Statistics of Substructures in Dark Matter Haloes
aa r X i v : . [ a s t r o - ph . C O ] N ov Mon. Not. R. Astron. Soc. , 1–13 (2011) Printed 16 October 2018 (MN L A TEX style file v2.2)
Statistics of Substructures in Dark Matter Haloes
E. Contini, , ⋆ G. De Lucia, S. Borgani, , , Dipartimento di Astronomia, Universit´a di Trieste, via G.B. Tiepolo 11, I-34131 Trieste,Italy INAF - Astronomical Observatory of Trieste, via G.B. Tiepolo 11, I-34143 Trieste, Italy INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy
16 October 2018
ABSTRACT
We study the amount and distribution of dark matter substructures within dark matterhaloes, using a large set of high-resolution simulations ranging from group size tocluster size haloes, and carried our within a cosmological model consistent with WMAP7-year data. In particular, we study how the measured properties of subhaloes varyas a function of the parent halo mass, the physical properties of the parent halo, andredshift. The fraction of halo mass in substructures increases with increasing mass:it is of the order of 5 per cent for haloes with M ∼ M ⊙ and of the orderof 10 per cent for the most massive haloes in our sample, with M ∼ M ⊙ .There is, however, a very large halo-to-halo scatter that can be explained only inpart by a range of halo physical properties, e.g. concentration. At given halo mass,less concentrated haloes contain significantly larger fractions of mass in substructuresbecause of the reduced strength of tidal disruption. Most of the substructure mass islocated at the outskirts of the parent haloes, in relatively few massive subhaloes. Thismass segregation appears to become stronger at increasing redshift, and should reflectinto a more significant mass segregation of the galaxy population at different cosmicepochs. When haloes are accreted onto larger structures, their mass is significantlyreduced by tidal stripping. Haloes that are more massive at the time of accretion(these should host more luminous galaxies) are brought closer to the centre on shortertime-scales by dynamical friction, and therefore suffer of a more significant stripping.The halo merger rate depends strongly on the environment with substructure in moremassive haloes suffering more important mergers than their counterparts residing inless massive systems. This should translate into a different morphological mix forhaloes of different mass. Key words: cosmology: dark matter - clusters: general - galaxies: evolution - galaxy:formation.
In the currently accepted ΛCDM paradigm for cosmic struc-ture formation, small dark matter haloes form first whilemore massive haloes form later through accretion of dif-fuse matter and mergers between smaller systems. Duringthe last decades, we have witnessed a rapid development ofnumerical algorithms and a significant increase in numeri-cal resolution, that have allowed us to improve our knowl-edge of the formation and evolution of dark matter struc-tures. In particular, the increase in numerical resolutionhas allowed us to overcome the so-called overmerging prob-lem , i.e. the rapid disruption of galaxy-size substructuresin groups and clusters (Klypin et al. 1999, and references ⋆ Email: [email protected] therein). If any, we are now facing the opposite problem,at least on galaxy scales, where many more substructuresthan visible dwarf galaxies are found (Ishiyama et al. 2009;Tikhonov & Klypin 2009, and references therein).According to the two stage theory proposed byWhite & Rees (1978), the physical properties of galaxies aredetermined by cooling and condensation of gas within thepotential wells of dark matter haloes. Therefore, substruc-tures represent the birth-sites of luminous galaxies, and theanalysis of their mass and spatial distribution, as well as oftheir merger and mass accretion histories provide importantinformation about the expected properties of galaxies in theframework of hierarchical galaxy formation models.Nowadays, a wealth of substructures are routinely iden-tified in dissipationless simulations, and their statisticalproperties and evolution have been studied in detail in the c (cid:13) E. Contini et al. past years. The identification of dark matter substructures,or subhaloes , remains a difficult technical task that can beachieved using different algorithms (see e.g. Knebe et al.2011). Each of these has its own advantages and weaknesses,and different criteria for defining the boundaries and mem-bership of substructures are likely leading to systematic dif-ferences between the physical properties of subhaloes iden-tified through different algorithms. However, these might beprobably corrected using simple scaling factors, as suggestedby the fact that different studies find very similar slopesfor the subhalo mass function, i.e. the distribution of sub-structures as a function of their mass. This is one of themost accurately studied properties of dark matter substruc-tures, although it remains unclear if and how it depends onthe parent halo mass. Moore et al. (1999) used one high-resolution simulation of a cluster-size halo and one high-resolution simulation of a galaxy-size halo, and found thatthe latter can be viewed as a scaled version of the former.Later work by De Lucia et al. (2004) used larger samples ofsimulated haloes, but found no clear variation of the sub-halo mass function as a function of the parent halo mass.Such a dependency was later found by Gao et al. (2004) andGao et al. (2011), who showed that the subhalo mass func-tion varies systematically as a function of halo mass and halophysical properties like concentration and formation time.Typically, only about 10 per cent of the total mass of adark matter halo is found in substructures. In addition, theirspatial distribution is found to be anti-biased with respectto that of dark matter (Ghigna et al. 2000; De Lucia et al.2004; Nagai & Kravtsov 2005; Saro et al. 2010). It is unclearif the radial distribution of substructures depends on theparent halo mass. De Lucia et al. (2004) found hints for asteeper radial number density profiles of substructures in lowmass haloes than in high mass haloes. They used, however,a relatively small sample of simulated haloes, that were runwith different codes and numerical parameters. In this study,we will re-address this issue by using a much larger sample ofsimulated haloes, all run with the same code and numericalparameters.Most previous work focusing on dark matter substruc-tures has studied their properties as a function of their present day mass. This quantity cannot be, however, sim-ply related to the luminosity of the galaxies residing in thesubstructures under consideration. Indeed, dark matter sub-structures are very fragile systems that are strongly affectedby tidal stripping (De Lucia et al. 2004; Gao et al. 2004).Since this process affects primarily the outer regions of sub-haloes, and galaxies reside in their inner regions, it is tobe expected that the galaxy luminosity/stellar mass is morestrongly related to the mass of the substructure at the timeof infall (i.e. before becoming a substructure) than at present(Gao et al. 2004; Wang et al. 2006; Vale & Ostriker 2006).In this paper, we will study the evolution of dark matter sub-structures splitting our samples according to different valuesof the mass at infall.In this paper, we take advantage of a large set of N-body simulations covering a wide dynamical range in halomass, and with relatively high-resolution. This will allowus to study how the statistical properties of substructuresvary as a function of halo mass, cosmic epoch, and physicalproperties of the parent halo. The layout of the paper is asfollows: in section 2, we introduce the simulation set and samples used in our study. In section 3, we study how thesubhalo mass function and subhalo spatial distribution varyas a function of halo mass, redshift and concentration. Inthe second part of our paper (section 4), we discuss the massaccretion and merging histories of subhaloes as a functionof their mass, accretion time, and environment. Finally, inSection 5, we discuss our findings and give our conclusions.
Our set of DM haloes is based on ‘zoom-in’ simulations of27 Lagrangian regions extracted around massive dark matterhaloes, originally identified within a low-resolution N-bodycosmological simulation. For a detailed discussion of thissimulation set, we refer to Bonafede et al. (2011, see alsoFabjan et al. 2011). The parent simulation followed 1024 DM particles within of a box of 1 h − Gpc comoving on aside. The adopted cosmological model assumed Ω m = 0 . bar = 0 .
04 for the con-tribution of baryons, H = 72 km s − Mpc − for the present-day Hubble constant, n s = 0 .
96 for the primordial spec-tral index, and σ = 0 . z = 0 within a top-hat sphere of 8 h − Mpc ra-dius. With this parameters choice, the assumed cosmogonyis consistent with constraints derived from seven-year datafrom the Wilkinson Microwave Anisotropy Probe (WMAP7,Komatsu et al. 2011).The selected Lagrangian regions were chosen so that13 of them are centred around the 13 most massiveclusters found in the cosmological volume, all havingvirial mass M ≃ h − M ⊙ . Additional regionswere chosen around clusters in the mass range M ≃ (5 − × h − M ⊙ . Within each Lagrangian region,we increased mass resolution and added the relevanthigh-frequency modes of the power spectrum, using theZoomed Initial Condition (ZIC) technique presented byTormen, Bouchet & White (1997). Outside the regions ofhigh–resolution, particles of mass increasing with distanceare used, so that the computational effort is concentratedon the cluster of interest, while a correct description ofthe large–scale tidal field is preserved. For the simulationsused in this study, the initial conditions have been gener-ated using m DM = 10 h − M ⊙ for DM particle mass in thehigh–resolution regions. This mass resolution is a factor 10better than the value used by Bonafede et al. (2011) andFabjan et al. (2011) to carry out hydrodynamic simulationsfor the same set of haloes.Using an iterative procedure, we have shaped each high–resolution Lagrangian region so that no low–resolution par-ticle ‘contaminates’ the central ‘zoomed in’ halo, out to 5virial radii of the main cluster at z = 0. In our simulations,each high resolution region is sufficiently large to containmore than one interesting massive halo, with no ‘contami-nants’, out to at least one virial radius. Our final sample con-tains 341 haloes with mass larger than 10 h − M ⊙ . We havesplit this sample into 5 different subsamples, as indicated Here we define the virial mass ( M ) as the mass containedwithin the radius R , that encloses a mean density of 200 timesthe critical density of the Universe at the redshift of interest.c (cid:13) , 1–13 tatistics of Substructures in Dark Matter Haloes Table 1.
Our simulation set has been split in five subsamples,according to the halo mass. In the first column, we give the nameof the subsample, while the second column indicates the rangeof M values corresponding to each subsample. The third andfourth columns give the number of haloes and mean number ofsubhaloes (with mass above 2 · h − M ⊙ ) within the virial ra-dius ( R ), respectively.Name Mass range N haloes ¯ N subs S1 > h − M ⊙
13 2943S2 [5-10] × h − M ⊙
15 1693S3 [1-5] × h − M ⊙
25 358S4 [5-10] × h − M ⊙
29 146S5 [1-5] × h − M ⊙
259 40 in Table 1, where we list the number of non-contaminatedhaloes for each sample and the mean number of substruc-tures per halo in each subsample.Simulations have been carried out using the Tree-PMGADGET-3 code. We adopted a Plummer–equivalent soft-ening length for the computation of the gravitational force inthe high–resolution region. This is fixed to ǫ = 2 . h − kpcin physical units at redshift z <
2, and in comoving units athigher redshift. For each simulation, data have been storedat 93 output times between z ∼
60 and z = 0. Dark mat-ter haloes have been identified using a standard friends-of-friends (FOF) algorithm, with a linking length of 0.16in units of the mean inter-particle separation in the high-resolution region. The algorithm SUBFIND (Springel et al.2001) has then been used to decompose each FOF groupinto a set of disjoint substructures, identified as locally over-dense regions in the density field of the background halo. Asin previous work, only substructures which retain at least20 bound particles after a gravitational unbinding proce-dure are considered to be genuine substructures. Given ournumerical resolution, the smallest structure we can resolvehas a mass of M = 2 × h − M ⊙ . To avoid being too closeto the resolution limit of the simulations, we will sometimesconsider only substructures that contain at least 100 parti-cles, i.e. we will adopt a mass limit of 1 × h − M ⊙ . In this section we will consider some basic statistics of thedark matter substructures in our sample. In particular, wewill address the following questions: what is the mass frac-tion in substructures? What is their mass and spatial dis-tribution? And how do these properties vary as a functionof the halo mass, or as a function of other physical proper-ties of the parent haloes? As discussed above, if subhaloesare to be considered the places where galaxies are located,these statistics provide us important information about thestatistical properties of cluster galaxy populations expectedin hierarchical cosmologies.
Previous work has found that only 5 to 10 per cent ofthe halo mass is contained in substructures, with most of it actually contained in relatively few massive subhaloes(Ghigna et al. 1998 2000; Springel et al. 2001; Stoehr et al.2003; Gao et al. 2004; De Lucia et al. 2004).Results for our simulation set are shown in Figure 1.The top left panel shows the cumulative mass fraction insubhaloes above the mass indicated on the x-axis, for thefive samples considered in our study. There is a clear trendfor an increasing mass in substructures for more massivehaloes. For our most massive sample (S1), about ten percent of the halo mass is contained in substructures moremassive than 2 × h − M ⊙ , and approximately ten percent of the mass in substructures is contained in the mostmassive ones. For less massive haloes, the mass fraction insubstructures decreases.Most of the substructures are located outside the cen-tral core of dark matter haloes. In particular, the top rightpanel of Figure 1 shows that the substructure mass fractionis smaller than ∼ ∼ . × r , and increasesto half its total (within r ) value at ∼ . × r . The re-sults shown can be explained by considering that haloes oflarger mass are less concentrated and dynamically youngerthan their less massive counterparts. As we will show below,and as discussed in previous studies, subhaloes are stronglyaffected by dynamical friction and tidal stripping. Less mas-sive haloes assemble earlier than their more massive coun-terparts, i.e. accrete most of the haloes that contribute totheir final mass at early times, so that there was enoughtime to ‘erase’ the structures below the resolution of thesimulation in these systems. In addition, haloes that wereaccreted earlier, and therefore suffered of tidal stripping forlonger times, are preferentially located closer to the centre(see Figure 15 in Gao et al. 2004).For haloes of the same mass, a relatively large range ofconcentrations is possible so that a range of mass fractions isexpected. This is confirmed in the bottom panels of Figure 1where we have considered only haloes in our least massivesample (S5), and split it into three different bins accordingto the halo concentration so as to have the same number ofhaloes for each bin. We approximate the concentration by V max /V , where V max is the maximum circular velocity,which is computed by considering all particles bound to agiven halo, while V = p GM /R . Interestingly, thelowest concentration bin contains substructure mass frac-tions that are, on average, very close to those of our mostmassive samples (S1 in the top panels). This confirms thatthe halo to halo scatter is very large, and that it can beexplained only in part by haloes in the same mass bin cov-ering a range of physical properties. In order to give an ideaof the intrinsic scatter of haloes in the same mass bin, wehave repeated the last point in the top right panel of Fig-ure 1, showing this time the median and the 25th and 75thpercentile of the distributions obtained at R/R = 1.
One of the most basic statistics of the subhalo population isprovided by the subhalo mass function, i.e. the distributionof dark matter substructures as a function of their mass.This has been analysed in many previous studies with theaim to answer the following questions: does the subhalo mass c (cid:13) , 1–13 E. Contini et al.
Figure 1.
Top panels: cumulative mass fraction in substructures as a function of subhalo mass (left) and normalized distance from thehalo centre (right), for the five samples used in this study (different symbols, as indicated in the legend). In the right panel the rightmostsymbols with error bars show the median, 25 th and 75 th percentile of the distributions at R/R = 1. Bottom panels: same as in thetop panels but using only haloes from our sample S5 (the least massive one), and splitting the sample in three different bins accordingto the concentration of the parent haloes. In all panels, symbols connected by lines show the mean values, while error bars show the rmsscatter around the mean. function vary as a function of the parent halo mass? Howdoes it vary as a function of cosmic time? And as a functionof halo properties (e.g. concentration, formation time, etc.)?First studies were based on very small samples of sim-ulated haloes, and claimed the ‘universality’ of the subhalomass function. E.g. Moore et al. (1999) compared the sub-structure mass distribution obtained for one simulated clus-ter of mass similar to that of the Virgo Cluster, and onesimulated galaxy-size halo, and argued that galactic haloescan be considered as ‘scaled versions’ of cluster-size haloes.De Lucia et al. (2004) used a sample of ∼
11 high resolutionresimulations of galaxy clusters together with a simulationof a region with average density. They argued that the sub- halo mass function depends at most weakly on the parenthalo mass and that the (nearly) invariance of the subhalomass function could lie in the physical nature of the dynam-ical balance between two opposite effects: the destruction ofsubstructures due to dynamical friction and tidal strippingon the one hand, and the accretion of new substructures onthe other hand. Contemporary work by Gao et al. (2004)and later work (e.g. Gao et al. 2011) has demonstrated thatthe subhalo mass function does depend on the parent halomass, as well as on the physical properties of the parent halo,in particular its concentration and formation time. We notethat Gao et al. (2004) used a sample of simulated haloesthat was not homogeneous in terms of resolution (typically c (cid:13) , 1–13 tatistics of Substructures in Dark Matter Haloes Figure 2.
Slope of the differential mass function measured forthe different samples considered in this study, at different cosmicepochs (solid line for z = 0, dotted for z = 0 .
5, dashed line for z = 1, and dash-dotted line for z = 2). Error bars are computedas the standard deviation of the slopes measured for each halowithin the sample. For reasons of clarity, a small shift has beenadded to the abscissa. lower than ours), cosmological parameters, and simulationcodes. The sample used in Gao et al. (2011) was insteadbased on a ho homogeneous set of cosmological parame-ters (consistent with WMAP first-year results) and includedsimulations with resolution higher than that of our sample.Their sample, however, did not include very massive haloes( ∼ h − M ⊙ ). It is therefore interesting to re-address thequestions listed above using our simulation sample.In Figure 2, we plot the slope of the differential massfunction obtained by fitting a power law to the mass func-tions of each sub-sample considered in our study. Follow-ing De Lucia et al. (2004), we have restricted the fit by dis-carding the most massive (and rarest) substructures (thosewith mass above 10 h − M ⊙ for the samples S1 and S2,and with mass above 10 . h − M ⊙ for the samples S3, S4and S5). We find that, albeit weakly, the slope of the sub-halo mass function depends on the parent halo mass, andthat there is a weak trend for shallower slopes with increas-ing lookback times. The best fit values we measure varyin the range between ∼ − .
65 and ∼ − .
8, in agreementwith results from previous studies (e.g. Ghigna et al. 2000;De Lucia et al. 2004; Gao et al. 2004). When including themost massive substructures in the fit, we obtain steeperslopes, ranging from ∼ − .
91 and ∼ − .
86 at redshift z = 0,but the trends shown in Figure 2 are not altered significantly.As explained by Gao et al. (2011), the dependence ofthe subhalo mass function on halo mass is a consequenceof the fact that more massive haloes are on average lessconcentrated and dynamically younger than their less mas-sive counterparts. Since the strength of tidal disruption de-pends on halo concentration, and since haloes of a given mass are on average less concentrated at higher redshift, wealso expect that the subhalo mass function depends on time.Figure 3 shows the cumulative subhalo mass function (nor-malized as in Gao et al. 2004) at four different redshifts inthe left panels and for different concentrations in the rightpanels (in these panels, only haloes identified at redshiftzero have been considered). Top and bottom panels referto the haloes in the mass range [1 − × h − M ⊙ and[1 − × h − M ⊙ respectively. We derive the three sub-samples by splitting the range of concentration in order tohave the same number of haloes in each subsample. Resultsshown in Figure 3 confirms previous findings by Gao et al.(2011), and extend them to larger parent haloes masses:haloes at higher redshift have significantly more substruc-tures than those of the same mass at later times. The figuresuggests that there is a significant evolution between z = 0and z ∼ .
5, but it becomes weaker at higher redshifts.We note that for the highest redshift considered, the sub-halo mass function does not significantly differ from thatfound at z ∼
1, but we note that this could be due to poorstatistics. Gao et al. (2011) find a similar trend for haloesof similar mass. At any given cosmic epoch, there is a largehalo-to-halo scatter which is due, at least in part, to internalproperties of the parent halo like concentration, as shown inthe right panels of Figure 3. For the ranges of mass shownin Fig. 3, low concentration haloes host up to an order ofmagnitude more substructures than haloes of the same massbut with higher concentration. The difference between thedifferent concentration bins are larger (and significant) forthe most massive substructures.In order to verify that the results of our analysis arerobust against numerical resolution, we have compared thecumulative sub-halo mass function obtained for the set ofsimulated halos presented here to that obtained for the samehalos simulated at a 10 times lower mass resolution. Wefind that the two distributions agree very well to each other,within the mass range accessible to both resolutions. Thisconfirms that both our simulations and the procedure of haloidentification are numerically converged.
Previous studies (Ghigna et al. 2000; De Lucia et al. 2004;Nagai & Kravtsov 2005; Saro et al. 2010) have shown thatsubhaloes are ‘anti-biased’ relative to the dark matter inthe inner regions of haloes. No significant trend has beenfound as a function of the parent halo mass, with only hintsfor a steeper profiles of subhaloes in low massive haloes(De Lucia et al. 2004).The analysis of our sample of simulated halos confirmsprevious findings that dark matter subhaloes are anti-biasedwith respect to dark matter, with no dependence on parenthalo mass. In fact, there is no physical reason to expect sucha trend. We note that De Lucia et al. (2004), who foundhints for such a correlation, used a smaller sample of simu-lated haloes, that were carried out using different simulationcodes and parameters. In contrast, our simulated haloes areall carried out using the same parameters and simulationcode. c (cid:13) , 1–13 E. Contini et al.
Figure 3.
Cumulative mass functions (CMF) in units of rescaled subhalo mass, and multiplied by M sub /M to take out the dominantmass dependence. Top and bottom panels are for haloes in the mass range [1 − × h − M ⊙ and [1 − × h − M ⊙ , respectively.In the left panel, results are shown for different redshifts (solid line for z=0, dotted line for z=0.5, dashed line for z=1 and dash-dottedline for z=2). In the right panel, only haloes identified at redshift zero have been considered, and they have been split in three bins,according to their concentration. Only subhaloes with more than 100 bound particles have been used to build these functions. Nagai & Kravtsov (2005) find that the anti-bias is muchweaker if subhaloes are selected on the basis of the massthey had at the time of accretion onto their parent halo.We confirm their results in Figure 4, where we show theradial distribution of substructures in our sample S1 (themost massive haloes in our simulation set). The top panelof Figure 4 shows the radial distribution of substructuresselected on the basis of their present day mass, while inthe middle panel the mass of the substructure at the timeof accretion (defined as the last time the halo was identi-fied as a central halo, see below) has been used. The figureshows that, in this case, selecting progressively more mas-sive substructures reduces the anti-bias between subhaloesand dark matter. The bottom panel of Figure 4 shows thatthe same is obtained by discarding substructures that are accreted recently. The two selections tend to pick up haloesthat suffered a stronger dynamical friction (i.e. haloes thatwere more massive at the time of accretion) or that sufferedof dynamical friction for a longer time (haloes that were ac-creted earlier). As a consequence, both selections tend topreferentially discard subhalos at larger radii, thus bringingthe radial distribution of subhaloes closer to that measuredfor dark matter.As shown above (see right panels of Figure 1), mostof the substructure mass is located at the cluster outskirts.De Lucia et al. (2004) showed that this distribution is de-pendent on the subhalo mass, with the most massive sub-structures being located at larger distances from the clustercentre with respect to less massive substructures. In partic-ular, De Lucia et al. (2004) split their subhalo population c (cid:13) , 1–13 tatistics of Substructures in Dark Matter Haloes Figure 4.
Radial distribution of dark matter substructures be-longing to haloes of the sample S1. In the top panel, different linescorrespond to different thresholds in the M sub /M ratio, basedon the present-day subhalo mass. In the middle panel, the sub-halo mass at the time of accretion has been considered, while inthe bottom panel different lines correspond to subhaloes accretedat different times. Figure 5.
Cumulative radial distributions for subhaloes with M sub /M > .
01 (solid line) and M sub /M < .
001 (dot-ted line) from all samples, at different redshifts. On the y-axis,we plot the total mass in subhaloes within a given distance fromthe centre, normalized to the total mass in subhaloes within R ,for each subhaloes population. in two subsamples by choosing a rather arbitrary mass ratiobetween the subhalo mass and the parent halo mass (theychose the value 0.01 for this ratio). Our simulations exhibitthe same trends, but we find that this can be more or less‘significant’ depending on the particular threshold adoptedto split the sample. In Figure 5, we show the radial dis-tribution of substructures with M sub /M > .
01 (solidlines) and M sub /M < .
001 (dashed lines). Our trendsare weaker than those found by De Lucia et al. (2004) atredshift zero, when the same division is adopted. We note,however, that these trends are dominated by the most mas-sive substructure and are, therefore, significantly affectedby low number statistics. Figure 5 also shows that the masssegregation becomes more important at increasing redshift.Considering that haloes of a given mass are less cen-trally concentrated and dynamically younger than theircounterparts at later redshift, the trend found can be ex-plained as follows: the ‘younger’ haloes have massive sub-haloes preferentially in their outer regions because strippinghas not had enough time to strip their outer material andeventually disrupt them. In more dynamically evolved clus-ters (those at present time), stripping has had more timeto operate and to wash out any difference between the twodistributions. In this picture, the balance between dynam-ical friction and stripping on one hand, and the accretionof new subhaloes on the other hand is such that the lattereffect is dominating over the former. This is in agreementwith the results shown above for the evolution of the cumu-lative mass function of substructures, whose normalizationincreases with increasing redshift.We stress that in Figure 5 we are considering subhaloesof different mass at the time they are identified. As dis- c (cid:13) , 1–13 E. Contini et al. cussed in Section 1, this cannot be simply related to themass and/or luminosity of the galaxies. So the trend shownin Figure 5 cannot simply be related a different spatial dis-tribution for galaxies in different luminosity bins, as donefor example in Lin et al. (2004, see their figure 8).
In this section, we study the evolution of substructures as afunction of time, focusing in particular on their mass accre-tion histories and merger histories. In order to obtain theseinformation, we have constructed merger histories for allself-bound haloes in our simulations, following the methodadopted in Springel et al. (2005) and the improvements de-scribed in De Lucia & Blaizot (2007).Briefly, the merger tree is constructed by identifyinga unique descendant for each substructure. For each sub-halo, we find all haloes that contain its particles in thefollowing snapshot, and then count the particles by givinghigher weight to those that are more tightly bound to thehalo under consideration. The halo that contains the largest(weighted) number of its particles is selected as descendant.Next, all the pointers to the progenitors are constructed. Bydefault, the most massive progenitor at each node of thetree is selected as the main progenitor . De Lucia & Blaizot(2007) noted that this can lead to ambiguous selectionswhen, for example, there are two subhaloes of similar mass.In order to avoid occasional failures in the merger tree con-struction algorithm, they modified the definition of the mainprogenitor by selecting the branch that accounts for most ofthe mass of the final system, for the longest time. We haveapplied this modification to our merger trees. In this sec-tion, we consider only substructures that contain at least100 bound particles, and in a few cases, we use particularmass ranges to ease the comparison with the literature.In this section we will also study if the accretion andmerger history of substructures depend on the environment,that we will approximate using the parent halo mass. It isworth stressing, however, that our haloes provide likely abiased sample for this analysis. In fact, excluding the mostmassive sample and some haloes that belong to the sampleS2, all the other haloes reside in the regions surrounding themost massive haloes, which might not represent the ‘typical’environment for halo in the same mass range.
In this section, we use the merger trees constructed forour cluster sample to study the mass accretion historiesof subhaloes of different mass and residing in differentenvironments. Several previous studies (Gao et al. 2004;De Lucia et al. 2004; Warnick et al. 2008) have pointed outthat once haloes are accreted onto larger systems (i.e. theybecome substructures), their mass is significantly reducedby tidal stripping. The longer the substructure spends in amore massive halo, the larger is the destructive effect of tidalstripping. Previous studies have found that the efficiency oftidal stripping is largely independent of the parent halo mass(De Lucia et al. 2004; Gao et al. 2004).We re-address these issues using all substructures re-siding within the virial radius of our haloes, and with mass larger than 10 h − M ⊙ at redshift z = 0 (in our simula-tions, these substructures contain at least 100 particles). Bywalking their merger trees, following the main progenitorbranch, we construct the mass accretion history (MAH) forall of these subhaloes, and record the accretion time ( z accr )as the last time the halo is a central halo, i.e. before it is ac-creted onto a larger structure and becomes a proper subhalo.Our final sample includes 39005 haloes, that we split in twobins of different mass by using either their present day massor their mass at the accretion time. We end up with 33576haloes with mass larger than 10 h − M ⊙ at present (25344when using the mass at the accretion time), and 5429 haloeswith mass lower than the adopted threshold (13661 if the ac-cretion mass is used). In order to analyse the environmentaldependence of the mass accretion history, we consider sep-arately subhaloes residing in our S5 and S1 samples (thesecorrespond to our lowest and largest parent halo mass, re-spectively).The top panels in Figure 6 show the distribution of theaccretion times for the two mass bins considered. Left andright panels correspond to a splitting in mass done on thebasis of the present day mass and of the mass at accre-tion, respectively. When considering the present day mass(left panel), the differences between the two distributionsare small, with only a slightly lower fraction of more mas-sive substructures being accreted very late, and a slightlylarger fraction of substructures in the same mass range be-ing accreted between z ∼ . z ∼
1. A larger differ-ence between the two distribution can be seen when con-sidering the mass at the time of accretion (right panel).Substructures that are less massive at the time of accre-tion have been accreted on average later than their moremassive counterparts. In particular, about 90 per cent ofthe substructures in the least massive bin considered havebeen accreted below redshift 0.5, while only 50 per cent ofthe most massive substructures have been accreted over thesame redshift range. The distribution obtained for the mostmassive substructures is broader, extending up to redshift ∼
2. This is largely a selection effect , due to the fact thatwe are only considering substructures that are still presentat z = 0. Once accreted onto larger systems, substructuresare strongly affected by tidal stripping so that, among thosethat were accreted at early times, only the most massiveones will still retain enough bound particles at present toenter our samples. The less massive substructures that wereaccreted at early times, have been stripped below the res-olution of our simulations and therefore do not show up inthe solid histogram that is shown in the top right panel ofFigure 6.The bottom panels of Figure 6 show the distribution ofthe ratios between present day mass and mass at accretionfor subhaloes of different present day mass (left panel) andfor different mass at accretion (right panel). Less massivesubhaloes, which were accreted on average more recently,lose on average smaller fractions of their mass compared tomore massive subhaloes for which the distribution is skewedto higher values. The difference between these distributionsbecomes more evident when one split the samples according c (cid:13) , 1–13 tatistics of Substructures in Dark Matter Haloes Figure 6.
Left panels: distribution of the accretion times (top panel) and of the fraction of mass loss since accretion (bottom panel)for subhaloes of different mass at present time (different linestyles, as indicated in the legend). Right panels show the same distributionsbut for subhaloes split according to their mass at the accretion time.
Figure 7.
Distribution of mass loss (ratio between the presentday mass and the mass at accretion) for two different accretionranges: solid line for z accr > z accr < to the mass at the time of accretion, as shown in the rightpanel. As explained above, however, this is affected by thefact that many of the least massive substructure will bestripped below the resolution of the simulation at z = 0.We have repeated the analysis done in Fig. 6 for subhaloes in each of the five samples used in our study, and we foundthere is no significant dependency on the environment.Fig. 7 shows that, as expected, substructures accretedearlier suffered significantly more stripping than substruc-tures that were accreted at later times. In particular, about90 per cent of subhaloes accreted at redshift larger than 1have been stripped by more than 80 per cent of their massat accretion. For haloes that have been accreted at redshiftlower than 1, the distribution is much broader, it peaks at ∼ . c (cid:13) , 1–13 E. Contini et al.
Figure 8.
Average mass accretion history for three ranges of accretion times. In the left panels, substructures are split according totheir present day mass, while in the right panels they are split according to their mass at the time of accretion. significant stripping due to tidal interactions with the par-ent halo. Once again, this entails the fact that luminositymust correlate stronger with the subhalo mass computed atthe time of accretion, i.e. before stripping had time to oper-ate. In Fig. 9 we plot the mean MAHs for subhaloes inthe two mass bins considered and for two different ‘envi-ronments’, parametrized as the mass of the parent halo. Inparticular, we consider the samples S5 and S1 (i.e. the leastand the most massive haloes used in our study). Dashedand long-dashed lines show the MAHs for subhaloes in thesample S5 with mass in the range [10 − ] h − M ⊙ andlarger than 10 h − M ⊙ , respectively. Solid and dotted linesshow the MAH for subhaloes in the same mass ranges butfor the sample S1. Here we consider the present day subhalomass. Computing the same plot by adopting the subhalomass at the time of accretion does not alter the results. Wefind that the environment does not significantly influencethe mass accretion history of substructures. In the bottompanel, the long dashed line (corresponding to substructuresmore massive than 10 h − M ⊙ in the sample S5) is likelyaffected by low number statistics. In the same panel a smalldifference can be seen for the less massive substructures that appear to be less stripped in the sample S5 than in S1 (com-pare dashed and solid lines). The difference, however, is notlarge, but this might be affected by the fact that our haloesall reside in the regions surrounding very massive clusters. In recent years, a large body of observational evidencehas been collected that demonstrates that galaxy interac-tions and mergers play an important role in galaxy evo-lution. In particular, numerical simulations have shownthat major mergers between two spiral galaxies of com-parable mass can completely destroy the stellar disk andleave a kinematically hot remnant with structural and kine-matical properties similar to those of elliptical galaxies(Mo, van den Bosch & White 2010, and references therein).Minor mergers and rapid repeated encounters with othergalaxies residing in the same halo ( harassment ; Moore et al.1996; Moore et al. 1998) can induce disk instabilities and/orthe formation of a stellar bar, each of which affects the mor-phology of galaxies falling onto clusters. As galaxy mergersare driven by mergers of the parent dark matter haloes, it c (cid:13) , 1–13 tatistics of Substructures in Dark Matter Haloes Figure 9.
Average mass accretion history for subhaloes in threedifferent ranges of accretion time, as a function of environment.Dashed and long-dashed lines show the MAH for subhaloes in thesample S5 with mass in the range [10 − ] h − M ⊙ and largerthan 10 h − M ⊙ , respectively. Solid and dotted lines show theMAH for subhaloes in the same mass ranges but for the sampleS1. is interesting to analyse in more detail the merger statisticsof dark matter substructures.The mass accretion history discussed in the previoussection does not distinguish between merger events (of dif-ferent mass ratios) and accretion of ‘diffuse material’. Inorder to address this issue, and in particular to study themerger rates of dark matter substructures, we have takenadvantage of the merger trees constructed for our sam-ples. We have selected all subhaloes with mass larger than10 h − M ⊙ at redshift zero, and have followed them backin time by tracing their main progenitor branch, and record-ing all merger events with other structures. In particular, wetake into account only mergers with objects of mass largerthan 10 h − M ⊙ , and mass ratios larger than 5 : 1. We notethat both these values are computed at the time the halo isfor the last time central (the mass of the main progenitorat the time of accretion is considered to compute the massratio).Fig. 10 shows the merging rate for all subhaloes that Figure 10.
Mean number of major mergers as a function of red-shift, for subhaloes in two different ranges of accretion time. Wetake into account only subhaloes with mass M > h − M ⊙ atredshift z = 0 and merger events that include systems with mass M > h − M ⊙ . Figure 11.
Mean number of major mergers as a function of red-shift, for subhaloes in different environments, quantified as themass of their parent halo. As in Fig. 10 we take into account onlysubhaloes with mass M > h − M ⊙ at redshift z = 0 andmerger events that include systems with mass M > h − M ⊙ .c (cid:13) , 1–13 E. Contini et al. satisfy the above conditions. We consider in this plot onlyobjects that experienced at least one merger event. The solidline shows the mean number of mergers for subhaloes thatwere accreted at z < .
5, while the dotted line shows theresulting merger rate for objects accreted between 0 . < z
1. The figure shows that in both cases, the slope of the linesbecome shallower close to the accretion time, i.e. mergersbetween substructures are suppressed because of the largevelocity dispersion of the parent haloes. Interestingly, haloesthat were accreted earlier experience, on average, one moremajor merger than haloes accreted at later times.We repeat the same analysis looking at the merging rateas a function of environment. Fig. 11 shows the cumulativenumber of mergers for subhaloes in our five samples. Themean number of mergers increases as a function of the parenthalo mass, although subhaloes in the sample S4 experienceon average fewer mergers than subhaloes in the sample S5.This is not surprising since subhaloes in the surroundings ofmore massive haloes have a larger probability to merge withother structures.
We have used a large set of high-resolution simulated haloesto analyse the statistics of subhaloes in dark matter haloes,and their dependency as a function of the parent halo massand physical properties of the parent halo. While some ofthe results discussed in this study confirm results from pre-vious studies, it is the first time that a systematic analysisof the properties and evolution of dark matter substructuresis carried out using a large simulation set carried out usingthe same cosmological parameters and simulation code. Ourmain results can be summarized as follows:(i) More massive haloes contain increasing fractions ofmass in subhaloes. This does not exceed ∼
10 per cent ofthe total mass, in agreement with previous studies. There is,however, a very large halo-to-halo scatter that can be par-tially explained by a range of halo physical properties, e.g.the concentration. Indeed, in more concentrated haloes sub-structures suffer of a stronger tidal stripping so that they arecharacterized by lower fractions of mass in substructures.(ii) We find that the subhalo mass function depends weaklyon the parent halo mass and on redshift. This can be ex-plained by considering that haloes of larger mass are lessconcentrated and dynamically younger than their less mas-sive counterparts, and that haloes of a given mass are onaverage less concentrated at higher redshift. Our findingsconfirm results from previous studies (Gao et al. 2011), andextend them to larger halo masses.(iii) As shown in previous work (e.g. Ghigna et al. 1998;De Lucia et al. 2004), subhaloes are anti-biased with respectto the dark matter in the inner regions of haloes. The anti-bias is considerably reduced once subhaloes are selected onthe basis of their mass at the time of accretion, or neglect-ing those that were accreted at later times. We also findthat the spatial distribution of subhaloes does not dependsignificantly on halo mass, as suggested in previous work byDe Lucia et al. (2004). The most massive substructures arelocated at the outskirts of haloes and this mass segregationis more important at higher redshift. (iv) Once accreted onto larger systems, haloes are stronglyaffected by tidal stripping. The strength of this stripping ap-pears to depend on the mass of the accreting substructures:those that are more massive at the time of accretion tend tobe stripped by larger fractions of their initial mass.(v) Mergers between substructures are rare events. Follow-ing the merger trees of substructures, however, we find thatthey have suffered in the past about 4-5 important (massratio 1:5) mergers. As expected, the number of mergers ex-perienced depends on the environment: subhaloes in moremassive systems have experienced more mergers than thoseof similar mass residing in less massive haloes.Dark matter substructures mark the sites where lumi-nous satellites are expected to be found, so their evolutionand properties do provide important information on thegalaxy population that forms in hierarchical models. As dis-cussed in previous studies, however, because of the strongtidal stripping suffered by haloes falling onto larger struc-tures, it is not possible to simply correlate the populationof subhaloes identified at a given cosmic epoch to that ofthe corresponding galaxies. The galaxy luminosity/stellarmass is expected to be more strongly related to the massof the substructure at the time of infall and, depending onthe resolution of the simulations, there might be a significantfraction of the galaxy population that cannot be traced withdark matter substructures because they have been strippedbelow the resolution limit of the simulation (the ‘orphan’galaxies - see for example Wang et al. 2006).Nevertheless, our results do provide indications aboutthe properties of the galaxy populations predicted by hierar-chical models. Tidal stripping is largely independent of theenvironment (we have parametrized this as the parent halomass), while the accretion rates of new subhaloes increasesat increasing redshift. The nearly invariance of the sub-halo mass function results from the balance between thesetwo physical processes. If the amount of dark matter sub-structures is tracing the fraction of recently infallen galax-ies, the fraction of star forming galaxies is expected to in-crease with increasing redshift (the ‘Butcher-Oemler’ effect,Butcher & Oemler 1978, Kauffmann 1995). In addition, ourfindings suggest that stronger mass segregation should befound with increasing redshift.There is a large halo-to-halo scatter that can be onlypartially explained by a wide range of physical properties.This is expected to translate into a large scatter in e.g. thefraction of passive galaxies for haloes of the same mass,with more concentrated haloes hosting larger fraction ofred/passive galaxies. Finally, there is an obvious merger biasthat is expected to translate into a different morphologicalmix for haloes of different mass. In future work, we plan tocarry out a more direct comparison with observational dataat different cosmic times, by applying detailed semi-analyticmodel to the merger trees extracted from our simulations.
ACKNOWLEDGEMENTS
We thank the anonymous referee for constructive commentsthat helped improving the presentation of the results. ECand GDL acknowledge financial support from the EuropeanResearch Council under the European Community’s Sev-enth Framework Programme (FP7/2007-2013)/ERC grant c (cid:13) , 1–13 tatistics of Substructures in Dark Matter Haloes agreement n. 202781. This work has been supported by thePRIN-INAF 2009 Grant “Towards an Italian Network forComputational Cosmology” and by the PD51 INFN grant.Simulations have been carried out at the CINECA NationalSupercomputing Centre, with CPU time allocated throughan ISCRA project and an agreement between CINECA andUniversity of Trieste. We acknowledge partial support by theEuropean Commissions FP7 Marie Curie Initial TrainingNetwork CosmoComp (PITN-GA-2009-238356). We thankA. Bonafede and K. Dolag for their help with the initialconditions of the simulations used in this study.This paper has been typeset from a TEX/ L A TEX file preparedby the author.
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