Draft version July 23, 2012Preprint typeset using LATEX style emulateapj v. 12/16/11

RHAPSODY: II. SUBHALO PROPERTIES AND THE IMPACT OF STRIPPING FROM A STATISTICALSAMPLE OF CLUSTER-SIZE HALOS

Hao-Yi Wu,1,2 Oliver Hahn,1 Risa H. Wechsler,1 Peter S. Behroozi,1 Yao-Yuan Mao1

1Kavli Institute for Particle Astrophysics and Cosmology; Physics Department, Stanford University, Stanford, CA, 94305SLAC National Accelerator Laboratory, Menlo Park, CA, 94025

2Physics Department, University of Michigan, Ann Arbor, MI 48109; hywu@umich.edu

Draft version July 23, 2012

ABSTRACT

We discuss the properties of subhalos of the Rhapsody simulations, a new high-resolution statisticalcluster sample introduced in Wu et al. (2012). We focus on subhalo populations that are associatedwith galaxies accessible with current and upcoming optical cluster surveys. We demonstrate that thecriteria applied to select subhalo samples have significant impact on the inferred properties of thesample, including scatter in the number of subhalos, velocity structures, and the correlation betweenthe number of subhalos and formation time. We find that the number of subhalos, when selected usingthe peak maximum circular velocity in their histories (a property expected to be closely related to thegalaxy luminosity), is uncorrelated with the formation time of the main halo. This is in contrast tothe previously reported correlation from studies where subhalos are selected by the current maximumcircular velocity, and this difference can be explained by the stripping of subhalos. In addition, wefind that the properties of the most massive subhalo, as well as the subhalo mass fraction, are stronglycorrelated with halo concentration and formation history. These correlations are important to takeinto account when interpreting results from cluster samples selected with different criteria. Our samplealso includes a fossil cluster which is presented separately and placed in context with the rest of thesample.Keywords: cosmology: theory — dark matter — galaxies: clusters: general — galaxies: halos —

methods: N-body simulations

1. INTRODUCTION

The statistics and distributions of galaxy clusters inthe universe play an essential role in precision cosmol-ogy, not only because galaxy clusters are sensitive toboth cosmic expansion and large-scale structure growthrate, but also because multi-wavelength cluster surveysprovide enormous statistical power and are complemen-tary to other cosmological probes (see, e.g., Allen et al.2011 and Weinberg et al. 2012 for reviews, and referencestherein). Several large galaxy cluster surveys will soonunleash the statistical power of galaxy clusters, but atthe same time the precision cosmology can be achievedby these surveys will be limited by the systematic effectsinvolved in these surveys (e.g., Cunha et al. 2009; Fedeliet al. 2011; Pillepich et al. 2012).

Optical surveys of galaxy clusters, in particular, havepresented its statistical power that allows compelling cos-mological constraints (e.g., Gladders et al. 2007; Rozoet al. 2010). Optical surveys usually define optical rich-ness as the number of cluster member galaxies selectedwith a certain criterion and use it as a mass tracer (e.g.,Koester et al. 2007). The relation between cluster rich-ness and halo mass has been empirically determined withimproved accuracy (e.g., Rozo et al. 2009). However, var-ious systematic errors can be involved in these analysisprocesses, including cluster identification and centering(e.g., Rykoff et al. 2012), richness–mass relation (e.g.,Rozo et al. 2009, 2011), orientation and projection ef-fects (e.g., Cohn et al. 2007; White et al. 2010; Erick-son et al. 2011), cross-comparison with multi-wavelengthdata (Rozo et al. 2012), uncertainties in theoretical cali-brations of halo statistics (Wu et al. 2010), etc. With the

advent of large optical survey (e.g., PanSTARRS1, DES2,Euclid3, LSST4), it is especially imperative to controlthese issues within the desired accuracy.

N-body simulations of galaxy clusters have been aninvaluable tool to help us understand the systematicsexpected in optical cluster surveys. By focusing on thegravitational interactions of dark matter in the universe,N-body simulations simulate the growth of density peaksinto dark matter halos, which represent the loci of galax-ies and galaxy clusters (e.g., Kravtsov et al. 2004; Zhenget al. 2005). In particular, dark matter halos form hier-archically; a cluster-size halo formed through numerousmerging processes of smaller halos. These small halos be-come subhalos and, if they are massive enough to allowthe formation of galaxies, will contribute to the satel-lite galaxies of a galaxy cluster (e.g., Ghigna et al. 1998;Moore et al. 1998, 1999). Therefore, studying the accre-tion history of subhalos can help us to infer the assemblyof galaxy clusters, and the subhalos population in cluster-size halos can help us to understand the galaxy contentof galaxy clusters.

To compare with results from deep, wide surveys, oneneeds to simulate a large cosmological volume. However,simulating cluster-size halos in a cosmological volumeusually requires compromise between the sample size andresolution, because the total number of particles one cansimulate is determined by the computational resources

1 The Panoramic Survey Telescope & Rapid Response System;http://pan-starrs.ifa.hawaii.edu/

2 The Dark Energy Survey; http://www.darkenergysurvey.org/3 http://sci.esa.int/euclid/4 The Large Synoptic Survey Telescope; http://www.lsst.org/

2 WU ET AL.

available (e.g., comparison between the recent Millen-nium XXL simulation [Angulo et al. 2012] and BolshoiSimulation [Klypin et al. (2011)]). In order to achievelarge sample size and high resolution simultaneously, wehave been applying the “zoom-in” or multi-resolutionsimulation technique to develop the Rhapsody sam-ple (Wu et al. 2012, thereafter Paper I). Rhapsodycurrently includes 96 halos of mass 1014.8±0.05h−1M,selected from a cosmological volume of side length 1h−1Gpc and with mass resolution 1.3×108h−1M. Thissample is currently unique in terms of its sample size andresolution and occupies a new statistical regime of simu-lations (see Figure 1 in Paper I). In this second paper, wefocus on the subhalo population of these galaxy clusters.We give particular attention to the various propertiescharacterizing the subhalos and how they are affected bythe formation history of the clusters.

The correlation between formation history and observ-able properties of clusters is important because it can in-troduce bias in the mass calibration process. For exam-ple, Wu et al. (2008) showed that if the richness of a clus-ter is correlated with the formation time, then richness-selected clusters will be impacted by assembly bias (earlyformed halos have higher halo bias), which will in turnimpact cluster mass self-calibration and cause systematicerrors in the inferred cosmological parameters. There-fore, a careful calibration of the possible correlation be-tween cluster observables and formation history is neces-sary for measuring cosmological parameters, not only forthe systematics this correlation may involve, but also forthe potential extra signal that can be derived from thecorrelation.

In this work, we present the properties of subhalos—including the subhalo number statistics, the scatter ofsubhalo number, as well as the shape of subhalo distri-bution and velocity ellipsoid—and put our measurementsinto the context of earlier simulations. We pay special at-tention to the effect of resolution and subhalo selectioncriterion on the inferred subhalo properties. We find thatinsufficient resolution and a selection method affected bystripping tend to introduce the extra-Poisson scatter ofthe number of subhalos.

We present how formation time can impact the proper-ties of subhalos, including the subhalo mass fraction, thedominance of the main halo over its subhalos, and sub-halo numbers. One of the main findings in this work, aspresented in §5, is that the correlation between subhalonumber and formation time sensitively depends on howsubhalos are selected. If we use a subhalo selection cri-terion that is insensitive to the stripping of dark matterparticles, subhalo number and halo formation time arenot correlated. This result implies that observationallythe galaxy number may not be a good indicator of theformation time of a galaxy cluster at a given mass.

This paper is organized as follows. In §2.1, we brieflysummarize the simulations and halo catalogs used in thiswork. §3 describes the properties of subhalos, includingthe subhalo mass function, scatter of subhalo number, aswell as the shape of the distribution and velocity ellip-soid. §4 discusses how subhalo properties, including thesubhalo mass fraction, and the dominance of the mainhalo, can be affected by formation time. In §5, we in-vestigate how halo formation time and subhalo stripping

affect the number of subhalos. We conclude in §6.

2. HALO CATALOGS

The Rhapsody sample includes 96 cluster-size halos ofmass Mvir = 1014.8±0.05h−1M, re-simulated from a cos-mological volume of 1 h−1Gpc. Each halo was simulatedat two resolutions: 1.3× 108h−1M (equivalent to 81923

particles in this volume), which we refer to as “Rhap-sody 8K” or simply “Rhapsody”; and 1.0×109h−1M(equivalent to 40963 particles in this volume), which werefer to as “Rhapsody 4K.” The simulation parametersare summarized in Table 1.

All simulations in this work are based on a ΛCDMcosmology with density parameters Ωm = 0.25, ΩΛ =0.75, Ωb = 0.04, spectral index ns = 1, normalizationσ8 = 0.8, and Hubble parameter h = 0.7.

2.1. The Simulations

The development of Rhapsody can be summarized asfollows:

1. Selecting the re-simulation targets: We start fromone of the 1 h−1Gpc volumes named Carmen fromLasDamas (McBride et al.) and selecting halos ina narrow mass bin 1014.8±0.05.

2. Generating initial conditions: We use the initialcondition generator Music (Hahn & Abel 2011)and implement the second-order Lagrangian per-turbation theory.

3. Performing gravitational evolution: We use thepublic version of Gadget-2 (Springel 2005).

4. Identifying halos and subhalos: We use the adap-tive phase-space halo finder Rockstar (Behrooziet al. 2011a), which can achieve high completenessin identifying subhalos.

5. Constructing merger trees: We use the gravita-tionally consistent merger tree by (Behroozi et al.2011b).

We kindly refer the reader to Paper I for more detailson the simulations, as well as the values and variances ofvarious key properties of the main halos in Rhapsody.

2.2. Subhalo selection methods

Subhalos in cluster-size halos are closely related to thesatellite galaxies of galaxy clusters in observations. Foreach main halo in Rhapsody, we consider the subha-los inside its virial radius, Rvir. For each subhalo, wefocus on the maximum circular velocity of dark matterparticles associated with it, vmax, defined as

vmax =

√GM(< rmax)

rmax. (1)

This quantity is often used as the proxy for subhalo mass,since the mass of a subhalo is often ambiguous to deter-mine because a subhalo is embedded in the high over-density of its main halo. We focus on the values of vmax

at two different epochs:

• v0: the value of vmax measured at z = 0, a quantityrelated to the current subhalo mass.

Subhalos from Rhapsody Cluster Simulations 3

Type Name Mass Resolution Force Resolution Number of Particles Number of Particles[ h−1M] [ h−1 kpc] in Simulation in Each Targeted Halo

Full Volume Carmen 4.94×1010 25 11203 12K

Zoom-inRhapsody 4K 1.0×109 6.7 5.4Ma /40963(equiv.) 0.63Mb

Rhapsody 8K 1.3×108 3.3 42Ma / 81923(equiv.) 4.9Mb

aThe mean number of high-resolution particles in each zoom-in region.bThe mean number of high-resolution particles within the Rvir of each targeted halo.

Table 1Simulation parameters for the Rhapsody resimulations.

• vpk: the highest vmax value in a subhalo’s history,a quantity related to the highest subhalo mass inits entire history.

The parameter vpk is more closely related to the lumi-nosity and stellar mass of satellite galaxies than v0 (e.g.,Reddick et al. 2012). This is because the stellar compo-nent of a galaxy is denser and less easily stripped thandark matter. Even though a galaxy could lose dark mat-ter particles at the outskirts of its halo, the stars in itscore can remain intact for a longer time. Therefore, aquantity that is unaffected by stripping provides a betterproxy for the stellar mass of galaxies. In addition, using asubhalo abundance matching scheme, one can assign lu-minosity and stellar mass to subhalos based on their vpk.Subhalos selected with an value of vmax less impactedby stripping provide a better match to observed galaxystatistics. For example, Nagai & Kravtsov (2005) andConroy et al. (2006) showed that using vac (vmax whensubhalos are accreted) in selecting subhalos can betterreproduce the statistics of observed galaxies. Reddicket al. (2012) further showed that an abundance match-ing model using vpk provides an excellent fit to the galaxytwo-point correlation function and the conditional stel-lar mass function for galaxies in groups as measured fromSDSS data.

In several cases, in order to compare with results fromother simulations, we also use the mass of subhalos attwo different epochs, which are impacted differently bystripping:

• M0: the current subhalo mass, defined by the par-ticles bound to the subhalo according to the imple-mentation of Rockstar.

• Mpk: the highest mass in a subhalo’s entire history.

In general, as presented below, M0 vs. Mpk shows a sim-ilar trend to v0 vs. vpk, while v0 and vpk are usually lessaffected by stripping than M0 and Mpk.

Finite resolution in N-body simulations leads to the“overmerging” effect (e.g., Klypin et al. 1999): small sub-halos tend to fall below the resolution limit before theymerge with the central object so that the stellar com-ponent associated with subhalos is expected to survivelonger than subhalos. A detailed discussion of this reso-lution dependence and the associated completeness limitsof the subhalo population will be presented in a separatepaper (Wu et al., in preparation).

3. SUBHALO STATISTICS AND DISTRIBUTIONS AT Z = 0

In this section we focus on the statistics and distribu-tions of subhalos. In particular, we compare the variousselection criteria described in the previous section. Wealso compare the results from the 8K (main) and 4K(with 8 times lower mass resolution) samples to studythe impact of resolution.

3.1. Subhalo mass function

The mass function of subhalos has been found to followa power law for low mass subhalos and an exponentialcutoff for massive subhalos (e.g., Gao et al. 2004; An-gulo et al. 2009; Giocoli et al. 2010). In this work, weinvestigate the validity of this form for different subhaloselections.

Figure 1 shows the number of subhalos above a giventhreshold of v0 (left), vpk (middle), and M0 (right). Ineach panel, the blue/red lines correspond to subhaloswithin Rvir of Rhapsody 8K/4K halos, while the thingrey curves correspond to individual halos in Rhapsody8K. The black curves correspond to all halos and subha-los within 7 h−1Mpc around the center of the main halo,a region where re-simulated halos are well resolved5. Theblue dashed lines indicate the best-fit slopes of the dis-tribution functions of 8K; here we restrict our fit wellinside the power-law regime. Our M0 function slope isslightly shallower than De Lucia et al. (2004); Boylan-Kolchin et al. (2010) and Gao et al. (2012), which couldbe attributed to a slightly different mass definition. Ourv0 function is in good agreement with the galactic sizehalo results of Boylan-Kolchin et al. (2010) and Wanget al. (2012).

The 4K and 8K subhalo mass functions deviate froma power law at different values of v0, vpk, and M0. Thisindicates that different selection methods have differentcompleteness limits. In addition, the gray curves showthe significant scatter of subhalo distribution functionsfor different main halos. Given this large scatter, wewould like to know how well the number of subhalos indifferent bins are correlated. In Figure 2, we assign sub-halos into bins of 50 km/s using v0 (left) or vpk (right),starting from 50 km/s (we note that the first bin in eithercase is incomplete, and the last bin includes subhalos be-yond 250 km/s). We compare each pair of bins and findthat the subhalo number between bins shows only a mod-erate or weak correlation6. This means that a halo thatis rich in massive subhalos is not necessarily also rich inlow-mass subhalos.

3.2. Scatter of subhalo number

Boylan-Kolchin et al. (2010) and Busha et al. (2011)have shown that for galactic halos (M ≤ 1013.5h−1M),

5 To decide the size of the well-resolved ambient region aroundthe main halo, we compare halos in the re-simulated region (com-posed of high-resolution particles only) with those in the originalCarmen simulation. At 7 h−1Mpc, the re-simulated regions re-cover the halo population in the corresponding region in the Car-men simulation. In addition, at 7 h−1Mpc, the number of low-resolution particles is less than 4%, although it varies greatly fromhalo to halo and is sometimes 0%.

6 Throughout this work, we use rank correlation, which makesour results insensitive to outliers.

4 WU ET AL.

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8K, subs4K, subs8K, within 7 Mpc/h

Figure 1. The number of subhalos above a given threshold v0 (left), vpk (middle), and M0 (right). In each panel, the blue/red curvesrepresent the mean number of subhalos inside Rhapsody 8K/4K halos, while the black curves represent all halos and subhalos within7 h−1Mpc around the center of the main halo in the re-simulation. The thin gray curves in the background indicate the subhalos forindividual Rhapsody 8K halos. The blue dashed line indicates the slope of the 8K sample in the regime where subhalos are complete.

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Figure 2. Correlation between the number of subhalos in different bins. Subhalos are binned by v0 (left) or vpk (right), with bin size 50km/s, starting from 50 km/s. The subhalo numbers between different bins have moderate or weak correlation.

the distribution function of the number of subhalos (N)deviates from the Poisson distribution when 〈N〉 in-creases. In this section, we explore whether the samebehavior can be measured in our cluster-size halos; weare able to test this over a larger dynamic range thanhas been possible with existing simulations.

Figure 3 shows the Poissonness of the scatter using dif-ferent subhalo selection methods. The left panel corre-sponds to the ratio between the measured sample scatterσ =

√Var[N] and the Poisson scatter σPoisson =

√〈N〉.

The right panel presents the second moment of the sub-

halo number distribution

α =〈N(N − 1)〉1/2

〈N〉 (α = 1 for Poisson) (2)

Both quantities are measures of how the distribution de-viates from the Poisson distribution.

The x-axis corresponds to the mean number of sub-halos for a given selection criterion. This choice allowscomparison of the subhalo populations that are selecteddifferently. Each curve corresponds to a different selec-tion criterion, and the corresponding threshold in maxi-mum circular velocity (km/s) or mass (log10 h

−1M) ismarked on each curve. For a low threshold or high 〈N〉,

Subhalos from Rhapsody Cluster Simulations 5

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Figure 3. Scatter of the number of subhalos for various selection methods. The left panel corresponds to the ratio between the samplescatter and the Poisson scatter, while the right panel corresponds to the second moment of the distribution. The x-axis is the mean numberof subhalos for each selection method, and the threshold is marked on each curve. Here we compare various cases: (1) v0 vs. vpk (redvs. blue), (2) M0 vs. Mpk (purple vs. green), (3) 4k vs. 8K (purple dash vs. purple solid). In each pair, the former selection method leadsto extra non-Poisson scatter. This trend indicates that stripping of subhalos or insufficient resolution can lead to extra non-Poisson scatterand might lead to a scatter greater than the values in observations.

the scatter deviates significantly from the Poisson scat-ter. When the threshold increases, the scatter is closerto Poisson. In addition, α has a trend that is similar tothat of σ/σPoisson and only slightly deviates from unity.As stated in Boylan-Kolchin et al. (2010), slight devia-tions of α from unity can correspond to large deviationsfrom the Poisson distribution.

In Figure 3, the trends with different selection criteriacan be summarized as follows:

• v0 vs. vpk (red vs. blue): vpk selection is closer tothe Poisson distribution, indicating that strippingof subhalos can introduce extra non-Poisson scat-ter.

• M0 vs. Mpk (purple vs. green): Mpk selection iscloser to the Poisson distribution, which can alsobe understood with stripping.

• vpk vs. Mpk (blue vs. green): similar. Both proper-ties are usually computed before a subhalo’s infalland behave similarly.

• M0 4K vs. 8K (purple dash vs. solid): 8K is closerto the Poisson distribution, indicating that insuf-ficient resolution can introduce extra non-Poissonscatter.

3.3. Subhalo spatial distribution and kinematics

In Paper I, we have discussed the shape and velocityellipsoid of dark matter particles for the entire subhalo.In this section, we present analogous measurements forsubhalos, which are selected with the two selection crite-ria discussed above (v0 and vpk).

The shape parameters are defined through the momentof inertia with respect to the halo center

Iij = 〈rirj〉 (3)

where ri is the ith component of the position vector r ofa dark matter particle with respect to the halo center.The eigenvalues of Iij are sorted as λ1 > λ2 > λ3, andthe shape parameters are defined as: a =

√λ1, b =

√λ2,

c =√λ3. We present the dimensionless ratios b/a and

c/a. In addition, the triaxiality parameter is defined as

T =a2 − b2a2 − c2 (4)

T ≈ 1 (a > b ≈ c) indicates a prolate halo, while T ≈ 0(a ≈ b > c) indicates an oblate halo. Intermediate valuesof T correspond to triaxial halos.

Analogously, the velocity ellipsoid is defined as (e.g.,White et al. 2010):

σ2ij = 〈vivj〉 (5)

Sorting the eigenvalues of the velocity ellipsoid as λ1 >λ2 > λ3, one can again define dimensionless ratios λ2/λ1

and λ3/λ1 to describe the anisotropy of the velocity el-lipsoid.

The scatter of the velocity dispersion along the line ofsight can be calculated as

〈σ2los〉=

1

3(λ1 + λ2 + λ3) (6)

(δσ2los)

2 =4

45(λ2

1 + λ22 + λ2

3 − λ1λ2 − λ2λ3 − λ3λ1) . (7)

We use subhalos within Rvir selected above a giventhreshold without imposing weighting.

6 WU ET AL.

0.4 0.5 0.6 0.7 0.8 0.9 1.0Shape Parameters

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(d)

Figure 4. The distribution of the shape and velocity ellipsoid parameters for dark matter particles and subhalos inside the main halos.The upper panels correspond to shape parameters, and the lower panels correspond to the velocity ellipsoid. As shown in Panels (a) and(b), dark matter particles tend to be more prolate than subhalos. Panel (c) shows that the velocity ellipsoid of subhalos tend to be moreelongated. Panel (d) shows that the statistical error of line-of-sight velocity dispersion measurement is larger for subhalos than for darkmatter particles.

Subhalos from Rhapsody Cluster Simulations 7

Figure 4 shows the cumulative distribution functionof the shape and velocity ellipsoid parameters of darkmatter particles (solid), as well as of subhalos selectedwith v0 > 100 km/s (dotted) and subhalos selected withvpk > 150 km/s (dashed). Panels (a) and (b) correspondto the shape parameters. Panel (a) shows b/a and c/a.Subhalos selected with v0 tend to have a distribution thatis more spherical than both subhalos selected with vpk

and dark matter particles. Panel (b) shows the triaxialityparameter T . We find that the dark matter distributiontends to be more prolate than the subhalo distribution.

Panels (c) and (d) correspond to the velocity ellipsoidparameters. Panel (c) shows the ratios of the eigenvaluesof the velocity ellipsoid (λ2/λ1 and λ3/λ1). We find thatthe velocities of subhalos tend to be more anisotropicthan those of the dark matter particles (based on λ3/λ1).Panel (d) shows the fractional scatter in velocity disper-sion measurements obtained along different lines of sight.This scatter is higher for subhalos than for dark matterparticles. This trend can be understood based on thefindings of White et al. (2010): the motions of subhalostend to be anisotropic because they can retain their infallvelocities for a long time. In contrast, the merged andstripped material is dynamically older and contributesto a well-mixed isotropic velocity distribution. It wouldbe interesting to investigate whether and, if so, how thedifference between the velocity ellipsoids depends on en-vironment (see also Faltenbacher 2010, who see a de-pendence of subhalo kinematics on environment in theMillennium simulation). An analysis of the environmen-tal dependence of halo properties will be deferred to afuture paper.

4. CORRELATION BETWEEN SUBHALO PROPERTIESAND FORMATION HISTORY

In Figure 5, we present the correlation of eight proper-ties we measured for our sample. The first four propertiesinvolve the properties of subhalos, two properties involvethe formation time, and two properties involve the haloprofile.

4.1. Formation history parameters

As discussed Paper I, we adopt an exponential-plus-power law model with two parameters (McBride et al.2009)

M(z) =M0(1 + z)βe−γz (8)

−d lnM

dz≈γ − β when z << 1 . (9)

Thus, γ − β can be used as a measure for the late-timeaccretion rate.

We use z1/2 as the formation time proxy throughoutthe paper. As presented in Paper I, z1/2 has a broaderdistribution than formation time proxies based on thefit and can better account for the scatter of halo prop-erties. In addition, we note that throughout this work,using formation time proxies based on fitting functions(as those studied in Paper I) leads to the trends that aresimilar to using z1/2.

In addition, we include the scale factor when the lastmajor merger occurs, almm.

We note that the formation history is often reflectedby the halo density profile, which we studied in detail in

Paper I. For the completeness of our correlation analy-sis, we include one of the density profile parameters, thehalo concentration defined by the Navarro–Frenk–White(NFW) profile (Navarro et al. 1997)

ρ(r)

ρcrit=

δc(r/rs)(1 + r/rs)2

(10)

The concentration parameter is defined as

cNFW =Rvir

rs. (11)

4.2. Subhalo mass fraction

The subhalo mass fraction can be defined as

fsub(Mth) =1

Mmain

∑

Msub>Mth

Msub (12)

(as studied in, e.g., De Lucia et al. 2004; Shaw et al.2006).

Here we choose Mth = 1010h−1M, which correspondsapproximately to our completeness limit (see Fig. 1). Wenote that the correlations presented below are insensitiveto the value of Mth.

The subhalo mass fraction can be used as an indica-tion of a recent major merger event. For example, if amassive halo accreted onto the main halo recently, it re-tains most of its mass and contributes to a large fsub.Therefore, we expect that fsub is correlated with haloformation time and can be used as an indicator of thestate of relaxedness.

In Figure 5, we present various subhalo properties andtheir correlation (or lack thereof) with several formationtime proxies. The third row and column correspond tofsub. The figure shows that fsub is strongly correlatedwith the formation time proxy z1/2, as well as a redshiftproxies that correlated with the late-time accretion rateγ − β and halo concentration cNFW . That is, halos ofhigher fsub tend to be late-formed, with high late-timeaccretion rate and low concentration.

The subhalo mass fraction is a also quantity of obser-vational interest. For example, it can be inferred fromgravitational lensing (e.g., Dalal & Kochanek 2002; Veg-etti et al. 2012; Fadely & Keeton 2012). An accuratemodeling of the subhalo mass fraction is essential for thestudy of the lensing flux ratios (e.g., Xu et al. 2009). Forthe massive systems considered here, we find that fsubis strongly correlated with the mass of the most massivesubhalo. Typical values of fsub counting only the mostmassive halo are 30% of the total fsub counting all halosmore massive than ∼ 10−5 of the main halo, but the cor-relation coefficient between the two values is ∼ 0.9. Thiscorrelation also implies that fsub is correlated with haloformation time and concentration, which is shown explic-itly in Figure 5. Because strong lensing clusters select asample of halos that have higher than average concentra-tions, it is important to take this correlation into accountwhen interpreting measures of the substructure fractionfrom strong lensing.

4.3. Mass contributed by merged subhalos

In the previous subsection, we addressed the mass con-tributed by present subhalo population. In this section,

8 WU ET AL.

60 100

N100(V0)

0.0

5.0

10.0

15.0

20.0

P

60 80 100 120

N100(Vpk)

0.73

0.1 0.3 0.5

fsub

0.46

1 2

Vmainpk /V1st sub

pk

-0.30

0.0 0.4 0.8 1.2

almm

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0.0 0.4 0.8 1.2

z1/2

-0.38

1 2 3γ − β

0.25

3 5 7 9

cNFW

60

100

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0)

-0.36

70 90 110 130

N100(Vpk)

0.0

5.0

10.0

15.0

20.0

P

0.18 -0.03 -0.22 -0.02 -0.04 60

80

100

120

N100(V

pk)

-0.09

0.1 0.3 0.5

fsub

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5.0

10.0

15.0

20.0

P

-0.78 0.34 -0.74 0.60 0.1

0.3

0.5

f su

b

-0.62

1.5 2.5

Vmainpk /V1st sub

pk

0.0

5.0

10.0

15.0

20.0

25.0

P

-0.48 0.73 -0.54 1

2

3

Vm

ain

pk/V

1st

sub

pk

0.59

0.2 0.6almm

0.02.04.06.08.0

10.012.014.016.0

P

-0.60 0.48 0.0

0.4

0.8

1.2

alm

m

-0.42

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0.0

5.0

10.0

15.0

P

-0.690.0

0.4

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z 1/2

0.70

1 2 3

γ − β0.0

5.0

10.0

15.0

P

1

2

3

γ−β

-0.58

3 5 7cNFW

0.0

5.0

10.0

15.0

P

Figure 5. Correlation (or lack thereof) between the subhalo content and formation time proxies. Top two rows show the subhalo numberfor subhalos selected on v0 and vpk. In each case the threshold is chosen to have an average of 100 subhalos per host. The following rowsshow various halo quantities that are related to the formation history of the halo. All correlations are weaker with subhalos selected onvpk than on v0.

Subhalos from Rhapsody Cluster Simulations 9

10−4 10−3 10−2 10−1 100

µ = Msub/Mmain0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

∑>µµ

merged with host (Mmerg)merged + surviving (Mac)surviving (M0)

Figure 6. The contribution of mass to the main halo from subha-los above a certain mass ratio µ = Msub/Mmain

0 . The blue curvecorresponds to the contribution from subhalos that have mergedwith the main halo, and we use the subhalo mass right before itmerges and gets lost Msub

merging ; the shaded area corresponds to the

standard deviation of the sample. The red curve includes mergedsubhalos and those that are surviving; for surviving subhalos weuse their mass at accretion Mac. The black curve corresponds tosubhalos that are still surviving today, for which M0 is used, andis equivalent to fsub(> µ).

we would like to address the mass was brought into themain halo by all merging events in a halo’s history. Fig-ure 6 shows the contribution to main halo mass frommerged subhalos. This has also been explored by, e.g.,Berrier et al. (2009) for lower mass systems. The x-axiscorresponds to the ratio of subhalo mass to main halomass, µ = Msub/Mmain

0 , and the y-axis corresponds tothe fraction of main halo mass contributed by subhalosabove a given µ.

Here we consider two types of subhalos. The first typeis those subhalos that have merged into the main haloand can no longer be identified; for this type of subhalo,we use its last measured mass before it lost its identity.These subhalos are represented by the blue curve, andthe blue shaded region corresponds to the standard de-viation. Our results indicate that, on average, 40% ofthe main halo’s mass comes from merged subhalos withµ > 10−4; however, this number varies greatly from haloto halo.

The second type is the subhalos that still survive to-day (i.e. can be identified by the halo finder at z = 0).For this type, we use its mass when it accretes onto themain halo, Mac. Since subhalos tend to lose a signifi-cant amount of mass due to tidal stripping of the mainhalos, using Mac ensures that we include the mass thatonce belonged to subhalos but later got stripped by themain halo. We do not explicitly count the subhalos thatmerge into other subhalos because their masses have al-ready been included in the surviving subhalos. The redcurve corresponds to the sum of the first and the secondtypes. We find that more than 70% of the mass can be

attributed to halos with µ > 10−4 that were accretedonto the main halo.

Finally, the black curve shows the subhalos that aresurviving today and their contribution to main halo. Weuse the current mass of these subhalos, M0, and thisquantity is equivalent to the subhalo mass fraction fordifferent mass thresholds, fsub(> µ). Subhalos with µ >10−4 constitute a little less than 20% of the mass of themain halo; this fraction is dominated by the mass in themost massive subhalo.

4.4. Dominance of the main halo

In this section, we study the difference between a mainhalo and its largest subhalo, a quantity motivated bythe definition of the so-called “fossil groups.” Obser-vationally, these systems are defined as having a largemagnitude gap between the brightest and second bright-est galaxies, in addition to being X-ray luminous (e.g.,Tremaine & Richstone 1977; Jones et al. 2003). Fossilsystems are often considered as a population of galaxygroups that have assembled in an early stage of the uni-verse and have not undergone many mergers at latertimes.

Since predicting the optical and X-ray properties of ourhalos is beyond the scope of this paper, we alternativelydefine a related property, the dominance of the centralgalaxy hosted in the main halo:

D =vmain

pk

v1st subpk

. (13)

We investigate the degree of the dominance of the mainhalo, and the extent to which this dominance correlateswith a halo’s formation history. We use the first subhalo(the subhalo with highest vpk) in our definition but notethat using the second subhalo leads to the same trend.In Figure 5, the fourth column and row show that Dis correlated with the formation time, late-time accre-tion, concentration, deviation from NFW, and subhalomass fraction. We also note that all our main halos havesimilar vmain

pk ; therefore, the BCG dominance is almost

completely determined by v1st subpk .

The trends observed in Figure 5 can be understoodas follows: Since v1st sub

pk indicates the maximum of thesubhalo mass that accretes to a main halo, a main halowith a low v1st sub

pk has fewer massive subhalos accret-

ing onto it (this is also reflected by its low fsub). Withrelatively fewer incoming subhalos, to achieve the samemass today, these halos must have obtained most of theirmass at early time and have undergone slow accretion atlate time, thus leading to the low γ − β and the highconcentration that we find.

While we were preparing this manuscript, we learnedabout the related work of Hearin et al. (2012), who havestudied how the “magnitude gap” of the two brightestcluster members, which is analogous to our dominanceparameter D. They have found that for SDSS groups ofgiven velocity dispersion, the magnitude gap is correlatedwith the richness, and this correlation can in turn help re-duce the scatter in the mass inferences using optical masstracers. In our case, for halos of the same mass, subhalosselected with vpk is not correlated with D; this lack ofcorrelation indicates that the direct comparison between

10 WU ET AL.

observations (velocity dispersion, luminosity, gap) andsimulations (mass, vpk, D) still involves uncertainties.

4.5. The curious case of Halo 572: An outlier and afossil cluster

In the Rhapsody sample, we found one peculiarhalo—Halo 572—which is a prominent outlier in forma-tion time (highest z1/2) and occupies the tail of manyhalo properties as well as the corner of several scatterplots (marked as stars in Figure 5). It has unusuallyhigh cNFW (2.7σ; deviation from the mean in unit ofthe standard deviation) and BCG dominance (3.2σ). Italso has one of the lowest late-time accretion rate γ − β(2.7σ), fsub (1.6σ), and subhalo numbers selected withseveral different criterions. This halo obtained most ofits mass at very early time and nearly stopped accretingmass at late time, leading to these extreme properties.Images of the evolution of Halo 572 are shown in Figure3 of Paper I, where it is evident that this halo had anatypical formation history. We find that Halo 572 doesnot live in an atypical environment on large scales.

The high dominance of the main halo indicates that,if such a halo is observed, it would likely have a large lu-minosity gap between the brightest and second brightestgalaxies, and its high concentration will make it X-rayluminous. Therefore, we expect that this halo will host acluster that satisfy the criteria of a “fossil.” In addition,it is a “real” fossil cluster in the sense that it has unusu-ally early formation history. Studies of fossil groups inboth simulations and observations have come to a rangeof conclusions, with debate about whether fossil groupshave distinct assembly history or are merely an inter-mediate state in galaxy formation (see, e.g., Cui et al.2011; Sales et al. 2007; and references therein). Halo 572presents a case of distinctively early formation historyand the consequential properties. It is highly probablethat “real fossils” exist in the universe but are very rare;thus, they require more stringent selection criteria to bedistinguished from some transient states of cluster for-mation.

5. CORRELATION BETWEEN FORMATION TIME ANDSUBHALO NUMBER: IMPACT OF SUBHALO

SELECTION

For halos of different masses, the variance of halo prop-erties is often correlated with halo mass; however, at afixed halo mass, significant scatter remains. The diver-sity of halo properties can sometimes be attributed todifferences in the formation history. In this section, weinvestigate in detail the correlation between formationtime and subhalo number (as previously explored by e.g.,Zentner et al. 2005; Wechsler et al. 2006; Giocoli et al.2010). We show that this correlation is mainly caused bysubhalo stripping and insufficient resolution. When wemodel the satellite galaxy population in clusters, how-ever, subhalo stripping becomes less relevant, and thiscorrelation no longer exists.

Zentner et al. (2005) found that early-formed halostend to have fewer subhalos. In their study, subhalosare selected with a threshold on v0 (vmax at z = 0).This trend has been explained by the fact that in early-formed halos, subhalos tend to accrete at early time andare more likely to be destroyed, which leads to a low num-ber of subhalos. However, we find that this correlation

is strongly dependent on how subhalos are selected. Inwhat follows, we explore the dependence of this correla-tion on various subhalo selection criteria and understandthe trend by investigating the accretion and stripping ofsubhalos.

Figure 7 demonstrates how subhalo selection based onv0 or vpk can lead to different correlation between for-mation time and subhalo number. The left panel corre-sponds to selecting subhalos with v0 > 100 km/s, and theright panel corresponds to vpk > 150 km/s. Each pointcorresponds to a main halo in our sample. The x-axescorrespond to the number of subhalos under either selec-tion criterion, and the y-axes correspond to the formationtime proxy z1/2.7 When subhalos are selected based onv0 (left panel), formation time and subhalo number aresignificantly anti-correlated, as shown in, e.g., Zentneret al. (2005). However, when subhalos are selected basedon vpk (right panel), this anti-correlation no longer existsin our sample.

This lack of anti-correlation can be understood as fol-lows. When subhalos are selected with a given thresholdof v0, the stripping of subhalos directly impacts the sub-halo number: for a halo that assembled earlier, its sub-halos experience stripping for a longer time and with ahigher intensity (because of the high halo concentration),and the subhalos’ masses and v0 tend to be greatly re-duced. Therefore, fewer subhalos remain above the v0

threshold. On the other hand, when we select subhalosusing vpk, the stripping of subhalos does not directly im-pact the subhalo number, as long as those subhalos arestill identifiable.

To support our argument above, we investigate theevolution of the subhalo population for these two subhaloselection criteria. Figure 8 shows subhalo number as afunction of the scale factor a (we choose to plot in log ato emphasize the late-time behavior). As in the previ-ous figure, the left/right panel corresponds to the v0/vpk

selection. In both panels, we plot the subhalo numberevolution for the highest z1/2 quartile (blue) and lowestz1/2 quartile (red).

In the left panel of Figure 8, we can see that for early-formed halos (blue), the subhalo accretion rate is high atearly time but suddenly declines after a ≈ 0.6. On theother hand, for the late-formed halos (red), their subhaloaccretion rate is high at late time. At a ' 1, the early-formed halos have fewer subhalos than late-formed halos.In the right panel, although the early-formed halos havedeclined in subhalo accretion rate at late time, their sub-halo number still grows at late times, and their subhalonumbers are similar to late-formed halos at a ' 1.

The different trends in both panels can be attributedprimarily to the stripping of subhalos. Figure 9 showsthe cumulative distribution of the fractional change ofvmax of subhalos

δvmax =vpk − v0

vpk(14)

This quantity can be used as a measure of the amountof stripping experienced by subhalos. Higher δvmax indi-cates that a subhalo has experienced stronger or longer

7 We note these two criteria correspond to roughly the samenumber of subhalos.

Subhalos from Rhapsody Cluster Simulations 11

40 50 60 70 80 90 100 110 120 130N(> V0)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

z 1/2

V0 >100 km/s; cor = -0.37, p = 0.00021

40 50 60 70 80 90 100 110 120N(> Vpk)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

z 1/2

Vpk >150 km/s; cor = -0.017, p = 0.87

Figure 7. The impact of subhalo selection on the correlation between subhalo number and formation time z1/2. The left panel corresponds

to selecting subhalos using v0 > 100 km/s, while the right panel corresponds to vpk > 150 km/s. Although the v0 selection presentssignificant anti-correlation between subhalo number and formation time, the vpk selection presents no such correlation. This trend can beexplained by the stripping of subhalos, as demonstrated by the following two figures.

10−1 100

a0

20

40

60

80

100

120

〈N〉

Selection: V0 > 100 km/s

High z1/2 quartileLow z1/2 quartile

10−1 100

a0

20

40

60

80

100

120

〈N〉

Selection: Vpk > 150 km/s

High z1/2 quartileLow z1/2 quartile

Figure 8. The evolution of subhalo number, split by quartiles of z1/2. Left: subhalos with v0 > 100 km/s; right: subhalos with

vpk > 150km/s. When subhalos are selected with v0, the subhalo number of early-formed (blue) and late-formed (red) halos split at z = 0;however, when subhalos are selected with vpk, there is no clear split of halo number at z = 0. This trend is reflected by the difference inthe correlation seen in Figure 7.

12 WU ET AL.

0.0 0.2 0.4 0.6 0.8 1.0δV = (Vpk − V0)/Vpk

0.0

0.2

0.4

0.6

0.8

1.0

P(>

δV

)

Vpk > 150 km/s

High z1/2 quartile

Low z1/2 quartile

Dashed: 4K

Figure 9. The cumulative distribution function of δvmax = (vpk−v0)/vpk, an indication of the amount of stripping experienced bysubhalos. We again split the main halo by the formation timez1/2. For early-formed halos (blue), their subhalos have on average

higher δvmax (stronger stripping) than late-formed ones (red). Ifsubhalos are selected with v0, highly-stripped subhalos tend to fallbelow the threshold, leading to a low subhalo number. This canexplain the correlation seen in the v0 selection in Figures 7 and 8.

102

〈N(> Vcut or > Mcut)〉

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

Cor(

N,z 1/2)

60 km/s

100 km/s

100 km/s

140 km/s

10.4

10.8

11.2

10.811.211.6

V0 (Dashed: 4K)Vpk (Dashed: 4K)M0

Mpk

Figure 10. The correlation between subhalo number and z1/2, fordifferent subhalo selection methods. When subhalos are selectedwith v0 or M0 (red and purple), an anti-correlation exists for allthresholds; when subhalos are selected with vpk or Mpk (blue andgreen), the anti-correlation is greatly reduced or non-existent. Inaddition, the comparison between 8K and 4K (solid and dashedof the same color) sample shows that the anti-correlation can beenhanced by insufficient resolution.

stripping and has lost more mass. For early-formed halos(blue), subhalos on average have higher δvmax , indicatingthat these subhalos experience more stripping and theirvmax is reduced more. As a result, if we select subhalosusing v0, we tend to exclude subhalos that have expe-rienced more stripping. These subhalos will however beincluded if we select subhalos using v0.

Therefore, the correlation between formation time andsubhalo number seen in a selection on v0 can be at-tributed to the exclusion of highly-stripped subhalos.Since subhalos selected with v0 have less observationalrelevance than those selected with vpk, our results implythat cluster richness is unlikely to be correlated with theformation time of the halo in observations.

So far we have been using two specific selection thresh-olds for v0 and vpk. Here, we investigate how our re-sults depend on the selection threshold. In Figure 10,we present the correlation between subhalo number andhalo formation redshift, Cor(N, z1/2), where N is thesubhalo number above some selection thresholds. Wediscuss four selection methods: v0, vpk, M0, and Mpk.For each selection threshold, we compute the mean num-ber of subhalos, 〈N〉. Using 〈N〉 as the x-axis allows usto put these curves on the same figure.

In Figure 10, the different magnitudes of correlationcan easily be seen. Here we compare four pairs of selec-tion methods:

• v0 vs. vpk (red vs. blue): the former has strongercorrelation with z1/2 due to subhalo stripping, asdiscussed above.

• M0 vs. Mpk (purple vs. green): the former hasstronger correlation, for the same reason as above.

• v0 vs. v0 4K (red solid vs. dashed): the latter hasstronger correlation, indicating that an unphysicalcorrelation can be introduced by insufficient reso-lution.

• vpk vs. vpk 4K (blue solid vs. dashed): the latterhas stronger correlation, indicating that using vpk

does not mitigate the impact of resolution. Forother quantities, comparisons between 8K and 4Kshow the same trend.

Since vpk is more relevant for observations than v0, thelack of correlation we find when selecting by vpk indicatesthat, observationally, the formation time of a galaxy clus-ter is unlikely to be inferred from the number of galaxiesalone. Thus, richness-selected galaxy clusters are un-likely to be biased in terms of their formation time, im-plying that the effect of assembly bias may be negligible(Wu et al. 2008).

We now return to the discussion of Figure 5. Therewe have shown that various observables, including thesubstructure fraction, the BCG dominance, and the con-centration, are highly correlated with formation time andthe amount of late-time accretion. Each of these are alsocorrelated with the number of subhalos selected by thecurrent maximum circular velocity, vmax. However, thesecorrelations largely disappear when selecting substruc-tures based on vpk, which is the property expected to bemore strongly correlated with galaxy luminosity or stel-lar mass (Conroy et al. 2006; Reddick et al. 2012). This

Subhalos from Rhapsody Cluster Simulations 13

reduces the likelihood that these observables provide ad-ditional mass information for richness-selected samplesof galaxy clusters.

6. SUMMARY AND DISCUSSION

We present the key properties of subhalos in the Rhap-sody cluster re-simulation project, a sample of 96 halosof mass 1014.8±0.05h−1M resolved with approximately5× 106 particles. We focus on how the impact of forma-tion time on subhalo properties; this impact is furthercomplicated by subhalo selection criteria and resolution,which need to be carefully taken into account to make ob-servationally relevant inferences. Our findings are sum-marized as follows:

1. Subhalo statistics: In §3.1, we have shown thesubhalo mass function for several subhalo massproxies, v0, vpk, and M0; their cumulative massfunctions have slopes -3.0, -3.1, and -0.87, respec-tively. We found that for a given halo, the numbersof large and small subhalos are only moderatelycorrelated with each other.

2. Scatter of subhalo number: In §3.2, we com-pare subhalos selected with v0 and vpk, and sub-halos in the main sample and the 4K sample. Wefound that subhalo numbers of v0 selection and the4K sample tend to show stronger non-Poisson scat-ter, indicating subhalo stripping and insufficientresolution can lead to extra non-Poisson scatter.

3. Shape of subhalo distribution and velocityellipsoid: In §3.3, we have compared these quan-tities measured from subhalos selected with v0 andvpk, as well as dark matter particles. For shapeof distribution, we find that dark matter particlestend to be more prolate than subhalos. For veloc-ity ellipsoid, we show that subhalos present higherline-of-sight velocity dispersion.

4. Formation history and subhalo properties:We have quantified the correlations between vari-ous subhalo properties and halo formation historyin §4 and in Figure 5. The fraction of mass in sub-halos and the dominance of the main halo are bothhighly correlated with formation time, late-timeaccretion rate, and concentration. These correla-tions have important implications for interpretinglensing-selected clusters and X-ray selected clus-ters.

5. A fossil cluster: Our sample includes a peculiaroutlier, Halo 572 (presented in §4.5), with excep-tionally high formation redshift, concentration, anddominance of the central halo; and exceptionallylow late-time mass accretion rate, subhalo number,and subhalo mass fraction. This finding indicatesthat halos of distinct formation history are likelyto be distinguishable observationally, if stringentselection criteria are used.

6. Subhalo stripping: In §5, we have demonstratedthat the subhalo number, when selected using vpk

(a more observationally relevant property), doesnot correlate with formation time. This is in con-trast to subhalos selected with v0, which has been

done in the past and shows that early formed haloshave fewer subhalos. We demonstrate that the cor-relation with v0 can be attributed to subhalo strip-ping and insufficient resolution and is thus largelyoverestimating the effect to be expected for clustersatellite galaxies. This finding implies that the for-mation history of clusters is unlikely to be directlyinferred from the number of their satellite galaxies.

In this work, we have demonstrated how various sub-halo properties can be affected by different selection cri-teria. This dependence on selection criteria is also relatedto the completeness of subhalos. In a separate paper, wewill address the issue of the completeness of subhalosin detail by comparing several simulations with differ-ent resolutions and by comparing with observations. Wewill also investigate the impact of completeness on themeasured velocity dispersion of subhalos.

We thank Eduardo Rozo for many helpful sugges-tions. This work was supported by the U.S. Departmentof Energy under contract numbers DE-AC02-76SF00515and DE-FG02-95ER40899, and by Stanford Universitythrough a Gabilan Stanford Graduate Fellowship to HWand a Terman Fellowship to RHW. Additional supportwas provided by SLAC-LDRD-0030-12. This work usesdata from one of the LasDamas simulations, which areavailable at http://lss.phy.vanderbilt.edu/lasdamas/ .

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