Formation times, mass growth histories and concentrations of dark matter haloes

1.3 MB·English

by  Carlo Giocoli

Mon.Not.R.Astron.Soc.000,1–15(2011) Printed20January2012 (MNLATEXstylefilev2.2)
Formation times, mass growth histories and concentrations
of dark matter haloes
Carlo Giocoli1,2(cid:63), Giuseppe Tormen3, Ravi K. Sheth4,5
2
1 1 INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, 40127, Bologna, Italy
0 2 INFN - Sezione di Bologna, viale Berti Pichat 6/2, 40127, Bologna, Italy
2 3 Dipartimento di Astronomia, Universita` di Padova, vicolo dell’Osservatorio 3, 35122, Padova, Italy
4 The Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
n
5 Center for Particle Cosmology, University of Pennsylvania, 209 S 33rd St, Philadelphia, PA 19104, USA
a
J

9
1

] ABSTRACT
O
We develop a simple model for estimating the mass growth histories of dark matter
C halos.Themodelisbasedonafittotheformationtimedistribution,whereformationis
. definedastheearliesttimethatthemainbranchofthemergertreecontainsafraction
h
f ofthefinalmassM.Ouranalysisexploitsthefactthatthemedianformationtimeas
p
a function of f is the same as the median of the main progenitor mass distribution as
-
o a function of time. When coupled with previous work showing that the concentration
r c of the final halo is related to the formation time t associated with f ∼ 0.04,
f
t
s our approach provides a simple algorithm for estimating how the distribution of halo
a concentrationsmaybeexpectedtodependonmass,redshiftandtheexpansionhistory
[
ofthebackgroundcosmology.Wealsoshowthatonecanpredictlog10cwithaprecision
2 of about 0.13 and 0.09 dex if only its mass, or both mass and tf are known. And,
v conversely,onecanpredictlog10tf frommassorcwithaprecisionof0.12and0.09dex,
7 approximately independent of f. Adding the mass to the c-based estimate does not
7 result in further improvement. These latter results may be useful for studies which
9 seek to compare the age of the stars in the central galaxy in a halo with the time the
6 core was first assembled.
.
1
Key words: galaxies: halos - cosmology: theory - dark matter - methods:
1
1
1
:
v
1 INTRODUCTION dependsonhowcisestimated.RecentlyPradaetal.(2011)
i
X report that, if c is determined by fitting to the circular ve-
One main focus of modern observational cosmology is the
r locityratherthanthedensityprofile,thenthec−M relation
a studyoftheuabundanceandstructuralpropertiesofgalaxy is no longer monotonic. Provided one knows which method
clusters: typically, the abundance is expressed as a func-
has been used to estimate c, measuring the M −c relation
tion of mass M, and the structural property of most inter-
atarangeofredshiftsprovidesimportantconstraintsonthe
est is the central concentration c of the mass density pro-
background cosmology (Eke et al. 2001; Dolag et al. 2004;
file. There are various ongoing and planned surveys which
Netoetal.2007;Maccio`etal.2008).)Thetightnessofthese
are capable of finding thousands of clusters using well-
constraintsrestsonaccuratelyestimatingthemassandcon-
controlled/calibratedselectioncriteria,soanumberoflarge
centration of each cluster.
homogeneous samples of galaxy clusters will be available in
thenearfuture(Laureijsetal.2011).Suchsampleshavethe In all cases, these relations are thought to be a con-
potentialtostronglyconstrainthefundamentalparameters sequence of the fact that the structure of a halo, and its
ofagivencosmologicalmodel(e.g.ΛCDM).Thisisinlarge concentrationcinparticular,dependuponitsassemblyhis-
part because of numerical simulations (Navarro et al. 1997, tory(Navarroetal.1996;Bullocketal.2001;Wechsleretal.
e.g.) which have shown that, at fixed mass, halos are less 2002;Zhaoetal.2003b,2009;Giocolietal.2010a).Roughly
concentratedifthematterdensity,Ωm,islower.Inaddition, speaking,cisrelatedtotheratioofthebackgrounddensity
foragivencosmology,thereisareasonablytightmonotonic at the time at which the mass within the core of the halo
correlation between M and c, although this M −c relation was first assembled to that at which the halo was identi-
fied(e.g.,today).Lowerformationredshiftstypicallyimply
smaller concentrations, but the exact dependence is a func-
(cid:63) E-mail:[email protected] tion of the expansion history of the background cosmology.
(cid:13)c 2011RAS
2 C. Giocoli et al. 2011
Recently Zhao et al. (2009) have provided a prescrip- 2 FORMATION AND THE MASS OF THE
tionforrelatingahalo’sconcentrationtoitsmassassembly MAIN PROGENITOR
history. This prescription requires knowledge of the time at
ThemasswhichmakesupahaloofmassM atz willbein
which a halo had assembled 4% of its mass: 0 0
manysmallerpiecesatz>z .Wewillfollowcommonprac-
0
ticeincallingthesepiecesthe‘progenitors’,andwewilluse
(cid:34) (cid:18) t (cid:19)8.4(cid:35)1/8 N(m,z|M0,z0)todenotethe‘massfunction’ofprogenitors:
cvir =4 1+ 3.751t (1) i.e., the mean number of progenitors of M0 that, at z >z0,
0.04 hadmassm.Wewillusem (z)todenotethemostmassive
1
ofthesetofprogenitorspresentatz.This‘mostmassivepro-
genitor’ may be different from what we will call ‘the main
where t denotes the time at which the halo had assembled
f
progenitor’, m (z), which is the most massive progenitor
a fraction f of its mass (in what follows we will use indis- MP
ofthemostmassiveprogenitorofthemostmassiveprogeni-
tinctly both t and t to denote the present day time, at
1 0
tor...andsoonbacktosomehighredshiftz>z .Indeed,it
which the halo had assembled all its mass). If this remark- 0
isalmostcertainlytruethatm (z)(cid:62)m (z),withequality
ablerelationwereexact,thenitwouldimplythattheM−c 1 MP
only guaranteed when m (z)>M /2. We quantify the dif-
relation, and its dependence on cosmology, comes entirely 1 0
ferencesbetweenthesetwoquantitiesintheAppendix.But
fromthefactthatt /t hasadistributionwhichdepends
0.04 1
because it is m which is relevant to equation (1) we will
onhalomass(recallthatmassivehalosassembletheirmass MP
use it in what follows.
later, so they typically have larger values of t /t ). This
0.04 1
The formation time of a halo, p (>z |M ,z ), is usu-
mass dependence itself depends on cosmology, as does the f f 0 0
allydefinedastheearliesttimewhenm (z)>fM .(Note
conversion between time and the observable, redshift. MP 0
that this is different from defining formation as the earliest
Unfortunately, estimating the mass and cosmology de-
timethatasingleprogenitorcontainsafractionf ofthefinal
pendenceoft /t isnotstraightforward.Analyticarguments
f 1 mass, which would correspond to requiring m (z) > fM .
for this are only available for f >1/2 (Lacey & Cole 1993; 1 0
WediscussthischoiceintheAppendix.)Ifp (m,z|M ,z )
Nusser & Sheth 1999), and these are not particularly accu- MP 0 0
denotes the probability that m (z)=m, then
rate (Giocoli et al. 2007). So the main goal of the present MP
workistoprovideasimpleprescriptionthatisaccurateeven p (>z |M ,z )=p (>fM ,z |M ,z ), (2)
f f 0 0 MP 0 f 0 0
when f (cid:28)1.
Althoughthecloserelationbetweencandassemblyhis- wherethesubscriptf onthelefthandsideisareminderthat
tory is one of the main motivations of the present work, we theshapeofthedistributiondependsonf.Thisisasimple
note that there is also interest in using the distribution of generalization, to arbitrary values of f, of the argument in
galaxyvelocitydispersions(Shethetal.2003)andtheweak Lacey&Cole(1993)whoassumedthatf =1/2,andNusser
lensingsignaltoconstraincosmologicalparameters(Fuetal. & Sheth (1999) who considered f >1/2.
2008; Schrabback et al. 2010). Both these observables are When f (cid:62) 1/2 then mMP = m1, since a halo cannot
expected to be related to halo mass and concentration, so havemorethanoneprogenitorwithmassgreaterthanfM0.
these studies will also benefit from a better understanding As a result, pMP(m,z|M0,z0) is simply related to the pro-
of the formation history of dark matter haloes. genitor mass function N(m,z|M0,z0). In particular,
Inaddition,thereisconsiderableinterestinquantifying p (>fM ,z |M ,z )=N(>fM ,z |M ,z ) forf (cid:62)1/2.
thedifferencebetweenthetimewhenthestarsinthecentral MP 0 f 0 0 0 f 0 0
(3)
galaxyinahaloformedandthetimewhenthosestarswere
This is fortunate, since simple analytic estimates of N(>
firstassembledintoasingleobject.Whereascrudeestimates
fM ,z |M ,z ) are available. Unfortunately, this simplic-
ofthestellaragecanbemadefromtheobservedcolors,there 0 f 0 0
ity is lost when f < 1/2, and so, although equa-
arecurrentlynoestimatesofthehaloassemblytime.Aby-
tion (2) still holds, there is no analytic estimate for p (>
product of our study of the joint distribution of halo mass, MP
fM ,z |M ,z ) in this case.
concentration and assembly history is an estimate of how 0 f 0 0
In what follows, we will simply measure p (>
precisely the mass and concentration of a halo predict its f
z |M ,z ), with z determined by requiring m (z ) (cid:62)
assembly time. f 0 0 f MP f
fM in simulations, and use it to determine the MP distri-
0
This paper is organized as follows. In Section 2 we de-
butionatanygivenz.Namely,whereastheusualprocedure
scribe the close connection between the formation time dis- (when f (cid:62)1/2) is to estimate the formation time distribu-
tributionandthemassofthemainprogenitoronwhichour
tion by differentiating the right hand side of equation (2)
model is built. Section 3 describes our simulation-based es-
with respect to z , we will instead estimate the MP distri-
f
timate of the formation time distribution for arbitrary f.
bution (when f <1/2) by differentiating the left-hand side
Italsoquantifiescorrelationsbetweenhalomass,concentra-
with respect to f (at fixed z). From this, one can derive an
tion and t , providing estimates of how precisely the con-
f estimateofthemeanvalueofm ateachz.Butnotethat
MP
centrationcanbeestimatedfromknowledgeofthemassand
this mean value can be got more directly from equation (2)
assembly history, as well as how well the mass and concen-
by using the fact that
trationofahalopredictitsformationtime.InSection4we
describe how to use our fitting formula to generate Monte- (cid:90) M0
dm p (>m ,z|M ,z )=(cid:104)m (z)|M ,z (cid:105).
Carlo merger histories from which to estimate halo concen- MP MP MP 0 0 MP 0 0
0
trations. Section 5 quantifies how halo assembly histories (4)
dependoncosmologicalparameters,andafinalsectionsum- Since the median m˜ at z is easy to estimate – it is that
MP
marizes. f atwhichequation(2)equals1/2–comparisonoftheme-
(cid:13)c 2011RAS,MNRAS000,1–15
Halo formation times and concentrations 3
dianandthemeanprovideasimpleestimateofhowskewed plots, it suffices to rewrite the expression for the median
p (m ,z|M ,z ) is. formation redshift as:
MP MP 0 0
In what follows, we will be more interested in the me- (cid:115) (cid:115)
δ (z ) S(M ) S(fM )
dian relation than in the mean. This is because the mean c f =1+w˜ 0 0 −1. (9)
δ (z ) f δ2(z ) S(M )
formation redshift as a function of the required mass of the c 0 c 0 0
main progenitor, (cid:104)z |f(cid:105), traces out a different curve than
f Toanexcellentapproximation,thelefthandsideisjustthe
does the mean main progenitor mass as a function of red-
ratio of linear theory growth factors at the two redshifts.
shift (cid:104)f|z(cid:105) (Nusser & Sheth 1999). In contrast, the median
E.g., for an Einstein-de Sitter model (itself an excellent ap-
z as a function of f traces out the same curve as does the
f proximation for redshifts greater than about 2), this ratio
medianf asafunctionofz:thisconvenientpropertyofthe would be (a /a ) = (t /t )2/3. And, for a power-law spec-
0 f 0 f
medianrelations,whichequation(2)makesobvious,hasnot
trumthefinaltermontherighthandsideisjustafunction
been highlighted before.
of f. Since w˜ is also a function of f only, the only mass
f
dependence comes from S(M )/δ2(z ). If we fix the mass,
0 c 0
then this term decreases as z increases, so the distribution
0
2.1 Scaled units
of (t /t ) is expected to shift towards unity as the redshift
f 0
Inprinciple,thedistributionp (>z |M ,z )dependsonf, at which the halos are identified increases.
f f 0 0
M and z . However, the analytic solution for f (cid:62) 1/2 is
0 0
actually a function of the scaled variable
3 A FITTING FORMULA FOR THE
δ (z )−δ (z )
w = c f c 0 , (5) FORMATION TIME DISTRIBUTION
f (cid:112)
S(fM )−S(M )
0 0
In what follows, we will use the GIF2 numerical simulation
where δc(z) is the initial overdensity required for spheri- (Gaoetal.2004;Giocolietal.2008)tocalibrateourfitting
cal collapse at z, extrapolated using linear theory to the formula for p (w ).
f f
present time and S(M) is the variance in the linear fluc-
tuation field when smoothed with a top-hat filter of scale
R = (3M/4πρ¯)1/3, where ρ¯is the comoving density of the 3.1 The simulation
background.Whenz isrescaledinthisway,thentheshape
f The GIF2 simulation data we analyze below are publicly
ofp(>w |M ,z )isessentiallythesameforallvaluesofM
f 0 0 0 available at http://www.mpa-garching.mpg.de/Virgo. The
and z , although the dependence on f is still strong. E.g.,
0 simulation itself is described in some detail by Gao et al.
for f (cid:62)1/2, the expected shape of this distribution is
(2004).ItrepresentsaflatΛcolddarkmatter(ΛCDM)uni-
(cid:18)1 (cid:19) (cid:18)w (cid:19) (cid:18) 1(cid:19)(cid:114)2 (cid:32) w2(cid:33) verse with parameters (Ωm, σ8, h, Ωbh2 ) = (0.3, 0.9, 0.7,
p (w )= −1 erfc √f + 2− exp − f 0.0196).Theinitialdensityfluctuationspectrumhadanin-
f f f 2 f π 2
dexn=1,withtransferfunctionproducedbycmbfast(Sel-
(6)
jak&Zaldarriaga1996).Thesimulationfollowedtheevolu-
(Lacey&Cole1993;Nusser&Sheth1999).Inwhatfollows, tionof4003particlesinaperiodiccube110Mpc/honaside.
weshowthatinoursimulationsthisremainstrueforsmaller Theindividualparticlemassis1.73×109M /h,allowingus
(cid:12)
values of f as well. Recent work (Paranjape et al. 2011;
to study the formation histories of a large sample of haloes
Musso&Sheth2012)suggeststhatw maynotbetheideal
f spanning a wide mass range. Particle positions and veloci-
choice for the scaling variable – this may account for some
ties were stored at 53 output times, mainly logarithmically
of thesmall departuresfrom self-similarity thatare evident
spaced between 1+z=20 and 1. Halo and subhalo merger
in the following plots. This means that we do not need to
trees were constructed from these outputs as described by
fit a different functional form for each value of f, M and
0 Tormen et al. (2004); Giocoli et al. (2007, 2008).
z . Rather we need only fit to p(w ), which is a function of
0 f Ateachsimulationsnapshothaloeswereidentifiedusing
f and w only. The expression above can become negative
f asphericaloverdensitycriterium.Ateachsnapshottheden-
at f < 1/2, so it is not a viable model for a probability
sityofeachparticleisassumedtobeinverselyproportional
distribution in this regime. Therefore, our main goal is to
to the cube of the distance to the tenth nearest neighbor.
provide a fitting formula which works even at small f.
We take as centre of the first halo the position of the dens-
Although we can use this fitting formula for the full
est particle. We then grow a sphere of matter around this
distribution p(w ) to estimate the mean (cid:104)w (cid:105), we will be
f f centre, and stop when the mean density within the sphere
moreinterestedinitsmedianw˜ .Thisisbecausethemedian
f falls below the virial value appropriate for the cosmologi-
formation redshift for fixed f satisfies
cal model at that redshift; we used the fitting formulae of
(cid:112) Eke et al. (1996) for the virial density. All the particles in-
δ (z )=δ (z )+w˜ S(fM )−S(M ) (7)
c f c 0 f 0 0
side this sphere are assigned to the new forming halo and
whereas the median main progenitor mass at fixed z > z0 areremovedfromthelist.Startingfromtheseconddensest
satisfies available particle we build the second halo, and so on until
[δ (z)−δ (z )]2 all the particles have been scanned. Particles not assigned
S(f˜M0)=S(M0)+ c w˜ 2c 0 . (8) to any collapsed objects are defined as field particles.
f Foreachhalomoremassivethan1011.5M /h(i.e.con-
(cid:12)
(Because both S(fM ) and w˜ depend on f, this last ex- taining at least ∼ 200 particles) in the five snapshots, cor-
0 f
pression must be solved to yield f˜.) In practice, these both responding to z = 0, 0.52, 1.05, 2 and 4.04 we build up
0
trace out the same relation, so for the purposes of making the merger history tree as follows. Starting from a halo at
(cid:13)c 2011RAS,MNRAS000,1–15
4 C. Giocoli et al. 2011
Figure 1. Dependence of rescaled formation time distribution p(w ) on the required fraction f of the final halo mass that must have
f
beenassembled.Differentsymbolsshowresultsfordifferentredshiftsz0 atwhichthehostswereidentifiedintheGIF2simulation.The
dashedcurveshowsequation(6),modifiedfollowingGiocolietal.(2007).Thesolidcurveshowsourequation(10).
z = z , we define its progenitors at the previous output, 3.2 The fitting formula
0
z = z +dz , as all haloes containing at least one particle
0 1 Figure 1 shows the cumulative distribution of rescaled for-
that at z=z will belong to that halo. The “main progeni-
0 mation redshifts. Each panel shows results for a different
tor”atz +dz isdefinedastheonewhichprovidesthemost
0 1 value of f, as indicated in the axis labels f = 0.9,0.5,0.1
masstothehaloatz .Thenwerepeatthesameprocedure,
0 and 0.01. In each panel, symbols show the measurements
now starting with the main progenitor at z =z +dz and
0 1 forhaloesmoremassivethan1011.5h−1M identifiedatfive
identifyitsmainprogenitoratz=z +dz +dz ,andsoon (cid:12)
0 1 2 different redshifts, approximately z = 0,1/2,1,2, and 4.
backwardintime.Sincewealwaysfollowthemainprogeni- 0
Clearly,scalingz tow producescurveswhichareapprox-
torhalo,theresultingmergertreeconsistsofamaintrunk, f f
imately independent of redshift.
whichtracesthemainprogenitorbackintime,m (z),and
MP The solid line shows the result of fitting
of “satellites”, which are all the progenitors which, at any
α
time, merge directly onto the main progenitor. P(>w )= f (10)
f ewf2/2+αf −1
to the measurements. This returns a value α for each f,
f
In what follows, to guarantee a well-defined sample of
whichisshownbythesymbolsinFigure2.Theα (f)rela-
f
relaxed haloes, we excluded all halos for which m (z) ex-
MP tion is well described by
ceeded the final host halo mass by more than 10%, as well
as all halos which were, in fact, unbound. α =0.815e−2f3/f0.707, (11)
f
(cid:13)c 2011RAS,MNRAS000,1–15
Halo formation times and concentrations 5
B =(1+0.63f−2/3)/β . For this model the mean is
0 f
(cid:90) Γ(B +1/β )
(cid:104)w (cid:105)= dw p(w )w =γ 0 f (16)
f f f f f Γ(B )
0
but one must estimate the median numerically.
Figure 3 compares the differential formation redshift
distributionforfourvaluesoftheassembledfractionf,with
our two fitting functions. The symbols show the measure-
mentsforhaloesmoremassivethan1011.5h−1M atfivedif-
(cid:12)
ferentz ,asinFigure1.Forequation(15)(dashedcurves),
0
the best-fit parameters γ and β satisfy
f
1
γ =0.12+ +4.3(f −0.24)2, (17)
f 290f1.4
and
e3.2γf
β =3.05e−0.6γf + . (18)
f 3800
Figure 4 shows these scalings. Despite the fact that equa-
tion (15) provides a more accurate description of the mea-
surements, we have found equation (12) to be more useful
because it comes with a simple expression for the median
relation.
Figure2.Dependenceofα ,thefreeparameterinequation(10),
f
onthemassfractionf =mMP/M0 requiredforformation.
3.3 Test of scaled units
whichisshownasthesolidcurveineachpanel.Theshaded Figure 5 shows the correlation between the formation red-
regions around each curve show the 1 and 2σ contours of shift(forf =0.9,0.5,0.1and0.01)andthehosthalomass
∆χ2, where χ2(αf) = (cid:80)i[Pi − P(> wi,f,αf)2], is com- for the same choices of z0 as before, in scaled units. I.e.,
puted for each mass and redshift bin. (Equation 10 con- the redshift is expressed as δc(zf)−δc(z0) and the mass
tinues to provide a good fit if we define formation using asS(fM0)−S(M0).Thedifferentsymbolsrepresentdiffer-
m1 rather than mMP, with the only difference being that ent z0 (same as Figure 1); they show the median of the
α =0.867e−2f3/f0.8. This α is larger than that for m correlation, i.e., the median of δc(zf) − δc(z0) in bins of
f f MP
by a factor of 1.06/f0.1: the median w is shifted to higher S(fM0)−S(M0), while the solid lines of the same color
f
show the first and third quartiles. The dotted and solid
redshiftsthanwhenformationisdefinedusingthemainpro-
lines in each panel show a least-squares fit to the correla-
genitor.)
tion, and the prediction of our model (equation 13). The
Figure 1 shows that equation (10) is reasonably accu-
dashed line in the two top panels shows the prediction,
rate at least between the first and third quartiles of the
N(> m,z |M ,z ) = 1/2, associated with equation (6).
distributions. The associated differential distribution is f 0 0
Note that, in the format we have chosen to present our re-
p(w )=−∂P(>wf) =− αfwew2/2 . (12) sults, the prediction boils down to a line of slope 1/2, with
f ∂wf (cid:2)ew2/2+αf −1(cid:3)2 zero-point equal to log10w˜f.
It has median
(cid:112) 3.4 The median mass accretion history
w˜ = 2 ln(α +1). (13)
f f
As described above, we have obtained the merger history
and the mean, computed via equation (4), is
treeforeachhalomoremassivethan1011.5h−1M atz ,for
(cid:12) 0
(cid:104)w (cid:105)=(cid:112)π/2 αf (cid:88)∞ (1−αf)k. (14) four different choices of z0. Figure 6 shows the mass of the
f 1−αf k1/2 mainhaloprogenitor,inunitsofthemassatz0,asafunction
k=1 of z. The different panels show results for parent halos of
For large values of f, αf (cid:28) 1, and (cid:104)wf(cid:105) → (cid:112)αfπ2/2 different masses, here expressed as fractions of M0∗ (where
whereas w˜f → (2/π)(cid:104)wf(cid:105). In this limit, the distribution is S(M0∗) ≡ δc2(0), so M0∗ corresponds to 8.9×1012M(cid:12)/h).
rather skewed. However, it becomes less skewed as f in- From top to bottom we show the halo accretion histories
creases. When αf = 1, then (cid:104)wf(cid:105) = (cid:112)π/2 = 1.25 whereas starting from different redshifts z0. The open circles and
w˜ =(cid:112)2 ln(2)=1.18. And when f =0.04 then α =8 so dashedcurvewhichenclosethemrepresentthemedian,the
f f
(cid:104)w (cid:105)=2.06 whereas w˜ =2.09. first and the third quartile of the tree extracted from the
f f
Since equation (12) is not very accurate in the tails of simulation. The solid curves enclose 95% of the trees.
the distribution, we have also fit p(w ) to Equation(2)showsthat,forthesemedianquantities,it
f
shouldnotmatterifweplotthemedianz atfixedm/M =
f
p(wf)=A0wf0.63f−2/3e−γfwfβf (15) f,orthemedianf atfixedz.Thisisindeedthecase,andis
incontrasttowhatwouldhavehappenedifwehadchosento
where A = βγB0/Γ(B ) is a normalization factor, with showthecorrespondingmeanvalues:thecurvewhichtraces
0 f 0
(cid:13)c 2011RAS,MNRAS000,1–15
6 C. Giocoli et al. 2011
Figure 3.Differentialformationtimedistribution.OpendiamondsshowtheresultsfromtheGIF2simulationforhaloesmoremassive
than 1011.5h−1M(cid:12) identified at five redshifts z0 =0, 0.52, 1.05, 2 and 4.04. The solid and dashed curves show the result of fitting to
equations(12)and(15).
Figure 4.Dependenceofthebest-fitfreeparametersinequation(15)onf;curvesshowthescalingsgivenbyequations(17)and(18).
Thebottompanelsshowtheresidualsfromtheserelations.
(cid:104)z |f(cid:105)asafunctionoff isslightlydifferentfromthatwhich showsourmodel,evaluatedasfollows:givenM atredshift
f 0
traces(cid:104)f|z (cid:105)asafunctionofz.Ourequation(2)showsthat z , we compute S(M ), δ (z ), and a table of S(fM ) for
f 0 0 c 0 0
thisisaconsequenceofthemeannotbeingthesameasthe any 0 < f < 1. Inverting equation (13) then gives z via
f
median: the distributions are skewed. equation (9). This agrees very well with the numerical sim-
The dot-dashed curve in each panel shows the predic- ulation results and therefore also with the model of Zhao
tion of Zhao et al. (2009): it describes our measurements et al. (2009). But note that our prescription is much easier
very well. The dashed curve (for z = 0 only), shows the to implement.
0
modelofvandenBosch(2002).Thesolidcurveineachpanel
(cid:13)c 2011RAS,MNRAS000,1–15
Halo formation times and concentrations 7
Figure 5.Correlationbetweenformationredshift–definedasthefirsttimethatthemainprogenitorcontainsafractionf ofthefinal
massM0 –andhosthalomass,expressedinscaledunits.Thedifferentpanelsshowresultsfordifferentchoicesoff (sameasprevious
figure).Thesymbols,alsosameaspreviousfigure,showresultsfordifferentz0.Thedotted,dashed,andsolidlinesshowaleast-squaresto
thedataatallredshifts(opendiamonds),therelationassociatedwithequation(6),andthepredictionofoursimplemodel(equation13).
Inthisformat,bothmodelspredictalineofslope1/2,sotheyonlydifferinthevalueofthezero-point.
3.5 Correlation between formation times 3.6 Halo concentrations from masses and
formation times
Equation (1) states that the formation time t of a halo
0.04
can be converted into an estimate of its concentration. To
checkthis,weestimatedc foreachhaloinGIF2byfitting
vir
the matter density profile to the NFW functional form,
ρ
Sofar,wehavemainlytestedifwf isindeedagoodscaling ρ(r|Mvir)= r/r (1+sr/r )2 , (19)
variable for the entire range of f. In what follows, it is in- s s
teresting to know if the different formation time definitions where
arecorrelated.E.g.,ifahaloisabovethemedianformation M (cid:20) c (cid:21)−1
redshiftforonevalueoff,isitalsolikelytohaveformedat ρ = vir ln(1+c )− vir (20)
s 4π(R /c )3 vir 1+c
above average w for another f? Figure 7 shows that this vir vir vir
f
correlation is weak, at least for f = 1/2 and 0.04: When and c ≡r /R . Figure 8 shows the correlation between
vir s vir
w −(cid:104)w (cid:105) and w −(cid:104)w (cid:105) are normalized by their c and t /t , for various M and z . The dot-dashed
0.5 0.5 0.04 0.04 vir 0 0.04 vir 0
rms values, then the correlation coefficient is ∼0.25. curve (same in each panel) shows equation( 1): if c were
vir
(cid:13)c 2011RAS,MNRAS000,1–15
8 C. Giocoli et al. 2011
Figure 6.MassgrowthhistoryofdarkmatterhaloesintheΛCDMGIF2cosmology.Thedifferentpanelsrefertovariousredshiftsand
host halo masses, as labeled. In each panel, open circles show the median z at fixed m/M, and the curves that enclose them show the
firstandthethirdquartilesofthedistributiondefinedbytheGIF2sample.Thesecurvesarethesameasthoseforthemedianm/M at
fixed z: as expected from equation (2). The dot-dashed, dashed and solid curves show the model of Zhao et al. (2009), van den Bosch
(2002)(validonlyforhaloesstartingfromz0=0)andourequation(9).
(cid:13)c 2011RAS,MNRAS000,1–15
Halo formation times and concentrations 9
Table 1. Precision of concentration estimates for halos with
Mvir >1013h−1M(cid:12).
rms(log10cvir|Mvir) 0.124
rms(log10cvir|τ0.04,Mvir) 0.120
rms(log10cvir|τ0.5,τ0.04,Mvir) 0.100
Table 2. Precision of formation time estimates for halos with
Mvir >1013h−1M(cid:12).
rms(τ0.5|Mvir) 0.123
rms(τ0.5|cvir) 0.088
rms(τ0.5|cvir,Mvir) 0.087
rms(τ0.04|Mvir) 0.176
rms(τ0.04|cvir) 0.111
rms(τ0.04|cvir,Mvir) 0.109
Figure 7.Correlationbetweenformationtimes,whenformation (cid:63) τf ≡log10(tf/t1)
isdefinedasthefirsttimethatthemainprogenitormMPcontains
0.5 and 0.04 of the total mass. We have scaled the variables by
their rms values, so the fact that the median w0.5/σ0.5 at fixed the first and third quartiles. Different panels show results
w0.04/σ0.04 is rather different from the median w0.04/σ0.04 at fordifferentredshifts.Thedottedlineshowsaleast-squares
fixed w0.5/σ0.5 indicates that the correlation between these two fit to the data, while the dot-dashed and solid curves show
isweak.
the relations which come from inserting the mass accretion
histories of Zhao et al. (2009) and our equation (9) into
equation(1).Bothapproachescapturethemassdependence
completely predicted by t /t then there should be no
0 0.04 of the concentration, and its flattening with redshift, very
scatter around this relation.
well.Butnotethatourestimateofthevalueoft /t that
0.04 1
In each panel, dots show the halos, and triangles with
isappropriateforagivenmassbin(e.g.,ourequation9),is
error bars show the median and first and third quartiles in
much more straightforward to implement.
log (c ) for narrow bins in t /t . The actual distribu-
10 vir 0 0.04 TheM−crelationformassesoforder106M isneeded
(cid:12)
tions of log (c ) and log (t /t ) are shown along the
10 vir 10 0 0.04 formodelingthemilli-lensingeffectsofgalaxy-sizehalosub-
leftaxisandacrossthebottom.(Notethat,foreachM bin,
0 structures (Keeton 2003; Keeton et al. 2003; Sluse et al.
the peak of the distribution of t /t shifts towards unity
0 0.04 2003;Metcalf&Madau2001;Metcalf&Amara2012).The
as z increases, for the reason given following equation 9.)
0 smallmassregimeisalsoimportantforestimatingthedark
Clearly,althoughequation(1)providesareasonabledescrip-
matter annihilation signal coming from the integrated ef-
tionofthetriangles,thescatteraroundthisrelationiscon-
fectofsubstructuresandfieldhaloesatmassessmallerthan
siderable.Quantitatively,thescatterinlog c is0.13dex
10 vir 106M (Pieri et al. 2005; Giocoli et al. 2008, 2009). How-
for z =0 halos with M (cid:62)1013h−1M . (cid:12)
0 vir (cid:12) ever, this is below the mass resolution of current numerical
The previous subsection showed that t and t of a
0.5 0.04 simulations. Therefore, Figure 10 shows the result of using
haloareonlyweaklycorrelated.Soitisreasonabletoaskif
ourmodel(equation(9,wheret andt areobtainedus-
0.5 0.04
some of the scatter in c can be removed if one includes
vir ingourmassaccretionhistorymodel)toestimatetheM−c
information about t0.5. E.g., one might reasonably expect relation at six different redshifts down to 3×104h−1M .
(cid:12)
the most concentrated halos to have small values of both
t andt .Wehavefoundthatthetypicalconcentration
0.5 0.04
scales as
3.7 Formation times from masses and
(cid:34) (cid:18) t (cid:19)1.15 (cid:18) t (cid:19)2.3(cid:35) concentrations
log c =log 0.45 4.23+ 1 + 1 .
10 vir 10 t0.04 t0.5 In real datasets, while we may hope to have a reasonable
(21) estimate of the mass of the host halo, and perhaps of its
For halos more massive than 1013h−1M , the rms scatter concentration, we are unlikely to know either t or t .
(cid:12) 0.04 0.5
around this relation (0.1 dex) is indeed reduced with re- So one might reasonably ask how well t can be predicted
f
spect to the case in which only t is known (0.12 dex). given knowledge of either M , c or both.
0.04 vir vir
Evidently, the remaining scatter is due to other processes. Since equation (10) gives the distribution of z given
f
Table 1 summarizes how this scatter decreases as more in- M and z , it can be manipulated to give the distri-
vir 0
formation is added. bution of t /t at fixed M . In practice, we can get a
f 1 vir
Oneofthemaingoalsofthispaperistoprovideasimple good idea of the rms scatter around the mean by estimat-
algorithmforgeneratingtheM−crelation.Figure9shows ing how far on either side of the median value one must
this relation in GIF2. The open circles show the median of go before 68% of the total probability has been enclosed.
(cid:112)
the distribution at fixed mass, and the error bar encloses Thesecorrespondtovaluesofw = 2ln(1+0.188α )and
f f
(cid:13)c 2011RAS,MNRAS000,1–15
10 C. Giocoli et al. 2011
Figure8.Correlationbetweenhosthaloconcentrationandthetimeatwhichthesystemassembled4%ofitsmass.Thedifferentpanels
(left, central and right) show results for different host halo masses M0, while the different windows show results for different z0 (same
asFigure9).Thehistogramsshowthedistributionsofcvir andt0/t0.04,andthesymbolswitherrorbarsshowthecorrelationbetween
them:themedianandfirstandthirdquartilesofthedistributionofcvir ateacht0/t0.04.Thedot-dashedcurveshowsequation(1).
Figure9.MedianmassconcentrationrelationintheGIF2simu- Figure 10.Massconcentrationrelationatsixdifferentredshifts
lation.Opencirclesshowhaloesinthesimulationatfivedifferent predicted using equation (9). The concentration prediction has
redshifts z0, as labeled in the panels. Error bars show the first beenextendeddowntosmallmasses:3×104M(cid:12)/h.
andthethirdquartilesofthedistributionforeachmassbin.The
dottedlineshowsaleast-squaresfittothedata;dot-dashedand
the solid curves show the predictions based on the median mass
growthhistoryofZhaoetal.(2009)andourequation(9).Ineach
panel we show the rms in log10(cvir) at fixed mass, for haloes define τf ≡ log10(tf/t0), then at any z0, (cid:104)τ0.5|cvir(cid:105) ≈
moremassivethan1013M(cid:12)/h. 0.175 − 0.503log10cvir whereas (cid:104)τ0.04|cvir(cid:105) ≈ −0.049 −
0.825log c . In both cases, the rms scatter around the
10 vir
fit is about 0.1 dex. Adding the mass does not reduce the
(cid:112)
w = 2ln(1+5.303α ). For f =0.04 we have α =8 so scattersignificantly,sowedonotreportfitstothiscase.Ta-
f f f
w =1.35and2.75respectively.ForM =8.9×1012h−1M , ble2summarizestheprecisionwithwhichahalo’sformation
f 0 (cid:12)
S(M )/δ2(z )=1,soδ (z )/δ (z )=2.81and4.68,making time can be predicted from its mass and concentration. In
0 c 0 c f c 0
(t /t ) = 0.18 and 0.08. Thus, the rms value of log(t /t ) Figure 11 we show the correlation between the halo con-
f 0 f 0
should be about 0.16 dex. This is indeed close to the value centration and τ , where f = 0.5 on the left and f = 0.04
f
0.17dexmeasureddirectlyinthesimulations.Asimilarana- on the right panel, for haloes with M (cid:62) 1013M /h and
vir (cid:12)
lyis for f =0.5 returns similarly good agreement. at any considered z . The data points with error bars show
0
The t − c relation is more complicated. If we themedian,withthequartiles,ofboththeconcentrationat
f vir
(cid:13)c 2011RAS,MNRAS000,1–15