SLAC-PUB-15331

21 Pages·2012·5.1 MB·English

SUBMITTEDTOAPJDECEMBER11,2012 SLAC-PUB-15331
PreprinttypesetusingLATEXstyleemulateapjv.11/10/09
THEROCKSTARPHASE-SPACETEMPORALHALOFINDERANDTHEVELOCITYOFFSETSOFCLUSTERCORES
PETERS.BEHROOZI,RISAH.WECHSLER,HAO-YIWU
KavliInstituteforParticleAstrophysicsandCosmology,DepartmentofPhysics,StanfordUniversity;
DepartmentofParticlePhysicsandAstrophysics,SLACNationalAcceleratorLaboratory;Stanford,CA94305
SubmittedtoApJDecember11,2012
ABSTRACT
Wepresentanewalgorithmforidentifyingdarkmatterhalos,substructure,andtidalfeatures.Theapproach
isbasedonadaptivehierarchicalrefinementoffriends-of-friendsgroupsinsixphase-spacedimensionsandone
timedimension,whichallowsforrobust(grid-independent,shape-independent,andnoise-resilient)trackingof
substructure; as such, it is named ROCKSTAR (RobustOverdensityCalculation using K-Space Topologically
Adaptive Refinement). Our method is massively parallel (up to 105 CPUs) and runs on the largest current
simulations (>1010 particles) with high efficiency (10 CPU hours and 60 gigabytes of memory required per
billion particles analyzed). A previous paper (Knebe et al. 2011) has shown ROCKSTAR to have excellent
recoveryofhaloproperties;weexpandonthesecomparisonswithmoretestsandhigher-resolutionsimulations.
Weshowasignificantimprovementinsubstructurerecoverycomparedtoseveralotherhalofindersanddiscuss
thetheoreticalandpracticallimitsofsimulationsinthisregard. Finally,wepresentresultswhichdemonstrate
conclusivelythatdarkmatterhalocoresarenotatrestrelativetothehalobulkorsubstructureaveragevelocities
andhavecoherentvelocityoffsetsacrossawiderangeofhalomassesandredshifts.Formassiveclusters,these
offsets can be up to 350 km s- 1 at z=0 and even higher at high redshifts. Our implementation is publicly
availableathttp://code.google.com/p/rockstar.
Subjectheadings:darkmatter—galaxies:abundances—galaxies:evolution—methods:N-bodysimulations
1. INTRODUCTION cretionhistories,seee.g.Wechsleretal.2002;Springeletal.
In the paradigm of Lambda Cold Dark Matter (ΛCDM), 2005; Faltenbacher et al. 2005; Allgood et al. 2006; Harker
etal.2006;Tweedetal.2009;Wetzeletal.2009).
the majority of the matter density of the universe does not
This level of uncertainty is unacceptable for current and
couple to electromagnetic fields, leaving it detectable only
future surveys, including data expected to come from the
throughitsgravitationalandpossiblyweakforceinteractions.
Baryon Oscillation SpectroscopicSurvey(BOSS), Dark En-
Nonetheless,theeffectsofdarkmatteronthevisibleuniverse
ergy Survey (DES), BigBOSS, Panoramic Survey Tele-
arespectacular,asthesteepgravitationalpotentialsofbound
scope and Rapid Response System (Pan-STARRS), Ex-
darkmatterhaloschannelbaryonstogether,formingthebirth-
tended Roentgen Survey with an Imaging Telescope Ar-
placesofvisiblegalaxies. Inthismodel,thelocations,sizes,
ray (eROSITA), Herschel, Planck, James Webb Space Tele-
and merging histories for galaxies are thus intricately con-
scope(JWST),andLargeSynopticSurveyTelescope(LSST)
nectedtothegrowthofbounddarkmatterstructures.
(Schlegel et al. 2009; The Dark Energy Survey Collabora-
Testing this model in detail requires extensive computer
tion 2005; Schlegel et al. 2011; Chambers 2006; Predehl
simulations,asthecomplicatednonlinearevolutionofstruc-
et al. 2010; Pilbratt et al. 2010; The Planck Collaboration
ture growth cannotbe fully evaluated by hand. The simula-
2006;Gardneretal.2006;LSSTScienceCollaborationsetal.
tions generally follow the evolution of a set of dark matter
2009),inordertofullyrealizetheconstrainingpowerofthese
particlesand outputthe positionsand velocitiesof the parti-
observations.Indeed,derivedquantitiessuchasthehalomass
cles at several discrete timesteps. These outputs must then
functionandautocorrelationfunctionsmustbeunderstoodat
bepostprocessedtodeterminethelocationsandpropertiesof
the one percentlevel in order for theoreticaluncertaintiesto
bounddarkmatterstructuresalsoknownas“halos”—namely,
beatthesamelevelasstatisticaluncertaintiesinconstraining,
thelocationsandpropertieswhichinfluencetheformationof
e.g.,darkenergy(Wuetal.2010;Cunha&Evrard2010,for
visiblegalaxies. Thispostprocessing(“halo-finding”)neces-
example).Althoughtheremaybealimittotheaccuracypos-
sarily involves both ambiguity and imprecision—ambiguity
siblewithdarkmattersimulationsalone,giventheimpactof
in the definitions (e.g., the center of a bound halo) and im-
baryonsondarkmatterhaloprofiles(Staneketal.2009)and
precisionindetermininghalopropertiesduetolimitedinfor-
the power spectrum (Rudd et al. 2008), different halo find-
mation (e.g., for halos consisting of only a few dark matter
ers running on the same simulation demonstrate that a large
particles,orfordeterminingparticlemembershipintwoover-
fractionof this uncertaintyis still due to the processof halo
lappinghalos). Currently,as a consequence,essential statis-
findingitself(Knebeetal.2011).
ticalmeasures(e.g.,thenumberdensityofhaloswithagiven
As previously mentioned, some of these uncertainties are
mass)areknowntoatbest5-10%evenforaspecificcosmol-
duetolimiteduseofinformation. Considerableprogresshas
ogy(Tinkeretal.2008). Moreover,halopropertiesarerarely
been made since the first generation of position-space halo
checked for consistency over multiple timesteps, which can
finders(Davis et al. 1985;Lacey & Cole 1994), which used
beaseriousproblemforrobustmodelingofgalaxyformation
only particle locations to determine bound halos. Currently,
theories(seehoweverBehroozietal.2011;thishasalsobeen
themostadvancedalgorithmsareadaptivephase-spacefind-
addressedinspecificcontextswhencreatingmergertrees,for
ers (e.g., Maciejewski et al. 2009; Ascasibar et al. 2010),
exampleimprovingsubhalo tracking and smoothermass ac-
whichmakeuseofthefullsixdimensionsofparticlepositions
Work supported in part by US Department of Energy contract DE-AC02-76SF00515.
2 BEHROOZI,WECHSLER,&WU
and velocities. At the same time, an often-overlooked(with 2.1. SummaryofPreviousApproaches
the possible exception of Tweed et al. 2009) aspect of halo
Previously published approaches to halo finding may be
findingistheextrainformationstoredinthetemporalevolu-
classified,withfewexceptions,intotwolargegroups. Spher-
tionofboundparticles. Inthis paper,we detailan advanced
ical overdensity (SO) finders, such as Amiga’s Halo Finder
adaptive phase-space and temporal finder which is designed
(AHF;Knollmann&Knebe2009),AdaptiveSphericalOver-
tomaximizeconsistencyofhalopropertiesacrosstimesteps,
density Halo Finder (ASOHF; Planelles & Quilis 2010),
ratherthanjustwithinasinglesimulationtimestep. Together
Bound Density Maxima (BDM; Klypin et al. 1999), SO
withacompanionpaper(Behroozietal.2011),whichdetails
(Lacey&Cole1994;Jenkinsetal.2001;Evrardetal.2002),
theprocessofcomparingandvalidatinghalocatalogsacross
parallel SO (pSO; Sutter & Ricker 2010), Voronoi Bound
timestepstocreategravitationallyconsistentmergertrees,our
Zones(VOBOZ;Neyrincketal.2005),andSplineKernelIn-
combinedapproachis the first to use particle informationin
terpolativeDenmax(SKID;Stadel2001)proceedbyidentify-
seven dimensions to determine halo catalogs, allowing un-
ingdensitypeaksintheparticledistributionandthenadding
precedentedaccuracyandprecisionindetermininghaloprop-
particles in spheres of increasing size around the peaks un-
erties.
til the enclosed mass falls below a predetermined density
Furthermore, in contrast to previous grid-based adaptive
threshold (a top-down approach). Friends-of-friends (FOF)
phase-spaceapproaches,oursisthefirstgrid-independentand
andHOP-based(Eisenstein&Hut1998)halofinders,suchas
orientation-independentapproach;itisalsothefirstpublicly-
FOF,SUBFIND,theLANLhalofinder,parallelFOF(pFOF),
availableadaptivephase-spacecodedesignedtorunonmas-
Ntropy-fofsv, and AdaptaHOP (Davis et al. 1985; Springel
sivelyparallelsystemsontheverylargesimulations(>1010 et al. 2001; Habib et al. 2009; Rasera et al. 2010; Gard-
particles)whicharenecessarytoconstrainstructureformation
neretal.2007;Tweedetal. 2009),collectparticlestogether
acrosstherangeofscalesprobedbycurrentandfuturegalaxy
which fall above a certain density threshold and then, if so
observations. Finally, we remark that our halo finder is the
designed, search for substructure inside these particle col-
first which is successfully able to probesubstructuremasses
lections (a bottom-up approach). Phase-space finders typi-
downto theverycentersofhosthalos(seealso Knebeetal.
cally extend these two approachesto include particle veloc-
2011),whichisessentialforafullmodelingofgalaxystatis-
ity information, either by calculating a phase-space density,
ticsandwillenablefuturestudiesoftheexpectedbreakdown
such as the Hierarchical Structure Finder (HSF; Maciejew-
betweenhalopositionsandgalaxypositionsduetotheeffects
ski et al. 2009) and the Six-DimensionalHierarchical Over-
ofbaryoninteractionsattheverycentersofgalaxies.
density Tree (HOT6D; Ascasibar et al. 2010) or by using a
Throughout,wehavepaidcarefulattentionnotonlytothe
phase-spacelinkinglength,asdoesSix-DimensionalFriends-
basictaskofassigningparticlestohalos,butalsotothepro-
of-Friends(6DFOF;Diemandetal.2006).
cess of estimating useful properties from them to compare
There are three notable exceptions to these algorithms,
with observations. While in many cases galaxy surveys are
namelytheORIGAMIhalofinder(discussedinKnebeetal.
not able to probe halo properties to the same precision as
2011),theHierarchicalBound-Tracingalgorithm(HBT;Han
halo finders in simulations, one significant counterexample
etal.2011),andSURV(Giocolietal.2010). ORIGAMIop-
exists. It is a common practice especially for halo finders
erates by examiningphase-space shell crossingsfor the cur-
based on the friends-of-friends algorithm (e.g., Davis et al.
rent particle distribution as compared to the initial particle
1985; Springel et al. 2001; Habib et al. 2009; Rasera et al.
distribution; shells which have crossed along three dimen-
2010; Gardner et al. 2007; see also §2.1) to calculate halo
sionsare consideredto be halos(as opposedto shells which
velocitiesbyaveragingallhaloparticlevelocitiestogetherto
have crossed along one or two dimensions, which would be
find a bulk velocity. Examinationof the differencebetween
consideredas sheets and filaments, respectively). HBT uses
velocitiesintheinnerregionsofhalosandthebulkaveraged
a friends-of-friendsapproach to find distinct halos and uses
velocity suggests that the bulk average velocity may be off-
particle lists fromdistinct halosat previoustimesteps to test
set by several hundred km s- 1 from the velocity at the ra- for the presence of subhalos. SURV is a very similar algo-
diuswheregalaxiesareexpectedtoreside;differencesatthis rithm,exceptthatdistincthalosareidentifiedusingspherical
scaleareeasilydetectableinclusterredshiftsurveysandmay overdensities. These algorithmsallrely heavilyon temporal
alsofactorininterpretingobservationsofthekineticSunyaev- informationintheirapproachtohalofinding.
Zel’dovicheffect. Asthisdifferencehasanimportantimpact
onthe usefulnessof derivedhalo properties,weadditionally
2.2. LimitationsofPreviousAlgorithms
performan investigationof the core-bulkvelocitydifference
inhalosacrossawiderangeofredshiftsandmasses. In order to develop an improved halo finder, it is impor-
Webeginthispaperwithasurveyofpreviousworkinhalo tantto understandsome ofthe shortcomingsofpreviousap-
findingaswellaspreviouslimitationsin§2. Wediscussour proaches. Forthevastmajorityofhalos,eventhemostbasic
improvedmethodologyin§3andconductdetailedtestsofour of algorithms (FOF and SO) do an acceptable job of deter-
approachin§4. Wepresentananalysisofthetheoreticaland mininghalopropertiesto10%accuracy(Knebeetal.2011).
practicallimitationsof simulationsin termsof trackingsub- However, recent interest in the detailed properties and his-
structure in §5. Finally, our results concerning the velocity tories of halos — e.g., precision mass functions and merger
offsets of cluster cores are presented in §6; we summarize trees and the shape of tidal structures — requires improve-
our conclusionsin §7. Multiple simulationsincludingslight mentstoolderapproaches;thishasresultedinaproliferation
variationsof cosmologicalparametersare considered in this ofnewcodesinthepastdecade(summarizedinKnebeetal.
paper; all simulations model a flat ΛCDM universe and we 2011).
always take the HubbleconstantH to be 70 km s- 1 Mpc- 1; The most significant improvements to halo finding have
0
equivalently,h=0.7. come from using the informationfrom six-dimensional(po-
sitionandvelocity)phasespace. Twotraditionalweakpoints
2. PREVIOUSHALOFINDERS for 3D (position-space) halo finders have been major merg-
TheROCKSTARHaloFinder 3
ers and subhalos close to the centers of their host halos. In halo concentrationsat a given halo mass. At the same time,
both cases, the density contrast alone is not enough to dis- whileTweedetal.(2009)successfullyresolvesthisproblem,
tinguishwhichparticlesbelongtowhichhalo: whentwoha- itnonethelessonlyfindshalosinpositionspace,andthushas
losarecloseenough,theassignmentofparticlestohalosbe- thesameweaknessesinidentifyingsubhaloandmajormerger
comesessentiallyrandomintheoverlapregion. However,as properties.
long as the two halos have relative motion, six-dimensional TheHBT and SURV algorithms(Han et al. 2011;Giocoli
halo finders can use particle velocity space information to etal.2010)usetheingeniousapproachoftracingsubhalosby
very effectively determine particle-halo membership. This, using the particles found in previously distinct halos, which
coupledwiththe abilityof6Dhalofindersto findtidalrem- couldpotentiallyalsosolvemanyoftheseproblems.Yet,they
nants(whicharecondensedinphasespacebutnotinposition bothalsoincludeassumptionsaboutaccretionontosubhalos
space), means that phase-space capabilities are required for (e.g., that subhalos cannot accrete background matter from
themostaccurateandinterestingstudiesofdarkmatterhalos. the host) which are untrue in HBT’s case for large linking-
Atthesametime,phasespacepresentsauniquechallenge. lengths (as halos will be identified as satellites far from the
While position space and velocity space have well-defined actualvirialradiusofthehost)andforbothhalofinderswith
distance metrics, there is not a single, unique way to com- majormergers(wheresatelliteandhostaremoreambiguous;
bine position and velocity distance into a single metric. For they can in any case easily trade particles with each other).
a useful definition of phase-space distance, one needs to be Theseassumptionsvastlysimplifythecodeatsomeexpense
abletodecide,e.g.,whetheranobjectjustpassingbytheori- tothecompletenessandaccuracyofthemassfunction.More
gin at 1 km s- 1 is closer or farther than an object at rest 1 seriously, the design of the algorithms requires temporal in-
kpcaway. Oneapproach,usedby6DFOF,istochooseinad- formation to find subhalos; in cases where simulation out-
vanceastaticconversionbetweenvelocityandpositionspace. puts are spaced very far apart or when only one timestep is
Whilesimple,thisapproachseemssomewhatself-limiting:if available,theycannoteffectivelyfindsubstructure. Theseis-
too short a linking-length is chosen, the full extent of sub- suesareinprinciplefixable:futureversionsofthealgorithms
structures cannot be found; if too large a linking-length is could easily combine advanced single-timestep substructure
chosen, then otherwisedistinct substructureswillbe merged findingwithchecksagainstprevioustimesteps’particlelists.
together.
A demonstrablysuperiorapproach,atleast in termsof re- 3. ANIMPROVEDAPPROACH:ROCKSTAR
coveringhaloproperties(Knebeetal.2011),istoadaptively
Ourprimarymotivationindevelopingahalofinderwasto
define a phase-space metric. Both HSF and HOT6D subdi-
improve the accuracy of halo merger trees that are required
vide the simulation space into six-dimensional hyperboxes
foran understandingofgalaxyevolution. Thedesignof our
containing (at the maximum refinement level) as little as a
halo finder was thus motivated by a requirementfor consis-
single particle each. For a given particle, the enclosing hy-
tent accuracy across multiple timesteps. This interest led to
perboxgivesalocalestimateofthephase-spacemetric,based
the developmentof a unique, adaptivephase-spacetemporal
ontherelativesizesofthehyperbox’sdimensionsinposition
algorithmwhich is provablyindependentof halo orientation
and velocity space. The usefulness of this estimate depends
and velocity relative to the simulation axes, and which also
heavilyonthemethodforpartitioningspaceintohyperboxes;
attempts to be highly preserving of particle-halo and halo-
HSF uses, for example, an entropy-basedapproachto deter-
subhalo membership across timesteps. In addition, we paid
mine whether more information is contained in the particle
special attention to the algorithm’sefficiency and paralleliz-
locationsforpositionspaceorvelocityspace.
ability,toallowittorunonthelargestcurrentdatasetsandso
These algorithms all give excellent results for identifying
thatitcouldeasilyscaletothenextgenerationofsimulations.
halo centers at a single timestep. However, consistent halo
Thusfar,wehaverunthehalofinder(andinmanycasesthe
catalogsacrosstimestepsareoftencompromisedbyafunda-
partner merger tree code) on the Bolshoi ( 1010 particles,
mentalambiguityin the definitionof a hosthalo. Formajor
∼
Klypinetal.2011)andLasDamassimulations(200boxesof
mergers, it is often unclear which halo center represents the
larger “host” or central halo, and which represents the sub- 1- 4 109particleseach, McBrideetal. inpreparation),on
halo. Phase-spacehalofindinghelpswhenthetwohalocen- sever×al20483 simulationsrun to create Dark EnergySurvey
tersarerelativelyfarapart(i.e.,weaklyinteracting),because (DES)simulatedskycatalogs,on onehundredhighresolu-
∼
thereexistsastrongcorrelationbetweenthevelocitiesofpar- tionhalossimulatedaspartoftheRHAPSODY project(Wu
ticles in the halo outskirtsand halo centers. However,when etal.2012a,b),andonhalosA-1throughA-5oftheAquarius
the centers come close enough to interact strongly, this cor- Simulation(upto1.4 109particlesinasinglehalo;Springel
×
relation is weakened, and it becomesmuchmore difficultto etal.2008).
accuratelyassignparticlestothehalos.Asaresult,itismuch As a first step, our algorithm performs a rapid variant of
more difficultto determinewhich of the halo centers should the3Dfriends-of-friends(FOF)methodtofindoverdensere-
beconsideredthehosthalo.Sincethedefinitionofhalomass gions which are then distributed among processorsfor anal-
often includes the mass of subhalos, this problemcan result ysis (§3.1). Then, it builds a hierarchy of FOF subgroups
inlargemassfluctuationsacrosstimestepsformerginghalos. in phase space by progressivelyand adaptively reducingthe
A number of solutions to this problem have been pro- 6D linking length, so that a tunable fraction of particles are
posedandexaminedwiththeAdaptaHOPhalofinder(Tweed capturedateachsubgroupascomparedtotheimmediatepar-
et al. 2009). Tweed et al. (2009) found that a temporal ap- ent group (§3.2). Next, it converts this hierarchy of FOF
proach(examiningthehostvs. subhaloassignmentatearlier groups into a list of particle memberships for halos (§3.3).
timesteps)wasmostsuccessfulatfixingthisproblem. Other It then computes the host halo/subhalo relationships among
methods, such as choosing the densest halo center to be the halos, usinginformationfromtheprevioustimestepif avail-
hosthalo,haveinherentinstabilitiesbecauseofthespreadin able(§3.4). Finally,itremovesunboundparticlesfromhalos
andcalculates halo properties, beforeautomaticallygenerat-
4 BEHROOZI,WECHSLER,&WU
1. The simulationvolumeisdivided Asthe 3D FOF groupsare used in ourmethodonly to di-
into 3D Friends-of-Friends groups vide up the simulation volume into manageable work units,
weinsteadmakeuseofamodifiedalgorithmwhichisanor-
for easy parallelization.
derofmagnitudefaster. Asisusual,neighboringparticlesare
assignedtobeinthesamegroupiftheirdistanceiswithinthe
linkinglength. However,ifaparticlehasmorethanacertain
2. For each group, particlepositions
numberofneighborswithinthelinkinglength(16,inourver-
and velocities are divided (normal-
sion),thentheneighbor-findingprocessforthoseneighboring
ized) by the group position and ve- particlesisskipped. Instead,neighborsfortheoriginalparti-
cle outto twice the originallinkinglength are calculated. If
locity dispersions, giving a natural
anyofthoseparticlesbelongtoanotherFOFgroup,thatcor-
phase-space metric.
respondinggroup is joined with that of the originalparticle.
Thus,althoughtheneighbor-findingprocesshasbeenskipped
3. A phase-space linking length is forthenearestparticles,groupswhichwouldhavebeenlinked
throughthoseintermediateneighborsarestilljoinedtogether.
adaptively chosen such that 70% of
Thisprocessthereforelinkstogetheratminimumthesame
the group’s particles are linked to-
particles as in the standard FOF algorithm—finefor our de-
getherin subgroups. sired purpose—but does so much faster: neighbors must be
calculated over a larger distance, but many fewer of those
4. The process repeats for each calculations must be performed. Indeed, rather than slow-
ing down as the linking length is increased, our variation of
subgroup: renormalization, a new
FOF becomesfaster. Because of this, we are free to choose
linking-length, and a new level of anexceptionallylargevalueforthelinkinglength.Avalueof
substructurecalculated. b=0.2istoosmall,asitdoesnotincludeallparticlesoutto
thevirialradius(Moreetal.2011);afterevaluatingdifferent
5.Oncealllevelsofsubstructureare choicesforb(see§4.5),wechoseb=0.28,whichguarantees
thatvirialsphericaloverdensitiescanbedeterminedforeven
found, seed halos are placed at the
themostellipsoidalhalos.
lowest substructure levels and par- The most important parallelization work occurs at this
ticles are assigned hierarchically to stage.Separatereadertasksloadparticlesfromsnapshotfiles.
Depending on the number of available CPUs for analysis, a
theclosestseedhaloinphasespace.
masterprocessdividesthesimulationregionintorectangular
boundaries, and it directs the reader tasks to send particles
6. Once particles have been as-
withinthoseboundariestotheappropriateanalysistasks.
signed to halos, unbound particles Eachanalysistask firstcalculates3DFOFsin itsassigned
are removed and halo properties analysis region, and FOFs which span processor boundaries
areautomaticallystitchedtogether. TheFOFgroupsarethen
(positions, velocities, etc.) are
distributed for further phase-space analysis according to in-
calculated. dividualprocessorload. Theload-balancingprocedureisde-
scribedinfurtherdetailinAppendixA. Currently,single3D
FIG.1.—Avisualsummaryoftheparticle-haloassignmentalgorithm. FOF groups are analyzed by at most one processor. Also,
in the currentimplementation, there is no supportfor multi-
ple particle masses, although support could easily be added
ingparticle-basedmergertrees(§3.5). A visualsummaryof
by varying the linking length depending on particle mass.
thesestepsisshowninFig.1.
Providedenoughinterest,supportformultiprocessoranalysis
of single large halos as well as support for multiple particle
3.1. Efficient,ParallelFOFGroupCalculation
massesmaybeaddedinafutureversionofROCKSTAR.
The 3D friends-of-friends (FOF) algorithm has existed
since at least 1985 (Davis et al. 1985). In principle, imple- 3.2. ThePhase-SpaceFOFHierarchy
mentation of the algorithm is straightforward: one attaches
For each 3D FOF group which is created in the previous
two particlesto the samegroupif theyarewithin a prespec-
step, the algorithmproceedsbybuildinga hierarchyof FOF
ified linking length. Typically, this linking length is chosen
subgroupsinphasespace. Deeperlevelsofsubgroupshavea
intermsofafractionbofthemeaninterparticledistance(in
tighter linking-lengthcriterion in phase space, which means
ourcode,asforothers,thecuberootofthemeanparticlevol-
that deeper levels correspond to increasingly tighter isoden-
ume);commonvaluesforgeneratinghalocatalogsrangefrom
sity contoursaroundpeaksin the phase-spacedensity distri-
b=0.15tob=0.2(Moreetal.2011).
bution. Thisenablesaneasywaytodistinguishseparatesub-
In practice, this meansthat one mustdetermine neighbors
structures—abovesomethresholdphase-spacedensity,their
foreveryparticlewithinasphereofradiusequaltothelink-
particle distributions must be distinct in phase space; other-
ing length. Even with an efficient tree code (we use a cus-
wise,itwouldbedifficulttojustifytheseparationintodiffer-
tombinaryspacepartitioningtree),thisrepresentsagreatdeal entstructures.1
of wasted computation,especially in dense cluster cores. In
Beginning with a base FOF group, ROCKSTAR adaptively
such cases, particles might have tens of thousandsof neigh-
borswithinalinkinglength,allofwhichwilleventuallyend 1Thiswouldnotbetrueif,e.g.,haloshadPlummerprofilesorotherwise
upinthesameFOFgroup. flatdensityprofilesintheircenters.
TheROCKSTARHaloFinder 5
chooses a phase-space linking length based on the standard in the autocorrelation function close to the simulation force
deviationsoftheparticledistributioninpositionandvelocity resolution.
space. Thatis,fortwoparticles p and p inthebasegroup, Foraparentgroupwhichcontainsonlyasingleseedhalo,
1 2
thephase-spacedistancemetricisdefinedas: all the particles in the groupare assigned to the single seed.
Foraparentgroupwhichcontainsmultipleseedhalos,how-
~x - ~x 2 ~v - ~v 2 1/2 ever, particles in the group are assigned to the closest seed
d(p1,p2)= | 1σ22| +| 1σ22| , (1) halo in phase space. In this case, the phase-space metric is
(cid:18) x v (cid:19) setbytheseedhaloproperties,sothatthedistancebetweena
whereσ andσ aretheparticlepositionandvelocitydisper- halohandaparticle pisdefinedas:
x v
sionsforthegivenFOFgroup;thisisidenticaltothemetricof
1/2
Gottlöber(1998). Foreachparticle,thedistancetothenear- ~x - ~x 2 ~v - ~v 2
est neighbor is computed; the phase-space linking length is d(h,p)= |rh2 p| +| hσ2p| (3)
thenchosensuchthataconstantfraction f oftheparticlesare dyn,vir v !
v
linkedtogetherwithatleastoneotherparticle.Inlargegroups r =v t = max (4)
dyn,vir maxdyn,vir
(>10,000particles),wherecomputingthenearestneighborfor 4πGρ
allparticlescanbeverycostly,thenearestneighborsareonly 3 vir
calculated for a random10,000-particlesubset of the group, q
whereσ istheseedhalo’scurrentvelocitydispersion,v is
v max
asthisissufficienttodeterminethelinkinglengthtoreason-
its current maximum circular velocity (see §3.5), and “vir”
ableprecision.
specifies the virial overdensity, using the definition of ρ
vir
The proper choice of f is constrained by two considera-
from Bryan & Norman (1998), which corresponds to 360
tions.Ifonechoosestoolargeavalue(f >0.9),thealgorithm
timesthebackgrounddensityatz=0,however,otherchoices
willtake muchlonger,anditcanalso findspurious(notsta-
ofthisdensitycaneasilybeapplied.
tistically significant)subgroups. Ifonechoosestoolow ofa
Usingtheradiusr astheposition-spacedistancenor-
value(f <0.5),thealgorithmmaynotfindsmallersubstruc- dyn,vir
malization may seem unusual at first, but the natural alter-
tures. Assuch, we usean intermediatevalue(f =0.7);with
native (using σ ) gives unstable and nonintuitive results. At
x
therecommendedminimumthresholdforhaloparticles(20).
fixedphase-spacedensity,subhalosandtidalstreams(which
In our tests of the mock NFW (Navarro et al. 1997) halos
havelowervelocitydispersionsthanthehosthalo)willhave
described in Knebe et al. (2011), this results in a false posi-
largerposition-spacedispersionsthanthehosthalo. Thus,if
tiverateof10%for20-particlegroupscomparedtoacosmo-
σ wereused,particlesintheoutskirtsofahalocouldbeeas-
x
logicalsubhalodistribution,whichdeclinesexponentiallyfor
ilymis-assignedtoasubhaloinsteadofthehosthalo. Using
larger group sizes. These false positives are easily removed
r , on the otherhand, preventsthis problemby ensuring
by the significance and boundedness tests described in §3.3 den,vir
thatparticlesassignedtosubhaloscannotbetoofarfromthe
and§3.5.3. maindensitypeakeveniftheyarecloseinvelocityspace.2 In-
Once subgroups have been found in the base FOF group,
tuitively,the largesteffectof usingr isthat velocity-space
vir
thisprocessisrepeated. Foreachsubgroup,thephase-space
informationbecomesthedominantmethodofdistinguishing
metric is recalculated, and a new linking-length is selected
particlemembershipwhentwo halosare within each other’s
such that a fraction f of the subgroup’s particles are linked virialradii.3
togetherinto sub-subgroups. Groupfindingproceedshierar-
This process of particle assignment assures that substruc-
chicallyinphasespaceuntilapredeterminedminimumnum-
ture masses are calculated correctly independently of the
ber of particles remain at the deepest level of the hierarchy.
choiceof f,thefractionofparticlespresentineachsubgroup
Here we set this minimum number to 10 particles, although
relativetoitsparentgroup.Inaddition,forasubhalocloseto
halopropertiesarenotrobustapproachingthisminimum.
thecenterofitshosthalo,itassuresthathostparticlesarenot
mis-assignedtothesubhalo—thecentralparticlesofthehost
3.3. ConvertingFOFSubgroupstoHalos willnaturallybecloserinphasespacetothetruehostcenter
ForeachofthesubgroupsatthedeepestleveloftheFOFhi- thantheyaretothesubhalo’scenter.
erarchy(correspondingtothelocalphase-spacedensitymax-
ima),aseedhaloisgenerated.Thealgorithmthenrecursively 3.4. CalculatingSubstructureMembership
analyzes higher levels of the hierarchyto assign particles to In addition to calculating particle-halo membership, it is
these seed halosuntilall particlesin the originalFOFgroup alsonecessarytodeterminewhichhalosaresubstructuresof
havebeenassigned.Topreventcaseswherenoisegivesriseto other halos. The most common definition of substructure is
duplicatedseedhalos,weautomaticallycalculatethePoisson a bound halo contained within another, larger halo. Yet, as
uncertainty in seed halo positions and velocities, and merge halo masses are commonly defined to include substructure,
thetwoseedhalosiftheirpositionsandvelocitiesarewithin the question of which of two halos is the largest (and thus,
10σ of the uncertainties. Specifically, the uncertainties are whichshouldbecalledasatelliteoftheother)canchangede-
calculated as µx =σx/√n and µv =σv/√n, where σx and σv pendingonwhichsubstructureshavebeenassignedtothem.
arethepositionandvelocitydispersions,andnisthenumber Thisisoneofthelargestsourcesofambiguitybetweenspher-
ofparticles,allforthesmallerofthetwoseedhalos. Thetwo ical overdensity halo finders, even those which limit them-
halosaremergedif selvestodistincthalos.
(x - x )2µ- 2+(v - v )2µ- 2<10√2. (2) 2Fordeterminationoftidalstreams,this“problem”becomesa“feature,”
1 2 x 1 2 v anduseofσxmaybepreferabletorvir.
Inourtests,qthisthresholdyieldsanear-featurelesshaloauto- 3 An alternate radius (e.g., r200b or r500c) could be used instead, but it
wouldhaveaneffectonlyonasmallfractionofparticlesinasmallfraction
correlationfunction;lowervaluesresultinaspuriousupturn ofhalos(majormergers).
6 BEHROOZI,WECHSLER,&WU
Webreaktheself-circularitybyassigningsatellitemember- notedin§6,thehalocorecanhaveasubstantialvelocityoff-
ship based on phase-space distances before calculating halo set from the halo bulk. Since the galaxy hosted by the halo
masses. Treatingeachhalocenterlikea particle, we use the will presumably best track the halo core, we calculate the
same metric as Eq. 3 and calculate the distance to all other mainvelocityforthehalousingtheaverageparticlevelocity
haloswithlargernumbersofassignedparticles. Thesatellite withintheinnermost10%ofthehaloradius. Forcalculating
haloinquestionisthenassignedtobea subhalooftheclos- thebound/unboundmassofthehalo(see3.5.2),however,we
estlargerhalowithinthesame3Dfriends-of-friendsgroup,if usethemoreappropriateaveragedhalobulkvelocityinclud-
oneexists. If the halo catalogatan earlier timestep is avail- ingsubstructure.
able, this assignment is modified to include temporal infor-
mation.Halocoresatthecurrenttimestepareassociatedwith 3.5.2. HaloMasses
halosattheprevioustimestepbyfindingthehaloattheprevi- Forhalomasses, ROCKSTAR calculatessphericaloverden-
oustimestepwiththelargestcontributiontothecurrenthalo sitiesaccordingtomultipleuser-specifieddensitythresholds:
core’sparticlemembership. Then,host-subhalorelationships e.g.,thevirialthreshold,fromBryan&Norman(1998),ora
are checked against the previous timestep; if necessary, the densitythresholdrelativetothebackgroundortothecritical
choiceofwhichhaloisthehostmaybeswitchedsoastopre- density. Asisusual, these overdensitiesarecalculatedusing
servethehost-subhalorelationshipoftheprevioustimestep. alltheparticlesforallthesubstructurecontainedinahalo.On
As explained above, these host-subhalo relationships are theotherhand,subhalomasseshavetraditionallybeenama-
onlyusedinternallyforcalculatingmasses:particlesassigned jorpointofambiguity(fordensity-spacehalofinders). With
tothehostarenotcountedwithinthemassofthesubhalo,but a phase-space halo finder, such as ROCKSTAR, the particles
particles within the subhalo are counted as part of the mass belongingto the subhalo can be more reliably isolated from
ofthe hosthalo. Thischoiceassuresthatthe massof a halo thehost,andthusless ambiguityexists: thesamemethodof
won’tsuddenlychangeasitcrossesthevirialradiusofalarger calculatingsphericaloverdensitiesmaybeappliedtojustthe
halo, and it provides more stable mass definitions in major particlesbelongingtothesubhalo.Intermsofthedefinitionof
mergers.Oncehalomasseshavebeencalculated,thesubhalo wherethesubhalo“ends,”Eq.3impliesthatthesubhaloedge
membershipisrecalulatedaccordingtothestandarddefinition is effectively where the distribution of its particles in phase
(subhalosarewithinr∆ofmoremassivehosthalos)whenthe spacebecomesequidistantfromthesubhaloanditshosthalo.
mergertreesareconstructed. Ifalternatemassdefinitionsarenecessary,thehalofindercan
Forclarity,itshouldbenotedthateverydensitypeakwithin output the full phase-space particle-halo assignments; these
the original FOF analysis group will correspond to either a maythenbepost-processedbytheusertoobtainthedesired
hosthaloorasubhalointhefinalcatalog.Ithasbeenobserved masses.
thatFOFgroupswill“bridge”or“premerge”longbeforetheir
correspondingSOhalocounterparts(e.g.,Klypinetal.2011). 3.5.3. UnbindingParticles
However, as we calculate full SO propertiesassociated with
By default, ROCKSTAR performs an unbinding procedure
eachdensitypeak,asingleFOFgroupisnaturallyallowedto
beforecalculating halo mass and v , althoughthis may be
max
containmultipleSOhosthalos;thusthebridgingorpremerg-
switched off for studies of e.g., tidal remnants. Because the
ingofFOFgroupsdoesnotaffectthefinalhalocatalogs.
algorithm operates in phase space, the vast majority of halo
particles assigned to central halos are actually bound. We
3.5. CalculatingHaloPropertiesandMergerTrees
findtypicalboundednessvaluesof98%atz=0;see§4.5and
Typically, several properties of interest are generated for Behroozietal.(2012a).Evenforsubstructure,unboundparti-
halocatalogs.Regardlessofthequalityofparticleassignment clestypicallycorrespondtotidalstreamsattheoutskirtsofthe
in the halo finder, carefulattention to halo propertycalcula- subhalo,makingacomplicatedunbindingalgorithmunneces-
tionisessentialforconsistent,unbiasedresults. sary. For this reason, as well as to improve consistency of
halomassesacrosstimesteps, we performonlya single-pass
3.5.1. HaloPositionsandVelocities unbindingprocedureusingamodifiedBarnes-Hutmethodto
accurately calculate particle potentials (see Appendix B for
For positions, Knebe et al. (2011) demonstrated that halo
details).4 Sincemanyuserswillbeinterestedinclassicalhalo
finders which calculated halo locations based on the maxi-
findingonlyasopposedtorecoveringtidalstreams,thecode
mumdensity peakwere more accuratethanFOF-based halo
by default does not output halos where fewer than 50% of
finders which use the averaged location of all halo particles
the particlesare bound; this thresholdis user-adjustable,but
(see also Gao & White 2006). The reason for this may be
changing it does not produce statistically significant effects
simply understood: as particle density rapidly drops in the
on the clustering or mass function until halos with a bound
outerreachesofahalo,thecorrespondingdispersionofparti-
fractionoflessthan15%areincluded(see§4.5).
clepositionsclimbsprecipitously. Consequently,ratherthan
Wenotethat,inmajormergers,amorecarefulapproachto
increasingthe statistical accuracyof the halo centercalcula-
unbinding must be used. In many cases where merging ha-
tion,includingtheparticlesatthehaloboundaryactuallyre-
los initially have large velocity offsets, particles on the out-
ducesit. The highest accuracyis instead achievedwhen the
skirts of the halos can mix in phase space before the halo
expected Poisson error (σ /√N) is minimized. As our halo
x cores themselves merge. This results in many particles be-
finderhasaccess(viathehierarchyofFOFsubgroups)tothe
ingunboundwithrespecttoeitherofthetwohalocores,even
innerregionsofthehalodensitydistribution,anaccuratecal-
thoughtheyareboundtotheoverallmergingsystem.Assuch,
culationofthecenterispossiblebyaveragingtheparticlelo-
anaiveunbindingoftheparticleswouldleadtothe merging
cationsfortheinnersubgroupwhichbestminimizesthePois-
halos’massesdroppingsharplyforseveraltimestepspriorto
sonerror.Typically,fora106particlehalo,thisestimatorends
upaveragingthepositionsoftheinnermost103particles. 4Providedenoughinterest,wemayaddtheoptionofamulti-passunbind-
The picture for halo velocities is not quite as simple. As ingprocedureinfutureversionsofthehalofinder.
TheROCKSTARHaloFinder 7
the final merger.5 To counter this effect in major mergers, To help with cluster studies, we calculate several halo re-
ROCKSTAR calculatesthe gravitationalpotentialofthe com- laxationparameters. Theseincludethecentralpositionoffset
binedmergingsystem,ratherthanforindividualhalos,when (X , defined as the distance between the halo density peak
off
determiningwhethertounbindparticles.Thisbehavioristrig- andthehalocenter-of-mass),thecentralvelocityoffset(V ,
off
geredbydefaultwhentwohaloshaveamergerratioof1:3or definedas the differencebetween the halo core velocity and
larger;thisvalueisuser-adjustable,buthaslittleeffectonthe bulkvelocity),andtheratioofkinetictopotentialenergy( T )
|U|
recoveredmassfunctionorclustering(see§4.5).
forparticleswithinthehaloradius. Wereferinterestedread-
ersto Netoetal. (2007)foradescriptionandcomparisonof
3.5.4. AdditionalHaloPropertiesandMergerTrees
methodsfordetermininghalorelaxedness.
Two more common outputs of halo finders are vmax, the We also calculate ellipsoidal shape parameters for halos.
maximum circular velocity and Rs, the scale radius. vmax is Following the recommendations of Zemp et al. (2011), we
simplytakenasthemaximumofthequantity GM(r)r- 1;it calculatethe massdistributiontensorforparticleswithinthe
shouldbenotedthatthisquantityisrobustevenforthesmall- haloradius,excludingsubstructure:
esthalosbecauseoftheextremelyshallowdepepndenceofv
max
onradius.ForR ,wedividehaloparticlesintoupto50radial 1
s M = xx (11)
equal-massbins(withaminimumof15particlesperbin)and ij N i j
directlyfitanNFW(Navarroetal.1997)profiletodetermine XN
themaximum-likelihoodfit.
The sorted eigenvalues of this matrix correspond to the
We also calculate the Klypin scale radius for comparison squares of the principal ellipsoid axes (a2 > b2 > c2). We
(Klypinetal.2011),whichusesv andM tocalculateR
max vir s output both the axis ratios (b and c) as well as the largest
undertheassumptionofanNFW(Navarroetal.1997)profile. a a
Inparticular,forNFWprofiles,theradiusR atwhichv ellipsoidaxisvector,A~.
max max
is measured is a constantmultiple b of the radius R , and is Finally,wementionthatourhalofinderautomaticallycre-
s
givenby: ates particle-based merger trees. For a given halo, its de-
d 1 scendant is assigned as the halo in the next timestep which
db b- 1ln(1+b)- 1+b =0 (5) has the maximum number of particles in common (exclud-
(cid:20) (cid:21) ingparticlesfromsubhalos). Whileitispossibletousethese
Thismaybesolvednumerically,withtheresultthat merger trees directly, we recommend instead to use the ad-
vanced merger tree algorithm discussed in Behroozi et al.
R =2.1626R (6)
max s (2011). This algorithm detects and corrects inconsistencies
Instead of using R directly (which is ill-determined for acrosstimesteps(e.g.,haloswhichdisappearandreappearas
max
small halos), we make use of the ratio of R /R to relate theycrossthedetectionthreshold)tofurtherimprovethetem-
max s
themassenclosedwithin2.1626R tov ,M ,andthecon- poralconsistencyofthemergertrees.
s max vir
centrationc=R /R :
vir s
4. TESTS&COMPARISONS
c R 2.1626
f(c) =v2maxGMvir f(2.1626) (7) TheROCKSTARalgorithmhasalreadyundergoneextensive
vir testing and comparison to other halo finders in Knebe et al.
x
f(x) ln(1+x)- (8) (2011). Intestswithgeneratedmockhalos, ROCKSTAR suc-
≡ 1+x cessfullyrecoveredhalopropertiesforhalosdownto20parti-
Thus, by numericallyinvertingthe functionon the left-hand cles,inmanycases(e.g.,forhalomassandbulkvelocity)bet-
sideofEq.7,cmaybefoundasafunctionofM andv , terthanallseventeenotherparticipatinghalofinders.Incases
vir max
and the Klypin scale radius R can be derived. The Klypin whereitdidnotperformbest,itwasoftenonlymarginallybe-
s
scaleradiusismorerobustthanthefittedscaleradiusforha- hindone ofthe otherphase-spacehalofinders. Notably, out
los with less than 100 particles; this is due not only to shot ofallthehalofinders,itwastheonlyonetofullysuccessfully
noise, butalsoduetothefactthathaloprofilesarenotwell- recoverallhaloproperties(mass,location,position,velocity,
determined at distances comparable to the simulation force andv )forasubhalocoincidingwiththecenterofitshost
max
resolution. halo. In addition, Knebe et al. (2011)compared mass func-
We additionally calculate the angular momentum of the tions, v functions, correlation functions (for r>2 Mpc),
max
halo(usingboundparticlesouttothedesiredhaloradius)and and halo bulk velocities for a cosmological simulation with
thehalospinparameter(λ),asintroducedbyPeebles(1969): 10243 particles; ROCKSTAR had results comparableto other
halofindersinalltheseresults,althoughonlytheotherphase-
J E
spacehalofindersshareditslowmasscompletenesslimit(
λ= | | (9)
GM2.5 25particlesforM ). ∼
pvir 200c
We thus focus on comparisonsbeyond those already cov-
whereJ isthemagnitudeofthehaloangularmomentumand
ered in Knebe et al. (2011). Our comparisons cover results
E is the total energyof the halo (potentialand kinetic). For
forseveraldifferentdarkmattersimulations, brieflysumma-
comparison, we also calculate the Bullock spin parameter
rizedin§4.1. Weprovideavisualdemonstrationofthealgo-
(Bullocketal.2001),definedas
rithm’s performancein §4.2, a detailed comparisonwith the
J J massandcorrelationfunctionsforotherhalofindersin§4.3,
λB= = (10) anevaluationofthedynamicalaccuracyofhalopropertiesin
√2MvirVvirRvir 2GMv3irRvir §4.4,andwepresentjustificationforourchoiceofthedefault
5 Withthe BDMhalofinder(Klypin etaql.2011), forexample, wehave parameters in §4.5. Finally, we show figures demonstrating
observedhalomasseswhichdropbyafactorofthreeonaccountofthisaffect. theexcellentperformanceofthehalofinderin§4.6.
8 BEHROOZI,WECHSLER,&WU
FIG.2.—ROCKSTARallowsrecoveryofevenveryclosemajormergers. Thisfigureshowsanexampleofamajormergerinvolving1013M⊙halosfromthe
Bolshoisimulation(Klypinetal.2011).Thetoppanelshowsthecompleteparticledistributionaroundthemerginghalos.Inthesecondrow,theleftpanelshows
thehostparticledistribution,andtherightpanelshowsthesubhaloparticledistribution,withparticlescoloredaccordingtosubhalomembership. (Theparticle
plottingsizehasbeenincreasedtoshowmoreclearlytheextentofthesmallsubstructuresintheright-handpanel). Thetwodifferentcolorsintheleft-hand
panelhintatthefactthatthereareindeedthreehalosinvolvedinthemajormerger,twoofwhichareextremelyclosetomerging.Theuniformsubhaloshapesin
theright-handpanelsuggestthatsubhaloparticlescanbedistinguishedwithoutbiasdespiteextremevariationsinthehostparticledensitybetweenthesubhalo
centersandthesubhalooutskirts. Thebottomrowshowsmoreclearlytheextremelyclosemajormerger. Thebottomleft-handpanelshowsthefullparticle
distributioninpositionspaceinasmallregionclosetothemerginghalocores;here,thebimodaldistributionisevident,butdistinguishingparticlemembership
isimpossiblebeyondtheimmediatevicinityofthecores. Ontheotherhand,thebottomright-handpanelshowsthesameparticlesinvelocityspace,wherethe
bimodalityfromthetwocoresshowsaclearvelocityseparation,allowingparticlestobereasonablyassignedeveninthehalobulk.
TheROCKSTARHaloFinder 9
10-1 Tinker et al. (2008)
10-2 420 Mpc h-1
10-3 1.1 640 Mpc h-1
-3-1c dex]1100--45 Tinker 1 12040000 MMppcc hh--11
Mp Tinker et al. (2008) f/
[10-6 420 Mpc h-1 f
f 10-7 640 Mpc h-1 0.9
1000 Mpc h-1
10-8 2400 Mpc h-1
10-9 0.8
1011 1012 1013 1014 1015 1011 1012 1013 1014 1015
Halo Mass [M ] Halo Mass [M ]
O• O•
10-1
BDM
10-2 1.1 Rockstar
Tinker et al. (2008)
-3-1c dex]1100--43 Rockstar 1
Mp BDM f /
f [10-5 RTionckkesrt aert al. (2008) f
0.9
10-6
10-7 0.8
109 1010 1011 1012 1013 1014 1015 109 1010 1011 1012 1013 1014 1015
Halo Mass [M ] Halo Mass [M ]
O• O•
FIG.3.—Thehalomassfunctionfordistincthalosisverysimilartopreviouslypublishedresults. ThetoprowshowscomparisonstotheTinkeretal.(2008)
centralhalomassfunctionforeachofthefourLasDamasboxes(seeTable1).Goodagreementisseenabove100particles.Thebottomrowshowsacomparison
betweentheROCKSTARandBDM(Klypinetal.2011)halofindersontheBolshoisimulation(20483particles,250Mpch- 1).Theleft-handplotsshowthefull
massfunctions,andtheright-handplotsshowtheresiduals,withPoissonerrorsshownfortheBolshoisimulation.AsnotedinTinkeretal.(2008),thecalibrated
massrangedoesnotextendbelow1011h- 1M⊙;furthermore,theauthorsstate“thebehaviorofthefittingfunctionatlowermassesisarbitrary.”Wethereforedo
notextrapolatetheTinkeretal.(2008)massfunctioninourcomparisons.
100 10-1
10-1 10-2
-1ex]10-2 -1ex]10-3
d d
-3c 10-3 -3c
p p
M BDM (all) M10-4 BDM (sats)
[10-4 Rockstar [ Rockstar
f f
10-5 10-5
10-6 10-6
100 1000 100 1000
v [km s-1] v [km s-1]
max max
FIG.4.—Thehalovelocityfunctionisalsoverysimilartopreviouslypublishedresults. Thisfigureshowscomparisonsbetweenvelocity(vmax)functionsfor
theROCKSTARhalofinderandBDMontheBolshoisimulation(20483particles,250Mpch- 1). Theleft-handplotshowsallhalos;theright-handplotshows
satellitehalosonly.
10 BEHROOZI,WECHSLER,&WU
4.1. SimulationParameters
We use five sets of simulations for this paper. The first
of these, Bolshoi, has already been extensively detailed in 105
Klypinet al. (2011). To summarizethe relevantparameters, BDM (v > 300 km s-1)
Mitipscah2- 10,4r8u3npuasritnicglethseimAuRlaTtisoinmouflactoiomnocvoindge(sKidreavletsnogvthet25al0. 104 Rockstarmax
1997) on the NASA Ames Pleiades supercluster. The as-
sumedcosmologyisΩm=0.27,ΩΛ=0.73,h=0.7,ns=0.95, 103
and σ = 0.82, similar to WMAP7 results (Komatsu et al.
x
2011); the effective force-softeninglength is 1 kpc h- 1, and 102
theparticlemassis1.36 108M h- 1.
⊙
×
We also use four simulations of different sizes from the 101
LasDamas project (McBride et al, in preparation).6 These
have11203 to14003 particlesincomovingregionsfrom420 100
Mpc h- 1 to 2400 Mpc h- 1 on a side, and were run using the 0.01 0.1 1 10
GADGET-2 code (Springel2005). Again, the assumed cos- r [Mpc h-1]
mology is similar to WMAP7, with Ωm =0.25, ΩΛ =0.75,
h=0.7, n =1.0, and σ =0.8. The effective force-softening
s
lengths range from 8 to 53 kpc h- 1, and the particle masses
rangefrom1.87 109M⊙ h- 1 to 4.57 1011h- 1M⊙. Details 106
× ×
ofallthesimulationsaresummarizedinTable1.
105 BDM (vmax > 150 km s-1)
Rockstar
4.2. VisualDemonstration
Inordertodemonstratehowthealgorithmperformsonha- 104
losinarealsimulation,Fig.2showsanexampleofhowpar-
ticles are assigned to halos in a major merger; this example 103
x
hasbeentakenfromtheBolshoisimulationatz=0.Fromthe
top image in the figure, one might expectthat two large ha- 102
losseparatedbyabout200kpch- 1 aremergingtogether,but
101
careful analysis reveals that the larger halo actually consists
ofanothermajormergerwhereinthehalocoresareseparated
by only 15 kpc h- 1. As shown in the bottom-left panel of 100
0.01 0.1 1 10
thefigure,theexistenceofthethirdmassivehaloisvisibleat r [Mpc h-1]
a moderate significance level in position space—however, a
position-spacehalofinderwouldhavenowaytocorrectlyas-
signparticlesbeyondtheimmediatelocalityofthetwocores.
Yet,inthebottom-rightpanel,theseparationofthetwohalo
106
coresisclearlydistinguishableinvelocityspace.Assuch,not
onlycan the distinct existenceof the close-to-merginghalos 105 BDM (vmax > 100 km s-1)
bereliablyconfirmed,butparticleassignmenttothetwohalos Rockstar
basedonparticlevelocitycoordinatescanbereliablycarried 104 Subfind
outaswell.
We also remark that phase-space halo-finding helps im- 103
prove the accuracy of subhalo shapes by removingthe need
x 102
to perform a position-space cut to distinguish host particles
fromsubstructureparticles. Satellite halosare usuallyoffset 101
invelocityspacefromtheirhosts,butjustasimportantly,they
usuallyalsohaveamuchlowervelocitydispersionthantheir 100
hosts. Thisimpliesthatsatellitesmaybereliablyfoundeven
inthedensecoresofhalos—althoughthepositionspaceden- 10-1
0.01 0.1 1 10
sityisveryhighforhostparticles,theaveragevelocitydisper- r [Mpc h-1]
sion is large enoughthat the lower-dispersionsubhalo parti-
clescanbereliablydistinguishedfromhostparticles. Conse-
FIG.5.—Two-pointcorrelation functions areverysimilartopreviously-
quently,asshowninthemiddle-rightpanelofFig.2,satellites
publishedresultsexceptrelativelyclosetothecentersofhalos. Thisfigure
arepickedouteveninthedensehalocenterswithoutbiasas showscorrelationfunctionsfortheROCKSTARhalofinderandBDMonthe
regardtoshapeorsize. Bolshoi simulation (20483 particles, 250 Mpch- 1). Thetop panel shows
thecorrelation function forvmax>300kms- 1 (10,000halos), themiddle
4.3. MassandCorrelationFunctionComparisons apnadnetlhsehboowtstotmhepcaonrreellsahtioownsfuthnectcioonrrefolartivomnaxfu>nc1ti5o0nkfmorsv-m1a(x1>001,00000khmalos-s1),.
Anextensivecomparisonofthemassfunctionandcorrela- ThebottompanelincludesacomparisonwithSubfind. Subfindhaloswere
tion functionbetween ROCKSTAR and otherhalo findershas onlyavailablefor1/125thofBolshoi(atotalof∼1000halosforthisvmax
threshold). Forafaircomparison,bothROCKSTARandBDMresultsforthe
bottompanelwerecomputedonthesameregionofBolshoi.
6http://lss.phy.vanderbilt.edu/lasdamas/