The Dynamical State of Dark Matter Haloes in Cosmological Simulations I: Correlations with Mass Assembly History
aa r X i v : . [ a s t r o - ph . C O ] S e p Mon. Not. R. Astron. Soc. , 1–13 () Printed 5 November 2018 (MN L A TEX style file v2.2)
The Dynamical State of Dark Matter Haloes in CosmologicalSimulations I: Correlations with Mass Assembly History
Chris Power , , Alexander Knebe & Steffen R. Knollmann International Centre for Radio Astronomy Research, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia Department of Physics & Astronomy, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom Departamento de F´ısica Te´orica, M´odulo C-15, Facultad de Ciencias, Universidad Aut´onoma de Madrid, 28049 Cantoblanco, Madrid, Spain
ABSTRACT
Using a statistical sample of dark matter haloes drawn from a suite of cosmological N -body simulations of the Cold Dark Matter (CDM) model, we quantify the impact of a sim-ulated halo’s mass accretion and merging history on two commonly used measures of itsdynamical state, the virial ratio η and the centre of mass offset ∆ r . Quantifying this relation-ship is important because the degree to which a halo is dynamically equilibrated will influencethe reliability with which we can measure characteristic equilibrium properties of the struc-ture and kinematics of a population of haloes. We begin by verifying that a halo’s formationredshift z form correlates with its virial mass M vir and we show that the fraction of its recentlyaccreted mass and the likelihood of it having experienced a recent major merger increaseswith increasing M vir and decreasing z form . We then show that both η and ∆ r increase withincreasing M vir and decreasing z form , which implies that massive recently formed haloes aremore likely to be dynamically unrelaxed than their less massive and older counterparts. Ouranalysis shows that both η and ∆ r are good indicators of a halo’s dynamical state, showingstrong positive correlations with recent mass accretion and merging activity, but we arguethat ∆ r provides a more robust and better defined measure of dynamical state for use in cos-mological N -body simulations at z ≃ . We find that ∆ r . . is sufficient to pick outdynamically relaxed haloes at z =0. Finally, we assess our results in the context of previousstudies, and consider their observational implications. Key words: methods: N -body simulations – galaxies: formation – galaxies: haloes – cos-mology: theory – dark matter – large-scale structure of Universe One of the fundamental assumptions underpinning modern theoriesof galaxy formation is that galaxies form and evolve in massive viri-alised haloes of dark matter (White & Rees 1978; White & Frenk1991). Characterising the properties of these haloes is an impor-tant problem, both theoretically and observationally, and its studyhas been one of the main objectives of cosmological N -body sim-ulations over the last two decades. The majority of these simula-tions have modeled halo formation and evolution in a purely ColdDark Matter (CDM) universe (cf. Springel et al. 2006), with the fo-cus primarily on their equilibrium structure (cf. Diemand & Moore2009). Various studies have revealed that CDM haloes in dy-namical equilibrium are triaxial structures (e.g. Bailin & Steinmetz2005) supported by velocity dispersion rather than rotation (e.g.Bett et al. 2007), with mass profiles that are divergent down to thesmallest resolvable radius (e.g. Diemand et al. 2008; Stadel et al.2009; Navarro et al. 2010) and an abundance of substructure (e.g.Diemand et al. 2007; Springel et al. 2008; Gao et al. 2011).The qualification that a halo is in dynamical equilibrium is a particularly important one when seeking to characterise the struc-ture and kinematics of simulated haloes in cosmological simu-lations. Previous studies have shown that dynamically unrelaxedhaloes tend to have lower central densities (see, for example,Tormen et al. 1997; Macci`o et al. 2007; Romano-D´ıaz et al. 2007)and higher velocity dispersions (see, for example, Tormen et al.1997; Hetznecker & Burkert 2006; D’Onghia & Navarro 2007)than their dynamically relaxed counterparts. This means that a dy-namically unrelaxed halo is likely to have a measurably lower con-centration c vir and higher spin parameter λ than its dynamicallyrelaxed counterpart (see, for example, Gardner 2001; Macci`o et al.2007), and so care must be taken to avoid contaminating halo sam-ples with dynamically unrelaxed systems when measuring, for ex-ample, spin distributions (e.g. Bett et al. 2007; Macci`o et al. 2007;D’Onghia & Navarro 2007; Knebe & Power 2008) and the correla-tion of halo mass and concentration c vir − M vir (e.g. Macci`o et al.2007; Neto et al. 2007; Gao et al. 2008; Prada et al. 2011).Yet haloes do not exist in isolation, and the degree to whichthey are dynamically relaxed or unrelaxed bears the imprint ofboth their environment and their recent mass assembly and merg- c (cid:13) RAS
Power, Knebe & Knollmann ing history. As previous studies have shown, dynamically unrelaxedhaloes tend to have suffered one or more recent significant merg-ers (e.g. Tormen et al. 1997; Hetznecker & Burkert 2006). For thisreason, it is common practice to use dynamical state and recentmerging history interchangeably, with the understanding implicitthat unrelaxed haloes are ones that have suffered one or more re-cent major mergers.However, it is important to establish this practice on a morequantitative footing and to assess how well a halo’s dynamical stateand its recent mass assembly history correlate. This is because ofthe need to identify robustly haloes that are in dynamical equilib-rium – or indeed disequilibrium – in cosmological simulations .The goal of this paper is to quantify this relationship using a sta-tistical sample of haloes drawn from cosmological N -body simu-lations of the CDM model. The CDM model is the ideal testbed forthis study because of the fundamental role merging plays in halomass assembly (e.g. Maulbetsch et al. 2007; Fakhouri & Ma 2008;McBride et al. 2009; Fakhouri & Ma 2010; Fakhouri et al. 2010),and because we expect massive haloes, which on average form laterthan their less massive counterparts, to have more violent recentmerging histories.Such an undertaking has practical implications. For example,if we want to robustly characterise the predicted variation of, say,concentration c vir with virial mass M vir on galaxy group and clus-ter mass scales ( M vir & M ⊙ ), then it is essential that we canidentify relaxed systems in a robust fashion. Should we use massassembly histories directly and select only haloes that have quies-cent recent merging histories, or are commonly used measures thatestimate dynamical state based on material within the halo’s virialradius r vir adequate? This is particularly important for comparisonwith observations that provide crucial tests of the theory, such asthe analysis the M vir − c vir relation for groups and clusters drawnfrom the Sloan Digital Sky Survey by Mandelbaum et al. (2008).In this paper, we examine how a halo’s mass assembly historyand dynamical state varies with its virial mass M vir and its for-mation redshift, and adopt simple measures to characterise a halo’srecent mass assembly and merging history – namely, the fraction ofmass assembled ( ∆ M/M ); the rate of change of mass with redshift /MdM/dz ; and the most significant merger δ max . We comparethese with two measures of the halo’s dynamical state – the virialratio η = 2 T / | W | , (1)where T and W are the kinetic and gravitational potential energiesof halo material (cf. Cole & Lacey 1996; Hetznecker & Burkert2006), and the centre-of-mass offset ∆ r = | ~r cen − ~r cm | /r vir , (2)where ~r cen and ~r cm are the centres of density and mass of halomaterial and r vir is the halo’s virial radius (cf. Crone et al. 1996;Thomas et al. 1998, 2001). Previous studies have shown thatboth η and ∆ r increase in the aftermath of a major merger (e.g.Hetznecker & Burkert 2006; Poole et al. 2006), and we will clarify Our focus is fixed firmly on haloes in cosmological simulations, butwe note that the relationship between dynamical state and recent massassembly history is equally important observationally. Here, for exam-ple, estimates of the dynamical masses of galaxy clusters require as-sume a population of dynamical tracers that are in dynamical equilib-rium (e.g. Piffaretti & Valdarnini 2008), while reconstructions of a galaxycluster’s recent merging history look for signatures of disequilibrium (e.g.Cassano et al. 2010). See § precisely how they relate to a halo’s mass assembly and mergingactivity in general. We note that our work develops earlier ideaspresented in Knebe & Power (2008), in which we investigated therelationship between halo mass M vir and spin λ , and it comple-ments that of Davis et al. (2011), who address related but distinctissues in their critique of the application of the virial theorem (cf. § §
2, we describe ourapproach to making initial conditions; finding and analysing darkmatter haloes in evolved outputs; constructing merger trees of ourdark matter haloes; and our criteria for defining our halo sample.In §
3, we examine the relationship between a halo’s virial mass M vir , its formation time z form and measures of its mass accretionand merging history. In §
4, we present commonly used measuresfor assessing the dynamical state of a dark matter halo – the virialratio η = 2 T / | W | (cf. § ∆ r = | ~r cen − ~r cm | /r vir (cf. § M vir and z form . In § § We have run a series of particle cosmological N -body sim-ulations following the formation and evolution of structure in theCDM model. We use a sequence of boxes of side L box varying be-tween h − Mpc and h − Mpc from z start =100 to z finish =0.In each case we assume a flat cosmology with a dark energy term,with cosmological parameters Ω = 0 . , Ω Λ = 0 . , h = 0 . , anda normalisation σ = 0 . at z = 0 . Various properties of thesesimulations are summarised in table 1.Initial conditions were generated using a standard procedurethat can be summarised as follows;(i) Generate the CDM transfer function for the appropriate cos-mological parameters ( Ω , Ω Λ , Ω b and h ) using the Boltzmanncode CMBFAST (Seljak & Zaldarriaga 1996). This is convolvedwith the primordial power spectrum P ( k ) ∝ k n , n = 1 , to obtainthe unnormalised power spectrum, which is normalised by requir-ing that the linear mass σ ( R ) equal σ on a scale of h − Mpc at z =0.(ii) Create a statistical realisation of a Gaussian random field ofdensity perturbations in Fourier space, whose variance is given by P ( k ) , where k = p k x + k y + k z and whose mean is zero.(iii) Take the inverse transform of the density field and computepositions and velocities using the Zel’dovich approximation.(iv) Impose these positions and velocities on an initial uniformparticle distribution such as a grid or “glass”.Note that throughout our we use a “glass”-like configuration as ourinitial uniform particle distribution (White 1996).All simulations were run using the parallel TreePM code GAD-GET2 (Springel 2005) with constant comoving gravitational soft-ening ǫ and individual and adaptive timesteps for each particle, ∆ t = η p ǫ/a , where a is the magnitude of a particle’s gravita- c (cid:13) RAS, MNRAS , 1–13 ynamical State of Dark Matter Haloes I Table 1. Properties of the Simulations . Each of the simulations contains particles. In addition, L box is the comoving box length in units of h − Mpc ; N run is the number of runs in the series; m part is the particlemass in units of h − M ⊙ ; ǫ is the force softening in comoving units of h − kpc ; and M cut is the halo mass corresponding to N cut =600 particles,in units of h − M ⊙ .Run L box N run m part ǫ M cut Λ CDM L20 . × . × Λ CDM L50 . × . × Λ CDM L70 . × Λ CDM L100 . × . × Λ CDM L200 . × . × Λ CDM L500 . × . × tional acceleration and η = 0 . determines the accuracy of thetime integration. Groups were identified using the MPI-enabledversion of
AHF , otherwise known as A MIGA’s H alo F inder (Knollmann & Knebe 2009). AHF is a modification of
MHF ( MLAPM ’s H alo F inder; see Gill et al. 2004) that locates groups aspeaks in an adaptively smoothed density field using a hierarchy ofgrids and a refinement criterion that is comparable to the force res-olution of the simulation. Local potential minima are calculated foreach of these peaks and the set of particles that are gravitationallybound to the peaks are identified as groups that form our halo cata-logue.For each halo in the catalogue we determine its centre-of-density ~r cen (using the iterative “shrinking spheres” method de-scribed in Power et al. 2003) and identify this as the halo centre.From this, we calculate the halo’s virial radius r vir , which wedefine as the radius at which the mean interior density is ∆ vir times the critical density of the Universe at that redshift, ρ c ( z ) =3 H ( z ) / πG , where H ( z ) and G are the Hubble parameter at z and the gravitational constant respectively. The correspondingvirial mass M vir is M vir = 4 π vir ρ c r . (3)We adopt a cosmology- and redshift-dependent overdensity crite-rion, which for a Λ CDM cosmology with Ω = 0 . and Ω Λ = 0 . gives ∆ vir ≃ at z =0 (c.f. Eke et al. 1998). Merger Trees
Halo merger trees are constructed by linking haloparticles at consecutive output times; • For each pair of group catalogues constructed at consecutiveoutput times t and t > t , the “ancestors” of “descendent”groups are identified. For each descendent identified in the cata-logue at the later time t , we sweep over its associated particlesand locate every ancestor at the earlier time t that contains a sub-set of these particles. A record of all ancestors at t that containparticles associated with the descendent at t is maintained. AHF may be downloaded from http://popia.ft.uam.es/AMIGA • The ancestor at time t that contains in excess of f prog of theseparticles and also contains the most bound particle of the descen-dent at t is deemed the main progenitor . Typically f prog = 0 . ,i.e. the main progenitor contains in excess of half the final mass.Each group is then treated as a node in a tree structure, whichcan be traversed either forwards, allowing one to identify a haloat some early time and follow it forward through the merging hi-erarchy, or backwards, allowing one to identify a halo and all itsprogenitors at earlier times. A degree of care must be taken when choosing which haloes to in-clude in our sample, to ensure that our results are not affected by thefinite resolution of our simulations. One of the key calculations inthis study is of a halo’s virial ratio η = 2 T / | W | (see § T and W are the kinetic and gravitational potential energies of ma-terial within r vir . The gravitational potential energy is particularlysensitive to resolution; if a halo is resolved with too few particles,its internal structure will not be recovered sufficiently accuratelyand the magnitude of W will be underestimated.We estimate how many particles are needed to recover W ro-bustly from a N -body simulation in Figure 1. Here we generateMonte Carlo N -body realisations of a halo whose spherically aver-aged mass profile is described by the Navarro et al. (1997) profile, ρ ( x ) ρ c = δ c cx (1 + cx ) ; (4)here x = r/r vir is the radius r normalised to r vir , c is the concen-tration parameter and δ c is the characteristic density, δ c = ∆ vir c ln(1 + c ) − c/ (1 + c ) . (5)The resulting gravitational potential energy is given by W = − π G ρ δ c (cid:16) r vir c (cid:17) × (cid:20) c c )(1 + c ) − ln(1 + c )(1 + c ) (cid:21) . (6)In a N -body simulation or realisation, we calculate W by randomlysampling particles within r vir and rescaling; this gives W = (cid:18) N − N vir N k − N k (cid:19) (cid:18) − Gm ǫ (cid:19) Σ N k − i Σ N k j = i +1 − K s ( | r ij | /ǫ ) , (7)where there are N vir particles in the halo, each of mass m p . Wesample N k particles from N vir , | r ij | is the magnitude of the sep-aration between particles i at ~r i and j at ~r j , and the prefactor ( N − N ) / ( N k − N k ) accounts for particle sampling. ǫ is thegravitational softening and K s corresponds to the softening kernelused in GADGET2 . For the Monte Carlo realisations in Figure 1 weset ǫ to be vanishingly small, but for the simulations we use ǫ as itis listed in table 1.Figure 1 shows | W | measured for Monte Carlo realisationsof a halo with c =
10 and r vir =200 kpc as a function of N vir . Forcomparison the horizontal dotted lines indicate the value of | W | ( ± ) we expect from equation (6). If N vir ≈ or fewer, themeasured | W | deviates from the expected | W | by greater than ;therefore we might regard N cut = 300 as the lower limit on N vir for a halo to be included in our sample. However, we adopt a moreconservative N cut = 600 in the remainder of this paper; this is be-cause the structure of simulated haloes are affected by finite grav-itational softening (cf. Power et al. 2003), they are are seldom (ifever) smooth and spherically symmetric (e.g. Bailin & Steinmetz c (cid:13) RAS, MNRAS , 1–13
Power, Knebe & Knollmann
Figure 1. How many particles are required to measure accurately thegravitational binding energy of a dark matter halo? . Here we gener-ate Monte Carlo realisations of a NFW halo and calculate the gravitationalpotential energy of material within the virial radius. If there are too fewparticles within r vir , the potential energy will be inaccurate. W as inspection of equation (6) reveals. In this section we establish quantitative measures for a halo’s massaccretion and merging histories, and we examine how these mea-sures relate to virial mass M vir and formation redshift z form . Quantifying Formation Redshift
We begin our analysis by ver-ifying the correlation between virial mass M vir and formationredshift z form for our halo sample. We adopt the convention ofCole & Lacey (1996) and define z form as the redshift at which themass of the main progenitor of a halo of mass M vir ( z ) identifiedat z first exceeds M vir ( z ) / . This is equivalent to z / , mb in thesurvey of halo formation redshift definitions examined by Li et al.(2008).Our expectation is that more massive CDM haloes will assem-ble more of their mass at later times than their lower mass counter-parts and this is borne out by Figure 2. Here we show the variationof z form with M vir for our halo sample; the filled circles and barsindicate the medians and upper and lower quartiles respectively,within logarithmic mass bins of width 0.5 dex. The relationship be-tween the mean and median z form with M vir can be well approxi-mated by h z form i ≃ − .
22 log M + 1 . , (8)and Med z form ≃ − .
23 log M + 1 . , (9)where M is M vir in units of h − M ⊙ . This is in very good Figure 2. Relationship between Virial Mass and Formation Redshift .Here we show how the formation redshift z form varies with virial mass M vir at z =0. We determine z form directly from a halo’s merger tree – for ahalo of M vir identified at z =0, we identify the redshift z form at which themass of its main progenitor first exceeds half its virial mass at z =0. Data arebinned using equally spaced bins in Log M vir ; filled circle and bars cor-respond to medians, upper and lower quartiles. The solid, upper and lowerdashed curves corresponds to the median z form and its upper and lowerquartiles predicted by extended Press-Schechter theory (cf. Lacey & Cole1993). agreement with the mean variation reported for the “Overall” sam-ple of haloes drawn from the Millennium and Millennium II simu-lations (cf. Springel et al. 2005 and Boylan-Kolchin et al. 2009 re-spectively) in Table 3 of McBride et al. (2009), who found h z form i = − .
24 log M + 1 . . We show also the variation predicted by extended Press-Schechter(EPS) theory for our choice of CDM power spectrum – see thesolid and dashed curves, indicating the median, upper and lowerquartiles of the distributions (cf. Lacey & Cole 1993). These curveswere generated using realisations of Monte-Carlo merger treesfor haloes with z =0 masses in the range M vir /h − M ⊙ . . We note a slight but systematic offset between the medi-ans evaluated from the simulated haloes and those predicted byEPS theory, such that the simulated haloes tend to form earlierthan predicted. This effect has been reported previously by bothvan den Bosch (2002) and Maulbetsch et al. (2007). Quantifying Recent Mass Accretion History
Because moremassive systems tend to form later than their less massive coun-terparts, it follows that the rate at which a halo assembles its massshould increase with increasing M vir and decreasing z form . Therecent comprehensive study by McBride et al. (2009) provides auseful fitting formula that captures the complexity of a halo’s massaccretion history and allows haloes to be categorised into differentTypes I to IV, which depend on their growth rates. However, weadopt two simple well-defined measures of a halo’s mass accretionrate that have a straightforward interpretation; c (cid:13) RAS, MNRAS , 1–13 ynamical State of Dark Matter Haloes I • (∆ M/M ) ∆ t , the fraction of mass that has been accreted by ahalo during a time interval ∆ t ; and • α = 1 /MdM/dz , the rate of fractional change in a halo’svirial mass with respect to redshift over a redshift interval ∆ z .Note that α is equivalent to the α free parameter used inWechsler et al. (2002). We find that (∆ M/M ) ∆ t and α are suf-ficient as simple measures of the mass accretion rate and we usethem in the remainder of this paper.For the fiducial timescale ∆ t , we use twice the dynamicaltimescale τ dyn estimated at the virial radius, τ dyn = √ r vir V vir = 2 . (cid:18) ∆ vir (cid:19) − / (cid:18) H ( z )70 (cid:19) − Gyrs (10)Note that τ dyn depends only on z and is the same for all haloes.For our adopted cosmological parameters, ∆( z ) ≃ at z =0,and so ∆ t = 2 τ dyn ≃ . Gyr which corresponds to a redshiftinterval of ∆ z ≃ . at z =0. Merging proceeds on a timescale τ merge & τ dyn , with τ merge → τ dyn as the mass ratio of the mergerdecreases. Our adopted timescale of ∆ t = 2 τ dyn for the responseof a halo to a merger is reasonable when compared to typical valuesof τ merge /τ dyn expected for haloes in cosmological simulations, asestimated by Boylan-Kolchin et al. (2008) .We determine both (∆ M/M ) τ dyn and α directly from eachhalo’s merger tree by tracking M vir ( z ) of its main progenitor overthe interval ∆ z ; α is obtained by taking the natural logarithm of theprogenitor mass at each redshift and estimating its value by linearregression. Haloes that have high mass accretion rates will have (∆ M/M ) ∆ t → and α → −∞ .In Fig. 3 we show how a halo’s mass accretion rate correlateswith its virial mass and formation time. (∆ M/M ) τ dyn ( α ) showsa steady monotonic increase (decrease) as M vir increases over therange h − M ⊙ . M vir . h − M ⊙ . For example, inspec-tion of (∆ M/M ) τ dyn reveals that . of the virial mass of ahalo with M vir ∼ h − M ⊙ has been accreted since z ≃ . ,compared to ∼ for haloes with M vir ∼ h − M ⊙ overthe same period. (∆ M/M ) τ dyn ( α ) shows a similar increase (de-crease) with decreasing z form although it’s interesting to note thatthe trend flattens off for haloes that form at z & .This analysis confirms our theoretical prejudice that moremassive haloes and haloes that formed more recently tend to bethe haloes with the measurably highest accretion rates. Reassur-ingly, our results are in good agreement with the findings of recentstudies. For example, McBride et al. (2009) examined the mass ac-cretion and merging histories of a much larger sample of haloesdrawn from the Millennium and Millennium-II simulations andfound that the mean instantaneous mass accretion rate varies withhalo mass as ˙ M/M ∝ M . ; this compares favourably with ourequivalent measure, (∆ M/M ) τ dyn ∝ M . . Maulbetsch et al.(2007) looked at halo accretion rates, normalised to their maximummasses, over the redshift interval z =0.1 to 0 for haloes with masses M vir /h − M ⊙ and found only a weak dependenceon halo mass, with higher mass haloes have higher rates. This isconsistent with with our results for α , whose median value changesby ∼ over the same range in halo mass. In particular, we refer to their equation 5 with values of j/j C ( E ) =0 . and r C ( E ) /r vir that are consistent with the results of cosmologicalsimulations. Here j is the specific angular momentum of a merging subhalo, j C ( E ) is the specific angular momentum of the circular orbit correspondingto the subhalo’s orbital energy E , and r C ( E ) is the radius corresponding tothis circular orbit. Figure 3. Relationship between Recent Mass Accretion History, HaloMass and Formation Redshift . For each halo of virial mass M vir andformation redshift z form identified at z =0, we follow its merger tree backfor one dynamical time τ dyn ( ≃ . Gyrs, ∆ z ≃ . ) and characteriseits mass accretion history using two measures. The first is (∆ M/M ) τ dyn ,the fraction of mass accreted over τ dyn (upper panels), and the second is α , the average mass accretion rate of Wechsler et al. (2002) (lower panels).Data points and bars correspond to medians and upper and lower quartiles.Note that we use equally-spaced logarithmic bins in M vir and z form . Figure 4. Frequency of Major Mergers and Dependence on Halo Massand Formation Redshift . Here we determine the most significant mergerof mass ratio δ max = M acc ( z i ) /M vir ( z f ) experienced by each halo since z =0.5, where z i and z f correspond to the initial and final redshifts. We thencompute the fraction of haloes f ( δ max ) at a given virial mass (left handpanel) and given formation redshift (right hand panel) that have experiencedmergers with mass ratios δ max in excess of (filled circles), (filledsquares) and (filled triangles).c (cid:13) RAS, MNRAS , 1–13
Power, Knebe & Knollmann
Quantifying Recent Merger Activity
Both (∆ M/M ) τ dyn and α provide useful insights into a halo’s total mass accretion rate, butthey cannot distinguish between smooth and clumpy accretion. InFig. 4 we focus specifically on a halo’s merger history by consid-ering the likelihood that a halo of a given M vir (left hand panel) or z form (right hand panel) has experienced at least one merger witha mass ratio δ max since z =0.6.Each halo identified at z =0 has a unique merger history, whichcharacterises not only how its M vir grows as a function of timebut also details of mergers it has experienced over time. Usingthis merger history, we construct the distribution of mass ratiosof mergers δ experienced by a halo of a given M vir or z form be-tween z . . . We define δ = M acc ( z i , z f ) /M vir ( z f ) where M acc ( z i , z f ) is the mass of the less massive halo prior to its merg-ing with the more massive halo, M vir ( z f ) is the virial mass of themore massive halo once the less massive halo has merged with it,and z i > z f and z f are the redshifts of consecutive simulationsnapshots. The maximum value of δ for a given halo gives us its δ max and we use this to compute the fraction of haloes of a given M vir or z form that have δ max in excess of (filled circles), (filled squares) and (filled triangles).Fig. 4 reveals that mergers with higher mass ratios (i.e. mi-nor mergers) are more common than mergers with lower mass ra-tios (i.e. major mergers), independent of M vir and z form , and thatmore massive (older) haloes tend to experience more mergers thantheir lower mass (younger) counterparts. For example, the likeli-hood that a h − M ⊙ galaxy-mass halo experiences a mergerwith δ max > is ∼ , compared to for δ max > . In contrast, the likelihood that a h − M ⊙ cluster-mass halo experiences mergers with δ max > , is ∼ , . Interestingly, we find that the fraction ofhaloes that have experienced a merger more significant than δ max increases with M vir approximately as f ( δ max ) ∝ M . .These results are broadly in agreement with the findings ofFakhouri et al. (2010). Inspection of the leftmost panel of theirFigure 7 shows the mean number of mergers with mass ratiosgreater than 1:10 and 1:3 between z =0 and z ∼ . increaseswith increasing halo mass, such that a (10 ) h − M ⊙ hasa likelihood of ∼ ( ∼ ) to have experienced a mergerwith δ max > , and a likelihood of ∼ ( ∼ ) to haveexperienced a merger with δ max > .In Fig. 5 we show the full (cumulative) distributions of δ max for haloes split into bins according to M vir (left hand panel)and z form (right hand panel); note that we consider only haloeswith δ max > . Interestingly this figure reveals that theprobability distribution of δ max is insensitive to M vir , but dependsstrongly on z form . For example, the median δ max , med ≃ . ,independent of M vir whereas it increases from δ max , med ≃ . for haloes with z form & . to δ max , med ≃ . for haloeswith . z form . and δ max , med ≃ . for haloes with z form . .Figs. 3 to 5 demonstrate that there is a strong correlation at z =0 between a halo’s virial mass M vir , its formation redshift z form and the rate at which it has assembled its mass through accretionand merging over the last τ dyn or equivalently ∆ z ∼ . . Weuse these results in §
5, where we investigate the degree to whicha halo’s M vir , z form and mass accretion rate affect the degree towhich it is in dynamical equilibrium. Figure 5. Cumulative Distribution of δ max as a Function of Halo Massand Formation Redshift . We show how the fraction of haloes whose mostsignificant merger’s mass ratio is less than δ max , as a function of virial mass(left hand panel) and formation redshift (right hand panel). Note that weselect only haloes that have δ max > , and we consider only mergersbetween z =0.5 and z =0. In the key, the numbers in brackets correspond tothe lower and upper bounds in M vir and z form . In this section we describe the two commonly used quantitativemeasures for a halo’s dynamical state, the virial ratio η and thecentre of mass offset ∆ r , and we examine their relationship withvirial mass M vir and formation redshift z form . η The virial ratio η is commonly used in cosmological N -body simu-lations as a measure of a halo’s dynamical state (e.g. Cole & Lacey1996; Bett et al. 2007; Neto et al. 2007; Knebe & Power 2008;Davis et al. 2011). It derives from the virial theorem, d Idt = 2 T + W + E S , (11)where I is the moment of inertia, T is the kinetic energy, W =Σ ~F .~r is the virial, and E S is the surface pressure integrated overthe bounding surface of the volume within which I , T and W areevaluated (cf. Chandrasekhar 1961). Provided the system is isolatedand bounded, the virial W is equivalent to the gravitational poten-tial energy. While not strictly true for haloes that form in cosmo-logical N -body simulations, the convention has been to evaluate W as the gravitational potential energy with this caveat in mind (e.g.Cole & Lacey 1996). We follow this convention and treat W as thegravitational potential energy computed using equation (7).If the system is in a steady state and in the absence of sur-face pressure, equation (11) reduces to T + W = 0 , which canbe written more compactly as T / | W | = 1 (e.g. Cole & Lacey1996). We refer to the ratio η = 2 T / | W | as the virial ratio and weexpect η → for dynamically relaxed haloes. However, we mightexpect E S to be important for haloes that form in cosmological c (cid:13) RAS, MNRAS , 1–13 ynamical State of Dark Matter Haloes I Figure 6. Correlation between Virial Ratios η and η ′ . We bin all haloesin our sample at z =0 according to their η and evaluate the median η ′ withineach bin. The upper and lower quartiles of the distributions in η and η ′ areindicated by bars. N -body simulations; in this case Shaw et al. (2006) have proposedmodifying the virial ratio to obtain η ′ = (2 T − E s ) / | W | . (12)We calculate both T and W using all material within r vir , whilewe follow Shaw et al. (2006) by computing the surface pressurecontribution from all particles that lie in a spherical shell with innerand outer radii of 0.8 and 1.0 r vir , P s = 13 V Σ i ( m i v i ); (13)here V corresponds to the volume of this shell and v i are theparticle velocities relative to the centre of mass velocity of thehalo. The energy associated with the surface pressure is therefore E s ≃ πr med P s where r med is the median radius of the shell.Figure 6 shows how the median η and η ′ for the haloes in oursample compare, with bars indicating the upper and lower quartilesof the distributions. We might expect that η ′ ∼ and insensitiveto variation in η ; however, this figure reveals that the relationshipbetween η and η ′ is not so straightforward. Haloes that we wouldexpect to be dynamically relaxed, with η ∼ , have values of η ′ ¡0,suggesting that E S tends to over-correct. Similar behaviour hasbeen noted in both Knebe & Power (2008) and Davis et al. (2011)for high redshift haloes ( z & ). The relation between the median η and η ′ is flat η . . but rises sharply from η ′ ∼ . to peakat η ′ ∼ . before declining sharply for η & . to a median of η ′ ∼ . in the last plotted bin. Interestingly, the width of the η ′ distribution increases with η ; if η tracks recent major merging ac-tivity as we expect, then this suggests that η ′ – and consequentlythe surface pressure correction term E S – is sensitive to mergersbut in a non-trivial way. Figure 7. Correlation between Centre-of-Mass Offset ∆ r and VirialRatios η and η ′ . We can clearly see the relation which is confirmed bymeasuring a Spearman rank coefficient of 0.45 whereas we find an anti-correlation with Spearman rank coefficient of -0.18 for η ′ . ∆ r Another commonly used measure of a halo’s dynamical state is thecentre-of-mass offset ∆ r , ∆ r = | ~r cen − ~r cm | r vir , (14)which measures the separation between a halo’s centre-of-density ~r cen (calculated as described in § r vir , normalised by r vir (cf.Crone et al. 1996; Thomas et al. 1998, 2001; Neto et al. 2007;Macci`o et al. 2007; D’Onghia & Navarro 2007). ∆ r is used as asubstructure statistic, providing an estimate of a halo’s deviationsfrom smoothness and spherical symmetry. The expectation isthat the smaller the ∆ r , the more relaxed the halo; for example,Neto et al. (2007) define dynamically relaxed haloes to be thosewith ∆ r . , while D’Onghia & Navarro (2007) adopt ∆ r . . Macci`o et al. (2007) favoured a more conservative ∆ r . based on a thorough analysis.We can get a sense of how well ∆ r measures the dynamicalstate of a halo by comparing it to η and η ′ . In Figure 7 we plot themedian η and η ′ (filled circles and squares respectively) against themedian ∆ r ; as before, bars indicate the upper and lower quartiles ofthe distributions. This figure shows that both η and η ′ correlate with ∆ r – but in different senses; as ∆ r increases, η increases while η ′ decreases. The increase (decrease) is a gradual one; for example,for ∆ r . . , the median η is flat with a value of ∼ . , but for ∆ r & . there is a sharp increase and ∆ r & . , η ∼ . . Al-though direct comparison is difficult, a similar trend can be gleanedfrom Figure 2 of Neto et al. (2007). We use the Spearman rank cor-relation coefficient to assess the strength of the correlation between ∆ r and η ( η ′ ) (cf. Kendall & Gibbons 1990), and find strong pos-itive and negative correlations for η (Spearman rank coeffecient r =0.97) and η ′ ( r =-0.95) respectively. c (cid:13) RAS, MNRAS000
5, where we investigate the degree to whicha halo’s M vir , z form and mass accretion rate affect the degree towhich it is in dynamical equilibrium. Figure 5. Cumulative Distribution of δ max as a Function of Halo Massand Formation Redshift . We show how the fraction of haloes whose mostsignificant merger’s mass ratio is less than δ max , as a function of virial mass(left hand panel) and formation redshift (right hand panel). Note that weselect only haloes that have δ max > , and we consider only mergersbetween z =0.5 and z =0. In the key, the numbers in brackets correspond tothe lower and upper bounds in M vir and z form . In this section we describe the two commonly used quantitativemeasures for a halo’s dynamical state, the virial ratio η and thecentre of mass offset ∆ r , and we examine their relationship withvirial mass M vir and formation redshift z form . η The virial ratio η is commonly used in cosmological N -body simu-lations as a measure of a halo’s dynamical state (e.g. Cole & Lacey1996; Bett et al. 2007; Neto et al. 2007; Knebe & Power 2008;Davis et al. 2011). It derives from the virial theorem, d Idt = 2 T + W + E S , (11)where I is the moment of inertia, T is the kinetic energy, W =Σ ~F .~r is the virial, and E S is the surface pressure integrated overthe bounding surface of the volume within which I , T and W areevaluated (cf. Chandrasekhar 1961). Provided the system is isolatedand bounded, the virial W is equivalent to the gravitational poten-tial energy. While not strictly true for haloes that form in cosmo-logical N -body simulations, the convention has been to evaluate W as the gravitational potential energy with this caveat in mind (e.g.Cole & Lacey 1996). We follow this convention and treat W as thegravitational potential energy computed using equation (7).If the system is in a steady state and in the absence of sur-face pressure, equation (11) reduces to T + W = 0 , which canbe written more compactly as T / | W | = 1 (e.g. Cole & Lacey1996). We refer to the ratio η = 2 T / | W | as the virial ratio and weexpect η → for dynamically relaxed haloes. However, we mightexpect E S to be important for haloes that form in cosmological c (cid:13) RAS, MNRAS , 1–13 ynamical State of Dark Matter Haloes I Figure 6. Correlation between Virial Ratios η and η ′ . We bin all haloesin our sample at z =0 according to their η and evaluate the median η ′ withineach bin. The upper and lower quartiles of the distributions in η and η ′ areindicated by bars. N -body simulations; in this case Shaw et al. (2006) have proposedmodifying the virial ratio to obtain η ′ = (2 T − E s ) / | W | . (12)We calculate both T and W using all material within r vir , whilewe follow Shaw et al. (2006) by computing the surface pressurecontribution from all particles that lie in a spherical shell with innerand outer radii of 0.8 and 1.0 r vir , P s = 13 V Σ i ( m i v i ); (13)here V corresponds to the volume of this shell and v i are theparticle velocities relative to the centre of mass velocity of thehalo. The energy associated with the surface pressure is therefore E s ≃ πr med P s where r med is the median radius of the shell.Figure 6 shows how the median η and η ′ for the haloes in oursample compare, with bars indicating the upper and lower quartilesof the distributions. We might expect that η ′ ∼ and insensitiveto variation in η ; however, this figure reveals that the relationshipbetween η and η ′ is not so straightforward. Haloes that we wouldexpect to be dynamically relaxed, with η ∼ , have values of η ′ ¡0,suggesting that E S tends to over-correct. Similar behaviour hasbeen noted in both Knebe & Power (2008) and Davis et al. (2011)for high redshift haloes ( z & ). The relation between the median η and η ′ is flat η . . but rises sharply from η ′ ∼ . to peakat η ′ ∼ . before declining sharply for η & . to a median of η ′ ∼ . in the last plotted bin. Interestingly, the width of the η ′ distribution increases with η ; if η tracks recent major merging ac-tivity as we expect, then this suggests that η ′ – and consequentlythe surface pressure correction term E S – is sensitive to mergersbut in a non-trivial way. Figure 7. Correlation between Centre-of-Mass Offset ∆ r and VirialRatios η and η ′ . We can clearly see the relation which is confirmed bymeasuring a Spearman rank coefficient of 0.45 whereas we find an anti-correlation with Spearman rank coefficient of -0.18 for η ′ . ∆ r Another commonly used measure of a halo’s dynamical state is thecentre-of-mass offset ∆ r , ∆ r = | ~r cen − ~r cm | r vir , (14)which measures the separation between a halo’s centre-of-density ~r cen (calculated as described in § r vir , normalised by r vir (cf.Crone et al. 1996; Thomas et al. 1998, 2001; Neto et al. 2007;Macci`o et al. 2007; D’Onghia & Navarro 2007). ∆ r is used as asubstructure statistic, providing an estimate of a halo’s deviationsfrom smoothness and spherical symmetry. The expectation isthat the smaller the ∆ r , the more relaxed the halo; for example,Neto et al. (2007) define dynamically relaxed haloes to be thosewith ∆ r . , while D’Onghia & Navarro (2007) adopt ∆ r . . Macci`o et al. (2007) favoured a more conservative ∆ r . based on a thorough analysis.We can get a sense of how well ∆ r measures the dynamicalstate of a halo by comparing it to η and η ′ . In Figure 7 we plot themedian η and η ′ (filled circles and squares respectively) against themedian ∆ r ; as before, bars indicate the upper and lower quartiles ofthe distributions. This figure shows that both η and η ′ correlate with ∆ r – but in different senses; as ∆ r increases, η increases while η ′ decreases. The increase (decrease) is a gradual one; for example,for ∆ r . . , the median η is flat with a value of ∼ . , but for ∆ r & . there is a sharp increase and ∆ r & . , η ∼ . . Al-though direct comparison is difficult, a similar trend can be gleanedfrom Figure 2 of Neto et al. (2007). We use the Spearman rank cor-relation coefficient to assess the strength of the correlation between ∆ r and η ( η ′ ) (cf. Kendall & Gibbons 1990), and find strong pos-itive and negative correlations for η (Spearman rank coeffecient r =0.97) and η ′ ( r =-0.95) respectively. c (cid:13) RAS, MNRAS000 , 1–13
Power, Knebe & Knollmann
Figure 8. Relationship between Dynamical State and Halo Mass andFormation Redshift.
For haloes identified at z =0, we plot the median η and η ′ versus M vir (left hand panel) and z form (right hand panel) using equallyspaced bins in Log M vir and z form . Data points and bars correspond tomedians and upper and lower quartiles. This is suggestive – as we show below, ∆ r correlates morestrongly with merging activity than either of η or η ′ (cf. Figure 13in § ∆ r and η increase with strength of merging activity,whereas η ′ appears to be over-corrected by E S (as we have notedabove). From this we conclude that E S (as we evaluate it) correlateswith significant merger activity, which is confirmed by a Spearmanrank coefficient of 0.38 for the correlation between E s and δ max .Interestingly Davis et al. (2011) examined the correlation be-tween ∆ r and η ′ for high redshift haloes ( z & ) and noted atendency for haloes with small values of η ′ to have larger values of ∆ r . Inspection of their Figure 4 shows that this is true for haloeswith . . ∆ r . ; for ∆ r . . the relation with η ′ is flat.Davis et al. (2011) argue that, because there is no systematic shiftin η ′ for ∆ r < . , ∆ r is not a useful measure of dynamical stateat high redshifts. M vir and z form In Figures 8 and 9 we examine how η , η ′ and ∆ r vary with M vir (left hand panels) and z form (right hand panels) for the halo pop-ulation at z =0. Haloes are sorted in bins of equal width in mass( ∆ log M =0.5 dex) and redshift ( ∆ z =0.25), and we plot the me-dian η / η ′ / ∆ r within each bin against the median M vir / z form ; barsindicate the upper and lower quartiles of the respective distribu-tions. For reference, we also plot a horizontal dotted line in eachpanel of Figure 8 to indicate a virial ratio of unity.Because more massive haloes tend to form at later times, andbecause these haloes tend to assemble a larger fraction of their massmore recently, we expect that η and ∆ r should increase with in-creasing M vir and decreasing z form . This is borne out in Figures 8and 9. We find that the mean and median ∆ r increases steadily withincreasing M vir as h log ∆ r i = − .
47 + 0 .
08 log M (15) Figure 9. Relationship between Centre-of-Mass Offset, Halo Mass andFormation Redshift.
For haloes identified at z =0, we plot the mediancentre-of-mass offset ∆ r versus M vir (left hand panel) and z form (righthand panel) using equally spaced bins in Log M vir and z form . Datapoints and bars correspond to medians and upper and lower quartiles. Figure 10. Distribution of virial ratios . Here we show the correlation be-tween halo mass and virial ratio η and η ′ at redshift z = 0 . and Med log ∆ r = − .
49 + 0 .
09 log M (16)where, as before, M is M vir in units of h − M ⊙ . This isconsistent with the result of Thomas et al. (2001, see their Figure9), who found a similar trend for ∆ r to increase with M for asample of cluster mass haloes ( . M / ( h − M ⊙ ) . )in a τ CDM model. Their typical values of ∆ r are offset to highervalues than we find, but this can be understood as an effect of Λ , c (cid:13) RAS, MNRAS , 1–13 ynamical State of Dark Matter Haloes I the merging rate being suppressed in the Λ CDM model comparedto the τ CDM model. Similarly, ∆ r varies strongly with z form ; for z form & we find that ∆ r ∝ (1 + z ) − . compared to ∆ r ∝ (1 + z ) − . for z form & .The mean and median η exhibit similar behaviour, increasingwith increasing M vir , albeit weakly, as, h log η i = 0 .
05 + 0 .
016 log M (17)and Med log η = 0 .
04 + 0 .
019 log M . (18)This means that η is systematically greater than unity for all M vir that we consider – η ∼ . for a typical h − M ⊙ halo, com-pared to η ∼ . for a typical h − M ⊙ halo. The same grad-ual increase in η with decreasing z form is also apparent.As we might have anticipated from inspection of Figures 6 and7, η ′ is systematically smaller than unity. Its variation with M vir is negligible ( ∝ M . ; a little surprising, when compared to ∝ M . at z =1, as reported by Knebe & Power 2008) but thereis a trend for the median η ′ to decrease with decreasing z form . Thismakes sense because E S increases with the significance of recentmergers and haloes that have had recent major mergers tend to havesmaller z form . This effect is also noticeable in the width of the η ′ distributions in each bin (as measured by the bars), which are largerthan than the corresponding widths of the η distribution.We look at this effect in more detail by plotting the distribu-tions of η and η ′ shown in Figure 10. Here it is readily apparentthat there is a systematic shift towards larger η as M vir increases.Interestingly the η ′ distribution remains centred on η ′ ∼ . , butit spreads with increasing M vir ; again, this suggests the sensitivityof η ′ to recent merging activity.Figs. 6 to 10 demonstrate that there is a strong correlationat z =0 between a halo’s virial mass M vir , its formation redshift z form and its dynamical state, as measured by the virial ratio η andthe centre-of-mass offset ∆ r . In contrast, the correlation with η ′ is more difficult to interpret, especially when η is large. In thesecases, we expect significant merging activity and as we note above,the correction by the surface pressure term E S increases the widthof the original η distribution by a factor of ∼ η ′ is systematically offset below unity. Forthis reason we argue that η ′ is not as useful a measure of a halo’sdynamical state as η . We have established quantitative measures of a halo’s mass assem-bly and merging history and its dynamical state in the previous twosections, and we have investigated how these relate separately toa halo’s virial mass M vir and its formation redshift z form . In thisfinal section we examine the relationship between a halo’s massassembly history and its dynamical state directly.In Figures 11 and 12 we show explicitly how a halo’s re-cent mass accretion and merging history impacts on its virial ra-tio. As in section 3, we quantify a halo’s mass accretion history by (∆ M/M ) τ dyn , the fractional increase in a halo’s mass over the pe-riod τ dyn (equivalent to a redshift interval ∆ z ≃ . at z =0), and α , the mean accretion rate over the period τ dyn ). We use δ max , themass ratio of the most significant merger experienced by the haloover τ dyn , to characterise a halo’s recent merging history. Figure 11. Relationship between Recent Mass Accretion and the VirialRatio . Here we investigate how (∆ M/M ) τ dyn , the fraction of mass ac-creted over τ dyn (left hand panel), and α , the mean accretion rate over τ dyn (right hand panel), correlate with the standard ( η , filled circles) andcorrected ( η ′ , filled squares) virial ratio. Data points correspond to mediansand bars correspond to the upper and lower quartiles. Figure 12. Relationship between Most Significant Recent Merger andthe Virial Ratio . Here we investigate how δ max , which measures the massratio of the most significant recent merger since z =0.5, correlates withthe standard ( η ) and corrected ( η ′ ) virial ratios respectively. Filled circles(squares) correspond to medians of η ( η ′ ), while bars indicate the upper andlower quartiles.c (cid:13) RAS, MNRAS000
019 log M . (18)This means that η is systematically greater than unity for all M vir that we consider – η ∼ . for a typical h − M ⊙ halo, com-pared to η ∼ . for a typical h − M ⊙ halo. The same grad-ual increase in η with decreasing z form is also apparent.As we might have anticipated from inspection of Figures 6 and7, η ′ is systematically smaller than unity. Its variation with M vir is negligible ( ∝ M . ; a little surprising, when compared to ∝ M . at z =1, as reported by Knebe & Power 2008) but thereis a trend for the median η ′ to decrease with decreasing z form . Thismakes sense because E S increases with the significance of recentmergers and haloes that have had recent major mergers tend to havesmaller z form . This effect is also noticeable in the width of the η ′ distributions in each bin (as measured by the bars), which are largerthan than the corresponding widths of the η distribution.We look at this effect in more detail by plotting the distribu-tions of η and η ′ shown in Figure 10. Here it is readily apparentthat there is a systematic shift towards larger η as M vir increases.Interestingly the η ′ distribution remains centred on η ′ ∼ . , butit spreads with increasing M vir ; again, this suggests the sensitivityof η ′ to recent merging activity.Figs. 6 to 10 demonstrate that there is a strong correlationat z =0 between a halo’s virial mass M vir , its formation redshift z form and its dynamical state, as measured by the virial ratio η andthe centre-of-mass offset ∆ r . In contrast, the correlation with η ′ is more difficult to interpret, especially when η is large. In thesecases, we expect significant merging activity and as we note above,the correction by the surface pressure term E S increases the widthof the original η distribution by a factor of ∼ η ′ is systematically offset below unity. Forthis reason we argue that η ′ is not as useful a measure of a halo’sdynamical state as η . We have established quantitative measures of a halo’s mass assem-bly and merging history and its dynamical state in the previous twosections, and we have investigated how these relate separately toa halo’s virial mass M vir and its formation redshift z form . In thisfinal section we examine the relationship between a halo’s massassembly history and its dynamical state directly.In Figures 11 and 12 we show explicitly how a halo’s re-cent mass accretion and merging history impacts on its virial ra-tio. As in section 3, we quantify a halo’s mass accretion history by (∆ M/M ) τ dyn , the fractional increase in a halo’s mass over the pe-riod τ dyn (equivalent to a redshift interval ∆ z ≃ . at z =0), and α , the mean accretion rate over the period τ dyn ). We use δ max , themass ratio of the most significant merger experienced by the haloover τ dyn , to characterise a halo’s recent merging history. Figure 11. Relationship between Recent Mass Accretion and the VirialRatio . Here we investigate how (∆ M/M ) τ dyn , the fraction of mass ac-creted over τ dyn (left hand panel), and α , the mean accretion rate over τ dyn (right hand panel), correlate with the standard ( η , filled circles) andcorrected ( η ′ , filled squares) virial ratio. Data points correspond to mediansand bars correspond to the upper and lower quartiles. Figure 12. Relationship between Most Significant Recent Merger andthe Virial Ratio . Here we investigate how δ max , which measures the massratio of the most significant recent merger since z =0.5, correlates withthe standard ( η ) and corrected ( η ′ ) virial ratios respectively. Filled circles(squares) correspond to medians of η ( η ′ ), while bars indicate the upper andlower quartiles.c (cid:13) RAS, MNRAS000 , 1–13 Power, Knebe & Knollmann
Figure 13. Correlation between centre-of-mass offset ∆ r and RecentMerging and Accretion History . Here we examine whether ∆ r correlateswith the fraction of mass accreted over τ dyn , (∆ M/M ) τ dyn , the meanaccretion rate α and the most significant merger δ max . Filled circles corre-spond to medians in the respective bins; bars represent the upper and lowerquartiles of the distribution. We expect that the standard virial ratio η should increase withincreasing mass accretion rate and decreasing mass ratio of mostsignificant merger, which is in good agreement with the behaviourthat we observe. In particular, the median variation of η with α and δ max can be well approximated by log η ≃ . − . α and η ≃ . δ . ; the corresponding variation of ∆ r can be wellapproximated by . − . α and . δ . .Interestingly we note that the median corrected virial ratio η ′ declines with increasing mass accretion rate and mass ratio of mostsignificant merger. Both correlations indicate that merger eventslead to a state that is less virialised, but, as we have noted al-ready, the inclusion of the surface pressure term over-corrects thevirial ratio. We see in Figure 11 that for (∆ M/M ) τ dyn . . ,both the median η and η ′ are flat; η ∼ . whereas the median η ′ ∼ . . Above (∆ M/M ) τ dyn ∼ . , the median η increasessharply whereas it is the width of the η ′ distribution that shows thesharp increase. Comparison with Figure 12 provides further insight– the median η ( η ′ ) shows a gradual increase (decrease) with in-creasing δ max , starting at η ∼ . ( η ′ ∼ . ) for δ max ∼ . .For δ max =0.1, η ∼ . ( η ′ ∼ . ). However, whereas the widthof the η distribution is largely insensitive to δ max , the width of the η ′ distribution increases rapidly, bearing out our observations in theprevious section.In Figure 13 we show how ∆ r varies with (∆ M/M ) τ dyn , α and δ max . This reveals that ∆ r increases with increasing mass ac-cretion rate and mass ratio of most significant recent merger, as wewould expect. Although the scatter in the distribution is large, wecan identify the remnants of recent major mergers ( δ max & )as haloes with ∆ r & . . Haloes that have had relatively qui-escent recent mass accretion histories ( (∆ M/M ) τ dyn . . , δ max . ) have ∆ r . . . Figure 14. Relationship between η , ∆ r and z δ max , the redshift of themost significant recent major merger . We identify all haloes in our sam-ple at z =0 with δ max & / and identify the redshift z δ max at which δ max occurred. Both η and ∆ r are evaluated at z=0. Filled circles and bars cor-respond to medians and upper and lower quartiles. Merging Timescale & Dynamical State
We conclude our analy-sis by investigating the timescale over which the effect of a mergercan be observed in the virial ratio η and the centre-of-mass offset ∆ r . In Figure 14 we investigate how a typical halo’s η (upperpanel) and ∆ r (lower panel), measured at z = 0 , correlates withthe redshift at which the halo suffered it’s most significant merger, z δ max . For clarity, we focus on haloes for which δ max > / , al-though we have verified that our results are not sensitive to theprecise value of δ max that we adopt; filled circles correspond tomedians and bars indicate upper and lower quartiles. The median η increases with decreasing z δ max for z δ max & before peak-ing at z δ max ≃ . and declining at lower z δ max . The median ∆ r shows a similar steady increase with decreasing z δ max below z δ max ∼ . although there is evidence that it peaks at z δ max ≃ . before declining at lower z δ max . The redshift interval correspond-ing to z δ max ≃ . represents a time interval of ∆ t ≃ . or ∼ . τ dyn .This is consistent with the finding of Tormen et al. (1997),who examined the velocity dispersion v rms of material within r vir of simulated galaxy cluster haloes (see their Figure 5). They notedthat merging leads to an increase in v rms of the main (host) halobecause the merging sub-halo acquires kinetic energy as it falls inthe potential well of the more massive main halo. The peak in v rms corresponds to the first pericentric passage of the subhalo, afterwhich v rms declines because subsequent passages are damped,and so the main halo relaxes. This will occur on a timescale oforder ∼ − τ dyn , which is consistent with the peak in η at z δ max ≃ . . We would expect to see a peak in ∆ r on roughly themerging timescale τ merge , which as we noted in § − τ dyn (cf. Boylan-Kolchin et al. 2008).We can take this analysis a little further by looking at the de-tailed evolution of η and ∆ r over time. In Figure 15 we plot the red- c (cid:13) RAS, MNRAS , 1–13 ynamical State of Dark Matter Haloes I Figure 15. Response of M vir , η and ∆ r to a major merger. We includeall haloes with δ max & / at z =0 and plot the redshift variation of M vir , η and ∆ r against the time since the major merger, normalised by the dy-namical time of the halo at the redshift at which the merger occurred. Filledcircles and bars correspond to medians and upper and lower quartiles, whilecurves correspond to the histories of 5 individual haloes. shift variation of M vir (normalised to its value at z =0; lower panel), η (middle panel) and ∆ r (upper panel) against the time since ma-jor merger, normalised by the dynamical time τ dyn estimated at theredshift at which the merger occurred, z δ max . Medians and upperand lower quartiles are indicated by filled circles and bars. For il-lustrative purposes, we show also the redshift variation of M vir , η and ∆ r for a small subset of our halo sample (red, blue, green, cyanand magenta curves). As in Figure 14, we adopt δ max & / .Our naive expectation is that both η and ∆ r should increasein response to the merger, peak after ∆ t ≃ τ dyn and then re-turn to their pre-merger values. If this behaviour is typical, thenwe expect pronounced peaks in the median values of η and ∆ r at ∆ t/τ dyn ≃ . However, it is evident from Figure 15 that there isno significant difference between the median η and ∆ r pre- andpost-major merger, and so our naive expectation is not borne out byour results.This is not surprising if one inspects histories for η and ∆ r forindividual haloes, in the spirit of Tormen et al. (1997); η and ∆ r in-crease following a major merger, but the behaviour is noisy (reflect-ing e.g. differences in orbital parameters of merging subhaloes, theredshift dependent virial radius, dependence on environment, etc...)and the timescale of the response varies from halo to halo – simplyaveraging or taking the median washes any signal away. Neverthe-less it is worth looking at this in more detail, which we shall do ina forthcoming paper. The aim of this paper has been to quantify the impact of a dark mat-ter halo’s mass accretion and merging history on two measures ofdynamical state that are commonly used in cosmological N -bodysimulations, namely the virial ratio η = 2 T / | W | (cf. Cole & Lacey1996) and the centre-of-mass offset ∆ r = | ~r cen − ~r cm | /r vir (cf. Crone et al. 1996; Thomas et al. 1998, 2001). The virial ratio η de-rives from the virial theorem and and the expectation is that η → for dynamically equilibrated haloes. The centre-of-mass offset ∆ r can be regarded as a substructure statistic (Thomas et al. 2001) thatprovides a convenient measure of how strongly a halo deviates fromsmoothness and spherical symmetry. We expect a halo’s dynam-ical state and its mass assembly history to correlate closely. Un-derstanding how is important because the degree to which a halois dynamically equilibrated affects the reliability with which wecan measure characteristic properties of its structure, such as theconcentration parameter c vir (e.g. Macci`o et al. 2007; Neto et al.2007; Prada et al. 2011), and kinematics, such as the spin parameter λ (e.g. Gardner 2001; D’Onghia & Navarro 2007; Knebe & Power2008). For this reason, it is desirable to establish quantitatively howwell they correlate and to assess how η and ∆ r can help us to char-acterise how quiescent or violent a halo’s recent mass assemblyhistory has been.Our key results are that η and ∆ r show strong positive corre-lations with each other (cf. Figure 7) – as η increases for a halo, sotoo does ∆ r – and that both are useful indicators of a halo’s massrecent mass accretion and merging history. For example, η and ∆ r correlate strongly with δ max , which measures the significance of ahalo’s recent merging activity; haloes with η . . (cf. Figure 12)and ∆ r . . (cf. Figure 13) will have quiescent recent mass as-sembly histories – they are unlikely to have suffered mergers withmass ratios greater than 1:10 over the last few dynamical times.In contrast, interpreting the corrected virial ratio η ′ = (2 T − E S ) / | W | , where E s is the surface pressure energy, is problem-atic (at least insofar as we have implemented it here, which fol-lows the prescription of Shaw et al. 2006 and has been applied inKnebe & Power 2008 and Davis et al. 2011). In principle, η ′ shouldaccount for the approximation that is made when we define a haloto be a spherical overdensity of ∆ vir times the critical density ata particular redshift. As we noted in §
2, haloes are more complexstructures than this simple working definition gives them credit for,and by defining the halo’s extent by the virial radius r vir the likeli-hood is that material that belongs to the halo will be neglected. Bycorrecting the virial ratio η for what is effectively a truncation of thetrue halo, the corrected virial ratio η ′ takes account of the “missing”kinetic energy. However, our results imply that the correction itself(the surface pressure energy E S ) is sensitive to a halo’s merginghistory, and that it increases with increasing δ max (cf. Figure 12).For this reason we would caution against the use of η ′ to identifydynamically relaxed haloes, at least in the form that is currentlyused.Interestingly, we find that systems with violent recent massassembly histories (most significant merger with a mass ratio δ max & / between . z . ) have values of η and ∆ r (as measured at z =0) that peak at z δ max ≃ . − . , whichcorresponds to a timescale of ∼ . τ dyn (cf. Figure 14). Thisis consistent with the earlier analysis of Tormen et al. (1997),who found that the velocity dispersion v rms of material withinthe virial radius – which is linked to the virial ratio η – peakson first closest approach of the merging sub-halo with the centreof the more massive host halo. This should occur on a timescaleof ∼ − τ dyn , after which v rms and η should dampen away.Similar arguments can be made for ∆ r . We note that thesearguments can be made in a statistical sense, but if we lookat the merging histories of individual haloes, the behaviour of η and ∆ r is much more complex, and as we demonstrate asimple timescale for their response to a major merger is difficult c (cid:13) RAS, MNRAS , 1–13 Power, Knebe & Knollmann to define (cf. Figure 15). We shall return to this topic in future work.What is the significance of these results? Structure formationproceeds hierarchically in the CDM model and so we expect tofind correlations between virial mass M vir and formation redshift z form (cf. Figure 2), which in turn result in positive correlations be-tween M vir / z form and η / ∆ r . (cf. Figures 8 and 9). This means thatmore massive haloes and those that formed more recently are alsothose that are least dynamically equilibrated, a fact that we shouldbe mindful of when characterising the halo mass dependence ofhalo properties that are sensitive to dynamical state (e.g. c vir and λ ). It’s worth noting that the correlation between M vir and η isstronger than the correlation between M vir and ∆ r ; the median η rises sharply with M vir and there is no overlap between the widthof the distributions of η is the lowest and highest mass bins. In con-trast, the median ∆ r in the highest mass bin lies in the high- ∆ r tailof the lowest mass bin.This is interesting because η as it is usually calculateddepends on W , which is sensitive to the precise boundary of thehalo. Correcting for the surface pressure term does not appearto help, as we point out – indeed, the surface pressure termitself correlates with merging activity. This points towards anambiguity in the use of η – as we note, it rarely if ever satisfies η =1. We discuss this point in a forthcoming paper, but we notethat even in ideal situations, what one computes for η dependson r vir (cf. Cole & Lacey 1996; Łokas & Mamon 2001) – and soapplying a flat cut based on a threshold in η alone risks omittingmassive haloes that might otherwise be considered dynamicallyequilibrated. For this reason we advocate the use of ∆ r in cosmo-logical N -body simulations as a more robust measure of a halo’sdynamical state; its calculation is computationally inexpensive, itis well defined as a quantity to measure, and its interpretation isboth clear and straightforward. We find that ∆ r . . , whichcorresponds to a δ max . . , should be sufficient to pick outthe most dynamically relaxed haloes in a simulation volume at z =0.Although our focus has been fixed firmly on haloes in cos-mological simulations, we note that our results have observationalimplications. Whether or not an observed system – for example, agalaxy cluster – is in dynamical equilibrium will affect estimatesof its dynamical mass if we assume a luminous tracer populationthat is in dynamical equilibrium (e.g. Piffaretti & Valdarnini 2008).Similarly, studies that seek to reconstruct a galaxy cluster’s re-cent merging history tend to use signatures of disequilibrium (e.g.Cassano et al. 2010). The most obvious measure of disequilibriumis the centre of mass offset ∆ r , or its projected variant. Althougha more careful study in which we mock observe our haloes (and aseeded galaxy population) is needed, our results suggest that ∆ r could be used to infer the redshift of the last major merger (cf. Fig-ures 13 and Figure 9, although care must be taken as Figure 15 re-veals). Observationally, this would require measurement of, for ex-ample, projected displacements between gas and dark matter fromgravitational lensing and X-ray studies. We note that Poole et al.(2006) have already tested this idea using idealised hydrodynamicalsimulations of mergers between galaxy clusters and found that thecentroid offset between X-ray and projected mass maps capturesthe dynamical state of galaxy clusters well, but it is interesting toextend this idea using cosmological hydrodynamical simulations ofgalaxy groups and clusters. This will form the basis of future work. ACKNOWLEDGMENTS
CP acknowledges the support of the STFC theoretical astrophysicsrolling grant at the University of Leicester. AK is supported by the
Spanish Ministerio de Ciencia e Innovaci´on (MICINN) in Spainthrough the Ramon y Cajal programme as well as the grants AYA2009-13875-C03-02, AYA2009-12792-C03-03, CSD2009-00064,and CAM S2009/ESP-1496. He further thanks the AluminumGroup for chocolates. SRK acknowledges financial support fromSwinburne University of Technology’s Centre for Astrophysics andSupercomputing’s visitor programme. He acknowledges supportby the MICINN under the Consolider-Ingenio, SyeC project CSD-2007-00050. The simulations presented in this paper were carriedout on the Swinburne Supercomputer at the Centre for Astrophysics& Supercomputing, the Sanssouci cluster at the AIP and the ALICEsupercomputer at the University of Leicester.This paper has been typeset from a TEX/ L A TEX file prepared by theauthor.
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