Assembly bias in the clustering of dark matter haloes

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by  Liang Gao

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev1.4)
Assembly bias in the clustering of dark matter haloes
⋆
Liang Gao 1 , Simon D. M. White2
1 Department of Physics, University of Durham, South Road, Durham, DH13LE
2 Max–Planck–Institut fu¨r Astrophysik, D-85748 Garching, Germany
5February2008
7
0
ABSTRACT
0
We use a very large simulation of structure growth in a ΛCDM universe – the Mil-
2
lennium Simulation – to study assembly bias, the fact that the large-scale clustering

n of haloes of given mass varies significantly with their assembly history. We extend
a earlier work based on the same simulation by superposing results for redshifts from
J
0 to 3, by defining a less noisy estimator of clustering amplitude, and by considering

2 halo concentration, substructure mass fraction and spin, as well as formation time,
2
as additional parameters. These improvements lead to results with less noise than

previous studies and coveringa wider range of halo masses and structural properties.
2
We find significant and significantly different assembly bias effects for all the halo
v
properties we consider, although in all cases the dependences on halo mass and on
1
redshift are adequately described as a dependence on equivalent peak height ν(M,z).
2
9 Theν-dependencesfordifferenthalopropertiesdifferqualitativelyandarenotrelated
1 as might naively be expected given the relations between formation time, concentra-
1 tion, substructure fraction and spin found for the halo population as a whole. These
6 results suggest that it will be difficult to build models for the galaxy populations of
0 dark haloes which can robustly relate the amplitude of large-scale galaxy clustering
h/ to that for mass clustering at better than the 10% level.
p
Key words: methods: N-body simulations – methods: numerical –dark matter –
-
o galaxies: haloes – galaxies:clustering
r
t
s
a
:
v 1 INTRODUCTION (2006)demonstratedadependenceofhaloclusteringonhalo
i
X concentration,notingittohavesimilarstrengthathighand
Gao, Springel & White (2005) used the very large Millen-
low mass, but opposite sign; concentrated haloes are the
r nium Simulation of Springel et al. (2005) to demonstrate
a most clustered at low mass, but the least clustered at high
that the standard ΛCDM paradigm predicts the clustering
mass.Thisisalsosurprisingsincethereisatightcorrelation
ofdarkmatterhaloestodependnotonlyontheirmassbut
between halo formation time and halo concentration (e.g.
also on their formation time. The effect is strong for low-
Navarro, Frenk & White 1997; Wechsler et al. 2001; Zhao
mass haloes but weak or absent at high mass. This result
et al. 2003) so one might expect a similar dependence on
is surprising since it is incompatible with the standard ex-
the two properties. More recently, Bett et al. (2007) stud-
cursion set theory for the growth of structure (e.g. Bond
ied clustering as a function of halo spin and shape, find-
et al. 1991; Lacey & Cole 1993, Mo & White 1996), and it
ing stronger clustering at given mass for larger spin and
contradicts a fundamental assumption of the halo occupa-
for smaller major-to-minor axis ratio. (See also Hahn et al.
tion distribution (HOD) models often used to study galaxy
(2007) foraslightlydifferentapproachtostudyingenviron-
clustering,namelythatthegalaxycontentofahaloofgiven
mental effects on halo spin.) Interestingly, the dependences
mass is statistically independent of its larger scale environ-
on shape and spin are stronger for high-mass haloes.
ment (e.g. Kauffmann, Nusser & Steinmetz 1997; Jing, Mo
We use the term “assembly bias” to describe all such
&Boerner1998;Peacock&Smith2000;Bensonetal.2000;
dependences, since they show that the spatial distribution
Berlind et al. 2003; Yang,Mo & van den Bosch 2003).
of haloes depends not only on their mass but also on the
Subsequent studies confirmed this result (Harker et al.
details of their assembly history. Such details are undoubt-
2006; Zhu et al. 2006; Wechsler et al. 2006; Wetzel et al.
edly reflected in the galaxy populations they host, so one
2007; Jing, Suto & Mo 2007). In addition, Wechsler et al.
may expect assembly bias to affect galaxy clustering in a
waywhichisnoteasilyrepresentedinasimpleHODmodel.
A first assessment of the strength of such effects was made
⋆ Email:[email protected] byCroton,Gao&White(2006)usingadirectsimulationof
(cid:13)c 0000RAS
2 L. Gao & S. D. M. White
galaxy formation within the Millennium Simulation, while (together with galaxy data from two independent galaxy
possible observational evidence for them has been cited by formationmodels)throughadatabaseathttp://www.mpa-
Yang,Mo&vandenBosch(2006),Blanton,Berlind&Hogg garching.mpg.de/millennium
(2007) and Berlind et al. (2007). Some recent theoretical
work has explored extensions of excursion set theory which
may allow a description of halo assembly bias (Wang, Mo 3 HALO ASSEMBLY BIAS
& Jing 2007, Sandvik et al. 2007, Zentner 2007) but it re-
Intheone-dimensionalexcursionsetmodelforstructurefor-
mainsunclearwhethertheseapproachescanaccountforthe
mation (e.g. Bond et al. 1991; Lacey & Cole 1993), the for-
results discussed above and in this paper.
mation history of a halo is encoded in the random walk at
In this Letter we extend the work of Gao, Springel &
higher mass resolution than that which defines the halo it-
White (2005), again using the Millennium Simulation, by
self,andisthusstatisticallyindependentoftheenvironment,
definingalessnoisyestimatorofclusteringstrength,bycom-
which is encoded in the random walk at lower resolution
bining results for a numberof redshifts, and byconsidering
(White1996). This is inconsistent with thedependenceson
clusteringasafunctionofhalopropertiesotherthanforma-
formationtimeandconcentrationfoundinsimulations(Gao
tion time. As a result, we can study assembly bias in more
et al. 2005; Wechsler et al. 2006). Here we extend the work
detail and over a substantially wider halo mass range than
ofGaoetal.(2005) bydefiningalessnoisymeasureofclus-
intheearlierpaper.InSection2wesummarisetherelevant
tering strength, by analysing data at z=1, 2 and 3 as well
properties of the simulation and of the halo database that
as z = 0, and by examining assembly bias as a function of
we study. In Section 3, we explore halo assembly bias as
properties other than formation time.
a function of halo mass and of a variety of halo structural
The specific halo properties which we will consider in
properties.Finallywegiveashortsummaryanddiscussion.
thisLetter are:
(1) FormationTime FollowingGaoetal.(2004,2005),
we define the formation time of a dark halo as the redshift
2 THE SIMULATION
when half of its final mass was first assembled into a single
The Millennium Simulation was carried out by the Virgo object. We use the stored merging tree to find the earliest
Consortium in 2004 on an IBM Regatta system at the time when its most massive progenitor had more than half
Max Planck Society’s supercomputer centre in Garching the final mass, then interpolate linearly between this and
(Springel et al. 2005). It adopted concordance values for the immediately preceding output to estimate the redshift
theparameters of aflat ΛCDMcosmological model,Ωdm = when this progenitor had exactly half the final mass. This
0.205, Ωb = 0.045 for the current densities in Cold Dark we defineas theformation time.
Matter and baryons,h=0.73 for thepresent dimensionless (2) Concentration We characterise the concentration
value of the Hubble constant, σ8 = 0.9 for the rms linear ofadarkhalobytheratioVmax/V200ofthepeakvalueofits
mass fluctuation in a sphere of radius 8h−1Mpc extrapo- circular velocity curve to the circular velocity at r200. Here
lated to z = 0, and n = 1 for the slope of the primordial Vc(r) = (GM(r)/r)1/2 and V200 = Vc(r200), where r200 is
fluctuation spectrum. The simulation followed 21603 dark the radius enclosing a mean overdensity 200 times the crit-
matter particles from z = 127 to the present-day within ical value. Vmax is the maximum value of Vc for r < r200,
a cubic region 500h−1Mpc on a side. The individual parti- evaluated using only particles bound to the main subhalo
cle mass is thus 8.6 108h−1M⊙. The gravitational force of the FOF halo. This definition has the advantages of be-
had a Plummer-equiv×alent comoving softening of 5h−1kpc. ing robust and of not requiring any model to be fit to the
Initial conditions were set using the Boltzmann code CMB- simulation data.
FAST (Seljak & Zaldarriaga 1996) to generate a realisation (3) Subhalo mass fraction within r200 As one mea-
of the desired power spectrum which was then imposed on sure of the amount substructure, we consider the fraction
a glass-like uniform particle load (White 1996). Fr200 ofthetotalmasswithinr200 intheformofself-bound
TheTREE-PMN-bodycodeGADGET2(Springel2005) substructures detected by SUBFIND. Note that we exclude
was usedto carry outthesimulation andthefull datawere the main subhalo, which we identify with the body of the
stored at 64 times spaced approximately equally in thelog- halo itself, and that SUBFIND only catalogues subhaloes
arithm of the expansion factor at early times and at ap- made up of 20 or more particles. In practice, we find that
proximately 200 Myr intervals after z = 1. At run time all thismeasurecanonlybeusedeffectivelytorankhaloeswith
collapsed haloes with at least 20 particles were identified FOFmassesabove5000particles.Wewillnotconsiderlower
using a friends-of-friends (FOF) group-finder with linking masshaloeswhenstudyingclusteringasafunctionofFr200.
parameter b=0.2 (Davis et al. 1985). Post-processing with (4) Mass fraction in the main subhalo As an al-
the substructure algorithm SUBFIND (Springel et al. 2001) ternative measure of the amount of substructure in a halo,
allowedavarietyofinternalstructuralpropertiestobemea- we use the ratio F of the mass of the main self-bound
FOF
sured for all these haloes and for their resolved subhaloes. subhalo to the mass of the original FOF halo. This defini-
This in turn allowed trees to be built which store detailed tion contrasts with the previous one in being sensitive to
assembly histories for every object and its substructure. In neighboringstructuresbeyondr200 whichare“fortuitously”
this paper we concentrate on haloes with small FOF mass joinedtothemainhalobytheFOFlinkingprocedure.With
corresponding to 65 particles or more. At redshifts 3, 2, 1 this measure we can study clustering as a function of sub-
and0thereare4.6 106,5.8 106,6.2 106and5.7 106such structurefraction to a FOF mass limit of 2000 particles.
× × × ×
FOF haloes, respectively. The halo/subhalo data and their (5) Halo spin Halo spin can be conveniently defined
associated structureformation treesarepublicallyavailable through λ= J~/(√2MVr200) where angular momentum J~
| |
(cid:13)c 0000RAS,MNRAS000,000–000
Halo assembly bias 3
Figure1.Biasfactorasafunctionofhalomassandhaloproperties.Halomassisgivenparametricallythroughtheequivalentpeakheight
ν(M,z)=δ(M)/δc(z). The additional properties inthe firstfive panels are: (a) formationredshift, (b) concentration, (c) substructure
massfractionwithinr200,(d)massfractioninthemainsubhalooftheFOFhalo,and(e) spin.Thesymbolsrepeatedinallfivepanels
are bias factors defined for all haloes of the relevant mass. The solid and dashed curves in these panels are bias factors for haloes in
the lowerandupper 20% tailsofthe distributions ofthe particular propertyconcerned. Panel (f) plots halomassas afunctionof ν at
the fourredshiftswe combined tomake these plots.Both the linesinthis panel and the symbolsinthe earlierpanels arecolour-coded
according to redshift, as indicated by the labels. The numbers in parentheses following the redshift labels in each panel give the rms
scatter in b among 100 bootstrap resamplings of the halo subsamples used to obtain b-values for the solid and dashed curves at the
centralandatthelargestν-valuesplottedforeachredshift.
(cid:13)c 0000RAS,MNRAS000,000–000
4 L. Gao & S. D. M. White
and mass M are the values within a sphere of radius r200 massesabove1015h−1M⊙ at z=0.Thereisahintthatthe
and the circular velocity V is also evaluated at this radius dependencemayreverseatthesemasses, withyounghaloes
(Bullock et al. 2001). beingmore clustered than old ones as argued byJing, Suto
& Mo (2007), but any such reversal is clearly very weak.
As in Gao et al. (2005), we examine assembly bias by
Theirsuggestion thatreversaloccursnearν 1.7 isclearly
comparingthespatialclusteringofasubsetofhaloesofgiven ∼
not supported by our data.
masstothatofthesetasawhole.Intheearlierpaperwees-
The dependence of assembly bias on concentration dif-
timatedabiasfactorbforeachsubsetofhaloesasthesquare
fers qualitatively from that on formation time. The most
root of the separation-averaged ratio of its spatial autocor-
concentrated haloes are the most clustered for ν < 1, but
relation function tothatofthedarkmatter.Thisestimator
theyaretheleastclusteredforν >1.Wehavecheckedthat
becomes quitenoisy when thenumberof haloes in thesub-
at each ν our halo samples obey the relation between con-
setissmall.Inthispaperweprefertoestimatebastheratio
centration and formation time first pointed out by Navarro
of the halo-mass cross-correlation to the mass autocorrela-
et al. (1997); concentrated haloes of a given mass typically
tion.Specifically,weestimatebastherelativenormalisation
formed earlier than less concentrated haloes. Nevertheless,
factor which minimizes the mean square of the difference
there is clearly a range of ν (roughly 1 < ν < 2) where
ltohgeξchomm−ovloigngbξsmepmafroartifoonurraenqugeal6whi−d1tMhbpicn<sinrl<og2r0shp−a1nMnipncg. clustering is significantly stronger for older yet also for less
concentrated haloes. Overall, our results for concentration
Thisestimatorhasimprovednoisecharacteristicsbecauseof
agree well with those of Wechsler et al. (2006) and Jing,
thelargenumberofdarkmatterparticlesavailable.Accord-
Suto&Mo(2007),thoughtheyarelessnoisybecauseofthe
ing to standard halo bias models it should be equivalent to
betterstatistics provided by theMillennium Simulation.
the earlier estimator (e.g. Mo & White 1996) and we have
Thedependenceofassemblybiasonsubstructureisdif-
verified that the two give consistent b values for our halo
ferentagain inshape,andmoreoverdiffersbetweenourtwo
data. In these models the large-scale bias depends on mass
substructuremeasures.Forboth,thedependencevariesonly
and redshift through theequivalent “peak height”
slowly with ν, and haloes with more substructure are al-
mostalwaysthemorestronglyclustered.Thedependenceis
ν(M,z)=δ (z)/σ(M), (1)
c
stronger, however, when substructure is measured by the
where σ(M) is the rms linear overdensity (extrapolated to fraction of FOF mass in the main subhalo (FFOF) than
z=0)withinaspherewhichinthemeancontainsmassM, when it is measured by the subhalo mass fraction within
and δc(z) is the linear overdensity threshold for collapse at r200 (Fr200),anditgetsweakerwithincreasingν inthefirst
redshiftz,again extrapolatedtothecorrespondingvalueat case, while it strengthens in the second. Indeed, thedepen-
z =0. σ(M) depends only on thepower spectrum of initial denceappearstoreverseatν <1intheFr200case,although
densityfluctuations,whileδ (z)dependsonthecurrentden- thisneedstobeconfirmedbyasimulation of higherresolu-
c
sities in gravitating matter and dark energy (e.g. Eke, Cole tion. Note that at all ν our halo samples obey the kind of
& Frenk 1996). Gao et al. (2005) showed that halo cluster- correlationofsubstructurewithformationtimeandconcen-
ingintheMillenniumSimulationobeysthisscalingoverthe tration pointed out out by Gao et al. (2004). Haloes with
redshiftrange0 z 5.Wewilluseittosuperposeresults moresubstructuretendtobeyoungerandtohavelowercon-
≤ ≤
from different redshifts. centrations. Thus the dependence of assembly bias on sub-
Halo assembly bias is shown in Fig. 1 as a function of structure is the opposite of what might naively have been
ν(M,z) and of the various halo properties discussed above. inferred from its dependence on formation time, and also
These plots combine results for redshifts 0, 1, 2 and 3. In differs qualitatively from that on concentration.
each of the first five panels we repeat bias values for all Assemblybiasalsodependsonhalospin;rapidlyrotat-
haloes of a given mass from Figure 1 of Gao et al. (2005). ing haloes cluster more strongly than slowly rotating ones,
The haloes in each ν bin are ranked in terms of each of the withlittledependenceonν.Thisagreesreasonablywellwith
five properties in turn, and bias values are then estimated theresultsofBettetal.(2007)whofindasomewhatstronger
using our cross-correlation method for subsets consisting of trend with halo mass. This may reflect their slightly differ-
haloes in the upper and lower 20% tails of the distribution ent(andcleaner)definitionofhalospin,theirdifferent(and
inthisadditionalproperty.Dashedlinesshowbiasvaluesfor noisier) estimator of bias, or the fact that they only used
thehaloes with thehighest formation redshifts, thehighest numerical data for z =0.
concentrations, the most substructure and the most spin. Theseplotsshowclearlythatassemblybiasdependson
Solid lines give bias values for the opposite tails. Colours the physical properties of haloes in a complex way. Typical
(both for symbols and for lines) denote the redshift of the effects are at the 10 to 30% level in b (20 to 70% in corre-
outputfrom whichthedataweretaken,asindicated bythe lationorpowerspectrumamplitude)anddependonseveral
labels.Notethegoodagreementbetweenresultsfordifferent additional parameters at fixed halo mass. Our results are,
redshifts where theseoverlap. Finally, thelast panelof Fig. however, consistent with the hypothesis that dependences
1 gives halo mass as a function of ν for the four redshifts onmassandredshiftcanbecombinedintoadependenceon
usedtomakethisfigure.Withthesecurvesonecanconvert thesingle parameter ν.
thex-axesin theother panels to halo mass for 0 z 3. Finally we note that because of the large size of the
≤ ≤
The results for formation time in Fig. 1 confirm those Millennium Simulation the formal errors on thebias curves
ofGao etal(2005) andextendthem toconsiderably higher of Figure 1 are very small. We can see this in several ways.
ν.Assemblybiasisstrongonlyforlow-masshaloes,withthe There is little fluctuation along each dashed or solid curve,
oldest haloes being themost clustered. It is weak or absent even though neighboring points refer to disjoint ν ranges
at the highest masses. The largest ν values correspond to with no halos in common, and so should have uncorrelated
(cid:13)c 0000RAS,MNRAS000,000–000
Halo assembly bias 5
sampling uncertainties. Then we have estimated uncertain- for checking our numerical estimates of cross-correlations.
ties for each dashed and solid curve by measuring the rms Databases containing halo/subhalo data at all times, as
scatter in b among 100 bootstrap resamplings of the halo well halo formation trees and galaxy properties for two
subsamplescorrespondingtothecentralandthelargestval- independent galaxy formation simulations are available at
ues of ν plotted. In every case the scatter at corresponding http://www.mpa-garching.mpg.de/millennium
points on the solid and dashed curves is consistent within
the estimation uncertainties. We thus average these values
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ACKNOWLEDGEMENTS
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We thank the referee for helpful comments. The authors ZentnerA.R.,2007,IJMPDinpress,astro-ph/0611454
are grateful to the Virgo Consortium, and in particular to Zhao D. H., Mo H. J., Jing Y. P., Boerner G., 2003, MNRAS,
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Volker Springel, for the tremendous amount of work they
ZhuG.,ZhengZ.,LinW.P.,JingY.P.,KangX.,GaoL.,2006,
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make its results available for analysis. GL also thanks Car-
los Frenk, Shaun Cole, Adrian Jenkins and Cedric Lacey
for stimulatingdiscussions. Weare grateful toEric Hayashi
(cid:13)c 0000RAS,MNRAS000,000–000