Serb. Astron. J. � 190 (2015), 11 - 23 UDC 524.7–857 : 524.8–17DOI: 10.2298/SAJ1590011M Original scientific paper

HALO STATISTICS ANALYSIS WITHIN MEDIUMVOLUME COSMOLOGICAL N-BODY SIMULATION

N. Martinovic

Astronomical Observatory, Volgina 7, 11060 Belgrade 38, SerbiaE–mail: nmartinovic@aob.rs

(Received: August 19, 2014; Accepted: January 9, 2015)

SUMMARY: In this paper we present halo statistics analysis of a ΛCDM N-body cosmological simulation (from first halo formation until z = 0). We studymean major merger rate as a function of time, where for time we consider bothper redshift and per Gyr dependence. For latter we find that it scales as thewell known power law (1 + z)n for which we obtain n = 2.4. The halo massfunction and halo growth function are derived and compared both with analyticaland empirical fits. We analyse halo growth through out entire simulation, making itpossible to continuously monitor evolution of halo number density within given massranges. The halo formation redshift is studied exploring possibility for a new simplepreliminary analysis during the simulation run. Visualization of the simulation isportrayed as well. At redshifts z = 0− 7 halos from simulation have good statistics

for further analysis especially in mass range of 1011 − 1014 M�/h.

Key words. dark matter – galaxies: halos – large-scale structure of Universe –methods: numerical

1. INTRODUCTION

Current dominant cosmological paradigm isΛCDM - cold dark matter coupled with dark en-ergy (Λ) (see e.g. Planck Collaboration 2013, La-hav and Liddle 2014, etc). Within that paradigm(as within every other cosmological paradigm) one ofthe major assignments is to determine formation andevolution of large scale structures. Although thereis a tendency to find analytical solutions, reachingthem is still beyond the scope of current apparatus.Thus, numerical simulations are one of the essentialtools for probing structure formation (Springel et al.2006).

From the ingenious work of Holmberg in 1941(galaxy collision simulated with light bulbs) and pio-neering simulations by von Hoerner (1963), Aarseth(1963), Peebles (1970) and White (1976), rise in com-

puter power has driven advances in computationalastrophysics (Dehnen and Read 2011). At presenttime, together with deep field observations and largedata surveys, cosmological simulations are irreplace-able part of new era of precision cosmology. They areused for large scale structure evolution (Millennium- Springel et al 2005, Millennium II - Boylan-Kolchinet al. 2009, Teyssier et al. 2009, Kim et al. 2011,Millennium XXL - Angulo et al. 2012, etc.), near-field cosmology research (CLUES project - for exam-ple: Gottloeber et al. 2010), and for galaxy evolution(ILLUSTRIS - Vogelsberger et al. 2014). Increasingmass resolution makes it possible to study evolutionof halos in greater details as it resolves morphologyand adds constraints to what simulations need to re-produce to match observations. Statistics obtainedfrom connection of observations, theory, and simula-tions will be used to test our simulation as well.

11

N. MARTINOVIC

Most of matter in the universe is dark andis initially smoothly distributed (White 1994). Fromsimulations we see that from this quasi smooth distri-bution, evolve array of sheets, filaments and clustersnamed cosmic web (Bond et al. 1996). Constituentsof these structures are in the form of dark matter ha-los: virialized dense clumps of dark matter (Coorayand Sheth 2002). Considering that halos representpotential wells into which matter falls in, it is as-sumed that galaxies form and evolve within them(White and Rees 1978). More interestingly, under-standing evolution of underlying dark matter densityfield represented by dark matter halos can help usto interpret clustering of galaxies (Kauffmann et al.1999). Of course, these halos and their propertiesare result of numerical simulations.

There are two ways we can study halos: wecan study distribution of halos, and we can studytheir internal structure (Harker et al. 2006). Distri-bution, which will be of more interest to us in thispaper, is an important tracer for evolution of halosand, thus, structure growth (e.g. Lukic 2008). Fur-thermore, within numerical simulations we are givenan opportunity to study assembly and growth of thehalos. By following temporal evolution of the haloand their statistical characteristics (growth, cluster-ing, formation times, etc), we can expand our knowl-edge on evolution of galaxies that reside within them.Structure in which we follow evolution of halos fromsmall clumps to large objects is referenced to as themerger tree (Kauffmann and White 1993, Lacey andCole 1993.).

One of the widespread halo distribution anal-ysis is the halo mass function, precisely due to itsimportance in connection between the dark halo andgalaxy properties. Also it is a very useful tool foranalysis and testing of numerical simulations. It issensitive to cosmological parameters (Lukic 2008).With it, for example, it can be seen that for WDM(warm dark matter) cosmologies it is predicted thatthere are fewer halos of lower mass than what is pre-dicted with the CDM (cold dark matter) counterpart(Colin et al 2000, Angulo et al. 2013).

In this paper we will present the medium vol-ume pure cosmological simulation that we have per-formed. Through halo catalogues and merger tree wewill study general halo characteristics. Apart fromconsiderations of major merger rate from simulation(Gottloeber et al. 2001, Fakhouri and Ma 2008, An-gulo et al. 2009, Wetzel et al. 2009, Genel et al.2009, Fakhouri et al. 2010, Hopkins et al. 2010), wewill concentrate on the formation redshift of halosshowing tendency, as predicted by bottom-up modeof structure formation, toward massive halos beingformed later in the life of Universe. We will addressa possibility for the new approach on using a simpleformation redshift analysis (hence requiring less com-puting resources) as a mean for preliminary analysisof halo formation during the simulation run.

We will compare halo mass function of our ha-los with both analytical and empirical fits, showingevolution of understanding of halo formation, start-ing from spherical collapse of Press and Schechter(1974), through elliptical correction of Sheth, Mo

and Tormen (2001), continuing with the refinementdone by numerical simulations of Warren et al.(2006) and Angulo et al. (2012).

Alternative path for general halo distributionand characteristics analysis will be presented in theform of halo growth function (Heitmann et al. 2006).We derive the halo growth function both for all therelated analytical and empirical fits and for our sim-ulation, thus allowing us to easily analyze the evolu-tion of halo number density with a very good timeresolution within the used mass bins.

We will show that the analysis of our halo dis-tribution characteristics is in good agreement withanalytical fits and other simulations. This simulationwill be used as a platform for subsequent research.

This paper is organized as follows. In Section2. the initial conditions of our simulation are pre-sented. In Section 3. the attributes of used computa-tional resources can be found. The software used foracquiring halo catalogues and creating merger treeof our simulation is presented in Section 4. Imple-mented analysis is presented in Section 5 with fo-cus on the mass function in subsection 5.1 and halogrowth function in subsection 5.2. Results are de-tailed in Section 6 with a short overview of visual-ization of halos. In Section 7 we engage in discussionand at the end of that section, short concluding re-marks are presented.

Convention used for presenting Hubble con-stant in this paper is: H = 100 h km

s Mpc .

2. INITIAL CONDITIONS

For initial conditions we have used Las Damascosmology (McBride et al. 2009) which assumesflat space and cosmological parameters with values:Ωm = 0.25, ΩΛ = 0.75, Ωb = 0.04, h = 0.7. For theinitial power spectrum, code CMBfast (Seljak andZaldarriaga 1996) was used, with power-law indexns = 1 for primordial power spectrum and σ8 = 0.8,which is the mean linear mass fluctuation in spheresof radius of 8 Mpc/h extrapolated at z = 0.

The linear expansion part of the simulationwas computed from the perturbed density field us-ing the 2nd order Lagrangian perturbation theory(Crocce et al. 2006) up to z = 599. The non-linear part of the simulation was calculated usingthe publicly available, highly parallelized, Tree-PMN-body code GADGET2 (Springel 2005) that usesthe hierarchical tree for calculations of short-rangeforces and particle-mesh algorithm for calculationsof long-range forces. The symulation was run up tothe present time (a = 1) in a periodic box with sizeof 130 Mpc/h, employing 5123 dark matter particlesand a comoving softening length of 8 kpc. Particlemass resolution is 1.14 ∗ 109 M�/h.

Large box suppressing finite volume artifactsand high initial redshift of the simulation provide ac-curate results for acquiring mass function (Lukic etal. 2007). Generally, both things are tuned towardreduction of systematic effects.

12

COSMOLOGICAL SIMULATION

Fig. 1. False color visualization of final snapshot(z ∼ 0). All particles are collapsed along one axis,projecting 3D box onto 2D representation. Brightercolors mark denser regions.

Simulation was run with a grid structurethat leaves possibility to zoom in resimulations upto mass resolution of 20483. In total there are84 snapshots taken during the simulation, start-ing with 6 snapshots between redshifts 599 and 9(z = [499, 249, 149, 99, 49, 19]) and the rest from theredshift 9 till the present time in the multiples of1.0304106 (in the scale factor space). Each snap-shot is approximately 3.5 GB and all are availablefor further analysis.1 Visualization of final snapshotis given in Fig. 1.

3. COMPUTING RESOURCES

Simulation was executed in PARADOX Clus-ter at Scientific Computing Laboratory of Instituteof Physics Belgrade. Cluster consists of 89 workernodes powered by 2 x quad core Xeon E5345 @ 2.33GHz and each with 8GB of RAM. Computing nodesare interconnected with the Gigabit Ethernet net-work. Total available storage is up to 50TB. Forthe simulation we have used 256 cores running about39000 hours in total computing time (152 hours perprocessor) with 34000 timesteps. Adaptive time-stepcalculations were used when necessary for individualparticles.

4. HALO CATALOGUESAND MERGER TREE

Analyses involving halos require that we firstidentify a halo, and then determine its mass (Lukicet al. 2007)

The formalism for identification of halos wasfirst introduced by Press and Schechter (1974) byassuming spherical collapse. They identified densitypeaks in matter distribution and then added parti-cles in layers around the center of the peaks until themean density within the particular sphere reachesthe theoretically derived density threshold.

Another approach, called friends-of-friends,was developed by Davies et al. (1985). They de-fined the halo by assuming that particles are boundtogether if each of them is within a predefined linkinglength from at least one other particle in that group.

For creation of the halo catalogue we haveused the halo finding software named ROCKSTAR(Behroozi et al. 2013a). Its primary focus is onconsistent accuracy over many timesteps. It worksby combining friends-of-friends (FOF), phase-space,and spherical overdensity analysis in locating halos.The analysis is started by classical 3D FOF algo-rithm (with large linking length, b = 0.28), and sub-sequent 6D iterations where the FOF hierarchy isestablished (by choosing that 70% of the group’s par-ticles are linked iteratively in subgroups).

From those substructures halos are formedby placing seeds in deepest substructures in hierar-chy, after which particles are recursively assigned tothe closest seeds based on their phase-space vicin-ity. This is repeated for all levels of the hierarchy.Then it uses the information from previous snapshotsto establish halo/subhalo connections. Tree codeis used to find the halo density peaks after whichit unbounds the non-virialized particles. Unbindingis done by calculating full particle potentials afterwhich halo masses and spherical overdensity are cal-culated following the rest of the halo properties aswell. The virial overdensity is defined by using thedefinition of ρvir from Bryan and Norman (1998),which corresponds to 360 times the average matteroverdensity (which is ρm = Ωm ρcrit). So, withinthe virial radius of the halo, the density of matter isequal or higher than the virial overdensity.

We have used minimum of 100 particles as alower limit for a halo, with the FOF linking lengthset b = 0.28 and the FOF linking fraction set to 0.7,both recommended settings. With such limits thefirst structures were discerned by ROCKSTAR onsnapshot which corresponds to redshift 9 (scale fac-tor 0.1). Total number of found halos is over 66000at redshift 0.

Merger tree was assembled using ConsistentMerger Tree (Behroozi et al. 2013b), a softwarepackage that is complementary with the ROCK-STAR halo finder and which is designed so thatfor halo properties, dynamical consistency is ensuredacross timesteps.

1Contact author for the full free and open access to snapshots, and/or halo catalogue, etc.

13

N. MARTINOVIC

If one or more particles of a halo identified ina snapshot become a gravitationally bound part of asingle halo in a following snapshot then such an inter-action is declared a merger. Interacting halos are de-clared progenitors and a resulting halo is called a de-scendant. Mergers are considered to be major whenthe descendant halo has progenitors whose mass ra-tio is > 0.3. Both halo and subhalo mergers areconsidered.

5. ANALYSIS

Apart from allowing us a simple analysis suchas total number of halos through time, halo cata-logues and merger tree can be used for analysis ofmany other characteristics of cosmological simula-tions.

One of the first analysis was done by deter-mining the total number density of halos in our sim-ulation across all redshifts. From our halo cataloguewe simply extract the number of halos per redshiftand divide it by volume. This will give us a briefview of the first appearance of halos and change intheir total number.

As major mergers are the essential part ofmany important events in the Universe, they are usu-ally the focus of analysis. We defined them as merg-ers where the ratio of halo masses involved (that arecoalescing) is above 0.3. We use the merger tree ofour simulation to derive the number of major mergersover number of all halos of same population both perunit of Gyr and per unit of redshift independently.We also use the merger tree to extract the number ofmajor merger events and trace their number densitythrough our simulation time. Both halo and subhalomergers are taken into consideration.

Another interesting thing that can be dis-cerned from the merger tree is the mean formationtime of halos at redshift 0 (Lacey and Cole 1993) incertain mass bins since we have the information onhow the halos evolve through time. It is derived bybinning halos according to mass on redshift 0, andthen searching for the mean redshift when the mainprogenitor mass was half of the mass at the final red-shift.

We have performed a complementary analysysof identification of the highest mass halo within onesnapshot which effectively identify when the first ha-los of certain mass appear in our simulation, andit points to possible significant merger events. Thisanalysis can be done for each snapshot as it is re-trieved, thus revealing where more extensive analysisshould be performed.

These analyses paint the picture of structureformation by exploring the halo evolution. For fur-ther halo statistics we will turn to the halo massfunction analysis and halo growth function.

5.1. Mass Function

The halo mass function is defined as numberof halos per volume of space, per unit of mass. Wetreat the mass function as represented in Lukic et al.(2007). Usually halos are binned together according

to mass or mass logarithms and for each bin (comov-ing) the number density is determined, in our case,we have chosen the latter case. Bins are selected insuch a way that they contain a significant popula-tion of halos, thus reducing the shot noise due to lowpopulation in each bin.

In the case of our halo mass function, halobins of width Δ log M = 0.5 dex were chosen, wherefrom hereafter dex = Δ log (M / (M�/h)), and eachdata point represents the center of its respective massbin.

So, for our mass logarithms, the mass functioncan be represented as:

F (M, z) =dn

d log M. (1)

Apart from calculating the halo mass function fromsimulations, we can estimate how it would look from:

F (M, z)anly =ρ0

Mf(σ)| d lnσ

d log M|; (2)

where f(σ) represents various analytical and empir-ical solutions for the mass function, σ is the massvariance of linear density field and ρ0 is the meandensity of the Universe. Generally, f(σ) is considerednot directly linked to the redshift, that is, the red-shift dependence is gained through σ(M, z), where itis achieved through the growth factor (Murray et al.2013). The mass variance is also sensitive to differentcosmologies.

For calculation of mass variance, the powerspectrum, transfer function, top hat function thesame method that is described in Murray et al.(2013) was used. The transfer function was calcu-lated by using the CMBfast code. As showed inMurray et al. (2013), the mass variance was calcu-lated using the top-hat filter, the linear power spec-trum is used and the redshift dependence of fittingfunctions was achieved through the mass variance, ormore precisely, through its connection to the growthfactor which is governed by the redshift dependence.

Historically, the first analytical model of themass function was developed by Press and Schechter(1974) (PS). Although finer fits were developed inthe meantime, especially for higher redshifts, the PSformalism has a good agreement in lower redshifts(z → 0) and it is a good starting point. It assumesthat the entire mass is in halos and it considers apurely spherical collapse. It was shown later thatit generally overestimates the number of lower masshalos and underestimates the number of high masshalos. Function is given as:

fPS(σ) =

√2π

δc

σexp

(− δ2

c

2σ2

), (3)

where δc = 1.686 is the critical value of density per-turbation (overdensity) of a dark matter sphere afterwhich it collapses into a virialized halo, and σ is themass variance, same as earlier.

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COSMOLOGICAL SIMULATION

Improvement over PS was done by Sheth, Moand Tormen (2001) (SMT, hereafter). They ex-tended the PS fit with an elliptical collapse model.Basically, they theoretically derived a fit that showedless discrepancy from mass functions calculated di-rectly from numerical simulations. Their function isgiven as:

fSMT(σ) = A

√2a

π

[1 +

( σ2

aδ2c

)p]

δc

σexp

[− aδ2

c

2σ2

],

(4)

where A = 0.3222, a = 0.707 and p = 0.3, andδc = 1.686. Interestingly, if we set: A = 0.5, a = 1and p = 0 we would get PS from the above formula.

Another very interesting halo mass functionfit was developed by Warren et al. (2006) (Warren,hereafter). We focused on it, apart from PS andSMT, because Warren computed their fit by using alarge number of cosmological simulations using thesame code and the same cosmology with statisticsspanning same mass range as we did in our simula-tion. Warren function is represented as:

fW(σ) = 0.7234 (σ−1.625 + 0.2538) exp[−1.1982

σ2

].

(5)

Last halo mass function fit we used for compari-son was derived by Angulo et al. (2012). It isderived from unprecedented cosmological simulationthat uses 67203 particles representing dark matterstructures in a periodic box of size 3 Gpc/h, thusallowing within the same simulation, to span even agreater mass range than Warren’s fit. It is given as:

fA(σ) = A

[( b

σ

)a

+ 1]

exp[− c

σ2

], (6)

where A = 0.201, a = 1.7, b = 2.08, and c = 1.172.

5.2. Halo growth function

Another path of analyzing the halo statisticsover the course of simulation was done by analyzingthe halo growth function (Heitmann et al. 2006).It is defined as the number of halos in given massbins throughout the simulation. From it we can ana-lyze how mass is distributed over halos through time.For that we extend our halo growth function over allavailable snapshots, giving us a finer time resolutionand making the analysis more continuous. We areusing it to focus more on how the abundance of ha-los within one mass bin is evolving through time andto get an insight into changing the ratio of these binsat different redshifts.

For construction of a halo growth function wehave chosen wide logarithmic mass bins consideringthat population of each bin varies across redshifts.Wide mass bins enable us to keep the number of lowpopulated data points to minimum, considering thatwithin a short period of time, the population num-ber of each bin rises to a significant level. On the

other hand, we still have enough information fromthese bins to analyze the evolution of number den-sity of low and high mass halos throughout the sim-ulation. Four bins covering masses from 1011M�/hto 1015M�/h were selected.

We derive an estimation of the halo growthfunction by integrating the mass function for givenmass bins over redshifts:

n(M1, M2, z) =∫ M2

M1

F (M, z) d logM (7)

here n(M1, M2, z) is the number density of halos ina mass bin limited by M1, M2 on a redshift z for agiven fit f(σ) on which F (M, z) is dependent, as seenin Eq. (1).

6. RESULTS

The number density of halos across all massesas a function of redshift in our simulation is repre-sented in Fig. 2. Each point represents the totalnumber density of dark matter halos in one snap-shot. As the simulation progresses and dark matterstarts to form more dark matter halos we see a risein their number density. Having in mind that ourhalo catalogue has halos above 1011M�/h, we seethat the total number density is starting to decreaseas the simulation approaches lower redshifts (towardz ∼ 0).

0 2 4 6 8 100.00

0.01

0.02

0.03

0.04

0 2 4 6 8 10Redshift

0.00

0.01

0.02

0.03

0.04

Tot

al h

alo

conc

entr

atio

n −

n [(

Mpc

/h)−

3 ]

Fig. 2. The total number density of identifiedhalos in the simulation as a function of redshifts.We see gradual formation of dark matter halos andsubsequent reduction of their number density towardz ∼ 0.

Major mergers, presented in Fig. 3, are con-sidered as another interesting analysis.

Top of Fig. 3 is a graph with two distinctplots, both plotted as a function of redshift, wherewe track merger rate of halos of mass greater than1012M�/h per unit of Gyr, dNmerge/dt (blue x); andper unit of redshift, dNmerge/dz (red diamonds). Ha-los with mass greater than 1012M�/h are chosen be-cause this enables us to compare our results withwide a range of previous results. As seen, both plots

15

N. MARTINOVIC

cover the same halos over the same time, but the dif-ference is, as noted in Fakhouri and Ma (2008), thatthe mean merger rate per Gyr increases with redshiftcontrary to the mean merger rate per redshift whichis constant. This is due to the cosmological factor,dt/dz, spanning shorter times at higher redshifts.

The black line is a power law fit (Genel et al.2009) showing that mean marger rate per Gyr in-creases with redshift as ∝ (1 + z)n, where we findn = 2.4 for the plot shown. Black asterisks rep-resent results from Gottloeber et al. (2001), wherethey consider the major merger rate for primary ha-los only. Two nearly horizontal dotted lines are re-sults for the major merger rate from the Millenniumsimulation from Fakhouri and Ma (2008, brown, up-per line) and from Angulo et al. (2009, green, lowerline).

0 1 2 3 4 5 60

1

2

3

4

5

6

7

0 1 2 3 4 5 6Redshift

0

1

2

3

4

5

6

7

Maj

or m

erge

rs −

n [1

0−5 (

Mpc

/h)−

3 ]

Fig. 3. Top: Mean merger rate as a function ofredshift. Red diamonds are per redshift dependence,results of Fakhouri and Ma (2008) are presented asbrown dotted line and Angulo et al. (2009) resultsare presented as green dotted line. Blue crosses areper Gyr dependence, overplotted are black asteriskswhich represent results of Gottloeber et al. 2001. Ascan be seen, the mean major merger per redshift isnearly constant, while per Gyr changes as ∝ (1+z)n

with n = 2.4 (black line). Error bars are Poissonstandard errors. Error bars for the right-most pointsare large and are omitted. Bottom: Number den-sity of major mergers as a function of redshift. De-rived by counting the number of a new major mergerevents from each snapshot normalized to the numberof identified halos in the snapshot. A clear peak isseen around z ∼ 2 which coincides with the maxi-mum quasar abundance.

The bottom graph in Fig. 3 is a total num-ber density of major mergers in our simulation asa function of redshift. Points represent the majormerger number density from each snapshot. Theyhave been derived by counting the number of a newmajor merger events for each snapshot (that is de-scendants which had progenitors merging from pre-vious snapshot) normalized to the total number ofidentified halos per snapshot. It can be noted thatthe number density of major mergers has the sametrends as the total number density of identified ha-los. We see the rise in number density as more darkmatter halos are created and become dynamicallyinvolved and its subsequent receding when the darkmatter halo pairs with similar mass become sparse.Calculated values of plotted number density are obvi-ously smaller due to lesser frequency of major merg-ers per volume (in comparison to the total halo num-ber density, for example), but a clear peak can beseen as well.

In Fig. 4 we see two plots in connection tothe formation time. The first one, labeled by red di-amonds, displays the redshift of the first appearanceof halos as a function of halo mass. We introduceit as a new way for a simple preliminary analysisof halo formation during cosmological runs. It is asimple visualization of the highest mass halos fromeach snapshot. We see practically a linear trend ofconstantly higher mass halos as the simulation wasprogressing. Points that are spread horizontally, i.e.the ones that point to the high mass gain for haloat the next snapshot, suggest a significant mergerevent, making this analysis inexpensive precursor ofthe merger tree. Also, it should be considered thatthis plot considers only a single, the most massive,halo from each snapshot.

11 12 13 14 150

2

4

6

8

10

11 12 13 14 15Mass of halo (log)

0

2

4

6

8

10

Red

shift

of f

irst a

ppea

ranc

e of

hal

o

Highest mass haloMean formation redshift

Highest mass haloMean formation redshift

Harker et al. 2006

Harker et al. 2006

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Red

shift

of m

ean

form

atio

n tim

e

Fig. 4. Redshift of the first appearance of ha-los as a function of halo mass (red diamond, lefty-axis). Within this plot we visualize mass of thehighest mass halos in each snapshot of simulation.The other curve is the mean formation time redshiftas a function of halo mass bins (blue triangle, rightaxis). From both we see that bigger halos identified atz ∼ 0 tend to form earlier in the simulation. Blackdotted line represents results of mean formation red-shift from the Millenium simulation as presented inHarker et al. (2006). Error bars are Poisson stan-dard errors. At largest halo masses, we omit last twomass bins because of the numerical noise due to thesmall halo population at these masses.

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COSMOLOGICAL SIMULATION

The second plot (blue triangles) is the meanformation time redshift as a function of halo massbins. It represents the mean value of redshift atwhich the most massive progenitors of a halo fromredshift 0 have half of their mass. In fact, that givesus average redshift at which all the halos of certainmass bin will form. The mass bin covers ranges from1011 to 1015 M�/h, has width of 0.2 dex and is rep-resented by blue triangles positioned at the centerof the bin range. From this we can see that it islinear (apart for bins with the highest mass), indi-

cating that massive halos from z ∼ 0 tend to formlater in the simulation (closer to z ∼ 0). It should benoted that the highest mass range in the mean for-mation time plot is stricken by the low populationof those bins heavily affecting averaging. The dottedline in the plot represents the empirical fit for themean formation time of halos from the Millenniumsimulation as seen in Harker et al. (2006). It canbe noted that the mean formation times for halos inboth simulations are in a good agreement.

11 12 13 14 15 11 12 13 14 15Mass of halos - MO • /h [log]

10-5

10-4

10-3

10-2

10-1

n [(

Mpc

/h)-3

] (lo

g)

z - 3

z - 3

10 11 12 13 14 Mass of halos - MO • /h [log]

10-5

10-4

10-3

10-2

10-1

n [(

Mpc

/h)-3

] (lo

g)z - 2

z - 2

10-5

10-4

10-3

10-2

10-1

n [(

Mpc

/h)-3

] (lo

g)

11 12 13 14 15

Mass of halos - MO • /h [log]

z - 1

z - 1

10-5

10-4

10-3

10-2

10-1

n [(

Mpc

/h)-3

] (lo

g)

10 11 12 13 14

Mass of halos - MO • /h [log]

Warren et al.Press & SchechterAnguloSheth-Mo-Tormen

Warren et al.Press & SchechterAnguloSheth-Mo-Tormen

z - 0

z - 0

Fig. 5. Plot of mass function: the number density of halos as a function of halo mass (binned), plotted for4 different redshifts (z ∼ 0, 1, 2 and 3). Squares represent centers of mass bins of width 0.5 dex. Results fromsimulation are compared with 4 different fits, 2 analytical: Press-Schechter (1974); Sheth, Mo and Tormen(2001); and 2 empirical: Warren et al. (2006) and Angulo et al. (2012). We see a good agreement betweenthe simulation and most of the fits within each redshift, especially z ∼ 1 and z ∼ 2. The Press-Schechter fitshows a significant divergence at higher redshifts. Fewer number of squares in the z ∼ 4 plot is a consequenceof massive halos not yet formed. Error bars are the Poisson standard errors.

17

N. MARTINOVIC

After computing the halo mass function fromour simulation as represented by Eq. (1), we compareit to several known analytical and empirical solutionsof interest. In Fig. 5 we can see the halo mass func-tion calculated from the executed simulation at 4 dif-ferent redshifts, (z � 0, 1, 2 and 3) with analytical fitsoverplotted as lines. Analytical and empirical fits cancover a wider mass range, but we have restricted it to1010 to 1015 M�/h to coincide with the mass rangeavailable from the simulation. Squares represent cen-ters of mass bins used to bin the halo masses foundin our halo catalogue, where bins have width 0.5 dex.As can be seen, there are less squares available athigher redshifts, because higher mass dark matterhalos have not formed yet at those redshifts. It canbe noticed that there is a good agreement betweenfits at all redshifts except for the Press-Schechter fitwhich diverges both at high and low mass ranges andacross all redshifts. A good agreement between thesimulation and other fits (except PS as noted) is alsonoticeable with only a slight disagreement betweenthe highest mass bins at redshift 0, but as mentionedearlier, that can be due to the lower number of haloswithin the bin.

In Fig. 6 the halo growth function for oursimulation is given against analytical and empiricalexpectations. It represents the number density of ha-los within a certain bin as a function of redshift. Thesame fits used for halo mass function are presentedhere with overplotted points derived from our simu-lation. It is clear that results are distributed within 4mass bins covering masses of 1011−1015M�/h span-ning the entire simulation period in which the haloswere identified. Using this plot we can determine red-

shifts around which halos of certain mass bin beginto form (except for the lowest mass bin). Again, wenotice a discrepancy between the Press-Schechter fitover other fits, amplified at higher redshifts. More-over, we see that at high redshifts all fits divergeto some extent. A good agreement at lower red-shift (0 < z < 1) between the simulation results andfits is noticeable across all mass bins, but at higherredshifts, some divergence appears. For the highestmass bins there is an overabundance of halos at z > 1but, as seen from other analysis, this is most likelydue to a low number of halos populating mass binsat those redshifts. The same problem affects othermass bins near their maximum redshifts.

If we compare fits and results for differentmass bins in Fig. 6, it becomes clear that mass binswith higher masses emerge later in the simulation,thus once again stressing the fact that higher massdark matter halos form later.

Visualization of data has been performed bothfor popularization purposes and for testing the in-tegrity of the halo finder. In Fig. 1, visualization ofthe final snapshot can be seen. Along with othersnapshots it has been performed by collapsing allparticles along one axis, leaving two dimensional dis-tribution of particles. After that, the density wascalculated by simple binning of particles, where fi-nally, the color was assigned proportionally to den-sity (darker color to low density, brighter colors tohigh density regions). For animation purposes parti-cle positions were extrapolated in between snapshotsin order to lengthen the animation for easier appre-hension of events within it.

Fig. 6. Plot of halo growth functions: number density of halos as a function of redshift presented for 4mass bins. Results of simulation are compared to 4 different fits, same ones that were used in halo massfunction analysis, presented in Fig. 5. A good agreement at lower redshift (0 < z < 1) between the simulationresults and fits is noticeable across all mass bins, but at higher redshifts, some divergence appears. Note thediscrepancy between fits themselves at higher redshifts. Error bars are the Poisson standard errors.

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−500 0 500 1000 −500 0 500 1000Distance (kpc)

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Fig. 7. Visualization of several identified dark matter halos from simulation. Black circles represent thevirial radius of identified halos from the halo finder. Note the different density and ellipticity of halos.

In Fig. 7, visualization of distinct halos is pre-sented. We have chosen halos that cover a wide rangeof properties (isolated, in clusters, etc) to visuallyconfirm the halo finder results. For each halo a 3Dbox around it was extracted and visualization wasdone by simply plotting particle positions sliced onone axis through the derived center of a halo. Over-plotted are the circles representing the virial radiusof identified halos from the halo finder. It can be seenthat there is no major discrepancies between the halofinder identified halos and the visual representationof them.

7. DISCUSSION AND CONCLUSION

In this paper we present results of pure N-body cosmological simulation of 130 Mpc/h periodicbox with 5123 particles performed with GADGET2.The halo catalogue and merger tree were obtainedusing the ROCKSTAR code and through their anal-ysis the dark matter halo distribution statistics ispresented. Visualization of simulation was describedand depicted. Special consideration was dedicated tohalo formation time, major merger rate, mass func-tion and halo growth function which were derivedwith good time resolution and compared with bothanalytical and empirical fits.

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A quick look at total number density of ha-los across all redshifts tells us that we have an initialrise in number of halos and that after certain redshifttheir number starts to recede, which is expected.

The merger tree gives us, among other things,insight into major mergers of our simulation. If weconcentrate on the mean merger rate per time plotswe notice that we have a good agreement betweenour plot and that of Gottloeber et al. (2001) whoconsider the primary halo major mergers (consis-tent with our masses of more than 1012M�/h), butthere is a discrepancy between our results and ma-jor mergers of Millennium simulations (Fakhouri andMa 2008, Fakhouri et al. 2010) where they consideronly the halo-halo mergers, while in our case, as al-ready mentioned, the subhalo mergers are consideredas well. It should also be noted that our simulationhas a lower statistical sample and a finer time res-olution, leading to lower number of major mergersat each point. The dependence on time resolutionwas shown by Gottloeber et al. (2001) where, as ex-pected, the scatter increases with finer resolution i.e.plots consist of more points that are less populated.Scatter due to low population is obvious for higherredshifts in our sample simply because the first halosof considered mass did not appear until z ∼ 6 mean-ing that the number of major mergers is low aroundthat redshift.

Angulo et al. (2009) have used the Millen-nium simulation (> 1012M�/h) in the same massrange as us and with the same considerations formergers (subhalo mergers as well). Their results arein a better agreement with our results than that ofFakhouri and Ma (2008), but still with slight off-set. Considering the already mentioned discrepan-cies with Fakhouri and Ma (2008) also from Millen-nium simulation, this points us to work of Hopkins etal. (2010). They tried to quantify the contributionof uncertainties and systematic effects for differentmeans of deriving merger rates and they came tothe conclusion that there is a discrepancy of factor∼ 2 − 3 for high-resolution dark matter simulationsif mergers are defined consistently (within the samecosmology).

Another test for validity of our merger ratecame from Genel et al. (2009) from which we usedthe fitting function, compared it with our meanmerger rate per Gyr and found that it scales withredshift as a power law ∝ (1 + z)n, where we foundthat n fits our data well if n = 2.4. This rate co-incides with the form of power law acquired fromobservations for scaling merger rates with redshiftfor: n = 2.43 (Bridge et al. 2009), n = 2.5 (Burkeyet al. 1994), n = 2.3 (Patton et al. 2002), n = 2.7(Le Fevre et al. 2000).

Looking at the plot that deals with total num-ber of major mergers across all redshifts we can seethat their number reaches a peak around z ∼ 2 whichcoincides with the highest number density of quasars,thus fueling theories that quasars are created duringsignificant merger events in which gas is efficientlysupplied to the super massive black hole (Kauffmannand Haehnelt 2000 and references thereafter). Majormergers within the mass range that can be found in

our simulation prove to be an excellent foundationfor further analysis.

Two plots linked with formation time of halosgive us yet another insight into evolution of halos.

The plot of the highest mass halos within thesimulation is explored as a very simple but power-ful tool for preliminary analysis of simulation. Wecan use this method to identify when the first ha-los of certain mass appear in our simulation whichgives us constraints on single halo mass expectationsat high redshifts. For example, it shows us that ha-los of greater mass form the later in the simulation- 1012 M�/h appear around z ∼ 6, halos of mass of1013 M�/h can first be discerned around z ∼ 4 etc.

We also investigate a simple derivation of thehighest mass halo within a snapshot as a new way forsimple preliminary analysis of halo formation duringcosmological runs. Modern simulations are becomingbigger (having a large number of particles) involvingmore data and stressing the need for less costly realtime analysis. Deriving the highest mass halo withina snapshot is far less complicated (therefore demand-ing less resources) than deriving the halo mass func-tion or halo growth function, but can be used as anindication on where those analysis should be of mostinterest for us. Halo finders are already performingon the fly analysis of simulations making this imme-diately available analysis. Small jumps in mass inseveral places on the plot of highest mass halos areobvious markers of significant merger events. Thosekind of information is important if the resources arelimited and we want to continue our analysis (butwithout the need for completion of simulation, forexample).

Mind due, this analysis can not be used fordetermining halo mass function or halo formationtime. For that we turn to the mean formation timeplot (blue triangles, Fig. 4). Number of halos at thefinal redshift is sufficient to determine mean forma-tion time of halos, in a good agreement with resultsof other simulations. For example, we notice a goodagreement between the empirical fit from Millenniumsimulation plotted over our results for the same anal-ysis. Figure 4 shows the tendency that higher masshalos form later in the simulation.

From these analysis inference can be madethat lower mass halos tend to form earlier and thathigher mass halos tend to form later in the simula-tion. Together with visualization and this analysiswe can also derive that a considerable clustering isin progress which, after certain time period, reducespossibility for major merger events, as observed inFig. 3.

For more details on general halo evolution ina specific mass range and across all redshifts we turnto a more specific analysis, such as the halo massfunction and halo growth functions.

One question is how much can different cos-mological parameters influence halo mass functions?Jenkins et al. (2001), Warren et al. (2006) and Tin-ker et al. (2008), amongst others, have tried to ad-dress and investigate the issue. Multiplicity func-tion has a universal form, where the same fitting pa-rameters are used as consistent across different cos-mologies. Jenkins et al. (2001) found consistency

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for different CDM cosmologies up to ∼ 15%. Butwithin different ΛCDM cosmologies Warren et al.(2006) showed that there is a probable nonnegligi-ble inconsistency while varying several parameters(Ωm, σ8, h), where the consistency can be up to5% for smaller variations. Results of Tinker et al.(2008) confirm that there is an inconsistency, butthat it is related to the large variations in cosmolo-gies (Ωm = 0.1−1). They point that the universalitybelow 5% is reachable (and desirable in an era of pre-cision cosmology) but that is an extremely challeng-ing task and beyond the scope of this paper. Con-sidering that all the mass functions referenced herehave a similiar cosmology (except Press-Schechterone) the consistency between them is sufficient forgeneral comparison.

Looking at halo mass function plots (Fig. 5)where we compare expectations with our data, wesee a good agreement with all fits for halos at red-shift z = 0. There is slight underabundance of high-est mass halos in our simulation, which is easily ex-plained by small statistical sample populating thisbin. A better agreement is easily seen at other red-shifts. It is also noticeable that SMT, Warren, andAngulo fits are consistent among themselves. Onthe other hand, we see that the PS fit is not in agood agreement for the entire mass range and atall redshifts. The smallest discrepancy is at redshift0, and it gets larger toward earlier redshifts, espe-cially underestimating the number of halos populat-ing higher mass bins. Considering that higher masshalos tend to diverge from sphericity due to violentnature of their formation (mergers, higher densityregions, etc), they tend to collapse sooner than PS isestimating, thus slightly increasing their abundanceas can be seen both from our simulation and fromother fits. Also, as pointed above, different cosmol-ogy used for Press-Schechter can also contribute tothe observed discrepancy.

If we divert our attention to the halo growthfunction, unlike in the previous work (Heitmann etal 2006, Lukic et al. 2007), we extend our simulationhalo growth function over whole simulation period inwhich the halos were identified. If we look at Fig. 6,it is immediately clear that all fits diverge at higherredshifts, and this is where we can see the true differ-ence between them. Note that Warren et al. (2006)and Angulo et al. (2012) fits have the highest consis-tency between them, both being derived purely fromnumerical simulations, slightly misaligning only athigh redshifts. At these redshifts it becomes quiteobvious how much PS diverges from other fits.

Interestingly, our simulation follows the SMTfit on lowest mass bin. The highest mass bin is againaffected by smaller number of halos populating it al-though, at redshifts between 0 and 1, all fits andour data converge. On other two remaining (mid-dle) bins, our data follow the Warren fit most consis-tently. This is expected considering that both War-ren simulations and our data suffer from a limitedperiodic box size, where Warren, during resimula-tions, had a limited number of realizations of theUniverse. Unlike them, Angulo had one sufficientlybig box with a superior mass resolution.

Curiously, since the first identification of ha-los of high masses (more than 1014 M�/h) we havea slight overabundance of them on earlier redshiftsas can be seen from the halo growth plot (Fig. 6)but we end up with a slight underabundance of thehighest mass halos as can be seen from the massfunction for redshift 0 (Fig. 5). It should be dullynoted though that their number hasn’t reached theplateau as it did for the halos in lower mass bins. It isobvious that mass functions have different crossovertimes depending on their mass.

As we see, the extending halo growth func-tion makes it possible to continuously monitor theevolution of halo number density within given massranges and to qualitatively compare it with fits athigher redshifts, an area of interest which is becom-ing more and more important in the era of precisioncosmology.

From all the analysis combined with halogrowth and mass function it is easily deduced, ashas been done before, that the most of dark matterhalos is migrating from smaller mass halos towardmore and more massive halos (as is predicted by“bottom-up” model of CDM - Peebles (1965), Pee-bles and Dicke (1968), Silk (1968), Gott and Reese(1975), etc.), implying more prominent clustering to-ward lower redshifts.

Overall, through these analysis, it is seen thatour simulation is in a good agreement with analyticaland empirical expectations and with previous numer-ical simulations. The simulation covers evolution ofhalos over 4 orders of magnitude (1011−1015 M�/h)and at redshifts of z = 0 − 7 halos from simulationhave a good statistics for halo analysis especially inthe mass range of 1011 −1014 M�/h and therefore itwill be used as a platform in subsequent studies.

Acknowledgements – Author would like to thank hismentor Miroslav Micic for guidance and devotedsupport during this project. Author would like tothank also Manodeep Sinha for useful advices anddiscussion. Gratitude goes to him and to KellyHolley-Bockelmann for help with setting initial con-ditions. During the work on this paper the authorwas financially supported by the Ministry of Educa-tion and Science of the Republic of Serbia throughthe project: 176021 ’Visible and invisible matter innearby galaxies: theory and observations’.

Numerical results were obtained on the PARADOXcluster at the Scientific Computing Laboratory ofthe Institute of Physics Belgrade, supported in partby the national research project ON171017, fundedby the Serbian Ministry of Education, Science andTechnological Development.

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STATISTIQKA ANALIZA REZULTATA KOSMOLOXKIH SIMULACIJAN TELA NA SKALI SREDNjIH ZAPREMINA, PRIMENjENA NA HALOE

N. Martinovic

Astronomical Observatory, Volgina 7, 11060 Belgrade 38, SerbiaE–mail: nmartinovic@aob.rs

UDK 524.7–857 : 524.8–17Originalni nauqni rad

U ovom radu je predstavljena statis-tika haloa tamne materije u svojstvu testi-ranja ΛCDM kosmoloxke simulacije N tela(od trenutka formiranja prvih haloa tamnematerije do z = 0). Ispitujemo normalizo-van broj velikih sudara u funkciji vremena,gde za funkciju vremena razmatramo i zavis-nost u funkciji crvenog pomaka i zavisnostu funkciji milijarde godina. Za potonju za-visnost nalazimo da je ona proporcionalnadobro poznatom stepenom zakonu (1 + z)n, gdesmo dobili da je n = 2.4. Funkcije mase

haloa i funkcije rasta haloa su dobijene iupore�ene i sa analitiqkim i sa empirijskimfitovima. Funkcija rasta haloa je proxi-rena tako da potpuno obuhvata sve periode usimulaciji u kojima su identifikovani haloitamne materije, omogu�avaju�i da se konti-nualno prati evolucija gustine haloa u za-datim rasponima mase. Vizualizacija simu-lacije je tako�e predstavljena. Na crvenimpomacima z = 0 − 7 statistika haloa je jakodobra za dalje analize, posebno u rasponu maseod 1011 − 1014 M�/h.

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