Dark matter halo abundances, clustering and assembly histories at high redshift

0.43 MB·

by  J. D. Cohn

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed1February2008 (MNLATEXstylefilev2.2)
Dark matter halo abundances, clustering and assembly
histories at high redshift
J.D. Cohn1 and Martin White2
1Space SciencesLaboratory,
2Departments of Physics and Astronomy,
University of California, Berkeley, CA 94720
8
0
0 1February2008
2

n
a ABSTRACT
J We use a suite of high-resolution N-body simulations to study the properties, abun-

4 danceandclusteringofhighmasshalosathighredshift,includingtheirmassassembly
historiesandmergers.Wefindthattheanalyticformwhichbestfitstheabundanceof

] halos depends sensitively on the assumed definition of halo mass, with common defi-
h
nitions of halo mass differing by a factor of two for these low concentration, massive
p
halos.Asignificantnumberofmassivehalosareundergoingrapidmassaccretion,with
-
o major merger activity being common. We compare the mergers and mass accretion
r histories to the extended Press-Schechterformalism.
t
s We consider how major merger induced star formation or black hole accretion
a may change the distribution of photon production from collapsed halos, and hence
[
reionization, using some simplified examples. In all of these, the photon distribution

2 for a halo of a given mass acquires a large scatter. If rare, high mass halos contribute
v significantlytothephotonproductionrates,thescatterinphotonproductionratecan
8 translate into additional scatter in the sizes of ionized bubbles.
0
1 INTRODUCTION estimates of the number and properties of virialized dark
2
0 matter halos at high redshift, the focus of this paper. At
. Observations of the anisotropy of the cosmic microwave z =10, halos with M > 109h−1M⊙ are expected to be bi-
06 bknacokwglreodugnedof(CthMeBve)rryadeaiartlyioUnnhiavveersegiavnendudsraumnaptrieccaelldyencotend- ased similarly to very massive clusters (M > 1015h−1M⊙)
today, with the most massive and recently formed halos
7 firmed the picture of large-scale structure as arising from
growing rapidly and merging frequently. We explore some
0 the gravitational amplification of small perturbations in a
: properties of these collapsed halos at a high redshift using
v Universe with a significant cold dark matter component
a suite of high resolution, collisionless, N-bodysimulations.
i (Smoot et al. 1992). In this model the ionization history
X We pay particular attention to merger rates and mass ac-
of the Universe has two main events, a ‘recombination’ at
cretionhistorieswithaneyetoapplicationsforreionization.
r z 103inwhichitwentfromionizedtoneutralanda‘reion-
a ∼ WealsocomparetheN-bodyresultswiththepredictionsof
ization’ duringz 7 12inwhich theradiation from early
∼ − theoft-used Press & Schechter(1974) formalism.
generationsofcollapsedobjectswasabletoionizetheinter-
If halo mergers are accompanied by a temporary in-
galactic medium. The former event is strongly constrained
crease in photon production (due either to starbursts
by the CMB. A new generation of instruments will soon
or increased black hole accretion e.g. Carlberg 1990;
allow us to probe this second event: “the end of the dark
Barnes & Hernquist 1991, 1996; Mihos & Hernquist 1994,
ages” (for reviews of reionization see e.g. Barkana & Loeb
1996; Kauffmann& Haehnelt 2000; Cavaliere & Vittorini
2001; Cooray & Barton 2006; Fan,Carilli & Keating 2006;
2000)weexpectreionization tobeinfluencedbythemerger
Furlanetto, Oh & Briggs 2006).
and accretion history of dark matterhalos, beyondjust the
Since at reionization a very small fraction of the mass
factthatmoremassivehalosemitmorephotons.Withasim-
affectedeachandeverybaryonintheUniverse,reionization
ple model of star formation we show that merger-induced
is particularly sensitive to the distribution and behavior of
scatter in photon production may be significant, with the
collapsed structure.Weexpectthattheionizingsourcesare
production rates acquiring a substantial tail to large pho-
situated in large (Tvir > 104K or M > 107h−1M⊙) dark ton production rates. Since the massive halos are relatively
matterhaloswherethegascancoolefficientlytoformstars1.
rare,thisindividualhaloscatterisexpectedtotranslateinto
Models for the sources of reionization thus often start with
a scatter of photon production rates inside ionized regions,
changing thebubbledistribution.
1 WewillonlyconsiderPopIIstarshere;PopIIIstars,whichcan Theoutlineofthepaperisasfollows. In 2wedescribe
§
formintheabsenceofmetalsinsmallerhalos,areexpectedtobe the N-body simulations. The basic halo properties are de-
lesslikelybyredshift10(Yoshida,Bromm&Hernquist2004). scribed in 3 along with the results for mergers and mass
§
(cid:13)c 0000RAS
2 Cohn & White
Figure 1.Illustrativemergertreesfortwohaloswithmassesof1.2(left)and2.9×1010h−1M⊙ (right).Timerunsupwardsinstepsof
10Myr,fromz=12.7(bottom) to z=10(top) and the ageof the Universe(inMyr)isshown atevery second step. Ateach timethe
area of the symbol is proportional to the halo mass, with masses decreasing to the right in each group, and lines show the progenitor
relationship.Theleftmostbranchshowsthemaintrunkofthetree.Thehaloatlefthasa(major)1:2mergeratthelasttimestep,while
themaintrunkofhaloatrighthasa1:2mergeratthefirsttimestep,a1:6twostepslaterandthenonlysmallermergersafterthat.
gains and the comparison to Press-Schechter. The conse- whichwewould,byeye,havecharacterized asseparate(see
quences of this merging in a simple model for photon pro- alsoDaviset al.1985;Cole & Lacey1996,forsimilardiscus-
ductionareelucidatedin 4andwesummarizeandconclude sion).Thisproblemismitigatedwithourmoreconservative
§
in 5. choice of b.
§
For each halo we compute a number of properties, in-
cluding the potential well depth,peak circular velocity, the
positionofthemostboundparticle(whichwetaketodefine
2 SIMULATIONS AND PARAMETERS
the halo center) and M , the mass interior to a radius,
180
Webaseourconclusionson5darkmatteronlyN-bodysim- r180, within which the mean density is 180 times the back-
ulationsofaΛCDMcosmology withΩ =0.25,Ω =0.75, ground density2. As discussed in White (2001, 2002) and
m Λ
h = 0.72, n = 0.97 and σ = 0.8, in agreement with a Hu & Kravtsov (2003), the choice of halo mass is problem-
8
widearrayofobservations.Theinitialconditionsweregener- atic and ultimately one of convention. We shall return to
ated atz =300usingtheZel’dovich approximation applied thisissue in the nextsection.
to a regular, Cartesian grid of particles. Our two highest Mergertreesarecomputedfromthesetofhalocatalogs
resolution simulations employed 8003 equal mass particles by identifying for each halo a “child” at a later time. The
(M = 2 106 and 1.7 107h−1M⊙) in boxes of side 25 childisdefinedasthathalowhichcontains,atthelatertime
and50h−×1Mpcwith Plu×mmerequivalentsmoothings of1.1 step, more than half of the particles in the parent halo at
and 2.2h−1kpc. They were evolved to z = 10 using the the earlier time step (weighting each particle equally). For
TreePM code described in White (2002) (for a comparison the purposes of tracking halos this simple linkage between
with other codes see Heitmann et al. 2007). We ran 3 ad- outputssuffices (notethat we do not attempt to track sub-
ditional, smaller simulations in a 20h−1Mpc box, one with halos within larger halos, which generally requires greater
6003 particles and two with 3003 particles (each started at sophistication). Two examples of the halo merger trees are
z =200). A comparison of the boxes allows us to check for given in Fig. 1, where we see a rich set of behaviours, in-
finite volume, finite mass and finite force resolution effects. cluding major and minor mergers and many body mergers.
Weshall comment on each where appropriate. From the merger trees it is straightforward to compute the
The phase space data for the particles were dumped timewhen a halo ‘falls in’ toa larger halo, thenumberand
at 15 outputs spaced by 10Myr from z = 12.7 to z = 10 masses of theprogenitors etc.
for all but the largest box. The lower resolution of the Due to finite computational resources, all N-body sim-
largest box makes it less useful for merger trees, so it was ulations must trade-off computational volume for mass res-
sampled for only subset of these output times, ending at olution. By running multiple simulations we can overcome
z = 10. For each output we generate a catalog of halos this to some extent, but not entirely. We have chosen to
using the Friends-of-Friends (FoF) algorithm (Davis et al. slightly under-resolve the low mass (Tvir 104K) halos in
≃
1985)withalinkinglength,b,of0.168timesthemeaninter- order to simulate a slightly larger volume, since our focus
particle spacing. This partitions the particles into equiv-
alence classes, by linking together all particle pairs sepa-
2 Notethisissimplyadefinitionofhalomass,notthehalofinder.
rated by less than b. The halos correspond roughly to par-
We still use FoF particles to define the group centers. However
ticles with ρ > 3/(2πb3) 100 times the background den-
≃ giventhecenterweusealloftheparticlesinthesimulationwhen
sity. We also made catalogs using a linking length of 0.2
determiningM180.OurM180 massesshouldthusbecomparable
times the mean inter-particle spacing, which we shall dis- tothesumoftheparticlesinanSO(180)group–acommondef-
cussfurtherbelow.WefoundthattheFoFalgorithmwitha initionthatemploysboththeSO(180)halofinderanddefinition
larger linking length had a tendency to link together halos ofmass.
(cid:13)c 0000RAS,MNRAS000,000–000
Dark matter halo abundances, clustering and assembly histories at high redshift 3
Figure 2. The peak height, ν – which governs the abundance, Figure 3. The mass functions for our box compared to the
clustering and merging behavior in analytic models – for z = 0, Warrenetal (2005) (dashed line), Jenkinsetal. (2001) (dash-
4, 7 and 10. For example, objects with ν = 3 have M ≃ 4× dotted line), Sheth&Tormen (1999) (dotted line) and Press-
108h−1M⊙ atz=10butM ≃6×1014h−1M⊙ atz=0. Schechter (solid lower line) mass functions for our cosmology.
N-body M180 results are plotted as solid symbols for the 8003
(25h−1Mpc, squares, 50h−1Mpc, hexagons) and 6003 (20h−1
willbeonthemoremassivehaloswhichhavemorefrequent Mpc, triangles) runs. Only masses where there are more than
major mergers. Under reasonable assumptions (see below) 10halosintheboxandwhereresolutioneffectsareunimportant
between 1 2 off all photon production occurs in halos are shown. Open symbols denote the analogous FoF(0.2) mass
more mas3siv−e t3han 109h−1M⊙ at z =10, and we easily re- functionsforthesamesimulations.
solvetheseobjectswiththe25h−1Mpcsimulationwhichwe
use for thebulk of thepaper.
We show the mass function(s) from our three high-
est resolution simulations in Fig. 3. If we use as our mass
estimator the sum of the particle masses in the FoF(0.2)
3 HALO PROPERTIES groups (open symbols) then we find good agreement with
the Sheth& Tormen (1999) or Jenkins et al. (2001) forms.
3.1 Halo abundance and clustering Thisistheprocedurefollowedbymostofthegroupsabove3.
The highest mass objects in our volume have mass However if we choose instead to use M180 as our mass es-
1010h−1M⊙ and radii of several tens of kpc. At z = 1∼0 timator (filled symbols) we find a different mass function.
these halos are analogous to rich clusters today, being re- Althoughthismass function shows amarked excessof high
centlyformedandrare:Fig. 2showsthemassasafunction masshaloscomparedtothePress & Schechter(1974)form,
of peak height, ν δ /σ(M), at z = 10, 7, 4 and 0. The it is a better fit than the alternate forms mentioned above.
c
threshold δc(t)is d≡efinedas1.686/D(t), whereD is thelin- Agreeably,for thescales plotted,theM180 mass function is
ear growth factor normalized to unity at z = 0 and σ2(M) independent of the initial FoF group catalog used to define
is thevarianceof themass computed usinglinear theory at thecentersaboutwhichM180 isdetermined.Thisisnottoo
z =0. In our cosmology δ (z =10) 13.8. Due to the flat- surprisingasthegroup centershardlychangeandthenum-
c
nessofthedimensionlesspoweronth≃escalesofinterest,the ber of “small” groups which split off of larger FoF groups
slightly red initial spectrum and the low clustering ampli- as the linking length is decreased is tiny compared to the
tude,thecharacteristicmass,M⋆,whereσ δc,is (1)M⊙ number of low-mass “field” halos. The differences in mass
at z=10, so all of the halos we consider ar∼e MO. functions then comes primarily from the definition of the
⋆
One of the most basic and useful q≫uantities we masses of the found objects. Comparing halo by halo the
can derive from the simulations is the mass function, FoF(0.2) masses are almost twice M180, though the differ-
the spatial abundance of halos as a function of mass. ence depends on mass. A similar difference was also noted
High redshift mass functions have been studied by many byReed et al. (2007) asa shift to lower abundanceat fixed
groups (e.g. Jang-Condell & Hernquist 2001; Reed et al. mass when comparing an FoF(0.2)-based mass function to
2005; Springelet al. 2005; Reed et al. 2007; Heitmann et al that of a different halo finder. We believe the primary is-
2006; Trac & Cen 2006; Iliev et al. 2005, 2006a; Maio et al
2006; Zahn et al. 2007; Lukicet al. 2007) and Lukicet al.
3 Duetothefinitesizeofourboxesthemassfunctionisslightly
(2007)offeracomprehensivesummaryofrecentwork.Most
suppressedathighmass.Wecanestimatethissuppressionusing
previous work finds mass functions which are better fit
(extended) Press-Schechter theory, assuming we have simulated
by the Sheth & Tormen (1999), Jenkins et al. (2001) or
theconditionalmassfunctionwithinaregionofexactlymeanden-
Warren et al(2005)form.Wefindthattheappropriatemass
sityonthemassscaleofthebox.Themassfunctionsplottedhave
function to use dependsprimarily on thedefinition of mass been corrected for this expected suppression, which ranges from
chosenanddefinitionswhichatz 0giveverysimilarmass <1% to 22% over the mass range plotted. See, e.g. Lukicetal.
≃
functions can give quitedifferent ones at z=10. (2007);Reedetal.(2007)forfurtherdiscussion.
(cid:13)c 0000RAS,MNRAS000,000–000
4 Cohn & White
Figure 4. Density profiles of the 5 most massive halos in the
8003 runatz=10.Themassesrangefrom1−3×1010h−1M⊙.
The2haloswiththeflatterprofiles(shortandlongdashedlines)
correspondtothe3rd and4th mostmassivehalosandbothhave
hadamajormerger(greaterthan1:6)withintheprevious10Myr.
Halosizes(r180)arebelow100h−1kpcforallofthehalosshown. Figure 5.(Top)Thehalo-dark-mattercrosscorrelation,ξhm(r)
Thesolidline,offset,showsanisothermalsphereprofile(ρ∝r−2) (top), forhalos with(comoving) numberdensity 10−2.0,10−1.5,
for comparison. This indicates why FoF(0.2) masses assuming 10−1.0 and 10−0.5h3Mpc−3 (open symbols from top to bot-
an isothermal profile may be expected to disagree with SO(180) tom) from our 50h−1Mpc simulation. The solid squares show
massesasdiscussedinthetext. the dark matter correlation function, ξmm(r). The ratio, b(r)≡
ξhm(r)/ξmm(r),isshowninthelowerpanel.
sue is not the halo finder, but the mass definition. Their
second halo finder assigns masses which are essentially our function upon the mass definition, and the ambiguity in
M180. The mass discrepancy is much larger for these halos this quantity in many analytic treatments, significant care
at z =10 than it is for group and cluster-sized halos at the must be taken when making predictions for the abundance
present day (e.g. Figure 11 in White 2002). of halos. Even if we decide to treat all halos as a simple
The mass differences are quite interesting. The his- 1-parameter family, it is likely preferable to make compar-
torical argument for choosing FoF(0.2) was that the FoF isons with some quantity more directly related to observ-
group finder selects particles approximately within a den- ables (such as circular velocity, halo virial temperature or
sity 3/(2πb3) 60 times the mean density. For a singu- potentialwelldepth)ortodiscussstatisticsasafunctionof
lar isothermal≃sphere profile (ρ r−2) and a critical den- numberdensity rather than mass.
∝
sity Universe the mean enclosed density is thus 180ρcrit, Like rich clusters we expect that these massive ha-
in accord with arguments based on spherical top-hat col- los, in the process of formation, will not lie on the usual
lapse (e.g. Peacock 1998). At z = 10 the Universe is close ‘vacuum’ virial relation 2KE=PE, where KE and PE re-
to critical density so we might expect the FoF(0.2) and fer to the potential and kinetic energy respectively. In
M180 mass functions to agree better than at lower z where fact we find that 2KE/PE 1.4 for halos in the range
180ρ¯≃45ρcrit. However,weare focusing on veryhigh mass 108−1010h−1M⊙, very sim≃ilar to the value found for rich
halos which have only recently formed at z =10. They are clusters today (Knebe& Muller 1999; Cohn & White2005;
therefore less centrally concentrated4 than a ‘typical’ halo. Shawet al.2007).Asimilar ‘excess’kineticenergywasalso
ThiscanalsobeseeninFig.4,wherehaloprofilesarecom- found by Jang-Condell & Hernquist (2001) for lower mass
pared to the isothermal sphere profile. As the halo profiles halos. The ratio is larger than unity because of the steady
are less steep than the isothermal sphere form assumed in accretion of material onto thecluster (Cole & Lacey 1996).
the argument above, this leads to the differences in mass Fig. 5 shows the clustering of the dark matter and the
between theFoF(0.2) and SO(180) definitions. halosfromour50h−1Mpcrun.Weplottheauto-correlation
Bycontrast,wefindthattheFoF(0.168) massfunction function of thedark matterand thecross-correlation of the
is very similar to the M180 points plotted, and a halo by halocenterswiththedarkmatterrespectively.Thelatteris
halo comparison shows that thetwomasses agree towithin bothlesssubjecttonoise5 fromoursmallsampleofmassive
20-30 per cent. As we go down the mass function, to more halos and more applicable to understanding how radiation
concentrated,lowermasshalos,weexpectFoF(0.2)tobetter from the halos would influence the surrounding mass. The
match M (e.g. Cole & Lacey 1996).
180
In general, given the strong dependence of the mass
5 Thereisessentiallynoshot-noiseforthedarkmatter,ξmm(r),
andjackknifeerrorsonthecross-correlation,ξhm,areafewper-
4 They correspond roughly to c ≃ 2−5 for halos of the form centforthesamplesshown.Jackknifedrasticallyunderestimates
proposedbyNavarro,Frenk&White(1997). theerrorsfromfinitevolumehowever.
(cid:13)c 0000RAS,MNRAS000,000–000
Dark matter halo abundances, clustering and assembly histories at high redshift 5
Figure6.Acomparisonofthelarge-scalebiasmeasuredforthe
mass thresholded samples of Fig. 5 with a number of theoret-
ical models: the bias of the Press-Schechter mass function (as
computed by Efstathiouetal. 1988; Cole&Kaiser 1989, solid),
theSheth-Tormenmassfunction(Sheth&Tormen1999,dashed)
and the fitting function of Sheth,Mo&Tormen (2001, dotted).
Although the mass function is in good agreement with that of
Sheth&Tormen (1999), their bias formula underestimates the
clusteringoftheraresthalos.
ratio of the cross- to auto-correlation functions defines the
scale dependentbias, b (r).
h
The mass auto-correlation function is in good agree-
mentbetweenthe25and50h−1Mpcboxesupto1h−1Mpc,
with ξ from the 25h−1Mpc box falling below that of the
50h−1Mpc box beyond this scale. The 20h−1Mpc box has
noticeably less power over a wide range of scales. For the Figure 7. (Top) The fraction of halos with M > 109h−1M⊙
masses where we can compare and for the range of linear whichhavehada1:10,1:5or1:3merger(toptobottom)backto
scales plotted, the halo-mass cross-correlation functions of thelookbacktimeshowninall4ofoursimulations.(Bottom)The
the 25 and 50h−1Mpc boxes are in excellent agreement, so fractionof haloswithM >109h−1M⊙ whichhavealargemass
we haveshown theresults only for the50h−1Mpc box. gain (mf/mi ≥ 1.1, 1.2, 1.33) vs. time. Here we show only the
two highest resolution simulations for clarity. Both plots would
Ourhalosamplesaremassthresholded,howeverbyus-
coincideifallmergerswere2-bodywithin10Myr.
ing number density as our marker we largely avoid the is-
sues of mass definition discussed earlier. The differences in
bias at fixed n¯ for the different mass choices, arising from
Similar trends for rare halos to have larger bias than
the scatter between different mass definitions, is only a few
the modern fits predict have been seen at lower red-
percent. Taking b (1.5h−1Mpc) as the asymptotic value,
h shift (e.g. Shen et al. 2008; White,Martini & Cohn 2008;
the large-scale bias is in good agreement with the mod-
Angulo, Baugh & Lacey2008,forrecentwork)butwemust
els of Press & Schechter (1974); Efstathiou et al. (1988); also remember that b (1.5h−1Mpc) is likely higher than
h
Cole & Kaiser(1989);Mo & White(1996);Jing(1998)and
b (r ) so the degree of overshoot is hard to quantify
h
30% higher than that of Sheth& Tormen (1999). Those → ∞
∼ precisely.Asexpected,theclusteringstrengthisanincreas-
of Sheth,Mo & Tormen (2001) and Tinker et al. (2005) lie
ing function of mass (Kaiser 1984; Efstathiou et al. 1988;
inbetween.(ThemodelofSeljak & Warren (2004)onlyex-
Cole & Kaiser1989),oradecreasingfunctionofhaloabun-
tendsupto100timesthenon-linearmass,whereb 3,and
∼ dance.
it not applicable to our results.) To make contact with the
earlier literature we plot in Fig. 6 the bias as a function of
peak height, ν, obtained from n¯ using theSheth & Tormen
3.2 Mergers and Mass Gains
(1999) mass function. When computing b(> ν) in the sim-
ulation we rank order the halos by FoF(0.2) mass in order Wenowconsiderthehierarchicalassemblyofthedarkmat-
tobestmatchthechosenmassfunction.Thismassfunction ter halos through merging and accretion. We shall use the
isthenusedwhenanalyticallycomputingthehalo-weighted 8003,25h−1Mpcsimulationsinceitprovidesbothhighmass
bias b(> ν) from each of the analytic forms which provide resolution and a representative volume. Since our progeni-
b(ν).Because ofthistheSheth & Tormen (1999)bias func- tor relationships are based on particles in the FoF groups,
tion is the only one which would give an average bias of we use the FoF(0.168) masses for consistency. As discussed
unitywhen integrated over ν. earlier,forourmassivehalosthesemassesarewithin20 30
−
(cid:13)c 0000RAS,MNRAS000,000–000
6 Cohn & White
per cent of M and none of ourconclusions depend sensi-
180
tivelyonthischoice.Fig. 7shows thefraction ofhaloswith
109 M 1010h−1M⊙ which have experienced at least
≤ ≤
one major merger as function of lookback time, in intervals
of 10Myrs. We show three different definitions of ‘major’
merger, where the largest two progenitors of the halo have
ratios below 1:10, 1:5 or 1:3. Mergers are frequent but not
ubiquitous – not all halos have had a major merger within
140Myrs,butmanyhave.Thefractiondecreasesforsmaller
mass ratios and for lower mass halos, as expected.
We can also consider mass gains between time steps,
often denoted in the literature as m /m where m is the
f i i
mass of the largest progenitor at the earlier time and m
f
is themass of the halo underconsideration. Mass gains are
sometimes used as a proxy for mergers. Fig. 7 shows those
haloswhosemassincreasedbyatleast 10,20or33percent
as a function of lookback time. The top and bottom panels
ofFig.7wouldbeidenticalifallmergersweretwobodyand Figure 8. The mass accretion history for halos in the range
therewasnosmoothaccretion.AscanbeseeninFig.1this (5−8)×108h−1M⊙ fromthe 8003 (dashed) and6003 (dotted)
is not the case; Fig. 7 quantifies this difference for major simulations and the functional form of Milleretal. (2006, solid)
mergers. basedonEPS.
ThePress-Schechtermodelpredictstheevolutionofthe
mass function, and it can be extended to make predictions
forthetimehistoryofhalos.This“excursionsetformalism”
is often called extended Press-Schechter (Bond et al. 1991;
Bower 1991; Lacey & Cole 1993, 1994; Kitayama & Suto
1996) and denoted EPS – see Zentner (2006) for a recent
review. Although it is analytically tractable, it has many
inconsistencies and does not compare particularly well to
N-bodysimulations(seee.g.Sheth & Pitman1997;Tormen
1998; Somerville et al. 2000; Cohn, Bagla & White 2001;
Benson, Kamionkowski & Hassani2005;Li et al.2006).For
example,Li et al.(2006)foundthatwithEPShalosofmass
1011 1014h−1M⊙ at z 0 formed later than in N-body
− ≃
simulations (but see Percival, Miller & Peacock 2000, for a
slightly different quantity). In Fig. 8 we compare the N-
bodymassaccretionhistoriesformassivehalosatz =10to
amodelbyMiller et al.(2006)basedonEPSwhichpredicts
almost exponential growth with redshift. (Other analytic
models also exist, see e.g. Neistein, van den Bosch & Dekel
Figure9.EPS(solid)andsimulation(dottedanddashed)results
(2006) for a summary and comparison, there are some dis-
crepancies between these which are not yet fully under- 1fo0r8thh−e1Mnu⊙mbaesraoffunpcrtoigoenniotforMsporfogh.alos with Mf = (4−4.5)×
stood.) We find that EPS predicts mass growth which is
too rapid also for the high mass, high redshift regime stud-
iedhere.TheN-bodymassaccretion historiesare relatively WeshowarepresentativeexampleofN(M )/N(M )
prog f
well fit by an exponential in z – a growth model also pro- forMf intherange(4 4.5) 108h−1M⊙inFig.9.Formost
posedbyWechsleret al.(2002)onthebasisofN-bodysim- of the range the agree−ment×is reasonably good. At the low
ulations of galaxy-sized halos at low z – but the coefficient massendEPSsignificantlyunderpredictsthenumberofpre-
predicted by Miller et al. (2006) is larger than measured in decessors found in our simulations (see also Percival 2001).
thesimulations. At the high mass end the EPS rate starts to climb rapidly,
Perhaps the most common use of EPS is to predict eventuallydivergingunphysically.Thesetrendsareindepen-
merger rates, and EPS has been used in this context in dentofthefinalmasschosen,orthedefinitionofmassused.
several recent models of reionization. To compare the EPS The EPS formula as progenitor mass goes to zero also di-
predictions with our simulations we computed merger rates verges, which we could not approach dueto our finitemass
using only our last (10Myr) time step, taking for any halo resolution, but the mass weighted EPS calculation is finite
withz=10masswithinMf toMf+∆Mf thedistribution at both ends7. There is another notable difference between
ofprogenitors6,Mprog.TheEPSpredictioncanbefoundin EPS and our simulation. Though it is relatively small, our
theAppendix. timestepisstilltoolargeforallmergerstobetruly2-body
(seeFig. 1),asimplicitly assumed byEPS.Alarge fraction
6 We thank D. Holz for suggesting this as a useful comparison
quantity. 7 WethankJunZhangforemphasizingthis.
(cid:13)c 0000RAS,MNRAS000,000–000
Dark matter halo abundances, clustering and assembly histories at high redshift 7
(20-50percent,dependingonM )ofthehalosareactually check these assumptions and significantly extend this work
f
produced in 3 (or more)-body mergers. because we have access to the detailed merger history of
Finallywealsolookedforevidencethatrecentlymerged each halo. This allows us to go beyond their analytic esti-
halos clustered differently than randomly chosen halos of mates to explicitly calculate the full distribution of photon
the same mass. The correlation function of 1:2 or 1:3 production for a halo of mass m, taking into account the
mergers appeared to be slightly (< 10%) enhanced at distribution of histories and theirassociated (and different)
1Mpc compared to the random sample, but the num- photon production rates for a fixed m.
ber of merged halos was too small for this to be statisti- From themerger treefor each halo at z=10 (t=t )
obs
cally meaningful. The effect thus appears to be modest, if weidentifywhichprogenitorshadatleastonemajormerger
present at all, just as was found for lower redshift, high- (greaterthan1:3or1:10),andthetimet theyoccurred.
merge
mass halos (e.g. Gottl¨ober et al. 2002; Percival et al. 2003; Weincludeallofthemergersinthetreeandweplacet
merge
Scannapieco & Thacker 2003; Wetzel et al 2007). This sug- at random within the 10Myr interval between the relevant
gests that the clustering of massive halos does not depend outputs.Eachofthesemergersisallowedtocontribute“ex-
strongly upon their recent merger history. This in turn sig- cess” photons beyond those which would automatically be
nificantly eases the modeling of merger-related processes, assignedtothehaloonthebasisofitsz=10mass,M ,but
h
such as enhanced photon production during reionization thenumberof photonscontributed is exponentially attenu-
which we now discuss. atedwithane-foldingtimeτ.The“excess”photonproduc-
tion is thusproportional to
α Mαe(tmerge−tobs)/τ , (2)
Ms ≡
4 REIONIZATION EFFECTS mXerge
The rate of photon production in a galaxy can be where the sum is over all halos which have undergone a
enhanced by mergers, which can trigger starbursts or major merger and we take α = 1 or 5/3. The exponential
possibly accretion onto a black hole which may be decayismotivatedbymodelingofstarbursts,e.g.Conselice
present(e.g.Carlberg1990;Barnes & Hernquist1991,1996; (2006),hencethesubscripts.Wealsoconsideranothervari-
Mihos & Hernquist 1994, 1996; Kauffmann & Haehnelt ant,includingallhaloswithmajormergerswithinτ oftobs,
2000;Cavaliere & Vittorini2000).Itisreasonabletoantici- with noattenuation:
patethatthemergersoflargedarkmatterhaloscouldhave
MΘ(t t τ) , (3)
bh obs merge
similar effects on thephoton production rate of thesources M ≡ − −
mXerge
within them. We will make this assumption, and then con-
sider the consequences of the merger rates computed above where Θ(x) = 1 if x > 0, 1/2 if x = 0 and zero otherwise.
for the photon production distribution. We denote this by a subscript bh, to indicate photon pro-
We frame our discussion in terms of a sim- duction byblack holes, which might havetheir photon pro-
ple but promising model for reionization proposed by ductionrateincreaseovertimeandthendecayoncethefuel
Furlanetto, Zaldarriaga & Hernquist (2004b), though our is exhausted. Assuming a step-like function is a crude first
result is true more generally. In these models, a halo of a approximationtothisuncertainphysics.Inallcaseswetake
given mass m (in units of some reference mass) is consid- the quiescent photon production to depend on the z = 10
ered a source of photons with rate halomasswiththesameindex,α,as s.Wenotethispre-
M
scription might cause some overcounting if many mergers
dn
γ =ζ (m)m . (1) occur within a short time period and the gas becomes de-
dt t
pleted from the earliest ones. A more refined model would
Usually ζt is taken to be mass independent, scale accountfortheevolvingbaryonbudgetwithinthehalo,but
as m2/3 or transition from m2/3 to m0 at M ourtreatment is sufficient for thepurpose of illustration.
1010h−1M⊙ (Furlanetto, McQuinn & Hernquist 2005, mo∼- The relative amplitudes of these two modes of photon
tivated by Kauffmann et al. 2003). A region around these productiondependonanumberofdifferentfactors(seee.g.
halos is taken to be ionized if the photons within it are Cohn & Chang (2007) for discussion and summary of esti-
sufficient to ionize all the interior mass. Some extensions matesattheseredshifts)butafactorβ 5isnotunreason-
∼
also give recombinations spatial and/or temporal depen- ablefor starbursts and could beeven larger for black holes.
denceandincorporatethisintofindingthebubbleproperties The total photon production is thusenhanced by a factor
(Furlanetto & Oh 2005; Furlanetto, McQuinn & Hernquist
Mα+β α
2005; Cohn & Chang 2007), or incorporate Eq. (1) into N- εmrg ≡ h MαMs (4)
body simulations (Iliev et al. 2006a; McQuinn et al. 2007; h
Zahn et al.2007).Undertheseassumptionsthemorphology forthe“starburst”prescription,oritsanalogue α
Ms →Mbh
ofionizedregionscanbecomputedfromthephotonproduc- forthe“blackhole”prescription.Inprinciplebothcancon-
tion rate and spatial distribution of dark matter halos. tribute.Weconsideredthetwoeffectsseparately,theircom-
A first step at including halo mergers within the bination is straightforward.
above formalism (and its generalizations) was presented in Figure 10 shows a typical example of the cumulative
Cohn & Chang(2007).Thosecalculationswerebasedonthe distribution of enhancement factors, Eq. (4). We took the
Press-Schechter formalism, and so could only provide aver- starburstform,1:3mergers,α=5/3,τ =75Myrandβ=5,
age numbers of mergers for halos in a given mass range; butothercasesareverysimilar.Theenhancementdistribu-
scatter was computed by assuming that the mergers had a tionisextended,withalongtailtohighε andapeakat
mrg
Poisson distribution. With our simulations we are able to thosehaloswhich havenot merged.Abouthalfof thehalos
(cid:13)c 0000RAS,MNRAS000,000–000
8 Cohn & White
tons. Even choosing ζ m0, such halos contribute 10
∝ ∼
per cent of the photons. The number density of such halos
is 0.03h3Mpc−3. Bubble radii in different models range
∼
overseveralordersofmagnitude.Amiddle-of-the-roadesti-
mate is 3h−1Mpc, which would contain about 3 halos with
M > 109h−1M⊙. The bubble radius would also be larger
thanthecorrelationlengthofourhalos,soclusteringisonly
expected to change this number by a factor of order unity.
A small number of halos contributing a large fraction of
the photons means that scatter in their photon production
should affect the properties of the bubbles.
Ourcalculation is relatively crude,butit suggests that
theinclusion of mergers intoa morerefined modelof reion-
ization could alter the distribution of ionized regions. For
modelsbasedonapproximatedynamics(e.g.McQuinn et al.
2007;Mesinger & Furlanetto2007;Zahn et al.2007),apos-
sible first step would be to assign a merger history to the
sources at random. This is accurate to the extent that re-
cently merged halos are not spatially biased with respect
to a random sample of halos of the same mass. For mod-
els which marry the analytic model to dark matter sim-
ulations8 the merger history is known, so only the pho-
Figure10.Thecumulativeprobabilityforenhancement ofpho- ton production rate needs to be modified. More complex
tonproduction over the quiescent case, εmrg, foronetoy model. simulations involving radiative transfer will need to follow
Inthisexample,mergers(1:3)enhancethequiescentphotonpro- the photon production history as the halos evolve, per-
duction rate (∼ m5/3) at z = 10 with β = 5 and τ = 75Myr haps using a semi-analytic model (such as in Benson, et al.
(see text). Roughly 80 per cent of these halos have some en- 2001; Ciardi, Stoehr & White2003;Benson, et al. 2006). A
hancement, shown for 2 bins in halo mass (with the range in full-blown simulation including radiative transfer and N-
log10(M/h−1M⊙)shownineachpanel).Thereare1195and136 body in a large enough volume is still out of reach (but
halos in the low- and high-mass bins, respectively. The peak at
see Sokasian et al. 2003; Kohler, Gnedin & Hamilton 2005;
εmrg = 1 comes from the 20 per cent of the halos which have Trac & Cen 2006; Iliev et al. 2006a,b, for recent progress).
had no 1:3 mergers in the previous 140Myr. Trends for other
parametersandparameterizations arediscussedinthetext.
5 SUMMARY AND CONCLUSIONS
havetwicethephotonproduction,while20percenthaveno
enhancement. Choosing a larger β increases the size of the Using 5 N-body simulations with different sized boxes and
enhancement, but does not qualitatively change the form particle loads we considered the abundance, clustering and
of the distribution. Similarly, changing α or τ changes the assembly histories of high mass halos at high redshift. We
detailed form of the distribution but not its character. Ha- present results specifically for z = 10, but the evolution of
los down to 108h−1M⊙ show a very similar distribution of the populations is smooth and the results will be similar
enhancements. By contrast the model for black hole accre- at slightly higher and lower redshift. Like the halos of rich
tionproducesabimodaldistribution,asthe“early”mergers groups or clusters today the halos we consider are in the
contributerelativelymorethaninthecaseofthestarbursts, process of forming, growing rapidly through accretions and
leading to a second peak. mergers.Wefoundthattheyhadlargervelocitydispersions
Even though the scatter in photon contributions from than a naive application of the virial theorem would pre-
halo to halo is large for a given mass, if a large number dict,duetoasurfacepressurefrominfallingmaterial.Being
of such halos are found in a bubble, their contributions to recently formed, the halos were not very centrally concen-
the photon numbers will tend to the mean, allowing the trated,leadingtoafactoroftwodifferencebetweenFoF(0.2)
distribution to be replaced by theaverage. Precisely count- masses and M . When measured against M we found
180 180
ing the number of halos of a given mass and the com- our halo abundances were closer to the Press-Schechter fit-
binedphotonscatterinsideatypicalbubbleisunfortunately ting formula than that of Sheth& Tormen (1999), though
self-referential:changingtheionizationproperties(including thesimulationshadmorehighmasshalosthantheanalytic
mergers) changes the bubble sizes and thus the number of form. If FoF(0.2) masses were used instead, the mass func-
haloswithin.Differentassumptions aboutthenatureofthe tion approached that of Sheth & Tormen (1999), in agree-
sourcesandtheirfeedbackcangivedrasticallydifferentbub- mentwith earlier work. This discrepancyindicates that an-
blesizes,andtherelativeimportanceofhighvs.lowermass alyticmodelswhichassign anobservabletohalosofagiven
halos (e.g. McQuinn et al. 2007; Zahn et al. 2007). Given size need to pay particular attention to the marker of halo
theseuncertaintiesweconsiderpropertiesinanaveragevol- size employed.
ume, for illustration.
For quiescent photon production and ζ m2/3, ana-
∝
lyticestimates such as Press-Schechtergive that halos with 8 Unfortunatelyoursimulationvolumesaretoosmalltoprovide
M > 109h−1M⊙ contribute between 13 − 23 of all pho- convergedanswersforthisstepwiththeexistingruns.
(cid:13)c 0000RAS,MNRAS000,000–000
Dark matter halo abundances, clustering and assembly histories at high redshift 9
Thehighmasshalosweresignificantlyclustered,andwe wherethenumberof halos of mass M, N(M,t),is given by
calculated the halo bias by taking the halo-mass cross cor- theusual mass function (Press & Schechter1974):
relation and dividing by the matter auto-correlation func-
1 ρ δ (t) dσ2 δ2(t)
tion. Our rare halos were more clustered than the recent N(M,t)= 0 c exp c (6)
models of Sheth & Tormen (1999); Sheth,Mo & Tormen √2πM σ3(M)(cid:12)(cid:12)dM(cid:12)(cid:12) (cid:20)−2σ2(M)(cid:21)
(2001); Tinkeret al. (2005) and closer to the models of with ρ0 thebackground de(cid:12)(cid:12)nsity.(cid:12)(cid:12)To get Eq. (5) thenumber
Efstathiou et al.(1988);Cole & Kaiser(1989);Mo & White ofhalosatagiventimeismultipliedbythefractionofhalos
(1996); Jing (1998). that have mass M at t and have jumped from mass M
i
Merging is common, though not ubiquitous, in high within δt:
mass halos at z=10. Major mergers, with progenitor mass
(M/M )P˙ (M M;t)dM δt=
rfoartiomsolreessththananh1a:l3f,oofchcuarlorsedwwitihthMin 1>40M10y9rh−o1fMz⊙=. W10e (M/Mi i)1(2π)i−→1/2(∆σ2)−3i/2[−dδdct(t)]|dσd2M(Mii)|dMiδt. (7)
looked at thefraction ofhalos undergoingmergers for ava-
where∆σ2 =σ(M )2 σ(M)2.ToproducetheEPSpredic-
riety of lookback times and progenitor ratios, finding more i −
tionsdiscussedin 3.2,wetakeδt=∆t=10Myr,assuming
mergers for more massive halos, longer lookback times or §
that this time is small.
less extreme merger events. Mass gains, parameterized by
m /m , showed similar trends even though not all merger
f i
eventsweretwobodywithinour10Myrtimestep.TheEPS
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