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DRAFTVERSIONFEBRUARY4,2008 PreprinttypesetusingLATEXstyleemulateapjv.04/03/99 ONTHECORRELATIONBETWEENSPINPARAMETERANDHALOMASS ALEXANDERKNEBE1,CHRISPOWER2 DraftversionFebruary4,2008 ABSTRACT WereportonacorrelationbetweenvirialmassMandspinparameterλfordarkmatterhalosformingatredshifts z>10. Wefindthatthespinparameterdecreaseswithincreasinghalomass. Interestingly,ouranalysisindicates 8 th∼athalosformingatlatertimesdonotexhibitsuchastrongcorrelation,inagreementwiththefindingsofprevious 0 studies.Webrieflydiscusstheimplicationsofthiscorrelationforgalaxyformationathighredshiftsandthegalaxy 0 populationweobservetoday. 2 Subjectheadings:galaxies:formation—cosmology:theory—cosmology:earlyUniverse—methods: n numerical a J 1. INTRODUCTION eter(Peebles1969), 9 J E Thephysicalmechanismbywhichgalaxiesacquiretheiran- λ= | |, (1) 2 gularmomentumisanimportantproblemthathasbeenthesub- GM5/2 p ject of investigationfor nearlysixty years(Hoyle 1949). This whereJisthemagnitudeoftheangularmomentumofmaterial ] h reflects the fundamentalrole playedbyangularmomentumof withinthevirialradius,M isthevirialmass,andE isthetotal p galacticmaterialindefiningthesizeandshapesofgalaxies(e.g. energy of the system. It has been shown that halos that have - Fall & Efstathiou1981). Yet despite its physicalsignificance, suffereda recentmajormergerwilltendto havea higherspin o r a precise and accurate understandingof the origin of galactic parameterλthantheaverage(e.g.Hetznecker&Burkert2006; t angular momentum remains one of the missing pieces in the Power,Knebe&Knollmann2008). Thereforeonecouldargue s a galaxyformationpuzzle. that within the framework of hierarchical structure formation [ Afundamentalassumptionincurrentgalaxyformationmod- that higher mass halos should have larger spin parameters on elsisthatgalaxiesformwhengascoolsandcondenseswithin average than less massive systems because they have assem- 1 v the potentialwellsofdarkmatterhalos(White&Rees 1978). bledalargerfractionoftheirmass(bymerging)morerecently. 3 Consequentlyitisprobablethattheangularmomentumofthe However, if we consider only halos in virial equilibrium, 5 galaxywillbelinkedtotheangularmomentumofitsdarkmat- should we expect to see a correlation between halo mass and 4 terhalo(e.g. Fall&Efstathiou1980;Mo,Mao&White1998; spin?Onemightnaïvelyexpectthatmoremassivesystemswill 4 Zavala, Okamoto & Frenk 2007). Within the context of hier- havehadtheirmaximumexpansionmorerecentlyandsothese 1. archical structure formation models, the angular momentum systemswillhavebeentidallytorquedforlongerthansystems 0 growth of a dark matter proto-halo is driven by gravitational that had their maximumexpansion at earlier times. This sug- 8 tidaltorquingduringtheearlystages(i.e. thelinearregime)of geststhatspinshouldincreasewithtimingofmaximumexpan- 0 itsassembly. This“TidalTorqueTheory”hasbeenexploredin sionandthereforehalomass. However,onefindsatbestaweak : detail;itisawell-developedanalytictheory(e.g.Peebles1969, correlationbetweenmassandspinforequilibriumhalosatz=0 v Doroshkevich 1979, White 1984) and its predictions are in (e.g. Cole&Lacey1996;Maccioetal. 2007;Bettetal. 2007, i X goodagreementwith theresultsofcosmologicalN-bodysim- hereafterB07),andthecorrelationisforspintodecreasewith r ulations (e.g. Barnes & Efstathiou 1987; Warren et al. 1992; increasinghalomass,contrarytoournaïveexpectation. a Sugerman, Summers& Kamionkowski2000; Porciani, Dekel Inthispaper,wereportona(weak)correlationbetweenspin & Hoffman 2002). However, once the proto-halo has passed and mass for equilibrium halos at redshift z=10. The trend is through maximum expansion and the collapse has become forhigher-masshalostohavesmallerspins,andisqualitatively non-linear, tidal torquing no longer provides an adequate de- similar to the one reported by B07 for the halo population at scription of the evolution of the angular momentum (White z=0. We presentthe main evidencein supportof this correla- 1984),whichtendstodecreasewithtime. Duringthisphaseit tion in Section 4 and we consider its implications for galaxy is likely thatmergerandaccretioneventsplayan increasingly formationinSection6. importantroleindeterminingboththemagnitudeanddirection oftheangularmomentumofagalaxy(e.g. Bailin&Steinmetz 2. THESIMULATIONS 2005). Indeed, a number of studies have argued that mergers Forthesimulationspresentedinthispaperwehaveadopted and accretion events are the primary determinants of the an- the cosmology as given by Spergel et al. (2003) (Ω = 0.3, 0 gular momenta of galaxies at the present day (Gardner 2001; ΩΛ =0.7, σ8 =0.9, and H0 =70km/sec/Mpc). Each run em- Maller,Dekel&Somerville2002;Vitvitskaetal.2002). ployed N =2563 particles and differed in simulation box-size L ,whichleadstotheparticlemassm differingbetweenruns box p Itiscommonpracticetoquantifytheangularmomentumof – m =ρ Ω (L /N)3, where ρ =3H2/8πG. This allows adarkmatterhalobythedimensionless“classical”spinparam- p crit 0 box crit 0 ustoprobearangeofhalomassesatredshiftz=10.Theprimary parametersofthesesimulationsaresummarizedinTable1. 1AstrophysikalischesInstitutPotsdam,AnderSternwarte16,14482Potsdam,Germany 2CentreforAstrophysics&Supercomputing,SwinburneUniversity,MailH31,POBox218,Hawthorn,VIC3122,Australia 1 2 Halos in all runs have been identified using the MPI paral- (2008)andourinterpretationisconsistentwiththecorrelations lelizedversionoftheAHFhalofinder3(AMIGA’s-Halo-Finder), wefindintheseruns. Thesecondisthatthetypicalcollapsing whichisbasedontheMHFhalofinderofGill,Knebe&Gibson massM∗ atz=10issmall–oforder103h- 1M –andbecause ⊙ (2004). Foreachhalowe computethevirialradiusR, defined weresolvemassscalesthathavecollapsedmoreorlesssimul- astheradiusatwhichthemeaninteriordensityis∆ timesthe taneously,weseeapopulationthathasyettorelax.Atz=1,the vir backgrounddensityoftheUniverseatthatredshift. Thisleads typicalcollapsingmassismuchlarger–oforder1011h- 1M – ⊙ tothefollowingdefinitionforthevirialmassM: andsoweresolveapopulationofhaloswhosemassaccretion histories are more diverse. The most massive systems tend to M= 4π∆ Ωρ R3. (2) be ones that have formed most recently, and are therefore the vir crit 3 leastdynamicallyrelaxed. Note that∆ is a functionof redshiftand amountsto ∆ We have used the following relation between Q and M to vir vir 210 at redshift z=10 , ∆ 230 at z=1, and the “usua≈l” classifydynamicallyrelaxedandunrelaxedsystems: vir ∆ 340atz=0(cf. Gross1≈997).Table1summarisestheto- vir talnu≈mberofhalos(Nhalos)recoveredbyAHF,whilezfinalgives Qallowed ∝ M- 0.015 ,forz=1 (4) theredshiftthatthesimulationhasbeenevolvedto. Qallowed const. ,forz=10 ∝ We addthatweranfiverealisationsofB20toredshiftz=10 We allow the Q valuesof halosin oursample to deviate from inordertohaveastatisticallysignificantsampleofhalosinthat thesescalingrelationsbynotmorethan particularmodel.However,wealsonotethatthefittingparam- eterspresentedinthefollowingSectionarerobustinthesense Q - Q Q Q +Q (5) that they do not depend on whether we stack the halos from allowed lim≤ ≤ allowed lim thosefiverunsorusethemindividually. withQlim=0.15(indicatedbythedashedlinesinFig.1). Fur- thermore, we consider only halos that contain at least N = min 3. THEHALOSAMPLE 600particleswithintheirvirialradiustoensurethatwearenot Weshowinacompanionpaper(Power,Knebe&Knollmann influenced by particle discreteness. Interestingly, when com- 2008)thatasubstantialfractionofthe halopopulationathigh puting spin, the tighest restriction on particle number comes redshiftisnotinvirialequilibrium. Becausewewishtoexam- not from the calculation of angular momentum but from the inethespindistributionofequilibriumhalos,itisimportantto calculationofthepotentialenergy.Bycomparinganalyticsolu- accountforunrelaxedsystemswheninvestigatingcorrelations tionswithMonteCarlorealisationsofNavarro,Frenk&White between spin and halo mass. For example, it has been shown (1997)haloes,wefindthatatleast600particlesarerequiredif that the spin can increase sharply in the aftermath of mergers theenergyistobecomputedtobetterthan10%. with massratiosas modestas5:1(e.g. Hetznecker& Burkert 4. THESPIN-MASSCORRELATION 2006),andthatthedegreetowhichahaloisindynamicalrelax- ation isas importantasrecentmerginghistoryin its influence Calculating the total energy E of a halo is computationally on spin (D’Onghia & Navarro 2007). To ensure that halos in expensive,andsocomputingλusingEq.(1)isalsoexpensive. oursampleareinvirialequilibrium,wecomputethevirialratio This prompted Bullock et al. (2001) to introduce a modified foreachhalo,whichwedefineas spinparameter 2T+S J Q= p+1. (3) λ′= , (6) U √2MVR Here T represents the kinetic energy, U the potential energy, whereV = GM/Rmeasuredthecircularvelocityatthevirial andS thesurfacepressureofagivenhaloofmassM. Byin- p radiusRandJ representstheabsolutevalueoftheangularmo- cluding S , we can accountfor the effect of infalling material p mentum. Wpe follow Bullock et al. and compute spin using on the dynamical state of the halo. Each of these quantities equation(6). are evaluatedusing allgravitationallyboundparticles, andwe InFigure2,weinvestigatethecorrelationbetweenhalospin adopttheformulaofShawetal.(2006)forthesurfacepressure λ′ and mass M. We show only halos that fulfill our selec- termS (cf. equations.(4)-(6)intheirstudy). p tioncriteria(individualdots)andbinthedatainfivemassbins InFigure1weshowthattherelationbetweenhalomassand equallyspacedinlog-spacebetweenM andM ofthecon- Q canvarywithmass. Thisisapparentatredshiftz=1, where min max sidered halos at the respective redshift. The values plotted as wefindatrendformoremassivehalostobelessvirialised. In histogramstherebyrepresenttheweightedmeanofallspinpa- contrast, high redshift halos are less virialised on average (as indicatedbytheincreasedaverage Q - 0.3),butwefindno rameters in the respective mass range where the weight is in- h i≈ verselyproportionaltotheerrorestimate apparenttrendwithmass. Why is there a mass dependence at z=1 but not at z=10? σ 0.2 Therearetwofactors.Thefirstisthathighredshifthalos“see” J = (7) aneffectiveslopeoftheinitialpowerspectrumofn - 3,and J λ′√N eff ≈ so the time at which a particular mass scale starts to collapse forthespinparameterofahaloconsistingofN particlesasde- is relatively insensitive to mass. Therefore we do not expect rivedin Bullock et al. (2001)(cf. equation(7) in that study). to find a strong correlation between virial state Q and mass. The error bars indicate the standard deviation of the spin pa- Wehavecheckedthishalopopulationsdrawnfromthesimula- rametervaluesinthebinfromtheweightedmean.4 tionsofscale-freecosmologiesofKnollmann,Power&Knebe Thebestfittingpower-lawstothesehistogramsrevealthat 3AHFisalreadyfreelyavailablefromhttp://www.aip.de/People/aknebe 4Weliketonoteinpassingthatwealsoperformedalloftheanalysisandstabilitychecksusingthemedianandthescatteraboutthemedianineachbin. The resultsremainunaffectedandwethereforedecidetoonlylistthemfortheweightedmeans. 3 – Q =0.5 lim λ′ Mα (8) ∝ Netoetal. (2007)criteria: with • α = - 0.002 0.149 ,forz=1 – Nmin=600 α = - 0.059±0.171 ,forz=10. (9) – x <0.07 ± off This indicates that there is a weak correlation at high red- – f <0.1 sub shiftsforspintodecreasewithincreasingmass,albeitstronger than the one at z=1. We compute Spearman rank correlation Here N is again the minimum number of particles in a min coefficientsatz=10(1)andfindRs=- 0.137(0.06). halo,QlimtheviriallimitasdefinedinEq.(5),xoffmeasuresthe As an alternative approach, we fit a lognormal function to distancebetweenthemostboundparticleandthecentreofmass eachofourhalosamplesatz=10andz=1, inunitsofthevirialradius,ρ isanindicatorofhowwellthe rms densityprofileofthehalocanbefittedbyaNavarro,Frenk& 1 ln2(λ′/λ′) White (1997) profile (cf. Eq. (2) in Maccio et al. 2007), and P(λ′)= exp - 0 . (10) λ′√2πσ0 2σ02 ! fsub is the fraction of mass in subhalos. The resulting power lawslopesα(cf. Eq.(8))forz=1andz=10arepresentedin TheresultingcurvesarepresentedinFig.3whereasthebest-fit Table6.Againwenotethatthereappearstobeamuchstronger parameters, median values for λ′ =median(λ′) and median correlationbetweenλ′ andM athigherredshift. Whilethere- med halomassesMmed aregiveninTable2. Inspectionofthebest- lation is consistent with zero at z=1 (as confirmed by Maccio fitparametersconfirmthatthemedianspindeclinesaswemove et al. 2007) spin and mass are correlated at z=10. How this fromlessmassivetomoremassiveobjectsathighredshift. resultrelatestotheBettetal. (2007)result–whofindaweak correlationatz=0–willbediscussedinthefollowingSection. 5. STABILITYOFRESULTS Becauseoftheweaknatureofthemeasuredcorrelationitis 6. CONCLUSIONSANDDISCUSSION vitallyimportanttocheckitscredibilitybyperformingastatis- We have performed a careful investigation of the relation ticalanalysis.Tothisextentweinvestigatethesensitivityofthe betweenvirialmassanddimensionlessspinparameterfordark logarithmicslopeαwithrespecttoanumberofparametersthat matterhalosformingathighredshiftsz>10inaΛCDMcos- enter into its determination, namely the number of bins Nbins mology. Theresultofourstudy,which∼isbasedona seriesof usedforthehistograms;thevirialisationcriterionparametrized cosmological N-body simulations in which box size was var- via Qlim; and the minimum number of particles Nmin within a iedwhilekeepingparticlenumberfixed,indicatesthatthereis halo’svirialradius. Notethatwevaryoneparameteratatime, a weak correlation between mass and spin at z=10, such that keepingtheothersattheirfixed“standard”values. Theresults the spin decreases with increasing mass. If there is a corre- arepresentedinTable3,Table4,and5. lation at z=1, we argue that it is significantly weaker than the Wefindthatbinnumberhaspracticallynoeffectontheslope. one we find at z=10; this is in qualitative agreement with the SimilarlywefindthatvaryingthevirialisationcriterionQlimhas findings of previous studies that focused on lower redshifts littleeffectontheslopeoftherelationbetweenmassandspin, (Maccio et al. 2007, Shaw et al. 2005, Lemson & Kauffmann regardless of redshift (Table 4). In contrast, we find that the 1999). minimumnumberof particleswithin a halo’svirialradiushas astrongandsystematiceffectontheresultatz=1–asNmin in- Interestingly, B07 find a weak correlation between me- creases, wefindthatthelogarithmicslopebecomesshallower. dian spin and halo mass at z=0 in the Millennium Simulation Thisdoesnotappeartobetrueathighredshift,althoughaswe (Springelet al. 2005), in the same spirit as the one presented go to earlier times we find that the number of massive haloes here for z=10: lower mass halos tend to have higher spins. becomesprogressivelysmallerandourdeterminationofαbe- However, as we show, the correlation between halo mass and comesincreasinglyunreliable. spinisweakeratz=1thanatz=10,whereasthecorrelationre- Thesetestsleadustobelievethatourresultsarebothstable ported in B07 for halos at z=0 is much stronger than the one andreliable,andourmainresultholds:thecorrelationbetween we find at z=1. This is not what one would expect, and so spinparamaterλ′ andhalomassM isoneorderofmagnitude it is important to try and understand the source of the differ- largeratredshiftz=10thanatz=1. encebetweenourresultatz=1andtheB07resultatz=0. B07 fitted a 3rd-order polynomial to the median spins of halos in Asafurthertestofthecredibilityofthecorrelationwemea- themassrange3 1011<M/(h- 1M⊙)<3 1014 atz=0. The sure between halo mass and spin, we use the criteria of three form of this poly×nomial∼is extremely s∼ensi×tive to the precise otherstudiestoselectourhalosample. Theseare: valuesofthebest-fitparameters(Bett,privatecommunication) and it is not straightforward to extrapolate its behaviour out- Maccioetal. (2007)criteria: side of the given mass range and redshift. We derive our es- • timates of the power-lawexponentsfrom the spin distribution – N =250 min withrespecttohalomassatz=1.Ourhaloslieinthemassrange – xoff<0.04 3 109<M/(h- 1M⊙)<5 1012. B07basetheirmedianspins × × – ρ <0.4 upon ∼1.5 million ha∼los with correspondingly small errors, rms ∼ and note that the weak nature of the trend of spin with mass Bettetal. (2007)criteria: makesithardtodetect. Thissuggeststousthatthecorrelation • betweenmassandspinatz=10isremarkablystrongratherthan – N =300 thecorrelationatz=1beingtooweak! min 4 When studying correlations between halo mass and spin, diskwithascalelengthdeterminedbythespecificangularmo- greatcaremustbetakenindefiningthehalosample. Inpartic- mentumofthegas,whichwewouldexpecttoberelatedtothe ular, wefindthatmassresolution(i.e. the numberofparticles angularmomentumofthehalo(e.g. Zavala,Okamoto&Frenk with which a halo is resolved) and the degree of virialisation 2007). of a halo can have a significant effect on the strength of the If more massive halos at high redshifts show a tendency to correlation(atleastatz=1, cf. Table5). This–atleast–isin havesmallerspinparameters,thegasdiskswillhavelowerspe- goodagreementwiththefindingsofB07. cificangularmomentaandthereforewillbemorecentrallycon- centrated.Ifstarformationratecorrelateswithsurfacedensity, We note that Power & Knebe (2006) demonstrated that the thenwemightexpectthestarformationratetobeenhancedin sizeofsimulationboxcanleadtoasuppressionofangularmo- moremassivehalos.Becausemassivehalostendtoformprefer- mentum in smaller boxes, due to the absence of longer wave- entiallyinhighdensity,highlyclusteredenvironmentsinwhich length perturbations in the initial conditions. This will lead themergerratealsotendstobeenhanced,thenwemightexpect to a bias in our estimate of λ (approximatelya 10%effect) star formation to proceed more rapidly and at earlier times in ∼ butwehaveverifiedthatthespindistributionsweobtainfrom thesehalos. Mightthisexplaintheeffectof“downsizing”(e.g. a B20 run truncated on scales larger than the longest wave- Cowie et al. 1996), the successive shifting of star formation length perturbation modelled in the B1 run produces results fromhigh-tolow-massgalaxieswithdecreasingredshift? We that are consistent. Indeed, we would expect the correlation shallpursuethisinamorequantitativemannerinfuturework. to be strengthenedif the B1 spinswere correctedfor boxsize effects. AK would like to thank Swinburne University for its hos- pitality where this work was initiated. AK further acknowl- Itisinterestingtospeculateontheconsequencesofthiscor- edges funding through the Emmy Noether programme of the relation for galaxy formation at high redshifts and the galaxy DFG(KN755/1). CPacknowledgesfundingthroughtheARC populationweobservetoday. Inthestandardpictureofgalaxy Discovery Projected funded “Commonwealth Cosmology Ini- formation,gascoolsontodarkmatterhalosandisshockheated tiative”, grantDP 0665574. The simulationspresented in this tothevirialtemperatureofthehalo.Theangularmomentumof paperwerecarriedoutontheBeowulfclusterattheCentrefor thegasandthedarkmattershould(initially)besimilarbecause Astrophysics&Supercomputing,SwinburneUniversityaswell theyaresubjecttothesametidalfield. Astheinnermostdens- astheSanssouciclusterattheAstrophysikalischesInstitutPots- estpartsofthegaseoushalocool,theywillsettleintoagaseous dam. REFERENCES Barnes,J.;Efstathiou,G.;1987,ApJ319,575 Maccio,A.V.;Dutton,A.A.;vandenBosch,F.C.;Moore,B.;Potter,D.;Stadel, Bett, P.; Eke, V.; Frenk, C.S.; Jenkins, A.; Helly, A.; Navarro, J.; 2007, J.;2007,MNRAS378,55 astro-ph/0608607 Maller,A.H.;Dekel,A.;Somerville,R.;2002,MNRAS329,423 Bullock,J.S.;Dekel,A.;Kolatt,T.S.;Kravtsov,A.V.;Klypin,A.A.;Porciani, Mo,H.J.;Mao,S.;White,S.D.M.;1998,MNRAS295,319 C.;Primack,J.R.;2001,ApJ555,240 NavarroJ.F.,FrenkC.S.,WhiteS.D.M.,1997,ApJ490,493 Cole,S.;Lacey,C.;1996,MNRAS281,726 NetoA.F.,etal.,astro-ph/0706.2919 CowieL.L.,SongailaA.,HuE.M.,ChoenJ.G.,1996,AJ112,839 PeeblesJ.;1969 Doroshkevich,1979 Power,C.B.;Knebe,A.;Knollmann,S.;2008,MNRASsubmitted FallS.M.&EfstathiouG.1980,MNRAS,193,189 Porciani;Dekel;Hoffman;2002 Gao,L.;White,S.D.M.;2007,MNRAS377,L5 Shaw,L.D.;Weller,J.;Ostriker,J.P.;Bode,P.;2006,ApJ646,815 Gardner,J.P.;2001,ApJ557,616 Spergel,D.;etal.;2003 Gill,S.P.D.;Knebe,A.;Gibson,B.K.;2004,MNRAS351,399 Springel,V.;etal.;2005,Nature435,629 GottloeberS.,YepesG.,2007,ApJ664,117 Sugerman;Summers;Kamionkowski;2000 Gross,M.A.K.,1997,PhDthesisUniv.California,SantaCruz Vitvitska, M.; Klypin, A.A.; Kravtsov, A.V.; Wechsler, R.H.; Primack, J.R.; Hetznecker,H.;Burkert,A.;2006,MNRAS370,1905 Bullock,J.S.;2002,ApJ581,799 KnollmannS.,PowerC.,KnebeA.,2008,MNRASaccepted White,S.D.M.&Rees,M.1978,MNRAS,183,341 Lemson,G.;Kauffmann,G.;1999,MNRAS302,111 White,S.D.M.;1984 ZavalaJ.,OkamotoT.&Frenk.C.S.astro-ph/0710.2901 5 TABLE 1 SUMMARY OF THECOSMOLOGICAL SIMULATIONS AND NUMBEROF HALOS. run L [h- 1Mpc] m [h- 1M ] z Nz=1 Nz=10 Nz=1 Nz=10 box p ⊙ final halos halos relaxedhalos relaxedhalos B01 1 4.9 103 10 — 8780 — 286 B02 2.5 7.8×104 10 — 7991 — 201 B05 5 6.2×105 1 16917 6532 832 109 B10 10 4.9×106 1 18589 4360 949 37 B20 20 4.0×107 1 20514 10947 995 27 B100 100 4.9×109 1 24696 — 978 — × TABLE 2 FITTING PARAMETERS FORP(λ). run λ′0 σ λ′med Mmed[h- 1M⊙] run λ′0 σ λ′med Mmed[h- 1M⊙] z=1 z=10 — B01 0.042 0.538 0.041 5.95e+06 — B02 0.040 0.539 0.037 8.19e+07 B05 0.033 0.521 0.031 7.92e+08 B05 0.036 0.516 0.035 5.63e+08 B10 0.037 0.527 0.034 6.15e+09 B10 0.033 0.540 0.029 4.14e+09 B20 0.038 0.543 0.036 5.40e+10 B20 0.030 0.251 0.027 3.33e+10 B100 0.037 0.535 0.035 5.25e+12 — TABLE 3 VARIATION OFαWITHNbins (Qlim=0.15,Nmin=600). Nbins αz=1 σα αz=10 σα ± ± 4 -0.001 0.169 -0.061 0.191 ± ± 5 -0.002 0.149 -0.059 0.171 ± ± 6 -0.005 0.137 -0.053 0.155 ± ± 7 -0.006 0.128 -0.056 0.144 ± ± 8 -0.003 0.120 -0.069 0.132 ± ± TABLE 4 VARIATION OFαWITHQlim (Nbins=5,Nmin=600). Qlim αz=1±σα αz=10±σα Nrze=la1xedhalos Nrze=la1x0edhalos 0.05 -0.007 0.148 -0.058 0.153 2123 264 ± ± 0.10 -0.004 0.148 -0.062 0.168 3365 497 ± ± 0.15 -0.002 0.149 -0.059 0.171 3811 660 ± ± 0.20 -0.005 0.151 -0.052 0.173 3994 739 ± ± 0.25 -0.002 0.153 -0.052 0.173 4165 774 ± ± 6 TABLE 5 VARIATION OFαWITHNmin (Qlim=0.15,Nbins=5). Nmin αz=1±σα αz=10±σα Nrze=la1xedhalos Nrze=la1x0edhalos 100 0.003 0.129 -0.037 0.158 19707 5478 ± ± 200 0.002 0.137 -0.038 0.167 10809 2534 ± ± 300 0.002 0.142 -0.035 0.174 7336 1526 ± ± 600 -0.002 0.149 -0.059 0.171 3811 660 ± ± 1000 -0.004 0.155 -0.058 0.196 2303 343 ± ± 2000 -0.011 0.164 -0.048 0.186 1154 120 ± ± TABLE 6 APPLYING DIFFERENTVIRIALISATION CRITERION (Nbins=5). criterion α α Nz=1 Nz=10 z=1 z=10 relaxedhalos relaxedhalos Netoetal. (2007) 0.000 0.149 -0.041 0.178 3486 429 ± ± Maccioetal. (2007) 0.001 0.146 -0.040 0.184 4275 512 ± ± Bettetal. (2007) 0.003 0.146 -0.040 0.175 8582 1955 ± ± 7 FIG.1.—RelationbetweenvirialparameterQ(cf.equation3)andhalomass.ThesolidlinesrepresenttheadoptedvirialisationcriteriaasgivenbyEq.(4).Note thatwealreadyappliedthemasscutof600particlesperhaloforthisplotandhencethenumberofhalosappearingdoesnotagreewiththenumbergiveninTable1. FIG.2.—CorrelationofspinparameterλwithmassM.Thebinneddata(histograms)hasbeenfittedtoapowerlaws(dashedline,cf.Eq.(9)). FIG.3.—Lognormaldistributionsofthespinparameterλ′.