxy clusters. The Rhapsody sample is in a mass regime similar to these observational programs and can provide a statistical description of the dark matter halos associated with these clusters. This paper presents the first results from the Rhapsody simulations. We focus on the 96 high-resolution main halos, quantifying their formation history using the large sample size and wide span of history, as well as the high spatial and temporal resolution. We then focus on how formation time impact the properties of halos. We then present halo density profiles, halo shapes, velocity ellipsoid, and radial velocity profiles, and assess the extent to which these properties are affected by the for- Number of halos (per 0.1 dex of mass) Consuelo Carmen Bolshoi MultiDark Millennium Millennium-XXL Rhapsody 4k Rhapsody 8k Phoenix Aquarius Number of particles per halo Figure 1. Comparison of the halo samples in various N-body simulations; Rhapsody is in a unique statistical regime of wellresolved massive halos. The number of halos (per 0.1 dex in mass) is shown as a function of number of particles inside the virial radius of the halo. Symbols correspond to halos in re-simulation projects; the Rhapsody 4K and 8K samples are shown as two colored stars (M vir = ±5 h 1 M ). Curves correspond to halos in different cosmological volumes, and black stars on these curves correspond to the number of halos of the same mass as Rhapsody. We note that Consuelo and Carmen both include 50 volumes, and only one volume is presented here. mation history. Our results are put into the context of results in the literature and are compared with results obtained from smaller samples or extrapolation from lower masses. We pay special attention to the the halo density profile, presenting the distribution of halo concentration based on several fitting functions, as well as the evolution of concentrations. In a second paper in this series (Paper II), we will present the subhalos in our sample and explore the impact of formation time on them. The impact of formation time on subhalos is more complex in the sense that the subhalo properties depends on the selection method of subhalos, the stripping experienced by a subhalo, and the resolution of the simulation. In Paper II, we will address these issues and present how we expect formation time will impact the properties of galaxies in clusters. This paper is organized as follows. In 2, we detail the simulations. We first present the formation history of the main halos in 3.1 to pave ways for studying the impact of formation history. In 4, we present various halo properties and how these properties are affected by formation history. We conclude in THE SIMULATIONS The Rhapsody sample includes 96 cluster-size halos of mass M vir = ±5 h 1 M, re-simulated from a cosmological volume of 1 h 1 Gpc. Each halo was simulated at two resolutions: h 1 M (equivalent to particles in this volume), which we refer to as Rhapsody 8K or simply Rhapsody ; and h 1 M (equivalent to particles in this volume), which we

3 Rhapsody Simulations of Massive Galaxy Clusters 3 Type Name Mass Resolution Force Resolution Number of Particles Number of Particles [ h 1 M ] [ h 1 kpc] in Simulation in Each Targeted Halo Full Volume Carmen K Zoom-in Rhapsody 4K M a / (equiv.) 0.63M b Rhapsody 8K M a / (equiv.) 4.9M b a The mean number of high-resolution particles in each zoom-in region. b The mean number of high-resolution particles within the R vir of each targeted halo. Table 1 Simulation parameters. refer to as Rhapsody 4K. These two sets allow detailed studies of the impact of resolution. The simulation parameters are summarized in Table 1. The initial conditions were generated with the multiscale initial condition generator Music (Hahn & Abel 2011). The particles were then evolved using the public version of Gadget-2 (Springel 2005). The halo finding was performed with the phase-space halo finder Rockstar (Behroozi et al. 2011a). Finally, merger trees were constructed with the gravitationally-consistent code of Behroozi et al. (2011b). We provide more details on our methods below. All simulations in this work are based on a ΛCDM cosmology with density parameters Ω m = 0.25, Ω Λ = 0.75, Ω b = 4, spectral index n s = 1, normalization σ 8 = 0.8, and Hubble parameter h = 0.7. Figure 2 shows images of 90 halos at z = 0 in the 8K sample. Halos are sorted by their concentration and subhalo number, as described in the following sections. In Figure 3, we show the evolution of 4 individual halos, selected as extremes in the distribution of concentration and subhalo number. Movies for several individual halos are available at The cosmological volume Our re-simulations are based on one of the Carmen simulations from the LArge Suite of DArk MAtter Simulations (LasDamas; McBride et al.). A Carmen simulation represents a cosmological volume of 1 h 1 Gpc with particles. Its initial conditions are based on the 2LPT code in Crocce et al. (2006), and the N-body simulation was run with the Gadget-2 code. Rhapsody uses the same cosmological parameters as Carmen. When selecting targets for re-simulation from the massive end of the halo mass function, we choose a mass bin that is narrow enough so that mass trends of halo properties are negligible but at the same time wide enough to include a sufficient number of halos for statistical analyses. Here we focus on a 0.1 dex bin surrounding log 10 M vir = This mass range allows us to select 100 halos in a narrow mass range, and is well-matched to the masses of the massive clusters studied in X-ray, SZ, and optical cluster surveys Initial conditions For each of the halos in our sample, we generate multiresolution initial conditions using the Music code (Hahn & Abel 2011). We use the same white noise field of Carmen ( of its modes) to generate large-scale perturbations consistent with Carmen. The equivalent resolution ranges from in the lowest resolution region to ( for the 4K sample) in the highest resolution region. In between, the mass resolution changes by factors of 8 every 8 times the mean interparticle distance. For each of our re-simulation targets, we choose a zoom-in volume that is 40% larger than the Lagrangian volume of the friends-of-friends halo at z = 0. This choice has been tested to provide a well converged dark matter density profile in our convergence tests. With this setting, no low-resolution particle was found within the virial radius of any targeted halo. The typical number of high resolution particles per simulation is thus about 42/5.4 million for 8K/4K with a standard deviation of 18%. In Music, particle displacements and velocities have been computed from the multi-scale density field using the second-order Lagrangian perturbation theory (2LPT) at a starting redshift of 49, in accordance with Carmen. The use of 2LPT is important for statistical studies of such massive systems since their masses depend on the accuracy of the initial conditions (e.g., Crocce et al. 2006; Tinker et al. 2008; Reed et al. 2012; Behroozi et al. 2012; and McBride et al., in preparation) Gravitational evolution After generating the initial conditions, we evolve each cluster-forming region using the public version of the Gadget-2 code (Springel 2005). Gravitational forces are computed using two levels of particle-mesh together with the force tree to achieve a force resolution of comoving 3.3/6.7 h 1 kpc in the Rhapsody 8K/4K for particles in the high resolution region. For each simulation, we save 200/100 snapshots logarithmically spaced in scale factor a between a = 75 and a = 1 for the 8K/4K sample. We note that the virial masses of the re-simulated halos change somewhat with the improved resolution. As a result, a fraction of the halos fall outside the narrow targeted mass range log 10 M vir = 14.8 ± 5. In most cases, the masses scatter upward. We discard those halos falling outside the targeted mass bin of Rhapsody to keep the the mass selection clean. In principle, to obtain all halos in the 14.8 ± 5 mass bin in the re-simulated sample, one needs to re-simulate a wider range of masses around 14.8 to include all halos that end up in the targeted bin. However, the large-suite of re-simulations thus required are beyond the scope of this work. Thus, we note that Rhapsody does not include the complete sample of halos within log 10 M vir = 14.8 ± 5 in either the original volume or the re-simulations. However, we do not expect this fact to affect the results presented in this paper, because the main approach in this paper is stacking all halos in Rhapsody for sufficient statistics and our sample should be unbiased. Global statistics for halos in this bin in the entire cosmological volume (for example, the two-point correlation function) are not used in the

4 4 WU ET AL. present work Halo and subhalo identification Our simulations are processed with the adaptive phasespace halo finder Rockstar (Behroozi et al. 2011a), which can achieve high completeness in finding substructures even in dense environments (also see Knebe et al. 2011). Based on the phase-space information, small subhalos passing through the dense central region of the main halo can be robustly identified. This feature is especially important for studying the subhalo populations, which we focus on in Paper II. We note that the algorithm is only applied to high-resolution particles in the simulations. Rockstar pays special attention to major merger events (two halos of similar mass merge with each other), which arise frequently in the formation history of Rhapsody halos (because of their high masses) and often cause difficulties in the construction of merger trees. During a major merger between two halos, a large fraction of dark matter particles appear as unbound to either of the merging halos, even though they are bound to the entire merging system. Therefore, regular unbinding procedures tend to result in ambiguities or inconsistencies in halo mass assignment. Rockstar addresses this issue by computing the gravitational potential of the entire merging system, thus making the mass evolution of halos self-consistent across time steps Merger trees Rhapsody 4K/8K has 100/200 output snapshots between z = 12.3 and z = 0, equally spaced in ln a. We apply the gravitationally-consistent merger tree algorithm developed by Behroozi et al. (2011b). The idea behind this new merger tree implementation is that the stochasticity in N-body simulations often leads to failures in halo finding. For example, the halo finder might find a spurious halo that is in reality a random density fluctuation at a certain time step, or the halo finder might miss a halo because it falls below the detection threshold at that particular time step. Given these limitations in halo finders, traditional implementations of merger trees often encounter problems in linking halos across different time steps. The gravitationally-consistent merger tree algorithm resolves this issue by comparing adjacent time steps to recover missing subhalos and remove spurious halos, thereby improving the completeness and purity of the halo catalogs and ensuring correct linking of halos across time steps. This algorithm compares two adjacent time steps and can be summarized as follows: (1) It takes the halos at the later time step and evolves their positions and velocities backward in time, deciding whether the progenitors are missing or incorrectly linked. (2) It takes the halos at the earlier time step and looks for its descendant in the later time step. If the descendant is missing, the algorithm decides whether a merger occurs or the current halo is spurious. For detailed implementation, we refer the reader to Behroozi et al. (2011b). 3. THE BUILDUP OF CLUSTER-SIZE HALOS 3.1. Mass accretion history In this work, halo definition is based on the spherical overdensity of virialization, vir, with respect to the critical density, ρ crit. We use the most-bound particle calculated by Rockstar as the center of a halo. Based on this center, we draw a sphere with radius R vir so that the mean overdensity enclosed is equal to vir. With the cosmological parameters used herein, vir = 94 with respect to the critical density at z = 0 (Bryan & Norman 1998); i.e. vir = 94c = 376m. The mass accretion history of dark matter halos provides a way to characterize or categorize the evolution of halos, which impacts various halo properties, including their clustering, internal structure, and subhalos (e.g., Wechsler et al. 2002; Zhao et al. 2003; Harker et al. 2006; Hahn et al. 2007; Maulbetsch et al. 2007; Li et al. 2008). Various parameterizations have been proposed to fit the mass accretion histories of halos in simulations (e.g., Wechsler et al. 2002; van den Bosch 2002; Tasitsiomi et al. 2004; Zhao et al. 2009; McBride et al. 2009). In this section, we provide fits to the mass accretion history of Rhapsody halos and discuss the goodness-of-fit as well as the distributions of formation time proxies obtained from various descriptions of halo growth. Figure 4 summarizes our study of the mass accretion history of the main halos in Rhapsody. Panel (a) presents the mass evolution of the main halos in Rhapsody. The gray curves show the evolution of M vir for individual halos, and the black curve shows the average of all halos. We note that the dispersion in log mass around the mean mass accetion history is roughly constant back to z = 12. In the upper sub-panel, we added a cyan line showing the expected pseudo evolution of a static halo caused by the evolution of vir and the critical density (Diemer et al. 2012). For each main halo identified at z = 0, we search through its merger tree to find the most massive progenitor at each redshift. Panel (b) shows the mass growth rate defined as d ln M vir (1) dz as a function of z. Our halo formation histories span almost 5 orders of magnitude in mass and cover the time between z = 12 and z = 0. This range has not previously been measured for such a large sample of halos. For the evolution of individual halos, we fit several parameterized mass accretion history models. We compare three models for the mass accretion history: 1. An exponential model (Wechsler et al. 2002) M(z) = M 0 e αz. (2) We note that this model assumes a constant mass accretion rate represented by the exponential growth index d ln M dz = α. (3) Panel (b) of Figure 4 effectively shows that d ln M/dz is not exactly constant, thus leading to deviations from a pure exponential model. In addition, we can define the formation time proxy as z α = ln 2/α. (4)

5 Rhapsody Simulations of Massive Galaxy Clusters 5 Figure 2. Images of 90 Rhapsody halos at z = 0. The halos are first sorted by concentration (high concentration on the upper rows). In each row, the halos are then sorted by the number of substructures (selected with v 0 > 100 km/s, high number of substructures on the left columns). Each image has a physical extent of 4 h 1 Mpc on a side, which is slightly larger than the average virial radius of 1.8 h 1 Mpc.

6 6 WU ET AL. Figure 3. Evolution of four Rhapsody halos. From top to bottom, the images show the progenitors of four halos at z = 3 and z = 1, and the halo at z = 0. The four halos chosen are the corners of Fig 2. From left to right, they have high concentration, high subhalo number [337]; low concentration, high subhalo number [377]; high concentration, low subhalo number [572]; low concentration, low subhalo number [653]. Halo 572 has the highest concentration, the least late-time accretion, and the most dominant BCG of our full sample. It is also the halo with the most massive progenitor at z = 3. Each panel has a comoving extent of 4 h 1 Mpc on a side.

7 Rhapsody Simulations of Massive Galaxy Clusters 7 Property Median Frac. Scatter Mean Frac. SD Def. M vir [h 1 M ] R vir [h 1 Mpc] σ v [km/s] 1, , v max [km/s] 1, , M 200m M 200c M 500m M 500c h 1 M ; 4.1 R 200m R 200c R 500m R 500c h 1 Mpc; 4.1 a lmm a 1/ a α γ β c NF W c NF W like c Einasto γ NF W like γ Einasto α Einasto b/a c/a T λ 2 /λ λ 3 /λ δ σ 2 los Table 2 Properties of Rhapsody halos at z = 0. The third column corresponds to the ratio of the 68% scatter to the median. The fifth column corresponds to the ratio of the standard deviation to the mean. 2. An exponential-plus-power law model with two parameters (McBride et al. 2009) We note that M(z) = M 0 (1 + z) β e γz. (5) d ln M dz γ β when z << 1. (6) Thus, γ β can be used as a measure for the latetime accretion rate. We can define (analogous to Eq. 4) z γ β = ln 2/(γ β). (7) In addition, we can define z βγ to be the redshift at which M(z βγ ) = M 0 /2 obtained by solving this equation numerically. We note that z βγ and γ β are completely anti-correlated with each other (rank correlation equals -1); thus, z βγ serves a formation time proxy that describes the late-time accretion. That is, a lower z βγ corresponds to a higher γ β and indicates a higher late-time accretion rate. 3. An exponential-plus-power law model with one parameter, motivated by Equation 5. When we perform the fit of Equation 5, we observe that the two parameters β and γ are highly correlated (see the inset of Panel [c]). We thus adopt a 1-parameter model M(z) = M 0 (1 + z) c0+c1γ e γz. (8) We note that c 0 and c 1 are fixed values for all halos, and only γ is the only parameter that varies in the fit. We search for the optimal values of c 0 and c 1, which is discussed further below.

8 8 WU ET AL Static halo exp exp+pl (2-para) exp+pl (1-para, optimal) Mvir [h 1 M ] dlnmvir/dz Mfit/Mtrue z (a) Mass evolution z (b) Mass growth rate Cumulative Distribution β γ exp 0.2 exp+pl (2-para) exp+pl (1-para) exp+pl (1-para, optimal) (RMS of Residual) (c) Comparison of models Cumulative Distribution z lmm z 1/2 z α = ln2/α z γ β = ln2/(γ β) z βγ z (d) Distribution of formation time proxies Figure 4. (a) Mass accretion history of the main halos in Rhapsody (gray: individual halos; blue: average). The virial mass of the most massive progenitor is shown at each output redshift. (b) The mass accretion rate of these halos, d ln M vir /dz as a function of z, averaged over every 3 output redshifts to reduce the noise. (c) Comparison of different forms of mass accretion history: exponential model (Eq. 2) with one free parameter α; the exponential-plus-power law model with two free parameters β and γ (Eq. 5) or one free parameter (Eq. 8). The one-parameter model with an optimal relation of the exponential and power law relation (the purple curve; also see the inset) provides a compelling description of the mass accretion history. (d) The cumulative distribution of several proxies for formation time: z 1/2 (half-mass redshift); z α = ln 2/α; z γ β = ln 2/(γ β); z βγ (half-mass redshift in Eq. 5). These different formation time proxies probe somewhat different epochs in a halo s history.

9 Rhapsody Simulations of Massive Galaxy Clusters 9 For each model, we minimize the target function 9 2 model = 1 N [log N 10 M(z i ) log 10 M model (z i )] 2. i=1 (10) Panel (c) of Figure 4 presents the cumulative distribution of RMS residuals, model (Eq. 10), for these model fits. The red curve corresponds to the exponential model, which has the largest residual; the green curve corresponds to the exponential-plus-power law model with two free parameters (β, γ), which has the smallest residual. In between these two models, we compare two 1- parameter models: the blue curve corresponds to using the linear fit between β and γ (β = 4.16γ 4.00) to eliminate one parameter; the purple curve corresponds to an optimization of the relation between β and γ to minimize the overall residuals, obtained with several iterations. The optimal relation is given by β = 5.27γ 4.61 (11) The 1-parameter model using this optimized relation performs almost as well as the 2-parameter model. In the bottom sub-panel of Panel (a), we compare the residual of these fitting functions, M model (z)/m true, which is averaged over the entire sample. The pure exponential model systematically deviates from the data, while the two exponential-plus-power law models show less residuals. Panel (d) of Figure 4 presents the cumulative distribution functions for several halo formation time proxies: z 1/2, the redshift that the halo first reaches half of its final mass. z α = ln 2/α, the redshift at which M(z α ) = M 0 /2 in the exponential fit. z γ β = ln 2/(γ β), analogous to z α, where β and γ comes from fitting the exponential-plus-power law model (Eq. 5). z βγ, obtained by numerically solving M(z βγ ) = M 0 /2 in the exponential-plus-power law model (Eq. 5). We observe that none of the formation time proxies obtained from the fitting functions describe the half-mass redshift well. The rank correlation between z α and z 1/2 is 0.55, and the rank correlation between z γ β and z 1/2 is This is not surprising; since these fitting functions are used to fit almost 5 orders of magnitude in mass growth (10 10 to h 1 M ), they do not provide accurate description for the final stages of mass evolution and are not flexible enough to describe small fluctuations in mass accretion rate at relatively low redshift. These different formation time definitions are useful in that they 9 We note that this function differs from what was used in McBride et al. (2009) N 2 model = 1 N i=1 [M(z i )/M 0 M model (z i )/M 0 ] 2 M(z i )/M 0. (9) Because our mass accretion history spans approximately 5 orders of magnitude in mass and starts from redshift 12, weighting by M(z i )/M 0 significantly underweights high redshift outputs. probe different epochs in a halo s history; z α tends to be a slightly earlier epoch, while z β γ and z βγ tend to be later. In fact, z β γ is equivalent to the value of z α measured with the only low redshift outputs. Although z βγ is completely correlated with z β γ, it corresponds to a slightly earlier redshift Merger rate Figure 5 presents merger rate measurements for Rhapsody halos. The left panel shows the cumulative number of merger events. The y-axis corresponds to the average number of merger events each main halo has experienced since a given redshift, which is specified by the x-axis. The different curves correspond to different merger mass ratios, µ = M merging /M main, and the shaded regions indicate the standard deviation of the sample. The right panel corresponds to the merger rate as a function of merger mass ratio µ. We plot the differential number of merger events each main halo has experienced, per dµ per dz, for a given merger ratio. The different curves represent different redshifts. The merger rate trends are almost the same for different redshifts. The blue shaded region corresponds to the scatter of the blue curve (z = 5) and indicates there is large variation from halo to halo. These trends are roughly consistent with the results of Fakhouri & Ma (2008) and Fakhouri et al. (2010), who have combined the results from Millennium simulations I and II. Our work have extended the mass and merger ratio over which merger rates can be quantified. 4. HALO PROPERTIES AT Z = 0 AND CORRELATION WITH FORMATION TIME In this section, we present various properties of the main halos in Rhapsody at z = 0. The median, mean, and standard deviation of various properties are summarized in Table 2. In Figure 6, we present the distributions of some of the properties discussed in this section, showing their histograms along with correlations between each other and with several formation history parameters presented in 3.1. The rank correlation between properties are shown in each panel; we note that throughout this work, we present rank correlation to avoid the impact of outliers. Correlations with higher absolute values are presented with larger fonts Halo masses and velocity dispersions In Table 2, we list two properties that are closely related to halo mass: the maximum circular velocity and the velocity dispersion of dark matter particles. The maximum circular velocities is defined at a radius r max that maximizes GM(< r)/r: GM(< r max ) v max =. (12) r max The velocity dispersion is calculated based on dark matter particles: σv 2 = v v 2 = 1 N p v i v 2 (13) N p i=1

10 10 WU ET AL. Average Number of Merger Events Since z µ >1 µ >3 µ >0.1 µ >0.3 Mergers per halo per dµ per dz slope = z = 5 z = 1.0 z = z µ = M merging /M main Figure 5. The merger rate of Rhapsody halos. Left: Cumulative merger rate as a function of redshift (per halo). The average number of merger events each main halo has experienced since z is shown for three different merger mass ratios µ = M merging /M main. Right: Differential merger rate as a function of merger mass ratio. The number of merger events per halo per dµ per dz is shown as a function of µ. The three different curves correspond to different redshifts, and the trend is independent of redshift. For both panels, each colored band corresponds the standard deviation of the curve of the same color. We note the correlation between M vir and σ v is 0.32, indicating that there is non-negligible residual mass velocity dispersion scaling despite the narrow mass range of our sample. In addition, we calculate halo masses and radii based on several different overdensity values that are commonly used: 200m = 50c, 200c, 500m = 125c, and 500c. The subscript c and m indicate that the overdensity is with respect to the critical density ρ crit and mean matter density Ω M ρ crit, where Ω M = These masses and radii are summarized in Table Density profiles and halo concentration The density profiles of dark matter halos have been shown to follow the universal Navarro Frenk White form (Navarro et al. 1997) and can be well characterized by a concentration parameter c. However, it is still an ongoing effort to characterize the scatter of the concentration parameter and its dependence on halo mass (e.g., Bullock et al. 2001; Neto et al. 2007; Macciò et al. 2008; Gao et al. 2008; Prada et al. 2011; Bhattacharya et al. 2011). The statistics of halo concentration is of increasing importance for interpreting observations. As mentioned in the introduction, the CLASH project has been a major effort of the Hubble Space Telescope and aims for detailed and unbiased measurements of the density profile of galaxy clusters, which are tests of both the ΛCDM paradigm and our understanding of the assembly of clusters. In addition, the modeling of the concentration mass relation impacts the interpretation of the weak lensing results (e.g., King & Mead 2011) and X-ray results (e.g., Ettori et al. 2010). Therefore, it is imperative for simulators to quantify the concentration of clusters accurately. Panel (a) of Figure 7 presents the density profiles of the main halos in Rhapsody. The blue curve corresponds to the mean density profile, while the gray curves correspond to individual halos. We use 32 bins between R vir and R vir that are equally spaced in log r for plotting these curves. We note that the 8K and 4K samples show perfect agreement in their density profiles. For each main halo, we use all dark matter particles between 13 h 1 kpc and R vir to fit the halo concentration, adopting the maximum-likelihood estimation without binning in radius. 10 We fit for three different parameterizations of density profile: 1. The Navarro Frenk White (NFW) profile (Navarro et al. 1997) ρ(r) δ c = ρ crit (r/r s )(1 + r/r s ) 2 (16) d ln ρ d ln r = 1 + 3(r/r s) 1 + (r/r s ) (17) which is characterized by one parameter c NF W = R vir /r s. 10 The procedure of finding the maximum-likelihood estimator is to maximize the log-likelihood function over a set of parameters, p. The log-likelihood function is defined as l(p) = 1 log(ν p(r i )), (14) N i where the summation runs over all the N particles, and ν p(r) = 1 N 4πr2 ρ p(r) (15) so that ν p(r)dr = 1. This approach is consistent with the radially-binned fitting method with a large number of particles and is more stable than the binned method when the halo has fewer particles.

11 Rhapsody Simulations of Massive Galaxy Clusters 11 a lmm γ β c NFW NFW c/a λ 3 /λ P z / P a lmm Numbers in red: Rank correlation z 1/2 : Half-mass redshift a lmm : Scale factor of last major merger γ β: Late-time accretion rate ( -dlnm/dz) c NFW : Concentration parameter from the NFW fit NFW : KS statistic for the NFW model c/a: Halo shape parameter λ 3 /λ 1 : Velocity ellipsoid parameter P γ β P c NFW P NFW 2 P c/a P z1/2 almm γ β 9 7 cnfw NFW c/a λ 3 /λ 1 Figure 6. Distributions of and correlations between main halo properties and formation history parameters. Rank correlation is presented throughout this work.

12 12 WU ET AL. 1.0 r 2 ρ(r)/ρcrit ρfit(r)/ρmean Mean of data NFW: c=5.45 NFW-like: c=5.52, γ=2.76 Einasto: c=4.93, γ= r [Mpc/h] (a) Cumulative Distribution NFW NFW like Einasto = max M(< r) M model (< r) /M vir (b) 1.0 K-S test for normal distribution 1.0 K-S test for log-normal distribution Cumulative Distribution c NFW p=0.920 c NFW like p=28 c Einasto p= Concentration = R vir /R 2 (c) Cumulative Distribution c NFW p=0.480 c NFW like p=0.338 c Einasto p= ln C (d) Figure 7. (a) Density profiles for the main halos in Rhapsody (gray: individual halos; blue: average). Comparison of three models (fit to the average profile) are shown in the bottom. (b) The Kolmogorov Smirnov statistics for these three models. The NFW-like and the Einasto models work equally well. (c, d) Concentration distributions based on: c NF W, c NF W like (with free outer slope), and c Einasto = R vir /R 2. Panel (c)/(d) shows the cumulative distribution of c/ln c and the corresponding best-fit of normal/log-normal distribution (dashed curve). Both models work well for c NF W and c Einasto while c NF W like prefers a log-normal distribution.

13 Rhapsody Simulations of Massive Galaxy Clusters An NFW-like profile with a free outer slope γ ρ(r) δ c = ρ crit (r/r s )(1 + r/r s ) γ 1 (18) d ln ρ d ln r = 1 + γ(r/r s) 1 + (r/r s ) (19) which reduces to the NFW profile when γ = 3. This profile can be characterized by two parameters, γ NF W like and c NF W like = (R vir /r s )(γ NF W like 2). The latter is defined so that R vir /c NF W like equals the radius at which the density slope is The Einasto profile (Einasto 1965) ( ) α d ln ρ r d ln r = 2 (20) r 2 This model is characterized by two parameters, r 2 and α. To compare with the previous models, we define c Einasto = R vir r 2 (21) γ Einasto = 2c α Einasto (slope at R vir ) (22) where γ Einasto is the slope of the log-log density profile at R vir and can be compared with γ NF W like. The best fit values of these parameters are summarized in Table 2. The mean value is in good agreement with Prada et al. (2011) and Bhattacharya et al. (2011). For the NFW fit, the standard deviations are σ(c)/c = 0.26 and σ(log 10 c) = 0.11, which are slightly smaller than the values quoted in Bhattacharya et al. (2011) (0.33 and 0.16 based on 200c ). This difference is presumably due the decreasing scatter in the concentration mass relation with increasing mass; as mentioned in Bhattacharya et al. (2011), their scatter is slightly smaller for massive halos (for M 200 > h 1 M, the scatter is σ(c)/c = 0.28, which is very close to our value). In Panel (a) of Figure 7, we add three curves corresponding to these three models, ρ model /ρ mean, in the bottom. These curves are chosen to fit the stacked binned density profile of all halos. We note that these values are slightly different from the average values of halos (shown in Table 2); i.e., the fit of the average results in slightly higher concentrations than the average fit to each halo. Among these three profiles, the Einasto profile fits to the stacked density profile best, deviating by up to 5%, whereas the NFW profile deviates by up to 10%. A similar trend of deviations has also been shown in the Phoenix simulations (Gao et al. 2012) and the Aquarius simulations (Navarro et al. 2010). Busha et al. (2005) showed that the asymptotic form of the halo density profiles becomes much steeper than NFW at large radii. Here, we find that these deviations are systematic and present in most individual halos at the present epoch as well. Panel (b) of Figure 7 presents the Kolmogorov-Smirnov statistics for these three models, where model = max M(< r) M model(< r) M vir (23) The two-parameter models, the NFW-like and the Einasto models, work almost equally well and are significantly better than the single-parameter NFW model. Panels (c) and (d) present the cumulative distribution functions of concentrations obtained from these fits. We explore whether a normal or a log-normal distribution provides a better description. In the middle/right panel, we show the cumulative distribution for c/ln c for the three different fits stated above. We also show the corresponding best fit normal/log-normal distributions and list the p-value based on a Kolmogorov-Smirnov test for goodness of fit. For the NFW and the Einasto profiles, both normal and log-normal distribution provide acceptable fits. For the NFW-like profile, a log-normal distribution provides a better description. Figure 6 presents c NF W and NF W. These two quantities are strongly correlated with z 1/2, a lmm, and γ β, and the correlation is strongest with a lmm. The correlation between concentration and formation time has been well known in the literature (e.g.,, Wechsler et al. 2002), and the standard explanation is that the concentration of halos reflects the density of the universe at the time the halo formed. The correlation between the deviation from NFW and formation time can be understood as the relaxedness of halos; late-formed halos tend deviate further from NFW due to the disturbance from recent mergers and late-time accretion. However, we note that c NF W and NF W only have a weak correlation; that is, highly concentrated clusters have only a slight tendency to be closer to NFW. Figure 8 presents the impact of formation time on halo concentration from the evolution of c NF W. Halos are split by their z 1/2, and the red/blue shows the mean of the concentration evolution of halos in the low/high z 1/2 quartile. Halo concentrations are measured every z = 0.5 using the maximum likelihood method. Above z = 1, both curves remain relatively constant and are similar to each other, indicating that the formation time defined at the present day only carries information about the recent evolution of the density profile. At low redshift, halos with high z 1/2 tend to have their concentration increasing steadily with time; this steady increase are presumably related to the lack of significant mass growth and higher degree or relaxedness Concentration evolution 4.4. Halo shapes N-body simulations have shown that dark matter halos have significant ellipticity and triaxiality (e.g., Jing & Suto 2002; Kasun & Evrard 2005; Allgood et al. 2006; Bett et al. 2007). Calibrations of halo shapes from simulations impact the accuracy of weak lensing mass calibration (e.g., Bett 2012) or even constraints on the self-interactions of dark matter particles (e.g., Miralda- Escudé 2002). The shape parameters are defined through the moment of inertia with respect to the halo center I ij = r i r j (24) where r i is the i th component of the position vector r of a dark matter particle with respect to the halo center, and the average is over all dark matter particles within R vir. Since all particles within R vir are of the same mass,

14 14 WU ET AL Low z 1/2 quartile High z 1/2 quartile 0.1 NFW Concentration Vr/Vvir (< r/rvir) z Low z 1/2 quartile High z 1/2 quartile r/r vir Figure 8. Evolution of concentration. The red/blue curves corresponds to the mean evolution of the halos of low/high formation redshift z 1/2. For late forming halos, the concentration remains relatively constant all the way from z=7 to the present. For early forming halos, the concentration tracks late forming halos from z=7 to z=1, but these halos have steadily increasing concentration with time for z < 1. These halos accrete very little mass over this time. no weighting by mass is needed. The eigenvalues of I ij are sorted as λ 1 > λ 2 > λ 3, and the shape parameters are defined as: a = λ 1, b = λ 2, c = λ 3. We use the dimensionless ratios b/a and c/a in our analysis. In addition, the triaxiality parameter is defined as T = a2 b 2 a 2 c 2 (25) T 1 (a > b c) indicates a prolate halo, while T 0 (a b > c) indicates an oblate halo. Intermediate values of T correspond to triaxial halos. We list the halo shape properties in Table 2. The distribution of b/a is presented in Figure 6. The shape parameter c/a is only weakly correlated with formation history. The correlation is even weaker for b/a and T. Allgood et al. (2006) shows that c/a is correlated with formation time (same as our z α ). These authors used a smaller radius 0.3R vir to measure the shape parameters, and they stated that the correlation is weaker for higher mass. When we measure halo shape parameters at a smaller radius R 500c, shapes are still only weakly correlated with formation time, indicating that the correlation is quite weak in this regime of mass and radius Velocity ellipsoid White et al. (2010) have demonstrated that the anisotropic motion of subhalos in clusters introduces significant scatter in velocity dispersion measured along different lines of sight. Here we follow the same procedure to measure the properties of the velocity ellipsoid of the dark matter particles of the Rhapsody sample. Analogous to the shape parameters, the velocity ellip- Figure 9. The radial velocity profiles of Rhapsody halos. The thin transparent curves show individual halos, whose profiles tend to be noisy. The red/blue curves correspond to halos in the low/high z 1/2 quartiles, and the thick curve correspond to the mean. Late-forming halos (red) tend to show strong infall or outflow, which could relate to coherent motions of subhalos. Earlyforming halos (blue) tend to show a more regular infall pattern. soid is defined as σij 2 = v i v j (26) Sorting the eigenvalues of the velocity ellipsoid as λ 1 > λ 2 > λ 3, one can again define dimensionless ratios λ 2 /λ 1 and λ 3 /λ 1 to describe the anisotropy of the velocity ellipsoid. The scatter of the velocity dispersion along the line of sight can be calculated as σ 2 los = 1 3 (λ 1 + λ 2 + λ 3 ) (27) (δσ 2 los) 2 = 4 45 (λ2 1 + λ λ 2 3 λ 1 λ 2 λ 2 λ 3 λ 3 λ 1 ). (28) We list the velocity ellipsoid parameters in Table 2. In Figure 6, we show the distribution of λ 3 /λ 1, which, like c/a, is only weakly correlated with formation time. The correlation is similar for the other two velocity ellipsoid parameters and their values measured at R 500c Radial velocity profile Figure 9 presents the cumulative radial velocity profile Vr (< r ) (29) V vir R vir of the dark matter particles that are bound to the main halos. The thin transparent curves in the background correspond to individual halos, while the black curve corresponds to the mean value. This profile indicates the infall and outflow structure of dark matter particles and have been used to define a static mass (Cuesta et al. 2008). However, we find that the radial velocity profiles of individual halos are too noisy to obtain robust estimate of the static radius, and the definition of the static

15 Rhapsody Simulations of Massive Galaxy Clusters 15 mass sensitively depends on how we set the criterion of being static. The detail exploration of the definition of static radius and mass is beyond the scope of the current work. We find that the radial velocity profile is impacted by formation time. In Figure 9, we further split halos by their z 1/2 and present the halos in the highest and lowest quartiles. Halos in the high/low z 1/2 quartile are presented with light blue/blue curves, and their mean is presented by the thick blue/red curve. The late-forming halos (red curve) tend to show strong infalls or outflows even within R vir, and on average the trend is outflow. These infalls and outflows are presumably related to the coherent motion of subhalos that are only recently accreted. On the other hand, the early-forming halos tend to show more regular infall patterns. We find it interesting that although formation history does not impact the velocity ellipsoid much, it clearly impacts the radial velocity profile. 5. SUMMARY AND DISCUSSION We have presented the first results of the Rhapsody project, which includes 96 cluster-size halos with mass M vir = ±5 h 1 M, re-simulated from 1 h 3 Gpc 3 with a resolution equivalent to particles in such a volume. In addition to achieving high resolution and large statistics simultaneously, Rhapsody is unique in its well-resolved subhalos and wide span of evolution history. Rhapsody also implemented the state-of-the-art algorithms for initial conditions (Music) and halo finding (Rockstar). Our findings are summarized as follows: 1. Summary of properties: In 4 and Table 2, we have presented the properties of the 96 main halos, including several commonly-used mass definitions, parameters of their density profiles, as well as the halo shapes and velocity ellipsoids. More detailed conclusions follow. 2. Mass accretion history: We have investigated the mass accretion history of the main halos in 3.1, tracking the most massive progenitors over 5 decades of mass growth. We compared various fitting functions for M vir (z) and found that the fractional mass growth rate d ln M vir /dz slightly decreases with redshift, which leads to deviations of M vir (z) from an exponential form. We have also shown that an optimal 1-parameter exponentialplus-power law model can precisely describe the mass accretion history of our halos. 3. Merger Rate: In 3.2, we have discussed the evolution of the subhalo population over cosmic time, the merger rate, and how mergers contribute to the mass assembly of our sample of massive cluster halos. We have shown that the differential merger rate (per dµ per dz, where µ is the merger mass ratio) follows a power law scaling with µ and is independent of redshift, in agreement with earlier work based on less massive halos or smaller samples. 4. Concentration: We have investigated various fitting functions for halo concentration in 4.2, including the NFW profile, an NFW-like profile (with a free outer slope), and the Einasto profile. We found that when fitting an NFW profile, the concentration distribution of our halos is consistent with a normal distribution and a log-normal distribution. The best-fit NFW profiles deviate by up to 10%, and the best-fit Einasto profiles deviate by up to 5%; these deviations appear to be systematic. 5. Concentration evolution: In 4.3, we show that the evolution of concentration is related formation time. Early-formed halos tend to have steadily increasing concentration at low redshift. 6. Halo shape and velocity ellipsoid: As discussed 4.4 and 4.5, the shape and velocity ellipsoid of our sample is uncorrelated with formation time proxies. 7. Radial velocity: In 4.6, we show that the radial velocities of halos are affected by formation time. Early-formed halos tend to show regular infall of dark matter particles, while late-formed halos tend to have large fluctuation of radial velocities. On average, late-formed halos have outflows of dark matter particles within R vir, which could be associated with the coherent motion of subhalos. The Rhapsody simulation suite can provide valuable information for other aspects of cluster cosmology. For example, the cluster-size halos in Rhapsody can be further used to study the covariances between mass tracers, for example, the galaxy content, the dynamics of galaxies, the weak gravitational lensing, the X-ray, and SZ effect. The formation history and environment of clusters can potentially impact these mass proxies systematically, either by altering the intrinsic properties of clusters or by affecting the observed properties through the impact of line-of-sight projection. As current multi-wavelength surveys will combine these different observables for cluster mass calibration, it is imperative to understand the covariances between these observables. In addition, our re-simulation technique can be applied to study the pink elephant clusters, which refer to a handful of massive clusters recently discovered at high redshift (e.g., Jee et al. 2009; Foley et al. 2011). These clusters have stimulated a great amount of discussion about whether they pose a challenge to the current ΛCDM paradigm of cosmology (Mortonson et al. 2011; Hoyle et al. 2011). To interpret the cosmological implications of these clusters correctly, it is important to understand their mass calibration. An extension of the current Rhapsody sample that includes a statistical sample for these massive clusters at high redshift will improve our understanding of these massive clusters. Understanding the covariances between different observable quantities and the potential biases in the mass measurements of these clusters can help us disentangle the astrophysical and cosmological implications of these clusters. In Paper II, we will focus on the subhalos in Rhapsody. In addition to the impact of formation time on subhalo properties, we find that there are intertwining effects from subhalo selection, stripping, and resolution of simulations.