The Birth and Growth of Neutralino Haloes
aa r X i v : . [ a s t r o - ph . C O ] J un Mon. Not. R. Astron. Soc. , 1–8 (2009) Printed 25 October 2018 (MN L A TEX style file v2.2)
The Birth and Growth of Neutralino Haloes
R. E. Angulo ∗ & S. D. M. White Max Planck Intitute fur Astrophysik, D-85741 Garching, Germany.
25 October 2018
ABSTRACT
We use the Extended-Press-Schechter (EPS) formalism to study halo assembly histo-ries in a standard ΛCDM cosmology. A large ensemble of Monte Carlo random walksprovides the entire halo membership histories of a representative set of dark matterparticles, which we assume to be neutralinos. The first generation halos of most par-ticles do not have a mass similar to the free-streaming cut-off M f.s. of the neutralinopower spectrum, nor do they form at high redshift. Median values are M = 10 to10 M f.s. and z = 13 to 8 depending on the form of the collapse barrier assumedin the EPS model. For almost a third of all particles the first generation halo has M > M f.s. . At redshifts beyond 20, most neutralinos are not yet part of any halobut are still diffuse. These numbers apply with little modification to the neutralinoswhich are today part of halos similar to that of the Milky Way. Up to 10% of theparticles in such halos were never part of a smaller object; the typical particle hasundergone ∼ Key words: cosmology:theory - large-scale structure of Universe.
The existence of “Cold Dark Matter” (CDM) is one of thepillars of our current understanding of the origin and evolu-tion of structure in our Universe. Many independent astro-physical observations have not only provided evidence of itsexistence but also indications of its properties; the CDM par-ticle should be non-baryonic, collisionless, neutral and withsmall initial velocities. In spite of the arguments supportingsuch a particle, no member of the standard model of particlephysics can fit the requirements. Fortunately, this problem is“solved” in supersymetric extensions where many candidateparticles for CDM emerge. In particular, the lightest neu-tralino, which should have a mass of about 100 GeV, is thecurrently favoured choice, as it is both weakly interactingand stable (see Bertone et al. 2005, for a review of currentcandidates).Some properties of the neutralino, such as its mass, havea direct impact on the formation and evolution of structurein the Universe. The finite temperature at which these par-ticles decouple from the radiation field before recombinationimprints features in the primordial power spectrum of fluc-tuations at very small scales. In particular, the free stream- ∗ [email protected] ing of neutralinos suppresses perturbations below ∼ . M f.s. ∼ − M ⊙ . Naturally,this will significantly affect the properties of the first objectsin the Universe, which must have masses comparable to, orlarger than, M f.s. . The free streaming of neutralinos mayalso modify the way in which much larger haloes grow. Forexample, the amount of mass that a halo accretes in a diffuseform depends on how much mass in the Universe is in col-lapsed objects. The objective of this paper is to study theseand other aspects of structure formation in a Λ-neutralinocosmology.The most accurate way to study the highly nonlineardynamics involved in the formation and evolution of haloesis via N-body simulations. However, resolving galactic haloesand objects with M ∼ M f.s. simultaneously, poses an ex-tremely hard problem that is currently impossible to solve.For example, a direct N-body simulation of the Milky Way’shalo would require at least 10 particles, almost 14 ordersof magnitude larger than the most sophisticated simulationsperformed so far. Extrapolating Moore’s law, such calcula-tion may become possible after the year 2050.So far, several different approaches have been used inthe literature to study the implications of neutralinos forearly structure formation. One consists in simulating scale- c (cid:13) Angulo et al. free cosmologies. There, the initial power spectrum is as-sumed to be a power law, with index similar to that ofCDM on very small scales. Although the range of scales is nolarger than in a standard CDM simulation, it superficiallyappears possible to study the formation of very small haloes,with masses similar to the free streaming cut-off mass (e.gWidrow et al. 2009). The weakness of such simulations isthat the results are not properly coupled to the evolutionof longer wavelength perturbation; as we shall see below,these are actually very important in a realistic CDM cos-mology. Another tactic is to resimulate extremely low den-sity regions at very high resolution in such way that theLagrangian region of the simulation is confined to a smallzone and the mass of each simulation particle can be verysmall (e.g Diemand et al. 2005). This approach is intrinsi-cally limited to unrepresentative regions of the very high-redshift Universe. Yet another tactic is to carry out a setof nested zoomed simulations (Gao et al. 2005b) althoughthis technique has yet to be extended all the way to thefree-streaming mass.In this paper we adopt a less accurate but self-consistentstrategy based on excursion set theory (Press & Schechter1974; Bond et al. 1991; Bower 1991; Lacey & Cole 1993).This formalism has been extremely successful in reproduc-ing many aspects of dark matter halo formation, as well asin fitting halo mass distributions and clustering. It providesa realistic model for halo growth over its entire history. Al-though the theory has not been tested on very small scales,it is currently the only way to compute the full mass as-sembly history of present-day haloes, i.e. starting from thefree-streaming mass, the very bottom of the CDM hierar-chy, and following growth up to cluster scales, the largestcollapsed objects in the Universe.This paper is organized as follows. First, we illustratehow the free-streaming of neutralinos modifies the the pri-mordial density field. In § §
4, where we investigate the typical red-shift at which dark matter particles become part of a halofor the first time, as well as the mass of those haloes. Wethen move to § § M = 0 . Λ = 0 .
75, normalizationof density fluctuations, expressed in terms of the extrap-olated linear amplitude of density fluctuations in spheresof radius 8 h − Mpc at the present-day, σ = 0 .
9, primor-dial spectral index n s = 1 and Hubble constant, H =73 km s − Mpc − . We will also, in general, adopt a standardneutralino with mass 100 GeV. In the classical ΛCDM model, the density fluctuations in-crease monotonically with wavenumber. However, this be-haviour is modified by neutralino streaming. The mass ofthe neutralino sets the temperature at which the populationbecomes nonrelativistic. This moment sets the primordialvelocity dispersion of neutralinos and characterizes the scalebelow which perturbations are suppressed. Roughly speak-
Figure 1.
Top :
The dimensionless power spectrum of the darkmatter density field at z = 0 predicted by linear perturbation the-ory. Bottom :
The logarithmic derivative of the power spectrum,i.e. the local power law index of the power spectrum. The solidline shows the power spectrum assuming that the dark matterparticles are neutralinos of mass 100 GeV whilst the dashed lineseffectively assume an infinite mass for the dark matter particle.
Figure 2.
The logarithm of the z = 0 mass variance S ( M )smoothed with a real space top-hat filter as a function of thesmoothing radius (mass) normalised to σ = 0 .
9. The solid lineslines take into account the cut-off in the power spectrum for 100GeV neutralinos. c (cid:13) , 1–8 he Birth and Growth of Neutralino Haloes ing, any perturbation smaller than the mean free path ofneutralinos over a Hubble time will be dissipated, produc-ing a cut-off in the primordial power spectrum.We can see this effect quantitatively in Fig.1. This plotshows the neutralino-CDM power spectrum predicted bylinear perturbation theory. The corresponding variance ofthe linear density field, extrapolated to z = 0, in spheresof different radius (and hence enclosed mass) is shown inFig.2. We have computed fluctuations on large scales usingthe Boltzmann code CAMB (Lewis et al. 2000) whilst on verysmall scales, the power spectrum is that predicted by theapproach of Green et al. (2004) which includes effects fromthe free streaming of neutralinos .In both figures, solid lines assume a neutralino of mass100 GeV and a temperature at kinetic decoupling of 33 MeV.As previously discussed, free streaming suppresses the poweron very small scales ( R f.s. ∼ . h − pc) generating an ex-ponential cut-off. The counterpart of the lack of power onsmall scales is a flattening of the variance for very smallsmoothing radii. Thus there is a finite maximum varianceof the field which, for our cosmological model, is equal to ∼ − ∼ − M ⊙ (1% of the mass of the Earth) and themost abundant haloes are predicted by excursion set theoryto have masses typically of order 10 − M ⊙ .The existence of a minimum halo mass in the Universe,does not, however, imply that most of the dark matter wasonce part of such an object. Indeed, as we will show in sub-sequent sections, only a small fraction of neutralinos wereever part of such haloes. Before we investigate this and re-lated issues in more detail, the following section gives a briefoverview of the techniques that will make our analysis pos-sible. We start this section by reviewing excursion set theory andthe various collapse models that are used as part of it. Wealso discuss our practical implementation of the formalism.Finally, we provide a description of the N-body simulationwith which we will test some of our results.
Dark matter haloes are highly nonlinear objects. In spiteof their complexity, many of their statistical properties canbe described surprisingly well by the simple analytical ar-guments of excursion set theory. In the following we willgive a brief summary of the main ideas behind the model, acomprehensive review is given by Zentner (2007).The classic implementation of excursion set theory(Bond et al. 1991; Bower 1991; Lacey & Cole 1993) starts by considering a random particle and then smoothing the den-sity contrast field around it on progressively smaller scalesuntil its smoothed overdensity crosses a threshold for col-lapse. The particle then is assumed to belong to a halo ofLagrangian mass equal to that enclosed in the smoothingwindow at threshold.In practice, the smoothed density contrast is generatedby convolving the density contrast field with a smoothingwindow, W , δ R ( ~x ) = Z δ ( ~x ′ ) W ( ~x − ~x ′ ; R )d x ′ . (1)On the same scale the total variance of the field, S ( R ),is: S ( R ) ≡ h δ R ( x ) i = 12 π Z d kk P ( k ) f W ( k ; R ) , (2)where P ( k ) is the dark matter power spectrum discussedin the previous section. An extremely interesting filter is atop-hat in Fourier space, where f W ( k ; R ) = 1 for all pointswith k R − and f W ( k ; R ) = 0 otherwise. In this casethe smoothed density contrast executes a Markov randomwalk in the S − δ plane as R decreases, since increasingthe window adds wavemodes that are independent of thosepreviously included.Once we have computed the trajectory δ ( S ) for eachparticle, the next step is to associate the particle to a col-lapsed object. The simplest criterion for collapse is that adark matter particle is considered part of a halo with mass M , such that the density contrast smoothed on the scale R ( M ) first exceeds some critical value. The relation between R and M is given by the volume enclosed in the smoothingwindow.The critical overdensity for collapse at a given redshiftmay be taken as constant or as function of the variance;these are known as constant and moving barriers respec-tively. The first is motivated by the spherical collapse model,which predicts that at the moment of virialisation the lin-early interpolated density contrast is δ sc ∼ . δ c ( S, z ) = √ qδ sc (cid:20) β (cid:18) Sqδ sc (cid:19) γ (cid:21) , (3)where, by using the expectation value for the shape of darkmatter haloes as a function of scale, a single condition (de-pending only on the variance of the field) can be applied toall trajectories at a given redshift.By analysing the criteria for collapse for an ensembleof walks as a function of redshift, the excursion set formal-ism predicts the probability distribution of halo assemblyhistories for an ensemble of DM particles.Despite the simplicity of the argument, the excursionset approach has been extremely successful in reproducingmany properties of DM haloes as determined from N-bodysimulations. In particular the clustering and number den-sity of haloes are well reproduced, as are the mass function c (cid:13) , 1–8 Angulo et al. of their progenitors. Nevertheless, the approach fails to re-produce all results from simulations, for example, the de-pendence of clustering on halo properties other than mass(Gao et al. 2005a; Wechsler et al. 2006; Gao & White 2007;Wetzel et al. 2007; Angulo et al. 2008a). This particular dis-agreement arises because the Markov nature of the over-density trajectories implies that there can be no correlationbetween the assembly history of a halo and its large-scaleenvironment (White 1996). Such a dependence is, however,found in simulations.
We have implemented the ideas described in the previoussubsection as follows. First we compute the smoothing radiifor which we will generate each random walk. These havea variable spacing in mass given by the condition M i =0 . M i − . This choice ensures that we can easily resolve allthe events where the halo associated with a random walkincreases its mass by a factor of two, allowing us to resolveall infall events and major mergers. The probability that ata given radius R i the smoothed field has a value between δ and δ + d δ is then given by exp[( δ − δ i − ) /S ( R i )]d δ .This approach, combined with the maximum finite vari-ance of the field (c.f. §
2) implies that only ∼ M ∗ halotoday.At each step of the random walks we check whethera collapse criterion is fulfilled or not. We considered boththe constant spherical collapse barrier and a square rootbarrier for which ( q, β, γ ) = (0 . , . , .
5) in Eq. 3. Notethat the latter set of parameters ensures that barriers atdifferent redshift will never intersect each other, as can hap-pen, for instance, for the ellipsoidal barrier that best repro-duces mass functions in N-body simulations (Sheth et al.2001; Mahmood & Rajesh 2005; Moreno et al. 2008).We have used two different starting points for the den-sity and variance. First we set (
S, δ ) i =0 = (0 ,
0) so the tra-jectories represent a random set of CDM particles. Thesewalks allow us to investigate general properties of dark mat-ter haloes which will be presented in §
5. The second startingpoint is (
S, δ ) i =0 = (4 . , . S, δ ) i =0 = (4 . , . § times to generate a largeensemble of random walks. This allow us to represent theprobability distribution of halo mass assembly histories forneutralinos in a ΛCDM Universe. The smallest haloes that can be resolved in current N-bodysimulations are many orders of magnitude more massivethan the smallest haloes expected in a neutralino-CDM Uni-verse. Nevertheless, by imposing on our random walks an artificial minimum halo mass that matches the resolutionof an N-body simulation, we can quantitatively assess theperformance of the excursion set approach.The N-body simulation we have chosen to compare ourresults with is the hMS. This calculation used 900 particles,each mass 1 . × M ⊙ , to solve the gravitational dynamicsof the dark matter distribution in a periodic box of side137 Mpc. The set of cosmological parameters is exactly thesame as those used in the Millennium Simulation (Springel2005).Every ∼
100 Gyr we have identified haloes using aFoF algorithm (Davis et al. 1985) as well as substructureswithin them using the
SUBFIND procedure described inSpringel et al. (2001). Furthermore, we have found the pro-genitor and descendants of every halo by following its 10%most bound particles across different snapshots, as describedin Angulo et al. (2008b). The combination of high-resolutionin mass and a considerable volume allows us to sample theevolution of dark matter particles over a wide range of halomass.
In this section we present statistics for the first haloes oc-cupied by typical DM particles based on our ensemble ofrandom walks. In particular, we will show the total massfraction in haloes of any mass, and the distribution of massand redshift at which typical neutralinos become part of ahalo for the first time.
Fig.3 shows as a function of redshift, the fraction of randomwalks that have ever crossed the threshold for collapse, aswell as the fraction that cross for a value of the smoothingmass greater than 2 . × M ⊙ . These curves correspond tothe fraction of all cosmic matter in halos of any mass, andin haloes above 2 . × M ⊙ .A common misconception is that in CDM models everyDM particle belongs to a halo of some mass. Fig.3 clearlycontradicts this. The fluctuation cut-off induced by neu-tralino streaming implies that some trajectories never crossthe critical threshold for collapse. As we follow trajectoriesdown to smaller smoothing masses, the fraction that havecrossed the threshold asymptotes to values smaller thanunity. The asymptotic value depends on the shape of thebarrier, the fraction of all matter predicted to be in diffuseform at z = 0, i.e. to be part of no clump, is 5% and 22%for constant and square root barriers, respectively.This effect is stronger at at high redshift where a muchlower fraction of DM is part of haloes. Less than half themass is in any halo beyond redshifts 7 . .
8, for theellipsoidal and spherical models, respectively. By z ∼
34 thefraction of diffuse mass has reached 90% in both models.Finally, beyond z ∼
60 less than 1% of the mass is in gravi-tationally bound structures.Black circles in Fig.3 show the fraction of DM parti-cles associated to FoF haloes (identified with at least 20particles) in the hMS simulation. This can be compared di-rectly with the red lines which indicate the fraction of ran-dom walks in haloes above the corresponding mass limit c (cid:13) , 1–8 he Birth and Growth of Neutralino Haloes Figure 3.
The fraction of mass in collapsed objects of any mass(black curves) and above a mass of 2 . × M ⊙ (red curves)as a function of redshift. The results displayed as solid lines arecomputed using a constant barrier while the dashed lines are com-puted using a square root barrier. Measurements from an N-bodysimulation are also displayed as black circles. Note that the blacklines assume no cut other than that imposed by the nature of theneutralino. M > . × M ⊙ . For the square root barrier 49 .
4% of therandom walks are associated with such haloes at z = 0, thisis very close to the fraction of the particles found in FoFhaloes (54 . In Fig.4 we present the distribution of the redshift at whichthe threshold is first crossed for the random walks that endup in some halo at z = 0. This corresponds to the differen-tial probability that a halo particle first becomes part of acollapsed object at redshift z .If we assume the ellipsoidal model for collapse (thedashed lines), the median value of the distribution is z =10 .
65, and 90% of the crossings occur at redshifts lower than35 .
2. For the spherical collapse (solid lines) these redshiftsare modified to z = 12 . z = 33 . z = 13. Less than 10% of themass in today’s haloes was already in a halo of any mass byredshift 36.Fig.5 shows the distribution of the smoothing massesat which these first barrier crossings occur. This plot thusdisplays the probability per logarithmic mass interval thata neutralino that is part of a halo today was first accretedin diffuse form onto a halo of mass M . This is equivalent to Figure 4.
The distribution of redshifts at which DM particlesare accreted onto a halo of any mass (black curves) or onto haloesmore massive than 2 . × M ⊙ . the mass distribution of the first haloes of randomly chosenDM particles from present-day haloes.Because of the shallow slope of the variance at smallradii (Fig. 2), the distribution of masses shown in Fig.5 isextremely flat; it varies by a factor of ∼ ∼
20 ordersof magnitude in mass. As a result, the first halo mass for DMparticles is very diverse. Median values of the distributionare ∼ − M ⊙ and 1 . M ⊙ for the constant and movingbarrier respectively. Most neutralinos were never part of ahalo of mass M < M f.s. . In fact, only 10% were ever partof an object smaller than Earth mass, the same mass fractionfor which the first halo was more massive than ∼ M ⊙ .Further results from our random walks are that the me-dian redshift of collapse of “first” objects of mass smallerthan 10 − M ⊙ is 24 .
1, only slightly larger than that of ob-jects in the range 10 − M ⊙ to 10 M ⊙ which is z = 15 . M ∼ M f.s. areefficiently disrupted by tidal forces as they merge into moremassive haloes.In both plots the measurements from the hMS simula-tion are compared with predictions from the excursion settheory, where we impose an artificial mass cut to match theresolution of the N-body simulation (red lines). The simu-lated redshift and mass distributions agree well with thosepredicted by the excursion set formalism. This motivates ourextrapolation of the predictions down to scales where thetheory has not yet been tested. Nevertheless, it is importantto keep in mind that this is an aggressive extrapolation. c (cid:13) , 1–8 Angulo et al.
Figure 5.
The mass distribution at first threshold crossing. Thesolid and dashed lines indicate the results for constant and mov-ing collapse barriers, respectively. Note that the statistics onlyincluded the trajectories that do eventually fulfill the criterionfor collapse, hence, the area below each curve is 1.
The particle-based nature of the excursion set formalismmakes it ideal to study the assembly history of dark matterhaloes. As we travel along a random walk we follow a DMparticle from the very bottom of the CDM hierarchy throughthe successive mergers it witnesses until it ends up, for ex-ample, in a Milky-Way mass halo today. For the statisticsin this section we have modified our ensemble of trajectoriesso that all of them end up in a halo of similar mass to theone that hosts our own galaxy ( M = 1 . × M ⊙ ). Fig.6 presents the distribution of the number of infall eventsfor all the trajectories in our ensemble of random walks.These are defined as an increase in mass by more than afactor of two, and so are the events where the halo thathosts a given particle is accreted onto a larger system.Black lines in Fig.6 follow DM particles over the fullhierarchy, while red lines follow them only from the momentthey are hosted by a halo more massive than 2 . × M ⊙ .The latter results can be compared directly to our N-bodysimulation. As in previous plots, results for both the spher-ical collapse barrier (solid lines) and the square root barrier(dashed lines) are shown in the figure.If most mergers were to occur between objects of similarmass, the typical number of infall events would be very large.The difference in mass between the smallest structures anda Milky-Way halo is ≃
20 orders of magnitude, so one mightexpect to see typically 20 log ∼
67 infall events for eachparticle. Fig.6 indicates that this is a poor representation ofstructure assembly in a CDM Universe. The median of the
Figure 6.
A histogram of the number of infall events for a ran-dom sample of all the dark matter particles that constitute a1 . × M ⊙ halo today. Sets of curves indicate the number ofinfalls since the DM particles were accreted onto any halo (blacklines) or onto a 2 . × M ⊙ halo (red lines). Solid and dashedlines assume a constant and moving barrier, respectively. histogram is between 4 and 5 events. Thus, a typical masselement in a Milky-Way halo has gone through a rather smallnumber of events where its host halo falls into somethingbigger than itself. Among all these infall episodes, typicallyonly one occurs while the particle is part of a halo largerthan 2 . × M ⊙ . This result suggests that minor mergersplay an important role in the growth of dark matter haloes.Another common misconception is that the growth ofdark matter haloes is due entirely to the accretion of otherhaloes. From our random walks we can assess this statementby computing the amount of mass that is formally accretedsmoothly, i.e. not via mergers of any type. This quantitycan be read off from Fig.6 as the fraction of mass that hassuffered zero infalls. We can see that, depending of the shapeof the barrier, 5 −
10% of the mass of a ∼ M ⊙ halo wasaccreted smoothly.Naturally by imposing a higher mass threshold theamount of unresolved accretion increases, reaching ∼ −
20% of their masssmoothly and their particles have typically suffered 1 − To close this section, in Fig.7 we turn our attention to thenumber of major mergers that a typical DM particle thatends up in a Milky-Way sized halo has experienced during c (cid:13) , 1–8 he Birth and Growth of Neutralino Haloes Figure 7.
A histogram of the number of major mergers in whichthe dark matter particles of a Milky-Way sized halo have been in-volved throughout their history. Black curves histogram the totalnumber of major mergers experienced by the haloes containinggiven dark matter particles while red curves include only thosemajor mergers where the particle is part of the larger object andgreen curves those where it is part of the smaller object. its life. We define a major merger as an event in which thehalo associated to a given trajectory changes its mass by afactor in the range [1.33, 4]. This represents mergers wherethe mass ratio of the haloes is less than a factor of three.The histograms displayed in Fig.7 show that most darkmatter particles have been involved in a relatively smallnumber of major mergers. In fact, 10 −
20% of all particlesnever witness a major merger. The median of the distribu-tion is 4 for the spherical collapse model and 3 for the squareroot barrier.We divide the trajectories into two categories accordingto the size of the halo before the merger. The first subgroup,displayed as red lines, represents the cases where the particlebelongs to the larger object (1 . M i < M f < M i ) whilegreen lines show the case where the particle belongs to thesmaller object (2 M i < M f < M i ). It is interesting to notethat these subsets have different distributions. This impliesthat major mergers where the particle is part of the smallerobject are more frequent than those where it is part of thelarger one.The mass cut-off induced by free streaming also pro-duces a correlation between the redshift and the mass ratioof the progenitors. Mergers at high redshift are more likelyto have low mass ratios, while mergers between very dissim-ilar haloes necessarily occur only at low redshift. We have used the excursion set formalism to examine haloformation and mass assembly in a standard ΛCDM cosmol- ogy where the DM is made of 100 GeV neutralinos. Ouranalytic treatment of the problem allows us to follow haloassembly histories for a large number of DM particles overmore than 25 orders of magnitude in halo mass. For the firsttime we are able to study halo assembly histories of DM par-ticles from the very bottom of the CDM hierarchy up to thelargest structures that exist today.The free streaming of neutralinos induces an exponen-tial cut-off at large wavenumbers in the primordial powerspectrum. For our neutralino model this sets 10 − M ⊙ as theminimum mass that any dark matter halo can have. Our re-sults suggest, however, that a very small fraction of the DMparticles were ever part of a halo of mass ∼ M f.s. . In fact,the mass of the first halo for a typical particle is comparableto the mass of the Sun. In addition, we have found that mostof the matter is not part of any halo at early times; the typ-ical redshift for first collapse is z = 14. Today, 5% to 10% ofthe dark matter is still not part of any clump, and beyondredshift 14 most of the mass was in diffuse form. These lateformation times imply that even very low mass haloes arenot expected to be strongly concentrated and thus that theyshould be relatively easily disrupted.The particle-based formulation of the excursion set the-ory allowed us to trace back to the very bottom of the CDMhierarchy, the particles that today form a Milky-Way sizedhalo. We found that there are rather few generations of ac-cretion/merger events. Typical particles experience three orfive of such episodes, only one of which occurs after the par-ticle is part of a 10 M ⊙ halo. About 10% of the mass ofMilky-Way sized haloes was accreted in diffuse form ratherthan as part of a smaller halo.Our results depend little on the exact mass of the neu-tralino or on the shape of the barrier for collapse. Compar-ison with an N-body simulation suggests that the excursionset formalism gives reliable results at least over the lim-ited mass range where a comparison can be made. Structuregrowth in the concordance cosmology is considerably lesshierarchical than is often thought. ACKNOWLEDGMENTS
We would like to thank Adrian Jenkins for providing us withthe N-body simulation used in this work. We also acknowl-edge useful conversations with Cedric Lacey, Chung Pei Ma,Jorge Moreno and Eyal Neistein.
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