Seeking Observable Imprints of Small-Scale Structure on the Properties of Dark Matter Haloes
aa r X i v : . [ a s t r o - ph . C O ] S e p Publications of the Astronomical Society of Australia (PASA)c (cid:13)
Astronomical Society of Australia 2018; published by Cambridge University Press.doi: 10.1017/pas.2018.xxx.
Seeking Observable Imprints of Small-Scale Structure onthe Properties of Dark Matter Haloes
C. Power ∗ International Centre for Radio Astronomy Research, University of Western Australia, 35 Stirling Highway, Crawley,Western Australia 6009, Australia
Abstract
The characteristic prediction of the Cold Dark Matter (CDM) model of cosmological structure formationis that the Universe should contain a wealth of small-scale structure – low-mass dark matter haloes andsubhaloes. However, galaxy formation is inefficient in their shallow potential wells and so we expect theselow-mass haloes and subhaloes to be dark. Can we tell the difference between a Universe in which theselow-mass haloes are present but dark and one in which they never formed, thereby providing a robusttest of the Cold Dark Matter model? We address this question using cosmological N -body simulations toexamine how properties of low-mass haloes that are potentially accessible to observation, such as theirspatial clustering, rate of accretions and mergers onto massive galaxies and the angular momentum contentof massive galaxies, differ between a fiducial ΛCDM model and dark matter models in which low-masshalo formation is suppressed. Adopting an effective cut-off mass scale M cut below which small-scale poweris suppressed in the initial conditions, we study dark matter models in which M cut varies between 5 × h − M ⊙ and 10 h − M ⊙ , equivalent to the host haloes of dwarf and low mass galaxies. Our resultsshow that both the clustering strength of low-mass haloes around galaxy-mass primaries and the rate atwhich they merge with these primaries is sensitive to the assumed value of M cut ; in contrast, suppressinglow-mass halo formation has little influence on the angular momentum content of galaxy-mass haloes – it isthe quiescence or violence of a halo’s assembly history that has a more marked effect. However, we expectthat measuring the effect on spatial clustering or the merger rate is likely to be observationally difficultfor realistic values of M cut , and so isolating the effect of this small-scale structure would appear to beremarkably difficult to detect, at least in the present day Universe. Keywords: methods: N -body simulations – galaxies: formation – galaxies: haloes – cosmology: theory –dark matter – large-scale structure of Universe One of the key questions facing fundamental physicsand cosmology at the turn of the 21 st century con-cerns the nature of the dark matter. Approximately80% of the matter content of the Universe appears tobe in the form of exotic, non-baryonic dark matter (cf.Planck Collaboration et al. 2013) whose clustering isbelieved to play a crucial role in the formation and sub-sequent evolution of galaxies (e.g. White & Rees 1978;White & Frenk 1991). This non-baryonic dark matter iswidely assumed to be cold – that is, dark matter parti-cles were non-relativistic at the time of decoupling – andcollisionless, and these properties of Cold Dark Matter(hereafter CDM) lead to a number of fundamental con-sequences. The first of these is that dark matter haloeshave central density cusps (cf. Tremaine & Gunn 1979;Moore 1994); the second is that the halo mass function – ∗ [email protected] the number of haloes of mass M per unit mass per unitcomoving volume – increases with decreasing mass as M − α where α ∼ . ∼ − M ⊙ (cf.Green et al. 2004).Based on the results of cosmological N -body sim-ulations, cuspy haloes and an abundance of small-scale structure – low-mass haloes and subhaloes – arenow well established as robust predictions of the CDMmodel (e.g. Springel et al. 2008). If we consider a Uni-verse in which low-mass halo and subhalo formationis suppressed, as in Warm Dark Matter (hereafterWDM) models, we find that haloes can form cores (al-beit small ones; cf. Villaescusa-Navarro & Dalal 2011)but they are likely to remain cuspy for plausibleWDM particle masses ( m WDM & . − C. Power be suppressed in WDM models (e.g. Dunstan et al.2011; Smith & Markovic 2011; Schneider et al. 2013;Benson et al. 2013; Pacucci et al. 2013), and so it canbe argued that it is the abundance of small-scale struc-ture, rather than central density cusps, that is the defin-ing characteristic of the CDM model .However, we expect few of these low-mass haloesto host galaxies because galaxy formation willbe inefficient in their shallow potential wells (e.g.Dekel & Silk 1986; Efstathiou 1992; Thoul & Weinberg1996; Benson et al. 2002). For example, supernovae(e.g. Dekel & Silk 1986) and photo-ionizing sources(e.g. Benson et al. 2002; Cantalupo 2010) can quenchgalaxy formation in low-mass haloes, while the likeli-hood that a low-mass halo hosts a satellite galaxy ap-pears to be stochastic (cf. Boylan-Kolchin et al. 2012;Garrison-Kimmel et al. 2013), suggesting that the pro-cess is highly sensitive to details of a galaxy’s assemblyhistory (i.e. environment, gas accretion and star forma-tion history, etc...). This raises the question, if low-masshaloes and subhaloes remain dark because galaxy for-mation is inefficient on these mass scales, how can wetell the difference between a Universe in which small-scale structure is present but dark and one in which itsformation is suppressed, as in WDM models?In this paper we use the results of cosmological N -body simulations to address this question, compar-ing systematically dark matter halo properties in afiducial ΛCDM cosmology and in ΛWDM-like darkmatter models, in which low-mass halo formationis suppressed by truncating the ΛCDM power spec-trum on small scales. Numerous studies have in-vestigated the halo mass function in WDM models(recent examples include Schneider et al. 2011, 2013;Pacucci et al. 2013; Benson et al. 2013) and associatedissues arising from discreteness effects in such simula-tions (e.g. Wang & White 2007; Schneider et al. 2013;Angulo et al. 2013; Hahn et al. 2013), but we note thatdirect measurement of the halo mass function obser-vationally is fraught with difficulty (cf. Murray et al.2013b). For this reason we focus on three mea-sures of the halo population that are potentially ac-cessible to observation – (i) the spatial clusteringof low-mass haloes around galaxy- and group-masshaloes (10 h − M ⊙ < ∼ M vir < ∼ h − M ⊙ ); (ii) the rateat which these haloes assemble their mass and at which There is a caveat here – we have assumed implicitly that darkmatter is collisionless. Recent papers have revived the possi-bility that dark matter is self-interacting (e.g. Loeb & Weiner2011; Vogelsberger et al. 2012; Peter et al. 2013; Rocha et al.2013), an idea that was first explored systematically using sim-ulations in the early 2000s (see, for example Yoshida et al.2000; Dav´e et al. 2001; Col´ın et al. 2002). This recent work hasshown that it is possible to form small cores ( r core ≃ they experience mergers; and (iii) their angular momen-tum content. Although we analyse properties of the halopopulation, we reason that they provide a baseline fortrends that we observe in the galaxy population.We choose the cut-off mass M cut , the mass scale be-low which halo formation is suppressed, to vary be-tween 5 × h − M ⊙ < ∼ M cut < ∼ h − M ⊙ . These val-ues of M cut are unrealistic in the sense that they aretoo large to be consistent with observational constraints(see, for example, the review of Primack 2009, assumingcorresponding filtering masses from Bode et al. 2001)but they allow us to experiment with the consequencesof progressively more aggressive truncations of the ini-tial power spectrum on the properties of haloes with M & M ⊙ .The structure of this paper is as follows. In § § §
4) and weexplore measures of halo angular momentum and spin( § § We have run a sequence of cosmological N -body simu-lations that follow the formation and evolution of darkmatter haloes in a box of side 20 h − Mpc from a start-ing redshift of z =100 to z =0. For each run we assume aflat cosmology with a dark energy term, and for conve-nience we adopt the cosmological parameters of Spergel(2007) – matter and dark energy density parameters ofΩ m = 0 .
24 and Ω Λ = 0 .
76, a dimensionless Hubble pa-rameter of h = 0 .
73, a normalisation of σ = 0 .
74 anda primordial spectral index of n spec =0.95. Each simula-tion volume contains 256 equal-mass particles, whichfor the adopted cosmological parameters gives particlemasses of m p = 3 . × h − M ⊙ .The respective runs differ in the spatial scale belowwhich small-scale power in the initial conditions is sup-pressed. We generate a single realisation of the ΛCDMpower spectrum appropriate for our choice of cosmolog-ical parameters and in the case of the truncated modelswe introduce a sharp cut-off in the ΛCDM power spec-trum at progressively larger spatial scales. This cut-offspatial scale is set by the mass scale below which wewish to suppress halo formation. Details about the trun-cated models are given in the next section.All of our simulations were run using the parallelTreePM code GADGET2 (Springel 2005) with a con-
PASA (2018)doi:10.1017/pas.2018.xxx bservable Imprints of Small-Scale Structure on Dark Matter Haloes ǫ =1 . h − kpcand individual and adaptive particle time-steps. Thesewere assigned according to the criterion ∆ t = η p ǫ/a ,where a is the magnitude of a particle’s gravitationalacceleration and η = 0 .
05 determines the accuracy ofthe time integration.
Truncating the Initial Power Spectrum
We are interested in models in which small scale poweris suppressed at early times. Physically suppressionarises because dark matter free streams, which acts asa damping mechanism to wash out primordial densityperturbations and to introduce a cut-off in the linearmatter power spectrum. If the dark matter particle isa thermal relic, the spatial scale at which this cut-offoccurs can be calculated (cf. Bergstr¨om 2000). The freestreaming scale λ fs can be expressed as λ fs = 0 . dm h ) / (cid:16) m dm keV (cid:17) − / Mpc , (1)where m dm is the dark matter particle mass mea-sured in keV and Ω dm is the dark matter density (cf.Boehm et al. 2005).Provided λ fs is small compared to the spatial scaleswe are interested in simulating, the power spectrum willdiffer little from the ΛCDM power spectrum (which it-self may have a cut-off on comoving scales of order 1pc; cf. Green et al. 2004). However, as λ fs increases andapproaches the scale that we wish to resolve, then itbecomes necessary to determine how the power spec-trum changes. The shape of the linear power spectrumfor collisionless WDM models has been calculated by anumber of authors (e.g. Bardeen et al. 1986; Bode et al.2001), and it can be recovered from the CDM powerspectrum by introducing an exponential cut-off at smallscales. The larger λ fs , the larger the mass scale M fs be-low which structure formation is suppressed and thesmaller the wave-number k at which the WDM andCDM power spectra differ, although the relationshipbetween λ fs and M fs is sensitive to the precise nature ofthe WDM particle.We do not wish to make any assumptions about theprecise nature of the dark matter other than that it iscollisionless and that low-mass halo formation is sup-pressed, and so we follow Moore et al. (1999) and trun-cate sharply the power spectrum at k cut , suppressingpower at wave-numbers k > k cut . We choose k cut byidentifying a mass scale M cut and estimating the co-moving length scale R cut , R cut = (cid:18) M cut π ρ (cid:19) / (2)where ρ is the mean density of the Universe. Modelling Free Streaming
Similarly we choose not to include the effect of freestreaming in our initial conditions – partly becausewe wish to avoid assumptions about the precise na-ture of the dark matter, and partly for pragamtic rea-sons, which we now explain. In practice free stream-ing is mimicked by assigning a random velocity compo-nent (typically drawn from a Fermi-Dirac distribution)to particles in addition to their velocities predicted bylinear theory (cf. Klypin et al. 1993; Col´ın et al. 2008;Macci`o et al. 2012). However, capturing this effect cor-rectly in a N -body simulation is difficult – it can leadto an unphysical excess of small-scale power in the ini-tial conditions if the simulation is started too early (seeFigure 1 of Col´ın et al. 2008 for a nice illustration ofthis problem).Precisely how early is too early has yet to be prop-erly quantified, but it will depend explicitly on darkmatter particle mass – the lower the mass, the longerthe free streaming scale, and the larger the random ve-locity component required. If this exceeds the typicalvelocity predicted by linear theory, a population of spu-rious haloes forms (e.g. Klypin et al. 1993) that can ex-ceed in number by factors of ∼
10 the population thatforms when no random velocity component is included,as studied by Wang & White (2007). This is unlikely tobe a problem for studying the mass profiles of haloes– for example, Col´ın et al. (2008) started their simula-tions at reasonably late times because the random ve-locity component damps away with decreasing redshiftwhile the velocities predicted by linear theory increase– but it is not clear how much of a problem it will befor our study, in which we study quantities that dependon spatial correlations between haloes. For this reasonwe do not include the effect of free streaming, deferringthis to a forthcoming study on discreteness effects inWDM-like simulations (Power et al., in preparation).
Generation of Initial Conditions
We follow the standard procedure (e.g. Power et al.2003) of generating a statistical realisation of the highredshift density field using the appropriate linear the-ory power spectrum, from which initial displacementsand velocities are computed and imposed on a suitableuniform particle load; for this study we adopt an initialglass distribution (cf. White 1994). We use the Boltz-mann code
CMBFAST (Seljak & Zaldarriaga 1996) togenerate the CDM transfer function for our choice ofcosmological parameters. This is convolved with the pri-mordial power spectrum ( P ( k ) ∝ k n , where n is theprimordial spectral index) to obtain the appropriateΛCDM power spectrum P ( k ). To obtain a truncatedmodel, we chop P ( k ) sharply at k cut = 2 π/R cut (where R cut is given by equation 2) and thereby suppress poweron scales k & k cut . PASA (2018)doi:10.1017/pas.2018.xxx
C. Power
Table 1 Truncated Models : Simulation Details
Model M cut R cut k cut h − M ⊙ h − Mpc h Mpc − A 0.5 0.26 24.01B 1.0 0.33 19.06C 5.0 0.56 11.15D 10.0 0.71 8.85We consider five cases – a fiducial ΛCDM model andtruncated models in which small scale power is sup-pressed at masses below M cut = 5 × , 10 , 5 × and 10 h − M ⊙ respectively. Note that the cut-offwave-number k cut is always less than the Nyquist fre-quency of the simulation, k Ny ≃ h Mpc − . Values forthe cut-off masses and wave-numbers are given in Ta-ble 1. Halo Identification
Groups are identified using
AHF , otherwise known as A MIGA ’s H alo F inder (cf. Knollmann & Knebe 2009). AHF locates groups as peaks in an adaptively smootheddensity field using a hierarchy of grids and a refinementcriterion that is comparable to the force resolution ofthe simulation. Local potential minima are calculatedfor each of these peaks and the set of particles thatare gravitationally bound to the peaks are identified asthe groups that form our halo catalogue. Each halo inthe catalogue is then processed, producing a range ofstructural and kinematic information.We adopt the standard definition of a halo such thatthe virial mass is M vir = 4 πρ crit ∆ vir r / , (3)where ρ crit = 3 H / πG is the critical density of theUniverse and r vir is the virial radius, which defines theradial extent of the halo. The virial over-density crite-rion, ∆ vir , is a multiple of the critical density, and cor-responds to the mean over-density at the time of viri-alisation in the spherical collapse model (the simplestanalytic model of halo formation; cf. Eke et al. 1996).In an Einstein-de Sitter Universe, ∆ vir ≃ λ CDM model ∆ vir ≃
92 at z =0.Defined in this way, the virial radius r vir pro-vides a convenient albeit approximate boundary for adark matter halo that can be estimated easily fromsimulation data. However, it is only approximate –haloes that form in cosmological simulations are rela-tively complex structures. They are generally aspher-ical (e.g. Bailin & Steinmetz 2005) and asymmetric(e.g. Gao & White 2006) with no simple boundary(e.g. Prada et al. 2006), and so defining an appropri- ate boundary is not straightforward. This presents dif-ficulties when calculating, for example, a halo’s angularmomentum and its binding energy (cf. Lokas & Mamon2001). Material bound to the halo can lie outside of r vir ,and this will distort the angular momentum and bind-ing energy one measures for the halo using only materialfrom within r vir . This issue has been touched on by pre-vious authors (e.g. Cole & Lacey 1996; Lokas & Mamon2001; Shaw et al. 2006; Power et al. 2012) in the con-text of identifying when a halo is in virial equilibrium.In a similar vein, the angular momentum one measuresusing only material from within r vir will be biased. Thisis an important caveat that we need to bear in mindwhen discussing our analysis of halo angular momen-tum in § Halo Merger Trees
Halo merger trees are constructed by linking halo par-ticles at consecutive output times; • For each pair of group catalogues constructed atconsecutive output times t and t > t , the ‘an-cestors’ of ’descendant’ groups are identified. Foreach descendent identified in the catalogue at thelater time t , we sweep over its associated parti-cles and locate every ancestor at the earlier time t that contains a subset of these particles. A recordof all ancestors at t that contain particles associ-ated with the descendent at t is maintained. • The ancestor at time t that contains in excess of f prog of these particles and also contains the mostbound particle of the descendent at t is deemedthe main progenitor. Typically f prog = 0 .
5, i.e. themain progenitor contains in excess of half the finalmass.Each group is then treated as a node in a tree structure,which can be traversed either forwards, allowing one toidentify a halo at some early time and follow it forwardthrough the merging hierarchy, or backwards, allowingone to identify a halo and all its progenitors at earliertimes. In our analysis we concentrate on the main trunkof the merger tree, in which we follow the evolution ofthe main progenitor of a halo to earlier times.
However, care must be taken when including haloeswith masses below M cut in any analysis. An unfor-tunate feature of simulations of cosmologies in whichsmall-scale power is suppressed at early times is theformation of unphysical low-mass haloes by the frag-mentation of filaments, driven by the discreteness of thematter distribution. These spurious haloes form prefer-entially in filaments, at regular intervals of order themean interparticle separation of the simulation, akin to
PASA (2018)doi:10.1017/pas.2018.xxx bservable Imprints of Small-Scale Structure on Dark Matter Haloes lim ∼ . m / p M cut2 / , where m p is the particle mass(Wang & White 2007).We wish to identify haloes in the truncated modelsthat have clearly identifiable counterparts in the fiducialΛCDM simulation. These haloes form the halo sampleupon which our analysis is based. By selecting haloes inthis way, we can track the merger trees of the counter-parts and study the merging and accretion histories ofindividual systems, correlating any differences in haloproperties with the details of their mass assembly. Wecan also avoid including in our analysis spurious (un-physical) haloes that form below the mass cut-off inthe truncated models (see, for example, Wang & White2007).To identify counterparts, we adapt our algorithm forlinking haloes across time slices when building mergertrees to link haloes between runs at a given time. • For each pair of group catalogues, we process eachgroup and compute “virial” quantities, namely thevirial mass and radius, and the set of particles thatbelong to each halo. • For each halo in the fiducial ΛCDM model at time t , we loop over its associated particles and deter-mine how many of these particles are present inhaloes in the corresponding truncated model cat-alogue. • The halo in the truncated model that contains inexcess of f count = 75% of these particles is identi-fied as a counterpart candidate. However, the can-didate halo can be part of a much larger structurein the fiducial ΛCDM model, so we also requirethat the mass of the candidate halo not differ fromits CDM counterpart by not more than a factor of25%. Haloes that satisfy these conditions are iden-tified as counterparts. As our starting point, we compare and contrast the spa-tial clustering of dark matter haloes in the ΛCDM andtruncated models respectively as a function of redshift.We expect differences between models to be apparentfor haloes with masses M ∼ M cut and to become morepronounced with increasing redshift, when M cut is alarger fraction of the typical collapsing mass M ∗ . Visual Impression
In Figure 1 we show the projected dark matter distribu-tion in thin slices (20 × × h − Mpc ) taken throughthe ΛCDM (upper panels), Truncated B (middle pan- els; hereafter TruncB) and D (lower panels; hereafterTruncD) at z = 0, 1 and 4 (from left to right). Eachslice is centred on the geometric centre of the simulationvolume and the grey-scale is weighted by the logarithmof projected density.Figure 1 is instructive because it provides a power-ful visual impression of the effect of suppressing smallscale power at early times. The filamentary network islargely unaffected and the positions of the most massivehaloes, which form at the nodes of these filaments, aresimilar in each of the models we have looked at. Whatis striking, however, is the impact on the abundanceof low-mass haloes, which appear as small dense knotsin projection. As M cut increases, the projected numberdensity of these low-mass haloes decreases markedly aswe go from the ΛCDM run to the TruncD run (topand bottom panels respectively). This is evident in theclustering around more massive haloes and the absenceof low-mass systems in the void regions. Furthermore,the contrast between the models becomes increasinglynoticeable with increasing redshift – compare z =0 and z =4. Note also the presence of the low-mass haloes dis-tributed along filaments in “beads-on-a-string” fashionin the truncated models. Spatial Clustering
In Figure 2 we investigate how the clustering strengthof haloes differs between the different dark matter mod-els and as a function of redshift. We quantify cluster-ing strength by the correlation function ξ ( r ), whichmeasures the excess probability over random that apair of haloes i and j will be separated by a distance r = |E r | = |E r i − E r j | . ξ ( r ) is estimated using ξ ( r ) = 1 + DD ( r ) RR ( r ) , (4)where DD ( r ) is the number of haloes at comoving sepa-ration r compared to the number in a random distribu-tion RR ( r ). Because our focus is on differences, we con-struct the ratio of N ( r ) = DD ( r ) = RR ( r )( ξ ( r ) − N ( r ) CDM for the fiducialΛCDM run.In the upper panel of Figure 2 we consider pairsof haloes in which the primary’s mass is M vir > h − M ⊙ and the secondary’s mass is M vir > × h − M ⊙ , while in the lower panel we consider pairsof haloes in which both the primary and secondarymasses M vir > h − M ⊙ . This reveals that the clus-tering strength of low-mass haloes around high masshaloes (i.e. M vir > h − M ⊙ ) decreases with increas-ing M cut , although the dependence on M cut does not ap-pear to be straightforward. In the TruncA and TruncBruns, we find that N ( r ) /N ( r ) CDM is close to unityout to r ≃ h − Mpc, never deviating by more than
PASA (2018)doi:10.1017/pas.2018.xxx
C. Power
Figure 1.
The projected density distribution in 2 h − Mpc slices taken through the centres of each of the boxes. We have smoothed theparticle mass using an adaptive Gaussian kernel and projected onto a mesh. Each mesh point is weighted according to the logarithmof its projected surface density, and so the “darker” the mesh point, the higher the projected surface density.
10% to within ∼ h − kpc at all redshifts. For theTruncC and TruncD runs, the suppression in clusteringstrength is quite marked – by ∼
40% for the TruncCrun and ∼
50% for the TruncD run. Large deviationsat small radii reflect the small numbers of very closepairs. In contrast, the clustering strength of massivehaloes (i.e. M > h − M ⊙ ) does not appear to be af-fected by M cut , as we inferred from Figure 1. The ratio N ( r ) /N ( r ) CDM is noisy – reflecting the lower numberdensity of massive haloes – but it is approximately unitybetween 0 . z . Suppressing small scale power at early times leads to areduction in the clustering of low-mass haloes aroundmassive haloes ( M vir & h − M ⊙ ) at z .
3, whichimplies that the number of likely minor mergers a typ-ical halo will experience during a given period shoulddecline with increasing M cut . We expect this to dependon both halo mass and epoch. At a given z , the merg-ing history of haloes with masses M vir ∼ M cut shouldbe more sensitive to the clustering of small scale struc-ture than haloes with masses M vir ≫ M cut . Similarly,at earlier times when the typical collapsing mass M ∗ is PASA (2018)doi:10.1017/pas.2018.xxx bservable Imprints of Small-Scale Structure on Dark Matter Haloes Figure 2. Evolution of Spatial Clustering with Redshift.
We examine how the clustering strength of haloes with respect tothe fiducial ΛCDM model varies across the runs TruncA (solidcurves), TruncB (short dashed curves), TruncC (long dashedcurves) and TruncD (dotted-dashed curves) at z =0, 1, 2 and3 by plotting the ratio N ( r ) /N ( r ) CDM – the number of haloeswith comoving halo separation r – as a function of r . In the up-per panel, we look at the clustering of all secondary haloes withmass M vir > × h − M ⊙ around primary haloes with mass M vir > h − M ⊙ , while in the lower panel we look at the clus-tering of only massive haloes, for which both the primary andsecondary masses are M vir > h − M ⊙ . Figure 3. Impact on Mass Accretion Rate.
For each halo at z =0, we follow the main branch of its merger tree to higher red-shifts and compute the difference in virial mass between progeni-tors at z and z > z . From this we compute the mass accretionrate with respect to time (in Gyrs), normalised by the virial massof the descendent halo at z =0. Within each of the mass bins wecompute the average mass accretion rate for haloes in the fidu-cial λ CDM run (red filled circles), TruncB ( M cut = 10 h − M ⊙ ;green filled squares),TruncC ( M cut = 5 × h − M ⊙ ; cyan filledtriangles) and TruncD ( M cut = 10 h − M ⊙ ; magenta crosses). smaller and M cut is a larger fraction of M ∗ , we wouldexpect the effect of suppressing small scale structure tobe more pronounced.When computing mass accretion and merging rates,we use merger trees for all haloes between 5 × h − M ⊙ ( ∼ h − M ⊙ at z =0.Note that we have a hard lower limit of 100 particles fora halo to be retained in our catalogues; this correspondsto a mass of ∼ . × h − M ⊙ , and so we cannot iden-tify minor mergers with mass ratios of less than ∼ Mass Accretion Rate
In Figure 3 we show how the mass accretion rate of themost massive progenitors of haloes identified at z =0evolves with redshift. Note that his accretion rate in-cludes both smooth accretion and minor and majormergers. The distinction between minor mergers andsmooth accretion may be a moot one in the CDM model– as the mass and force resolution of the simulation in-creases, we continue to resolve increasing numbers oflow-mass haloes – but this is not necessarily the case inthe truncated models that we consider. PASA (2018)doi:10.1017/pas.2018.xxx
C. Power
Figure 4. Impact on Merger Rate.
For each halo at z =0,we follow the main branch of its merger tree to higher red-shifts and determine the number of mergers with mass ra-tios in excess of 6% experienced by the halo between z and z > z . From this we compute the merger rate per unit red-shift per unit time. Within each of the mass bins we com-pute the average merger rate for haloes in the fiducial λ CDMrun (red filled circles), TruncB ( M cut = 10 h − M ⊙ ; green filledsquares), TruncC ( M cut = 5 × h − M ⊙ ; cyan filled triangles)and TruncD ( M cut = 10 h − M ⊙ ; magenta crosses). From upper to lower panels, we show the av-erage accretion rate as a function of redshift forhaloes with virial masses at z =0 in the range5 × M vir /h − M ⊙ (filled circles),10 M vir / h − M ⊙ × (filled squares),5 × M vir /h − M ⊙ (filled triangles) and M vir /h − M ⊙ (crosses). Note that we measurethe accretion rate as the change in virial mass (∆ M )per unit redshift (∆ z ) per unit time (∆ t ), normalisedby the final (i.e. z =0) virial mass. Bars indicate r.m.s.scatter.Figure 3 shows that haloes accrete their mass atsimilar rates across the different models, regardless ofwhether or not small scale power is suppressed at earlytimes. On average, less massive haloes tend to havehigher accretion rates at z & z ∼ z =0 (see also Figure 9 for detailedmass accretion histories for individual haloes). In con-trast, more massive haloes accrete their mass at a steadyrate. We find that our accretion rates for ΛCDM haloesare in good agreement with those consistent with thoseof, for example, Maulbetsch et al. (2007). Figure 5. Distribution of Most Significant Mergers.
Foreach halo at z =0, we compute the mass ratio of the most sig-nificant merger δ max that it has experienced since z ≃ δ max for the respectivemodels. Merger Rates
In Figure 4, we focus on the merger rate ∆ N/ ∆ z/ ∆ t and its variation with redshift, where ∆ N is thenumber of mergers per unit redshift per unit time. Heredifferences between runs are immediately apparent andin the sense that we expect – for halo masses close to M cut increases, the merging rate decreases. Note thatthe estimated merger rate is quite noisy in the lowestmass bin (upper panel), especially at early times – inthis case, the lower limit of 100 particles imposed byour halo catalogues corresponds to a merger of pro-gressively greater mass ratio with increasing redshift.For this reason, we focus on haloes with masses at z =0 in excess of 10 h − M ⊙ . For haloes with massesbetween 10 M vir /h − M ⊙ × , we find thatthe average merger rate in the TruncC (TruncD) modelis a factor of ∼ .
5) smaller than that in the fiducialΛCDM model, and this is approximately constant withredshift. The difference is less pronounced for haloeswith masses between 5 × M vir /h − M ⊙ ,and for haloes with masses in excess of 10 h − M ⊙ there is no discernible difference in the merging rateswith redshift.In Figure 5 we assess how major mergers are affectedby suppression of small scale power at early times. Thisdemonstrates that the likelihood that the mass ratioof the most significant merger experienced by a halo PASA (2018)doi:10.1017/pas.2018.xxx bservable Imprints of Small-Scale Structure on Dark Matter Haloes z ≃ . δ max = M acc /M vir expe-rienced by each halo (identified at z =0) since ∼ . ,split according to virial mass at z =0.There are a number of interesting trends in this Fig-ure. The first is that most significant mergers withlarge mass ratios (i.e. minor mergers) are relatively un-common; the probability distribution increases approx-imately as a power law with δ max as δ . . The secondis that, in the CDM model, the likelihood that a haloexperiences a most significant merger with a given δ max does not depend strongly on its mass. For example, ahalo with virial mass of 10 h − M ⊙ is as likely to haveexperienced a major merger with mass ratio of ∼ h − M ⊙ halo – approximately 20%. The thirdis that there is some evidence that haloes in the massrange 5 × M vir /h − M ⊙ are less likely toexperience major mergers with mass ratios in excess of ∼
50% (compare TruncB and TruncD).
Suppressing small scale power at early times impactson both the clustering strength of low-mass haloes andthe rate at which mergers and accretions at later times.Do we see a corresponding influence on the angular mo-mentum content of haloes at later times?
Spin Parameter
We begin by considering the spin parameter λ , whichquantifies the degree to which the halo is supportedby rotation and which we define using the “classical”definition of Peebles (1969), λ = J | E | / GM / . (5)Here J and E are the total angular momentum andbinding energy respecively of material with r vir and G is the gravitational constant. We impose a lower limit of600 particles within r vir ( M vir > × h − M ⊙ ) whenmeasuring λ ; this ensures that both J and E are unaf-fected by discreteness effects (cf. Power et al. 2012).In Figure 6 we show how the median spin of the halopopulation evolves with redshift. In the upper panelwe focus on the haloes with M vir > × h − M ⊙ ,while in the lower panel we consider haloes with M vir > h − M ⊙ . Filled circles, squares, triangles and crossesrepresent the median spin of the halo populations inthe ΛCDM, TruncB, TruncC and TruncD runs, and er-ror bars indicate the 25 th and 75 th percentiles of thedistribution. This figure suggests that the behaviour This redshift interval corresponds to ∼ Figure 6. Variation of Median λ with Redshift. We showhow the median spin parameter λ med varies with redshift. In theleft hand panel we consider all haloes with virial masses in excessof M vir > . × h − M ⊙ , while in the right hand panel weconsider all haloes that satisfy M vir > h − M ⊙ . Lower andupper error bars represent the 25 th and 75 th percentiles. The filledcircles, squares, triangles and crosses correspond to the fiducialΛCDM, TruncB, TruncC and TruncD runs respectively. of the distribution of λ is sensitive to M cut – sys-tematic differences are apparent in the TruncC andTruncD runs when we include all haloes with M vir > × h − M ⊙ , whereas the distributions are statisti-cally similar when we include only haloes with M vir > h − M ⊙ . PASA (2018)doi:10.1017/pas.2018.xxx C. Power
This figure is interesting because we include alarge population of haloes in the TruncC and TruncDruns with M vir ≤ M cut when we include haloes with M vir > × h − M ⊙ , and so the apparent differ-ences are to be expected. In contrast, we do not seeany evident differences when we include haloes with M vir > h − M ⊙ . This is also interesting because itreveals that the median λ increases with decreasingredshift at approximately the same rate – in proportionto (1 + z ) − . – regardless of whether or not we includehaloes with masses below M cut .In Figure 7 we focus on individual haloes, showinghow λ and the specific angular momentum j = J/M vary with redshift z for a selection of haloes with quies-cent and violent merging histories, drawn from haloeswith M vir > h − M ⊙ over the redshift interval 0 ≤ z ≤
3. For each halo we determine the most significantmerger δ max that it has experienced since z =1, wherewe define δ max as the mass ratio of the most majormerger experienced by the main progenitor of a haloidentified at z =0 during the redshift interval 0 ≤ z ≤ δ max and we identify haloes in the upper (lower) 20% of thedistribution as systems with violent (quiescent) merginghistories. For ease of comparison, we focus on the ex-tremes – the ΛCDM and TruncD runs (top and bottomrespectively).There are a few of points worthy of note in relationto the evolution of the spin parameter with redshift.First, the spin parameter for a given halo is a verynoisy quantity but if we consider the average behaviourof haloes in the respective samples, we do not find anyclear correlation between spin and redshift (based ontheir Spearman rank coefficient). Second, there is a clearoffset between median spins in the quiescent and violentsamples – haloes with violent merging histories tend tohave higher spins (by factors of ∼ ∼ . .
29 for haloes in the quiescent sample and ∼ . .
42 in the violent sample. Importantly, third,it is the dynamical state and merging history of a halothat has greater impact on its instantaneous spin andspecific angular momentum – the influence of the darkmatter is a secondary effect at best.Note that we also compare the growth of angular mo-mentum and spin for three sets of cross matched haloesacross dark matter models – shown in Figures 8 and 9.From our cross matched catalogues we identified blindlya set of three haloes with M vir ≃ (7 . , . , . × h − M ⊙ , which are approximately 1, 10 and 100times the threshold mass of M cut =10 h − M ⊙ .Projections of the density distribution in cubes ap-proximately 2 r vir on a side and centred on the haloesare shown in Figure 8 – high, intermediate and low mass CDM
Figure 7. Variation of λ and j with Redshift for Relaxedand Unrelaxed Haloes. We use the merging histories of haloesto identify two samples of haloes, one with a quiescent merginghistory ( δ max . . z =3.0; left hand panels) and one witha violent merging history ( δ max & . z =0 satisfies M vir > h − M ⊙ ( ∼ z =0), the specific angular mo-mentum j = J/M (normalised to its value at z =0, j ) and di-mensionless spin parameter λ = J | E | / /GM / as a function ofredshift z .PASA (2018)doi:10.1017/pas.2018.xxx bservable Imprints of Small-Scale Structure on Dark Matter Haloes P ( k ) thekey difference between the models. There are small dif-ferences in the orientation of the intermediate and low-mass haloes (compare, for example, the intermediatemass halo in the TruncC and TruncD runs) and in thepositions of subhaloes (compare, for example, the lowmass halo in the TruncB and TruncD runs), but suchdifferences are to be expected at the mass and forceresolution of our simulations.Figure 9 shows in detail how the virial mass (upperpanels), specific angular momentum (middle panels),and spin parameter (lower panels) grows over time foreach of the three sets of haloes. For the most massivehalo, the mass assembly histories are indistinguishable,while the specific angular momentum and spin growthare in very good agreement with each other. For the in-termediate mass halo, there are differences in the massassembly histories at z &
1, with the TruncC and Truncdeviating from the ΛCDM and TruncB cases, but theyare negligible; the specific angular momentum and spingrowth show small differences but they are in goodbroad agreement. For the low mass halo, it’s notice-able that the mass growth is in good general agreementacross the models at z .
3, but the mass of the halo inthe TruncD case has to grow rapidly to catch up with itscounterparts in the ΛCDM, TruncB and TruncC runsat z &
3. This has a knock-on effect in the growth of itsspecific angular momentum and spin parameter; how-ever, the mass, specific angular momentum and spinparameter growth are in very good agreement for z . Specific Angular Momentum Profiles
There does not appear to be any systematic differencein the bulk angular momenta of haloes, i.e. the total an-gular momentum of material within r vir . What of thedistribution of angular momentum within r vir ? We fo-cus on the specific angular momentum profile, whichquantifies the fraction of material within the virial ra-dius that has specific angular momentum of j or less.Figure 10 shows the average specific angular momentumprofile M ( < j ) of haloes in each of our models.We compute specific angular momentum profiles us-ing the method presented in Bullock et al. (2001, 2002).In brief, we compute the total angular momentum of thehalo and define this as the z -axis; then we sort particlesinto spherical shells of equal mass and increasing radius,and we assign shell particles to one of three equal vol-ume segments determined by the particle’s angle withrespect to the z -axis; finally, we compute both the totaland z -component of the specific angular momentum ineach segment. This allows us to compute the fraction ofhalo mass with specific angular momentum of j (and its Figure 9. Direct Comparison of Haloes: Redshift Evo-lution of Spin and Specific Angular Momentum Evolu-tion.
Upper/middle/lower panel show growth of virial mass (nor-malised to M vir at z =0), specific angular momentum (normalisedto value at z =0) and spin parameter λ as function of 1 + z .PASA (2018)doi:10.1017/pas.2018.xxx C. Power x [Mpc/h ] y [ M p c / h ] x [Mpc/h ] y [ M p c / h ] x [Mpc/h ] y [ M p c / h ] x [Mpc/h ] y [ M p c / h ] x [Mpc/h ] y [ M p c / h ] x [Mpc/h ] y [ M p c / h ] x [Mpc/h ] y [ M p c / h ] x [Mpc/h ] y [ M p c / h ] x [Mpc/h ] y [ M p c / h ] x [Mpc/h ] y [ M p c / h ] x [Mpc/h ] y [ M p c / h ] x [Mpc/h ] y [ M p c / h ] Figure 8. Direct Comparison of Haloes: Projected Dark Matter Density Maps.
From left to right, haloes with virial massesat z =0 of M vir ≃ (7 . , . , . × h − M ⊙ in the CDM, TruncB, TruncC and TruncD (from top to bottom).PASA (2018)doi:10.1017/pas.2018.xxx bservable Imprints of Small-Scale Structure on Dark Matter Haloes Figure 10. Specific Angular Momentum Profiles.
We usethe method of Bullock et al. (2001, 2002) to determine the frac-tion of halo mass that has a total specific angular momentumof j or less. Note that we consider only haloes that satisfy M vir > h − M ⊙ . z component j z ) or less. Note that we scale our profilesby j max , the maximum specific angular momentum thatwe measure in our data; this is distinct from the j max used in Bullock et al. (2001, 2002), who estimate j max by fitting their universal angular momentum profile.In Figure 10 we show the specific angular momentumprofile for the total angular momentum j , although the j z behaviour is similar. For ease of comparison, we haveapplied small offsets to the data points from the trun-cated models. There are a few points worthy of notein this Figure. The profile gently curves towards shal-lower logarithmic slopes with increasing j ; we find that M ( < j ) ∼ j / for the lowest angular momentum ma-terial and M ( < j ) ∼ j / for the highest angular mo-mentum material. It is interesting that there is a sys-tematic trend for lower angular momentum material inthe ΛCDM and TruncB runs to have on average lowervalues of j than the TruncC and TruncD runs – thedifference is of order 25% at most. This trend is not ev-ident when one looks at the projected specific angularmomentum ( j z ) profile. However, the r.m.s. scatter islarge for a given M ( < j ) or M ( < j z ) in all our models,and for interesting values of M cut ∼ h − M ⊙ (com-parable to our TruncA run) there is no statistically sig-nificant difference. Figure 11. Angular Momentum at Turnaround.
We trackthe material associated with each halo identified at z =0 and com-pute the radial extent and angular momentum of this material asa function of redshift in the ΛCDM, TruncB and TruncD runs.When the material has reached its maximum radial extent, wedenote the epoch at which this occurs as turnaround and look atthe ratio of the magnitude of angular momentum of the materialat this redshift z t , J ( z t ), with respect to the magnitude of the an-gular momentum of this material at z =0. In the upper panel weshow the variation of this ratio with halo mass at z =0; in the lowerpanel, we show the cumulative distribution D ( < J ( z t ) /J ( z = 0)).PASA (2018)doi:10.1017/pas.2018.xxx C. PowerAngular Momentum of the Lagrangian Volume
In Figure 11 we investigate the angular momentum ofthe Lagrangian region corresponding to the virialisedhalo at z =0 and determine how it evolves with timefor haloes with masses in excess of 5 × h − M ⊙ at z =0. In other words, we track the angular momentumof all the material that contributes to the final halo at z =0. We identify particles at z =9 that reside within thevirial radius at z =0 and compute their angular momen-tum E J using their centre of mass and centre of massvelocity. In addition we estimate the mean radial veloc-ity of this material with respect to the centre of massvelocity and determine the redshift at which it changessign from positive to negative (i.e. from expansion tocontraction); this defines the redshift of turnaround z t .This is typically between 0 . . z . z ) − / (cf. White 1984). Therefore,we expect the angular momentum at turnaround to beclose to its maximum value and the frequency distri-bution of angular momenta should be similar in eachof the models we have looked at. Linear perturbationtheory no longer provides a good description of angularmomentum growth subsequent to turnaround and non-linear processes (i.e. mergers) are believed to becomemore important drivers of angular momentum evolu-tion during this phase. Therefore, if there are differencesbetween the models, we would expect them to be ap-parent in the ratio of the ‘peak’ angular momentum atturnaround to the final angular momentum at z =0.In the upper panel of Figure 11 we show the dis-tribution of J ( z t ) /J ( z = 0) versus halo mass, while inthe lower panel we show the cumulative distribution D ( < J ( z t ) /J ( z = 0)) for all haloes with masses in ex-cess of 5 × h − M ⊙ at z =0. For clarity we consideronly the λ CDM (filled circles, solid curves), TruncB(filled squares, dashed curves) and TruncD (crosses,dotted-dashed curves) runs. The upper panel revealsthat, on average, the ratio of J ( z t ) /J ( z = 0) does notvary appreciably with mass and that it is slightly lessthan unity (approximately 0.8). In other words, themagnitude of the total angular momentum of the ma-terial at turnaround is on average smaller than at z =0. Sugerman et al. (2000) have shown that the angular momen-tum continues to grow ‘quasi-linearly’ after turnaround until firstshell crossing, at which point it reaches its maximum value.
These Figures reveal that the small differencesthat we observe in the spin distributions are alsopresent in the specific angular momentum. The median J ( z t ) /J ( z = 0) differs by ∼
10% between the λ CDMmodel and the TruncD run.
The focus of this paper has been to determine the ex-tent to which suppressing the formation of small-scalestructure – low-mass dark matter haloes – affects ob-servationally accessible properties of galaxy-mass darkmatter haloes. Using cosmological N -body simulations,we have investigated the spatial clustering of low-masshaloes around galaxy-mass haloes, the rate at whichthese haloes assemble their mass and at which they ex-perience mergers, and their angular momentum contentin a fiducial ΛCDM model and in truncated (ΛWDM-like) models. The main results of our study can be sum-marised as follows; Large Scale Structure:
Visual inspection of the den-sity distribution reveals that the structure that forms intruncated models is indistinguishable from that in theΛCDM model on large scales but differs on small scales.Precisely how small this scale is depends on M cut , themass scale below which low-mass halo formation is sup-pressed, which we varied between 5 × h − M ⊙ and10 h − M ⊙ . For M cut = 5 × h − M ⊙ the differenceswith respect to the ΛCDM model are negligible, butthey become significant for M cut = 10 h − M ⊙ . Spatial Clustering:
These visual differences are ap-parent in the clustering strength of lower-mass sec-ondary haloes around galaxy-mass primaries. Fixing theprimary mass at M vir = 10 h − M ⊙ , we found that theclustering strength of secondaries around primaries de-pends strongly on M cut and the minimum secondarymass. If we include secondaries with masses M vir ≥ × h − M ⊙ , the differences are as great as ∼ M cut =10 h − M ⊙ . Unsurprisingly, we found nodependence on M cut if secondaries are restricted tohaloes with masses M vir ≥ h − M ⊙ . Mass Accretion and Merger Rates:
The sensitivity ofthe clustering strength to M cut has immediate conse-quences for the frequency of minor mergers. The effect ismost striking for models with M cut ≥ × h − M ⊙ ,when the rate of all mergers with mass ratios in ex-cess of ∼
6% is suppressed across all redshifts by fac-tors of ∼ M vir . × h − M ⊙ . This effect must be driven by a reduc-tion in the number of minor mergers because the fre-quency of major mergers does not depend on M cut otherthan in haloes with masses M vir ∼ M cut . Interestingly PASA (2018)doi:10.1017/pas.2018.xxx bservable Imprints of Small-Scale Structure on Dark Matter Haloes M cut at all. Halo Angular Momentum:
Minor mergers appear tohave little influence on the angular momentum contentof galaxy-mass haloes.1. We computed the spin parameter λ and found noobvious dependence on M cut but a strong depen-dence on mass accretion history has – a markedsystematic offset is evident between the averagespins of haloes with violent mass accretion his-tories and those with quiescent histories – by afactor of ∼ M cut . The spinof individual haloes evolve in an almost stochasticfashion over time and on average do not show anyobvious evolution with redshift.2. We examined the angular momentum distributionwithin haloes by constructing specific angular mo-mentum profiles, which quantify the fraction ofmaterial within a halo that has specific angularmomentum of j or less. We found a weak trendfor halo material in truncated models with valuesof M cut greater than 10 h − M ⊙ to have on av-erage smaller values of j by 25% at most, but ther.m.s scatter is large for a given M ( < j ) in all ourmodels and the differences have a low statisticalsignificance.3. We investigated the angular momentum of theLagrangian region corresponding to the virialisedhalo at z =0 and determined how it evolves withtime. We calculated J ( z t ) /J ( z = 0), the ratio ofthe angular momentum of the material at theturnaround redshift z t to z =0. Again the differ-ences between the models are small, at most 10%.These results indicate that small-scale structure haslittle impact on the angular momentum content ofgalaxy-mass haloes, in broad agreement with those ofWang & White (2009), who studied halo formation inHot Dark Matter models, and Bullock et al. (2002)and Chen & Jing (2002), who looked at WDM models.These results show that there are differences in thespatial clustering and merger rates of low-mass haloesbetween our fiducial ΛCDM model and the truncatedmodels – but that they are evident only in the mostextreme truncated models, with M cut in excess of10 h − M ⊙ . As we noted in the introduction, this isinconsistent with astrophysical constraints on the puta-tive WDM particle mass. Therefore, measuring the ef-fect on spatial clustering or the merger rate is likely tobe observationally difficult for realistic values of M cut ,equivalent to our TruncA runs, and so isolating the ef-fect of this small-scale structure would appear to be remarkably difficult to detect, at least in the presentday Universe.However, there are important caveats. The effect maynot be so subtle in the high redshift Universe, during theearliest epoch of galaxy formation, and so we might ex-pect marked differences in the abundances of low-masssatellites between our fiducial ΛCDM model and WDMor truncated models. This may have observable conse-quences for the ages and metallicities of the oldest starsin galaxies (e.g. Frebel 2005), the abundance of metalpoor globular clusters and the assembly of galaxy bulgesand stellar haloes. In addition, there is no compellingreason to expect that the efficiency of galaxy forma-tion will differ between a ΛCDM model and a WDM ortruncated model, and so it may be the case that galaxyformation in a WDM(-like) model is easier to reconcilewith the observed galaxy population than galaxy forma-tion in the fiducial ΛCDM model (see also Menci et al.2012; Benson et al. 2013). We shall return to these ideasin forthcoming work. Acknowledgments
CP thanks the anonymous referee for their thoughtfulreport. This work was supported by ARC DP130100117and by computational resources on the EPIC super-computer at iVEC through the National ComputationalMerit Allocation Scheme. The research presented in thispaper was undertaken as part of the Survey SimulationPipeline (SSimPL; http://ssimpl-universe.tk/ ). REFERENCES
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