N-body Dark Matter Haloes with simple Hierarchical Histories
MMon. Not. R. Astron. Soc. , 1–23 (2014) Printed 10 July 2018 (MN L A TEX style file v2.2)
N-body Dark Matter Haloes with simple HierarchicalHistories
Lilian Jiang, John C. Helly, Shaun Cole, Carlos S. Frenk
Institute for Computational Cosmology, Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK
10 July 2018
ABSTRACT
We present a new algorithm which groups the subhaloes found in cosmological N-body simulations by structure finders such as subfind into dark matter haloes whoseformation histories are strictly hierarchical. One advantage of these ‘Dhaloes’ overthe commonly used friends-of-friends (FoF) haloes is that they retain their individualidentity in cases when FoF haloes are artificially merged by tenuous bridges of particlesor by an overlap of their outer diffuse haloes. Dhaloes are thus well-suited for modellinggalaxy formation and their merger trees form the basis of the Durham semi-analyticgalaxy formation model, galform . Applying the Dhalo construction to the ΛCDMMillennium-2 simulation we find that approximately 90% of Dhaloes have a one-to-one, bijective match with a corresponding FoF halo. The remaining 10% are typicallysecondary components of large FoF haloes. Although the mass functions of both typesof haloes are similar, the mass of Dhaloes correlates much more tightly with thevirial mass, M , than FoF haloes. Approximately 80% of FoF and bijective andnon-bijective Dhaloes are relaxed according to standard criteria. For these relaxedhaloes all three types have similar concentration– M relations and, at fixed mass,the concentration distributions are described accurately by log-normal distributions. Key words: methods: numerical - galaxies: haloes - cosmology: theory - dark matter.
In hierarchical dark matter dominated cosmologies, such asstandard ΛCDM, galaxy formation is believed to be inti-mately linked to the formation and evolution of dark matterhaloes. Baryonic gas falls into dark matter haloes, cools andsettles into centrifugally supported star forming discs (Bin-ney 1977; Rees & Ostriker 1977; White & Rees 1978; White& Frenk 1991; Kauffmann & White 1993; Cole et al. 1994;Somerville & Primack 1999; Benson et al. 2003). Thus theevolution of the galaxy population is driven by the evolutionof the population of dark matter haloes which grow hierar-chically via mergers and accretion. Thus to model galaxyformation one must first have an accurate model of the evo-lution of dark matter haloes.The formation and evolution of dark matter haloes fromcosmological initial conditions in large representative vol-umes can now be routinely and reliably simulated using avariety of N-body codes (e.g. Springel 2005a). In contrast,simulations of the evolution of the baryonic component aremuch more uncertain with gross properties depending onthe details of uncertain sub-grid physics as well as on thelimitations of numerical hydrodynamics (Schaye et al. 2010;Creasey et al. 2011). Hence a useful and complementaryapproach is semi-analytic galaxy formation (e.g. White & Frenk 1991; Cole 1991; Lacey & Silk 1991; Kauffmann &White 1993; Cole et al. 1994, 2000; Somerville & Primack1999; Somerville et al. 2008; Bower et al. 2006; Benson &Bower 2010) in which one starts with the framework pro-vided by the dark matter halo evolution and uses analyticmodels to follow the processes of galaxy formation that oc-cur within these haloes. The key starting point for this ap-proach is halo merger trees which quantify the hierarchicalgrowth of individual dark matter haloes.In ΛCDM the first self-bound objects to form are haloeswith masses of around an Earth mass corresponding to thesmall scale thermal cut off in the CDM power spectrum(Green, Hofmann, & Schwarz 2004). In a cosmological N-body simulation the mass scale of the first generation ofhaloes is instead set by the mass resolution of the simulation.Subsequent generations of haloes form by mergers of earliergenerations of haloes plus some smoothly accreted mate-rial. The merging process does not produce a completelyrelaxed smooth halo and the remnants of the earlier gener-ation of haloes are often detectable as self-bound substruc-tures (subhaloes) within the new halo. Thus it is impor-tant to distinguish between haloes and the subhaloes thatthey contain which are the remnants of early generations ofnow merged haloes. A variety of algorithms which can iden- c (cid:13) a r X i v : . [ a s t r o - ph . C O ] F e b Jiang et al. tify these subhaloes in N-body simulations has been devised(Onions et al. 2012). These substructure finders are capableof detecting arbitrary levels of nested subhaloes within sub-haloes and in most cases also identify the background massdistribution in a halo as a subhalo. In this work we referto all of the groups identified by such substructure findersas “subhaloes” and merger trees constructed by identifyinga descendant for each subhalo as “subhalo merger trees”.Srisawat et al. (2013) compare a range of methods for theproduction of subhalo merger trees. The algorithm we useto determine subhalo descendants in this paper is includedin the comparison under the name
D-Trees .To construct the halo merger trees needed by semi-analytic models it is not sufficient to just track subhaloes be-tween simulation outputs (e.g. by tracking their consituentparticles), one also needs to identify their host haloes. Forinstance when a galaxy cluster forms it is normally assumedthat while the galaxies remain in their individual subhaloesthe diffuse gas surrounding them and gas blown out of thegalaxies by SN driven winds is not retained by the indi-vidual subhaloes but instead joins the common intra-clustermedium of the surrounding halo of the galaxy cluster. An-other issue that has to be addressed when building mergertrees for use in galaxy formation models is that structure for-mation for the collisionless material in N-body simulations isnot strictly hierarchical. Hence occasionally when two haloesmerge the subhalo resulting from the smaller progenitor canpass straight through the main halo and escape to beyond itsvirial radius. For the galaxy formation process to be followedit is necessary to retain the association between these twoseparated subhaloes so that an appropriate physical modelcan be applied to their diffuse collisional gas which wouldnot have separated after the merger. Merger trees that areuseful for galaxy formation modelling have to take accountof these considerations (Knebe et al. 2013). The Dhalo al-gorithm which we present produces a set of haloes which isstrictly hierarchical in the sense that once a subhalo becomesa component of a Dhalo it never subsequently demerges.It is now quite common for semi-analytic models to usehalo merger trees extracted directly from N-body simula-tions (Springel et al. 2001; Helly et al. 2003; Hatton et al.2003; Bower et al. 2006; Mu˜noz et al. 2009; Koposov etal. 2009; Busha et al. 2010; Macci`o et al. 2010; Guo et al.2011). There are many choices to be made both in definingthe halo catalogues and in constructing the links betweenhaloes at different times. Knebe et al. (2011) and Knebeet al. (2013) have found significant differences in even themost basic properties (e.g the halo mass function) of halocatalogues constructed with different group finding codes.Additionally, these halo catalogues can often be modifiedby the procedure of constructing the merger trees as someof the algorithms break up or merge haloes together in orderto achieve a more consistent membership over time (Helly etal. 2003; Behroozi et al. 2013). So, for example, even if onestarts with standard Friends-of-Friends (FoF) groups (Daviset al. 1985) the process of building the merger trees can alterthe abundance and properties of the haloes.Semi-analytic models such as galform have the op-tion of using information extracted directly from an N-bodysimulation or using Monte Carlo methods (see Jiang & vanden Bosch 2013, for a comparison of different algorithms)which make use of statistical descriptions of N-body results such as analytic halo mass functions (e.g. Sheth & Tormen1999; Jenkins et al 2001; Evrard et al. 2002; White 2001;Reed et al 2003; Linder & Jenkins 2003; Lokas, Bode &Hoffmann 2004; Warren et al. 2006; Heitmann et al. 2006;Reed et al 2007; Lukic et al 2009; Tinker et al. 2008; Boylan-Kolchin et al. 2009; Crocce et al. 2010; Courtin et al. 2011;Bhattacharya et al. 2011; Watson et al. 2013) and modelsfor the distribution of the concentrations of halo mass pro-files (e.g. Navarro, Frenk & White 1995, 1996; Bullock et al.2001; Eke, Navarro & Steinmetz 2001; Macci`o, Dutton, &van den Bosch 2008). These statistical descriptions are oftenbased on the abundance and properties of FoF haloes andso may not be directly applicable to the catalogues of haloesthat result from the application of a specific merger tree al-gorithm. The internal structure of the dark matter haloesstrongly influences galaxy formation models. Often the gasdensity profiles within dark matter haloes are assumed to berelated to the dark matter profile, e.g. through hydrostaticequilibrium and these influence the rate at which gas coolsonto the central galaxy. In addition the central potential ofthe dark matter halo effects the size and circular velocityof the central galaxy which in turn can have a strong effecton the expulsion of gas from the galaxy via SN feedback.Hence for semi-analytic galaxy formation modelling it is im-portant to adopt models of the individual haloes that areconsistent with the haloes that appear in the merger treesused by semi-analytic model.In this paper we present a detailed description of the lat-est N-body merger tree algorithm that has been developedfor use with the semi-analytic code galform . The algorithmis an improvement over the earlier version, described in Mer-son et al. (2012), which was run on the Millennium simu-lation (Springel 2005a) and widely exploited in a range ofapplications (Bower et al. 2006; Font et al. 2008; Kim et al.2011; Merson et al. 2012). The resulting differences betweenthe two algorithms are very small when applied to relativelylow resolution simulations such as the Millennium, but theimprovements in the new algorithm do a better job of track-ing halo descendants in high resolution simulations such asthe Millennium II (Boylan-Kolchin et al. 2009) and Aquar-ius simulations (Springel et al. 2008). The starting point forour merger trees are FoF haloes that are decomposed intosubhaloes, distinct self-bound structures, by the substruc-ture finder, subfind (Springel et al. 2001). Subhaloes aretracked between output times and agglomerated into a newset of haloes, dubbed Dhaloes, that have consistent member-ship over time in the sense that once a subhalo is accretedby a Dhalo it never demerges. In this process we also splitsome FoF haloes into two or more Dhaloes when subfind substructures are well separated and only linked into a singleFoF halo by bridges of low density material.Our paper is structured as follows. In Section 2 webriefly outline the new merger tree algorithm (full detailsare given in Appendix A) and its application to the Millen-nium II simulation. In Section 3 we compare and contrastthe properties of the resulting Dhaloes with the more com-monly used FoF haloes (Davis et al. 1985). We show specificrare examples where Dhaloes and their matched FoF coun-terparts exhibit gross differences either one FoF halo beingdecomposed into several Dhaloes or vice versa. We also ex-amine the distribution of mass ratios for matching Dhalo andFoF pairs. Then in Section 4 we compare statistical proper- c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes M halo /h − M fl F r a c t i o n Bijective(Dhalo)Bijective(FoF) Unbound(FoF)Remerged(FoF)
Figure 1.
The upper two curves, with bootstrap error bars, showthe fraction of Dhalo (red) and FoF haloes (blue) in the MSIIcatalogues that have a bijective (a unique one-to-one) match asa function of their respective Dhalo or FoF halo mass. The lowertwo curves show the fraction of FoF haloes that do not containa self-bound substructure (cyan) and the fraction whose mainsubhaloes are remerged by the Dhalo algorithm to form part of amore massive Dhalo (green). ties of halo populations including halo mass functions andtheir concentration–mass relation. We conclude in Section 5.
Immediately below, Section 2.1, we summarise the specifi-cation of the Millennium II simulation which we use to testand illustrate the application of our merger tree algorithm.We then give a brief outline of the construction of the mergertrees and their constituent haloes with the complete speci-fication detailed in Appendix A.
The Millennium-II (MSII) simulation (Boylan-Kolchin etal. 2009) was carried out with the gadget3 N-body code,which uses a “TreePM” method to calculate gravitational The Millennium-II simulation data will be availablefrom an SQL relational database that can be accessed athttp://galaxy-catalogue.dur.ac.uk:8080/Millennium . forces. The MSII is a cosmological simulation of the stan-dard ΛCDM cosmology in a periodic box of side L box =100 h − Mpc containing N = 2160 particles of mass 6 . × h − M (cid:12) . The cosmological parameters for the MSII are:Ω m = 0 .
25, Ω b = 0 . h = 0 .
73, Ω Λ = 0 . n = 1 and σ = 0 .
9. Here Ω m denotes the total matter density in unitsof the critical density, ρ crit = 3 H / (8 πG ). Ω b and Ω Λ de-note the densities of baryons and dark energy at the presentday in units of the critical density. The Hubble constant is H = 100 h km s − Mpc − , n is the primordial spectral in-dex and σ is the rms density fluctuation within a sphere ofradius 8 h − Mpc extrapolated to z = 0 using linear theory.These cosmological parameters are consistent with a com-bined analysis of the 2dFGRS (Colless et al. 2001; Percivalet al. 2001) and first year WMAP data (Spergel et al. 2003;Sanchez et al. 2006). The first step in building our merger trees is the construc-tion of catalogues of both FoF haloes (Davis et al. 1985) andtheir internal self-bound substructures , subhaloes, as iden-tified by subfind (Springel et al. 2001). The second step isto build subfind merger trees by tracking particles betweenoutput snapshots to determine the descendant of each sub-halo. Occasionally subfind fails to find a substructure as ittransits through the core of a larger halo. To avoid this re-sulting in the premature merging of substructures we havedeveloped an algorithm (Appendix A2) that looks severalsnapshots ahead to robustly link progenitor and descendantsubhaloes. A similar approach was adopted by Behrooziet al. (2013) to construct self-consistent merger trees forthe Bolshoi simulations (Klypin, Trujillo-Gomez, & Primack2011). The third step is to partition these subfind mergertrees into discrete branches. A new branch begins whenevera new subhalo forms and continues for as long as the subhaloexists in the simulation. When a merger occurs we decidewhich of the progenitor subhaloes survives the merger bydetermining which progenitor contributed the most boundpart of the descendant (see Appendix A2.1 ). The branchcorresponding to this progenitor continues, while the otherprogenitor’s branch ends. The final step is to bundle thesebranches together to define the composite Dhaloes and theirmerger trees. Here our algorithm (described in full in Ap-pendix A3) defines collections of subhaloes embedded hier-archically within each other as a single Dhalo, but excludesneighbouring subhaloes that may be part of the same FoF Here we identify the subhaloes using the subfind algorithm(Springel et al 2001). However this is not the only option andthere is now a large literature (see Onions et al. 2012) on alterna-tive methods of identifying self-bound structures. Some of theseare highly sophisticated and use full 6D phase space informationto disentangle spatial coincident subhaloes (Diemand, Kuhlen, &Madau 2006; Behroozi, Wechsler, & Wu 2012). As an example, wehave experimented with building Dhaloes by applying the Dhaloalgorithm from Appendix A3 onwards but with subfind subhalomerger trees replaced by those defined by the Hierarchical BoundTracing (HBT) algorithm of (Han et al. 2012). We find that theproperties of the Dhalo merger trees and the galaxies that resultafter they are processed by galform are extremely similar.c (cid:13)000
9. Here Ω m denotes the total matter density in unitsof the critical density, ρ crit = 3 H / (8 πG ). Ω b and Ω Λ de-note the densities of baryons and dark energy at the presentday in units of the critical density. The Hubble constant is H = 100 h km s − Mpc − , n is the primordial spectral in-dex and σ is the rms density fluctuation within a sphere ofradius 8 h − Mpc extrapolated to z = 0 using linear theory.These cosmological parameters are consistent with a com-bined analysis of the 2dFGRS (Colless et al. 2001; Percivalet al. 2001) and first year WMAP data (Spergel et al. 2003;Sanchez et al. 2006). The first step in building our merger trees is the construc-tion of catalogues of both FoF haloes (Davis et al. 1985) andtheir internal self-bound substructures , subhaloes, as iden-tified by subfind (Springel et al. 2001). The second step isto build subfind merger trees by tracking particles betweenoutput snapshots to determine the descendant of each sub-halo. Occasionally subfind fails to find a substructure as ittransits through the core of a larger halo. To avoid this re-sulting in the premature merging of substructures we havedeveloped an algorithm (Appendix A2) that looks severalsnapshots ahead to robustly link progenitor and descendantsubhaloes. A similar approach was adopted by Behrooziet al. (2013) to construct self-consistent merger trees forthe Bolshoi simulations (Klypin, Trujillo-Gomez, & Primack2011). The third step is to partition these subfind mergertrees into discrete branches. A new branch begins whenevera new subhalo forms and continues for as long as the subhaloexists in the simulation. When a merger occurs we decidewhich of the progenitor subhaloes survives the merger bydetermining which progenitor contributed the most boundpart of the descendant (see Appendix A2.1 ). The branchcorresponding to this progenitor continues, while the otherprogenitor’s branch ends. The final step is to bundle thesebranches together to define the composite Dhaloes and theirmerger trees. Here our algorithm (described in full in Ap-pendix A3) defines collections of subhaloes embedded hier-archically within each other as a single Dhalo, but excludesneighbouring subhaloes that may be part of the same FoF Here we identify the subhaloes using the subfind algorithm(Springel et al 2001). However this is not the only option andthere is now a large literature (see Onions et al. 2012) on alterna-tive methods of identifying self-bound structures. Some of theseare highly sophisticated and use full 6D phase space informationto disentangle spatial coincident subhaloes (Diemand, Kuhlen, &Madau 2006; Behroozi, Wechsler, & Wu 2012). As an example, wehave experimented with building Dhaloes by applying the Dhaloalgorithm from Appendix A3 onwards but with subfind subhalomerger trees replaced by those defined by the Hierarchical BoundTracing (HBT) algorithm of (Han et al. 2012). We find that theproperties of the Dhalo merger trees and the galaxies that resultafter they are processed by galform are extremely similar.c (cid:13)000 , 1–23
Jiang et al. group, but are only linked in by a bridge of low density mate-rial or subhaloes that are beginning the process of mergingbut have not yet lost a significant amount of mass. Sub-haloes are grouped into Dhaloes in such a way that once asubhalo becomes part of a Dhalo it remains a component ofthat Dhalo’s descendants at all later times at which the sub-halo survives, even if it is a satellite component that takesit temporarily outside the corresponding FoF halo. All of aDhalo’s subhaloes which survive at a later snapshot mustbelong to the same Dhalo at that snapshot. We take thisto be the descendant of the Dhalo. This defines the Dhalomerger trees. The mass of a Dhalo is simply the sum of themasses of its component subhaloes. The properties of FoF haloes, especially those defined bythe conventional linking length parameter of b = 0 . b times the mean inter-particleseparation), are well documented in the literature (e.g. Frenket al. 1988; Lacey & Cole 1994; Summers, Davis, & Evrard1995; Audit et al 1998; Huchra & Geller 1982; Press & Davis1982; Einasto et al. 1984; Davis et al. 1985; Frenk et al. 1988;Lacey & Cole 1994; Klypin et al. 1999; Jenkins et al 2001;Warren et al. 2006; Eke et al. 2004; Gottl¨ober & Yepes 2007)and such haloes are widely used as the starting point forrelating the dark matter and galaxy distributions(Peacock &Smith 2000; Seljak 2000; Berlind & Weinberg 2002). Thus asthe semi-analytic model galform (Bower et al. 2006; Fontet al. 2008, 2011; Lagos et al. 2011) instead uses Dhaloes asits starting point, it is interesting to contrast the propertiesof haloes defined by these two algorithms.As described in Section 2, FoF haloes are decomposedby subfind into subhaloes and those are then regrouped intoDhaloes. Hence for every FoF halo, we can find its matchingDhalo by finding which Dhalo contains the most massivesubhalo from the FoF group. We can perform this matchingthe other way round by finding the FoF halo containing themost massive subhalo from the Dhalo. In cases where themost massive subhalo of a FoF halo is also the most massivesubhalo of a Dhalo, these two matching procedures produceidentical associations. We refer to such cases as bijectivematches.Before comparing the properties of this subset of bi-jectively matched Dhaloes and FoF haloes we first quantifyhow representative they are by looking at the fraction ofeach set of haloes that have these bijective matches. Thetwo upper curves in Fig. 1 show the dependence of the bi-jective fraction of Dhaloes on Dhalo mass and FoF haloeson FoF mass. The first thing to note is that the fraction ofbijectively matched Dhaloes is large, being 90% or greaterover the full range from 10 to 10 h − M (cid:12) and so to afirst approximation there is a good correspondence betweenFoF and Dhaloes. Above 3 × h − M (cid:12) about 10% of the As a subhalo can, by definition, only belong to one Dhalo and asparticles can only belong to one subfind subhalo this means thatDhaloes are exclusive in the sense that no particles can belong tomore than one Dhalo.
Dhaloes do not have a bijective match which means theyinstead represent secondary fragments of more massive FoFhaloes that the Dhalo algorithm has split into two or moresubhaloes. Below 3 × h − M (cid:12) this non-bijective fractiondrops indicating that lower mass FoF haloes are less likelyto be split into two or more comparable mass Dhaloes. Thisbehaviour is consistent with the results of Lukic et al (2009)who found that 15-20% of FoF haloes are irregular structuresthat have two or more major components linked together bylow density bridges and that this fraction is an increasingfunction of halo mass. This is also to be expected in the hier-archical merging picture as the most massive haloes formedmost recently and so are the least dynamically relaxed.For the FoF haloes with mass above 10 h − M (cid:12) the bi-jectively matched fraction is unity, indicating that the mostmassive subhalo of such FoF haloes together with the sub-haloes embedded within it always gives rise to a Dhalo.Below 10 h − M (cid:12) the the bijective fraction begins to de-crease steadily with decreasing mass. This happens becauseas the FoF mass decreases there is an increasing probabilitythat the progenitor of this FoF halo has previously passedthrough a more massive neigbouring halo and this results inthe Dhalo algorithm remerging the FoF halo with its moremassive neighbour. This fraction of FoF haloes that are re-merged to form part of a more massive Dhalo is shown bythe green curve in Fig. 1. As one approaches 10 h − M (cid:12) ( ∼
15 particles) the bijective fraction plummets as at verylow masses many of the FoF haloes are not self-bound andso do not contain any subhaloes from which to build a Dhalo.The fraction of FoF haloes which do not contain a self-boundsubstructure is shown by the cyan curve in Fig. 1 and canbe seen to reach 50% at at a FoF mass of 20 particles.
It is conventional to define the virial mass, M vir , and associ-ated virial radius, r vir , of a dark matter halo using a simplespherical overdensity criterion centred on the potential min-imum. M vir = 43 π ∆ ρ crit r (1)where ρ crit is the cosmological critical density and ∆ is thespecified overdensity. In applying this definition we adopt∆ = 200 and include all the particles inside this sphericalvolume, not only the particles grouped by the FoF or Dhaloalgorithm, to define the enclosed mass, M , and associatedradius r . This choice is largely a matter of convention buthas been shown to roughly correspond to boundary at whichthe haloes are in approximate dynamical equilibrium (e.g.Cole & Lacey 1996).If the halo finding algorithm has succeeded in parti-tioning the dark matter distribution into virialized haloeswe would expect to see a good correspondence between thegrouped mass of the halo and M . For instance, as FoFhaloes are essentially bounded by an isodensity contour,whose value is set by the linking parameter (Davis et al.1985), then if they have relaxed quasi-spherical configura-tion a tight relation between M halo and M is inevitable.The only way M halo (cid:29) M is if the halo has multiplecomponents which have been spuriously linked together asillustrated in the typical example shown in the lower pan- c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes M FoF /h − M fl l o g M F o F / M FoF /M =2M FoF /M =3 9 10 11 12 13 14log M Dhalo / h − M fl l o g M D h a l o / M Dhalo /M =2M Dhalo /M =3 Figure 2.
The left panel shows the median, 1, 5, 20, 80, 95, 99 percentiles of the distribution of the mass ratios between FoF halo mass,M
FoF and virial mass, M as a function of FoF halo mass for haloes identified using the FoF group finder. The right panel shows thesame percentiles for the distribution of the mass ratio between Dhalo mass, M
Dhalo and virial mass, M , as a function of Dhalo massfor haloes identified using the Dhalo group finder. The blue dashed line in both panels shows where M
Halo /M =2.0 and the black oneM Halo /M =3.0. els of Fig. 4. M halo (cid:28) M could indicate cases wherethe group finder has split a virialized object into small frag-ments. Hence it is interesting to look at the distribution of M halo /M for both the FoF and Dhalo algorithms to sim-ply see how M halo compares to the conventional M defini-tion of halo mass and to give an indication of the frequencyof over linking and fragmentation.The two panels of Fig. 2 quantify the distribution of M halo /M for both the standard FoF haloes and for haloesdefined by the Dhalo algorithm. We immediately see thatthe distribution is much tighter for the Dhalo definitionthan for FoF haloes. For FoF haloes 5% of the haloeshave M FOF /M ∼ > M FOF /M ∼ >
3. In contrast forDhaloes only 5% have M Dhalo /M ∼ > . M Dhalo /M >
2. In the Dhalo panel only Dhaloesthat are bijectively matched with FoF haloes are included.Since such pairs of haloes contain the same most massivesubhalo, the centres used for calculating M are identical These grossly non-virialized multi-component systems are notalways detected by more often used relaxation criteria (Neto et al2007; Power, Knebe, & Knollmann 2012, and see Section 4.2), assuch criteria focus on the mass within r which can be in equil-librium even if diffusely linked to secondary mass concentrations. and result in the same M . Furthermore, since Fig. 1 in-dicates that all FoF haloes more massive than 10 h − M (cid:12) have a bijectively matching Dhalo, then above 10 h − M (cid:12) we are comparing the same population of haloes and usingthe same values of M . Consequently the wider distribu-tion of M halo /M for FoF is directly caused by the widerspread in M FoF masses. For the cases where M FoF (cid:29) M there is one or more substantial components of the FoF halothat lies outside r . We will see in Fig. 4 that these aregenerally secondary mass concentrations that are linked bytenuous bridges of quite diffuse material. The Dhaloes havea tighter distribution of M halo /M as in this algorithmthese secondary concentrations are successfully split off andresult in separate distinct Dhaloes.Our results for FoF haloes are consistent with earlier in-vestigations. Harker et al. (2006); Evrard et al. (2008); Lukicet al (2009) found that approximately 80-85% of FoF haloesare isolated haloes while 15-20% of FoF haloes have irregu-lar morphologies, most of which are described in Lukic et al(2009) as “bridged haloes”. The distribution of M FoF /M for “bridged haloes” given in figure 7 of Lukic et al (2009)is very similar to the 20% tail of our distribution above M FoF /M = 1 .
5, while the isolated haloes in Lukic et al(2009) have a distribution similar to the remaining 80% ofour distribution. c (cid:13) , 1–23 Jiang et al.
Figure 4.
Three examples of the relationship between FoF haloes and Dhaloes. In each panel all the points plotted are from a singleFoF halo. First all the FoF particles were plotted in green and then subsets belonging to specific Dhaloes were over-plotted. The magentapoints are those belonging to the bijectively matched Dhaloes. Other colours are used to indicate particles belonging to other non-bijective Dhaloes with a unique colour used for each separate Dhalo. Two projections of each halo are shown. The left panels showthe X-Y and right the X-Z plane. The black circle marks r of the FoF halo and the cyan circle marks twice the half mass radiusof the main subhalo of the FoF halo. The top row shows a typical case where M FoF ≈ M Dhalo . Here M FoF = 2 . × h − M (cid:12) , M = 1 . × h − M (cid:12) , and r = 0 . h − Mpc. The middle panel shows and example where the mass ratio M FoF /M Dhalo = 1 . M FoF = 1 . × h − M (cid:12) , M = 1 . × h − M (cid:12) and r = 0 . h − Mpc. The bottom row shows an extreme examplewhere M FoF (cid:29) M Dhalo and the FoF halo is split into many Dhaloes. Here M FoF = 1 . × h − M (cid:12) , M = 7 . × h − M (cid:12) and r = 0 . h − Mpc c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes Figure 5.
Examples of three typical Dhaloes showing how a single Dhalo can be composed of more than one FoF halo. In each panelall the points plotted are from a single Dhalo. First all the Dhalo particles were plotted in green and then subsets belonging to specificFoF haloes were over plotted. The magenta points are those belonging to the bijectively matched FoF halo. Other colours are usedto indicate particles belonging to other FoF haloes with a unique colour used for each separate FoF halo. Two projections of eachhalo are shown. The left panels show the X-Y and right the X-Z plane. From top to bottom the Dhalo masses of these examples are M Dhalo = 4 . × h − M (cid:12) , M Dhalo = 6 . × h − M (cid:12) and M Dhalo = 5 . × h − M (cid:12) . In all cases the majority of the Dhalomass is contained in the single bijectively matched FoF halo and the secondary FoF haloes are typically 100 times less massive.c (cid:13)000
Examples of three typical Dhaloes showing how a single Dhalo can be composed of more than one FoF halo. In each panelall the points plotted are from a single Dhalo. First all the Dhalo particles were plotted in green and then subsets belonging to specificFoF haloes were over plotted. The magenta points are those belonging to the bijectively matched FoF halo. Other colours are usedto indicate particles belonging to other FoF haloes with a unique colour used for each separate FoF halo. Two projections of eachhalo are shown. The left panels show the X-Y and right the X-Z plane. From top to bottom the Dhalo masses of these examples are M Dhalo = 4 . × h − M (cid:12) , M Dhalo = 6 . × h − M (cid:12) and M Dhalo = 5 . × h − M (cid:12) . In all cases the majority of the Dhalomass is contained in the single bijectively matched FoF halo and the secondary FoF haloes are typically 100 times less massive.c (cid:13)000 , 1–23 Jiang et al. l o g M F o F / h − M fl Bijective M Dhalo /h − M fl M F o F / M D h a l o Figure 3.
In the top panel, the 1, 5, 20, 50, 80, 95 and 99 per-centiles of the distribution of FoF halo mass, M FoF , is plottedagainst M
Dhalo for the bijectively matched pairs of haloes. In thebottom panel, the same percentiles of the distribution of the massratio M
FoF / M
Dhalo is plotted as a function of Dhalo mass. Theblack dashed lines are where M
FoF /M Dhalo =0.8, 1, 1.5 and 2.5.
We now turn to directly comparing the mass assigned toFoF haloes and their corresponding Dhaloes. Fig. 3 com-pares the distributions of these two masses and their ratiofor bijectively matched FoF and Dhaloes, i.e. haloes whichcontain the same most massive subhalo. First we see thatthe median of the distribution is very close to the one-to-one line. Furthermore on one side the distribution cuts offvery sharply with far fewer than 1% of haloes having FoFmasses significantly lower than their corresponding Dhalomass. In principal M Dhalo > M
FoF can occur as one aspectof the Dhalo algorithm is that includes satellite subhaloesthat previously passed through the main halo even if theyare now sufficiently distant so as not to be linked into thecorresponding FoF halo. However, such subhaloes are typi-cally much less massive than the main subhalo and the massgained in this way is out weighed by other sources of massloss. On the other side of the distribution there is a signif-icant tail of haloes for which M FoF > M
Dhalo . We see thatapproximately 5% have M FoF > . M Dhalo and 1% have M FoF > M Dhalo . These fractions are largely independentof Dhalo mass. The main reason for this tail is the pres-ence of FoF haloes that have a significant secondary mass concentration, often linked by a low density bridge, thatthe Dhalo algorithm succeeds in splitting off. For these bi-jectively matched haloes M FoF is unlikely to significantlyexceed 2 M Dhalo as if a single secondary mass concentrationhad a subhalo of mass greater than that of the most massivesubhalo in the Dhalo we would not have a bijective match.However, in rare instances M FoF > M Dhalo can occur whenthe FoF halo contains several massive secondary mass con-centrations.To illustrate the relationship between FoF and Dhaloeswe show three examples in Fig. 4 that have been chosen tobe representative of different points in the M FoF – M Dhalo dis-tribution. The halo shown in the top row is representativeof the majority of cases, namely those with M FoF ≈ M Dhalo .Here the only particles from the FoF halo that are not in-cluded in the Dhalo are a diffuse cloud of unbound particlesand the particles in a couple of subhaloes whose centres lieoutside twice the half mass radius of the main subhalo. Westress that these small differences are what is typical forcorresponding FoF and Dhaloes.The middle row of Fig. 4 shows an example where M FoF /M Dhalo = 1 .
5, which corresponds to the 95th per-centile of the distribution shown in Fig. 3. Here the FoF halois split into three well separated Dhaloes. The main Dhalo isdominant, but there two secondary Dhaloes, one a lot moremassive than the other, laying outside the r of the mainDhalo. For the purposes of semi-analytic galaxy formationmodels such as galform the three separate haloes given bythe Dhalo definition is clearly a better description than thesingle FoF halo as one would not expect the gas reservoirsassociated with these distinct haloes to have merged at thisstage and so each should be able to provide cooling gas totheir respective central galaxies.The bottom row of Fig. 4 shows a rare example with M FoF /M Dhalo ≈
2, the 99th percentile of the distribution,in which a single FoF halo is split into several substantialDhaloes. In this and the previous example the FoF halois clearly far from spherical and a large proportion of theFoF halo mass lies outside the virial radius that is definedby centring on the potential minimum of the most massivesubstructure. Clearly characterising such haloes by a NFWprofile fit just to the mass within the virial radius would bean inadequate description of the halo. In fact, in most studiesof halo concentrations, including our analysis present in Sec-tion 4.2, these haloes would be deemed to be unrelaxed andexcluded from subsequent analysis. In contrast, the Dhaloesin each of the examples presented are much closer to beingspherical with only a small amount of mass outside their re-spective virial radii. Each of the primary Dhaloes in Fig. 4,including the one in the bottom panel, are sufficiently sym-metrical and virialized to pass the relaxation criteria thatwe employ in Section 4.2 even though the correspondingFoF haloes in the bottom two panels are not.In the example shown in the bottom row of Fig. 4 wealso see case of a Dhalo that has two distinct components.Here the two clumps of black points are a single Dhalo dueto the fact that they passed directly through each otherat a redshift z = 0 .
89. This extreme example must havebeen a high speed encounter and so any galaxies they con-tained would have been unlikely to merge, but their ex-tended hot gas distributions would have interacted and pos- c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes log M FoF /h − M fl l o g M D h a l o / h − M fl Dhalo in FoF M FoF /h − M fl M D h a l o / M F o F log M FoF /h − M fl l o g M D h a l o / h − M fl M FoF /h − M fl M D h a l o / M F o F Figure 6.
In the left hand panels, we plot the median, 1, 5, 20, 80, 95 and 99 percentiles of the distribution of Dhalo mass, M
Dhalo (upper), and mass ratio M
Dhalo /M FoF (lower) against M
FoF for all the Dhalo matches to each FoF halo. The black dashed lines in eachpanel mark where M
Dhalo /M FoF =1. In the right hand panel, we plot the same quantities but only for secondary Dhaloes in each FoFhalo. sibly merged. It is for this reason that it is useful in thesemi-analytic models to associate them as a single halo.The Dhalo algorithm quite frequently merges severalFoF haloes together into a single Dhalo as a consequence ofthe way it avoids splitting up subhaloes which at an earliertimestep were in a single Dhalo. However unlike the extremeexample we have just seen the typical masses of subhaloeswhich pass through a Dhalo and then emerge to once againbecome a distinct FoF halo are much lower than the massof the main FoF halo. This is illustrated in Fig. 5, wherewe show the particles of three typical Dhaloes of a range ofmasses colour coded by their FoF halo membership. In eachcase we immediately see that the vast majority of the Dhaloparticles also belong to the (bijectively) matched FoF halo.However in addition there are isolated clumps of particles inthe outskirts of each Dhalo which belong to much smallerdistinct FoF haloes. There are also similar nearby clumpsof particles which due to surrounding diffuse material arelinked into the main FoF halo. In all cases each of theseclumps are typically less than one percent of the mass of themain halo. From the perspective of semi-analytic galaxy for-mation models it makes sense to treat each of these clumpsequally. For instance, they have all been within twice thehalf mass radius of the main Dhalo and could therefore havebeen ram pressure stripped of their diffuse gaseous haloes.In galform satellite galaxies move with the subhalo within which they formed (or if the descendant of the subhalo dropsbelow the 20 particle threshold with the particle that waspreviously the potential minimum of its subhalo) and sothe satellite galaxy positions reflect the spatial distributionof these subhaloes even if they move far from the halo towhich they are associated.
So far we have just compared FoF–Dhalo pairs which forma bijective match, that is their most massive subhaloes areidentical. However there other cases such as the examples ofsecondary Dhaloes in Fig. 4 in which the main subhalo of theDhalo is not the most massive subhalo in the correspondingFoF halo and conversely examples such as the secondaryFoF haloes in Fig. 5 in which the main subhalo of the FoFhalo is not the most massive subhalo in the correspondingDhalo. We will refer to this former set of matches as Dhaloin FoF halo and the latter as FoF in Dhalo matches. Notethat the bijective matches are a subset of both of these sets,i.e. they are the intersection of the two sets of matches. Tohave a complete census of the correspondence between FoFand Dhaloes it is important that we include non-bijectivelymatched haloes in our comparison. We compare the Dhaloto FoF halo masses for these two sets of pairings in Fig. 6and 7. c (cid:13) , 1–23 Jiang et al. log M Dhalo /h − M fl l o g M F o F / h − M fl FoF in Dhalo M Dhalo /h − M fl M F o F / M D h a l o log M Dhalo /h − M fl l o g M F o F / h − M fl M Dhalo /h − M fl M F o F / M D h a l o Figure 7.
As Fig. 6 but with the role of FoF and Dhalo reversed. In the left hand panels, we plot the median, 1, 5, 20, 80, 95 and 99percentiles of the distribution of FoF halo mass, M
FoF (upper), and mass ratio M
FoF /M Dhalo (lower) against M
Dhalo for all the FoFhalo matches to each Dhalo. The black dashed lines in each panel mark where M
FoF /M Dhalo =1. In the right hand panel, we plot thesame quantities but only for secondary FoF in each Dhalo.
The left hand panels of Fig. 6 show for all Dhalo inFoF halo matches the dependence of the mass, M Dhalo , andthe mass ratio, M Dhalo /M FoF on the FoF halo mass. Theright hand panel shows the same quantities but only for sec-ondary Dhalo in FoF halo haloes, i.e. excluding the bijectivematches. Focusing first on the right hand panels, we see thatthe percentiles of the distribution of secondary M Dhalo val-ues are all horizontal lines at high M FoF , indicating thatin this regime the distribution of M Dhalo is independent of M FoF . This suggests that the secondary Dhaloes that arelinked into high mass FoF haloes by bridges of diffuse mate-rial are essentially drawn at random from the Dhalo popula-tion. We note that in this way the FoF halo can be hundredsor more times more massive than many of the Dhaloes incontain. In these same panels, we see that at lower massesthe distribution of Dhalo masses is sharply truncated at M Dhalo = M FoF /
2. This is essentially by construction as ifa Dhalo with mass greater than M FoF / M Dhalo and M FoF . With increas-ing FoF mass there are more and more secondary Dhaloesper FoF halo. They increasingly dominate over the bijectivematches and so the contours tend to their values in the righthand panel.Fig. 7 shows the distribution of FoF halo mass for theFoF in Dhalo matches. Again the right hand panes show thedistribution for just the secondary matches while the lefthand panels also include the primary or bijective matches.Comparing the right hand panels of Fig. 7 and Fig. 6 we seethat the corresponding contours are shifted to lower masses.Thus it is rarer for a Dhalo to contain massive secondaryFoF halo than it is for FoF halo to contain massive sec-ondary Dhalo. The secondary Dhaloes arise from the re-merging step in the Dhalo algorithm whereby two subhaloesthat have passed through each other (the smaller has comewithin twice the half mass radius of the larger) are deemedthereafter always to be part (or satellite components) of thesame Dhalo even if they subsequently separate sufficientlyto become distinct FoF haloes. This occurs reasonably fre-quently, but as in the examples shown in Fig 4 the secondaryFoF haloes are typically much less massive than the primary c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes M Dhalo / h − M fl l o g M D h a l o / M Dhalo /M =2M Dhalo /M =1M Dhalo /M =0.5 Figure 8.
Like the right hand panel of Fig. 2, but for non-bijective Dhaloes. The curves show the median, 5, 20, 80, 95,99 percentiles of the ratio between the Dhalo mass, M
Dhalo ,and the virial mass, M . The horizontal dashed lines indicateM
Dhalo /M = 0.5, 1.0, 2.0. and contribute little to the total mass of the halo. Interest-ingly the near horizontal contours in the upper right handpanel Fig. 7 indicate that the mass distribution of this popu-lation of secondary FoF haloes is approximately independentof M Dhalo for high Dhalo masses. As these FoF haloes areoften heavily stripped by their passage through the mainDhalo this is not a trivial result. The contours begin todip at lower masses reflecting the fact it is unlikely for amatched FoF halo to have a mass greater than about one halfof M Dhalo without it being the primary or bijective match.This expectation is violated for M Dhalo < h − M (cid:12) , butthis is a resolution effect because at such low masses secon-daries with M FOF (cid:28) M Dhalo fall below the 20 particle limitof the catalogue and so their absence biases the distributiontowards higher ratios.The left hand panels of Fig. 7 are for all the matchesof FoF in Dhalo, including the bijective matches. These dis-tributions can be understood as a superposition of the dis-tributions in the right hand panels with the distribution forbijective matches shown in Fig. 3. At low masses the bijec-tive halo matches dominate whereas at large M Dhalo thereare many FoF haloes matched to each Dhalo. Thus, for ex-ample, at M Dhalo ≈ . h − M (cid:12) we transition from 50%of the matched FoF haloes being primary to 50% of thembeing much lower mass ( M FoF ≈ . h − M (cid:12) ) secondaryFoF haloes.In section 3.1.1, we examined the distribution of the Figure 9.
An example of one FoF halo split by the Dhalo al-gorithm into several Dhaloes. All the points plotted are from asingle FoF halo. First all the FoF particles are plotted in green andthen subsets belonging to specific Dhaloes are over plotted. Themagenta points are those belonging to the bijectively matchedDhalo. Other colours are used to indicate particles belonging toother Dhaloes with a unique colour used for each separate Dhalo.The black circle around the magenta points marks r of the FoFhalo and is also the r of the bijective Dhalo. The concentriccyan circle marks twice the half mass radius of this main subhalo.The other black circles show r locations for the non-bijectiveDhaloes, while the concentric blue circles indicate twice the halfmass radius of the corresponding subhalo. M Dhalo /M ratio for the bijectively matched haloes. Weare also interested this distribution for the non-bijectiveDhaloes shown in Fig. 8. We immediately notice the distri-bution is shifted towards lower values than the correspond-ing distribution for the bijective haloes shown in Fig. 2. Theorigin of this shift can be understood by reference to Fig. 9which shows an example of a FoF halo which is split intoseveral Dhaloes. The Dhalo whose particles are plotted inmagenta is the bijective match of the FoF halo and theDhaloes plotted in other colours are non-bijective matches.The black circles in Fig. 9 show the location of r foreach of the Dhaloes, while the other circles show the lo-cation of the half-mass radius of each Dhalo. For bijectivelymatched Dhaloes, the majority of which are isolated, r istypically slightly smaller than the half-mass radius. In con-trast we see in Fig. 9 that for many of the non-bijectivelymatched Dhaloes the half mass radius is much smaller than r . This is a consequence of the subfind algorithm whichdetermines the extent of a subhalo by finding saddle pointsin the density distribution (Springel et al. 2001). Hence asa subhalo enters a dense environment the mass assigned toit by subfind is decreased. This environmentally dependenteffect both lowers M Dhalo relative to M and increases thescatter in this relation. c (cid:13)000
An example of one FoF halo split by the Dhalo al-gorithm into several Dhaloes. All the points plotted are from asingle FoF halo. First all the FoF particles are plotted in green andthen subsets belonging to specific Dhaloes are over plotted. Themagenta points are those belonging to the bijectively matchedDhalo. Other colours are used to indicate particles belonging toother Dhaloes with a unique colour used for each separate Dhalo.The black circle around the magenta points marks r of the FoFhalo and is also the r of the bijective Dhalo. The concentriccyan circle marks twice the half mass radius of this main subhalo.The other black circles show r locations for the non-bijectiveDhaloes, while the concentric blue circles indicate twice the halfmass radius of the corresponding subhalo. M Dhalo /M ratio for the bijectively matched haloes. Weare also interested this distribution for the non-bijectiveDhaloes shown in Fig. 8. We immediately notice the distri-bution is shifted towards lower values than the correspond-ing distribution for the bijective haloes shown in Fig. 2. Theorigin of this shift can be understood by reference to Fig. 9which shows an example of a FoF halo which is split intoseveral Dhaloes. The Dhalo whose particles are plotted inmagenta is the bijective match of the FoF halo and theDhaloes plotted in other colours are non-bijective matches.The black circles in Fig. 9 show the location of r foreach of the Dhaloes, while the other circles show the lo-cation of the half-mass radius of each Dhalo. For bijectivelymatched Dhaloes, the majority of which are isolated, r istypically slightly smaller than the half-mass radius. In con-trast we see in Fig. 9 that for many of the non-bijectivelymatched Dhaloes the half mass radius is much smaller than r . This is a consequence of the subfind algorithm whichdetermines the extent of a subhalo by finding saddle pointsin the density distribution (Springel et al. 2001). Hence asa subhalo enters a dense environment the mass assigned toit by subfind is decreased. This environmentally dependenteffect both lowers M Dhalo relative to M and increases thescatter in this relation. c (cid:13)000 , 1–23 Jiang et al.
Having thoroughly compared individual Dhaloes with theircorresponding FoF haloes, we now turn to the statisticalproperties of the Dhaloes. We first look at the Dhalo massfunction and then the statistics of their density profiles ascharacterised by fitting NFW profiles (Navarro, Frenk &White 1995, 1996, 1997).
For many applications it is extremely useful to have ananalytic description of the number density of haloes as afunction of halo mass. A relevant example for us is whensemi-analytic galaxy formation models are constructed usingMonte-Carlo methods (Parkinson, Cole, & Helly 2008; Coleet al. 2000) of generating dark matter merger trees. In thiscase, in order to construct predictions of galaxy luminosityfunctions or any other volume averaged quantity(Cole et al.2000; Berlind et al. 2003; Baugh et al. 2005; Neistein & Dekel2008; Bundy, Ellis, & Conselice 2005; Giocoli, Pieri, & Tor-men 2008; Moreno, Giocoli, & Sheth 2008; van den Bosch,Tormen, & Giocoli 2005), one needs knowledge of the halomass function in order to know how many of each type oftree one has per unit volume. It has become common prac-tice to assume the halo mass function is given by analyticfitting functions which have been fitted to the abundance ofhaloes found by the FoF or other group finding algorithms(Davis et al. 1985; Lacey & Cole 1994; Knollmann & Knebe2009) in suites of cosmological N-body simulations. Murray,Power, & Robotham (2013) compare all the currently pro-posed fitting functions. In our semi-analytic modelling wewould like to achieve consistent results when using Monte-Carlo merger trees or when using merger trees extracted di-rectly from N-body simulations using the Dhalo algorithm.Hence it is important to directly determine the Dhalo massfunction and to compare it to such fitting formulae.We do this in Fig. 10 which compares the Dhalo andFoF mass functions that we measure in the MSII simulationswith various analytic prescriptions (Jenkins et al 2001; Sheth& Tormen 2002; Warren et al. 2006; Reed et al 2007; Tinkeret al. 2008; Watson et al. 2013). The left hand panel showsthe number density of haloes per unit logarithmic intervalof mass from the nominal 20 particle mass resolution of thesimulation up to 10 h − M (cid:12) which is the mass of the mostmassive haloes in the simulation. In constructing these massfunctions the halo mass we use is simply the aggregatedmass of all the particles assigned to each halo. Thus in theFoF case this is all particles linked to the halo by the FoFalgorithm while in the Dhalo case it is the sum of the massesof the subhaloes that compose an individual Dhalo. Alsoshown on this panel are the predictions of various analyticprescriptions. To evaluate these we use σ ( M ), the varianceof the density fluctuations as a function of mass (using atop hat filter), corresponding to the input power spectrumof the MSII propagated to the output time of the simulationusing linear theory. They are all clearly very similar and soin the left hand panel we expand the dynamic range of thecomparison by plotting each mass function divided by theprediction of the Sheth & Tormen (2002) model.The first thing that we note is that despite the some-times quite large differences (see §
3) in the masses of indi- vidual FoF and Dhaloes their two mass functions agree towithin 5% for all masses greater than 10 h − M (cid:12) . In therange 10 ∼ 100 particles) the sharp up turn in the FoF massfunction relative to that of Dhaloes is due to an increasingfraction of the FoF haloes not containing a self-bound sub-halo and so having no corresponding Dhalo (see Fig. 1).Thus this portion of the mass function is strongly affectedby the resolution of the simulation.The Jenkins et al (2001) fitting formula is within 10%of both the FoF and Dhalo mass functions for masses above2 × h − M (cid:12) . However below this mass it strongly underpredicts the number density of low mass haloes. Note thatwe only plot this fit and that of Watson et al (2013) over themass ranges used to constrain them in the original papers.The Watson et al (2013) mass function is only defined atvery high masses where we have poor statistics. It lies some-what below but is still compatible with our noisy estimates.The Warren (2006) model has the best agreement with ourFoF mass function, fitting it well all the way down to 40 par-ticles, beyond which we expect our limited resolution meansthat our FoF mass function is contaminated by spuriousunbound chance groupings of particles. However the Reed(2007) mass function does a better job of matching the lowmass end of our Dhalo mass function. The Sheth & Tormenmass function is intermediate at low masses between that ofWarren (2006) and Reed (2007), but systematically belowthe other models and our FoF and Dhalo mass function athigh masses, though still only at the 15% level. The Tin-ker (2008) mass function predicts halo abundances that areabout 5 to 10% higher than Warren (2006) and our esti-mated FoF abundances.In summary, the Dhalo and FoF mass functions are verysimilar and only differ by more than 5% below 10 h − M (cid:12) .As a result the established analytic mass function modelsfit the Dhalo mass function almost as well as they do thestandard FoF mass function. The differences between thedifferent analytic fitting formulae are greater than the dif-ference between the FoF and Dhalo mass functions. TheReed (2007) model is a slightly better description of theDhalo mass function due to it predicting a slightly lowerabundance at low masses. We now turn to the density profiles of the haloes as theseare an important ingredient in semi-analytic models such as galform where they influence the rate at which gas cools c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes M halo /h − M fl -5 -4 -3 -2 -1 d n / d l o g M / ( h − M p c ) Sheth&Tormen2002FoFDhalo M halo /h − M fl d n / d n S T Jenkins 2001Sheth&Tormen2002Warren 2006Reed 2007Tinker 2008Watson 2013FoFDhalo Figure 10. The left hand panel shows the differential mass functions for both FoF (linking length b = 0 . 2) haloes (blue line) and Dhaloes(red points) in the MSII simulation. We plot this down to ∼ h − M (cid:12) , the mass corresponding to 20 particles in the MSII simulationand we also plot the Sheth and Tormen (2002) mass function as a comparison. To expand the dynamic range, the right hand panel showsthe corresponding prediction of various analytic mass functions(Jenkins et al 2001,Warren et al 2006, Reed et al 2007, Tinker et al 2008,Watson et al 2013) as indicated in the legend but now relative to the Sheth and Tormen(2002) prediction. The FoF Dhalo data are nowshown as the heavy blue and red lines. and set the gravitational potential well in which galaxiesform. We choose to fit the halo density profiles using NFW(Navarro, Frenk & White 1996, 1997) profiles ρ NFW ( r ) ρ crit = δ c r/r s (1 + r/r s ) ( r (cid:54) r ) , (2)where δ c is the characteristic density contrast, and r s is thescale radius. We define the virial radius, r , as the radius atwhich the mean interior density equals 200 times the criticaldensity, ρ crit = 3 H / (8 πG ). The concentration is defined as c ≡ r /r s . The definition of r implies that δ c and c mustsatisfy δ c = 2003 c ln(1 + c ) − c/ ( c + 1) . (3)Our choice of NFW profiles is motivated by their accu-racy as a model of CDM haloes (Navarro, Frenk & White1996, 1997), their widespread use and so that our results canbe compared to those in Neto et al (2007) who studied thestatistics of NFW concentrations for FoF haloes identifiedin the Millennium Simulation (Springel 2005a). To allow usto compare directly with Neto et al (2007) we have followedtheir fitting procedure.For each halo, we have computed a spherically-averageddensity profile by binning the halo mass into 32 equallyspaced bins in log ( r ) between the virial radius andlog ( r/r ) = − . 5, centred on the potential minimum. We fit the two free parameters, δ c and r s by minimising themean square deviation σ = 1 N bin − N bin (cid:88) i [log ρ ( r i ) − log ρ NFW ( r i | δ c , r s )] (4)between the binned ρ ( r ) and the NFW profile. As in Netoet al (2007), we perform the fit over the radial range 0 . 1. In order to be consistent with the original NFWwork, we express the results in terms of fitted virial mass, M , and a concentration, c ≡ r /r s . We note thatwhile the fitted value of M used here and the directlymeasured M used earlier (e.g. in Fig. 2) are not identicalthey in general agree very accurately with an rms scatter ofless than 3%.Neto et al (2007) distinguished relaxed haloes fromhaloes that were not in dynamical equilibrium due to re-cent or ongoing mergers. They found that relaxed haloeswere well fit by NFW profiles while the profiles of unrelaxedhaloes were lumpier and yielded poorer fits with systemati-cally lower concentrations. Hence to compare to Neto et al(2007) we use the following three objective criteria to assesswhether a halo has reached equilibrium Neto et al (2007);Gao et al. (2008); Power, Knebe, & Knollmann (2012).(i) The fraction of mass in resolved substructures whosecentres lie inside r : f sub = (cid:80) N sub i (cid:54) =0 M sub ,i /M . We re-quire f sub < . c (cid:13)000 1. In order to be consistent with the original NFWwork, we express the results in terms of fitted virial mass, M , and a concentration, c ≡ r /r s . We note thatwhile the fitted value of M used here and the directlymeasured M used earlier (e.g. in Fig. 2) are not identicalthey in general agree very accurately with an rms scatter ofless than 3%.Neto et al (2007) distinguished relaxed haloes fromhaloes that were not in dynamical equilibrium due to re-cent or ongoing mergers. They found that relaxed haloeswere well fit by NFW profiles while the profiles of unrelaxedhaloes were lumpier and yielded poorer fits with systemati-cally lower concentrations. Hence to compare to Neto et al(2007) we use the following three objective criteria to assesswhether a halo has reached equilibrium Neto et al (2007);Gao et al. (2008); Power, Knebe, & Knollmann (2012).(i) The fraction of mass in resolved substructures whosecentres lie inside r : f sub = (cid:80) N sub i (cid:54) =0 M sub ,i /M . We re-quire f sub < . c (cid:13)000 , 1–23 Jiang et al. l o g ρ / ρ c=5.070 fsub=0.026 s=0.064 2T/U=1.65 NFWBijective halo log r/r l o g ρ / ρ c=6.78 fsub=0.062 s=0.028 2T/U=1.40 NFWnobijective1 c=8.52 fsub=0.027 s=0.011 2T/U=1.06 NFWnobijective2 log r/r c=8.00 fsub=0.023 s=0.009 2T/U=1.03 NFWnobijective3 r/r c=4.730 fsub=0.040 s=0.030 2T/U=1.20 NFWnobijective4 r/r l o g ρ / ρ c=4.050 fsub=0.010 s=0.260 2T/U=1.20 NFWnobijective5 Figure 11. Density profiles, ρ ( r ), for each of the Dhaloes shownin Fig. 9. The colour of the fitted NFW curve matches the colourcoding of the individual Dhaloes in Fig. 9. The two-parameter, δ c and r s , NFW least-square fits were performed over the radialrange 0 . < r/r < 1, shown by the black circles in Fig. 9. Theminimum fit radius r/r = 0 . 05 is always larger than the con-vergence radius derived by Power et al (2003), which we indicateby the solid vertical line in each panel. (ii) The centre of mass displacement, i.e. the differencebetween the position of the potential minimum and the cen-tre of mass, s = | r c − r cm | /r (Thomas et al. 2001). Notethat, the centre of mass is calculated using all the particleswithin r , not only those belonging to the FoF or Dhalo.We require s < . 07 for relaxed haloes.(iii) The virial ratio, 2 T / | U | , where T is the total kineticenergy of halo particles within r and U is their gravita-tional potential self energy. We require 2 T / | U | < . 35 forour relaxed haloes. (For haloes with more than 5000 parti-cles we use a random subset of 5000 particles to estimate U .) Fig. 9 shows a single FoF halo and its componentDhaloes which we use to illustrate the application ofthese selection criteria and ability of NFW profiles to fitsecondary/non-bijective Dhaloes. The spherically averageddensity profiles and our NFW fits to each of these Dhaloesare shown in Fig. 11 along with the values of the three se-lection parameters f sub , s and 2 T / | U | . The top left panel ofFig. 11 shows the density profile and NFW fit for the maincomponent of the FoF halo, which can be identified by thecyan circle in Fig. 9 which marks twice the half mass radius the most massive substructure in the FoF halo. In previ-ous analyses of FoF haloes, such as Neto et al (2007), thiswould be the only density profile fitted to the mass distribu-tion shown in Fig. 9. The bijectively matched Dhalo has thesame centre as the FoF halo and the NFW fit is performedon all the mass within r , (indicated by the concentricblack circle) consequently the density profile and NFW fitof the bijectively matched Dhalo is necessarily identical tothat or the corresponding FoF halo. Examining this regionin Fig. 9, we can clearly see that the mass distribution isasymmetric and has several distinct substructures indica-tive of a recent merger. This halo is not relaxed accordingto the above selection criteria as it fails to satisfy the cuton 2 T / | U | . Also its value of the centre offset, s , comes closeto the threshold. The NFW fit to its density profile can beseen to have significant deviations at both large and smallradii.We are also interested in whether NFW profiles pro-vide acceptable fits to the other Dhaloes found within thissingle FoF halo. These are shown in the remaining panelsof Fig. 11. According to the selection criteria three of theseDhaloes (those in the right-hand column) are relaxed. Theseare the blue, red and black Dhaloes in Fig. 9 and their den-sity profiles are shown, respectively, in the top, middle andbottom right-hand panels of Fig. 11. In all cases we see thatthe NFW fits provide a good description of the mass pro-file of these relaxed Dhaloes. The remaining two Dhaloesfail one or other of the selection criteria. The yellow Dhaloof Fig. 9, whose density profile is shown in the middle-leftpanel of Fig. 11, marginally fails the cut on 2 T / | U | . The cyanDhalo of Fig. 9, whose density profile is shown in the bottom-left panel of Fig. 11, which strongly exceeds the thresholdon s , can be seen to be very poorly fit by the NFW pro-file and have a particularly low concentration. This Dhalois very close to being within twice the half mass radius ofthe most massive substructure of the FoF halo, marked bythe cyan circle in Fig. 9. This being the radius used by theDhalo algorithm as part of its criteria to determine whethertwo subhaloes should be considered as two distinct haloes orcomponents of the same halo. It is this proximity to a mergerthat both creates the large offset, s , between the potentialminimum and the centre of mass within r and distorts theobject’s density profile. We also note that this Dhalo has themost extreme ratio of r to twice its half mass radius. InFig. 4 we saw that for isolated haloes r and twice the halfmass radius were very comparable, but in contrast we see inFig. 9 that the r of secondary Dhaloes can be significantlyboosted by the density of the surrounding environment.This systematic difference in the ratio of Dhalo mass to M for bijective and non-bijective Dhaloes is illustrated inFig. 8 which should be contrasted with the right-hand panelof Fig. 2. We see that the scatter in the ratio of M Dhalo /M is considerably larger for the non-bijective Dhaloes than itis for bijective Dhaloes. For bijective Dhaloes the 5 to 95%range of the distribution spans only a 30% range in the ratioof M Dhalo /M , while this is increased to approximately afactor of two for the non-bijective Dhaloes. In addition themedian M Dhalo /M ratio is reduced from 1 . ≈ . 95 for non-bijective Dhaloes. These differ-ences are principally caused by the way the subfind algo-rithm (Springel et al. 2001) is effected by the local environ-ment. subfind locates the edge of a substructure by search- c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes ing for a saddle point in the density distribution. Hence if thesame sub-structure is placed in a denser environment thiswill move the saddle point in and reduce the mass that sub-find associates with the sub-structure (see Muldrew, Pearce,& Power 2011, for a detailed discussion). As a Dhalo massis simply the sum of the masses of the subhaloes from whichit is composed this in turn reduces the mass assigned to theDhalo. This systematic dependence of Dhalo mass on envi-ronment is one of the reasons why instead of directly usingthe Dhalo mass as input to galform semi-analytic modelwe instead force the halo masses in the halo merger treesto increase monotonically so that they do not artificiallydecrease, just prior to mergers, due to such environmentaleffects. Here we compare the mass-concentration relation for FoFhaloes that we find in the high resolution MSII simulationwith that found by Neto et al (2007) in the lower resolutionMillennium Simulation. We then go on to compare thisrelation with the relation we find for the secondary/non-bijective Dhaloes. There is no need to separately look atthe bijective Dhaloes as their M and c are necessarilythe same as that of the corresponding FoF haloes as theyhave the same centre and all the surrounding mass is usedin the fit. As in Neto et al (2007) the mass we use in theserelations is the M of the NFW fit rather than the directlymeasured value. Fig. 12 shows concentration as a function ofmass for the range 10 . < M /h − M (cid:12) < . for ourcatalogue of FoF haloes. The top panel is for our relaxed FoFhalo sample, while the bottom panel shows results for all theFoF haloes, including systems that do not meet our equilib-rium criteria. In each case we find a significant spread inconcentration at fixed mass with a weak trend for decreas-ing concentration with increasing mass. This is generallyinterpreted (Navarro, Frenk & White 1995, 1996, 1997; Bul-lock et al. 2001; Eke, Navarro & Steinmetz 2001; Neto etal 2007; Gao et al. 2008) as reflecting the typical formationtime of the halo with the lowest mass haloes forming earli-est and having high density cores which reflect the densityof the universe at the time they formed. The dependenceof the median concentration of FoF haloes on mass is welldescribed by the power-law fit c = 5 . (cid:0) M / h − M (cid:12) (cid:1) − . , (5)for relaxed haloes and by c = 5 . (cid:0) M / h − M (cid:12) (cid:1) − . (6)for all haloes. These fits were performed only over themass range 10 . < M /h − M (cid:12) < . due to poorstatistics at higher masses and are shown by the blue solidlines in Fig. 12. Also shown on Fig. 12 is the fit for the As a precise test of our methods we first applied our analysisto FoF haloes in the milli-MillenniumII simulation, which has thesame volume, initial conditions and data format as MillenniumII(Boylan-Kolchin et al. 2009), but lower mass resolution, equal tothat of the Millennium Simulation (Springel et al. 2005b) analysedby Neto et al (2007). We found precise agreement with the mass-concentration relationship published in Neto et al (2007). median concentration for relaxed haloes found by Netoet al (2007). We plot these green lines only for M > /h − M (cid:12) corresponding to the resolution limit of theirstudy. We see that over the overlapping mass range ourmedian concentrations agree very well with those of Netoet al (2007) indicating that the mass profiles over the fit-ted radial range, − . < log( r/r ) < 0, are not affectedby mass resolution. Our fit is also similar to the relation c = 5 . M / h − M (cid:12) ) − . found by Macci`o et al.(2007) for relaxed haloes. The small difference could be be-cause they fit the mean rather than median of the relationor due to differences in the criteria used to select relaxedhaloes. Like us and Neto et al (2007), Macci`o et al. (2007)find unrelaxed haloes have systematically lower concentra-tions.Having demonstrated that for FoF haloes we recover amass-concentration relation which is in very accurate agree-ment with previous work (Neto et al 2007; Macci`o et al.2007), we now want to compare mass-concentration rela-tions for our bijective and non-bijective Dhaloes. The mass-concentration relation we find for the bijective Dhaloes ispractically identical that of the FoF haloes plotted in Fig. 12and so we have chosen not to effectively repeat the same plot.The similarity is inevitable as Fig. 1 shows that for massesgreater than 10 . h − M (cid:12) , for which we can measure con-centrations, the fraction of FoF haloes that have bijectivematches with Dhaloes is greater than 95% and these bijec-tivey matched haloes have identical centres and so identicalfitted NFW mass profiles.In Fig. 13 we show the mass-concentration for relaxedand all non-bijective Dhaloes. These haloes are all secondaryfragments of FoF haloes and so are a completely disjointcatalogue of haloes to those represented in the FoF mass-concentration relations of Fig. 12. To aid in comparing thetwo sets of relations we plot the power-law fits to the medianmass-concentration relations of Fig. 12 as dashed lines inFig. 13. It can be seen that these are very similar to thepower-law fits to the median relations c = 4 . (cid:0) M / h − M (cid:12) (cid:1) − . , (7)for relaxed and c = 5 . (cid:0) M / h − M (cid:12) (cid:1) − . (8)for all the non-bijective Dhaloes which are shown by thesolid lines in Fig. 13.Comparison of the bars and whiskers in Fig. 12 andFig. 13 show that the not only do the median mass-concentration relations for FoF and non-bijective Dhaloesagree very well, but the distribution of concentrations aboutthe medians are also quite similar. The large number ofhaloes we have in the MII simulation enables us to look atthese distributions in more detail and in Fig. 14 we show his-tograms of the concentration, distributions along with log-normal approximations P (log c ) = 1 √ π σ exp (cid:34) − (cid:18) log c − (cid:104) log c (cid:105) σ (cid:19) (cid:35) , (9)for two mass bins centred on 10 and 10 h − M (cid:12) . We seein all cases that the non-bijective Dhaloes have a very sim-ilar distribution of concentrations as the distribution of thecorresponding FoF sample and that both are approximated c (cid:13)000 0, are not affectedby mass resolution. Our fit is also similar to the relation c = 5 . M / h − M (cid:12) ) − . found by Macci`o et al.(2007) for relaxed haloes. The small difference could be be-cause they fit the mean rather than median of the relationor due to differences in the criteria used to select relaxedhaloes. Like us and Neto et al (2007), Macci`o et al. (2007)find unrelaxed haloes have systematically lower concentra-tions.Having demonstrated that for FoF haloes we recover amass-concentration relation which is in very accurate agree-ment with previous work (Neto et al 2007; Macci`o et al.2007), we now want to compare mass-concentration rela-tions for our bijective and non-bijective Dhaloes. The mass-concentration relation we find for the bijective Dhaloes ispractically identical that of the FoF haloes plotted in Fig. 12and so we have chosen not to effectively repeat the same plot.The similarity is inevitable as Fig. 1 shows that for massesgreater than 10 . h − M (cid:12) , for which we can measure con-centrations, the fraction of FoF haloes that have bijectivematches with Dhaloes is greater than 95% and these bijec-tivey matched haloes have identical centres and so identicalfitted NFW mass profiles.In Fig. 13 we show the mass-concentration for relaxedand all non-bijective Dhaloes. These haloes are all secondaryfragments of FoF haloes and so are a completely disjointcatalogue of haloes to those represented in the FoF mass-concentration relations of Fig. 12. To aid in comparing thetwo sets of relations we plot the power-law fits to the medianmass-concentration relations of Fig. 12 as dashed lines inFig. 13. It can be seen that these are very similar to thepower-law fits to the median relations c = 4 . (cid:0) M / h − M (cid:12) (cid:1) − . , (7)for relaxed and c = 5 . (cid:0) M / h − M (cid:12) (cid:1) − . (8)for all the non-bijective Dhaloes which are shown by thesolid lines in Fig. 13.Comparison of the bars and whiskers in Fig. 12 andFig. 13 show that the not only do the median mass-concentration relations for FoF and non-bijective Dhaloesagree very well, but the distribution of concentrations aboutthe medians are also quite similar. The large number ofhaloes we have in the MII simulation enables us to look atthese distributions in more detail and in Fig. 14 we show his-tograms of the concentration, distributions along with log-normal approximations P (log c ) = 1 √ π σ exp (cid:34) − (cid:18) log c − (cid:104) log c (cid:105) σ (cid:19) (cid:35) , (9)for two mass bins centred on 10 and 10 h − M (cid:12) . We seein all cases that the non-bijective Dhaloes have a very sim-ilar distribution of concentrations as the distribution of thecorresponding FoF sample and that both are approximated c (cid:13)000 , 1–23 Jiang et al. log M /h − M fl l o g c relaxed FoF halosNeto et al 2007(relaxed fof halo) log M /h − M fl l o g c FoF halosNeto et al 2007(relaxed fof halo) Figure 12. The mass-concentration relation for relaxed FoFhaloes in MSII (top panel) and for all the FoF haloes (bottompanel). The boxes represent the 25% and 75% centiles of the dis-tribution, while the whiskers show the 5% and 95% tails. Thenumbers on the top of each panel indicate the number of haloesin each mass bin. The median concentration as a function of massis shown by the blue solid line and is well fit by the linear rela-tions given in equations 5 and 6. The green lines in each panelcorrespond to fits of Neto et al (2007). accurately by log-normal distributions. Note that in bothcases we are binning haloes by the M of their fitted NFWprofile and so we are affected by the Dhalo mass being per-turbed and suppressed in non-bijective Dhaloes. We recallthat the FoF sample is essentially the same as the sample ofbijectively matched Dhaloes and so we conclude that con-centration distribution is essentially the same for both theprimary Dhaloes and those that are secondary fragmentsof FoF haloes. In all cases the concentration distributionsfor the relaxed samples have slightly higher median con-centrations and smaller dispersions than the correspondingcomplete mass selected samples.Also of interest is the fraction of both FoF haloes andnon-bijective Dhaloes that satisfy the equilibrium criteria.From the number of objects per mass bin given in the labelson Figs. 12 and 13 this can be seen to be in the range of80 to 85% for both FoF and Dhaloes. One might at first ex-pect that many multi-nucleated FoF haloes would fail boththe threshold on the asymmetry, s , and the fraction of massin sub-structures, f sub . However as these statistics are eval-uated only using the mass within r and not across thewhole FoF halo, ∼ > 98% of FoF haloes pass the substructurethreshold and ∼ > 88% the asymmetry threshold. The first ofthese numbers is slightly lower for the non-bijective Dhaloes,i.e. only ∼ > 93% pass the substructure threshold. Howeverthose passing the more stringent asymmetry threshold ismore comparable at ∼ > ∼ > 93% pass the criterion that the virial log M /h − M fl l o g c relaxed non-bijective halosrelaxed FoF halos log M /h − M fl l o g c Non-bijective halos FoF halos Figure 13. The mass-concentration relation for relaxed non-bijective Dhaloes in MSII (top panel) and for all the non-bijectiveDhaloes (bottom panel). The boxes represent the 25% and 75%centiles of the distribution, while the whiskers show the 5% and95% tails. The numbers on the top of each panel indicate thenumber of haloes in each mass bin. The median concentrationas a function of mass is shown by the blue solid line and is wellfit by the linear relations given in equations 7 and 8. The bluedashed line in each panel repeats the fits to the median mass-concentration relation for FoF haloes shown in Fig. 12 ratio 2 T / | U | < . 35. Consequently the fraction of the non-bijective Dhaloes that pass the relaxation criteria is verysimilar to that for the FoF or bijective Dhaloes. Hencein both cases the mass-concentration distributions that wehave quantified are representative of the vast majority of thehaloes. We have used the high resolution Millennium SimulationII cosmological N-body simulation to quantify the proper-ties of haloes defined by the Dhalo algorithm. This algo-rithm is designed to produce merger trees suitable for usewith the semi-analytic galaxy formation model, galform .We have included a full description of the Dhalo algorithmwhich produces a set of haloes, and the merger trees thatdescribe their hierarchical evolution, that are consistent be-tween subsequent snapshots of the N-body simulations. Wehave presented the properties of the Dhaloes by comparingthem with the corresponding properties of the much morecommonly used FoF haloes (Davis et al. 1985).We have shown that unlike the FoF algorithm the Dhaloalgorithm is successful in avoiding distinct mass concentra-tions being prematurely linked together into a single halowhen their diffuse outer haloes touch. We have also illus-trated how some Dhaloes can be composed of more thanone FoF halo. This occurs as structure formation in CDM c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes log c . . . . . . . . P ( l o g c ) M (cid:12) /hRelaxed σ log c NB 0 . 129 0 . . 111 0 . NBFoF . . . . . log c . . . . . . . P ( l o g c ) M (cid:12) /hALL σ log c NB 0 . 145 0 . . 133 0 . NBFoF log C M (cid:12) /hRelaxed σ log c NB 0 . 109 0 . . 096 0 . NBFoF . . . . . log c M (cid:12) /hALL σ log c NB 0 . 134 0 . . 127 0 . NBFoF Figure 14. The distribution of concentrations for haloes in the two mass bins 10 . < log M /h − M (cid:12) < . 25 and 11 . < log M /h − M (cid:12) < . 25. The upper panels are for samples of relaxed haloes while the bottom panels are for all haloes whether ornot they satisfy the relaxation criteria. In each panel the blue histogram is for FoF haloes and the red histogram is for Dhaloes that donot have bijective matches to FoF haloes. The smooth curves are log-normal approximations with the same log c and second moment, σ , as the measured distributions. The corresponding values of log c and σ are given in the legend. models is not strictly hierarchical and occasionally a halo,after falling into a more massive halo, may escape to beyondthe virial radius of the more massive halo. For the purposesof the galform semi-analytic model it is convenient to con-sider such haloes as remaining as satellites of the main halo.We find that such remerged FoF haloes are not uncommon, but contribute very little mass to the larger haloes to whichthey are (re)attached.Despite the complex mapping between FoF andDhaloes, which results in a significant fraction of FoF haloesbeing broken up into multiple Dhaloes while other FoFhaloes get (re)merged into a single Dhalo, we find that the c (cid:13)000 25. The upper panels are for samples of relaxed haloes while the bottom panels are for all haloes whether ornot they satisfy the relaxation criteria. In each panel the blue histogram is for FoF haloes and the red histogram is for Dhaloes that donot have bijective matches to FoF haloes. The smooth curves are log-normal approximations with the same log c and second moment, σ , as the measured distributions. The corresponding values of log c and σ are given in the legend. models is not strictly hierarchical and occasionally a halo,after falling into a more massive halo, may escape to beyondthe virial radius of the more massive halo. For the purposesof the galform semi-analytic model it is convenient to con-sider such haloes as remaining as satellites of the main halo.We find that such remerged FoF haloes are not uncommon, but contribute very little mass to the larger haloes to whichthey are (re)attached.Despite the complex mapping between FoF andDhaloes, which results in a significant fraction of FoF haloesbeing broken up into multiple Dhaloes while other FoFhaloes get (re)merged into a single Dhalo, we find that the c (cid:13)000 , 1–23 Jiang et al. overall mass functions of the two sets of haloes are verysimilar. The mass functions of our Dhalo and FoF halo cat-alogues are both reasonably well fit over the mass range of10 to 10 . h − M (cid:12) by currently popular analytic massfunctions such as those of Warren et al (2006) and Reed etal. (2007).Approximately 90% of the Dhaloes have a unique one-to-one, bijective, match with a corresponding FoF halo. Forthis subset of haloes the mass of the Dhalo, M Dhalo , corre-lates much more closely with the standard virial mass, M ,than does the FoF mass. The median M FoF /M = 1 . . M Dhalo /M = 1 . . 3. The larger scatter in the FoF case is often causedby secondary mass concentrations that lie outside the r radius of the main substructure and are linked into the FoFhalo by particle bridges in overlapping diffuse haloes. Thenon-bijective Dhaloes have a wider distribution, with 90%of the distribution spanning a factor 2.2 and with the me-dian ratio reduced to M Dhalo /M = 0 . 95. This is due tothe subfind substructure finder, which is part of the Dhaloalgorithm, assigning less mass to subhaloes when they moveinto overdense environments. When utilised in galform thissystematic loss of mass is not an issue as the merger treesare preprocessed and mass is added back in to ensure thebranches of the galform merger trees always have mono-tonically increasing masses.The high resolution of the Millennium II simulation hasallowed us to study the density profiles and concentrationsof both FoF and Dhaloes over a wide range of mass. Toavoid contaminating our samples with unrelaxed haloes forwhich fitting smooth spherically symmetric profiles is inap-propriate we exclude unrelaxed haloes using the relaxationcriteria from Neto et al (2007). We find that 80% of bothFoF and Dhaloes are relaxed according to these criteria. ForFoF haloes we accurately reproduce the mass–concentrationdistribution found by Neto et al (2007) at high masses andextend the distribution to much lower masses. Combiningour results with those of Macci`o et al. (2007) and Netoet al (2007), we find that a single power law reproduces themass-concentration relation for over five decades in mass.We also find that the mass-concentration distributions forDhaloes agree very accurately with those for FoF haloes.This is true even for non-bijective Dhaloes which are sec-ondary components of FoF haloes. The properties of suchhaloes have generally been overlooked in previous studies.We show that the distributions of concentrations aroundthe mean mass-concentration relation are well described bylog-normal distributions for both the FoF and Dhaloes. ACKNOWLEDGEMENTS REFERENCES Audit E., Teyssier R., Alimi J., 1998, A&A, 333, 779Baugh C. M., Lacey C. G., Frenk C. S., Granato G. L., SilvaL., Bressan A., Benson A. J., Cole S., 2005, MNRAS, 356,1191Behroozi P., Wechsler R., Wu H.-Y., 2012, ascl.soft, 10008Behroozi P. S., Wechsler R. H., Wu H.-Y., Busha M. T.,Klypin A. A., Primack J. R., 2013, ApJ, 763, 18Benson A. J., Bower R. G., Frenk C. S., Lacey C. G., BaughC. M., Cole S., 2003, ApJ, 599, 38Benson A. J., Bower R., 2010, MNRAS, 405, 1573Berlind A. A., Weinberg D. H., 2002, ApJ, 575, 587Berlind A. A., et al., 2003, ApJ, 593, 1Bhattacharya S., Heitmann K., White M., Luki´c Z., Wag-ner C., Habib S., 2011, ApJ, 732, 122Binney J., 1977, ApJ, 215, 483Bower R. G., Benson A. J., Malbon R., Helly J. C., FrenkC. S., Baugh C. M., Cole S., Lacey C. G., 2006, MNRAS,370, 645Boylan-Kolchin. M, Springel. V, White S. D. M, JenkinsA., Lemson. G., 2009, MNRAS, 398, 1150BBullock J. S., Kolatt T. S., Sigad Y., Somerville R. S.,Kravtsov A. V., Klypin A. A., Primack J. R., Dekel A.,2001, MNRAS, 321, 559Bundy K., Ellis R. S., Conselice C. J., 2005, ApJ, 625, 621Busha M. T., Alvarez M. A., Wechsler R. H., Abel T.,Strigari L. E., 2010, ApJ, 710, 408Cohn J. D., White M., 2008, MNRAS, 385, 2025Cole S., 1991, ApJ, 367, 45Cole S., Aragon-Salamanca A., Frenk C. S., Navarro J. F.,Zepf S. E., 1994, MNRAS, 271, 781Cole S., Lacey C. G., Baugh C. M., Frenk C. S., 2000,MNRAS, 319, 168Cole S., Lacey C., 1996, MNRAS, 281, 716Colless M., et al., 2001, MNRAS, 328, 1039Courtin J., Rasera Y., Alimi J., Corasaniti P., Boucher V.,F¨uzfa A., 2010, MNRAS, 410, 1911CCreasey P., Theuns T., Bower R. G., Lacey C. G., 2011,MNRAS, 415, 3706Crocce M., Fosalba P., Castander F. J., Gazta˜naga E.,2010, MNRAS, 403, 1353Davis M., Efstathiou G., Frenk C. S., White S. D. M., 1985,ApJ, 292, 371Diemand J., Kuhlen M., Madau P., 2006, ApJ, 649, 1Eke V. R., Navarro J. F., Steinmetz M., 2001, ApJ, 554,114Eke V. R., et al., 2004, MNRAS, 355, 769Einasto J., Klypin A. A., Saar E., Shandarin S. F., 1984,MNRAS, 206, 529Evrard, A.E., et al., 2002, APJ, 573, 7Evrard A. E., et al., 2008, ApJ, 672, 122Font A. S., et al., 2008, MNRAS, 389, 1619Font A. S., et al., 2011, MNRAS, 417, 1260Frenk C. S., White S. D. M., Davis M., Efstathiou G., 1988,ApJ, 327, 507Gao L., Navarro J. F., Cole S., Frenk C. S., White S. D. M., c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes Springel V., Jenkins A., Neto A. F., 2008, MNRAS, 387,536Gill S. P. D., Knebe A., Gibson B. K., 2005, MNRAS, 356,1327Giocoli C., Pieri L., Tormen G., 2008, MNRAS, 387, 689Gottloeber.S, 1997, ApJ, 557, 616Gottl¨ober S., Yepes G., 2007, ApJ, 664, 117Guo Q., et al., 2011, MNRAS, 413, 101Green A. M., Hofmann S., Schwarz D. J., 2004, MNRAS,353, L23Han J., Jing Y. P., Wang H., Wang W., 2012, MNRAS,427, 2437Harker G., Cole S., Helly J., Frenk C., Jenkins A., 2006,MNRAS, 367, 1039Hatton S., Devriendt J. E. G., Ninin S., Bouchet F. R.,Guiderdoni B., Vibert D., 2003, MNRAS, 343, 75Helly J. C., Cole S., Frenk C. S., Baugh C. M., Benson A.,Lacey C., 2003, MNRAS, 338, 903Heitmann K., Luki´c Z., Habib S., Ricker P. M., 2006, ApJ,642, L85Huchra J. P., Geller M. J., 1982, ApJ, 257, 423Jenkins A., Frenk C. S., White S. D. M., Colberg J. M.,Cole S., Evrard A. E., Couchman H. M. P., Yoshida N.,2001,MNRAS, 321,372Jiang F., van den Bosch F. C., 2013, arXiv, arXiv:1311.5225Kauffmann G., White S., 1993, MNRAS, 261, 921Kim H.-S., Baugh C. M., Benson A. J., Cole S., Frenk C. S.,Lacey C. G., Power C., Schneider M., 2011, MNRAS, 414,2367Klypin A. A., Trujillo-Gomez S., Primack J., 2011, ApJ,740, 102Knebe A., et al., 2011, MNRAS, 415, 2293Knebe A., et al., 2013, MNRAS, 435, 1618Klypin A., Gottlber, S., Kravtsov, A V., Khokhlov A. M.,1999, ApJ, 516.530KKnollmann S. R., Knebe A., 2009, ApJS, 182, 608Koposov S. E., Yoo J., Rix H.-W., Weinberg D. H., Macci`oA. V., Escud´e J. M., 2009, ApJ, 696, 2179Lacey C., Silk J., 1991, ApJ, 381, 14Lacey C., Cole S., 1993, MNRAS, 262, 627Lacey C., Cole S., 1994, MNRAS, 271, 676Linder E. V., Jenkins A., 2003, MNRAS, 346,573LLokas E. L., Bode P., Hoffmann Y., 2004, MNRAS,349,595LLagos C. D. P., Baugh C. M., Lacey C. G., Benson A. J.,Kim H.-S., Power C., 2011, MNRAS, 418, 1649Lukic Z., Reed D., Habib S., Heitmann K., 2009, ApJ, 692,217Ludlow A. D., Navarro J. F., Springel V., Jenkins A., FrenkC. S., Helmi A., 2009, ApJ, 692, 931Macci`o A. V., Dutton A. A., van den Bosch F. C., MooreB., Potter D., Stadel J., 2007, MNRAS, 378, 55Macci`o A. V., Dutton A. A., van den Bosch F. C., 2008,MNRAS, 391, 1940Macci`o A. V., Kang X., Fontanot F., Somerville R. S., Ko-posov S., Monaco P., 2010, MNRAS, 402, 1995Merson A. I., et al., 2012, arXiv, arXiv:1206.4049Moreno J., Giocoli C., Sheth R. K., 2008, MNRAS, 391,1729Muldrew S. I., Pearce F. R., Power C., 2011, MNRAS, 410,2617Mu˜noz J. A., Madau P., Loeb A., Diemand J., 2009, MN- RAS, 400, 1593Murray S. G., Power C., Robotham A. S. G., 2013, MN-RAS, 434, L61Murray S., Power C., Robotham A., 2013, arXiv,arXiv:1306.6721Navarro J. F., Frenk C. S., White S. D. M., 1995, MNRAS,275, 720Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462,563Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490,493Navarro J. F., Hayashi E., Power C., Jenkins A. R., FrenkC. S., White S. D. M., Springel V., Stadel J., Quinn T.R.,2004, MNRAS, 349, 1039NNeistein E., Dekel A., 2008, MNRAS, 388, 1792Neto A., Gao L., Bett P., Cole S., Julio F. N., Frenk C. S.,White S. D. M., Springel V., Jenkins A., 2007, MNRAS,381, 1450NOkabe N., Takada M., Umetsu K., Futamase T., SmithG. P., 2010, PASJ, 62, 811Onions J., et al., 2012, MNRAS, 423, 1200Parkinson H., Cole S., Helly J., 2008, MNRAS, 383, 557Peacock J. A., Smith R. E., 2000, MNRAS, 318, 1144Percival W. J., et al., 2001, MNRAS, 327, 1297Power C., Navarro J. F., Jenkins A., Frenk C. S., WhiteS. D. M., Springel V., Stadel J., Quinn T., 2003, MNRAS,338, 14Power C., Knebe A., Knollmann S. R., 2012, MNRAS, 419,1576Press W. H., Davis M., 1982, ApJ, 259, 449Reed D., Gardner J., Quinn T., Stadel. J, Fardal M., LakeG., Fabio G., 2003, MNRAS,346,565RReed D., Bower R., Frenk C. S., Jenkins. A, Theuns, T.,2007, MNRAS,374,2RRees M. J. & Ostriker J. P., 1977, MNRAS, 179, 541Snchez1 A. G., Baugh C. M. , Percival W. J, PeacockJ. K, Padilla N. D, Cole S., , Frenk C. S., Norberg P.,2006, MNRAS, 366, 1Schaye J., et al., 2010, MNRAS, 402, 1536Seljak U., 2000, MNRAS, 318, 203Sherth R. K., Tormen G., 1999, MNRAS,308.119SSheth, R. K., & Tormen, G. 2002, MNRAS, 329, 61Somerville R. S., Primack J. R., 1999, MNRAS, 310, 1087Somerville R. S., Hopkins P. F., Cox T. J., Robertson B. E.,Hernquist L., 2008, MNRAS, 391, 481Spergel D. N., et al., 2003, ApJS, 148, 175Springel V., White S. D. M., Tormen G., Kauffmann G.,2001, MNRAS, 328, 726Springel V., 2005b, MNRAS, 364, 1105Springel V., et al., 2008, MNRAS, 391, 1685Springel V., et al., 2005a, Nature, 435, 629Srisawat C., et al., 2013, MNRAS, 2357Summers F. J., Davis M., Evrard A. E., 1995, ApJ, 454, 1Tinker J., Kravtsov A. V., Klypin A., Abazajian K., War-ren M., Yepes G., Gottlber S., Holz D. E., 2008, ApJ, 688,709TThomas P. A., Muanwong O., Pearce F. R., CouchmanH. M. P., Edge A. C., Jenkins A., Onuora L., 2001, MN-RAS, 324, 450van den Bosch, F. C. 2002, MNRAS, 331, 98van den Bosch F. C., Tormen G., Giocoli C., 2005, MNRAS,359, 1029 c (cid:13)000 Audit E., Teyssier R., Alimi J., 1998, A&A, 333, 779Baugh C. M., Lacey C. G., Frenk C. S., Granato G. L., SilvaL., Bressan A., Benson A. J., Cole S., 2005, MNRAS, 356,1191Behroozi P., Wechsler R., Wu H.-Y., 2012, ascl.soft, 10008Behroozi P. S., Wechsler R. H., Wu H.-Y., Busha M. T.,Klypin A. A., Primack J. R., 2013, ApJ, 763, 18Benson A. J., Bower R. G., Frenk C. S., Lacey C. G., BaughC. M., Cole S., 2003, ApJ, 599, 38Benson A. J., Bower R., 2010, MNRAS, 405, 1573Berlind A. A., Weinberg D. H., 2002, ApJ, 575, 587Berlind A. A., et al., 2003, ApJ, 593, 1Bhattacharya S., Heitmann K., White M., Luki´c Z., Wag-ner C., Habib S., 2011, ApJ, 732, 122Binney J., 1977, ApJ, 215, 483Bower R. G., Benson A. J., Malbon R., Helly J. C., FrenkC. S., Baugh C. M., Cole S., Lacey C. G., 2006, MNRAS,370, 645Boylan-Kolchin. M, Springel. V, White S. D. M, JenkinsA., Lemson. G., 2009, MNRAS, 398, 1150BBullock J. S., Kolatt T. S., Sigad Y., Somerville R. S.,Kravtsov A. V., Klypin A. A., Primack J. R., Dekel A.,2001, MNRAS, 321, 559Bundy K., Ellis R. S., Conselice C. J., 2005, ApJ, 625, 621Busha M. T., Alvarez M. A., Wechsler R. H., Abel T.,Strigari L. E., 2010, ApJ, 710, 408Cohn J. D., White M., 2008, MNRAS, 385, 2025Cole S., 1991, ApJ, 367, 45Cole S., Aragon-Salamanca A., Frenk C. S., Navarro J. F.,Zepf S. E., 1994, MNRAS, 271, 781Cole S., Lacey C. G., Baugh C. M., Frenk C. S., 2000,MNRAS, 319, 168Cole S., Lacey C., 1996, MNRAS, 281, 716Colless M., et al., 2001, MNRAS, 328, 1039Courtin J., Rasera Y., Alimi J., Corasaniti P., Boucher V.,F¨uzfa A., 2010, MNRAS, 410, 1911CCreasey P., Theuns T., Bower R. G., Lacey C. G., 2011,MNRAS, 415, 3706Crocce M., Fosalba P., Castander F. J., Gazta˜naga E.,2010, MNRAS, 403, 1353Davis M., Efstathiou G., Frenk C. S., White S. D. M., 1985,ApJ, 292, 371Diemand J., Kuhlen M., Madau P., 2006, ApJ, 649, 1Eke V. R., Navarro J. F., Steinmetz M., 2001, ApJ, 554,114Eke V. R., et al., 2004, MNRAS, 355, 769Einasto J., Klypin A. A., Saar E., Shandarin S. F., 1984,MNRAS, 206, 529Evrard, A.E., et al., 2002, APJ, 573, 7Evrard A. E., et al., 2008, ApJ, 672, 122Font A. S., et al., 2008, MNRAS, 389, 1619Font A. S., et al., 2011, MNRAS, 417, 1260Frenk C. S., White S. D. M., Davis M., Efstathiou G., 1988,ApJ, 327, 507Gao L., Navarro J. F., Cole S., Frenk C. S., White S. D. M., c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes Springel V., Jenkins A., Neto A. F., 2008, MNRAS, 387,536Gill S. P. D., Knebe A., Gibson B. K., 2005, MNRAS, 356,1327Giocoli C., Pieri L., Tormen G., 2008, MNRAS, 387, 689Gottloeber.S, 1997, ApJ, 557, 616Gottl¨ober S., Yepes G., 2007, ApJ, 664, 117Guo Q., et al., 2011, MNRAS, 413, 101Green A. M., Hofmann S., Schwarz D. J., 2004, MNRAS,353, L23Han J., Jing Y. P., Wang H., Wang W., 2012, MNRAS,427, 2437Harker G., Cole S., Helly J., Frenk C., Jenkins A., 2006,MNRAS, 367, 1039Hatton S., Devriendt J. E. G., Ninin S., Bouchet F. R.,Guiderdoni B., Vibert D., 2003, MNRAS, 343, 75Helly J. C., Cole S., Frenk C. S., Baugh C. M., Benson A.,Lacey C., 2003, MNRAS, 338, 903Heitmann K., Luki´c Z., Habib S., Ricker P. M., 2006, ApJ,642, L85Huchra J. P., Geller M. J., 1982, ApJ, 257, 423Jenkins A., Frenk C. S., White S. D. M., Colberg J. M.,Cole S., Evrard A. E., Couchman H. M. P., Yoshida N.,2001,MNRAS, 321,372Jiang F., van den Bosch F. C., 2013, arXiv, arXiv:1311.5225Kauffmann G., White S., 1993, MNRAS, 261, 921Kim H.-S., Baugh C. M., Benson A. J., Cole S., Frenk C. S.,Lacey C. G., Power C., Schneider M., 2011, MNRAS, 414,2367Klypin A. A., Trujillo-Gomez S., Primack J., 2011, ApJ,740, 102Knebe A., et al., 2011, MNRAS, 415, 2293Knebe A., et al., 2013, MNRAS, 435, 1618Klypin A., Gottlber, S., Kravtsov, A V., Khokhlov A. M.,1999, ApJ, 516.530KKnollmann S. R., Knebe A., 2009, ApJS, 182, 608Koposov S. E., Yoo J., Rix H.-W., Weinberg D. H., Macci`oA. V., Escud´e J. M., 2009, ApJ, 696, 2179Lacey C., Silk J., 1991, ApJ, 381, 14Lacey C., Cole S., 1993, MNRAS, 262, 627Lacey C., Cole S., 1994, MNRAS, 271, 676Linder E. V., Jenkins A., 2003, MNRAS, 346,573LLokas E. L., Bode P., Hoffmann Y., 2004, MNRAS,349,595LLagos C. D. P., Baugh C. M., Lacey C. G., Benson A. J.,Kim H.-S., Power C., 2011, MNRAS, 418, 1649Lukic Z., Reed D., Habib S., Heitmann K., 2009, ApJ, 692,217Ludlow A. D., Navarro J. F., Springel V., Jenkins A., FrenkC. S., Helmi A., 2009, ApJ, 692, 931Macci`o A. V., Dutton A. A., van den Bosch F. C., MooreB., Potter D., Stadel J., 2007, MNRAS, 378, 55Macci`o A. V., Dutton A. A., van den Bosch F. C., 2008,MNRAS, 391, 1940Macci`o A. V., Kang X., Fontanot F., Somerville R. S., Ko-posov S., Monaco P., 2010, MNRAS, 402, 1995Merson A. I., et al., 2012, arXiv, arXiv:1206.4049Moreno J., Giocoli C., Sheth R. K., 2008, MNRAS, 391,1729Muldrew S. I., Pearce F. R., Power C., 2011, MNRAS, 410,2617Mu˜noz J. A., Madau P., Loeb A., Diemand J., 2009, MN- RAS, 400, 1593Murray S. G., Power C., Robotham A. S. G., 2013, MN-RAS, 434, L61Murray S., Power C., Robotham A., 2013, arXiv,arXiv:1306.6721Navarro J. F., Frenk C. S., White S. D. M., 1995, MNRAS,275, 720Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462,563Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490,493Navarro J. F., Hayashi E., Power C., Jenkins A. R., FrenkC. S., White S. D. M., Springel V., Stadel J., Quinn T.R.,2004, MNRAS, 349, 1039NNeistein E., Dekel A., 2008, MNRAS, 388, 1792Neto A., Gao L., Bett P., Cole S., Julio F. N., Frenk C. S.,White S. D. M., Springel V., Jenkins A., 2007, MNRAS,381, 1450NOkabe N., Takada M., Umetsu K., Futamase T., SmithG. P., 2010, PASJ, 62, 811Onions J., et al., 2012, MNRAS, 423, 1200Parkinson H., Cole S., Helly J., 2008, MNRAS, 383, 557Peacock J. A., Smith R. E., 2000, MNRAS, 318, 1144Percival W. J., et al., 2001, MNRAS, 327, 1297Power C., Navarro J. F., Jenkins A., Frenk C. S., WhiteS. D. M., Springel V., Stadel J., Quinn T., 2003, MNRAS,338, 14Power C., Knebe A., Knollmann S. R., 2012, MNRAS, 419,1576Press W. H., Davis M., 1982, ApJ, 259, 449Reed D., Gardner J., Quinn T., Stadel. J, Fardal M., LakeG., Fabio G., 2003, MNRAS,346,565RReed D., Bower R., Frenk C. S., Jenkins. A, Theuns, T.,2007, MNRAS,374,2RRees M. J. & Ostriker J. P., 1977, MNRAS, 179, 541Snchez1 A. G., Baugh C. M. , Percival W. J, PeacockJ. K, Padilla N. D, Cole S., , Frenk C. S., Norberg P.,2006, MNRAS, 366, 1Schaye J., et al., 2010, MNRAS, 402, 1536Seljak U., 2000, MNRAS, 318, 203Sherth R. K., Tormen G., 1999, MNRAS,308.119SSheth, R. K., & Tormen, G. 2002, MNRAS, 329, 61Somerville R. S., Primack J. R., 1999, MNRAS, 310, 1087Somerville R. S., Hopkins P. F., Cox T. J., Robertson B. E.,Hernquist L., 2008, MNRAS, 391, 481Spergel D. N., et al., 2003, ApJS, 148, 175Springel V., White S. D. M., Tormen G., Kauffmann G.,2001, MNRAS, 328, 726Springel V., 2005b, MNRAS, 364, 1105Springel V., et al., 2008, MNRAS, 391, 1685Springel V., et al., 2005a, Nature, 435, 629Srisawat C., et al., 2013, MNRAS, 2357Summers F. J., Davis M., Evrard A. E., 1995, ApJ, 454, 1Tinker J., Kravtsov A. V., Klypin A., Abazajian K., War-ren M., Yepes G., Gottlber S., Holz D. E., 2008, ApJ, 688,709TThomas P. A., Muanwong O., Pearce F. R., CouchmanH. M. P., Edge A. C., Jenkins A., Onuora L., 2001, MN-RAS, 324, 450van den Bosch, F. C. 2002, MNRAS, 331, 98van den Bosch F. C., Tormen G., Giocoli C., 2005, MNRAS,359, 1029 c (cid:13)000 , 1–23 Jiang et al. Vitvitska M., Klypin A. A., Kravtsov A. V., WechslerR. H., Primack J. R., Bullock J. S., 2002, ApJ, 581, 799Warren M. S., Abazajian K., Holz D. E., Teodoro L., 2006,ApJ, 646, 881Watson W. A., Iliev I. T., D’Aloisio A., Knebe A., ShapiroP. R., Yepes G., 2013, MNRAS, 433, 1230White S. D. M., Frenk, C. S., 1991,ApJ.379..52WWhite S. D. M., Rees, M. J., 1978,MNRAS.183..341WWhite M., 2001, ApJS, 143, 241 APPENDIX A: CONSTRUCTING DhaloMERGER TREES Here we describe in detail the algorithm used to producethe Dhalo merger trees. These merger trees are intendedto be used as input to the galform semi-analytic model ofgalaxy formation. The need for consistency between the halomodel used in the semi-analytic calculation and the N-bodysimulation imposes some requirements on the constructionof the merger trees.The galform galaxy formation model makes the ap-proximation that mergers between haloes are instantaneousevents and assumes that haloes, once merged, do not frag-ment. However, in N-body simulations halo mergers take afinite amount of time and it is not uncommon for a halofalling into another, more massive halo to escape to wellbeyond the virial radius after its initial infall (Gill, Knebe,& Gibson 2005; Ludlow et al. 2009). We therefore need tochoose when to consider N-body haloes to have merged inthe semi-analytic model and define our haloes such that theyremain merged at all later times. We also wish to define thehaloes used to construct the trees such that, as far as possi-ble, they resemble the spherically symmetric, virialised ob-jects assumed in the galaxy formation model. Quantifyingthe extent to which we have achieved this is one of the mainaims of this paper. A1 Halo catalogues The first step in building the merger trees is to use the FoF(Davis et al. 1985) and subfind algorithms (Springel et al.2001) to identify haloes and subhaloes in all of the simu-lation snapshots. The subfind algorithm decomposes eachFoF halo into subhaloes by identifying self bound densitymaxima. Usually the most massive subhalo contains most ofthe mass of the original FoF halo. Secondary density maximagive rise to additional subhaloes. Compared to the FoF halothe most massive subhalo does not include any of the massassigned to other subhaloes (a simulation particle can onlybelong to one subhalo) nor does it include particles that arenot gravitationally bound to it. Some of the lowest mass FoFhaloes have no self bound subhaloes and most FoF haloeshave at least some “fuzz” of unbound particles which belongto no subhalo. FoF haloes with no self-bound subhaloes arenot used in the construction of the merger trees. A2 Building the subhalo merger trees Before we can construct the Dhalo merger trees, it is nec-essary to define subhalo merger trees by identifying the de- scendant of each subhalo. The code we use to do this wasincluded in the merger trees comparison project carried outby Srisawat et al. (2013) under the name D-Trees . Theproject concluded that it was desirable feature for a mergertree code to use particle IDs to match haloes between snap-shots and have the ability to search multiple snapshots fordescendants. The latter requirement was due to the tendencyof the AHF group finder (Knollmann & Knebe 2009) usedin the project to temporarily fail to detect sub-structuresduring mergers.Since subfind suffers from a similar problem, we allowfor the possibility that the descendant of a subhalo maybe found more than one snapshot later. Our approach is todevise an algorithm which can identify the descendant ofa halo at any single, later snapshot, apply it to the next N step snapshots (where N step = 5), and pick one of these N step possible descendants to use as the descendant of thesubhalo in the merger trees.Alternative solutions to this problem include allowingthe merger tree code to modify the subhalo catalogue toensure consistency of subhalo properties between snapshots( ConsistentTrees , Behroozi et al. 2013) and using infor-mation from previous snapshots to define the subhalo cata-logue (HBT, Han et al. 2012).In common with all but one of the merger tree codes inthe comparison ( Jmerge , which relies entirely on aggregateproperties of the haloes), we identify descendants by find-ing subhaloes at different snapshots which have particles incommon. A2.1 Identifying a descendant at a single, later snapshot To find the descendant at snapshot j , of a halo which existsat an earlier snapshot, i , the following method is used. Foreach halo containing N p particles the N link most bound areidentified, where N link is given by N link = min( N linkmax , max( f trace N p , N linkmin )) (A1)with N linkmin = 10, N linkmax = 100 and f trace = 0 . i , descendant candi-dates are found by locating all haloes at snapshot j whichreceived at least one particle from the earlier halo. Then,a single descendant is chosen from these candidates as fol-lows. If any of the descendant candidates received a largerfraction of their N link most bound particles from the progen-itor halo than from any other halo at the earlier snapshot,then the descendant is chosen from these candidates onlyand the halo at snapshot i will be designated the main pro-genitor of the chosen descendant; otherwise, all candidatesare considered and the halo will not be the main progenitorof its descendant. The descendant of the halo at snapshot i is taken to be the remaining candidate which received thelargest fraction of the N l ink most bound of the progenitorhalo. For each halo at snapshot j , this method identifieszero or more progenitors of which at most one may be amain progenitor. Note that it is not guaranteed that a mainprogenitor will be found for every halo.By following the most bound part of the subhalo, weensure that if the core of a subhalo survives at the latersnapshot it is identified as the descendant irrespective of howmuch mass has been lost. It also means that in cases wherean object at the later snapshot has multiple progenitors we c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes Snapshot iSnapshot jAll particles in the group Nlink most bound particlesa) A single group which survivesat the next snapshot A BCb) Two groups merge. Group A is considered to have survived c) A satellite group which isstripped of much of its massHost haloAB ABC Figure A1. Schematic examples illustrating the method used to link subfind subhaloes between pairs of snapshots i and j , where i < j .The green circles represent subfind subhaloes. The most bound N link particles in each subhalo at the later time are shown in red. Fromleft to right are a) a single, isolated subhalo which still exists at the next snapshot, b) a merger between subhaloes A and B where moreof the most bound partciles of the merged halo C come from halo A than from any other halo and therefore halo A is considered to bethe main progenitor of halo C , and c) a satellite subhalo orbiting within a background halo which loses a large fraction of its particlesto its host halo at the next snapshot but is still identified by subfind . Arrows between green circles show the location of the majority ofthe particles in the subhalo at the later snapshot. Arrows starting from red circles show the location of the majority of the most boundparticles at the earlier snapshot. can determine which one of the progenitors contributed thelargest fraction of the most bound core of the descendantobject. We consider this main progenitor to have survivedthe merger while the other progenitors have merged onto itand ceased to exist as independent objects.Fig. A1 shows three examples of this linking procedure.In the simplest case (left) a single, isolated subhalo B atsnapshot j is identified as the descendant of subhalo A whichexists at the earlier snapshot i . Since more of the most boundparticles of subhalo B come from subhalo A than from anyother subhalo, we conclude that A is the main progenitor of B . In the second case (centre) two subhaloes A and B mergeto form subhalo C at the later snapshot. Subhalo A is de-termined to be the main progenitor because it contributedthe largest fraction of the most bound particles of the de-scendant, C . In the third example (right) a satellite subhalo A exists within a more massive host halo. In this case, par-ticles from the subhalo A are split between subhalo B andthe host halo C at the later snapshot. While a large fraction(or even the vast majority) of the particles from subhalo A may belong to the host halo at the later snapshot, we choosesubhalo B as the descendant because its most bound partcame from subhalo A . A2.2 Searching multiple snapshots for descendants If a subhalo is not found to be the main progenitor of itsdescendant, this may indicate that the subhalo has mergedwith another subhalo and no longer exists as an indepen-dent object. However, it is also possible that the substruc-ture finder has simply failed to identify the object at thelater snapshot because it is superimposed on the dense cen- All particles in the groupNlink most bound particlesHost haloSnapshot iSnapshot i+1Snapshot i+2 AC BDE Figure A2. A schematic example of a case where the descen-dant of a subhalo is found to be more than one snapshot later.The green circles represent a satellite subfind subhalo within alarger host halo which is represented by the blue circles. Threeconsecutive snapshots are shown.c (cid:13)000 A schematic example of a case where the descen-dant of a subhalo is found to be more than one snapshot later.The green circles represent a satellite subfind subhalo within alarger host halo which is represented by the blue circles. Threeconsecutive snapshots are shown.c (cid:13)000 , 1–23 Jiang et al. tral parts of a larger subhalo. Typically this phase lasts for asmall fraction of the host halo dynamical time (Behroozi etal. 2013) which in turn is much shorter than the usual inter-val between the snapshots of cosmological N-body simula-tions. Hence by looking one snapshot ahead we will normallyfind the missed subhalo, but one can be unlucky and catchit half an orbit later when again it is hidden by the densecore of the more massive subhalo in which it is orbiting.Hence looking several snapshots ahead exponentially supp-reses this possibility. Thus in order to distinguish betweensubhalo mergers and subhaloes which are just temporarilylost it is necessary to search multiple snapshots for descen-dants.In our algorithm for each snapshot i in the simulationdescendants are identified at later snapshots in the range i +1to i + N step using the method described in section A2.1. Foreach subhalo at snapshot i this gives up to N step possibledescendants. One of these descendants is picked for use inthe merger trees as follows: if the subhalo at snapshot i isthe main progenitor of one or more of the descendants, theearliest of these descendants which does not have a mainprogenitor at a snapshot later than i is chosen. If no suchdescendant exists, the earliest descendant found is chosenirrespective of main progenitor status.Descendants more than one snapshot later are only cho-sen in cases where the earlier subhalo is the main progenitor— i.e. where the group still survives as an independent ob-ject. If the subhalo does not survive we have no way to deter-mine whether it merged immediately or if subfind failed todetect it for one more snapshots prior to the merger, so wesimply assume that the merger happened between snapshots i and i + 1.Fig. A2 shows a case where a descendant more thanone snapshot later is chosen. Subhalo A exists at snapshot i . Its descendant at snapshot i + 1 is found to be the sub-halo D . However, the most bound particles of D were notcontributed by subhalo A , but by another progenitor, sub-halo C . This means that A is not the main progenitor of itsdescendant at snapshot i + 1 and so it is necessary to con-sider possible descendants at later snapshots. Two subhaloesat snapshot i + 2 ( B and E ) receive particles from subhalo A . Since the most bound particles of subhalo B came fromsubhalo A , A is the main progenitor of B and subhalo B istaken to be the descendant of A . A3 Constructing a halo catalogue At this point we have a descendant for each subhalo. Thisis sufficient to define merger trees for the subhaloes. These subfind trees can be split into “branches” as follows. Anew branch begins whenever a new subhalo forms (i.e. thesubhalo has no progenitors). The remaining subhaloes thatmake up the branch are found by following the descendantpointers until either a subhalo is reached that is not themain progenitor of its descendant, a subhalo is reached thathas no descendant, or the final snapshot of the simulation isreached. Each of these branches represents the life-time ofan independent halo or sub-halo in the simulation. We con-struct haloes and halo merger trees by grouping togetherthese branches of the subhalo merger trees using methodswhich will be described below. We refer to the resulting col-lections of subhaloes as “Dhaloes”. Fig. A3 is an example of a Dhalo merger tree with the subhalo merger tree branchesmarked. In this case there are three branches. Branch A isa single, massive halo which exists as an independent haloat all four snapshots. Branch B is a smaller halo which be-comes a satellite subhalo within halo A , but continues toexist. Branch C is another small halo which briefly becomesa satellite before merging with A .For each subhalo in a FoF halo we identify the leastmassive, more massive “enclosing” subhalo in the same FoFhalo. Subhalo A is said to enclose subhalo B if B ’s centrelies within twice the half mass radius of A . A pointer to theenclosing subhalo is stored for each subhalo that is enclosed.This produces a tree structure which is intended to representthe hierarchy of haloes, sub-haloes, sub-sub-haloes etc. inthe FoF halo. Any subhalo which is not enclosed by anyother becomes a new Dhalo. Any subhaloes enclosed by thissubhalo are assigned to the new Dhalo.We then iterate through the snapshots from high red-shift to low redshift. For each subhalo we find the maximumnumber of particles it ever contained while it was the mostmassive subhalo in its parent FoF halo. If a satellite subhaloin a Dhalo retains a fraction f split of its maximum isolatedmass then it is split from its parent Dhalo and becomes anew Dhalo. Any subhaloes enclosed by this subhalo are as-signed to the new Dhalo too. We usually set f split = 0 . 75, sothat when a halo falls into another, more massive halo thetwo haloes will only be considered to have merged into oneonce the smaller halo has been stripped of some of its mass.This is to ensure that haloes artificially linked by the FoFalgorithm are still treated as separate objects.In some cases a subhalo may escape from its parenthalo. This happens to halo B in Fig. A3. For the purposesof semi-analytic galaxy formation modelling, we would liketo continue to treat such subhaloes as satellites in the parenthalo so that each in-falling halo contributes a single branchto the halo merger tree. This is done by merging such ob-jects back on to the Dhalo they escaped from; the subhalois recorded as a satellite within the original Dhalo at alllater times regardless of its spatial position. Any subhaloesit encloses will also be considered to be part of this Dhalo.In practice the re-merging is carried out in the followingway. For each Dhalo A we identify a descendant Dhalo B bydetermining which later Dhalo contains the descendant ofthe most massive subhalo in A which survives at the nextsnapshot. In every case where a subhalo in A survives, weassign the descendant of the subhalo to Dhalo B . We repeatthis process for all Dhaloes at each snapshot in decreasingorder of redshift. This ensures that if any two subhaloes arein the same Dhalo at one snapshot, and both survive at thenext snapshot, they will both be in the same Dhalo at thenext snapshot.This process produces a Dhalo catalogue for each snap-shot. Each Dhalo contains one or more subhaloes and eachsubhalo may have a pointer to a descendant at some latersnapshot. Any subhaloes in a Dhalo which survive at thenext snapshot are guaranteed to belong to the same Dhaloat the next snapshot. This provides a simple way to identifya descendant for each Dhalo and defines the Dhalo mergertrees. Fig. A3 shows an example of a Dhalo merger tree.The two smaller haloes B and C merge with a larger halo A . Halo C survives as a satellite for one snapshot beforemerging with the descendant of A . Halo B also becomes a c (cid:13) , 1–23 ierarchial N-body Dark Matter Haloes Snapshot iSnapshot i+1Snapshot i+2Snapshot i+3SubgroupDhaloSubgroup merger tree branchAB C Figure A3. An example of a Dhalo merger tree showing two lessmassive haloes falling into another, more massive halo. Subhaloesare shown in green. Red areas indicate subhaloes which belong tothe same Dhalo. The black arrows show branches of the subhalomerger tree. satellite sub-halo and then temporarily escapes from the par-ent halo before falling back in. At all times after the initialinfall it is considered to be part of the parent Dhalo. c (cid:13)000