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Mon.Not.R.Astron.Soc.000,1–15(2008) Printed3January2009 (MNLATEXstylefilev2.2) Merger history trees of dark matter haloes in moving barrier models Jorge Moreno1, Carlo Giocoli2 & Ravi K. Sheth1 ⋆ 1Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6396, USA 2Dipartimento di Astronomia, Universitadegli Studi di Padova, Vicolo dell’osservatorio 2 I-35122 Padova, Italy 9 0 ABSTRACT 0 We present an algorithm for generating merger histories of dark matter haloes. The 2 algorithm is based on the excursion set approach with moving barriers whose shape n is motivated by the ellipsoidal collapse model of halo formation. In contrast to most a othermerger-treealgorithms,ourstakesdiscretesteps inmass ratherthantime.This J allows us to quantify effects which arise from the fact that outputs from numerical 3 simulations are usually in discrete time bins. In addition, it suggests a natural set of scalingvariablesfordescribingtheabundanceofhaloprogenitors;thisscalingisnotas ] h generalasthatassociatedwithasphericalcollapse.Wetestouralgorithmbycompar- p ing its predictions with measurements in numerical simulations. The progenitor mass - fractions and mass functions are in good agreement, as is the predicted scaling law. o We also test the formation-redshift distribution, the mass distribution at formation, r t and the redshift distribution of the most recent major merger; all are in reasonable s agreement with N-body simulation data, over a broad range of masses and redshifts. a [ Finally, we study the effects of sampling in discrete time snapshots. In all cases, the improvementoveralgorithmsbasedonthesphericalcollapseassumptionissignificant. 3 v Key words: galaxies: halo - cosmology:theory - dark matter - methods: numerical 0 0 8 3 . 1 INTRODUCTION which haloes were assumed to form from a spherical col- 2 lapse, were developed in the 1990s (Kauffmann & White 1 Most models of galaxy formation in a hierarchical universe 1993; Somerville & Kolatt 1999; Sheth & Lemson 1999; 7 assumethatthemergerhistoryofthesurroundingdarkmat- Cole et al. 2000) (see Zhanget al. (2008b) for a review). 0 terhalo playsan important role in determining theproper- : However, spherical collapse overpredicts (underpredicts) v ties of a galaxy (e.g. White & Rees (1978); White & Frenk the abundance of haloes in the low (high) mass regime. i (1991);Baugh(2006)andreferencestherein).Althoughhalo X To address these issues, Sheth & Tormen (1999) extended mergerhistoriescanbemeasuredusingN-bodysimulations, the excursion set framework to include ellipsoidal collapse r thesecanbetimeconsumingandcomputationally intensive a (Sheth,Mo & Tormen 2001; Sheth & Tormen 2002). This (Springel et al. 2005).Thishasfueled considerablestudyof clearlyshowedthatmerger-treeswhichassumesphericalcol- the formation and merger histories of dark matter haloes lapse are inadequate. from a Monte Carlo perspective. Monte Carlo merger trees Hiotelis & Del Popolo(2006)describeamergertreeal- have the advantage of being fast and one may easily probe gorithmwhichextendssomeoftheolderalgorithmstoincor- mass regimes inaccessible to current N-body simulations. porateaspectsoftheellipsoidalcollapseresults.Inaddition, Moreover, unlike N-body experiments, the cosmology and a number of new algorithms have recently been published initial conditions may beeasily modified. (Parkinson et al.2008;Neistein & Dekel2008);althoughef- The excursion set framework (Bond et al. 1991; ficient and accurate, such methods side-step the idea of el- Lacey & Cole 1993), which is motivated by the pioneering lipsoidal collapse altogether. Moreover, these methods are workofPress & Schechter(1974),providesthebasisforcur- callibrated to match N-body simulations, and are therefore rent models of halo assembly. Initially, this framework was limited by theaccuracy and scope of these simulations. based on the assumption that haloes form from a spher- The aim of the present work is to provide a merger ical collapse, of the type first described by Gunn & Gott history tree algorithm which is based explicitly on the (1972). Fast algorithms for generating halo merger trees, in excursion-set formalism with ellipsoidal collapse. The most significant difference between the algorithm we derive and ⋆ Email: jmoreno,[email protected], all theothers described aboveis that it takes discrete steps [email protected] in mass rather than time. This feature allows us to study a c 2008RAS (cid:13) 2 J. Moreno et. al. few problems which are more difficult to address with the an ensemble of random walks which start from the origin: other methods. (S,δ)=(0,0). The excursion set approach maps the distri- After we completed this project, Zhanget al. (2008b) butionofS,thenumberofstepsarandomwalkmusttaketo presented an alternative algorithm with ellipsoidal collapse firstcrosssuchabarrier,tothefractionofmassinm-haloes (anddiscrete-timesnapshots).Thisalgorithmgeneratespro- at redshift z. This quantity is associated with the so-called genitors across many mass bins and then assigns them unconditionalmassfunction.Theconditionalmassfunction to final haloes. In this sense, it is very similar to the of high redshift progenitors of a more massive final M-halo Kauffmann & White(1993)mergertree,butsolvesmanyof atsomelowerredshiftZ ismodeledusingwalkswhichstart itsproblems.Althoughthistreeisquitedifferentfromours, from (SM,δc(Z)) instead. italsoattemptstodescribethemergerhistoryofahalofrom The shape of the mass functions (unconditional and the excursion-set formalism. These two approaches show conditional) is determined by the shape of the barrier, en- thatgeneratingmergertreeswithouttuningtoN-bodysim- coded in the (q,β,γ) parameters. The spherical collapse ulations is a quitenon-trivial, yet interesting, challenge. model is associated with (q,β,γ) = (1,0,0), whereas ellip- A review of background material and a description of soidal collapse has (0.707,0.47,0.615). When γ > 1/2, not ouralgorithmaregiveninSection2.Section3comparesour allwalksareguaranteedtocrosstheellipsoidalcollapsebar- algorithm with excursion-set theory predictions, and with rier (Sheth& Tormen 2002). Moreover, the barriers associ- measurementsin N-bodysimulations. Thetestsincludethe ated with two different times may intersect; of course, this progenitormassfractionsandmassfunctions,theformation- never happens for the spherical collapse barriers. Sheth & redshiftdistribution,themassdistributionatformation,and Tormensuggestthatthisintersectionofbarriersmayrepre- the redshift distribution of the most recent major merger. sent the possibility that haloes can fragment. This compli- Section4summarisesourfindingsanddiscussespossibleap- cates the algorithm we describe below, so we instead study plications of our algorithm, an outline of which is in Ap- thelimitingcaseof‘square-root’barriersforwhichγ =1/2: pendix A. B(S,δc)=√qδc+β√S. (3) The predicted halo abundances associated with (q,β,γ) = 2 MERGER TREES IN THE EXCURSION SET (0.55,0.5,0.5) are very similar to those in simulations APPROACH (Mahmood & Rajesh 2005;Moreno et. al. 2008).Moreover, the first crossing distribution of this barrier is known In the excursion set approach, the problem of estimating (Breiman 1967). halo abundances is mapped to one of estimating the distribution of the number of steps a Brownian-motion random walk must take before it first crosses a barrier of specified 2.2 A conditional scaling symmetry height (Bond et al. 1991). In this approach, the height of Recallthattheunconditionalmassfunctionisassociated to thebarrier plays a crucial role. the first crossing distribution associated with walks which startfromtheorigin:(S,δ)=(0,0).Whenconstantbarriers are use, it can beexpressed self-similarly as 2.1 Constant and moving barriers ThePress-Schechtermass function isassociated with barri- f(m|z)dm=f(S|δc)dS =f(ν)dν, (4) ersofconstantheight-suchbarriersarisenaturallyinmod- whereν δ2/S.Theconditionalmassfunctionofm-haloes ≡ c elsinwhichhaloes form from asphericalcollapse model.In atz thatendupinboundobjectsofmassM >matZ <z constrast, in the ellipsoidal collapse model, barriers of the is given by f(m,zM,Z)dm= f(Sm,δ1SM,δ0)dSm, where | | form whereδ1 =δc(z),δ0 =δc(Z).Inotherwords,theconditional S γ mass function is associated with the first crossing distribu- B(S,δc)=√qδc 1+β»qδ2– ff. (1) tion of a barrier of height δ1 by random walks with origin c at(SM,δ0).Because astraight-line isstraight whateverthe are more natural. Here δc is the overdensity required for origin of the coordinate system, the conditional mass func- sphericalcollapse-itisamonotonically increasingfunction tion,inthesphericalcollapsemodelhasthesamefunctional of redshift, and it is given by δc0/D(z), where δc0 δc(z = form as that of the unconditional mass function, provided ≡ 0) ∼ 1.686 and D(z) is the linear growth factor. S is a one sets ν =(δ1−δ0)2/(Sm−SM). monotonically decreasing function of halo mass, given by: However, for the square-root barrier, a walk which 1 starts from (√qδ0 + β√SM,SM) must cross a barrier of S(m)≡σ2(m)= 2π2 Z dkk2P(k)W˜2(kR), (2) shape √q(δ1 −δ0)+β√Sm−SM +SM. This is not quite of the same form as equation (3). As a result, the condi- where P(k) is the initial power spectrum of density fluc- tional mass function is not simply a rescaled version of the tuations, linearly evolved to the present time, W˜ is the unconditional one. Rather, in this model, Fourier transform of W(x) = (3/x3)(sinx xcosx), R = (3m/4πρ¯)1/3,andρ¯isthecomovingbackgro−unddensity.At f(m,zM,z0)dm=f(Sm/SM η)d(Sm/SM), (5) | | largeR,theoverdensitycontainedintheassociated volume where is practically zero. As R (and m(R)) decreases, S(R) increases,andδR executesarandomwalk.Wereferthereader η≡ δ√1−S δ0. (6) to Lacey & Cole (1993) for more details. M Consider a barrier B(S,δc(z)), as in equation (1), and (see Breiman (1967) or Giocoli et al. (2007) for the exact c 2008RAS,MNRAS000,1–15 (cid:13) Moving barrier merger trees 3 Figure 1. A random walk and its associated mass history. The Figure 2.The same random walkas inFigure1,but now with dark filled circles represent the history of a halo of mass M at square-root rather than constant barriers, illustrating that the redshift z = 0. A merger (m′,m m′) m at redshift z is massaccretionhistorydependsonthebarriershape.Inouralgo- dasespoiccitaetdedbywtihthe S(mm−→mS′m)′isjucmonpnae−tctheedigahtt→(δSc(mz−).mA′,nδce(wz)b).raTnhche (rSitmhm−m, t′h,e√nqδecw(zo)b+jeβctpwSimth−mm′a)s.s (m−m′) is now connected at light-filledcirclesdenote themasshistoryofthisobject. dom walk steps under such jumps are not part of the mass expressions). Thus, final halos of different masses will have history (e.g., the gray portion with Sm<S <Sm′). similarprogenitormassfunctionswhenexpressedintermsof Thekeytoourmergertreealgorithm,whichisdescribed S /S , provided they have similar values of η. While this in detail in Appendix A, is to recognize that these jumps m M scaling is like that for the constant barrier model, in the mean that there are a set of other walks which one might square-root barrier model, the progenitor mass function is associatewiththisone–oneforeachjump.Onesuchwalkis not a function of the combination ν2 = η2/(S /S 1). illustratedbythesecondjaggedcurve,whichstartsatabout m M − This is interesting, because Sheth & Tormen (2002) have themiddleofthepanel.IfthejumpfromSmtoSm′ occurred shown that the conditional mass function in simulations whenthebarrierheightwasδc(z),thenthisotherwalkstarts is not well-fit by a function of ν. In what follows, we will from (Sm−m′,δc(z)). The ‘merger history’ associated with present evidence that it is, however, a function of η and thisnewbranchisrepresentedbythelight-shadefilledcircles S /S separately, so the qualitatively different scaling as- in Figure 1.For everysuch jump,anew random walk must m M sociated with the square-root barrier is indeed seen in sim- be drawn. For each jump within each of those new walks, ulations. the same process applies – more walks must be drawn. In summary,thebundleofsuchwalksencodestheentiremerger historyofapresent-dayobject.Noticethatjumpscanoccur atanyz–thereisnoconstraintthattheyhappenatdiscrete 2.3 Mass histories and merger trees times. However, if one is interested in the mass function of Figure 1 illustrates how the mass growth history of an ob- progenitors at some fixed z, one simply reads-off the list of jectisencodedintheexcursionsetapproachifobjectsform valuesof S at which this bundleof walks first cross δc(z). from asphericalcollapse (alsoseeFigure1ofLacey & Cole So far, we have discussed how to generate trees in (1993)). The jagged line shows a random walk which starts the spherical collapse model. Figure 2 shows the same fromtheorigin:(S,δ)=(0,0).Imaginedrawingahorizontal walk as before, but now the mass growth history associ- linewith heightδc0 =1.686 andmarkingthesmallestvalue ated with the walk is given by its intersection with square- of S at which the walk intersects this ‘barrier’ of constant root barriers of gradually increasing height. This shows height (δc0 corresponds to the present time and δc > δc0 clearly that the jumps in mass, and the times at which corresponds to higher redshifts). The dotted horizontal line they occur, are modified. But the overall logic remains the denotes such barrier. Then increase the height of this bar- same. Each jump gives rise to a new walk that starts from rier, and record how this value of S changes as δ increases. (Sm−m′,B(Sm−m′,δc(z))),whereBisgivenbyequation(3). Such mass history points are depicted as dark filled circles The natural generalisation to spherical collapse is in Figure 1. The dashed lines show that S will occasionally to incorporate the original γ > 1/2 barrier of jump from a small valueto a larger one. Since S is a proxy Sheth,Mo & Tormen (2001). Such a choice would compli- for mass, and δc for time, such a jump is a proxy for an cate this algorithm significantly. First of all, as Figure 2 instantaneouschangein mass: amerger.Notethattheran- illustrates, theshapeof thesquare-root barrier remains the c 2008RAS,MNRAS000,1–15 (cid:13) 4 J. Moreno et. al. Figure3.Theprogenitormassfraction(left)andmassfunction(right)atredshiftsz=(0.5,1,2,3,5),forhaloesofmassM/M⋆=0.06 atz=0.FilledcirclesshowmeasurementsintheGIF2simulation,andopencirclesarefromoursquare-roottrees.Thesmoothsolidand dashedcurvesshowtheexactsquare-rootbarriersolution,andtheseriesapproximation,respectively. Thelong-dashedcurveshowsthe ellipsoidalcollapsemodelwithγ>1/2,andtheshort-dashedcurveistheconstantbarrierprediction.Valuesofthescalingparameterη (equation 6)arealsoshown(seeFigure6). same with different redshifts, except for an overall vertical 3 COMPARISON WITH SIMULATIONS shift. This is not the case with γ > 1/2. As δc increases In this section, we compare the statistical properties of (increasing redshift), the term δc1/2−2γ makes the barrier in our merger history trees with expectations from the ex- equation (1) increase less rapidly with S. A consequence of cursion set theory which they are supposed to reproduce, this is that the barriers associated with different redshifts and with measurements in the GIF21 N-body simulation cross.Intheabsenceofcrossing-barriers (e.g.,constantand (Gao et al. 2004b). The simulation followed the evolution square-root barriers), one may uniquely map any point in of 4003 particles in a periodic cubic box 110h−1Mpc on a the (S,δ) plane to (m,z). The crossing of barriers invali- sideinaflatΛCDMbackgroundcosmologywithparameters datesthisproperty,makingtheidentificationofjumpswith (Ωm,σ8,h,Ωbh2,n) = (0.3,0.9,0.7,0.0196,1). Fifty simula- mergers at a given time ill-defined. tion snapshots were output, equally spaced in log(1+z). 1 GermanIsraelFund. c 2008RAS,MNRAS000,1–15 (cid:13) Moving barrier merger trees 5 Figure 4.SameasFigure3,butwithM/M⋆=0.6. Ateach snapshot,haloes were identifiedusingthespherical minimum mass considered was mdust = M/1000, and the overdensitycriterion,adoptingforvirialmassthedefinition mergerhistoriesofhaloeswithmassbelowthiswerenotfol- of Eke et al. (1996) (i.e., with virial density at 324ρ¯ at lowed (we call thisminimum mass the‘branching-mass res- redshift zero). The particle mass is mp =1.73 1∼09h−1M⊙ olution’). Weused random walks with 105 stepsin between × and only objects with at least ten particles are considered. SM and Sdust to ensure that the mass change between the M⋆(z), defined by δc2(z) = S(M⋆(z)), is the typical mass steps was less that mdust. Having a small step size is es- scale at redshift z. It is common practice to express halo sentialtofaithfully reproducerandom walksin acomputer. masses in terms of M = M (z = 0) (the z-dependence Moreover, if the step size is too large, we run the risk of ⋆ ⋆ is suppressed for the present time). For this cosmology missing branches. Numerical tests showed that the outputs and initial power spectrum, M⋆ = 8.7 1012h−1M⊙ converged to our results for small step sizes. See the Ap- × ≃ 5030mp. The simulation data and halo catalogues are pendixformoredetailsontheimplementationofourMonte available at http://www.mpa-garching.mpg.de/Virgo. See Carlo tree. Recall that ourtreedoes not takediscrete steps Giocoli et al. (2008a) for more details regarding the post- intime.Nevertheless,forfaircomparisonwiththemeasure- processing of the simulation. mentsfromtheGIF2simulation,thetreedatawerestoredin thesamediscreteredshiftbinsaswereoutputfromthesim- TocompareourmergerhistorieswiththoseintheGIF2 ulation. We use ‘Cont’ to denote the original tree data and simulation, we generated 2000 realisations of our tree for ‘Snap’forthedatastoredinredshiftsnapshots.Sections3.3 each final halo mass bin M of interest. In all cases, the c 2008RAS,MNRAS000,1–15 (cid:13) 6 J. Moreno et. al. Figure 5.SameasFigures3,butwithM/M⋆=6. and3.4studysomemerger-relatedquantitieswhicharesen- dashed curve shows the constant barrier (β,q) = (0,1) sitive to the differences between these two ways of storing prediction associated with spherical collapse (Lacey & Cole trees (Figures 8, 9 and 10). 1993). The solid curves show the exact square-root barrier solution with (β,q,γ) = (0.5,0.55,0.5); this is a com- plicated affair, involving sums of Parabolic Cylinder func- 3.1 The progenitor mass function tions (Breiman 1967). Dashed curves show the consider- ably simpler approximation to the solution which is due to Figures 3-5 show the progenitor mass fractions and mass Sheth& Tormen(2002);thisapproximationisexcellentover functions at five different redshifts (z = 0.5,1,2,3,5), for the entire range of interest. The long-dashed curves show haloes identified at z = 0 with final masses given by thissame approximation for theellipsoidal collapse barrier: M/M = 0.06,0.6 and 6. The corresponding values of η ⋆ (β,q,γ) = (0.707,0.47,0.615). The square-root barrier pre- (equation 6) are shown in each panel. In all three figures, diction agrees well with the γ = 0.615 curve, except in the filled circles show measurements in the GIF2 simulation, high-mass regime. This discrepancy becomes evident when and open circles show results from our square-root trees. η>1 and it is amplified with increasing η (see below). We probe the m<mp regime with our trees to verify con- sistencywithanalyticexcursion-setpredictions(thesmooth Before weaskhowourmergertreealgorithm compares curves in all the panels). The expressions we use are given with simulations, we note that it produces progenitor mass explicitlyinAppendixAofGiocoli et al.(2007).Theshort- functions that are well-described by the theory curves over c 2008RAS,MNRAS000,1–15 (cid:13) Moving barrier merger trees 7 Figure 6.Theη-symmetry.Differentcombinations ofM andz withsimilarη (equation 6andTable3.1).N-bodysimulationmeasure- mentsandtheSheth&Tormen(2002)resultwithγ>1/2areshown. a wide range of masses and redshifts. At high redshifts, M/M⋆ z η — M/M⋆ z η our tree data lie slightly below the theory curves at both highandlowm/M,andslightlyaboveinbetween,although 0.06 1 0.3 — 6 0.5 0.31 wherethecross-overpointsoccurdependsonzandM.Inall 0.06 2 0.65 — 6 1 0.66 otherregimes,ourMonteCarlotreesmatchthesquare-root barrier predictions. Any additional disagreement with the 0.6 3 1.45 — 6 2 1.44 GIF 2 simulation measurements (compare open and filled circles) is dueto limitations of the γ =1/2 model. Table1.Theη-symmetry(equation6)usedforthecomparisons showninFigure6. Finally, recall that the square-root and constant bar- riermodelsmakespecificpredictionsforhowtheconditional mass functions should scale with final halo mass and time. Table3.1 lists pairs with similar η,yetquitedistinct values the right-hand side (high final masses). Notice that the of M and z. Figure 6 compares the associated conditional curves are remarkably similar to one another, as are the massfunctions.Theblacksquaresandlong-dashedlinesre- symbols. This is true despite the fact that the values of fer to the left-hand side of Table 3.1 (low final masses), η are not perfectly identical, and that f(m,zM,Z)dm = | whereas the gray triangles and short-dashed lines refer to f(S /S η)d(S /S ) f(m/M η)d(m/M). The results m M m M | ≃ | c 2008RAS,MNRAS000,1–15 (cid:13) 8 J. Moreno et. al. Figure 7. Distribution of formation redshifts. Filled circles show simulation data, open circles and triangles show results from the square-root and constant barrier trees. Smooth curves show equation (7) with q = 1, 0.707 and 0.55 (short-dashed, long-dashed, and dashed), correspondingtothepredicteddistributionforconstant barriers(sphericalcollapse), moving(ellipsoidalcollapse) andsquare- rootbarriers. for low-mass haloes (black squares) are truncated at higher barriersprovidesasimpleexpressionforthisdistributionof m/M than they are for larger M (gray triangles), simply formation times: because only haloes with at least ten particles are consid- p(ω )dω =2ω erfc(ω /√2)dω , (7) ered. Evidently, the conditional mass functions are indeed F F F F F functions of η and S /S separately, rather than of the where m M combination ν. ωF δc(zF)−δc(z0) = η , (8) ≡ S(M/2) S(M) S(M/2)/S(M) 1 − − p p with η given byequation (6). 3.2 The distribution of formation redshifts ThefilledcirclesinFigure7showtheformationredshift Following Lacey & Cole (1993), a halo is said to have distributions for haloes with masses in the range 0.9M 6 ‘formed’ when it first acquires half of its final mass. For M 6 1.1M with log10(M/M∗) = {1,...,−1.5} in steps of a given halo mass, there is a distribution of formation red- 0.5intheGIF2simulation.Weusethefirstsnapshotwhen − shifts. Thisdistribution isexpectedtopeakat earlier times atleast halfthemassisinasingleprogenitorastheforma- forlowermasshaloes.Theexcursionsetmodelwithconstant tion time, and make no attempt to interpolate our simu- c 2008RAS,MNRAS000,1–15 (cid:13) Moving barrier merger trees 9 Figure 8. Distribution of the mass at formation for several final masses. The left half of each panel shows the mass just prior to formation, whereas the right half shows the mass just after formation. Filled circles show simulation data, open circles and triangles arefromthesquare-root andconstant barriertrees.The solidcurveshows µq(µ) (righthalf)andµp(µ) (left half)(equations 10 and9 respectively). Thedashedcurveshowsthesesameexpressionswithµ 1/4µ. → lation formation redshifts between these discretely spaced tion (7) only holds for white-noise initial conditions. Nev- output times (Harkeret al. 2006; Giocoli et al. 2007). The ertheless, as pointed out Lacey & Cole (1993), it remains a opencirclesandtrianglesshowthecorrespondingformation reasonable approximation to CDM case. Furthermore, note timedistributionsfromoursquare-rootandconstantbarrier that it provides an excellent description of the formation trees. Recall that we do not discretise redshift in our tree, times generated by our trees. However, no choice of q pro- so the question of interpolation does not arise. videsparticularlygoodagreementwiththeGIF2simulation, a discrepancy noted by previous authors (Lin et al. 2003; For the ellipsoidal collapse model, Giocoli et al. (2007) Hiotelis & Del Popolo 2006; Giocoli et al. 2007). This is showed that the formation redshift is well-described by likelyaconsequenceoftheexcursionsetassumptionthatdif- equation (7) if one replaces ωF → √qωF. The smooth ferent steps in the walk are uncorrelated (Sheth & Tormen curves show this with q = 1, 0.707 and 0.55 (short-dashed, 2002). See Pan et al. (2008) and references therein for how long-dashed, and dashed), which represent the (constant, one might improve on this. γ = 0.615, and square-root) barrier predictions. For higher values of q, the peaks are located at lower redshifts and the widths of the curves decrease. Strictly speaking, equa- c 2008RAS,MNRAS000,1–15 (cid:13) 10 J. Moreno et. al. Figure 9. Same as Figure 8, but showing only the region around m/M = 1/2. The peak in the simulations (filled symbols) is less pronouncedthaninthemergertrees(jaggedlines).Opencirclesshowtheresultofsamplingthemergertreesatthesameredshiftsasthe simulationsnapshots: thismakes adramaticdifferencearound0.496µ60.51,suggestingthat thesharpcusppredictedbythetheory willalsobepresentinsimulationswithsufficientlycloselyspacedoutputs. Thesmallerdiscrepanciesfurtherfromthe peakremain. 3.3 The mass distribution at formation andµ m/M,andthedistributionjustbeforeformation is ≡ The previous subsection studied halo formation, where for- 1 µ dµ q(µ)dµ= 1 2µ , (10) mthaetipornogweansitdoersfineexdceaesdsthheafilfrstthetimtoetatlh.aTthtehreefmoraes,stohfisomneasosf π(1−µ)“r1−2µ −p − ” µ2 can haveanyvaluebetween1/2 and1timesthefinalmass, where 1/4 6 µ 6 1/2. We have found that, to a very good and one can study the distribution of masses at, and just approximation, µp(µ) µq(µ) if one replaces µ 1/4µ → → priorto,formation.Theexcursionsetconstantbarriermodel (solid and dashed curvesin Figure 8, respectively). Let m B makes a prediction for this distribution (Nusser& Sheth and m denote the masses before and after formation, re- A 1999). The mass distribution at formation is expected to spectively.Roughly speaking, thesymmetry about M/2 in- be dicates that having a specific ratio of m to M/2 before B formation is equaly likely to having the same ratio of M/2 to m after formation. A 2 1 µ dµ Although these expressions were derived assuming a p(µ)dµ= − , where1/26µ61, (9) πr2µ 1 µ2 white-noise power spectrum, they are expected to be rela- − c 2008RAS,MNRAS000,1–15 (cid:13)