The Mass Profile and Accretion History of Cold Dark Matter Halos

13 Pages·2013·3.23 MB·English

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed4December 2013 (MNLATEXstylefilev2.2)
The Mass Profile and Accretion History of Cold Dark
Matter Halos
3 Aaron D. Ludlow1,⋆, Julio F. Navarro2, Michael Boylan-Kolchin3, Philip E. Bett1,
1
0 Rau´l E. Angulo4, Ming Li5,6, Simon D. M. White5, Carlos Frenk7, Volker Springel8,9
2
1Argelander-Institut fu¨r Astronomie, Auf dem Hu¨gel71, D-53121 Bonn, Germany
c
e 2Dept. of Physics and Astronomy, Universityof Victoria, Victoria, BC, V8P 5C2, Canada
D 3Centerfor Galaxy Evolution, 4129 ReinesHall, Universityof California, Irvine, CA 92697, USA
4Kavli Institute for Particle Astrophysics and Cosmology, Stanford University,SLAC National Laboratory, Menlo Park, CA 94025, USA
3 5Max-Planck-Institut fu¨r Astrophysik, Karl-Schwarzschild-Straße 1, 85740 Garching bei Mu¨nchen, Germany
6Purple Mountain Observatory, West Beijing Rd. 2, 210008 Nanjing, China
] 7Institute for Computational Cosmology, Dept. of Physics, Univ. of Durham, South Road, Durham DH1 3LE, UK
O 8Heidelberg Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany
C 9Zentrumfu¨r Astronomie der Universita¨tHeidelberg, ARI, M¨onchhofstr. 12-14, 69120 Heidelberg, Germany
.
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ABSTRACT
a
[ We use the Millennium Simulation series to investigate the relation between the ac-
cretion history and mass profile of cold dark matter halos. We find that the mean
2 inner density within the scale radius, r−2 (where the halo density profile has isother-
v
mal slope), is directly proportional to the critical density of the Universe at the time
8
8 when the virial mass of the main progenitor equals the mass enclosed within r−2.
Scaled to these characteristic values of mass and density, the average mass accretion
2
0 history, expressed in terms of the critical density of the Universe, M(ρcrit(z)), resem-
. bles that of the enclosed density profile, M(hρi), at z = 0. Both follow closely the
2
NFW profile, which suggests that the similarity of halo mass profiles originates from
0
the mass-independence of halo accretion histories. Support for this interpretation is
3
providedbyoutlierhaloswhoseaccretionhistoriesdeviatefromtheNFWshape;their
1
mass profiles show correlated deviations from NFW and are better approximated by
:
v Einasto profiles. Fitting both M(hρi) and M(ρcrit) with either NFW or Einasto pro-
i
X files yield concentration and shape parameters that are correlated, confirming and
extending earlier work that has linked the concentration of a halo with its accretion
r
a history.Thesecorrelationsalsoconfirmthathalostructureisinsensitivetoinitialcon-
ditions: only halos whose accretionhistories differ greatly from the NFW shape show
noticeable deviations from NFW in their mass profiles. As a result, the NFW profile
provides acceptable fits to hot dark matter halos, which do not form hierarchically,
and for fluctuation power spectra other than CDM. Our findings, however, predict a
subtle but systematic dependence of mass profile shape on accretion history which, if
confirmed, would provide strong support for the link between accretion history and
halo structure we propose here.
Key words: cosmology:dark matter – methods: numerical
1 INTRODUCTION posed by Navarro et al. (1995, 1996, hereafter NFW). The
NFW profilehas fixedshape, and is characterized by a log-
Numerical simulations have shown that the equilibrium
arithmic slope that steepens gradually from thecenter out-
structureofcolddarkmatter(CDM)halosisapproximately
wards. As such, it may be fully specified by just two pa-
self-similar. Spherically averaged density profiles, in partic-
rameters,usuallychosentobeeitherthevirialradiusanda
ular,arewellapproximatedbyscalingasimpleformulapro-
characteristic density or, equivalently, the halo virial mass
andaconcentrationparameter.(SeeSec.3.1forasummary
of relevant formulae and definitions.)
⋆ E-mail:[email protected]
2 Ludlow et al.
The gently-varying slope halos selected from the three Millennium Simulations, MS-
of the NFW profile confounded early theoretical expecta- I(Springelet al.2005),MS-II(Boylan-Kolchin et al.2009),
tions, which had envisioned a simple power-law behaviour and MS-XXL (Anguloet al. 2012), collectively referred to
(e.g.,Fillmore & Goldreich 1984;Hoffman & Shaham1985; hereafter as MS. These are amongst the largest cosmologi-
Quinn et al. 1986; Crone et al. 1994), and has motivated calN-bodysimulationsavailable,andprovideuswiththou-
a number of proposals to explain its origin (see, for a sandsofwell-resolvedhalosspanningmorethanfourdecades
recent review, Frenk & White 2012). Most attempts have inmass.Mergertreesareavailableforallthesesimulations,
taken as guidancethesecondary-infall model first proposed making them an ideal dataset to explore the relation be-
by Gunn & Gott (1972), complemented by various conjec- tween accretion history and mass profiles. In addition, the
tures about the role of mergers (e.g., Salvador-Sole et al. numerical homogeneity and sheer size of the volumes sur-
1998), dynamical friction (e.g., Nusser & Sheth 1999), an- veyedbytheMSallowustocombinelargenumbersofhalos
gular momentum (e.g., Williams et al. 2004), or adiabatic withsimilarpropertiestosmoothoutstatisticalfluctuations
invariants (e.g., Avila-Reese et al. 1998; Dalal et al. 2010), andidiosyncrasies ofindividualsystemsthatmightobscure
or else have argued that entropy generation during virial- the general trends. Our analysis reveals a subtle but well-
ization might be behind the halo structural similarity (see, defined relation between mass profile and accretion history
e.g., Taylor & Navarro2001; Pontzen & Governato 2013). that offers valuable new clues to the origin of the structure
Nogeneralconsensushasyetemerged,however,reflect- of CDM halos.
ingthedifficultythatallofthesemodelsfacewhentryingto The plan of this paper is as follows. We describe the
explainwhythesameNFWprofileseemstofitthestructure simulations in Sec. 2 and the analysis procedure in Sec. 3.
ofhalosformedthroughhierarchicalclusteringregardlessof We present our main results in Sec. 4 and summarize our
powerspectrum(Navarroet al.1997),aswellasthatofhot main conclusions in Sec. 5.
dark matterhalos or of systemsformed through monolithic
collapse (e.g., Huss et al. 1999;Wang & White 2009).
In addition, none of these models provides a thor-
2 NUMERICAL SIMULATIONS
oughexplanationfortheredshift-dependentcorrelationsbe-
tween mass and concentration seen in simulations, their Our analysis focuses on dark matter halos identified in the
scatter, or their dependence on cosmological parameters threeMillenniumSimulations.Weprovidehereabriefsum-
and power spectra. Halo concentration, which depends maryofthesesimulations andoftheirassociated halocata-
only weakly on mass, was originally linked to halo col- logs.Wereferthereadertotheoriginalpapersforextensive
lapsetime(Navarro et al.1997),butattemptstoreproduce details on each of the MS runs.
the simulation results with simple prescriptions based on
that proposal have met with limited success (Bullock et al.
2001; Eke et al. 2001; Neto et al. 2007; Maccio` et al. 2008; 2.1 The Millennium Simulation suite
Gao et al. 2008).
AllMS runsadoptaflat,WMAP1-normalized LCDMcos-
Better results have been obtained with empirical mod-
els that relate concentration to halo mass accretion history mology with the following cosmological parameters: Ωm =
and,inparticular, tothetimewhenthemain haloprogeni- 0.25, ΩΛ = 1−Ωm = 0.75, h = 0.73, n = 1 and σ8 = 0.9.
Here Ω is the present-day contribution of component i to
tor switches from a period of “fast growth” to one of “slow i
the total matter energy density in units of the critical den-
growth” (Wechsleret al. 2002; Zhao et al. 2003; Lu et al.
2006).Thesuccessofthesemodelsisnot,however,unquali- sity for closure, ρcrit; σ8 is the rms mass fluctuation in
8 h−1 Mpc spheres, linearly extrapolated to z = 0; n is
fied.Zhao et al.(2009),forexample,arguethathaloconcen-
thespectral index of primordial density fluctuations, and h
tration isdeterminedatthetimewhenthemain progenitor
istheHubbleparameter.Inadditiontousingthesamecos-
first reaches 4% of the final mass, but there seems to beno
mological parameters, the MS runs also adopted the same
naturaljustificationforwhyconcentrationshouldberelated
sequence of outputs in order to facilitate comparisons be-
tothisparticular,ratherarbitrarytimeofahalo’sassembly
tween the runs.
history.
MS-II follows thedark matterdistribution using 21603
Further complicating matters, there is now convinc-
ing evidence that a number of halos have density profiles particles of mass mp =6.89×106h−1M⊙ in a 100h−1Mpc
periodic box. MS-Ihas thesame total particle number,but
that deviate slightly, but significantly, from the NFW pro-
follows the evolution of structure in a comoving box of
file (Navarroet al. 2004). Accounting for these deviations
500h−1 Mpc on a side; each particle in MS-I is thus 125×
requires the introduction of an additional shape parameter,
thus breaking the structural similarity of CDM halos. One more massive than in MS-II, or mp = 8.61×108h−1M⊙.
MS-XXListhelargest of thethreesimulations in both box
parameterization that resultsin excellentfitsistheEinasto
size and particle number; it follows 67203 particles of mass
profile, where the logarithmic slope is a simple power law
of radius, dlnρ/dlnr ∝ (r/r−2)α: the shape parameter, mp =6.17×109h−1M⊙ in a 3h−1Gpc box.
α, is the exponent of the power law. This finding has now
been verified by additional work (Merritt et al. 2005, 2006;
2.2 Halo Catalogs
Navarro et al.2010;Gao et al.2008;Hayashi & White2008;
Stadel et al. 2009; Ludlow et al. 2011) but there is no clear A friends-of-friends (FOF) group finder (Daviset al. 1985)
understanding of what breaks the similarity or what deter- wasrunontheflyforeachsimulationoutputusingalinking
mines thevalue of α for a particular halo. length of b = 0.2 times the mean inter-particle separation
We explore these issues here using a large ensemble of and a minimum particle number Nmin = 20. The subhalo
The Mass Profile and Accretion History of Cold Dark Matter Halos 3
Figure 1. Halo density profiles and accretion histories. Left panel: Median density profiles of MS-II relaxed halos in the mass range
1.24<logM200/(1010h−1M⊙)<1.54(correspondingtoparticlenumbersintherange2.5×104<N200 <5×104),selectedaccording
totheirconcentration(seeboxesinthetoppanelofFig.2).Densitiesareshownscaledtoρ0,thecriticaldensityatz=0,andweighted
byr2inordertoenhancethedynamicrangeoftheplot.Radiiarescaledtothevirialradius,r200.Thebest-fitEinastoprofilesareshown
by the thin solidcurves, with parameters listed in the legend. Dot-dashed curves indicate NFW profiles (whose shape is fixed inthese
units) matched at the scale radius, r−2, where the r2ρ profiles peak. Arrows indicate the half-mass radius, r1/2. Right panel: Median
mass accretion histories (MAH) of the same set of halos chosen for the leftpanel. Haloaccretion history is defined as the evolution of
themassofthemainprogenitor,expressedinunitsofthemassofthehaloatz=0.Theheavy circlesindicatetheredshift,z−2,when
theprogenitor’smassequalsthemass,M−2,enclosedwithinthescaleradiusatz=0.Starredsymbolsindicatethehalf-massformation
redshift.
findersubfind,(Springel et al. 2001)wasthenruntoiden- 20% ofall halos withvirial mass oforder1012M⊙ and25%
tify self-bound substructurewithin each FOFhalo. of∼1013M⊙ halos.Onlyatverylargehalomasses,suchas
Subfind dissects each FOF halo into one dominant cluster-sized ∼1014M⊙ systems, theunrelaxedfraction ex-
structure (the main halo) and a number of subhalos that ceeds50%.WereferthereadertoNeto et al.(2007)forfur-
trace the self-bound remnants of accreted systems. We will therdiscussionofthesecriteria,andtoLudlow et al.(2012)
focus our analysis on main halos identified at z = 0 that for a discussion of how the inclusion of out-of-equilibrium
contain at least N200 = 5000 particles within their virial systemsmayimpactthemass-concentrationrelationatlarge
radius1 halo masses.
Since dark matter halos are dynamical systems, tran-
sients induced by mergers or ongoing accretion can lead to
rapid fluctuationsin thestructureof a halothat are poorly 3 ANALYSIS
captured with simple fittingformulae. We therefore impose
3.1 Fitting Formulae
threecriteria toflag systemsthat areclearly out of equilib-
rium. We consider a halo to be dynamically “relaxed” if it
We consider two different formulae to fit halo density pro-
simultaneously satisfies all threeof the following conditions
files. The NFW profile is given by
(Netoet al. 2007): (i) fsub < 0.1, (ii) doff < 0.07 and (iii)
2T/|U| < 1.35. Here fsub is the fraction of the halo’s virial ρ(r) = δc , (1)
mass contributed by subhalos, doff =|rp−rCM|/r200 is the ρcrit (r/rs)(1+r/rs)2
distancebetweenthehalobarycenterandthelocationofits where rs is a scale radius, ρcrit ≡ 3H2/8πG is the critical
potentialminimum,expressedinunitsofr200;and2T/|U|is density,and δ is the halo dimensionless characteristic den-
c
the virial ratio of kinetic to potential energies, measured in
sity.Thesetwoparameterscanalsobeexpressedintermsof
the halo rest frame. None of our conclusions are heavily af-
the halo virial mass, M200, and a concentration parameter,
fectedbytheserestrictions.Unrelaxedsystemsmakeuponly
c=r200/rs, which is related to δc by
200 c3
1 We define the virial radius, r200, of a halo as the radius of a δc = 3 [ln(1+c)−c/(1+c)]. (2)
sphere centered at the potential minimum that encloses a mean
densityof200×ρcrit.Weidentifyallvirialquantities (i.e.,mea- Note that for given mass the NFW profile has a single free
suredwithinr200)witha“200”subscript.Notethatallparticles parameter, the concentration. This profile can also be ex-
areusedtocomputer200,notjustthoseboundtothemainhalo. pressedintermsoftheenclosedmeandensity,M(hρi),where
4 Ludlow et al.
M(<r) 200Y(cx)
hρi(r)= (4π/3)r3 = x3 Y(c) ρcrit, (3)
where x=r/r200 and Y(u)=ln(1+u)−u/(1+u).
The Einasto profile (Einasto 1965) has an extra free
parameter, the shape parameter α, and may bewritten as
α
ρ 2 r
ln E =− −1 . (4)
(cid:18)ρ−2(cid:19) α(cid:20)(cid:18)r−2(cid:19) (cid:21)
Theparameter r−2 marks theradiuswhere thelogarithmic
slopeofthedensityprofileisequalto−2.Thesameproperty
holdsfortheNFWscaleradius,r ,andtherefore,forshort,
s
weshallhereafterrefertothescaleradiusofeitherprofileas
r−2.Quantitiesmeasuredat(orwithin)r−2 willbedenoted
by a “−2” subscript; e.g., hρ−2i=hρi(r−2). Of course, like
NFW,theEinastoprofilemayalsobeexpressedintermsof
its enclosed mean density profile, M(hρ i).
E
Wenotethat,forgivenconcentration,anEinastoprofile
withα≈0.18resemblescloselyanNFWprofileoverroughly
two decades in radius or enclosed mass. Profiles with other
valuesofαdeviatesystematicallyfromtheNFWshape(see,
e.g., Navarroet al. 2004, 2010).
3.2 Profile Fitting
Our analysis deals primarily with the spherically averaged
density profiles of relaxed CDM halos identified at z = 0
in each MS. We construct radial profiles using 32 concen-
tric bins, equally spaced in logr, spanning the radial range
−2.56logr/r200 60.
TheEinastoprofilehasthreefreeparameters:ρ−2,r−2,
and α. These are simultaneously adjusted in order to min-
imize its rms deviation from the binned density profiles. In
practice, we definea figure of merit,
Nbin
1
ψ2= Nbin Xi=1[lnρi−lnρE(ρ−2;r−2;α)]2, (5)
Figure 2. Mass dependence of the best-fit Einasto parameters
for all halos in our sample at z = 0. Only relaxed halos with whichisminimizedtoobtain thebest-fittingsetofparame-
morethan 5000 particles withinthe virialradiusareconsidered. tersforanygivenhalo. Equation 5weightsequally allloga-
Thetopandbottompanelsshow,respectively,theconcentration, rithmicradial binsand,foragivenradialrange, isapproxi-
c = r200/r−2, and shape parameter, α, as a function of halo matelyindependentofthenumberofbinsused.Itmeasures
virial mass. Individual points are colored according to the third deviationsofthetrueprofilefrom themodelcausedbysys-
parameter (see color bar on the right of each panel). Connected tematicshapedifferencesaswellasbytransientfeaturesin-
symbolstracethemedianvaluesforeachMillenniumSimulation
ducedby,forexample,substructuresortidalstreams.These
(see legend in the top panel); thin solid lines delineate the 25
featureslead tohighlycorrelated bin-to-bindeviationsthat
to 75 percentile range. The dashed curves indicate the fitting
typically dominate over the Poisson noise in the individual
formulae proposed by Gaoetal. (2008). For clarity only 10,000
radialbins.Forthisreasonwehavedecidedtoweightallbins
halos per simulation are shown in this figure. Halos shown in
grey are systems where the best-fit scale radius is smaller than equally (see Navarro et al. 2010, for further discussion).
the convergence radius; these fits aredeemed unreliableand the In practice, the parameters ρ−2 and r−2 can be ex-
correspondinghalosarenotincludedintheanalysis.Greyvertical pressed in a variety of equivalent forms, such as virial mass
barshighlightthreedifferentmassbinsusedtoexploreparameter and concentration (M200,c), or the magnitude and location
variations at fixed halo mass (see Sec. 4.3 and 4.4). Small boxes ofthepeakinthecircularvelocitycurve(Vmax,rmax).Inor-
indicate halos in each of those bins with average, higher-than- dertoeasecomparisonswithpreviouswork,wecharacterize
average,andlower-than-averagevaluesofα(bottompanel)orof
thedarkmatterhalomassprofileintermsofitsvirialmass
theconcentration (toppanel).
M0 = M200(z = 0), its concentration c = r200/r−2, and its
Einasto“shape”parameter,α.TheEinastoprofileprovides
anexcellent description ofthedensityprofileofrelaxed MS
halos: the median value of ψ is just 0.073+0.014, where the
−0.011
range represents the25th and 75th percentiles.
AnanalogousprocedureisusedwhenNFWfitsneedto
beperformed;inthiscase,thetwoparametersestimatedby
The Mass Profile and Accretion History of Cold Dark Matter Halos 5
Figure 3. Relation between mass profiles at z = 0 and accretion histories for relaxed, well-resolved halos (N200 > 2.5×104) in our
sample. Individual halos are colored by mass, according to the color bar at the top of the plot. Left panels: Mean enclosed densities
withintheradii,r1/4,r1/2,andr3/4,containing,respectively,25%,50%and75%ofthevirialmass,shownasafunctionofthe(critical)
densityoftheUniverseatthetimewhentheprogenitor’svirialmassequalsthemassenclosedwithineachofthoseradiiatz=0.These
densities are correlated, as expected if denser halos collapse earlier.However, the dependence varies with radius and is generally quite
weak. This explains, for example, why measures of halo density (such as the concentration) correlate only poorly with the half-mass
formationtime.Medians,quartiles,and10/90percentilesareindicatedbythebox-and-whiskersymbols.Rightpanels:Astheleftpanels,
butforradiiequaltohalf,one,andtwotimesthescaleradius,r−2.Thedottedlineindicatesdirectproportionality,scaledverticallyto
best fit the data of each panel (fit parameters given inthe legends). Theexcellent agreement between this simplescalingand the data
impliesthat, expressed inunits of thescaleradius,the shape ofthe massprofileofahalois intimatelyrelatedto that ofthe accretion
historyofitsmainprogenitor.
the fit can also be expressed as thevirial mass and concen- 3.3 Mass Profiles and Accretion Histories
tration.
The fits are carried out over a radial range rmin <r < The left panel of Fig. 1 illustrates the role of c and α in
r200. The fitting procedure yields robust estimates for ρ−2, describing the density profile. This figure shows the den-
r−2 and α, provided rmin is chosen to be the minimum of sity profile of MS-II halos selected in a narrow range of
either rconv or 0.05×r200. Here, rconv is the convergence mass, 1.24 < logM200/1010h−1M⊙ < 1.54. (Densities are
radiusdefinedbyPower et al.(2003),wherecircularvelocity weightedbyr2inordertoenhancethedynamicrangeofthe
profiles converge to betterthan ∼10%. plot.)Eachprofilecorrespondstodifferentsystems,grouped
6 Ludlow et al.
byconcentration:greensquarestrackthemedian2 profileof concentration at given mass. Interestingly, halos of average
halos with average concentration for that mass; bluecircles concentration have approximately the same shape parameter
and red triangles correspond to halos with concentration (α ≈ 0.18, i.e., quite similar to NFW), regardless of mass.
∼50% higher and lower than theaverage, respectively (see Halos with higher-than-average concentration have smaller
boxes in thetop panel of Fig. 2). valuesofαandviceversa.Thissuggeststhatthesamemech-
In the scaled units of Fig. 1 the scale radius, r−2, sig- anism responsible, at given mass, for deviations in concen-
nals the location of the maximum of each curve, and dif- tration from the mean might also be behind the different
ferent concentrations show as shifts in the position of the mass profile shapes at z = 0 parameterized by α. We ex-
maxima, which are indicated by large filled circles. In ad- plore this possibility next.
dition totheirdifferentconcentrations, theprofilesdifferas
wellinα,whichincreaseswithdecreasingconcentration(see
legends in Fig. 1). Arrows indicate the half-mass radius of
4.2 Characteristic densities and assembly times
each profile. Dot-dashed curves show NFW profiles (whose
shape is fixed in this plot) with the same concentration as As pointed out by Navarroet al. (1997) and confirmed by
the best Einasto fit (solid lines). The density profile curves subsequentwork(see,e.g.,Jing2000),thescatterinconcen-
moregentlythanNFWforα∼< 0.18andlessgraduallythan tration is closely related to the accretion history of a halo:
NFW for α∼> 0.18, respectively. the earlier (later) a halo is assembled the higher (lower) its
The(median)massaccretionhistoriescorrespondingto concentration.
the same sets of halos are shown in the right-hand panel of ThisisclearfromtheassemblyhistoriesshowninFig.1,
Fig. 1. We define the mass accretion history (MAH) of a which illustrate as well that defining “formation time” in a
halo as the evolution of the virial mass of the main pro- waythatcorrelatesstronglyandunequivocallywithconcen-
genitor3, usually expressed as a function of the scale factor tration is not straightforward. For example, the often-used
a = 1/(1+z), and normalized to the present-day value, half-massformationredshift,z ,variesonlyweaklywithc,
1/2
M0 = M200(z = 0). As expected, more concentrated ha- makingitanunreliableproxyforconcentration(Neto et al.
los accrete a larger fraction of their final mass earlier on. 2007). An ideal definition of formation time would result in
Filledstarsindicatethe“half-massformationredshift”,z1/2, a natural correspondence between the characteristic density
whereas filled circles indicate z−2, the redshift when the of a halo at z = 0 and the density of the Universe at the
mass of the main progenitor first reaches M−2, the mass time of its assembly.
enclosed within r−2 at z=0. Weexploretwopossibilities inFig3.Hereweshowthe
mean density enclosed within various characteristic radii at
z=0versusthecritical densityof theUniverseat thetime
when the main progenitor mass equals the mass enclosed
4 RESULTS
within thesame radii.
4.1 The mass-concentration-shape relations The left panels correspond to radii enclosing 1/4, 1/2,
and3/4ofthevirialmassofthehalo.Dotsindicateindivid-
The top panel of Fig. 2 shows the mass-concentration rela-
ualhaloscolored byhalomass,asshowninthecolorbarat
tionforoursampleofrelaxedhalosatz=0.Concentrations
thetop.Boxesandwhiskerstracethe10th,25th,75th,and
are estimated from Einasto fits, and are color coded by the
90th percentiles in bins of ρcrit. Note the tight but rather
shape parameter, α, as indicated by the color bar. Open
weak(andnon-linear)correlationbetweendensitiesatthese
symbols track the median concentrations as a function of
radii. This confirms our earlier statement that “half-mass”
mass. Thin solid lines trace the 25th and 75th percentiles
formation times are unreliable indicators of halo character-
of the scatter at fixed mass. Different symbols are used for
isticdensity:haloswithverydifferentz maynevertheless
the different MS runs, as specified in the legend. Note the 1/2
havesimilar concentrations.
excellent agreement in the overlapping mass range of each
The right-hand panels of Fig. 3 show the same den-
simulation, which indicates that ourfittingprocedureisro-
bust to theeffects of numerical resolution. sity correlations, but measured at various multiples of r−2,
the scale radius of the mass profile at z = 0. The mid-
The bottom panel of Fig. 2 shows the mass-α relation,
colored this time by concentration. The trend is again con- dle panel shows that the mean density within r−2, hρ−2i=
sistentwithearlierwork;themedianvaluesofαarefairlyin- M−2/(4π/3)r−32, is directly proportional to the critical den-
sity of theUniverseat thetime when thevirial mass of the
sensitivetohalomass,exceptatthehighestmasses,whereit
increases slightly. The mass-concentration-shape trends are mainprogenitorequalsM−2.Intriguingly,thisisalsotrueat
consistent with earlier work; for example, the dashed lines r−2/2(toprightpanel)andat2×r−2(bottomrightpanel),
although with different proportionality constants (listed in
correspond to the fitting formulae proposed by Gao et al.
thefigure legends).
(2008) and reproduce theoverall trendsvery well.
This means that there is an intimate relation between
Fig. 2 illustrates an interesting point already hinted at
the mass profile of a halo and the shape of its mass ac-
in Figure 1: the shape parameter seems to correlate with
cretion history, in the sense that, once the scale radius is
specified, the MAH can be reconstructed from the mass
profile, and vice versa. Since mass profiles are nearly self-
2 Median profiles are computed at each radius after scaling all
individualprofilesasinFig.1. similar when scaled to r−2, this implies that accretion his-
3 The main progenitor of a given dark matter halo is found by tories must also be approximately self-similar when scaled
tracingbackwardsintimethemostmassivehaloalongthemain appropriately.TheMAHself-similarity has beenpreviously
branchofitsmergertree. discussed by van den Bosch (2002), but its relation to the
The Mass Profile and Accretion History of Cold Dark Matter Halos 7
Figure4.Averagemassprofilesatz=0andaccretionhistoriesforhalosinthreedifferentmassbins(seeshadedregionsinthebottom
panelofFig.2).Topleft:Averagemassprofilesofallhalosineachbin,plottedasenclosedmass(inunitsofM200),versusinnerdensity
(inunitsof200×thecriticaldensity).Dashedlinesarebest-fitNFWprofiles,whichhaveasingleadjustableparameter,theconcentration,
c=r200/r−2. Heavy filled symbols indicate the enclosed mass, M−2, and density, hρ−2i, at the scale radius of each profile. Residuals
fromthebestfitsareshowninthebottominset.Topright:Sameastop-leftpanel,butscaledtotheenclosedmass,M−2,andoverdensity,
hρ−2i, at the scale radius. Scaled in this manner, halo mass profiles all look alike and are very well approximated by an NFW profile
(dashed curve). Bottom left: Average accretion histories of the same halos shown in the top panels. The plots show the growth of the
virialmassofthemainprogenitor,normalizedtothefinalmassatz=0,asafunctionoftime,expressedintermsofthecriticaldensity
ofthe Universeateach redshift.Thedashed curves arenot fits tothe data. Rather, theyindicate accretion histories parameterized, as
inthe top panel, by anNFW profileinthis M-ρplane. The singleadjustable parameter to these profiles isfullyspecified by the filled
heavysymbols,whichindicateM−2,chosentomatchthatofthemassprofiles(top-leftpanel)andbyρcrit(z−2),computedas776hρ−2i
following the correlation shown in the middle panel of Fig. 3. The light-colored heavy symbols indicate the scale mass and density of
thepredictedNFWprofile;darkfilledsymbolsmarkthelocationofthehalocharacteristicmassandthecorrespondingformationtime.
Bottom right:Sameaccretionhistoriesasinthebottom-leftpanel,butscaledtothecharacteristicvalues oftheMAH:M−2 andhρ−2i
(lightheavy symbols inthe bottom-leftpanel). Note the remarkablesimilarityinthe shapeof thehalomassprofiles atz=0andthat
oftheaccretionhistoriesoftheirmainprogenitors.
8 Ludlow et al.
Figure 7. As Fig. 4 but for halos with higher-than-average (blue), average (green), or lower-than-average (red) values of the Einasto
parameter α (see boxes in the bottom panel of Fig. 2). Left, middle, and right panels correspond to each of the three mass bins, as
indicatedinthelegends.Toppanels:Averagemassprofilescomparedwiththebest-fitNFWprofileforallhalosofthesamemass(seetop
leftpanel ofFig.4).Residualsfromthatprofileareshownatthebottom ofeachpanel.Notethesimilaritybetween theresidualcurves
ofsimilarcoloratallmasses.Differentvaluesofαimplydifferentprofileshapes,anddeviatesystematicallyfromNFW.Bottom panels:
Average mass accretion histories corresponding to the same halos as in the top panels. The dashed curves indicate the average “NFW
accretionhistories”foreachmassbin,asshowninthebottom-leftpanelofFig.4.Residualsfromthisaveragehistoryareshowninthe
bottom inset of each panel. Note the similarity between the shape of the residual curves of similar colors in all panels. This indicates
that the mass accretion history is intimately linked to the mass profile at z =0. Halos that, at z =0, have mass profiles that deviate
fromNFWinaparticularwayhaveaccretionhistoriesthatdeviatefromtheNFWshapeinasimilarway.
shapeofthemassprofile,ashighlightedhere,hassofarnot profile,thisisjusttheenclosedmass-meaninnerdensityre-
been recognized. lation, M(hρi) (see Sec. 3.1). For theMAH, this reduces to
expressing the virial mass of the main progenitor in terms
of thecritical density,ratherthantheredshift, M(ρcrit(z)).
4.3 NFW accretion histories and mass profiles In what follows, we shall scale all masses to the virial mass
ofthehaloatz=0,M0;ρcrit(z)tothevalueatpresent,ρ0;
WeexplorefurthertherelationbetweenMAHandmasspro-
and hρi to 200ρ0.
filebycastingbothinawaythatsimplifiestheircomparison,
i.e., intermsofmassversusdensity.Inthecase ofthemass Thetop-leftpanelofFig.4shows,inthesescaledunits,
The Mass Profile and Accretion History of Cold Dark Matter Halos 9
Figure 8.Concentrations and shapeparameters ofEinastoprofiles fitted to either accretion historiesor massprofiles atz=0.Heavy
symbols correspond to well-resolved halos grouped according to the c and α parameters of their mass profile (see details in the text).
GreydotscorrespondtoindividualhalosinthesamethreemassbinschosenfinFig.7.Theleftpanelshowsthattheshapeofthemass
accretionhistoryandthatofthemassprofilearecorrelated.ThepanelontherightisanalogoustoFig.5andshowsthatthesameapplies
totheconcentrations.Inthiscase,therelationdependsonthevalueofα,asshownbythecoloredlineslabelledinthelegend.Theheavy
symbolsareofthesametypeasintheleftpanel,butcoloredbyα(seeinset).Notethatthecorrelationsarerelativelyshallow,implying
thatevenlargedepartures fromNFW-likemassaccretionhistoriesleadonlytominordeviations fromNFWinthemassprofiles.
the average M(hρi) profile for halos in three different nar- Table 1. Parameters obtained for best-fits of eq. 6 to the
row mass bins (indicated by grey vertical bars in the bot- concentration-concentration relations for NFW profiles and for
tom panel of Fig. 2). These mean profiles are computed by Einastoprofileswithseveralvaluesoftheshapeparameterα.For
averaging halo masses, for given hρi, after scaling all indi- allcasesprovided,fitsareaccuratetobetterthan ∼<3%overthe
vidualhalosasindicatedabove.Asexpected,eachprofileis range−0.56log c[MAH]61.5
wellfitbyanNFWprofilewheretheconcentrationincreases
graduallywithdecreasingmass.Theheavysymbolsoneach NFW
profile indicate the value of M−2 and hρ−2i. The top-right a1 a2 a3
panel shows the same data, but scaled to these character- 2.521 0.729 0.988
istic masses and densities. Clearly the three profiles follow
Einasto
closely thesame NFW shape, which is fixedin these units.
α a1 a2 a3
The corresponding MAHs, computed as above by av- 0.10 4.124 0.849 0.833
eraging accretion histories of scaled individual halos, are 0.15 3.365 0.692 0.899
showninthebottom-leftpanelofFig.4.Theheavysymbols 0.20 2.946 0.614 0.953
0.25 2.697 0.557 1.003
on each profile again indicate the value of M−2 (as in the
0.30 2.504 0.530 1.042
above panel), as well as ρcrit(z−2) = 776hρ−2i, computed
0.35 2.322 0.528 1.068
usingtherelation showninthemiddle-rightpanelofFig.3.
0.40 2.154 0.543 1.084
Inthesescaledunits,asinglepointcanbeusedtospec-
ifythe“concentration”ofanNFWprofile,whichisshownby
thedashed curves.Interestingly,these provideexcellent de- Fig. 5. The three symbols in the same figure correspond to
scriptionsoftheMAHs:rescaled totheirown characteristic the three average profiles and MAHs shown in Fig. 4 and
density and mass they all look alike and also follow closely clearly follow the same relation. The dotted line in Fig. 5
the NFW shape (bottom-right panel of Fig. 4). The mass shows the best-fitrelation of theform
accretion histories and mass profiles of CDM halos are not
only nearly self-similar: they both have similar shapes that c [Mhρi]=a1 (1+a2×c [MAH])a3, (6)
may be approximated very well by the NFW profile. which is accurate to better than 3% over the range −0.56
This implies that the concentration of the mass pro- log c[MAH]61.5. Thebest-fitparameters are providedin
file just reflects the “concentration” of the MAH. Indeed, Table 1.
assuming that the NFW shape holds for both, the relation
ρcrit(z−2) =776hρ−2i delineates a unique relation between
the two concentrations, which is shown as a dashed line in Note that this is consistent with earlier claims that
10 Ludlow et al.
halo concentration is linked to the time when halo
growth switches from a fast- to a slow-accretion phase
(Wechsleret al. 2002; Zhao et al. 2003). In our interpreta-
tion, since both the MAH and the mass profile follow the
same NFW shape the scale radius of one tracks that of the
other: the “curvature” of the MAH is therefore reflected in
thatofthemassprofile.Noteaswellthattherelationshown
in Fig. 5 is rather weak; in other words, even large changes
intheMAHmapontoasmallrangeofconcentrationsinthe
massprofiles.Thisisattherootoftheweakcorrelation be-
tweenconcentrationandvirialmassreportedinearlierwork
(see, e.g., Neto et al. 2007).
4.4 Einasto accretion histories and mass profiles
Thestrikingsimilarity betweentheshapesoftheMAHand
massprofilediscussedabovesuggestsanexplanationforwhy
halos that are outliers in the mass-concentration relation
tendtohavemassprofilesthatdiffermoresignificantlyfrom
NFW(i.e.,theyhaveαparametersthatdifferfrom0.18,see
Fig.2).Inthisinterpretation,outliersinM200-chaveMAH
shapes that differ systematically from the mean, NFW-like
Figure 5. Relation between concentration parameters obtained shape.
from NFW fits to the average accretion histories and mass pro-
Inordertotestthis,wemayusetheEinastoformulato
files shown in Fig. 4. The dashed curve indicates the expected
fit both MAH and mass profiles. Fig. 6 shows Einasto M-
concentration-concentration dependence given the correlations
ρ profiles for various values of α, and compares them with
shown in the middle-right panel of Fig. 3, assuming an NFW
anNFWprofileofthesameconcentration.(Thisfigureuses
profile. The dotted line shows the best fit obtained using Eq. 6;
the parameters of the fit are provided in Table 1. Note that the thesamescalingsasFig.4.)Asstatedearlier,overtherange
relation is rather shallow, indicating that even halos whose ac- of mass and density plotted here the NFW profile is essen-
cretionhistoriesdiffergreatlymayhavesimilarconcentrations,a tially indistinguishable from an α = 0.18 Einasto profile,
result consistent with the weak mass-concentration dependence butsystematicdeviationsbecomeapparentforothervalues
reportedinearlierwork. of α. InterpretingFig. 6 as a mass accretion history, we see
thatα>0.18correspondstohalosthatareassembledmore
rapidly than expected from the NFW shape. The opposite
holds for α < 0.18. This behaviour is clearly seen in the
residuals from NFW, which are shown in the bottom inset
of thefigure.
The top panels of Fig. 7 show the average M(hρi) pro-
files of halos in three different narrow mass bins chosen to
have different values of α. These are halos whose Einasto
parameters fall in the boxes drawn in the bottom panel of
Fig. 2. The best-fit NFW profile for each mass bin (as in
Fig. 4) is indicated byadashed curvein each panel. Devia-
tionsfromtheNFWcurveareshownintheresidualspanels.
As expected, the residuals have different shapes depending
on the valueof theirshape parameter α.
The bottom panels of Fig. 7 show the corresponding
averagemassaccretionhistoriesandcomparethemwiththe
meanpredictedMAHsshowninthebottompanelsofFig.4.
The latter are the NFW MAHs that result from the hρ−2i-
ρcrit(z−2) correlation shown in Fig. 3. The residuals from
this predicted MAH are clearly similar in shape to those in
thetoppanels:inotherwords,onaverage,haloswhosemass
profilesdeviatefromNFWhavemassaccretionhistoriesthat
deviate from the NFW shape in a similar way.
Quantitatively,thisimpliesthatthebest-fitEinastopa-
Figure 6. Mass accretion histories, M(ρcrit), corresponding to
Einastoprofiles,comparedwithNFW.NotethatNFWresembles rameters of both MAHs and mass profiles must be corre-
closely anEinasto profile withα∼0.18 or so. Larger or smaller lated.Sinceweexpectthecorrelationstobeweak(see,e.g.,
values of α correspond to halos that have been assembled more Fig. 5) we group halos by mass, concentration, and shape
orlessrapidlythantheNFWcurve,respectively.Residualsfrom parameter(asmeasuredfromtheirmassprofiles)beforefit-
NFWareshowninthebottom panel. ting Einasto profiles to their corresponding average mass
accretion histories. To prevent possible biases induced by