Angular momentum evolution in Dark Matter haloes: a study of the Bolshoi and Millennium simulations
aa r X i v : . [ a s t r o - ph . C O ] S e p Mon. Not. R. Astron. Soc. , 1–12 (2016) Printed 27 September 2018 (MN LaTEX style file v2.2)
Angular momentum evolution in Dark Matter haloes: a study of theBolshoi and Millennium simulations
S. Contreras , N. Padilla , , C.D.P. Lagos , . Instit´uto Astrof´ısica, Pontifica Universidad Cat´olica de Chile, Santiago, Chile. Centro de Astro-Ingenier´ıa, Pontificia Universidad Cat´olica de Chile, Santiago, Chile International Centre for Radio Astronomy Research (ICRAR), M468, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia. Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), 44 Rosehill Street Redfern, NSW 2016, Australia.
27 September 2018
ABSTRACT
We use three di ff erent cosmological dark matter simulations to study how the orientation ofthe angular momentum vector (AM) in dark matter haloes evolve with time. We find thathaloes in this kind of simulations are constantly a ff ected by a spurious change of mass, whichtranslates into an artificial change in the orientation of the AM. After removing the haloesa ff ected by artificial mass change, we found that the change in the orientation of the AMvector is correlated with time. The change in its angle and direction (i.e. the angle subtendedby the AM vector in two consecutive timesteps) that a ff ect the AM vector has a dependenceon the change of mass that a ff ects a halo, the time elapsed in which the change of mass occursand the halo mass. We create a Monte-Carlo simulation that reproduces the change of angleand direction of the AM vector. We reproduce the angular separation of the AM vector sincea look back time of 8.5 Gyrs to today ( α ) with an accuracy of approximately 0.05 in cos( α ).We are releasing this Monte-Carlo simulation together with this publication. We also create aMonte Carlo simulation that reproduces the change of the AM modulus. We find that haloesin denser environments display the most dramatic evolution in their AM direction, as well ashaloes with a lower specific AM modulus. These relations could be used to improve the waywe follow the AM vector in low-resolution simulations. Key words: large-scale structure of Universe - statistical - data analysis
Semi-analytic models (SAMs) are an e ffi cient and accu-rate way to populate large volume simulations with galaxies(Lagos, Cora & Padilla 2008; Lagos et al. 2012; Baugh 2006), butin most cases, this hampers their ability to resolve halo and galaxyproperties that are important for small-scale physical processes,such as star formation and feedback. For example, some modelsrelate the star formation rate and the mass loading of stellar-drivenwinds to the density of gas in the galaxy disc (Croton et al. 2006;Lagos, Lacey & Baugh 2013), which critically depend on the abil-ity of models to compute the sizes of galaxies. Typically models dothe latter by assuming conservation of angular momentum (AM),from which a good measurement of the AM of dark matter haloesis crucial (e.g. Cole et al. 2000). This is usually done following Mo,Mao & White (1998, MMW) who used detailed numerical simula-tions of the process of gas cooling, to find a simple analytic re-lation between the sizes of galaxies and the specific AM (sAM)of halos that could be used by simpler models. Thus, this rela-tion implies that we need to be able to measure the AM of haloes,which cannot be accurately performed for structures traced by lessthan ∼ ffi ciently high-resolution simulation to limit theSAM to populate haloes with at least 1000 particles, as it is done byGuo et al. (2011). But in large volume simulations, this would im-ply very poor sampling of the dark matter halo mass function, withthe smallest haloes having masses of ≈ M ⊙ (e.g. in the Bolshoisimulation, Klypin, Trujillo-Gomez & Primack 2011). Another so-lution is to use Monte-Carlo simulations to follow the evolution ofthe AM of haloes, which in turn can be used to obtain the AM ofgalaxy discs using MMW. This approach was adopted by Padillaet al. (2014, see also Lagos et al. 2014, 2015), where a SAM usesthe merger trees of haloes defined with as few as 10 particles, butdoes not attempt to use the measured halo AM. It rather adopts aMonte-Carlo approximation relating changes in the AM to changesin the mass of the halo, also separating halo mass accretion betweensmooth accretion and mergers. This approximation provides an ac- c (cid:13) S. Contreras et al. curate prediction of the relative orientations of gas and stellar discsin observations of early-type galaxies (Lagos et al. 2014, 2015).In this work, we expand the analysis of the evolution of AMof haloes including not only the change in direction of the AMvector of haloes, but also the relation between changes takingplace at di ff erent times throughout the life of a halo. We pro-vide relations that can be used to improve the modelling of theAM of haloes in simplified models of galaxy formation, includ-ing SAMs, but also applicable to the subhalo abundance matching(SHAM, Kravtsov, Gnedin & Klypin 2004; Vale & Ostriker 2006;Conroy, Wechsler & Kravtsov 2006) or the halo occupation dis-tribution (HOD, Jing, Mo & B¨orner 1998; Peacock & Smith 2000)type modelling. In this work we use three di ff erent dark matter simulations. TheBolshoi simulation (Klypin, Trujillo-Gomez & Primack 2011), theMillennium simulation (Springel et al. 2005, hereafter MS-I) andthe Millennium-II simulation (Boylan-Kolchin et al. 2009, here-after MS-II). The properties of these simulations are listed intable 1. The Bolshoi simulation was run with an Adaptive-Refinement-Tree (ART) code, which is an Adaptive-Mesh-Refinement (AMR) type code (see Kravtsov, Klypin & Khokhlov1997 and Kravtsov 1999 for a description of the code), while theMS-I and the MS-II were run with GADGET-2 and GADGET-3 (Springel, Yoshida & White 2001; Springel et al. 2005), respec-tively, and a TreePM code.The Bolshoi simulation adopted a WMAP-5 cosmology ( Ω b = Ω M = Ω Λ = = H / = n s = σ = Ω b = Ω M = Ω Λ = = H / = n s = σ = ff erence in cosmology between the modelsshould not cause significant di ff erences in the results we presenthere.The most relevant di ff erence between the halo catalogues fromthese simulations is the halo finder algorithm and the merger treesused. The Millennium simulations use a Friends-of-Friends (FoF)group-finding algorithm (Davis et al. 1985). The haloes are identi-fied in each simulation output (snapshot) and contain a minimumof 20 particles. The halos in consecutive snapshots are connected tobuild merger trees. The Bolshoi simulation uses the Rockstar halofinder algorithm (Behroozi, Wechsler & Wu 2013) and ConsistentTrees (Behroozi et al. 2013) to build merger trees. The latter codesare thought to improve both the completeness (through detectingand inserting otherwise missing halos) and purity (through detect-ing and removing spurious objects) of both merger trees and halocatalogues (Behroozi et al. 2013). These latter qualities are desir-able for studies of the evolution of the AM vector.Another important di ff erence between the two halo cataloguesis the amount of snapshots available and length of the timesteps.The MS-I and MS-II have 63 and 67 snapshots available, respec-tively, between z =
127 to z =
0. The length of the timesteps varies toallow a better time resolution at low redshifts. The Bolshoi simula-tion has 181 snapshots from z =
14 to z =
0. In Bolshoi, these snap-shots are normally separated by ∆ a = .
03 (the expansion factor)between a = a = . ∆ a = .
06 at earlier snapshots formost cases.The haloes from the Millenium simulations were obtained
Simulation N P m P / h − M ⊙ L / h − MpcBolshoi 2048 . × . × . × Table 1.
Parameters of the Bolshoi, Millennium and Millennium II darkmatter simulations. N P represent the number of particle for each simula-tion, m P the mass of these particles and L is the length of the simulations’periodic boxes. from the German Astrophysical Virtual Observatory (GAVO) ,while the haloes from the Bolshoi simulation were obtained fromBehroozi’s personal webpage . The AM vector of a dark matter halo is computed summing thecross product of the dark matter particles position with respect thehalo center of mass ( ~ r − ~ r COM ) and their momentum vector withrespect to the halo center of mass ( m ( ~ v − ~ v COM ), where m is themass of the particle), ~ J = X i m i ( ~ r i − ~ r COM ) × ( ~ v i − ~ v COM ) . (1)To ensure an accurate measurement, we impose a minimumnumber of particles per halo of 1000 particles. Whenever we fol-low a halo through di ff erent snapshots, we require the halo to sat-isfy this limit at all redshifts, e ff ectively increasing the final, lowredshift halo mass of samples of haloes followed through longerperiods of time. A less conservative value for the minimum num-ber of particles required to robustly measure the angular momen-tum of dark matter haloes has been recently proposed by Benson2017 (i.e., 40,000 particles). To show the e ff ect of a higher cut inthe number of particles, most of the plots in this work will includethe results for halo samples with di ff erent number of particles.Before a halo undergoes a merger, and when it is not thelargest halo in the merger (i.e. it is not part of the main progenitorbranch of its descendants) or when the halo su ff ers a flyby, it loses aconsiderable number of dark matter particles. These losses, whichcan be driven by numerical errors due to the halo finder rather thanto an actual mass loss, can seriously a ff ect the AM direction of thehalo. In several cases, this mass loss is followed by a mass gain asthe halo orbits away from the main halo (Jiang et al. 2014). Thismass gain is also not of a physical origin. From this point on, wewill refer to this type of mass variations as spurious mass changes.In this work, we are only interested in the change of angle of theAM caused by smooth accretion of individual dark matter parti-cles, or by mergers with smaller haloes. To ensure this we onlyuse haloes that are part of the main progenitor branch and have adescendant at z =
0. We acknowledge that halo mass can decreasedue to di ff erent e ff ects such as tidal stripping and flybys, and thesee ff ects will also be included in our analyses. http://gavo.mpa-garching.mpg.de/portal/ c (cid:13) , 1–12 ngular momentum evolution in haloes Figure 1.
Schematic showing examples of the change in direction of theAM of haloes due to real mass increase, for example, due to smooth massaccretion (top panel), and spurious loss of mass for a snapshot, that is laterrecovered by the halo (bottom panel). The angles α n → n + and α n → n + areshown in the middle and right hand circles. Here each circle represents aDM halo at a given time and its size represents mass, while the solid arrowsshow their AM vector. The dotted lines in the middle and right hand circlesrepresent the AM vector at the snapshot n . In this section we study the change of the AM of haloes in pairs ofconsecutive snapshots, n, n +
1. We use the angle subtended by theAM in the snapshots n and n + ∆ α , as a way to detect spuriouschanges. If the particles a halo appears to lose are recovered later,the measured AM will show a sudden change at the moment ofparticle loss, to then go back practically all the way to its previousvalue once the particles are regained by the halo. In the top panelof Fig. 1 we show schematics of what we would expect for a realchange in the direction of the AM vector in dark matter halos due toa systematic change in mass, while in the bottom panel of Fig. 1 theschematic shows the e ff ect of spurious mass loss for a timestep, thatis later recovered, causing the AM vector to return to its originalvalue.Fig. 2 shows the e ff ect of spurious mass changes in the changein direction of the AM of halos. The left panel shows the anglesubtended by ~ J between two non-consecutive snapshots (leavingone snapshot in between). We refer to this angle as α n → n + . Weshow its value as a function of the angle subtended by the AM inthe two pairs of consecutive snapshots that result from using theintermediate one, α n → n + and α n + → n + . The results are shown forthe Bolshoi and the MS-I simulations (top and bottom panels). Theresults for MS-II look very similar to those in MS-I.The left panels show very small values of the angle subtendedby the AM vector at n and n +
2, showing that the large variationsbetween snapshots n and n + n + n +
2, indicating a spurious change of the AM vector,which is not as apparent once more than two consecutive snapshotsare examined.This is the case for all the simulations studied here (includingMS-II not shown in the figure) and all halo mass ranges. We findthat the haloes that su ff er from this e ff ect tend to lose mass and thenrecover it back again usually after one or two snapshots. However, we notice that there are extreme cases where haloes can lose andrecover mass several snapshots later.This loss and later recovery of matter are caused when the halofinder algorithm stops associating a portion of dark matter particlesas part of the main halo, a problem that can be present throughoutseveral snapshots. The inverse process (an introduction and laterextraction of particles) is also common and also produces this ef-fect. This can be caused by flybys, tidal stripping, and virialization.As was mentioned above, the ROCKSTAR algorithm tries to avoidthis e ff ect by looking at a number of snapshots to the past and fu-ture to avoid including (or removing) particles that do not (do) be-long to the halo. Indeed, haloes in the Bolshoi simulation (wherethe ROCKSTAR algorithm was applied) are less a ff ected by thisproblem.To avoid the spurious e ff ects driven by these haloes in ourstudy, we create a halo sample where haloes that have su ff ered asignificant change in mass and then are seen to recover it (or loseit again) are removed. We refer to this sample as the “clean halosample”. To make it to this sample haloes must satisfy the relation, | ∆ M n → n + + ∆ M n + → n + | > C ( | ∆ M n → n + | + | ∆ M n + → n + | ) , (2)where C is constant set equal to 0 . . ∆ M n → n + ∼ − ∆ M n + → n + , we remove the halofrom the sample. To avoid removing haloes that had no accretion,and whose change in mass is only due to numerical noise in the halofinder algorithm we use the condition that both, ∆ M n → n + / M n + and ∆ M n + → n + / M n + are below 10 − to include a halo in the sample.These values are selected so as to minimise the impact of the re-jected haloes in the main results presented throughout this paper.The middle panels of Fig. 2 show the results for the cleanedhalo sample. We notice that for both simulations, the angles of theAM vector change in the diagonal are now larger than in the leftpanels, e ff ect this is particularly strong for the Bolshoi simulation.This can be compared to the right panels of the figure where wechoose a random orientation for the change of the AM vectors ofhaloes (i.e. with no coherence in the direction in which the AMvectors are moving towards between consecutive snapshots). Thesimilarity between the middle and right panels indicates that mostof the changes of AM in the haloes appear to be mostly uncorre-lated. However, as can be seen by in Fig. 3, where we comparethe middle and right panels of Fig. 2, there are deviations betweenthe two, as the angle between angular momenta of snapshots n and n + In this section, we search for statistics of the rate of change of AMof haloes both, in their amplitude and direction. Our aim is to pro-duce a set of lookup tables that can then be adopted in simple mod-els of galaxy formation such as SAMs, to include a physical evolu-tion of the AM of galaxy host halos that is also numerically robust.We study the change of direction of the AM in two steps. First wewill concentrate on the angle subtended by the AM of a halo inpairs of snapshots. Then we will study how correlated the changesin the AM are throughout the lives of DM haloes. c (cid:13) , 1–12 S. Contreras et al.
Figure 2.
Left panels: The angle between the AM of halos in consecutive snapshots n + n +
2, as a function of the angle between the AM of halosin snapshots n and n +
1. Pixels are coloured by the median angle subtended by the change of the AM vector su ff ered by haloes between a fixed snapshot“n” and their descendants in two future snapshots ( α n → n + , see the colour bar for the scale), for the Bolshoi (top panels) and the Millennium (bottom panels)simulations. The n + =
0. Middle panels: same as left panels, but taking out all haloes that in snapshot n + Figure 3.
The ratio between the top middle panel of Fig. 2 ( α n → n + forthe cleaned halo sample) with the top right panel of Fig. 2 ( α n → n + for therandom halo sample). The top panel of Fig. 4 shows the median cosine of the angle sub-tended by ~ J measured at a lookback time of 8.5 Gyrs and at a later lookback times “t”. Coloured lines represent a di ff erent num-ber of particles per halo as indicated in the figure description. Theblack line represents the full galaxy population. This choice of linecolours will be used in several other figures of this paper. As can beseen, the most massive haloes tend to deviate more from its origi-nal AM compared to the less massive samples. This measurementis not a ff ected by the spurious mass changes discussed in the lastchapter since we measure di ff erence among several snapshots andnot consecutive ones.The grey line shows the predictions of the random orientationscase. In this case, the angular separation is lower than measured inthe simulation. This confirms that the change in the direction of ~ J iscorrelated in time, and not randomly oriented. The case of randomorientations for low and high halo masses have a similar behaviour(not shown in the figure). The ratio between the median angularseparation of haloes and their counterpart of the random case isshown in the bottom subpanel of Fig. 4.Fig. 5 is similar to Fig. 4, except that we only show haloes withno spurious mass changes in any of the 121 snapshots at lookbacktime 8 . ff ect the total change of an-gle (as is shown in Fig. 5) but a ff ect the mock random sample by c (cid:13) , 1–12 ngular momentum evolution in haloes Figure 4.
The top panel shows the median change of direction of the AMvector su ff ered by haloes between a lookback time t = ff erent numbers of particles per halo, N3, N3.5, N4, N4.5 and N5, cor-responding to 10 − . , 10 . − , 10 − . , 10 . − and morethan 10 particles respectively at lookback time t = = = increasing total angle separation in time. The results for haloes ofdi ff erent masses show a larger angular separation than in the caseof using all haloes (Fig. 4). Even when adopting di ff erent param-eters for Eq. 2, and also trying di ff erent recipes to clean spuriousmass changes in the sample, the result remains qualitatively similar.Cleaning the cases of spurious mass changes a ff ect primarily theresults for the low mass haloes, making them depart more stronglyfrom their initial AM direction.Fig. 6 shows the median change of angle of ~ J for the MS-I (top) and the MS-II (bottom). For both simulations, the randomcase (presented also in grey line) shows a larger angular separa-tion than measured for low mass haloes (and for the full sample).This could be easily misinterpreted as the change of angle of ~ J inthese simulations being uncorrelated (or even anticorrelated). Butthe cause behind this result is the spurious mass change of haloesdue to numerical noise a ff ecting the random sample. Once the massis recovered the AM mostly goes back to its original direction, ane ff ect that is not present in the random case. This is the same e ff ectthat caused Fig. 5 (the angular separation of the cleaned sample ofthe Bolshoi simulation) to show larger di ff erences than Fig. 4 (sameas Fig. 5, but for the full halo population).The question remains on whether the change of direction ofthe AM vector is related to the mass of the haloes, or a numericale ff ect produced by the di ff erence in a number of particles samplingthe mass distribution in the haloes. To answer this we also showin the bottom panel of Fig. 6 on dashed lines the haloes from the Figure 5.
Same as Fig. 4 but for a selection of haloes that are not a ff ectedartificial mass changes (cleaned halo sample, see section 3.1 for more de-tails). Figure 6.
Same as the main panel of Fig. 2 but for the MS-I (Top) andMS-II (Bottom). The dashed lines in the bottom panel show the MS-I re-sults for the same mass ranges in the three most massive bins in MS-II, forcomparison.
MS-I simulation, for ranges of particles that correspond to equalmass ranges of MS-II samples. Since the volume of the MS-II issmaller, this can only be done for the three most massive samples.To be able to do this comparison we use haloes with at least 80particles. Also, the most massive sample includes MS-I haloes withmasses corresponding to 10 -10 . particles per halo in the MS-II.There is no agreement between the behaviour of haloes of similarmass in the MS-I and MS-II simulations, regardless of the massrange studied. In the case of the most massive sample shown in the c (cid:13) , 1–12 S. Contreras et al.
Figure 7.
The median change of direction of the AM vector as a function of three factors: the halo mass ( M Halo ), the di ff erence in M ∗ ( ∆ log ( M ∗ )) and thechange of mass su ff ered by the halo between two redshifts ( ∆ M / M ). The left panels show the dependence with respect to M Halo and ∆ M / M , the middle panelsshow the dependence with M Halo and ∆ log ( M ∗ ), and the right panels show the dependence with respect to ∆ M / M and ∆ log ( M ∗ ). Whenever we compare twodi ff erent properties, we select haloes with a value of the third halo property as close as possible to its median value, except for ∆ log ( M ∗ ) where we selecthaloes with a value between 0.2 and 0.3. Haloes from all redshift ranges are used for these plots. bottom panel for MS-I, the minimum number of particles is highenough to obtain accurate measurements of AM direction (a totalof 800).A possible reason for the di ff erence is the environment, aswhile in MS-II the sample corresponds to the most extreme haloes,which probably live in the densest environments, in the MS-I simu-lation these haloes are actually the smallest ones which live in sev-eral di ff erent types of environments. This could indicate that massis not the only property of the halo that is relevant for the evolutionof ~ J , a subject we will come back to in Section 6.Fig. 7 shows ∆ α as a function of combinations of halo mass( M Halo ), the di ff erence between the snapshots of the logarithm ofthe characteristic mass at redshift z ( ∆ log M ∗ ( z )) and the changeof mass su ff ered by the halo in that time ( ∆ M / M , calculated as( M final − M initial ) / M final ) for the Bolshoi simulation. We allow pairsof snapshots separated by up to 30 snapshots to allow larger red-shift di ff erences. Whenever we show values of ∆ α as a function ofany two di ff erent properties, we do so by fixing the remaining vari-able to it corresponds to the median value, except for M ∗ , wherewe select haloes with ∆ log M ∗ ( z ) between 0.2 and 0.3 (this is anarbitrary cut; we test other values finding similar trends as the onesshown here).We use ∆ log M ∗ instead of elapsed time, since we find thatthere is less redshift dependence of the results, i.e. the evolution oftwo haloes that evolve in a constant ∆ log M ∗ at di ff erent redshift aresimilar compared to the evolution of two haloes that evolve duringthe same time at di ff erent redshifts. We also test other variableslike the redshift, the expansion factor( a ) the growth factor ( g ) theamplitude of the growing mode ( D ), but no property work as goodas M ∗ .To calculate M ∗ , we follow a similar procedure of the one pre-sented in Rodr´ıguez-Puebla et al. 2016 to calculate the characteris-tic mass of halos just collapsing at redshift z ( M C ). This is shownin Appendix A.As can be seen ∆ α shows a strong dependence on the rangeof M ∗ and the mass variation, where a higher ∆ α takes place whenthe timestep is longer or the change of mass is larger, as expected.The halo mass shows little dependence with ∆ α , where for smalltimesteps, high halo masses undergo a small change of angle. Forlong timesteps the relation reverses. This is consistent with the evo- lution seen in ∆ α and the dependence on halo mass in the casewhere random orientations are applied. One of the aims of this paper is to find a way to follow the evolu-tion of the AM vector in halo samples in low-resolution (or largevolume) dark matter simulations. To do this it is not enough to re-produce the amplitude of the change of AM direction because, aswe see with the mock random sample, this would correspond tothe random case shown in Figures 4, 5 and 6, and it does not re-produce the behaviour followed by the AM vector of the haloesin our simulations. In addition to the amplitude of the change, wealso need to reproduce the persistence of the direction of change inorder to reproduce the evolution of ~ J in the long run. We need toquantify if when there is a change of AM between a given pair ofsnapshots, the following changes occur in a direction that is relatedto the previous one. That is, whether the projection of ~ j m ′ − ~ j n and ~ j m − ~ j m ′ over a plane perpendicular to ~ J m ′ of di ff erent pairs of snap-shots n < m ′ < m are parallel or not. We define the direction (DIR)parameter as follows: cos ( DIR ) = ˆ ∆ J · ˆ ∆ J , (3) ∆ J = ˆ J n cos ( α ) − ˆ J n − , (4) ∆ J = ˆ J n + − ˆ J n cos ( α ) , (5)where α and α are the angle between ˆ J n − − ˆ J n and ˆ J n − ˆ J n + ,respectively.A value of DIR = o indicates that the change of AM contin-ues in the same direction (if we have a change of angle of 3 o and 1 o ,the total change of angle will be 4 o ). A value DIR = o indicatesanticorrelation (if we have changes of 3 o and 1 o , the total change ofangle will be 2 o ). If the median DIR is 90 o , the change of angles israndom (as in our random case).Fig. 8 shows DIR as a function of combinations of the halomass, the change of mass, and the change of the logarithmic of ∆ log ( M ∗ ), and as can be seen, it has a low dependence on the halomass, where more massive haloes have smaller values of DIR (i.e.changes in angle are less correlated). The direction has also a strongdependence on ∆ log ( M ∗ ) and ∆ M / M . Larger ∆ z and lower values c (cid:13) , 1–12 ngular momentum evolution in haloes Figure 8.
The median change angle between consecutive direction changes of the AM vector as a function of three factors: the halo mass ( M Halo ), the di ff erencein the logarithmic of M ∗ ( ∆ log ( M ∗ )) and the change of mass su ff ered by the halo between two redshifts ( ∆ M / M ). Panels are as in Fig. 7. of ∆ M / M imply larger DIR values, but the relation is not as clearas in Fig. 7. A simple way to reproduce the median change of direction in a halosample is to limit the allowed range of change of direction in therandom case,
DIR < h DIR i ≡ φ, (6)where φ is the maximum allowed angle of DIR, which by con-struction follows a uniform distribution (the mean of a random anduniform sample, h DIR i , is simply half the value of φ ). Fig. 9 showsthe instantaneous change of direction of the AM vector in consec-utive snapshots as a function of lookback time. The grey horizontallines show the value corresponding to random changes with di ff er-ent values of φ . Notice the jump in measured halo AM changes att = ff erent halo mass ranges show roughly con-stant values of DIR as a function of redshift, which indicates thatthe level of persistence in the direction of change of the AM vectoris similar throughout the life of a halo, and much stronger for highmass objects. We will use the measured changes of
DIR to inter-polate a value of φ to apply to our random cases, in order to tryand reproduce the observed evolution of the AM in a Monte-Carlofashion.We use the median change of angle and direction as a functionof halo mass, interval of redshift and change in halo mass to attemptto reproduce the evolution of the AM vector. We develop a Monte Carlo mock random sample (MC-Halo samplefrom this point on), to predict the change in the angle and directionof the AM vector of a dark matter halo, using the Bolshoi simula-tion. We later test the resulting model on the MS-I.To create this Monte-Carlo simulation, we measure the me-dian change of angle and direction that a ff ect a halo in the Bolshoisimulation as a function of its mass, its change of mass in a given Figure 9.
The angle between two consecutive changes of AM direction(taken throughout three consecutive snapshots) as a function of the look-back time (t) for di ff erent halo mass ranges as labelled. The halo mass andthe timestep between snapshots have a strong influence in the change of di-rection; this is easily seen at ∼ Gyr (also in Fig. 7). The grey lines showthe predictions of the random samples with a banning angle ( φ ) as labelledin the right part of the plot. By definition, the median direction of a mockrandom sample with a fixed banning angle will be not a ff ected by the lengthof the timestep or the halo mass, and behaves as h DIR inst . i = φ/ timestep and the change of M ∗ in that snapshot. To avoid the con-tamination of haloes that su ff er from spurious mass changes, weuse the cleaned halo sample to create this simulation.The predictions of the angular separation since a look backtime of 8.5 Gyrs for the full halo sample and for the cleaned halosamples for the are shown in the top and bottom panel of Fig. 10. Tocalculate the change of angle and the change of direction, we usetime intervals in the Bolshoi simulation that are equal to 7 consecu-tive snapshots (similar to one MS-I snapshot time interval) to avoidthe noise of the income and outcome of mass that typically a ff ecthaloes at short timescales. We manage to reproduce the evolutionto 0.05 accuracy in cos ( α ) for most cases. We also test the perfor-mance of the MC-Halo sample run with tables that use all haloes c (cid:13) , 1–12 S. Contreras et al.
Figure 10. (Top) Similar as the main panel of Fig. 4. The predictions ofthe MC-Halo sample are shown as dotted lines. The change of angle anddirection are calculated using the change of mass and redshift in consecutivesnapshots. (Bottom) Same as the top panel, for but for the cleaned halosample. (and not only the cleaned halo sample) finding similar results to theones presented here.We test the performance of these tables by following galax-ies at other initial times. The predictions of the angular separationsince a look back time of 6.5 and 11.2 Gyrs for the cleaned halosamples are shown in the top and bottom panel of Fig. 11, respec-tively. We succeed at reproducing the evolution to 0.05 accuracyin cos ( α ) for an initial look back time of 6.5 Gyrs, and to 0.1 foran initial look back time of 11.2 Gyrs. Finally, in Fig. 12 we testour MC-simulation over the cleaned halo sample of the MS-I. TheMC-Halo sample reproduces the same trends in the evolution of theangular separation of the real halo sample, but over predicts the an-gular separation by ∼ . α . This di ff erence could be causedbecause the clean mechanism does not work properly on the MS-I,as it shows in Fig. 2. A constant income and outcome of mass willartificially move the AM vector of the MC-Halo sample from itsoriginal position, making halos appear to have DIR closer to ran-dom than they truly have, which is consistent with what is shownhere. Nevertheless, we find that a di ff erence of 0.1 in cos α in 8.5Gyrs of evolution is acceptable considering that this is a di ff erentsimulation from the one we used to create the tables, with a dif-ferent volume, resolution and that the haloes are identified with adi ff erent halo finding algorithm.To provide a full evolutionary model of the AM vector of darkmatter haloes, we also develop a Monte Carlo simulation that pre-dicts the evolution of the modulus of the AM as a function of thesame parameters used before. In addition, we use a modificated ver-sion of the relationship found by Catelan & Theuns (1996, hereafterC&T96) that links the modulus of the specific AM (sAM), | j | , withit halo mass. The model for the growth of the angular momentum ofhalos of C&T96 has been tested against modern dark matter simu-lations (e.g. Zavala, Okamoto & Frenk 2008; Book et al. 2011) andfor hydrodynamic simulations (eg. Zavala et al. 2016), and proved Figure 11.
Similar as main to the bottom panel of Fig. 10, but for an initialtime of 6.5 (top) and 11.2 Gyrs (bottom).
Figure 12.
Similar as Fig. 10, but for the MS-I to be a good approximation. This relation assumes that, for a fixedredshift | j / h − M pc km / s | = p ( M / h − M ⊙ ) q , (7)with p and q constants. We can add a temporal dependence to p and q , | j / h − M pc km / s | = p ( a ) ( M / h − M ⊙ ) q ( a ) , (8)with log ( p ( a )) = ( log ( p ) if a > a log ( p ) ( a − a ) + log ( p ) if a a , (9)and c (cid:13) , 1–12 ngular momentum evolution in haloes Figure 13.
The top panel shows the median value of the sAM for haloedsample between a lookback time t = ff erence between the predicted modulus and the real modulusof the sAM vector. q ( a ) = ( q if a > a q ( a − a ) + q if a a , (10)Here, a is the expansion factor and p , p , q , q and a areconstants. This is slightly di ff erent to what is proposed in C&T 96,where they assume the value of q constant, close to 2 / | j |− M relation for di ff erent values of the scale factor, and we foundthat log ( p ), log ( p ), q , q and a are: -8.01, -0.77, 0.68, 0.09 and0.71. This fit was done for values of a between 0.3 and 1. Liao et al.(2015) also look into the evolution of p and q finding similar trendsto the ones found by us.To create the MC-Halo sample, with either the tables or usingthe analytic expression, we assume that: | j ( t + ∆ t , M + ∆ M ) | = | j pred ( t + ∆ t , M + ∆ M ) || j pred ( t , M ) | | j ( t , M ) | , (11)where j is the sAM of the halo and j pred the predicted sAM usingthe method explained above. By doing this, we can maintain thedispersion of the | j | − M relation as we evolve the AM throughtime. This will also preserve most of the original spin distributionof haloes. The evolution of | j | since a look back time of 8.5 Gyrsand the prediction of the MC-Halo sample using the tables and themodify expression of C&T 96 is shown in the top panel of Fig. 13.We successfully reproduce the evolution of the median value of thesAM modulus of Bolshoi halos to an accuracy better than 10% forthe modified version of C&T 96, and 20% for the MC-Halo sample.When we compare the evolution obtained via these two methodswith individual haloes (instead of the median growth), we noticethat our prediction di ff ers on average by 60% (50%) in 8.5 Gyrs ofevolution for the MC-Halo (C&T 96) samples, as is shown in thebottom panel of Fig. 13. This is acceptable if we consider that bothtechniques follow the modulus of | j | in a statistical way. A public version of the tables used in all these calculations,and the code necessary to create an MC-Halo sample are now pub-licly available in https://github.com/hantke/J_Tables As was noted previously when analysing Fig. 6, there is a strongdi ff erence in the evolution of the direction of the AM of haloes ofequal mass in simulations with di ff erent resolution / volumes. Thehaloes in the MS-I showed a stronger change of direction than thosein MS-II of similar mass. We pointed out that this could be due tothe environment since in the MS-II such haloes correspond to themost massive ones, which probably lie in knots of the cosmic web,whereas in MS-I the haloes being compared can lie on more diverseenvironments.In this section we study this issue using the Bolshoi simula-tion, classifying the environment of haloes into voids, walls, fila-ments and knots, using the T-Web algorithm (Forero-Romero et al.2009; Ho ff man et al. 2012) publicly available in the CosmoSim vir-tual observatory . The T-Web is defined as the Hessian of the grav-itational potential, and is computed on a 256 cubic grid, which inthe case of Bolshoi corresponds to cells with ∼ h − M pc a side.The tensor has three real eigenvalues in each grid. If 3, 2, 1 or 0of those eigenvalues are above a given threshold, then the grid isclassified as a knot, filament, wall or void, respectively. We choose0 . =
0, the environment of the haloes was assigned depending on thelocation of haloes at z = z = ~ J are a ff ected by thedi ff erent definition of environment available in the literature. Apart from the environment, we are interested in revealing otherdependencies on intrinsic properties, such as the amplitude of theAM. Fig. 15 shows the angular separation of haloes between a look-back time of 8.5 Gyrs and their descendant at a time “t” for haloesin the lower and higher 20 percentiles of | ~ J | (shown in dotted anddashed lines, respectively). The angular separation of ~ J betweenthese two snapshots shows a strong dependence on its modulus.We find di ff erences between these two bins of | ~ J | of around 0.4 in cos ( α ) regardless of halo mass. This dependence is also valid forthe sAM as we are comparing haloes of equal mass. Haloes with c (cid:13) , 1–12 S. Contreras et al.
Figure 14.
Similar to the main panel of Fig. 4. Haloes that at z = Figure 15.
Similar to the main panel of Fig. 4. Halos that have the 20%lowest and higher AM modulus are shown in dotted and dashed lines re-spectively. lower AM show a higher angular separation with their initial po-sition compared to the haloes with a higher AM. Along with theprevious tables of α and the direction as function of the halo mass,the redshift interval and the change of halo mass, we publish α and the direction as a function of these properties plus the modu-lus of the sAM so that this can be used in the generation of mockhalo samples using the spin distribution of the dark matter haloes(Gardner 2001). SUMMARY AND CONCLUSIONS
We studied how the orientation and the modulus of the AM vectorevolve in time. We use three dark matter simulations (the Bolshoisimulation, the MS-I and the MS-II) and select haloes with a min-imum of 1000 particles to have a good measurement of the AMvector. We found that there exist physical and numerical factorsthat a ff ect the evolution of the AM vector. Here we summarise themain conclusions of our work: • Haloes are a ff ected by spurious changes of mass. This ispresent in all the simulations studied here, but strongly on the MS-Iand the MS-II (where haloes were identified using solely the po-sition distribution of particles). These changes of mass cause ar-tificial slews on the AM vector of haloes. By cleaning the haloesa ff ected by spurious mass changes (using Eq. 2), we also removefrom the sample many low mass haloes that have a more passiveevolution of their AM vector direction. We test other ways of clean-ing these haloes, but in all cases, the resulting e ff ect is the removalof haloes that have a passive evolution. These e ff ects (the spuri-ous mass changes and the biased cleaning method) should be takeninto account before using the AM direction coming from any darkmatter simulation. • We measure the angle separation that a ff ects an AM vectorin time. We found that this angle separation is larger than the oneobtained from uncorrelated direction changes in consecutive snap-shots (i.e. random walks; we refer to this as the mock random sam-ple). This means that the change in the direction of the AM vectoris correlated in time. We also found that more massive haloes havea stronger correlation in the evolution of their AM direction. • The change of angle that a ff ects the AM vector has a strongdependence on the mass change su ff ered by the halo, as well as thetime interval in which this change of angle occurs (expressed bythe change in M ∗ ). The angle of change of the AM vector displaysa weak dependence on halo mass. • The change of direction that a ff ects the AM vector has a strongdependence on the change of mass that a ff ects the halo, the changeof M ∗ in which this change of angle occurs and also the halo mass. • Using the dependence mentioned above and the Bolshoi sim-ulation, we create a Monte Carlo mock random sample (MC-Halosample), capable of inferring the change of angle and the changeof direction of the AM vector. With this simulation, we reproducethe angular separation since a look back time of 8.5 Gyrs to todaywith an accuracy of 0.05 in cos ( α ) for most cases. The performanceof this MC-Halo sample is much better than the case in which wefollow the change of angle between consecutive snapshots in an ex-act way, but without making use of the information of the changeof orientation (equivalent to the randomly oriented case of Fig. 4,Fig. 5, Fig. 6). In this case one underpredicts the total evolution ofthe orientation of the angular momentum and and can hardly dis-tinguish between the evolution of haloes of di ff erent masses. • We test our MC-Halo sample on the MS-I simulation. We wereable to reproduce the general behaviour of the evolution of the an-gular momentum vector with an accuracy of ∼ cos ( α ). Thesepredictions are not as good as for the Bolshoi simulation, but this isdue to the merger trees, halo mass range, resolution and halo finderalgorithm applied to this simulation being di ff erent to the Bolshoisimulation, the one used to make the MC look-up tables. However,the predicted evolution of the vector is still very well reproduced,and we consider it to be a better alternative to using the angularmomentum vector measured with a small number of particles (as ithas been traditionally done in semi-analytic models). • We also created a MC-Halo sample that reproduces the evolu- c (cid:13) , 1–12 ngular momentum evolution in haloes tion of the modulus of the SAM of haloes, along with an analyticexpression based on the work of C&T96. We reproduce the modu-lus of ~ j since a look back time of 8.5 Gyrs to the present day witha precision of more than a 90% for the cleaned halo sample. • The environment has a strong role in the evolution of theAM vector direction. Haloes in denser environment show a wilderevolution in the direction of their AM vector ( cos ( α ) = .
42 ina 8.5 Gyrs evolution), compared to the median angle separation( cos ( α ) = .
59) and the change su ff ered by halos in low densityenvironments ( cos ( α ) = . • The evolution of the AM vector direction is also strongly cor-related with the amplitude of the AM vector. Haloes with higherAM modulus show a weaker evolution of their direction. For an 8.5Gyrs evolution, we found a di ff erence in the angular separation ofthe AM vector of around 0.4 in cos ( α ) regardless of halo mass. • The main trends reported here for haloes with di ff erent num-bers of particles appear to be the same. There is a dependence inmost of the relations with halo mass, but this dependence is smoothwith no obvious discontinuities that suggest resolution and / or nu-merical limitation. Thus, if we apply a more conservative cut inthe number of particles than the one used throughout the paper (eg.40,000 particles as suggested by Benson 2017), the main conclu-sions of this work would remain unchanged.The results found in this work show that the evolution of thedirection of the AM vector is quite complex but that can be well de-scribed by the relative mass change and time in which the accretionhappens. Using that dependence, we constructed look up tables thatcan be used to produce Monte-Carlo simulations of the evolution ofthe AM vector of halos in low-resolution simulations. This modelcan be added to any model that makes use of halo merger trees,such as SAMs, HODs and SHAMs, regardless of the halo finder al-gorithm or the minimum number of particles used for identificationin their dark matter simulations. This method provides an alterna-tive to using direct measurements performed with small numbers ofparticles ( < , ff ersless from shot noise (to the extent that the mass of halos su ff ers),and is able to describe the evolution of the angular momentum am-plitude and direction, including AM flips and slews that are impor-tant for the evolution of disk sizes and associated phenomena, i.e.star formation rates, disk instabilities, merger driven starbursts andstellar feedback. ACKNOWLEDGEMENTS
This work was possible thanks to the e ff orts of Gerard Lem-son and colleagues at the German Astronomical Virtual Observa-tory (GAVO) in setting up the Millennium Simulation database inGarching. We thank the people in charge of the CosmoSim webpage and the responsible of uploading the T-Web to that server.We thank Cedric Lacey, Carlton Baugh, Peder Norberg, and JaimeForero-Romero for many useful discussions. We acknowledge sup-port from the European Commission’s Framework Programme 7,through the Marie Curie International Research Sta ff ExchangeScheme LACEGAL (PIRSES-GA-2010-269264). SC further ac-knowledges support from CONICYT Doctoral Fellowship Pro-gramme. NP & SC acknowledge support from a STFC / Newton-CONICYT Fund award (ST / M007995 / REFERENCES
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APPENDIX A: CALCULATION OF M ∗ Here we present the procedure to calculate M ∗ , following the workof (Rodr´ıguez-Puebla et al. 2016) for the calculation of M C : σ ( M ∗ , a ) = , (A1)where a is the expansion factor and σ ( M ) is the amplitude of per-turbations calculated as σ ( M ) = D ( a ) D (1) ! π Z ∞ k P ( k ) W ( k , M ) dk . (A2)Here, P ( k ) is the power spectrum of perturbations, W ( k , M ) is theFourier transform of the real-space top-hat filter corresponding to asphere of mass M , and D ( a ) is the amplitude of the growing mode D ( a ) ≡ a g ( a ) = Ω M , Ω Λ , ! / √ + x x / Z x x ′ / [1 + x ′ ] / dx ′ , (A3) x ≡ Ω Λ , Ω M , ! / a . (A4)Here, Ω Λ , and Ω M , are the density contributions of matter and thecosmological constant at z = g ( a ) is the linealgrowth factor (Hamilton 2001; Klypin, Trujillo-Gomez & Primack2011). c (cid:13)000