natural downsizing in hierarchical galaxy formation · a downsizing effect can actually appear in...

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Mon. Not. R. Astron. Soc.000, 1–16 (2006) Printed 6 June 2018 (MN LATEX style file v2.2)

Natural Downsizing in Hierarchical Galaxy Formation

Eyal Neistein1, Frank C. van den Bosch2, and Avishai Dekel11 Racah Institute of Physics, The Hebrew University, Jerusalem, Israel2 Max-Planck-Institute for Astronomy, Konigstuhl 17, D-69117 Heidelberg, Germanye-mails: eyal [email protected]; [email protected]; [email protected]

ABSTRACTStellar-population analyses of today’s galaxies show “downsizing”, where the stars in moremassive galaxies tend to have formed earlier and over a shorter time span. We show thatthis phenomenon is not necessarily “anti-hierarchical” but rather has its natural roots in thebottom-up clustering process of dark-matter haloes. Whilethe main progenitor does indeedshow an opposite effect, the integrated mass in all the progenitors down to a given minimummass shows a robust downsizing that is qualitatively similar to what has been observed. Theseresults are derived analytically from the standard extended Press Schechter (EPS) theory, andare confirmed by merger trees based on EPS or drawn fromN -body simulations. The down-sizing is valid for any minimum mass, as long as it is the same for all haloes at any given time,but the effect is weaker for smaller minimum mass. If efficient star formation is triggered byatomic cooling, then a minimum halo mass arises naturally from the minimum virial temper-ature for cooling,T ≃ 10

4K, though for such a small minimum mass the effect is weakerthan observed. Baryonic feedback effects, which are expected to stretch the duration of starformation in small galaxies and shut it down in massive haloes at late epochs, are likely toplay a subsequent role in shaping up the final downsizing behaviour. Other appearances ofdownsizing, such as the decline with time of the typical massof star-forming galaxies, maynot be attributed to the gravitational clustering process but rather arise from the gas processes.

Key words: cosmology: theory — dark matter — galaxies: ellipticals — galaxies: haloes —

1 INTRODUCTION

A key issue in the study of galaxy formation is the anti-correlationbetween the stellar mass of a galaxy and the formation epoch of thestars in it, which is referred to in general terms as “downsizing”.In its most pronounced form, this is simply the fact that ellipti-cal galaxies consist of old stellar populations and tend to be moremassive while disc galaxies have younger stars and are typicallyless massive. However, a similar correlation between stellar massand age is detected within each of the two major classes of galax-ies, whether they are classified morphologically as ellipticals ver-sus spirals or by colour as “red sequence” versus “blue sequence”galaxies. These trends are quite robust, e.g., they are insensitive tohow luminosity is translated to stellar mass and colour to stellarage.

A downsizing effect can actually appear in different formswhich refer to different phenomena, involving different types ofgalaxies and different epochs in their histories. One form,whichis the main focus of the current paper, is the fact that the star for-mation histories inferred from present-day galaxies usingsyntheticstellar evolution models correlate with galactic stellar mass. Thestars in more massive galaxies tend to form at an earlier epoch andover a shorter time span. This phenomenon is termed “archaeo-logical downsizing” (ADS, following Thomas et al. 2005). Usingobserved line indices and abundance ratios, ADS has been detected

in elliptical galaxies (Thomas et al. 2005; Nelan et al. 2005), andin a large sample of galaxies from the Sloan Digital Sky Survey(Heavens et al. 2004; Jimenez et al. 2005).

The other face of downsizing is the fact that the sites of activestar formation shift from high-mass galaxies at early timesto lowermass systems at lower redshift. We term this phenomenon “down-sizing in time” (DST). It has first been detected by Cowie et al.(1996), who found that the maximum rest-frameK-band luminos-ity of galaxies undergoing rapid star formation has been decliningsmoothly with time in the redshift rangez = 0.2 − 1.7. This DSTphenomenon has been confirmed by numerous subsequent stud-ies (Guzman et al. 1997; Brinchmann & Ellis 2000; Kodama et al.2004; Juneau et al. 2005; Bell et al. 2005; Bundy et al. 2005).

It is important to realize that these two forms of downsizingcan be very different, and possibly even orthogonal to each other.The archaeological analysis of local galaxies highlights the forma-tion epoch of the majority of their stars, which at least in the case ofellipticals occurs at high redshifts,z ∼ 2−5. In contrast, downsiz-ing in time refers to the specific star-formation rate (SSFR)at rela-tively low redshifts,z . 1, and therefore focuses on later phases ofstar formation, which may involve only a small fraction of the starsin the galaxy. Unless the stellar-mass ranking of present day galax-ies is the same as that of their progenitors at higher redshifts, thesetwo forms of downsizing do not necessarily reflect the same phe-nomenon. Since hierarchical clustering in general does notpreserve

c© 2006 RAS

http://arxiv.org/abs/astro-ph/0605045v2

2 E. Neistein, F. C. van den Bosch, & A. Dekel

this mass ranking, the two forms of downsizing should be treatedas two different phenomena. Indeed, as we will demonstrate below,the understanding of one does not imply an understanding of theother.

In the standardΛCDM cosmological scenario, dark-matterhaloes are built hierarchically bottom up. This is obvious for theevolution of individual haloes, as they are constructed by the grad-ual gravitational assembly of smaller progenitor haloes that havecollapsed and virialized earlier on. The bottom-up clustering canalso be inferred statistically from the power spectrum of initialdensity fluctuations, which indicates that the mass distribution ofcollapsing systems is shifting in time from small to large masses.These hierarchical aspects of the clustering process have led to themisleading notion that one expects big haloes to “form” later thansmall haloes, without distinguishing between the dynamical col-lapse or assembly of these haloes and the formation epoch of thestars in them. The observed downsizing is therefore frequently re-ferred to in the literature as “anti-hierarchical”, and thus as posinga severe challenge to the standard model for structure formation.However, when comparing the histories of different haloes,the evo-lution may be interpreted as bottom-up or top-down depending onhow “formation” is defined.

The evolution of dark matter (DM) haloes has tradition-ally been studied through the histories of themain progenitors(Lacey & Cole 1993; Eisenstein & Loeb 1996; Nusser & Sheth1999; Firmani & Avila-Reese 2000; van den Bosch 2002b;Wechsler et al. 2002; Li et al. 2005). The main-progenitor assem-bly history is constructed by following back in time the mostmassive progenitor in each merger event. We termMmain(z) themain progenitor mass at redshiftz. The corresponding formationredshiftzmain of a halo of massM0 at z = 0 is commonly definedas the time at which the main progenitor contained one half ofto-day’s mass,Mmain(zmain) = M0/2. According to this definition,more massive haloes indeed form later. The formation redshift ofthe main progenitor has been computed by Lacey & Cole (1993)based on the Extended Press-Schechter (EPS) formalism, andthetrend with mass has been confirmed for various cosmologies (seefor example van den Bosch 2002b, hereafter vdB02). This has alsobeen tested using trees extracted from cosmologicalN -body sim-ulations (Lacey & Cole 1994; Wechsler et al. 2002). We confirmthis behaviour below using a new analytic estimation of the fulltime evolution ofMmain(z), based on the EPS formalism itselfwithout the need to construct merger-tree realizations.

However, the history of the main progenitor of a given halodoes not represent the history of the whole population of progen-itors in which the stars of a present-day halo have formed. Per-haps more directly relevant for the stellar population at any givenepoch is the sum over the masses ofall the virialized progenitorsin that specific tree at that time, which we termMall(z). If thissummation is performed down to a zero minimum mass, we haveby definitionMall(z) = M0. However, when a non-zero mini-mum massMmin(z) is applied, the same for all haloes, we find arobust archeological downsizing behaviour.1 We demonstrate thiseffect analytically based on the EPS formalism and confirm itusingMonte-Carlo EPS merger trees as well as trees extracted fromN -body simulations. We prove that this phenomenon is valid foranyrealistic power-spectrum shape and for any choice ofMmin(z), aslong as it is the same for all haloes at a given time. We note that

1 a similar point has been made in parallel by Mouri & Taniguchi(2006)using a very different methodology.

0

0.3

0.6

0.9

1.2

Log(

1+z)

M0 ~ 10 M

minM

0~100 M

min

0.040.1

0.4 0.99

Figure 1. An illustration of the upsizing ofMmain versus the downsiz-ing of Mall in dark-halo merger trees. Compared are random trees drawnfrom the EPS probabilities for haloes of current massesM0 ∼ 10Mmin

and∼ 100Mmin. The mass of the main progenitor versus the total massin all the progenitors aboveMmin are shown atz = 3. The progenitors,of massM , are marked by circles of sizes and spacings proportional to(M/M0)1/3. The values ofMmain/M0 andMall/M0 are indicated. Themain progenitor, along the left branch, is more massive in the less massiveM0, showing upsizing. The integrated mass in all the progenitors down toMmin is larger in the massiveM0, demonstrating “downsizing”.

a similar trend has been found by Bower (1991) for an Einstein-deSitter cosmology and a power-law power spectrum.

The difference betweenMmain and Mall is illustrated inFig. 1, which compares thez = 3 progenitors above a givenMmin

in random realizations of merger trees corresponding to currenthaloes ofM0 ∼ 10Mmin and∼ 100Mmin. The downsizing be-haviour forMall is apparent, while forMmain the familiar oppositetrend stands out (we term this trend as “upsizing”). The averagedistributions of relative masses inz = 3 progenitors, derived usingEPS (see below) for the same two values ofM0 as in Fig. 1, areshown in Fig. 2. The upsizing ofMmain is indicated by the excessof massive progenitors for the smaller current halo. The downsizingof Mall is demonstrated by the excess of the overall integral downtoMmin/M0 for the more massive current halo.

A realistic and necessary condition for star formation is thatthe gas is able to cool efficiently. This is only possible if the gasresides in a halo whose virial temperature exceeds a critical thresh-old of T ∼ 104K, above which atomic cooling becomes efficient.This provides a natural thresholdMmin(z) for Mall(z). If star for-

c© 2006 RAS, MNRAS000, 1–16

Natural Downsizing in Hierarchical Galaxy Formation 3

Log[ M/M0 ]

dN/d

(logM

) ×

M/M

0

−2 −1.5 −1 −0.5 00

0.05

0.1

0.15

0.2

M0 ~ 100 M

min

M0 ~ 10 M

min

z=3

Figure 2. Upsizing ofMmain versus downsizing ofMall in the distribu-tion of mass in progenitors atz = 3. The area under each curve, fromlog(M/M0) to 0, is the total mass in progenitors aboveM relative toM0.The excess of mass in massive progenitors for the smaller current halo indi-cates upsizing ofMmain. The excess in total mass down toMmin/M0 forthe more massive current halo demonstrates downsizing ofMall.

mation is of the maximum possible efficiency, namely if all the gasin haloes aboveMmin(z) turns into stars on a free-fall time-scale,then the ADS in the stellar population emerges naturally from theADS of Mall(z).

In reality, however, the star formation rate is likely to beslowed down by a variety of baryonic processes, especially by“feedback” effects. As a result, the star-formation history mayor may not maintain the ADS seeded byMall(z) of the DMhaloes. This should in principle be modeled by semi-analytic mod-els (SAMs) of galaxy formation, which attempt to incorporate thebaryonic physical processes in merger trees of DM haloes. Unfor-tunately, early SAMs failed to reproduce the ADS of ellipticalsas we know it today (e.g., Kauffmann 1996; Baugh et al. 1996;Kauffmann & Charlot 1998; Thomas 1999; Thomas & Kauffmann1999), probably due to an inadequate treatment of feedbackeffects. SAMs also failed to recover the similar global trendobeyed by blue-sequence galaxies in color-magnitude diagrams(van den Bosch 2002a; Bell et al. 2003), thus highlighting the ap-parent discrepancy between theory and observation. However,more recent models (e.g., Bower et al. 2005; De Lucia et al. 2006;Croton et al. 2006; Cattaneo et al. 2006) do succeed in reproduc-ing an ADS behaviour, largely because of an improved treatmentof the feedback effects. The early SAMs only included super-nova feedback, which is efficient in slowing down star forma-tion preferentially in smaller galaxies below a virial velocity of∼ 100 km sec−1 (Dekel & Silk 1986). The problem is that thisprocess only causes adelay in the star formation: the gas is onlyprevented from forming stars until the halo has grown sufficientlymassive that supernova feedback is no longer efficient. Becausethis results in relatively late star formation, even in massive galax-ies, the SAMs were unable to predict the correct stellar ages.The main success of the more modern SAMs is the inclusion ofAGN feedback and shock heating physics, which causes a shut-down, rather than a delay, of star formation at relatively late times(e.g., Birnboim & Dekel 2003; Binney 2004; Croton et al. 2006;Scannapieco et al. 2005; Cattaneo et al. 2006; Dekel & Birnboim

2006). Although the details of AGN feedback are still poorlyun-derstood, it has been argued that it is the main mechanism that ex-plains the “anti-hierarchical” nature of the relation between stellarmass and stellar age of galaxies.

However, we show below that the simulated star formationhistories of elliptical galaxies (De Lucia et al. 2006) are qualita-tively similar to the histories predicted byMall(z) of dark matterhaloes. This indicates that the roots of the observed ADS canbefound already in the natural downsizing of the dark matter haloes.Apparently, the complex feedback processes affecting the star for-mation do not change the general trend and only provide fine-tuning to the ADS effect. We conclude that ADS should not be re-garded as a surprising “anti-hierarchical” phenomenon of complexgas physics — it is rather the most natural, expected behaviour inthe hierarchical clustering scenario.

On the other hand, we find that the downsizing in time as ob-served at relatively low redshifts cannot be easily traced back tothe bare properties of the dark matter merger trees. The massdis-tribution of late-type efficient star formers at late times must bestrongly affected by feedback or other gas processes and thereforethe modeling of this aspect of downsizing should involve more re-alistic star formation rates. We show that only whenMmin(z) isproperly increasing with redshift, possibly mimicking therequiredbaryonic effects, the star-formation rate associated withMall(z)can be forced to a qualitative agreement with the observed down-sizing in time.

The paper is organized as follows. In the following introduc-tory section,§2, we spell out the relevant items from the EPS for-malism and describe how we generate Monte-Carlo merger treesthat serve us as a reference when needed. In§3, we address the av-erageMmain(z), derive an analytic approximation for it, and con-firm that it behaves opposite to downsizing. In§4 we study the aver-ageMall(z), compute it analytically from the EPS formalism, anddemonstrate that it shows a robust ADS behaviour. We also studythe mutual correlation between the formation times associated withMmain andMall. In §5 we compute the EPS formation rate of DMhaloes of a given mass, and compare it with star-formation historiesin semi-analytic simulations and in observations. In§6 we addressthe downsizing in time of the SSFR as observed at different red-shifts out toz ∼ 1. In §7 we summarize our results and discussthem.

Throughout this paper we use a flatΛCDM cosmology, withthe standard power spectrumP (k) = kT 2(k). The transfer func-tion (Bardeen et al. 1986) is

T (k) =ln(1 + 2.34q)

2.34q× (1)

[1 + 3.89q + (16.1q)2 + (5.46q)3 + (6.71q)4

]−1/4.

Hereq = k/Γ, with k in hMpc−1, andΓ is the power spectrumshape parameter (Sugiyama 1995)

Γ = Ωmh exp[−Ωb(1 +

√2h/Ωm)

], (2)

where Ωb = 0.044 throughout the paper. Unless specificallystated otherwise, we use the standard cosmological parameters,with ΩΛ = 0.7, Ωm = 0.3, σ8 = 1.0, andh = 0.7 (wheneverwe modifyΩm or h we recomputeΓ according to the above defi-nition).

c© 2006 RAS, MNRAS000, 1–16


2 EXTENDED PRESS-SCHECHTER THEORY

2.1 The Formalism

In the standard model for structure formation the initial densitycontrastδ(x) = ρ(x)/ρ − 1 is considered to be a Gaussian ran-dom field, which is therefore completely specified by the powerspectrumP (k). As long asδ ≪ 1 the growth of the perturbationsis linear andδ(x, t2) = δ(x, t1)D(t2)/D(t1), whereD(t) is thelinear growth factor. Onceδ(x) exceeds a critical thresholdδ0critthe perturbation starts to collapse to form a virialized object (halo).In the case of spherical collapseδ0crit ≃ 1.68. In what follows wedefineδ0 as the initial density contrast field linearly extrapolatedto the present time. In terms ofδ0, regions that have collapsed toform virialized objects at redshiftz are then associated with thoseregions for whichδ0 > δc(z) ≡ δ0crit/D(z).

In order to assign masses to these collapsed regions, thePress-Schechter (PS) formalism considers the density contrast δ0smoothed with a spatial window function (filter)W (r;Rf ). HereRf is a characteristic size of the filter, which is used to compute ahalo massM = γf ρR

3f/3, with ρ the mean mass density of the

Universe andγf a geometrical factor that depends on the particularchoice of filter. Theansatzof the PS formalism is that the fractionof mass that at redshiftz is contained in haloes with masses greaterthanM is equal to two times the probability that the density con-trast smoothed withW (r;Rf ) exceedsδc(z). This results in thewell known PS mass function for the comoving number density ofhaloes:

dn

d lnM(M, z) dM =

√2

πρ

δc(z)

σ2(M)

∣∣∣∣dσ

dM

∣∣∣∣ exp[− δ2c (z)

2σ2(M)

]dM (3)

(Press & Schechter 1974). Hereσ2(M) is the mass variance of thesmoothed density field given by

σ2(M) =1

2π2

∫∞

0

P (k) W 2(k;Rf ) k2 dk , (4)

with W (k;Rf ) the Fourier transform ofW (r;Rf ).The extendedPress-Schechter (EPS) model developed by

Bond et al. (1991), is based on the excursion set formalism. Foreach point one constructs ‘trajectories’δ(M) of the linear densitycontrast at that position as function of the smoothing massM . Inwhat follows we adopt the notation of Lacey & Cole (1993, here-after LC93) and use the variablesS = σ2(M) andω = δc(z) tolabel mass and redshift, respectively. In the limitRf → ∞ one hasthatS = δ(S) = 0, which can be considered the starting pointof the trajectories. IncreasingS corresponds to decreasing the filtermassM , andδ(S) starts to wander away from zero, executing arandom walk (if the filter is a sharpk-space filter). The fraction ofmatter in collapsed objects in the mass intervalM ,M+dM at red-shift z is now associated with the fraction of trajectories that havetheir first upcrossingthrough the barrierω = δc(z) in the intervalS, S + dS, which is given by

f(S, ω) dS =1√2π

ω

S3/2exp

[−ω2

2S

]dS (5)

(Bond et al. 1991; Bower 1991, LC93). After conversion to num-ber counting, this probability function yields the PS mass functionof equation (3). Note that this approach does not suffer fromthearbitrary factor two in the original Press & Schechter approach.

Since for random walks the upcrossing probabilities are inde-pendent of the path taken (i.e., the upcrossing is a Markov process),

the probability for a change∆S in a time step∆ω is simply givenby equation (5) withS andω replaced with∆S and∆ω, respec-tively. This allows one to immediate write down theconditionalprobability that a particle in a halo of massM2 at z2 was embed-ded in a halo of massM1 atz1 (with z1 > z2) as

P (S1, ω1|S2, ω2) dS1 = f(S1 − S2, ω1 − ω2)dS1 =

1√2π

(ω1 − ω2)

(S1 − S2)3/2exp

[− (ω1 − ω2)

2

2(S1 − S2)

]dS1 . (6)

Converting from mass weighting to number weighting, one obtainsthe average number of progenitors atz1 in the mass intervalM1,M1 + dM1 which by redshiftz2 have merged to form a halo ofmassM2:

dN

dM1(M1, z1|M2, z2) dM1 =

M2

M1P (S1, ω1|S2, ω2)

∣∣∣∣dS

dM

∣∣∣∣ dM1 . (7)

2.2 Constructing Merger-Trees

The conditional mass function can be combined with Monte-Carlotechniques to construct merger histories (also called merger trees)of dark matter haloes. If one wants to construct a set of progenitormasses for a given parent halo mass, one needs to obey two require-ments. First, thenumberdistribution of progenitor masses of manyindependent realizations needs to follow (7). Second, massneeds tobe conserved, so that in each individual realization the sumof theprogenitor masses is equal to the mass of the parent halo. In princi-ple, this requirement for mass conservation implies that the proba-bility for the mass of thenth progenitor needs to be conditional onthe masses of then − 1 progenitor haloes already drawn. Unfor-tunately, these conditional probability functions are unknown, andone has to resort to an approximate technique for the constructionof merger trees.

The most widely adopted algorithm is theN -branch treemethod with accretion developed by Somerville & Kolatt (1999,hereafter SK99). This method is more reliable than for examplethe binary-tree method of LC93. In particular, it ensures exact massconservation, and yields conditional mass functions that are in goodagreement with direct predictions from EPS theory (i.e., the methodis self-consistent).

The SK99 method works as follows. First a value for∆S isdrawn from the mass-weighted probability function

f(∆S,∆ω) d∆S =1√2π

∆ω

∆S3/2exp

[− (∆ω2)

2∆S

]d∆S (8)

[cf. equation (6)]. Here∆ω is a measure for the time step usedin the merger tree, and is a free parameter (see below). The pro-genitor mass,Mp, corresponding to∆S follows from σ2(Mp) =σ2(M) +∆S. With each new progenitor it is checked whether thesum of the progenitor masses drawn thus far exceeds the mass ofthe parent,M . If this is the case the progenitor is rejected and anew progenitor mass is drawn. Any progenitor withMp < Mmin

is added to the mass componentMacc that is considered to be ac-creted onto the parent in a smooth fashion (i.e., the formation his-tory of these small mass progenitors is not followed furtherbackin time). HereMmin is a free parameter that has to be chosen suf-ficiently small. This procedure is repeated until the total mass left,Mleft = M −Macc −

∑Mp, is less thanMmin. This remaining

mass is assigned toMacc and one moves on to the next time step.As all other methods for constructing merger trees (e.g., LC93;

c© 2006 RAS, MNRAS000, 1–16


Kauffmann & White 1993), the SK99 algorithm is only an approx-imation. In particular, it is based on themass-weighted progeni-tor probability function (8), rather than on thenumberdistribution(7), and mass conservation is enforced ‘by hand’, by rejecting pro-genitor masses that overflow the mass budget. Consequently,thenumber distribution of the first-drawn progenitor masses isdiffer-ent from that of the second-drawn progenitor masses, etc. Some-what fortunately, the sum of these distributions closely matches thenumber distribution (7) of all progenitors, but only if sufficientlysmall time steps∆ω are used (see SK99 and vdB02). In princi-ple, since the upcrossing of random walks through a boundaryisa Markov process, the statistics of progenitor masses should be in-dependent of the time steps taken, indicating that the method isnot perfectly justified. Consequently, not all statistics of the mergertrees thus constructed are necessarily accurate, something that hasto be kept in mind.

In this paper we adopt a time step of

∆ω =

√∣∣∣∣dS

dM

∣∣∣∣ 10−3M

[b+ a log10

(M

Mmin

)]−1

, (9)

wherea = 0.3, b = 0.8. As shown in SK99, this time step yieldsnumber distributions of progenitor masses that are in good agree-ment with (7). The average number of progenitors per time stepis ∼ 1.5 for 109h−1M⊙ 6 M 6 1014h−1M⊙ andMmin =108h−1M⊙.

3 GROWTH OF THE MAIN PROGENITOR

The full merger history of any individual dark matter halo isacomplex structure containing a lot of information. It has there-fore been customary to define amain progenitor history, some-times termedmass accretion history(Firmani & Avila-Reese 2000;Wechsler et al. 2002, vdB02) ormass assembly history(Li et al.2005), which restricts attention to the main “trunk” of the mergertree. This main trunk is defined by following the branching ofamerger tree back in time, and selecting at each branching pointthe most massive progenitor. We denote byMmain(z) the mass ofthis main progenitor as a function of redshiftz. Note that with thisdefinition, the main progenitor is not necessarily the most massiveprogenitor of its generation at a given time, eventhough it neveraccretes other haloes that are more massive than itself.

3.1 Analytical Derivation

Using EPS merger trees and cosmologicalN -body simulations,vdB02 and Wechsler et al. (2002) have obtained simple fittingfor-mulae for the main progenitor history. We show here that onecan actually derive a useful analytical approximation for the av-erageMmain(z), defined at each redshift as the average mass ofMmain(z) over an ensemble of merger-trees. We derive it directlyfrom the EPS formalism, without the need to construct Monte-Carlo merger trees. As shown in the Appendix,Mmain(z) obeysthe differential equation

dMmain

dω= −

√2

π

Mmain√Sq − S

. (10)

HereS = S(Mmain), Sq = S(Mmain/q), and the value ofq isbetween 2 and a maximum valueqmax. We show in the Appendixthat the uncertainty inq is an intrinsic property of the EPS theory;different algorithms for constructing merger-trees may correspond

0 0.3 0.6 0.9 1.2−2

−1.5

−1

−0.5

0

Log(1+z)

Log[

Mm

ain/M

0]

5×109

5×1013

Figure 3. Growth of the main progenitor for haloes of different present-day masses. The massMmain(z) is the average at a fixed redshiftz. Thecurves are normalized to match atz = 0. The halo masses, from top tobottom, range from5×109 to 5×1013 h−1M⊙ equally spaced in the log.The symbols refer to the averages over Monte-Carlo merger trees and thecurves represent our analytic results. The upsizing of the main progenitor isobvious.

to differentq within the allowed range. The maximum value,qmax,depends slightly on cosmology and mass. For example,qmax is be-tween2.1 and2.3 for flat cosmogonies withΩm between 0.1 and0.9 and halo masses between108 and1015 h−1M⊙.

Solving the differential equation forMmain we come up witha useful fitting formula:

Mmain(z) =Ωm

Γ3F−1q

[g(32Γ)

σ8(ω(z)−ω0)+Fq

( Γ3

ΩmM0

)].(11)

Hereg andFq are analytic fitting functions motivated by the shapeof the power-spectrum (see Appendix for their definition, and rangeof accuracy). For theΛCDM concordance cosmology, we find thatthe standard algorithm of SK99 for constructing random mergertrees yields anMmain(z) which is well fitted by eq. (11) withq =2.2. We therefore adopt this value below. Varyingq between 2 and2.3 (the maximum range allowed) gives rise to a relatively smallchange inMmain; nearMmain = 0.5M0 this change is∼ 8%.

Fig. 3 showsMmain(z)/M0, the average, main progenitor his-tory for haloes of different masses today, all normalized totoday’smass. The figure compares our analytic estimate based on equa-tion (11) with the averages over histories computed from Monte-Carlo merger tree realizations described in§2.2. We see that theanalytic estimate reproduces the results from the realizations quitewell, although there is a slight mismatch at highz. This differencemay either reflect the allowed intrinsic uncertainty withinthe EPSformalism or it may be due to other inaccuracies in the SK99 algo-rithm used to construct the trees.

3.2 Archaeological Upsizing

We see in Fig. 3 that the average growth curve of the main progen-itor is shifted toward later times in more massive haloes, implyingthe opposite of downsizing, termed here as upsizing. One waytoquantify the downsizing behaviour is via the quantity

c© 2006 RAS, MNRAS000, 1–16


0 0.3 0.6 0.9 1.2−2

−1.5

−1

−0.5

0

Log(1+z)

Log[

Mal

l/M0]

5×109

5×1013

Figure 4. Growth of the total mass in all the progenitors,Mall(z), forhaloes of different present-day masses. The minimum progenitor mass is∼ 109M⊙, specified in equation (20) as a function of redshift. The masses,curves and symbols are the same as in Fig. 3. A downsizing behaviour isclearly seen. It is more pronounced at small masses which arecloser toMmin.

Dmain(z | z0,M0) ≡ d

dM0

[Mmain(z)

M0

]. (12)

Positive values ofDmain mark an archeological downsizing be-haviour, negative values refer to upsizing, and|Dmain| measuresthe strength of the effect. As is clear from Fig. 3,Dmain(z) < 0at all z, indicating that the main progenitor histories of dark matterhaloes reveal upsizing.

Is this upsizing a generic feature ofMmain(z)? To answer this,we write the average main progenitor mass of a halo of massM0 asmall time step∆ω ago as

Mmain(∆ω)

M0=

∫ Sq−S0

0

f(∆S,∆ω)d∆S , (13)

(see Appendix). Differentiating with respect toM0 while keeping∆ω fixed yields the ADS strength

Dmain(z | z0,M0) = (14)

f(Sq − S0,∆ω)

[1

q

dS

dM(M0/q) −

dS

dM(M0)

].

Whether this is negative or not depends on the shape ofS(M). Fora self-similar power spectrum,S ∝ M−α, we have thatDmain < 0as long asα > 0. We have also verified numerically thatDmain <0 for the standardΛCDM power spectrum at all masses. While theabove expression forDmain is valid only for small∆ω, its sign isthe same at allz. A more accurate expression forDmain at anyzcan be obtained by differentiating equation (11) above.

3.3 Assembly Time of the Main Progenitor

Following numerous other studies (see§1), we define the assem-bly redshiftzmain of a halo of massM0 at timeω0 according toMmain(zmain) = M0/2. Using equation (11) we obtain

ωmain ≡ ω(zmain) = (15)

ω0 +σ8

g(32Γ)

[Fq

(Γ3

Ωm

M0

2

)− Fq

(Γ3

ΩmM0

)].

9 11 130

1.5

3

4.5

6

Log(M0) [h−1 M

sun]

Red

shift

, z

zall

zmain

Figure 5. Formation redshifts, when the mass was one half of today’s mass,for the main progenitor (zmain, solid lines, circles) and for all the progen-itors (zall, dashed lines, squares) versus today’s halo mass. The symbolsand error-bars refer to an ensemble of random EPS merger-trees. The thick-ness of thezmain curve refers to the allowed range obtained by varyingqbetween 2 andqmax in equation (15). The dashed line is the theoretical pre-diction (23) forzall. zmain shows upsizing whilezall shows downsizing.

In the case of scale-free initial conditions, were the power-spectrumis a pure power law,P (k) ∝ kn, we have thatS ∝ M−α withα = (n+ 3)/3. In this case the expression simplifies to

ωmain = ω0 +

√2π(qα − 1)

α

(√Sq −

√S0

). (16)

We note that Lacey & Cole (1993) computed a related expressionfor the average assembly redshift of the main progenitor (see theAppendix for more details).

Fig. 5 shows the average assembly redshift of the main pro-genitor,zmain, as a function of the present-day halo mass, for theΛCDM concordance cosmology. The thickness of the curve corre-sponds to the allowed range of intrinsic uncertainty inq in equa-tion (15), as computed in the Appendix. The ADS strength,Dmain,associated with the slope ofzmain(M), does not change signifi-cantly with halo mass. The theoretical EPS curve shows an excel-lent agreement with thezmain obtained from an ensemble of ran-dom EPS merger histories (circles with error bars). The error barscorrespond to the standard deviation inzmain over the individualmerger-trees.

Fig. 5 demonstrates again the upsizing behaviour of the mainprogenitor. This has been one of the reasons for interpreting theobserved downsizing as “anti-hierarchical”.

The distribution ofzmain in our ensemble of EPS merger treesis plotted in Fig. 6 for three different masses. One of them iscom-pared to the theoretical prediction by LC93,

Q(z) = − d

dz

∫ S2

S0

M0

Mf(S − S0, ω(z)− ω0)dS , (17)

wheref is defined in equation (5). As discussed in the Appendix,the theoretical distribution agrees with the random realizations atlow z, and any deviations are due to the limitations of the SK99algorithm used.

c© 2006 RAS, MNRAS000, 1–16


0 1 2 3 4 50

0.5

1

1.5

Redshift, z

dN/d

z

zmain

zall

5×1013

5×1012

5×1011

Figure 6. The distribution ofzmain andzall for different halo masses. Halomasses are5×1011, 5×1012 and5×1013 h−1M⊙ (dotted, dashed-dottedand solid lines). The solid thick line is the theoretical prediction for thedistribution ofzmain, equation (17), for a halo mass of5× 1012 h−1M⊙.The minimum progenitor mass is∼ 109M⊙, specified in equation (20) asa function of redshift.

4 GROWTH OF ALL THE PROGENITORS

Mmain(z) defined above only describes the mass growth historyof the main trunk of the full merger tree. It is unlikely, however,that this is an honest estimator of the star formation histories ofthe associated galaxies. After all, star formation can occur in allprogenitors that obey the necessary physical conditions, and is notrestricted to themain progenitor. Since gas needs to cool beforeit can form stars, and since the cooling time is primarily a func-tion of halo mass and redshift, we assume that star formationoc-curs in haloes with a mass above a threshold mass,Mmin(z). Thisprompts us to define the formation historyMall(z) of a present-daydark matter halo as the sum of the masses of all progenitors thatobey this condition. Supporting evidence for the possible successof such a model comes from the finding that the integral ofMall(z)over the entire present-day halo mass function provides a usefulbackbone for understanding the observed, universal star formationhistory (Hernquist & Springel 2003).

4.1 Analytical Derivation

The construction ofMall(z) for individual dark matter haloes re-quires a full merger tree, with a mass resolution that exceedsMmin(z). However, the formation history of a halo of massM0,averaged over many merger trees per each redshiftz, can be derivedstraightforwardly from the EPS formalism. It should equal the inte-gral over the progenitor mass function in the rangeM = Mmin(z)toM0:

Mall(z) = M0

∫ M0

Mmin(z)

P (S, ω|S0, ω0)

∣∣∣∣dS

dM

∣∣∣∣ dM , (18)

whereP (S,ω|S0, ω0) is defined in equation (6). Performing theintegral we obtain

Mall(z)

M0= 1− erf

(ω(z)− ω0√

2Smin(z)− 2S0

), (19)

whereSmin(z) ≡ S[Mmin(z)]. Note thatMall(z)/M0 dependson M0 throughS0, so that equation (19) cannot be written in anexplicit form.

We see that for a given cosmology, the average formation his-tory of a halo of massM0 is completely specified byMmin(z).As a first attempt we associateMmin with the halo mass that cor-responds to a virial temperature ofTvir = 104 K, the temperatureabove which atomic gas is able to cool and subsequently form stars.For a completely ionized, primordial gas this yields

Mmin(z) = 1.52 × 109h−1M⊙

(∆vir

101

)−1/2 (H(z)

H0

)−1

, (20)

where ∆vir(z) is the average overdensity of a virialized haloat redshift z relative to the critical density at that redshift(Bryan & Norman 1998), andH(z) is the Hubble parameter. Un-less specifically stated otherwise, we use this minimum thresholdmass in what follows.

The lines in Fig. 4 showMall(z) for several halo masses basedon equation (19). We now see that when “formation” is definedby Mall(z) we obtain ADS, with more massive haloes formingearlier. Also plotted in Fig. 4 are the results of the merger-trees(symbols). We see that while there is a fair, qualitative agreementbetween the histories extracted from the Monte-Carlo realizationsand the exact EPS predictions, the level of agreement becomes pro-gressively worse for more massive haloes (relative toMmin). Themerger trees predict an earlier formation time than what followsdirectly from EPS. A similar behaviour has been noticed by SK99(their Fig. 7), when comparing the empirical total mass containedin haloes above a minimum mass to the theoretical value. Thesedeviations arise from the approximations made in the algorithm forconstructing the Monte-Carlo merger trees (see discussionin §2.2).

Figure 7 shows theMall(z) histories of individual haloes, ob-tained from Monte-Carlo merger tree realizations (thin lines), com-pared to the average formation historyMall(z), calculated fromeq. 19 (thick dashed line). The scatter is higher for lower masshaloes. This can be crudely understood as a Poisson noise asso-ciated withNall, the number of progenitors aboveMmin at everygiven redshift. For the massive halo,M0 = 5×1013 h−1M⊙,Nall

is indeed quite large at all redshifts, leading to a small scatter. Forthe less massive halo,M0 = 5×1010 h−1M⊙, we haveNall < 20at all redshifts, which results in a larger scatter.

4.2 Archaeological Downsizing

The all-progenitor historiesMall(z) depend on the definition of thethreshold massMmin(z). Here we investigate the necessary con-ditions for these threshold masses in order forMall(z) to revealADS. Similar to what was done in§3.2, we define the “downsizingstrength”,Dall, as the derivative ofMall/M0 with respect toM0. Inorder to study theM0 dependence, we rewrite equation (18) usingdifferent variables,

Mall(ω)

M0=

∫ Smin−S0

0

f(∆S, ω − ω0)d∆S , (21)

where the functionf is defined in equation (5). This enables us todifferentiateMall/M0 with respect toM0, while keepingω fixed,which yields

Dall(ω |ω0,M0) = −f(Smin − S0, ω − ω0)dS

dM(M0) > 0 . (22)

Sincef is a probability function, anddS/dM < 0 for all M , wehave thatDall is always positive. This implies that ADS occurs

c© 2006 RAS, MNRAS000, 1–16


0 0.3 0.6 0.9 1.2−2

−1.5

−1

−0.5

0

Log(1+z)

−1.5

−1

−0.5

0

Log[

Mal

l/M0]

5×1010

5×1013

Figure 7. Individual realizations of all-progenitor histories (thin solidcurves) compared to the average at fixedz as calculated analytically (thickdashed curve). The curves are all normalized toM0 at z = 0. The upperand lower panels are forM0 = 5 × 1013 and5 × 1010 h−1M⊙ respec-tively. Mmin is specified in equation (20).

for any choice of the threshold massesMmin(z), and for any cos-mological power spectrum of fluctuations. The only assumptionsused are (i) that the threshold is global, i.e., thatMmin does not de-pend on the specific halo massM0, and (ii) that the excursion-settrajectories are Markovian, which allows the change of variablesleading from equation (18) to (21). Note that the downsizingaspectof Mall(z) does not depend on the actual shape off , which impliesthat ADS will occur for non-Gaussian density fluctuation fields aswell.

The opposite effect of upsizing could in principle occur if theMarkovian assumption of the EPS random walks breaks down, sothat the mass-weighted probability distributionP (S, ω|S0, ω0) de-pends onS0 rather than being a function ofS − S0 only. An ad-ditional requirement in this case is that the probability has a highercontribution from the low-mass end for largerM0. Therefore, theMarkovian nature of the random walks is a sufficient, but not anec-essary condition for ADS to occur.

4.3 Formation Time of All progenitors

Following the definition of assembly redshift, we define the forma-tion redshift of dark matter haloes,zall, by Mall(zall) = M0/2.Using equation (19) we obtain

ωall ≡ ω(zall) = ω0 + β√Smin − S0 , (23)

whereβ =√2/erf(1/2) ≃ 0.6745 (see also Bower 1991). The

dashed curve in Fig. 5 showszall as a function of halo mass com-puted using equation (23). The solid square with errorbars repre-sent the average and scatter as obtained from a large ensemble ofEPS merger trees. Note that these deviate significantly fromthedirect theoretical EPS prediction, especially at largeM0. This isin stark contrast to the case ofzmain(M0), where the merger treeresults agree well with the direct theoretical predictions. This sug-gests that the discrepancy inzall(M0) must originate in the statis-tics of smaller progenitors with masses< M0/2. As shown bySK99, theN -branch tree method with accretion used for the con-struction of the EPS merger trees slightly overpredicts thenum-

ber of small progenitors at high redshifts. Fig. 5 shows thatthiscan have a significant impact onzall; consequently, semi-analyticalmodels for galaxy formation that are based on such EPS mergertrees might actually overestimate the star formation ratesat highredshifts.

For haloes withM0 ≫ Mmin we have thatzall > zmain:typically the progenitors of a massive halo will have grown suffi-ciently massive to allow for star formation much before the finalhalo has assembled half its present-day mass into a single halo.Note thatzall − zmain decreases with decreasing halo mass. WhenM0 ≃ 2Mmin we have thatzall = zmain, by definition, whilezall < zmain for haloes withMmin < M0 < 2Mmin. Finally,for haloes withM0 < Mmin the formation redshiftzall is not de-fined. This systematic increase ofzall − zmain with increasing halomass may have interesting implications for galaxy formation, as itprovides a very natural means to break the self-similarity betweenhaloes of different masses, and their associated galaxies.

Although dynamical friction may delay the merging of galax-ies with respect to the epoch at which their host haloes merged,to first order we may associatezmain with the redshift belowwhich the haloes and their associated galaxies no longer experi-ence major mergers (i.e., belowzmain the main progenitor nevermerges with another halo of similar mass). In massive haloes, withM0 ≫ Mmin we expect that the majority of the stars have al-ready formed much before these last major mergers, and this ma-jority of the stars will thus have experienced one or more majormergers since their formation. Consequently, the majorityof thestars are most likely to reside in a spheroidal component, and thegalaxy is an early-type with relatively old stars. Contrary, in lowmass haloes, most of the progenitor haloes that are being accretedby the main progenitor atz < zmain will have massesM < Mmin,and will thus not have formed stars. The gas associated with theseprogenitors can only start to form stars once they become part ofthe main progenitor: star formation and galaxy assembly occur vir-tually hand-in-hand, with the stars being born in-situ in what is tobecome the final galaxy atz = 0. Since the system has not under-gone a major merger since roughly half the stars formed, the systemis likely to resemble a disk galaxy.

Although this is clearly severly oversimplified, it is interest-ing that some of the most pronounced scaling relations of galaxies,namely the relations between halo mass, stellar age, and galaxymorphology, may well have their direct origin in the backbone ofhalo formation histories combined with a simple halo mass thresh-old for star formation.

Finally, Fig. 6 shows the distribution ofzall for haloes of dif-ferent masses, as obtained from our EPS merger trees. Note that thescatter inzall is smaller for more massive haloes, as expected fromthe Poisson statistics discussed in§4.1

4.4 Comparison with N -body Simulations

While the merger trees analyzed thus far are based on the EPS for-malism, one can alternatively extract merger trees from cosmolog-ical N -body simulations. Here the gravitational dynamics is moreaccurate, limited only by numerical resolution effects. However, itshould be kept in mind that the identification of virialized haloes,and especially connecting them to construct merger trees, is a non-trivial enterprise involving several significant uncertainties.

We computeMall(z) from merger trees extracted from aΛCDM cosmologicalN -body simulation kindly provided by RisaWechsler. The simulation followed the trajectories of2563 colddark matter particles within a cubic, periodic box of comoving size

c© 2006 RAS, MNRAS000, 1–16


60 h−1Mpc from redshiftz = 40 to the present. The particle massis 1.1 × 109 h−1M⊙, and the minimum halo mass dictated by theresolution is2.2× 1010 h−1M⊙ (see Wechsler et al. 2002, for de-tails). For the construction ofMall(z) we imposeMmin values of5×1010 and5×1011 h−1M⊙, and we compare the resulting, aver-age formation histories to those computed from the EPS formalismusing the same threshold masses. The results are shown in Fig. 8,where symbols correspond to the formation histories extractedfrom theN -body simulations, while the lines show the direct, theo-retical predictions based on the EPS formalism (equation 19). Over-all the agreement is very satisfactory, although theN -body simula-tions predict a somewhat later formation whenMmin ≪ M0. Notethat the EPSmerger treesyield formation times that areearlierwith respect to the analytical formula (Fig. 4). Thus, the differencebetweenN -body simulations and EPS merger trees is larger thanthe difference between theN -body simulations and equation (19).Despite these discrepancies, theN -body results clearly confirm theEPS prediction thatMall of more massive haloes grows earlier. Wetherefore conclude that the ADS aspect ofMall is not an artifactof the EPS approximation, but is a generic property of DM mergertrees.

4.5 The Correlation between Formation Time and AssemblyTime

Sincezall increases with increasing halo mass (ADS), whilezmain

decreases (‘upsizing’), we have thatzall and zmain are anti-correlated when considering haloes of different masses. But whatabout the relation betweenzall and zmain for haloes of a fixedmass?

Fig. 9 shows the correlation betweenzall andzmain for haloesof given masses, withMmin = 5×1010. Results are shown for boththe numerical simulations (solid dots) and for EPS merger trees(contours). For a5×1011 h−1M⊙ halo, the number of progenitorsis small, and the full merger tree is not much more than the maintrunk. As a result, the values ofzall andzmain are not very differ-ent and they exhibit a rather strong correlation. When the mass getslarger, the scatter inzall tends to zero while the scatter inzmain re-mains large. Consequently, the correlation strength betweenzmain

andzall at fixed halo mass vanishes at largeM0.This has important implications. Using a large numerical sim-

ulation, Gao et al. (2005) and Harker et al. (2006) have foundapositive correlation betweenzmain and the environment density:i.e., haloes in an overdense region assemble earlier than haloesof the same mass in an underdense region. If galaxy properties,such as stellar age, is correlated withzmain, this means that haloesof a given mass host galaxies with different properties, depend-ing on their large scale environment. The results shown heresug-gest that this may be the case for relatively low mass haloes withM0 ≃ Mmin, since these systems reveal a positive correlationbetweenzmain and zall. In more massive haloes, however, withM0 ≫ Mmin, no such correlation is present, suggesting that thecorrelation betweenzmain and environment will not create a simi-lar correlation between stellar age and environment.

The positive correlation betweenzmain andzall at fixed massarises from their dependence on the merger tree properties at lowredshifts. For example, assume that the merger tree for somehalo issuch that the mass atz = 0.5 is the same as atz = 0. In this case,we can use the analytical expressions forzall andzmain starting atz0 = 0.5 , and not atz0 = 0. The correspondingzmain andzallwill refer to z0 = 0.5, so both will be delayed by the same amountof time, thus establishing a positive correlation.

0 0.3 0.6 0.9 1.2−2

−1.5

−1

−0.5

0

Log(1+z)

Log[

Mal

l/M0]

Mmin

= 5×1010

Mall

(N−body)

5×1013

5×1011

0 0.3 0.6 0.9 1.2−2

−1.5

−1

−0.5

0

Log(1+z)

Log[

Mal

l/M0]

5×1013

5×1012

Mall

(N−body)

Mmin

= 5×1011

Figure 8. All-progenitor histories drawn fromN -body simulations (sym-bols) compared to the EPS predictions (curves). The imposedmin-imum mass isMmin = 5 × 1010 and 5 × 1011M⊙ in thetop and bottom panels respectively. The mass bins in log massare[11.62, 11.78], [12.48, 13.00], [13.48, 14.00] where mass units areh−1M⊙. The number of haloes within each bin is 479, 205 and 23 re-spectively. The EPS theoretical curves corresponding to each mass bin areaverages over the same distribution of masses.

When we increase the halo mass, the ratioMmain(z)/Mall(z)decreases (this is true for any specific redshiftz). This implies thatthe fraction of mass incorporated in the main trunk is smaller, andas a result, there is more mass left in progenitors that belong to otherbranches. The scatter inMall comes from all the tree branches,where each branch contribute its own random behaviour. WhenMmain(z) is small with respect toMall(z) most of the contribu-tion to the scatter inMall comes from branches other than themain. This explains why the correlation betweenzall and zmain

gets poorer for high halo mass.

c© 2006 RAS, MNRAS000, 1–16


0 0.2 0.4 0.60

0.2

0.4

0.6

Log(1+zall

)

Log(

1+z m

ain)

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8

M=5×1013

M=5×1011 M=5×1012

All halos

Figure 9. Correlations betweenzmain and zall for random merger treeswith the same final mass. The contours, equally spaced in the log, referto the joint distribution from the EPS random merger trees. The minimummass isMmin = 5×1010 . The points are fromN -body merger-trees, withthe same mass bins as in Fig. 8.

5 HALO FORMATION RATES

We define the halo formation rate as the rate of change inMall.Using equation (19), this can be written as

R(ω |ω0,M0) ≡ d

dt

[Mall(ω)

M0

]= (24)

−√

2

π

1√Smin − S0

exp

[− (ω − ω0)

2

2Smin − 2S0

]dω

dt.

If we make the naive assumption that all the baryonic mass insidehaloes withM > Mmin forms stars instantaneously, than this ratereflects the star formation history of galaxies that at timeω0 arelocated in a halo of massM0: these formation rates basically reflectthe maximum possible star-formation efficiency.

The upper panel of Fig. 10 shows the star-formation histo-ries of elliptical galaxies as a function of the mass of the halo inwhich these galaxies are located atz = 0, from the semi-analyticalsimulations of De Lucia et al. (2006, their Figure 3). Note that thismodel predicts that ellipticals in more massive haloes formed theirstars earlier, and over a shorter period of time, in good, qualita-tive agreement with the observational data of Thomas et al. (2005)and Nelan et al. (2005). The lower panel of Fig. 10 shows the cor-responding formation rates of the dark matter haloes, as definedby equation (24). Here we have adopted the same cosmologicalparameters as in De Lucia et al. (i.e.,ΩΛ = 0.75, Ωm = 0.25,σ8 = 0.9, h = 0.73) and we have used a constant threshold massof Mmin = 1.72 × 1010 h−1M⊙, corresponding to the mass reso-lution of the numerical simulation used by these authors.

The formation rates of DM haloes, as seen in Fig. 10, reveal aqualitatively similar ADS behavior as for elliptical galaxies in theSAM of De Lucia et al. (2006) and in the observational data (e.g.,Thomas et al. 2005). Although the agreement is extremely good forthe massive haloes, the SAM predicts a significantly later forma-tion in lower mass haloes, indicating that the downsizing strengthis larger in the SAM. This highlights the crudeness of our simpli-fied model for star formation, while assumes that stars form instan-taneously as soon as the halo mass exceedsMmin. The comparison

9 10 11 12 130

0.1

0.2

0.3

Lookback time (Gyr)

For

mat

ion

rate

, R [G

yr−

1 ]

0

0.2

0.4

0.6De Lucia et al (galaxies)

DM HalosM

min=1.72×1010 h−1M

sun

Figure 10. Simulated star formation rate versus EPS halo formation rateR for different halo masses. Top panel: mean SSFR of elliptical galaxiestaken from the semi-analytic model of De Lucia et al. (2006).Galaxies arebinned by their halo mass atz = 0. Bottom panel: maximum SFR as im-plied by our simplified model, namelyR of equation (24), for different halomasses atz = 0. The curves in the two panels refer to halo masses of1012 ,1013, 1014, and1015 M⊙(solid, dashed, dotted, dashed-dotted lines re-spectively).Mmin is 1.72× 1010 h−1M⊙, the minimum halo mass in DeLucia et al. (2006). In the lower panel we add a curve for a halomass of1012 M⊙ with Mmin set to1011 M⊙ (thin solid line), as an example forthe halo formation rate whenMmin is only one order of magnitude belowthe halo mass.

with the SAM suggests that this is a fairly accurate assumption inmassive haloes. In low mass haloes, however, the baryonic feed-back processes modelled in the SAM must have caused a signif-icant delay in the formation of the stars. Indeed, the efficiency ofsupernova feedback to cause such a delay in larger in lower masshaloes (Dekel & Silk 1986). In principle we can increase the down-sizing strength for the dark matter haloes by increasingMmin. Forexample, the thin solid line plots the formation rates for a halo of1012M⊙ but with a higherMmin of 1011M⊙. This brings the for-mation rates in better agreement with the specific star formationrates of elliptical galaxies in haloes of1012M⊙ in the SAM. Thus,one may mimic the delays in star formation due to supernova feed-back effects by an increase in the star formation threshold massMmin, eventhough we do not necessarily consider this very phys-ical. We conclude that the ADS in galaxies has its natural originin the ADS ofMall, while the baryonic physics associated withcooling, star-formation, and feedback merely causes a shifting andstretching of the relative formation histories. The main trend withhalo mass, however, simply relates to the dark matter formationhistories.

We define the mean formation epoch of a dark matter halo as

ωR ≡∫ t00

R(ω)ωdt∫ t00

R(ω)dt(25)

If, for simplicity, we keep the star formation threshold mass con-stant, i.e.,Mmin(z) = Mmin, then this reduces to

ωR = ω0 +

√2

π(Smin − S0) . (26)

where we have used the fact that the denominator of equation (25)is equal to unity. Note that this mean formation epoch is very

c© 2006 RAS, MNRAS000, 1–16


10 11 125

7

9

11

Look

back

tim

e (G

yr)

Log(M0/30) [M

sun]

Highdensity

Thomas et al.Low density

DM Halos, Mmin

=109

Mmin

=1011

Figure 11. Effective formation epoch versus mass in EPS theory versusobservations. The formation epoch for dark matter haloes ofpresent massM0, based on equation (26), is plotted forMmin = 109 and1011M⊙

(solid curves). Halo masses are divided by 30 in order to roughly translateDM into stellar masses. The epoch for star formation as deduced from localellipticals by Thomas et al. (2005) is shown (shaded area) between the twodashed lines which refer to galaxies in low-density and high-density envi-ronments. Hereh = 0.75 as in Thomas et al. (2005). There is a downsizingbehaviour in both cases.

similar to ωall of equation (23), but withβ ≃ 0.67 replaced by√2/π ≃ 0.8.

Figure 11 compares our analytic estimates for the mean for-mation epoch of DM haloes to the star-formation histories deducedfrom nearby elliptical galaxies by Thomas et al. (2005), both for el-lipticals in low-density and high-density environments2. The solidlines correspond to our estimates of equation (26) forMmin = 109

and1011M⊙, as indicated. Note that we have divided the dark mat-ter masses by 30 to obtain a very rough proxy for the stellar mass.A comparison with the data of Thomas et al. is only trully mean-ingful if (i) all gas in haloes withM > Mmin is turned into starsinstantaneously, and (ii) haloes host only one galaxy whosestellarmass is equal toM0/30. Although neither of these is likely to becorrect, the data and ‘model’ are in qualitative agreement in that themore massive structures have formed earlier, i.e., the model showsADS. If Mmin = 109h−1M⊙, the DS strength is too weak, accrossthe mass range of interest, compared to the data. This indicates thatthe baryonic physics needs to delay and or suppress star formationrelatively more in lower mass haloes. Alternatively, ifMmin is sig-nificantly larger (∼ 1011h−1M⊙), the DS strength at fixed halomass is stronger, and there is less need to delay or suppress star for-mation in order to globally match the data. However, this requiresa yet unknown physical mechanism that can prevent star formationin all haloes below this mass limit.

Yet another way to view the ADS aspect of halo formationhistories is via the mean epoch at which a halo of massM0 at timeω0 has a progenitor of massM . The number density,dN(ω), ofprogenitors with masses in the intervalM to M + dM at timeωis given by equation (7). Using it to weight the averaging ofω weobtain

2 The density is defined as the number of galaxies within a one degreeradius, see Thomas et al. (2005) for details

ωp ≡∫dN(ω)ωdω∫dN(ω)dω

= ω0 +

√2

π(S − S0) . (27)

This resemblesωR in equation (26), meaning that the mean for-mation epoch for a givenMmin is equivalent to the mean epoch forprogenitors to be of a given mass,M = Mmin. The ADS behaviouris apparent in equation (27) from the fact thatωp increases withM0

(via S0). This implies that progenitorsof a given massappear ear-lier in the merger tree of a more massive present-day halo. This issimilar to the result obtained by Mouri & Taniguchi (2006), whoalso argue that downsizing is a natural prediction of hierarchicalformation scenarios.

6 DOWNSIZING IN TIME

As mentioned in§1, there is another observed downsizing effect,different from the archeological downsizing dealt with so far, whichrefers to the decrease with time of the characteristic mass of thegalaxies with the highest specific star formation rates. We showhere that, unlike the archeological downsizing, this downsizing intime (DST) is not in general rooted in the hierarchical clusteringof dark matter haloes. The dark haloes show such an effect only ifMmin is decreasing with time in a sufficiently steep pace.

The symbols in Fig 12 show the DST obtained from the data inBrinchmann & Ellis (2000). For a given SSFR we select from theirdata the stellar mass and redshift of a galaxy with that SSFR.Thesolid squares, connected by a dotted curve, plot the stellarmass ofobjects forming their stars with a SSFR of 1 Gyr−1, correspondingto a doubling time-scaleτc = 1 Gyr. Note that the characteristicstellar mass of systems forming stars at this rate is lower atlowerredshift; this is DST. The other symbols correspond to lowerSS-FRs of 0.1 Gyr−1 (solid dots connected by dashed curve) and 0.05Gyr−1 (stars connected by solid curve). Note that each of thesecurves reveals DST, and that more massive systems have lowerSS-FRs, at each redshift.

In order to compare this with dark matter haloes, we definethe “current”, specific formation rate of dark matter haloesas therate of change ofMall normalized toMall. This rate is obtainedby settingω = ω0 in the general expression forR(ω |ω0,M0) ofequation (24):

R(ω |ω,M0) = −√

2

π

1√Smin(z)− S(M0)

dω

dt(z) . (28)

For a fixed rateR = Rc, one can solve forMc(z), the mass ofhaloes that are formed with the rateRc:

S [Mc(z)] = Smin(z)−2

π

(dω

dt

)2

τ 2c , (29)

whereτc ≡ R−1c is the corresponding time-scale. The curves with-

out symbols in Fig 12 show theMc(z) relations thus obtainedfor four different time-scalesτc, as indicated. In order to allowfor a comparison with the data, we have divided the halo massesby 30, as a rough proxy for stellar mass. The first thing to no-tice is that these ‘model predictions’ have almost nothing in com-mon with the data. First of all, allMc(z) seem to converge to thesame mass at lowz, independent ofτc. This owes to the fact thatR → ∞ if M0 → Mmin; the specific formation rate becomesinfinite atMmin. Secondly, for high specific formation rates (lowτc), theMc(z) decreases with increasingz, opposite to the DSTobserved. This simply owes to the fact that lowτc implies thatMc(z) ∼ Mmin(z), which, according to equation (20) decreases

c© 2006 RAS, MNRAS000, 1–16


0 0.2 0.4 0.66

7

8

9

10

11

12

Log(1+z)

Log[

Mc/3

0]

[Msu

n h−

1 ]

τc=10 Gyr

τc=1 Gyr

τc=0.1 Gyr

τc=20 Gyr

Figure 12. The mass of haloes which form at a given rateRc atz,Mc(R =Rc, z) from equation (29). The curves are marked byτc = R−1

c . Forhigh enoughτc, the massMc depends solely ondω/dt, while for low τcit is given byMc(z) = Mmin(z). The connecting symbols refer to theobserved star-formation time scale for galaxies of a given stellar mass fromFig. 3 of Brinchmann & Ellis (2000), for the corresponding values ofτc.TheMc values for the dark haloes were divided by 30 in order to allowacrude comparison with the stellar mass of the galaxies.

with increasing redshift. Whenτc is sufficiently high (& 10 Gyr),however, the dark haloes show a qualitative DST, in thatMc in-creases with redshift. This basically owes to the fact that the contri-bution from thedω/dt term in equation (29) becomes dominantover the term governed by theMmin(z) behavior. We concludethat, in general, the formation histories of dark matter haloes donot show a DST effect as observed for galaxies. It is clear that DSTmust be driven by baryonic processes, which must strongly decou-ple the star formation rates from the halo formation rates. The chal-lenge for the models will be to do so while maintaining a fairly tightcoupling at highz, which, as we have shown, is required in orderto explain the ADS.

Another measure of DST for DM haloes is the time evolutionof the average halo mass at which dark matter is being added tovirialized progenitors, namely

MR =

∫R(ω |ω,M)M dn

dMdM∫

R(ω |ω,M) dndM

dM. (30)

HereR is given by equation (28) anddn/dM is the number densityof haloes per comoving volume (e.g., from Sheth & Tormen 2002).

In Fig. 13 we plotMR for several different growth rates ofMmin(z), all normalized to coincide with our standard value ofMmin at z = 0. Similar toMc defined above, the characteristicmassMR is decreasing with redshift when we setMmin to cor-respond to a constant virial temperatureTvir = 104 K (see equa-tion [20]), or whenMmin is constant in time. Only whenMmin isincreasing with redshift roughly as1 + z or faster does the massMR show a DST behaviour.

7 CONCLUSIONS

We have introduced a new quantity to quantify the growth of adark matter halo merger tree,Mall(z), the sum of the masses of

0 0.2 0.4 0.6

9

10

11

Log(1+z)

Log(

MR)

[M

sun h

−1 ]

Mmin

(z) ∝ T=104

Mmin

(z) = const

Mmin

(z) ∝ 1+z

Mmin

(z) ∝ (1+z)2

Figure 13. Average halo massMR weighted by halo formation rateR as afunction of redshift for differentz-dependences ofMmin(z). Downsizingin time is seen only whenMmin(z) is increasing linearly with(1 + z) orfaster.

all the virialized progenitors at redshiftz down to a minimum halomassMmin(z). We have shown, using EPS theory, that this quan-tity reveals an “archeological downsizing” behavior in that Mall(z)of more massive haloes grew earlier and on shorter time-scales.This behaviour is present for any choice of non-zeroMmin(z) andany cosmology. The only two conditions are (a) that the thresholdmassMmin(z) is independent of the massM0 of the present-dayhalo, and (b) that the progenitor mass function,P (M1, z1|M0, z0)(equation. [6]) either depends onS(M0) − S(M1) alone (i.e., thetrajectoriesδ(S) are Markovian), or is such that the fraction ofmass in progenitors belowMmin decreases with increasingM0.The fact that a similar archeological downsizing effect is revealedby EPS merger trees and in N-body simulations indicates thattheseconditions are at least approximately valid. One should note thatthe first condition, although quite robust, might be violated in cer-tain circumstances. For example, today’s halo massM0 could beinterpreted at highz as reflecting the local environment density,and if the threshold mass is somehow affected by its environmentthis could introduce a dependence ofMmin(z) onM0.

Using the EPS formalism, we have analytically formulatedthe virial mass growth curveMall(z), the corresponding forma-tion redshiftzall, and the formation rate. The latter is found to bequalitatively similar to the formation rate of stars in elliptical galax-ies, indicating that the observed archeological downsizing in thesesystems has its roots in the formation histories of the dark matterhaloes. However,Mall(z) is only a good tracer of the star forma-tion histories of galaxies if all the gas in haloes withM > Mmin

forms stars instantaneously. In reality, this will not be the case, ascooling and various feedback processes can delay and/or preventthe formation of stars, even in haloes withM ≫ Mmin. What isclear from our study, however, is that the halo mass dependence ofthese baryonic processes has to be such that it does not undo themass dependence already encoded inMall(z).

We have also studied the more common halo assembly histo-ries, defined as the mass growth histories,Mmain(z), of the mainprogenitor of the merger tree. We have developed an analytical ap-proximation for it based on EPS theory, and confirmed the known“upsizing” behaviour of this assembly history. We have shown that

c© 2006 RAS, MNRAS000, 1–16


it depends in principle on the shape of the power-spectrum, but it isvalid for all power-law spectra as well as for the CDM power spec-trum. The formation timeszmain andzall, for a sample of equal-mass haloes, were found to be correlated in a way that can be un-derstood in terms of the mass growth at low redshifts.

The downsizing in time, namely the decline with time of themass of star-forming galaxies, cannot be easily traced backto theproperties of the dark matter halo merger trees. With our idealizedrecipe of rapid star formation in virialized haloes aboveMmin(z),downsizing in time can be reproduced only ifMmin(z) is rapidlyincreasing withz. Otherwise, this kind of downsizing is most likelya result of feedback effects on star formation, which requires a moresophisticated modeling of the baryonic processes. The lesson isthat the different faces of “downsizing” reflect different phenom-ena, one naturally rooted in the hierarchical dark matter clusteringprocess and the other determined by non-trivial baryonic processes,which are yet to be properly modeled.

ACKNOWLEDGMENTS

We thank Risa Wechsler for providing the merger trees constructedfrom cosmologicalN -body simulations based on work supportedby NASA ATP NAG5-8218 (Primack and Dekel). We acknowledgestimulating discussions with Itai Arad and Aaron Dutton. This re-search has been supported by ISF 213/02 and by the German-IsraelEinstein Center at HU. AD acknowledges support from a BlaisePascal International Chair in Paris.

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APPENDIX: ANALYTICAL FORMULAE FOR Mmain

We useMmain(z) to denote the main progenitor mass at redshiftz, averaged (at fixedz) over many individual merger trees for thesame parent massM0. Here we use the EPS formalism to derive ananalytical estimate forMmain(z).

Basic Equation

Let’s start with a halo of massM0 at timeω0, and take a smalltime-step,∆ω, back in time. At the timeω0+∆ω we want to com-pute the average mass of the main progenitor. This requires the fullprobability distribution,P (Mmain|M0,∆ω), that a halo of massM0 atω0 has a main progenitor of massMmain at timeω0 +∆ω.For Mmain > M0/2, one has thatP (Mmain|M0,∆ω) is equalto the total progenitor distributiondN/dM given by equation (7),simply because any progenitor whose mass exceedsM0/2 must bethemainprogenitor. ForMmain < M0/2, however, the only validcondition is thatP 6 dN/dM , which is not sufficient to predictP (Mmain|M0,∆ω).

As a first, naive approximation we assume thatP (Mmain|M0,∆ω) = 0 for Mmain < M0/2, so that themain progenitor always has a massMmain > M0/2. Using

c© 2006 RAS, MNRAS000, 1–16


this approximation, the average mass of the main progenitor,Mmain(∆ω), can be written as:

Mmain(∆ω) =

∫ M0

M0/2

P (M |M0,∆ω)MdM . (31)

Using the definition ofdN/dM this reduces to

Mmain(∆ω) = M0

[1− erf

(∆ω√

2S2 − 2S0

)], (32)

with S2 = S(M0/2) andS0 = S(M0).We assume thatM0 is just the main progenitor of the previous

time-step3. The rate of change,dMmain/dω, can then be computedas

dMmain

dω= lim

∆ω→0

Mmain(∆ω)−M0

∆ω(33)

= −M0 lim∆ω→0

1

∆ωerf

(∆ω√

2S2 − 2S0

).

Using thaterf(x) → 2x/√π whenx → 0 this yields:

dMmain

dω= −

√2

π

Mmain√S2 − S

. (34)

In the case of scale free initial conditions, were the power-spectrum is a pure power law (S ∝ M−α) we can solve forMmain

analytically:

Mmain(ω) =[M

−α

2

0 + cα(ω − ω0)]− 2

α

, (35)

wherecα = α [2πS(M = 1) (2α − 1)]−1/2.The above derivation is based on the assumption that the main

progenitor always has a massMmain > M0/2 in the limit ∆ω →0. However, as we show below, when∆ω → 0 the probability thatMmain < M0/2 decrease like∆ω. Consequently, this will give anon-negligible effect for sufficiently largeω.

Towards better accuracy

The dot-dashed line in Fig. 14 shows the distribution of the mainprogenitor masses of a halo of massM0 = 1012 h−1M⊙ in a singletime step∆ω = 0.1, obtained from10.000 realizations based onthe SK99 algorithm. One can clearly see thatP (Mmain|M0,∆ω)has a non-negligible tail forMmain < M0/2. The following anal-ysis aims to find the solution forMmain taking this low-mass tailinto account.

The correct shape ofP (Mmain|M0,∆ω) can be constrainedby the following conditions:

(i) The integral ofP (Mmain|M0,∆ω) over all masses shouldequal unity, for all time-steps∆ω.

(ii) P (Mmain|M0,∆ω) = dN/dM , for Mmain > M0/2(equation [7]), andP (Mmain|M0,∆ω) 6 dN/dM , for Mmain <M0/2

(iii) P (Mmain|M0,∆ω) should not depend on the time-stepsubdivisions. This can be written as:

P (Mmain|M0,∆ω1 +∆ω2) = (36)∫P (M1|M0,∆ω1)P (Mmain|M1,∆ω2)dM1 .

3 Equation (32) becomes linear in∆ω for small enough∆ω, and thisgives:Mmain(∆ω1 +∆ω2|M0) = Mmain(∆ω2|Mmain(∆ω1|M0))

0 0.2 0.4 0.6 0.8

10−2

100

102

104

M/M0

dN/d

M ×

M0

Figure 14. Average number of progenitors of massM one time-step∆ω =0.1 before the present for halo of massM0 = 1012 h−1M⊙. The solidcurve is the theoretical prediction. The dot-dashed lines indicates that massdistribution of the main progenitor (defined as the most massive progenitor),as obtained from10.000 Monte-Carlo EPS realizations based on the SK99algorithm. Note that the probability distribution for the mass of the mainprogenitor, equalsdN/dM down toM0/2, but does not vanish down toM ∼ 0.25M0. Finally, the dashed line indicates the mass distribution ofthe least massive progenitors obtained in these10.000 realizations.

In what follows we estimate the limits onP (Mmain|M0,∆ω) us-ing conditions (i) and (ii), and show that they give a narrow rangefor Mmain(z). Condition (iii) does not force the solution to beunique, hence it enables a set of solutions, each of them is validwithin the EPS formalism. Because condition (iii) is more difficultto implement, we do not compute its effect onMmain, and assumethat it will not significantly affect the range of solutions.

The first condition onP (Mmain|M0,∆ω) is that its in-tegral equals unity. We definentail as the integral overP (Mmain|M0,∆ω) from Mmain = 0 toMmain = M0/2:

ntail = 1− M0√2π

∫ S2

S0

1

M

∆ω

∆S1.5exp

[−∆ω2

2∆S

]dS , (37)

where∆S = S − S0 = S(M) − S(M0).We can estimate the possible effectany tail will have on

Mmain, by computing the effect of the most extreme tails possible.The first extreme is to concentrate all the tail in a small range nearM = 0. In this case, the integral ofMmainP (Mmain|M0,∆ω)over the range0 6 Mmain < M0/2 will be zero. As a result,the Mmain(z) that corresponds to this extreme is given by equa-tion (32). The second extreme is that all the tail is concentratednearM0/2, that is,P (Mmain|M0,∆ω) has its maximum values(= dN/dM ) down to a lower mass limit,M/qmax, which is set bythe requirement that

ntail =M0√2π

∫ Sq+

S2

1

M

∆ω

∆S1.5exp

[−∆ω2

2∆S

]dS , (38)

whereSq+ = S(M/qmax). If we focus our attention on small time-steps∆ω, then we can use thatdP (Mmain|M0,∆ω)/dMmain ≃0 nearM0/2 to approximately write that

Sq+ ≃ S2 +

√π

2

(S2 − S0)1.5

∆ωntail . (39)

c© 2006 RAS, MNRAS000, 1–16


This enables us to use a simple equation forSq+, combined withthe definition ofntail in equation (37).

What remains is to find an appropriate expression forntail

which is valid in the limit of small time-steps∆ω. We thereforesplit the integral in equation (37) into two parts. The first one (n1)is for the range0 < ∆S < ∆Sǫ, where∆ω ≪ ∆Sǫ ≪ 1 and wecan make the approximationM ∼ M0:

n1 ≃ 1√2π

∫ S0+∆Sǫ

S0

∆ω

∆S1.5exp

[−∆ω2

2∆S

]dS (40)

= 1− erf

[∆ω√2∆Sǫ

]≃ 1−

√2

π

∆ω√∆Sǫ

.

The second range (n2) is for∆S > ∆Sǫ where the approximationexp

[−∆ω2/(2∆S)

]≃ 1 is valid:

n2 ≃ M0√2π

∫ S2

S0+∆Sǫ

1

M

∆ω

∆S1.5dS . (41)

Combining equations (37), (40) and (41) then yields :

ntail =

√2

π∆ω[∆S−0.5

ǫ − M0

2

∫ S2

S0+∆Sǫ

1

M

dS

∆S1.5

]. (42)

Finally we take the limit∆Sǫ → 0, and obtain

ntail =

√2

π∆ω

[1

2

∫ S2−S0

0

M −M0

M

d∆S

∆S1.5+

1√S2 − S0

], (43)

Substitution in (39) then yields

Sq+ ≃ 2S2−S0+(S2 − S0)

1.5

2

∫ S2−S0

0

M −M0

M

d∆S

∆S1.5, (44)

independent of∆ω. For the standardΛCDM cosmology this yields2.1 < qmax < 2.3 for 0.1 < Ωm 6 0.9 and108h−1M⊙ 6 M0 6

1015h−1M⊙. This implies that although there is a negligible prob-ability that the main progenitor has a massM0/2.3 < Mmain <M0/2 when∆ω → 0, this probability behaves like∆ω and itcannot be neglected. We can take this into account by rewritingequation (34) as

dMmain

dω= −

√2

π

Mmain√Sq − S

. (45)

with Sq = S(M0/q) and2 6 q . 2.3. In principle, any valueof q in the range above is allowed. In particular, merger trees con-structed using different algorithms may have different values ofqin the above range, as long as the algorithms adopt a sufficientlysmall time-step∆ω.

In Fig. 15 we showMmain for a halo of mass5 ×1012 h−1M⊙. The triangles show the results obtained from manyindependent EPS merger trees, constructed using the SK99 algo-rithm. Note that the averaging is done overMmain(z) at fixedz, which is the same as done in the analytical estimates (equa-tion [31]). The dashed, solid and dotted lines correspond tothe ana-lytical prediction of equation (45) forq = 2, q = 2.2 andq = 2.5,respectively. The curve forq = 2.2 is in excellent agreement withthe EPS merger trees. Note that this value forq is within the ex-pected range. In Fig. 3 we show that our analytical formula withq = 2.2 also accurately fits the merger-tree results for other valuesof M0.

0 0.2 0.4 0.6 0.8−2

−1.5

−1

−0.5

0

Log(1+z)

Log[

Mm

ain/M

0]

q = 2q = 2.2q = 2.5Merger−treesAverage over z

Figure 15. The mass of the main progenitor as computed in different waysfor M0 = 5× 1012 h−1M⊙. Big circles are from vdB02. Squares are theaverage of EPS merger trees, averaged overz at a fixedM . Triangles arefrom the same merger trees but averaged overM at a fixedz. Dashed linesshow the theoretical limits of the analytic formula (q = 2 andq = 2.5).The solid line is the analytic formula withq = 2.2.

A Universal Fitting Function

Equation (45) can be solved to obtain a direct, analytical formulafor Mmain(z). We use the fitting function forS(M) given invdB02:

S(M) = g2[

cΓ

Ω1/3m

M1/3

]· σ2

8

g2(32Γ), (46)

wherec = 3.804 × 10−4, andg(x) is an analytical function:

g(x) = 64.087[1 + 1.074x0.3 (47)

−1.581x0.4 + 0.954x0.5 − 0.185x0.6]−10

.

In terms of the new set of variables

M = MΓ3

Ωm, ω = ω

g(32Γ)

σ8, g(x) = g(c · x1/3) . (48)

equation (45) does not depend on cosmology, and can be easilysolved to give

M = F−1q [ω − ω0 + Fq(M0)] , (49)

where

Fq(M) = −√

π

2

∫ M

0

√g2(M ′/q)− g2(M ′)

M ′dM ′ . (50)

Finally we write the solution in the original variables:

Mmain(ω) =Ωm

Γ3F−1q

[g(32Γ)

σ8(ω − ω0) + Fq

(Γ3

ΩmM0

)].(51)

The analytical fitting function forFq(u) with q = 2.2 is

Fq(u) = −6.92 × 10−5 ln4 u+ 5.0× 10−3 ln3 u+ (52)

+8.64× 10−2 ln2 u− 12.66 ln u+ 110.8 ,

which is accurate to better than one percent over the range1.6 ×104 < MΓ3Ω−1

m < 1.6× 1013 h−1M⊙

c© 2006 RAS, MNRAS000, 1–16


An Alternative Method

We now present an alternative method to computeMmain(z),which is based on the method originally introduced by LC93.The LC93 argument is as follows: at a specific redshiftz, onecan compute the probability for having a progenitor with a masslarger thanM0/2. This probability equals the probability that atree will have itszmain greater thanz (wherezmain is defined by:Mmain(zmain) = M0/2). Although LC93 claim their formula isonly an approximation, we have not found any gap in their argu-ment, so we think this should be an accurate prediction.

Lets defineQ(ω1|M0, ω0) as the probability that a halo withmassM0 at timeω0 will have its merger-tree obeyω(zmain) =ω1. The probability for having a progenitor with mass bigger thanM0/2 then equals :∫ S2

S0

M0

Mf(S − S0, ω − ω0)dS =

∫∞

ω

Q(ω1|M0, ω0)dω1 , (53)

where f is defined in equation (5),S0 = S(M0), and S2 =S(M0/2). The distributionQ(ω|M0, ω0) is obtained by differenti-ating the above equation with respect toω and multiplying it by -1.In order to compute the mean formation time, we need to averageω over the probability distributionQ:

ωmain,1 =

∫∞

ω0

ωQ(ω|M0, ω0)dω = (54)

−∫

∞

ω0

ωdω∂

∂ω

∫ S2−S0

0

M0

M(S0 +∆S)f(∆S,∆ω)d∆S .

Here ωmain,1 is obtained by averaging over allω (time) possiblefor getting a massM0/2. This is different from the method usedabove, whereωmain was computed by averaging the main progeni-tor masses at a fixed time. The two methods should give slightlydifferent results, even if both are accurate. This is illustrated inFig. 15 where the triangles indicate the average, main progenitorhistory obtained by averaging overMmain at fixed z, while thesquares show the results obtained when averaging overz at fixedMmain.

So far we have repeated the analysis in LC93. Now, insteadof computing the derivative of equation (53), we simplify equa-tion (54) by a simple integration by parts:

ωmain,1 = −ω

[∫ S2−S0

0

M0

Mf(∆S,∆ω)d∆S

]∞

ω0

+ (55)

+

∫∞

ω0

dω

∫ S2−S0

0

M0

M(S0 +∆S)f(∆S,∆ω)d∆S .

The left part is justω04. We can switch the integrals on the right

hand side of the equation, and compute the integral overω first.Finally we have:

ωmain,1 − ω0 =M0√2π

∫ S2−S0

0

d∆S

M(S0 +∆S)√∆S

. (56)

For a self-similar power spectrum withS ∝ M−α the integralcan be done analytically:

ωmain,1 − ω0 =

√2

π(S2 − S0) 2F1

(1

2,− 1

α,3

2, 1− 2α

), (57)

4 One should take care when assigning the integral lower limit. The in-tegral over∆S should first be computed, similar to the analysis done forequation (37).

0 0.1 0.2 0.3 0.4 0.5

−0.4

−0.3

−0.2

−0.1

0

Log(1+z)

Log[

Mm

ain,

1/M0]

5×109

5×1013

Figure 16. Mmain,1(z) for several halo masses. Symbols are from themerger trees of vdB02. The halo masses range from5 × 109 to 5 ×1013h−1 M⊙, spaced by a decade. Smoothed lines are the results of theanalytic formula, Eq. 56, replacingS2 with S(Mmain,1). The equation isvalid only forMmain,1 > M0/2.

where2F1 is the Gauss Hypergeometric function. We can see that(ωmain,1 − ω0)/

√S2 − S0 has the same value for all masses, and

thus it is the natural variable to choose (as was done in LC93).On the other hand, we showed in equation (16) that(ωmain −ω0)/(

√S2 −

√S0) is a also constant, when the averaging of the

trees is made along the mass axis.The analysis above is still valid, if we replaceS2 with

S(Mmain,1), so that we can easily generalize this result to obtainMmain,1 in the rangeMmain,1 > M0/2. For masses belowM0/2,however, we cannot computeMmain,1 since this requires the proba-bility for getting a main progenitor with mass lower thanM0/2. Asdiscussed above, this part of the probability function is unknown.

In Fig. 16 we compare results from EPS merger tree re-alization (vdB02, plotted as symbols) to the analytical formula(smoothed lines). There are some deviation between the two,pre-sumably because the averaging is done over a large range in red-shift, where the SK99 algorithm may have slight inaccuracies. Thiseffect can be seen in Fig. 6 and in vdB02 (Fig. 4): the distributionof formation times is only accurate for low values ofωmain. Ourprevious method for computingωmain was not affected by this in-accuracy because it was derived using the limit of small time-stepsbehaviour. This, combined with the fact that this method canonlybe used to computeMmain(z) down toM0/2, and the fact thatequation (56) cannot be generalized easily sinceS0 is buried insidethe integrand, clearly favors the method discussed at the beginningof this appendix over the one discussed here.

c© 2006 RAS, MNRAS000, 1–16

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