The accretion history of dark matter halos II: The connections with the mass power spectrum and the density profile

1.6 MB·English

by  Camila A. Correa

Mon.Not.R.Astron.Soc.000,1–??(2013) Printed4August2015 (MNLaTEXstylefilev2.2)
The accretion history of dark matter halos II:
The connections with the mass power spectrum and the
density profile
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2 Camila A. Correa1⋆, J. Stuart B. Wyithe1, Joop Schaye2 and Alan R. Duffy1,3

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1 School of Physics, Universityof Melbourne, Parkville, VIC 3010, Australia
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2 LeidenObservatory,Leiden University,PO Box 9513, 2300 RA Leiden, The Netherlands
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3 Centre forAstrophysics and Supercomputing, Swinburne Universityof Technology, Melbourne, VIC 3122, Australia

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] 4August2015
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ABSTRACT
.
h We explore the relation between the structure and mass accretion histories of dark
p matterhalosusinga suite ofcosmologicalsimulations.We confirmthatthe formation
- time,definedas the time whenthe virialmassofthe mainprogenitorequalsthe mass
o
r enclosedwithin the scale radius,correlatesstronglywith concentration.We provide a
t semi-analytic model for halo mass history that combines analytic relations with fits
s
a to simulations. This model has the functional form, M(z) = M0(1+z)αeβz, where
[ the parameters α and β are directly correlated with concentration. We then combine

this model for the halo mass history with the analytic relations between α, β and the
2
linear power spectrum derived by Correa et al. to establish the physical link between
v
2 halo concentration and the initial density perturbation field. Finally, we provide fit-
8 ting formulae for the halo mass history as well as numerical routines†, we derive the
3 accretionrate as a function of halo mass,and demonstratehow the halo mass history
4 depends on cosmology and the adopted definition of halo mass.
0
. Key words: methods: numerical - galaxies: halos - cosmology:theory.
1
0
5
1
: 1 INTRODUCTION sityprofilethatcanbedescribedbyasimpleformulaknown
v
i as the‘NFW profile’ (NFW).
X Dark matter halos provide the potential wells inside which
galaxiesform.Asaresult,understandingtheirbasicproper- The origin of this universal density profile is not fully
ar ties,includingtheirformationhistoryandinternalstructure, understood. One possibility is that the NFW profile re-
is an important step for understandinggalaxy evolution. It sults from a relaxation mechanism that produces equilib-
is generally believed that the halo mass accretion history rium and is largely independent of the initial conditions
determines dark matter halo properties, such as their ‘uni- andmergerhistory(Navarro et al. 1997).However,another
versal’densityprofile(Navarro et al.1996,hereafterNFW). popular explanation, originally proposed by Syer& White
Theargumentisasfollows. Duringhierarchical growth,ha- (1998), is that the NFW profile is determined by the halo
losformthroughmergerswithsmallerstructuresandaccre- mass history,and it is then expectedthat halos should also
tionfromtheintergalacticmedium.Mostmergersareminor follow a universal mass history profile (Dekelet al. 2003;
(with smaller satellite halos) and do not alter the structure Manriqueet al. 2003; Sheth& Tormen 2004; Dalal et al.
of the inner halo. Major mergers (mergers between halos 2010; Salvador-Sol´e et al. 2012; Giocoli et al. 2012). This
of comparable mass) can bring material to the centre, but universal accretion history was recently illustrated by
they are found not to play a pivotal role in modifying the Ludlowet al. (2013), whoshowed that thehalo mass histo-
internal mass distribution (Wang& White 2009). Halo for- ries,ifscaledtocertainvalues,followtheNFWprofile.This
mationcanthereforebedescribedasan‘insideout’process, wasdonebycomparingthemassaccretionhistory,expressed
whereastronglyboundcorecollapses,followedbythegrad- in termsof thecritical densityof theUniverse,M(ρcrit(z)),
ual addition of material at the cosmological accretion rate. withtheNFWdensityprofile,expressedinunitsofenclosed
Through this process, halos acquire a nearly universal den- mass and mean density within r, M(hρi(<r)) at z = 0, in
a mass-density plane.
In this work we aim to provide a model that links the
⋆ E-mail:[email protected] halo mass history with the halo concentration, a parameter
† Availableathttps://bitbucket.org/astroduff/commah that fully describes the internal structure of dark matter
2 C.A. Correa, J.S.B. Wyithe, J. Schaye and A.R. Duffy
halos.Bydoingso,wewillgaininsightintotheoriginofthe
Table 2.Cosmologicalparameters.
NFW profileand itsconnection with thehalo mass history.
We also aim to find a physical explanation for the known
Simulation Ωm ΩΛ h σ8 ns
correlationbetweenthelinearrmsfluctuationofthedensity
field,σ, and halo concentration. DMONLY−WMAP1 0.25 0.75 0.73 0.90 1.000
Thispaperisorganizedasfollows.Webrieflyintroduce DMONLY−WMAP3 0.238 0.762 0.73 0.74 0.951
oursimulationsinSection2,whereweexplainhowwecalcu- DMONLY−WMAP5 0.258 0.742 0.72 0.796 0.963
latedthemergerhistorytreesanddiscussthenecessarynu- DMONLY−WMAP9 0.282 0.718 0.70 0.817 0.964
mericalconvergenceconditions.Thenweprovideamodelfor DMONLY−Planck1 0.317 0.683 0.67 0.834 0.962
thehalomasshistory,whichwerefertoasthesemi-analytic
model. This semi-analytic model is described in Section 3,
along with an analysis of theformation timedefinition.For ticles)whichassumevaluesforthecosmological parameters
this model we use the empirical McBride et al. (2009) for- derived from the different releases of Wilkinson Microwave
mula.ThisfunctionalformwasmotivatedbyEPStheoryin Anisotropy Probe (WMAP) and the Planck mission. Table
acompanionpaper(Correa et al.2015a;hereafterPaperI), 2 lists the sets of cosmological parameters adopted in the
andwecalibratethecorrelationbetweenitstwo-parameters different simulations.
(α and β) using numerical simulations. As a result, the Halo mass histories are obtained from the simulation
semi-analytic model combines analytic relations with fits outputs by building halo merger trees. We define the halo
to simulations, to relate halo structure to the mass accre- mass history as the mass of the most massive halo (main
tion history. In Section 3.6 we show how the semi-analytic progenitor) along the main branch of the merger tree. The
model for the halo mass history dependson cosmology and method used to create the merger trees is described in de-
the adopted definition for halo mass. In Section 4 we pro- tailinAppendixA1.Whileanalysingthemergertreesfrom
vide a detailed comparison between the semi-analytic halo thesimulations,welook foranumericalresolution criterion
mass history model provided in this work, and the analytic under which mass accretion histories converge numerically.
modelpresentedinPaperI.Theparametersinthisanalytic Webeginbyinvestigatingtheminimumnumberofparticles
modeldependonthelinearpowerspectrumandhalomass, halosmustcontainsothatthemergertreesleadtoaccurate
whereas in the semi-analytic model the parameters depend numerical convergence. We find a necessary minimum limit
on concentration and halo mass. We therefore combine the of 300 dark matter particles, which corresponds to a mini-
two models to establish the physical relation between the mumdarkmatterhalomassofMhalo ∼2.3×1011M⊙ inthe
linear power spectrum and halo concentration. We will ex- 100h−1Mpc box, Mhalo ∼ 2.6×1010M⊙ in the 50h−1Mpc
pand on this in a forthcoming paper (Correa et al. 2015c, box,and Mhalo ∼3.4×109M⊙ in the 25h−1Mpc box.
hereafter Paper III), where we predict the evolution of the In a merger tree, when a progenitor halo contains less
concentration-mass relation and its dependence on cosmol- than 300 dark matter particles, it is considered unresolved
ogy.Finally,weprovideasummaryof formulaeanddiscuss anddiscarded from theanalysis. Asaresult,thenumberof
ourmain findingsin Section 5. halos in the sample that contribute to the median value of
themasshistorydecreaseswithincreasingredshift.Remov-
ing unresolved halos from the merger tree can introduce a
bias. When the number of halos that are discarded drops
2 SIMULATIONS
tomorethan50% oftheoriginal sample, aspuriousupturn
In this work we use the set of cosmological hydrodynam- in the median mass history occurs. To avoid this bias, the
ical simulations (the REF model) along with a set of median mass history curve is only built out to the redshift
dark matter only (DMONLY) simulations from the OWLS at which lessthan 50% of theoriginal numberof halos con-
project (Schayeet al. 2010). These simulations were run tributetothemedian mass value.
with a significantly extended version of the N-Body Tree- Fig. 1 shows the effects of changing the resolution for
PM,smoothedparticlehydrodynamics(SPH)codeGadget3 thedarkmatteronlyandreferencesimulations.Wefirstvary
(last described in Springel 2005). In order to assess the ef- theboxsizewhilekeepingthenumberofparticlesfixed(left
fects of the finite resolution and box size on our results, panel).Then wevarythenumberof particles while keeping
most simulations were run using the same physical model theboxsizefixed(rightpanel).Theleft panel(rightpanel)
(DMONLY or REF) but different box sizes (ranging from ofFig.1showsthemasshistoryasafunctionofredshiftfor
25h−1Mpc to 400h−1Mpc) and particle numbers (ranging halosineleven(seven)differentmassbinsfortheDMONLY
from 1283 to 5123). The main numerical parameters of the (REF) simulation. All halo masses are binned in equally
runs are listed in Table 1. The simulation names contain spaced logarithmic bins of size ∆log M = 0.5. The mass
10
strings of the form LxxxNyyy,where xxx is the simulation histories are computed by calculating the median value of
box size in comoving h−1Mpc, and yyy is the cube root thehalomassesfrom themergertreeatagivenoutputred-
ofthenumberofparticlesperspecies(darkmatterorbary- shift, the error bars correspond to 1σ confidence intervals.
onic).Formoredetailsonthesimulationswereferthereader Thedifferentcolouredlinesindicatethedifferentsimulations
to AppendixA. fromwhichthehalomasshistorieswerecalculated.Thehor-
Our DMONLY simulations assume the WMAP5 cos- izontal dash-dotted lines in thepanels show the300×m
dm
mology, whereas the REF simulations assume WMAP3. To limitforthesimulationthatmatchesthecolour.Halosinthe
investigatethedependenceontheadoptedcosmological pa- simulationwithmasseslowerthanthisvalueareunresolved,
rameters, we include an extra set of five dark matter only andhencetheirmasshistoriesarenotconsidered.Themass
simulations(100h−1Mpcboxsizeand5123darkmatterpar- histories from halos whose main progenitors have masses
The accretion history of dark halos II 3
Table 1. List of simulations. From left-to-right the columns show: simulation identifier; comoving box size; number of dark matter
particles (there are equally many baryonic particles); initial baryonic particle mass; dark matter particle mass; comoving (Plummer-
equivalent)gravitational softening;maximumphysicalsoftening;finalredshift.
Simulation L N mb mdm ǫcom ǫprop zend
(h−1Mpc) (h−1M⊙) (h−1M⊙) (h−1kpc) (h−1kpc)
REF−L100N512 100 5123 8.7×107 4.1×108 7.81 2.00 0
REF−L100N256 100 2563 6.9×108 3.2×109 15.62 4.00 0
REF−L100N128 100 1283 5.5×109 2.6×1010 31.25 8.00 0
REF−L050N512 50 5123 1.1×107 5.1×107 3.91 1.00 0
REF−L025N512 25 5123 1.4×106 6.3×106 1.95 0.50 2
REF−L025N256 25 2563 1.1×107 5.1×107 3.91 1.00 2
REF−L025N128 25 1283 8.7×107 4.1×108 7.81 2.00 2
DMONLY−WMAP5−L400N512 400 5123 − 3.4×1010 31.25 8.00 0
DMONLY−WMAP5−L200N512 200 5123 − 3.2×109 15.62 4.00 0
DMONLY−WMAP5−L100N512 100 5123 − 5.3×108 7.81 2.00 0
DMONLY−WMAP5−L050N512 50 5123 − 6.1×107 3.91 1.00 0
DMONLY−WMAP5−L025N512 25 5123 − 8.3×106 2.00 0.50 0
Figure1.MedianhalomasshistoryasafunctionofredshiftfromsimulationsDMONLY(leftpanel)andREF(rightpanel)forhalosin
elevenandsevendifferentmassbins,respectively.Thecurvesshowthemedianvalue,andthe1σerrorbarsaredeterminedbybootstrap
resampling the halos from the merger tree at a given output redshift. The different colour lines show the mass histories of halos from
different simulations. We find that a necessary condition for a halo to be defined, and mass histories to converge, is that halos should
have a minimum of 300 dark matter particles. The horizontal dashed dotted lines show the 300×mdm limit for the simulation that
matches the colour, where mdm is the respective dark matter particle mass. When following a merger tree from a given halo sample,
somehalosarediscardedwhenunresolved.Thisintroducesabiasandsoanupturninthemedianmasshistory.Therefore,masshistory
curves arestopped once fewerthan50% ofthe originalsampleofhalos areconsidered. Simulations intheREFmodel with25h−1Mpc
comovingboxsizehaveafinalredshiftofz=2,thereforethehalosmasshistoriesbeginatthisredshift.
4 C.A. Correa, J.S.B. Wyithe, J. Schaye and A.R. Duffy
lower than 1012M⊙ at z = 0 were computed from simula- Table 3. Notation reference. Unless specified otherwise, quanti-
tions with 50h−1Mpc and 25h−1Mpc comoving box sizes. tiesareevaluatedatz=0.
In the right panel, where the mass histories from the REF
modelareshown,allmass historycurvesobtained from the Notation Definition
REF simulation with a 25h−1Mpc comoving box size have
afinalredshiftofz =2.Therefore,thesehalomasshistories M200 Mr(r200),total halomass
begin at this redshift. r200 Virialradius
r−2 NFWscaleradius
c NFWconcentration
Mz M(z),totalhalomassatredshiftz
3 SEMI-ANALYTIC MODEL FOR THE HALO Mr(r) M(<r),massenclosedwithinr
MASS HISTORY x r/r200
hρi(<r−2) Meandensitywithinr−2
Inthefollowingsubsectionswestudydarkmatterhaloprop- Mr(r−2) M(<r−2),enclosedmasswithinr−2
erties and provide a semi-analytic model that relates halo z−2 Formationredshift,whenMz equalsMr(r−2)
structure to the mass accretion history. We begin with the ρcrit,0 Criticaldensity
ρcrit(z) Criticaldensityatredshiftz
NFW density profile and derive an analytic expression for
ρm(z) Meanbackgrounddensityatredshiftz
themean innerdensity,hρi(<r−2), within thescale radius,
r−2.Wethendefinetheformationredshift,andusethesim-
ulations to find the relation between hρi(< r−2) and the thatthehalomassisdefinedasallmatterwithintheradius
criticaldensityoftheuniverseattheformationredshift.We r (see Table 3 for reference).
200
discuss the universality of the mass history curveand show The NFW profile is characterized by a logarithmic
howwecanobtainansemi-analyticmodelforthemasshis- slope that steepens gradually from the centre outwards,
torythatdependsononlyoneparameter(asexpectedfrom and can be fully specified by the concentration param-
ourEPSanalysispresentedinacompanionpaper).Wethen eter and halo mass. Simulations have shown that these
calibrate this single parameter fit using our numerical sim- two parameters are correlated, with the average concentra-
ulations. Finally, we show how the semi-analytic model for tion of a halo being a weakly decreasing function of mass
halomasshistorydependsoncosmologyandhalomassdef- (e.g.NFW;Bullock et al. 2001;Eke et al. 2001;Shaw et al.
inition. 2006;Netoet al.2007;Duffyet al.2008;Maccio` et al.2008;
Dutton& Maccio` 2014; Diemer & Kravtsov 2015). There-
fore, the NFW density profile can be described by a single
3.1 Density profile free parameter, the concentration, which can be related to
virialmass.ThefollowingrelationwasfoundbyDuffyet al.
An important property of a population of halos is their
(2008) from a large set of N−body simulations with the
sphericallyaverageddensityprofile.BasedonN-bodysimu-
WMAP5 cosmology,
lations,Navarroet al.(1997)foundthatthedensityprofiles
ofCDMhaloscanbeapproximatedbyatwoparameterpro- c=6.67(M200/2×1012h−1M⊙)−0.092, (3)
file,
for halos in equilibrium (relaxed).
ρ δ TheNFWprofilecanbeexpressedintermsofthemean
ρ(r)= crit c , (1)
(r/r−2)(1+r/r−2)2 internaldensity
wathewrehicrh itshethleogararidtihums,icr−d2ensisitythselopcheairsac−te2r,istρic r(azd)iu=s hρi(<r)= (4Mπ/r(3r))r3 = 2x030YY((ccx))ρcrit, (4)
crit
3H2(z)/8πG is the critical density of the universe and δc where x is defined as x = r/r200 and Y(u) = ln(1+u)−
is a dimensionless parameter related to the concentration c u/(1+u). From this last equation we can verify that at
by r=r , x=1 and hρi(<r )=200ρ .
200 200 crit
200 c3 Evaluating hρi(<r) at r=r−2, we obtain
δ = , (2)
c 3 [ln(1+c)−c/(1+c)] hρi(<r−2)= (M4πr/(3r)−r23) =200c3YY((1c))ρcrit. (5)
which applies at fixed virial mass and where c is defined −2
as c = r200/r−2, and r200 is the virial radius. A halo is of- Fromthislastexpressionweseethatforafixedredshiftthe
ten defined so that the mean density hρi(< r) within the meaninnerdensityhρi(<r−2)canbewrittenintermsofc.
halo virial radiusr∆ is a factor ∆ times thecritical density Bysubstitutingeq.(3)into(5),wecanobtainhρi(<r−2)as
of the universe at redshift z. Unfortunately, not all authors afunction of virial mass. Finally, wecan computethemass
adoptthesamedefinition,andreadersshouldbeawareofthe enclosed within r−2. From eq.(5) we obtain
difference in halo formation history and internal structure
Y(1)
whendifferentmassdefinitionsareadopted(seeDuffyet al. Mr(r−2)=M200Y(c), (6)
2008;Diemeret al.2013).WeexplorethisinSection3.6.2to
which thereader isreferred tofor furtherdetails. Through- where we used M =(4π/3)r3 200ρ .
200 200 crit
outthisworkweuse∆=200. WedenoteM ≡M (z)as Although the NFW profile is widely used and gen-
z 200
thehalomassasafunctionofredshift,M ≡M(<r)asthe erally describes halo density profiles with high accuracy,
r
halo mass profilewithin radius r at z =0, r as thevirial it is worth noting that high resolution numerical simu-
200
radius at z = 0 and c as the concentration at z = 0. Note lations have shown that the spherically averaged density
The accretion history of dark halos II 5
theoverdensityofhalosvarieswiththeirformationredshift.
SubsequentinvestigationshaveusedN-bodysimulationsand
empiricalmodelstoexploretherelationbetweenconcentra-
tion and formation history in more detail (Wechsler et al.
2002; Zhao et al. 2003, 2009). A good definition of forma-
tion time that relates concentration to halo mass history
wasfoundtobethetimewhenthemainprogenitorswitches
from a period of fast growth to one of slow growth. This is
basedontheobservationthathalosthathaveexperienceda
recentmajormergertypicallyhaverelativelylowconcentra-
tions,whilehalosthathaveexperiencedalongerphaseofrel-
atively quiescent growth have larger concentrations. More-
over, Zhao et al. (2009) argue that halo concentration can
be very well determined at thetime the main progenitor of
thehalo has 4% of its finalmass.
Thevariousformationtimedefinitionseachprovideac-
curate fits to the simulations on which they are based and,
at a given halo mass, show reasonably small scatter. How-
ever, our goal is to adopt a formation time definition that
has a natural justification without invoking arbitrary mass
fractions.Tothisend,wegobacktotheideathathalosare
Figure 2. Relation between the mean density within the NFW
formed ‘inside out’, and consider the formation time to be
scaleradiusatz=0andthecriticaldensityoftheuniverseatthe
defined as the time when the initial bound core forms. We
haloformationredshift,z−2,forDMONLYsimulationsfromthe
followLudlowetal.(2013)anddefinetheformationredshift
OWLSproject.ThesimulationsassumetheWMAP-5cosmolog-
icalparametersandhaveboxsizesof400h−1Mpc,200h−1Mpc, asthetimeatwhichthemassofthemainprogenitorequals
100h−1Mpc,50h−1Mpcand25h−1Mpc.Theblacksolidlinein- themass enclosed within thescale radius at z=0, yielding
dicatestherelationshownineq.(8),whichonlydependsoncos-
mologythroughthemass-concentrationrelation.Theblack(red) z−2=z[Mz =Mr(r−2)]. (7)
starsymbolsshowthemeanvaluesofthecomplete(relaxed)sam- Fromnowonwedenotetheformationredshiftbyz−2.Inter-
pleinlogarithmicmassbinsofwidthδlog M =0.2.Theblack
10 estingly, Ludlowet al. (2013) found that at this formation
dashedandsolidlinesshowtherelationsfoundbyfittingthedata
redshift, the critical density of the universe is directly pro-
ofthe complete andrelaxedsamples,respectively. Thefilledcir-
portionaltothemeandensitywithinthescaleradiusofhalos
clescorrespondtovaluesofindividualhalosandarecolouredby
massaccordingtothecolourbaratthetopoftheplot. at z =0:ρcrit(z−2)∝hρi(<r−2). A possible interpretation
ofthisrelationisthatthecentralstructureofadarkmatter
halo (contained within r−2) is established through collapse
andlateraccretionandmergersincreasethemassandsizeof
profiles of dark matter halos have small but systematic
thehalowithout addingmuch material toits innerregions,
deviations from the NFW form (e.g. Navarroet al. 2004;
thusincreasingthehalovirialradiuswhileleavingthescale
Hayashi& White 2008; Navarroet al. 2010; Ludlow et al.
radiusand its innerdensity (hρi(<r−2)) almost unchanged
2010;Diemer & Kravtsov2014).Whilethereisnoclearun-
(Husset al. 1999; Wang& White 2009).
derstanding of what breaks the structural similarity among
halos,analternativeparametrizationissometimesused(the
Einastoprofile),whichassumesthelogarithmicslopetobea 3.3 Relation between halo formation time and
simplepowerlawofradius,dlnρ/dlnr∝(r/r−2)α(Einasto concentration from numerical simulations
1965). Recently, Ludlowet al. (2013) investigated the rela-
tion between the accretion history and mass profile of cold Wenowstudytherelationbetweenρcrit(z−2)andhρi(<r−2)
using a set of DMONLY cosmological simulations from the
dark matter halos. They found that halos whose mass pro-
OWLSproject thatadopt theWMAP-5cosmology. Webe-
files deviate from NFW and are better approximated by
ginbyconsideringtwosamplesofhalos.Ourcompletesam-
Einasto profiles also have accretion histories that deviate
plecontainsallhalosthatsatisfyourresolutioncriteriawhile
from theNFWshapeinasimilarway.However,theyfound
our ‘relaxed’ sample retains only those halos for which the
theresidualsfrom thesystematicdeviationsfromtheNFW
separation between the most bound particle and the centre
shape to be smaller than 10%. We therefore only consider
ofmassof theFriends-of-Friends(FoF)haloissmaller than
theNFW halo density profilein thiswork.
0.07R ,where R istheradiusfor which themean inter-
vir vir
nal density is ∆ (as given by Bryan & Norman 1998) times
the critical density. Netoet al. (2007) found that this sim-
3.2 Formation redshift
ple criterion resulted in the removal of the vast majority of
Navarroet al. (1997) showed that the characteristic over- unrelaxed halos and as such we do not use their additional
density (δ ) is closely related to formation time (z ), which criteria.Atz=0,ourcompletesamplecontains2831halos,
c f
theydefinedasthetimewhenhalfthemassofthemainpro- while our relaxed sample is reduced to 2387 (84%).
genitor was first contained in progenitors larger than some Tocomputethemeaninnerdensitywithinthescalera-
fraction f of the mass of the halo at z = 0. They found dius, hρi(< r−2), we need to fit the NFW density profile
that the ‘natural’ relation δ ∝ Ω (1+z )3 describes how to each individual halo. We begin by fitting NFW profiles
c m f
6 C.A. Correa, J.S.B. Wyithe, J. Schaye and A.R. Duffy
Figure 3.Relation between formationredshift, z−2, and halo concentration, c (left panel), and between formationredshiftand z =0
halo mass, M200 (right panel). The different symbols correspond to the median values of the relaxed sample and the error bars to 1σ
confidence limits.The solidlineinthe left panel isnot a fit but a prediction of the z−2−c relationfor relaxed halos given by eq. (9).
Similarly,thedashedlineisthepredictionforthecompletesampleofhalos,assumingaconstantofproportionalitybetweenhρi(<r−2)
andρcritof854,ratherthanthevalueof900usedfortherelaxedsample.Thegreyareashowsthescatterinz−2plottedinFig.A1(right
panel). Similarly,the solidlineinthe rightpanel is aprediction of the z−2−M200 relation given by eqs. (9)and (3). The dashed line
alsoshows the z−2−M200 relationassuminghρi(<r−2)=854ρcrit and the concentration-mass relation calculated usingthe complete
sample.
to all halos at z = 0 that contain at least 104 dark matter 25h−1Mpc. The hρi(< r−2)-ρcrit(z−2) values are coloured
particleswithinthevirialradius.Foreachhalo,allparticles by mass according to the colour bar at the top of the plot.
in the range −1.25 6 log (r/r ) 6 0, where r is the The black (red) star symbols show the mean values of the
10 200 200
virial radius,are binnedradially in equally spaced logarith- complete(relaxed)sampleinlogarithmicmassbinsofwidth
micbinsofsize∆log r=0.078.Thedensityprofileisthen δlog M =0.2. As expected when unrelaxed halos are dis-
10 10
fit to these bins by performing a least square minimization carded (e.g. Duffy et al. 2008), the relaxed sample contains
of the difference between the logarithmic densities of the onaverageslightlyhigherconcentrations(byafactorof1.16)
modelandthedata,usingequalweighting.Thecorrespond- and so higherformation times (by afactor of 1.1).
ing mean enclosed mass, Mr(r−2), and mean inner density InFig.2thebest-fittothedatapointsfromtherelaxed
atr−2,hρi(<r−2),arefoundbyinterpolatingalongthecu- sample is shown by the solid line, while the dashed line is
mulative mass and density profiles (measured while fitting the fit to the complete sample. The ρcrit(z−2)−hρi(<r−2)
theNFWprofile)fromr=0tor−2=r200/c,wherecisthe correlation clearly shows that halos that collapsed earlier
concentration from the NFW fit. Then we follow the mass have denser cores at z = 0. Using the mean inner density-
historyofthesehalosthroughthesnapshots,andinterpolate critical density relation for therelaxed sample,
to determinetheredshift z−2 at which Mz =Mr(r−2).
We perform a least-square minimization of the quan- hρi(<r−2) = (900±50)ρcrit(z−2). (8)
tity ∆2 = 1 N [hρ i(ρ ) − f(ρ ,A)], to obtain
N i=1 i crit,i crit,i
the constant of proportionality, A. We find hρi(< r−2) =
(900±50)ρcrit(zP−2)fortherelaxedsample,andhρi(<r−2)= We replace hρi(< r−2) by eq. (5) and calculate the depen-
(854 ± 47)ρcrit(z−2) for the complete sample. The 1σ er- denceof formation redshift on concentration,
rorwasobtainedfrom theleastsquaresfit.Forcomparison,
Ludlow et al. (2014) found a constantvalueof 853 for their
200 c3 Y(1) Ω
relaxedsampleofhalosandtheWMAP-1cosmology.Fig.2 (1+z−2)3 = 900Ω Y(c) − ΩΛ. (9)
showstherelation betweenthemean innerdensityat z=0 m m
and the critical density of the universe at redshift z−2 for Thislast expressionistestedinFig.3(leftpanel)wherewe
various DMONLY simulations. Each dot in this panel cor- plot the median formation redshift as a function of concen-
responds to an individual halo from the complete sample tration using different symbols for different sets of simula-
in the DMONLY−WMAP5 simulations that have box sizes tions from the OWLS project. The symbols correspond to
of 400h−1Mpc, 200h−1Mpc, 100h−1Mpc, 50h−1Mpc and themedianvaluesoftherelaxedsample, andtheerrorbars
The accretion history of dark halos II 7
indicate 1σ confidence limits. The grey solid line shows the
z−2−crelationgivenbyeq.(9),whereasthegreydashedline
shows thesamerelation assumingaconstant of854 instead
of 900 (as obtained for the complete sample). Similarly, us-
ing the Duffy et al. (2008) concentration-mass relation we
obtain the formation redshift as a function of halo mass at
z=0(rightpanelofFig.3).Itisimportanttonotethatthe
z−2−candz−2−M relationsarevalidinthehalomassrange
1011−1015h−1M⊙,atlowermassestheconcentration-mass
relation begins to deviate from power laws (Ludlowet al.
2014).
In Appendix B we analyse the scatter in the forma-
tion time−mass relation and show that it correlates with
the scatter in the concentration−mass relation. Thus con-
cluding that the scatter in formation time determines the
scatter in the concentration. Also, we investigate how the
scatter in halo mass history drives the scatter in formation
time.
Figure 4. Mass histories of halos, obtained from different
DMONLY−WMAP5 simulations, as indicated by the colours.
3.4 The mass history
The bottom left legends indicate the halo mass range at z = 0,
selectedfromeachsimulation.Forexample,weselectedhalosbe-
Fig. 4 shows the mass accretion history of halos in differ-
entmassbinsasafunctionofthemeanbackgrounddensity. tween 109−1011M⊙ from the DMONLY−WMAP5−L025N512
simulation, divided them in equally spaced logarithmic bins of
ThemasshistoriesarescaledtoMr(r−2)andthemeanback- size ∆log M = 0.2, and calculated the median mass histories.
grounddensitiesarescaledtoρm(z−2)=ρcrit,0Ωm(1+z−2)3. The differ1e0nt curves show the median mass history of the main
Thefigureshowsthatallhalomasshistorieslookalike.This progenitors,normalizedtothemedianenclosedmassofthemain
is in agreement with Ludlowet al. (2013), who found that progenitorsatz=0,Mr(r−2).Themasshistoriesareplottedas
themassaccretionhistory,expressedintermsofthecritical afunctionofthemeanbackgrounddensityoftheuniverse,scaled
density of the Universe, M(ρcrit(z)), resembles that of the tothe meanbackground density atz−2.Thebluedashed lineis
enclosed NFW density profile, M(hρi(<r)). The similarity afit of expression(18) to the different mass historycurves. The
in the shapes between M(ρ (z)) and M(hρi(< r)) is still medianvalueoftheonlyadjustableparameter,γ,isindicatedin
crit
thetop-rightpartoftheplot.
notclear,butitsuggeststhatthephysicallymotivatedform
M(z)=M (1+z)αeβz,whichisaresultof rapidgrowth in
0
the matter dominated epoch followed by a slow growth in
thedarkenergyepoch,producesthedoublepower-lawofthe and hence
NFWprofile(seee.g.Lu et al.2006).Weusethisfeatureto
find a functional form that describes this unique universal ln MMzr((zr−=20)) −βz−2
α = . (13)
curveinordertoobtainanempiricalexpressionforthemass (cid:16) ln(1+z(cid:17)−2)
accretion history at all redshifts and halo masses.
From this last equation we see that α can be written as a
WearemotivatedbytheextendedPress-Schechteranal-
ysisofhalomasshistoriespresentedinPaperI.Inthatwork, function of β, Mr(r−2), Mz(z = 0) and z−2. However, as
we showed through analytic calculations, that when halo Mr(r−2) is a function of concentration and virial mass (see
mass histories are described by a power-law times an expo- eq.6),wecanwriteαintermsofβ,concentrationandz−2,
nential,
M(z)=M (1+z)αeβz, (10) α = ln(Y(1)/Y(c))−βz−2. (14)
0 ln(1+z−2)
andthattheparametersαandβareconnectedviathepower The next step is to find an expression for β. We find
spectrum of density fluctuations. In this work however, we β(z−2) by fitting eq. (10) to all the data points plotted in
aim to relate halo structure to the mass accretion history. Fig.4.WenowneedtoexpressM(z)(eq.10)asafunctionof
Wethereforefirstdeterminethecorrelationbetweenthepa- themeanbackgrounddensity.Todothis,wereplace(1+z)
rameters α and β and concentration. To this end, we first by (ρ (z)/ρ /Ω )1/3 and divide both sides of eq. (10)
m crit,0 m
findthe α−β relation that results from theformation red- byMr(r−2), yielding
shift definition discussed in the previous section. Thus, we
evaluatethehalo mass at z−2, α/3
M (z) M (z=0) ρ (z)
z = z m
Mz(z−2)=Mr(r−2) = Mz(z=0)(1+z−2)αeβz−2. (11) Mr(r−2) Mr(r−2) (cid:18)Ωmρcrit,0(cid:19)
1/3
ρ (z)
Taking thenaturallogarithm, we obtain, × exp β m −1 . (15)
Ω ρ
"(cid:18) m crit,0(cid:19) #!
ln MMzr((zr=−20)) = αln(1+z−2)+βz−2, (12) Multiplyingbothdenominatorsandnumeratorsbyρm(z−2),
(cid:18) (cid:19) we get, after rearranging,
8 C.A. Correa, J.S.B. Wyithe, J. Schaye and A.R. Duffy
followed by a slower growth at late times. The changefrom
α/3 α/3 rapidtoslowaccretioncorrespondstothetransitionbetween
Mz(z) = Mz(z =0) ρm(z−2) ρm(z) themassanddarkenergydominatederas(seePaperI),and
Mr(r−2) Mr(r−2) (cid:18)Ωmρcrit,0(cid:19) (cid:18)ρm(z−2)(cid:19) depends on the parameter β in the exponential (as can be
1/3 seen from eq. 10). The dependence of β on the formation
× exp β ρm(z−2) −1 redshiftisgivenineq.(21),whichshowsthatamorerecent
Ω ρ
"(cid:18) m crit,0(cid:19) #! formation time, and hence a larger halo mass, results in a
1/3 larger value of β, and so a steeper halo mass history curve.
ρ (z)
× exp γ m −1 , (16) ThislastpointcanbeseeninFig.5fromthemasshistories
"(cid:18)ρm(z−2)(cid:19) #! of halos in different mass bins (coloured lines shown in the
panels).Thepanelontheleftshowsthemasshistorycurves
where, we have defined γ ≡ fromtheDMONLYsimulationoutputs(colouredlinesinthe
β(ρm(z−2)/Ωm/ρcrit,0)1/3 = β(1 + z−2). The term background),and themass histories predicted by eqs. (10),
α/3 1/3 (14) and (21) (red dashed lines). From these panels we see
Mz(z=0) ρm(z−2) exp β ρm(z−2) −1
Mr(r−2) Ωmρcrit,0 Ωmρcrit,0 that, (i) the mass history formula works remarkably well
in eq.(16(cid:16))is equal(cid:17)tounity,(cid:18)wh(cid:20)ic(cid:16)h canbe(cid:17)seen by(cid:21)re(cid:19)placing when compared with the simulation, and (ii) the larger the
ρm(z−2)/Ωmρcrit,0 = (1 + z−2)3 and comparing with eq. massofahaloatz=0,thesteeperthemasshistorycurveat
(11). Henceeq. (16) becomes early times. Incontrast, themass history of low-mass halos
is essentially governed by thepower law at late times.
ThehalomasshistoriesplottedinFig.5comefromthe
α/3
Mz(z) = ρm(z) completesampleofhalos(relaxedandunrelaxed).Wefound
Mr(r−2) ρm(z−2) no significant difference in mass growth when only relaxed
(cid:18) (cid:19)
halos are considered. We therefore conclude that the fact
1/3
ρ (z)
× exp γ m −1 . (17) thatahaloisunrelaxedataparticularredshiftdoesnotaf-
"(cid:18)ρm(z−2)(cid:19) #! fectitshalomasshistory,providedtheconcentration−mass
relation fit from therelaxed halo sample is used. This is an
Thus, based on eq. (17), the functional form to fit the interesting result because while deriving the semi-analytic
massaccretionhistoriesfromthesimulationscanbewritten modelofhalomasshistory,weassumedthatthehalodensity
as profile is described by the NFW profile at all times. There-
forewhiletheNFWisnotagoodfitforthedensityprofileof
unrelaxedhalos (Netoet al. 2007),oursemi-analytic model
f(z˜,γ) = α(z−2,c,γ)z˜/3+γ(ez˜/3−1), (18) (basedonNFWprofiles)isagood fitallhalos(relaxed and
unrelaxed).
TherightpanelofFig.5showshalomasshistoriesfrom
where f(z˜,γ) = ln Mz(z) and z˜ = ln ρm(z) . From
Mr(r−2) ρm(z−2) theREFhydrodynamicalsimulations.Wecomputethehalo
eq. (14) we see that(cid:16)the para(cid:17)meter α is a fuhnction oif z−2, c mass as the total mass (gas and dark matter) within the
and γ, virial radius (r ). We find that the inclusion of baryons
200
steepens the mass histories at high redshift, therefore the
α = ln(Y(1)/Y(c))−γz−2/(1+z−2). (19) btheestcdonesccernitprtaiotinono-fmMa(szs)reislagtiivoennfrboymeqtsh.e(c1o0m),p(l1e4t)e,s(a2m1)p,laenodf
ln(1+z−2) halos, c=5.74(M/2×1012h−1M⊙)−0.097.
Therefore, γ isnowtheonlyadjustableparameter. Weper-
form a χ2-likeminimization of thequantity
3.5 The mass accretion rate
N
1
∆2 = N [log10(Mz(zi)/Mr(r−2))−f(z˜i,γ)]2, (20) The accretion of gas and dark matter from the intergalac-
Xi=1 ticmediumisafundamentaldriverofboth,theevolutionof
darkmatterhalosandtheformationofgalaxieswithinthem.
and find the value of γ that best fits all halo accretion his-
tories. The sum in the χ2-like minimization is over the N Forthatreason,developingatheoreticalmodelforthemass
accretion rate is the basis for analytic and semi-analytic
available simulation output redshifts at z (i = 1,N), with
i
z˜ =ln ρcrit,0Ωm(1+zi)3 . models that study galaxy formation and evolution. In this
i ρm(z−2) sectionwelookforasuitableexpressionforthemeanaccre-
Figh. 4 shows halo miass histories (with Mz(z) scaled to tion rate of dark matter halos. To achieve this, we take the
Mr(r−2)) for our complete halo sample as afunction of the derivative of the semi-analytic mass history model, M(z),
mean background density [ρm(z) scaled to ρm(z−2)]. The given byeq.(10) with respect totime and replacedz/dt by
bluedashed line is thefit of expression (18) to all the mass −H [Ω (1+z)5+Ω (1+z)2]1/2, yielding
0 m Λ
history curves included in the figure. Here, the only ad-
justable parameter is γ. We obtained γ = −3.01 ± 0.08,
dM
yielding dt = 71.6M⊙yr−1M12h0.7[−α−β(1+z)]
β=−3(ρm(z−2)/Ωm/ρcrit,0)−1/3 =−3/(1+z−2). (21) ×[Ωm(1+z)3+ΩΛ]1/2, (22)
Fig. 4 shows that the halo mass histories have a char- where h0.7 = h/0.7, M12 = M/1012M⊙ and α and β are
acteristic shapeconsisting of a rapid growth at early times, given by eqs. (14) and (21), respectively. As shown in the
The accretion history of dark halos II 9
Figure 5.MasshistoriesofallhalosfromsimulationsDMONLY−WMAP5(leftpanel)andREF(rightpanel).Halomassesarebinned
inequallyspacedlogarithmicbinsofsize∆log M =0.5.Themasshistoriesarecomputedbycalculatingthemedianvalueofthehalo
10
massesfromthemergertreeatagivenoutputredshift,theerrorbarscorrespondto1σ.Thedifferentcolourlinesshowthemasshistories
ofhalosfromdifferentsimulationsasindicatedinthelegends,whilethereddashedcurvescorrespondtothemasshistoriespredictedby
eqs.(10),(19)and(21).
previous section, the parameters α and β depend on halo estimate given byeq.(23).Asexpected,thelarger thehalo
mass(throughtheformationtimedependence).Wefindthat mass, thelarger thedark matter accretion rate.
thismassdependenceiscrucialforobtaininganaccuratede-
scriptionforthemasshistory(asshowninFig.5).However,
3.5.1 Baryonic accretion
the factor of 2 (3) change for α (β) between halo masses
of 108 and 1014M⊙ is not significant when calculating the Next, we estimate the gas accretion rate and compare
accretion rate. Therefore, we provide an approximation for our model with similar fitting formulae proposed by
the mean mass accretion rate as a function of redshift and Fakhouriet al. (2010) and Dekel& Krumholz (2013). The
halo mass, by averaging α and β over halo mass, yielding bottom panel of Fig. 6 shows the gas accretion rate as a
hαi=0.24, hβi=-0.75, and functionredshiftforarangeofhalomasses(log10M/M⊙ =
11.2 − 12.8). The grey circles correspond to the gas ac-
cretion rate measured in REF−L100N512. In this case we
hddMt i = 71.6M⊙yr−1M12h0.7 tchome pmuetregetrhetrteoetsa,lamndastshegnrowestthim(aMte=theMggaass +acMcreDtMio)nfrroamte
×[−0.24+0.75(1+z)][Ωm(1+z)3+ΩΛ]1/(22.3) by multiplying the total accretion rate by the universal
baryon fraction f = Ω /Ω . The green solid line corre-
b b m
Fig. 6 (top panel) compares the median dark matter sponds to our gas accretion rate model (given by Ωb/Ωm
accretion ratefordifferenthalomasses asafunctionofred- times eq. 23). The blue dot-dashed line is the gas accre-
shift(solidlines)tothemeanaccretionrategivenbyeq.(23) tionrateproposedbyDekel& Krumholz(2013)(dMb/dt=
(greydashedlines).Fromthemergertreesofthemainhalos, 30M⊙yr−1fbM12(1+z)5/2), who derived the baryonic in-
wecomputethemassgrowthrateofahaloofagivenmass. flow on to a halo dMb/dt from the averaged growth rate
We do this by following the main branch of the tree and of halo mass through mergers and smooth accretion based
computing dM/dt = (M(z )−M(z ))/∆t, where z < z , ontheEPStheoryofgravitationalclustering(Neistein et al.
1 2 1 2
M(z ) is the descendant halo mass at time t and M(z ) is 2006;Neistein & Dekel2008).Lastly,wecompareourmodel
1 2
the most massive progenitor at time t−∆t. The median with the accretion rate formula from Fakhouriet al. (2010)
valueofdM/dtforthecompletesetofresolvedhalosisthen [dMb/dt = 46.1M⊙yr−1fbM112.1(1 + 1.11z)(Ωm(1 + z)3 +
plottedfordifferentconstanthalomasses.Wefindverygood Ω )1/2], plotted as the purple dashed line. Fakhouriet al.
Λ
agreement betweenthesimulation outputsandtheanalytic (2010) constructed merger trees of dark matter halos and
10 C.A. Correa, J.S.B. Wyithe, J. Schaye and A.R. Duffy
3.6 Dependence on cosmology and mass definition
We have developed a semi-analytic model that relates the
innerstructureofahaloatredshiftzerotoitsmasshistory.
ThemodeladoptstheNFW profile,computesthemean in-
ner density within the scale radius, and relates this to the
criticaldensityoftheuniverseattheredshiftwherethehalo
virial mass equals the mass enclosed within r−2. This re-
lation enables us to find the formation redshift-halo mass
dependence and to derive a one-parameter model for the
halo mass history. In this section we consider the effects of
cosmology and mass definition on thesemi-analytic model.
3.6.1 Cosmology dependence
The adopted cosmological parameters affect the mean in-
ner halo densities, concentrations, formation redshifts and
halo mass histories. To investigate the dependence of halo
masshistoriesoncosmology,wehaverunasetofdarkmat-
teronlysimulationswithdifferentcosmologies. Table2lists
thesetsofcosmologicalparametersadoptedbythedifferent
simulations. Specifically, we assume values for the cosmo-
logical parameters derived from measurements of the cos-
mic microwave background by the WMAP and the Planck
missions (Spergel et al. 2003, 2007; Komatsu et al. 2009;
Hinshawet al. 2013;Planck Collaboration et al. 2014).
It has been shown that halos that formed earlier are
moreconcentrated(Navarroet al.1997;Bullock et al.2001;
Ekeet al. 2001; Kuhlenet al. 2005; Maccio` et al. 2007;
Netoet al. 2007). Maccio` et al. (2008) explored the depen-
dence of halo concentration on the adopted cosmological
model for field galaxies. They found that dwarf-scale field
halos aremore concentrated byafactor of 1.55 in WMAP1
compared to WMAP3, and by a factor of 1.29 for cluster-
sized halos.Thisreflectsthefact that halosofafixedz=0
mass assemble earlier in a universe with higher Ω , higher
m
Figure 6. Mean accretion rate of dark matter (top panel) and σ8 and/or higherns.
gas (bottom panel) as a function of redshift for different halo Thehaloformationredshiftcanberelatedtothepower
masses. Top panel: the accretion rate obtained from the simu- at the corresponding mass scale, and therefore depends on
lation outputs up to the redshift where the halo mass histories bothσ andn .Theparameterσ setsthepoweratascale
8 s 8
are converged. Grey dashed lines show the accretion rate esti- of 8h−1Mpc, which corresponds to a mass of about 1.53×
mated usingeq. (23). Bottom panel: gas accretion rate obtained 1014h−1M⊙(Ωm/Ωm,WMAP5),andawavenumberofk8.This
fromtheREF−L100N512simulation(greycircles),fromΩb/Ωm lastquantityisgivenbytherelationM =(4πρ /3)(2π/k)3.
m
timeseq.(23)(greensolidline),andfromvariousfittingformulae For a power-law spectrum P(k)∝kn, the variance can be
takenfromtheliterature.
written as σ2(k)/σ2 =(k/k )n+3. Therefore, the change in
8 8
σbetweenWMAP5andWMAP1foragivenhalomassthat
corresponds toa wavenumberk is
σWMAP1(k) = σ8,WMAP1 k (ns,WMAP1−ns,WMAP5)/2.(24)
σ (k) σ k
WMAP5 8,WMAP5 8
(cid:16) (cid:17)
quantified their merger rates and mass growth rate using A halo mass of 1012M⊙ corresponds to a wavenumber of
the Millennium and Millennium II simulations. They de- k1.3 ∼6k8.Thetotalchangeinthemeanpowerspectrumat
finedthehalomassasthesumofthemassesofallsubhalos this mass scale is σWMAP1(k1.3) =1.27. This is proportional
σWMAP5(k1.3)
withinaFoFhalo.Weseethatouraccretionratemodelisin tothe changeof theformation redshift,
excellent agreement with the formulae from Fakhouriet al.
(1+z )=1.27(1+z ). (25)
(2010) and Dekel& Krumholz (2013). We find that the f,WMAP1 f,WMAP5
Fakhouriet al.(2010)formulagenerallyoverpredictsthegas Next, we test how this change affects the halo mass
accretionrateinthelow-redshiftregime(e.g.itoverpredicts history. We showed that the mass history profile is well
it by a factor of 1.4 at z = 0 for a 1012M⊙ mass halo). described by the expression M(z)=Mz(z =0)(1+z)αeβz,
TheDekel& Krumholz(2013)formulaunderpredicts(over- where α and β both depend on the formation redshift. In
predicts) the gas accretion rate in the low- (high-) redshift themasshistorymodelpresentedinSection4therearetwo
regime for halos with masses larger (lower) than1012M⊙. best-fitting parameters that can be cosmology dependent.