The structure of dark matter halos in hierarchical clustering theories

奇怪的事实英语作文The Strangest Fact About the Universe.The universe is a vast and mysterious realm filled with countless wonders and baffling facts. One such fascinating yet odd fact is the existence of dark matter. Dark matteris a hypothetical type of matter that does not emit or absorb light and is thus undetectable through electromagnetic radiation. Despite its invisibility, astronomers believe that dark matter makes up a significant portion of the universe's mass, possibly even more than the visible matter we know of.The concept of dark matter was first proposed in the early 20th century to explain certain observed phenomena in cosmology, such as the rotation speeds of galaxies and the clustering of matter in the universe. Astronomers noticed that the visible matter in galaxies was not enough to account for the observed gravitational effects, leading them to hypothesize the existence of a hidden, massivecomponent that was not emitting light.One of the strangest things about dark matter is its nature. We don't know what dark matter is made of or how it interacts with other matter. It doesn't emit light or any other type of electromagnetic radiation, so we can't see it directly. We can only infer its existence through its gravitational effects on visible matter.Another odd fact about dark matter is its distribution in the universe. Astronomers have found that dark matter tends to cluster together in large halos around galaxies, creating a web-like structure that connects galaxies and other large-scale structures in the universe. This clustering suggests that dark matter may have played a crucial role in the formation and evolution of galaxies and other cosmic structures.The search for dark matter has been an ongoing challenge for physicists and astronomers. Experiments such as the Large Hadron Collider at CERN and underground detectors like XENON1T are designed to detect dark matterparticles interacting with ordinary matter. However, despite years of effort, we still haven't found direct evidence of dark matter particles.The existence of dark matter raises many intriguing questions about the universe and our understanding of it. How could such a significant component of the universe remain undetected for so long? What is the composition of dark matter, and how does it interact with other matter? How did dark matter influence the formation and evolution of galaxies and other cosmic structures?The answers to these questions could revolutionize our understanding of the universe. The study of dark matter could lead to new theories and models that explain the observed phenomena in cosmology and provide insights into the fundamental nature of matter and energy.In conclusion, the existence of dark matter is one of the strangest facts about the universe. Its hypothetical nature, distribution, and role in the formation of cosmic structures make it a fascinating subject of study. Thequest to understand dark matter and its implications for the universe is an ongoing scientific adventure that continues to captivate and challenge physicists and astronomers. As we delve deeper into the mysteries of dark matter, we may uncover new secrets and revelations about the vast and wonderful universe we inhabit.。

毕业设计（论文）英文摘要撰写指南为进一步规范本科毕业设计（论文）英语摘要的写作，提高撰写质量，特制定《毕业设计（论文）英文摘要撰写指南》，供撰写英文摘要时参考。

一、英文摘要撰写的几条原则1．汉英对译原则中英文摘要的一致性主要是指内容方面的一致性，从毕业生所撰写的英语摘要来看，存在两个误区，一是认为内容“差不多就行”或为了“好写”，在英文摘要中随意删去中文摘要的重点内容，或随意增补中文摘要所未提及的内容，造成文摘重心转移；二是认为英文摘要是中文摘要的硬性对译，对中文摘要中的每一个字都不敢遗漏，这往往造成英文摘要用词累赘、重复，拖沓、冗长。

英文摘要应严格、全面的表达中文摘要的内容，不能随意增删，但也不意味着一个字也不能改动，可有所出入。

2. 行文风格英文摘要的内容要求与中文摘要一样，一般包括目的、方法、结果和结论四部分。

英文摘要的撰写应简明扼要，不属于上述“四部分”的内容不必写入。

比如，在撰写英文摘要时可写出方法名称，而不必描述其操作步骤。

另外，在写作英文摘要时，要避免过于笼统的、空洞无物的一般论述和结论。

要尽量利用文章中的最具体的语言来阐述你的方法、过程、结果和结论，这样既可以给读者一个清晰的思路，又可以使你的论述言之有物、有根有据，使读者对你的工作有一个清晰、全面的了解。

3. 特殊字符特殊字符主要是指各种数学符号、上下脚标及希腊字母。

在摘要中尽量少用或不用特殊字符及由特殊字符组成的数学表达式，改用文字表达或文字叙述。

缩写字及首字母缩写词，如radar、laser、CAD等为大家所熟知的，可以直接使用。

对于那些仅为同行所熟悉的缩略语，应在其首次出现时用全称。

4. 时态与语态英文摘要时态的运用也以简练为佳，常用一般现在时、一般过去时，少用现在完成时、过去完成时，进行时态和其他复合时态基本不用；用于说明研究目的、叙述研究内容、描述结果、得出结论、提出建议或讨论等多用一般现在时。

不管是过去还是现在，涉及到公认事实、自然规律、永恒真理等，要用一般现在时；叙述过去某一时刻(时段)的发现、某一研究过程(实验、观察、调查等过程)等，要用一般过去时。

a r X i v :a s t r o -p h /0602394v 1 17 F eb 2006Mon.Not.R.Astron.Soc.000,000–000(0000)Printed 5th February 2008(MN L A T E X style file v2.2)MACHOs in dark matter haloesJanne Holopainen 1,Chris Flynn 1,Alexander Knebe 2,Stuart P.Gill 3,Brad K.Gibson 41Tuorla Observatory,V¨a is¨a l¨a ntie 20,Piikki¨o ,FIN-21500,Finland2AstrophysikalischesInstitut Potsdam,An der Sternwarte 16,14482Potsdam,Germany3Columbia University,Department of Astronomy,550West 120th Street,New York,NY 10027,USA 4Centre for Astrophysics,University of Central Lancashire,Preston,PR12HE,United Kingdom5th February 2008ABSTRACTUsing eight dark matter haloes extracted from fully-self consistent cosmological N -body simulations,we perform microlensing experiments.A hypothetical observer is placed at a distance of 8.5kpc from the centre of the halo measuring optical depths,event durations and event rates towards the direction of the Large Magellanic Cloud.We simulate 1600microlensing experiments for each halo.Assuming that the whole halo consists of MACHOs,f =1.0,and a single MACHO mass is m M =1.0M ⊙,thesimulations yield mean values of τ=4.7+5.0−2.2×10−7and Γ=1.6+1.3−0.6×10−6events star −1yr −1.We find that triaxiality and substructure can have major effects on the measured values so that τand Γvalues of up to three times the mean can be found.If we fit our values of τand Γto the MACHO collaboration observations (Alcock et al.2000),we find f =0.23+0.15−0.13and m M =0.44+0.24−0.16.Five out of the eight haloes under investigation produce f and m M values mainly concentrated within these bounds.Key words:gravitational lensing –Galaxy:structure –dark matter –methods:N -body simulations1INTRODUCTIONExperiments such as MACHO (Alcock et al.2000)and POINT-AGAPE (Calchi Novati et al.2005)have detected microlensing events towards the Large Magellanic Cloud (LMC)and the M31,supporting the view that some fraction of the Milky Way dark halo may be comprised of massive as-tronomical compact halo objects (MACHOs).The fraction of the dark halo mass in MACHOs,as well as the nature of these lensing objects and their individual masses can be constrained to some extent by combining observations of microlensing events with analytical models of the dark halo (e.g.Kerins 1998;Alcock et al.2000;Cardone et al.2001).However,the real dark halo might not be as simple as in an-alytical models,which typically have spherical or azimuthal symmetry and,in particular,smoothly distributed matter.Cosmological simulations indicate that dark matter haloes are neither isotropic nor homogeneous but are rather triax-ial (e.g.Warren et al.1992)and contain a notable amount of substructure (e.g.Klypin et al.1999).Even for the Milky Way it is still unclear if the enveloping dark matter halo has spherical or triaxial morphology:a detailed investigation of the tidal tail of the Sgr dwarf galaxy supports the notion of a nearly spherical Galactic potential (Ibata et al.2001;Majewski et al.2003)whereas that the data are also claimed to be consistent with a prolate or oblate halo (Helmi 2004).The shape of the halo has an effect on the predicted numberof microlenses along different lines-of-sight,for comparison with the results of microlensing experiments.In this paper,we examine the effects of dark halo mor-phology and dark halo clumpiness on microlensing surveys conducted by hypothetical observers located in eight N -body dark matter haloes formed fully self-consistently in cosmological simulations of the concordance ΛCDM model.The eight haloes are scaled to approximately match the Milky Way’s dark halo in mass and rotation curve prop-erties,and hypothetical observers are placed on the surface of a “Solar sphere”(i.e.8.5kpc from the dark halo’s centre)from where they conduct microlensing experiments along lines-of-sight simulating the Sun’s line-of-sight to the LMC.The effects on microlensing of triaxiality and clumping in the haloes are examined.The paper expands upon work in a study by Widrow &Dubinski (1998)although we take a slightly dif-ferent point of view.Instead of studying the errors caused by different analytical models fitted to an N -body halo,we analyse the microlensing survey properties of the simulated haloes directly.That is,we do not try to gauge the credibil-ity of analytical microlensing descriptions but rather use our self-consistent haloes for the inverse problem in microlensing (Cardone et al.2001).In other words,the free parameters (the MACHO mass and the fraction of matter in MACHOs)c0000RAS2Holopainen et al.are determined from the observations via the optical depth, event duration and event rate predictions of the models.This not only allows us to make predictions for these parameters based upon fully self-consistent halo models but also to in-vestigate the importance of shape and substructure content of the haloes.The paper is structured as follows.We describe the N-body haloes and their preparation in Section2,the sim-ulation details are covered in Section3,equations for the observables are derived in Section4,results are given in Section5,some discussion is presented in Section6,final conclusions are in Section7.2THE HALOES2.1Cosmological simulation detailsOur analysis is based on a suite of eight high-resolution N-body simulations(Gill,Knebe&Gibson2004a)carried out using the publicly available adaptive mesh refinement code MLAPM(Knebe,Green&Binney2001)in a standard ΛCDM cosmology(Ω0=0.3,Ωλ=0.7,Ωb h2=0.04,h= 0.7,σ8=0.9).Each run focuses on the formation and evo-lution of galaxy cluster sized object containing of order one million collisionless,dark matter particles,with mass reso-lution1.6×108h−1M⊙and force resolution∼2h−1kpc(of order0.05%of the host’s virial radius).The simulations have sufficient resolution to follow the orbits of satellites within the very central regions of the host potential( 5–10%of the virial radius)and the time resolution to resolve the satel-lite orbits with good accuracy(snapshots are stored with a temporal spacing of∆t≈170Myr).Such temporal resolu-tion provides of order10-20timesteps per orbit per satellite galaxy,thus allowing these simulations to be used in a pre-vious paper to accurately measure the orbital parameters of each individual satellite galaxy(Gill et al.2004b).The clusters were chosen to sample a variety of en-vironments.We define the virial radius R vir as the point where the density of the host(measured in terms of the cosmological background densityρb)drops below the virial overdensity∆vir=340.This choice for∆vir is based upon the dissipationless spherical top-hat collapse model and is a function of both cosmological model and time.We fur-ther applied a lower mass cut for all the satellite galax-ies of1.6×1010h−1M⊙(100particles).Further specific details of the host haloes,such as masses,density pro-files,triaxialities,environment and merger histories,can be found in Gill,Knebe&Gibson(2004a)and Gill et al. (2004b).Table1gives a summary of a number of rele-vant global properties of our halo sample.There is a promi-nent spread not only in the number of satellites with mass M sat>1.6×1010h−1M⊙but also in age,reflecting the dif-ferent dynamical states of the systems under consideration.2.2DownscalingAs we intend to perform microlensing experiments directly comparable to the results of the MACHO collaboration we need to scale our haloes to the size of the Milky Way.For this purpose we follow Helmi,White&Springel(2003)and Table1.Summary of the eight host dark matter haloes.The superscript cl indicates that the values are for the unscaled clus-ters.Column2shows the virial radius,R vir;Column3the virial mass,M vir;Column4the redshift of formation,z form;Column5 the age in Gyr;and thefinal column an estimate of the number of satellites(subhaloes)in each halo,N sat.apply an adjustment to the length scale,requiring that den-sities remain unchanged.Hence the scaling relations are:r=γr cl(1) v=γv cl(2) m=γ3m cl(3) t=t cl(4)where r is any distance,v is velocity,m is mass,t is time and the superscript cl refers to the unscaled value.Because the haloes are downscaled,γwill always be in the range γ∈[0,1].Tofind a suitableγ,the maximum circular velocity of each halo is required to be220km s−1.The resulting(scaled) rotation curves for all eight haloes are shown in Figure1. Thisfigure highlights that Halo#8is a special system—its evolution is dominated by the interaction of three merging haloes(Gill et al.2004b).We do not consider Halo#8to be an acceptable model of the Milky Way,but we include it into the analysis as an extreme case.One might argue that our(initially)cluster sized ob-jects should not be used as models for the Milky Way for they formed in a different kind of environment with less time to settle to(dynamical)equilibrium(i.e.the oldest of our systems is8.3Gyr vs.∼12Gyr for the Milky Way). However,Helmi,White&Springel(2003)showed that more than90%of the total mass in the central region of a realis-tic Milky Way model was in place about1.5Gyr after the formation of the object.Our study focuses exclusively on this central region(i.e.the inner15%in radius)boosting our confidence that our haloes do serve as credible models of the Milky Way for our purposes.In Table2we list the scaled radii and masses,as well as the scaling factorγ,for each halo.Note that we have adopted a Hubble constant of h=0.7,which applies throughout the paper.The circular velocity(rotation)curves for the scaled haloes are shown in Figure1and demonstrate how the scaled mass M(<r)is accumulated out to400kpc.Two vertical lines mark the position of observers at the Solar circle and the distance of the LMC from the halo centre; within this region,we note that the mass profiles are similarc 0000RAS,MNRAS000,000–000MACHOs in dark matter haloes3Table 2.Properties of the haloes after downscaling to Milky Way size.Column 2shows the scale factor for each halo,γ;Column 3shows the scaled virial radius,R vir ;Column 4the scaled virial mass,M vir ;and the final column the scaled particle mass,m p .20406080100120140160180200220240110100V c(k m s -1)r (kpc)Halo 1Halo 2Halo 3Halo 4Halo 5Halo 6Halo 7Halo 8Figure 1.Circular velocity curves of the downscaled haloes(V c =4Holopainen et al.Table3.Substructure within the inner70kpc of the(down-scaled)haloes.Column1shows the name of the halo;Column2 the distance from the centre of the subhalo to the centre of the host,l sat;Column3the truncation radius of the subhalo(within the truncation radius,the density of the subhalo is larger than the background density of the host halo),r sat;Column4the number of bound particles in the subhalo,N sat;and thefinal column the mass of the subhalo,M sat.fixed at89.1◦(the original angle between( r⊙, r LMC− r⊙)in Galactocentric coordinates).This configuration(illustrated in Figure3)gives1600individual sightlines,and microlens-ing cones,which sample different sets of particles within a halo.The choice for the number of sightlines is a compro-mise between computation time and sampling density.As seen in Figure3,1600sightlines sample the volume quite sufficiently,even when the sightlines are drawn as lines in-stead of actual cones.The microlensing cone itself is defined by the observer and the source.Figure2illustrates that we treat the source (i.e.LMC)as a circular disk with diameter d S.The parti-cles inside the cone are considered to be gravitational lenses. However,this kind of setup can lead to a handful of parti-cles(those closest to the observer)contributing most to the optical depth.This leads to high sampling noise,and is es-sentially due to the limited resolution of the cosmological simulation(see Figure4).It follows that the particle distri-bution has to be smoothed in some manner to suppress this noise;we describe this and the related issue of the MACHO masses in the next section.3.2Mass resolution vs.MACHO massIn order to compute microlensing optical depths and other properties from the particles in the simulations,we need to get from a mass scale of order106M⊙,i.e.that of the particles in the simulation,down to the mass scale of the MACHOS,of order1M⊙;i.e.we account for the fact that a typical particle in the simulations represents of order106 MACHOs.To overcome the limitation imposed by the mass reso-lution of the simulation we recall that individual particles inFigure2.The microlensing sightline is defined by the locations of the observer and the source and the cone by the sightline and the source diameter.We use l S=50.1kpc and d S=4.52kpc. The particles inside the cone are treated as gravitational lenses. Typically∼100particles from the cosmological simulation are found inside a cone.This number is increased to∼10000by breaking them up into subparticles according to the recipe out-lined in Section3.2.the simulation are not treated asδ-functions but have afi-nite size.The extent of a particle(i.e.its size)is determined —in our case—by the spacing of the grid,as we are us-ing an adaptive mesh refinement code(i.e.MLAPM).In MLAPM the mass of each particle is assigned to the grid via the so-called triangular-shaped cloud(TSC)mass-assignment scheme(Hockney&Eastwood1981)which spreads every in-dividual particle mass over the host and surrounding33−1 cells.The corresponding particle shape(in1D)reads as fol-lows:S(x)= 1−|x|MACHOs in dark matter haloes5Figure3.Sightlines used in the microlensing simulation.The image shows all1600sightlines of the40observers.The observers are located on a sphere with a radius of r⊙=8.5kpc.The sphere can be seen as the“dark ball”in the centre of the sightlines.The radius of the sphere which holds the sources is49.5kpc.Every observer has40sources(LMCs)at a distance of50.1kpc.The sightlines are not uniformly distributed because there is a limited number of observers and the angle( r obs, r LMC− r obs)is kept fixed at89.1◦.This is the original angle between( r⊙, r LMC− r⊙) in Galactocentric coordinates.the optical depth measurements obtained when the particles within the cones have been resampled to“subparticles”.In the un-resampled case,the handful of particles which hap-pen to reside close to the observer are found to dominate the calculation of the optical depth and other quantities of interest such as event duration and event rate.Variations in the optical depth from sightline to sightline can vary by up to a factor of three in the un-resampled sample cones,sim-ply because a handful of particles dominate the microlens-ing.Convergence experiments have shown that breaking the original simulation particles down to100subparticles is suf-ficient to reduce the noise from this source to be negligible; if we were to use more subparticles,there is no further gain in accuracy.The subparticles introduced have masses of order104 M⊙,so we are unfortunately still orders of magnitudes away from actual MACHO masses.In the formulae introduced later(Section4),each subparticle is expressed as a sin-gular concentration of MACHOs.That is,a subparticle is treated as a set of MACHOs which have the same posi-tion and velocity as the subparticle itself,and for conve-nience in the simulations the MACHO mass is chosen to be m M=1.0M⊙(although this is later relaxed by scaling).Fig-ure6demonstrates the hierarchy of particles starting from the cosmological simulation particle down to the individual MACHOs.123456789100102030405060708090100110120130τcone1−7Cone indexOriginal particles in useTSC subparticles in useparison between the use of cosmological simula-tion particles and triangular-shaped clouds containing100sub-particles.For the illustration,we distributed132observers on a perimeter of a disk with a radius of8.5kpc,located on the xy-plane.The volumes of consecutive cones overlap by half.Without any resolution enhancements the variation in optical depth from cone to cone can be as large as a factor of three!This is due to the insufficient mass resolution of the cosmological simulation. We reached the Poissonian noise level,in which cone-to-cone vari-ations are<20%,with100subparticles,by experimenting.The large scale trend is due to triaxiality of the halo.4THE MICROLENSING EXPERIMENTS4.1DefinitionsBefore going to the microlensing equations,we define what we mean by certain terms:A microlensing source is a circular region,oriented perpendicularly to the line-of-sight of the observer,in which uniformly distributed background source stars are located. Because of the statistical nature of the source model,the number of background stars affects only the microlensing event rate.The region with a diameter d S and a distance l S,is shown in Figure2.A source star is a star located somewhere in the disk of the microlensing source.A microlens is also a circular region,oriented perpen-dicularly to the line-of-sight of the observer and centered on a dark particle.The region has a radius r E,which de-pends on the location and mass of the particle.Lenses are always inside a microlensing cone between the source and the observer.A microlensing event is a detectable amplification of a source star caused by a microlens.Detectable means that the light of a source star is amplified by a factor larger than1.34.This occurs when the sightline to a source star passesa particle within the lens radius,also known as the Einstein radius.4.2Optical depthThe Einstein radius r E of a gravitational lens is defined as r2E=4Gml−1l−16Holopainen etal.Figure 5.For this illustrative example of the subparticle clouds,we placed 222cosmological simulation particles onto the xy -plane and broke each particle down to 10,000subparticles.To get the subparticle positions,we used the triangular-shaped cloud den-sity profile identically to the actual cosmological simulation.The subparticles form a cube (of a size of the corresponding MLAPM grid cell)around the position of the cosmological simulation par-ticle.In the real microlensing simulation,the whole volume under investigation is filled with intersecting cubes and the void areas between the cubes seen here are not present.Figure 6.Splitting the cosmological simulation particle to 100subparticles and a subparticle to N M MACHOs.In the first phase,we break the cosmological simulation particle to 100subparticles.The resulting subparticle represents an order of N M ∼104MA-CHOs with a mass of 1M ⊙.The subparticle is treated as a MA-CHO concentration where the represented MACHOs all have a single location (l M ≡l sub ),mass (m M ≡m sub /N M )and velocity (v M ≡v sub ).where m is the mass of the lens,l is the distance to the lens,l S is the distance to the source and A =4Gl sub−1ωS,(8)where ωis the solid Einstein angle of the lens and ωS is the solid angle of the source.When the solid angles are small,this can be approximated to τ≃r Er S2=Am Θ−21l S,(9)where r S is the radius of the source and Θ−2=(l S /r S )2.Note that Θis approximately the half opening angle of the cone.The optical depth of a subparticle with a mass m sub is defined as τsub =A Θ−2m sub1l S.(10)We take this value to be the total optical depth from all the ∼104MACHOs the subparticle represents.Optical depth is additive,as long as the lenses do not overlap or cover the whole source,and so the total optical depth in a microlensing cone which contains N subparticles is τcone =N iτ(i )sub=A Θ−2m subN i1l S(11)From Equation 11it follows that the optical depth in a cone depends mainly on the distances of the subparticles re-lated to the observer.The closer the particles are the larger is the optical depth.Section 3.2covered the details of par-ticle breaking,which also has a large effect on τcone .Note that τcone does not depend on the MACHO mass m M .4.3Event durationThe event duration of a lens describes the typical duration of the amplifying event the lens would produce.The detected event durations in the MACHO experiment are the order of 100days.Event duration depends on the tangential velocity with which a lens would seem to pass a source star and on the Einstein radius of the lens.The equation for an individual subparticle ist sub =πv sub ,(12)where r sub is the lens radius of the subparticle and v sub the apparent tangential velocity difference between the sub-particle and the source,respective to the observer.c0000RAS,MNRAS 000,000–000MACHOs in dark matter haloes7 The termπ2m1/2subl subl sub−12m1/2Ml subl sub−1t M=N Mτsub/N Mˆtsub,(15)whereΓM=τMπA1/2Θ−2m−1/2Mm subv subl sub−1πA1/2Θ−2m−1/2Mm subNi v(i)sub l(i)sub−18Holopainen et al.Table4.The mean optical depths and event rates,with their standard deviations,over the1600cones for each halo.See Figure 9for the corresponding(differential)distribution functions—the numbers represented here are the integrated values of those func-tions.We also list the modes of the distributions:ˆt exp columns give the most probable event duration for the respective distri-bution.Error limits show the variations between cones(i.e.mi-crolensing experiments)in a halo.mean4.7+5.0−2.1861.6+1.3−0.668MACHOs in dark matter haloes90 50 100 150 200 250 300 350 400 450 500 5101520253035404550a p p a r e n t t a n g e n t i a l v e l o c i t y (k m s -1)distance from observer (kpc)Figure 8.An example which shows the anticorrelation between apparent tangential velocities and locations of subparticles inside a microlensing cone.This anticorrelation explains why triaxiality does not show up in Γas clearly as in τ.The size range of the triangular-shaped clouds also shows up clearly in the plot.is done for all cones,we bin the observables and calculate the upper and lower fractiles so that they hold 65%of the values.Especially the optical depth fractiles reveal clear tri-axiality signal.In the first panel in Figure 10,the mean optical depth is largest when the sources are close to the z -axis (i.e.the major triaxial axis),as one would expect.For haloes #1,#2and #7,the optical depth on opposite sides of the xy -plane differs somewhat,demonstrating that matter is not neces-sarily distributed with azimuthal symmetry in the simulated (or real)haloes.There are high mean optical depth values near the pos-itive y -axis for Halo #1in the second panel.This anomaly is due to the subhalo discussed in previous Section 5.2.Now we can see that the subhalo is located close to the positive y -axis.This is confirmed by the true location of the subhalo,which is found to be l sat =(−3.37,19.8,1.36)kpc.The lowest panel is different from the two previous ones because we are measuring tri axiality.The optical depth val-ues from sources near the x -axis seem to be the lowest amongst all panels.These low values (τcone ∼4×10−7)can also be seen in the second panel as the lower fractiles at α∼90◦.Thus,by these “observations”we can verify that the x -axis is the minor axis.In the lowest panel,the fractiles bend upwards at α∼90◦because the sources near both the z -and y -axis produce larger values than the sources near the x -axis.Triaxiality can not be seen as clearly in Figure 11as in Figure 10because Γis affected by the lens velocities whereas τis not.Apparently,the velocities do not carry enough infor-mation about the triaxiality and the correlation between the source location and the observables are somewhat “washed out”.This is confirmed in Figure 8where we show that there is virtually no correlation between the apparent tangential velocity and the location of the subparticles inside a cone.Nevertheless,the most striking features,i.e.the z -axis ma-jority and the subhalo in Halo #1,are still clearly visible in Γ.For example,a suitable subhalo can produce roughly twice as many events as a similar area without one,based on the second panel in Figure 11.5.4Estimating MACHO mass and halo mass fraction in MACHOsThe MACHO collaboration used several analytical halo models to estimate the typical,individual MACHO mass,m M ,and the mass fraction of MACHOs in the halo,f ,re-sponsible for the lensing events observed.For example,oneof their models fits m M =0.6+0.28−0.20M ⊙and f =0.21+0.10−0.07.We calculate our own estimates for m M and f to see how our N -body halo models compare to the analytical ones in making these predictions.As the observational constraint,we use the blending corrected event durations chosen by the MACHO collab-oration’s criterion “A”,ˆtst (A ),from Table 8in Alcock et al.(2000).These values are not corrected for the observa-tional efficiency function E (ˆt)nor have they been reduced for the events caused by the known stellar foreground pop-ulations,given in Table 12in Alcock et al.(2000).We adopt N exp (A )=2.67for known population events and ar-rive at the observational values of τobs =3.38×10−8and Γobs =1.69×10−7events star −1yr −1.We then compare these values with our simulation produced values,τcone and Γcone ,to get m M and f .The simulation values are correctedwith the efficiency function E (ˆt)prior to calculating the pre-dicted values.The calculations are based on the following equations.First,we assume that f ∝τ,(18)which simply describes that that the probability of ob-serving an event is proportional to the number of lenses in the whole halo.From Equation 17,and from the fact that the number of observed events also follows the total number of lenses,we get 1Γ∝fm −1/2M.(19)These two equations permit us to estimate m M and f .From Equation 18,we get f obsτcone.(20)For the microlensing simulations we assumed f cone =1.0and thus,f =f obs =τobsm cone M=f obsΓobs2,(22)where f cone =1.0and m cone M=1.0M ⊙.Thus,we get the equation for the MACHO mass,m M =m obs M =f obsΓcone10Holopainen et al.Table 5.Same as Table 4buthereweusedtheMACHOcol-laboration’s efficiency function A.These numbers should corre-spond to the E (ˆt)–uncorrected observations if the MACHO frac-tion would satisfy f =1.0and the MACHO mass m M =1.0M ⊙.The distributions are not shown because they are essentially sim-ilar to Figure 9except for the smaller integrated values (given in Tables 5and 6).mean1.7+1.9−0.8900.6+0.5−0.272Haloτcone ˆt exp Γcone ˆtexp 10−7days10−6daysevents star −1yr −1#12.7+2.9−1.3840.9+0.8−0.466#22.9+3.6−1.3841.0+1.0−0.466#32.1+1.9−0.9840.7+0.5−0.364#42.4+2.7−1.2860.8+0.7−0.374#52.0+1.6−1.1940.6+0.4−0.369#62.4+2.5−0.9990.8+0.6−0.371#72.3+3.2−1.2890.7+0.8−0.361#81.3+1.0−0.51040.4+0.3−0.181educated guesses about the shape in form of some given pa-rameters (e.g.triaxiality,density profile,etc.).Widrow &Dubinski (1998)used a so called microlens-ing tube to get better number statistics for their event rates,and as they note,this “distorts the geometry of a realistic microlensing experiment”.However,our solution (the intro-duction of subparticles in accordance to the TSC mass as-sigment scheme)to the insufficient mass resolution of the cosmological simulation preserves the geometry.The downsides of using N -body haloes are:(i)We as-sume that the spatial and velocity distribution of MACHOs follows dark matter.(ii)The clusters we use are not as old as the Milky Way.Our models are based upon pure dark matter simula-tions which may not be appropriate if a significant fraction of the dark matter is composed of MACHOs,since they are composed of baryons,and baryonic physics has been ex-plicitly ignored!We thus implicitly assume that the spa-tial and velocity distribution of MACHOs follows the re-spective distributions of the underlying dark matter.Based upon these assumptions dark satellites comprised of MA-CHOs would be detectable through excess optical depth values and event duration anomalies in microlensing exper-iments.Simulations suggest that the majority of these dark satellites can be located as far as 400kpc from the halo cen-tre.Thus,multiple background sources at distances over 400kpc would be needed to detect possible dark subhaloes in a Milky Way sized dark halo.For example,POINT-AGAPE (Belokurov et al.2005,Calchi Novati et al.2005)is a sur-vey that is in principle able to detect even a dark MACHO satellite,on top of “free”MACHOs in the dark halo,because its source,M31,is distant enough.From the results,we can see that Halo #8is too young to be used as a Milky Way dark halo model,but all the other haloes seem to behave well.The fact that Halo #8consists of three merging smaller haloes explains the peculiar values it produces.The other haloes are not experiencing any violent dynamical changes —a requirement we would expect a model of the Milky Way dark halo to fulfill.Obviously,our MACHO mass function is a δ-function.A more complex function would force us to calculate event durations and rates for individual MACHOs instead of sub-c0000RAS,MNRAS 000,000–000。

a r X i v :a s t r o -p h /0305300v 1 16 M a y 2003Mon.Not.R.Astron.Soc.000,1–9(2003)Printed 2February 2008(MN L A T E X style file v2.2)Extended Press-Schechter theory and the density profilesof dark matter haloesNicos Hiotelis ⋆†1st Experimental Lyceum of Athens,Ipitou 15,Plaka,10557,Athens,Greece,E-mail:hiotelis@ipta.demokritos.grAccepted ...............Received ................;in original form ...........ABSTRACTAn inside-out model for the formation of haloes in a hierarchical clustering scenario is studied.The method combines the picture of the spherical infall model and a mod-ification of the extended Press-Schechter theory.The mass accretion rate of a halo is defined to be the rate of its mass increase due to minor mergers.The accreted mass is deposited at the outer shells without changing the density profile of the halo inside its current virial radius.We applied the method to a flat ΛCDM Universe.The resulting density profiles are compared to analytical models proposed in the literature,and a very good agreement is found.A trend is found of the inner density profile becoming steeper for larger halo mass,that also results from recent N-body simulations.Addi-tionally,present-day concentrations as well as their time evolution are derived and it is shown that they reproduce the results of large cosmological N-body simulations.Key words:cosmology:theory –dark matter –galaxies:haloes –structure –for-mation1INTRODUCTIONNumerical studies (Quinn,Salmon &Zurek (1986);Frenk et al.(1988);Dubinski &Galberg (1991);Crone,Evrard &Richstone (1994);Navarro,Frenk &White (1997),here-after NFW;Cole &Lacey (1996);Huss,Jain &Steinmetz (1999);Fukushige &Makino (1997);Moore et al.(1998),hereafter MGQSL;Jing &Suto (2000),hereafter JS:Hern-quist (1990),hereafter H90,Kravtsov at al.(1998),Klypin at al.(2001))show that the density profiles of dark matter haloes are fitted by models of the formρf (r )=ρcd r=λ+µν(r/r s )µ2Nicos Hiotelisfor stydying the formation of structures.Recently,such a modified PS approximation was combined with a spherical infall model picture of formation by Manrique et al.(2003), MRSSS hereafter.Their results are in good agreement with those of N-body simulations.In this paper we use the formalism of MRSSS,with justified modifications,and the same model parameters as in BKPD.We compare the characteristics of the resulting structures with those in N-body results.In Section2, we discuss the modified PS theory and its application to the calculation of the density profile.In Section3,the characteristics of the resulting dark matter haloes are presented.A discussion is given in Section4.2EXTENDED AND MODIFIEDPRESS-SCHECHTER THEORYOne of the major goals of the spherical infall model is the PS approximation.It states that the comoving density of haloes with mass in the range M,M+d M at time t is given by the relation:N(M,t)d M= πδc(t)M e−δc2(t)d M|d M(3)whereσ(M)is the present-day rms massfluctuation on co-moving scale containing mass M and is related to the power spectrum P by the following relationσ2(M)=23πρb0R3=Ωm0H20d t=H i g13f i∆i(7) andΩΛ,i is the initial values of the quantityΩΛ(a)=Λ/(3H2(a)).Eq.7is derived under the assumption that the initial velocity v i of the shell is v i=H i r i−v pec,i where the initial peculiar velocity,v pec,i,is given according to the linear theory by the relation v pec,i=170[1−Ωm,i(1+Ωm,i)](Lahav et al.(1991)).The radius of the maximum expansion is r ta=s ta r i,where s ta is the root of the equation g(s)=0that corresponds to zero velocity(d s/d t=0).The sphere reaches its turn-around radius at timet ta=12(s)d s(8) and then collapses at time t c=2t ta.The scale factor a of the Universe obeys the equation: d a2(a)(9) whereX(a)=1+Ωm,0(a−1−1)+ΩΛ,0(a2−1)(10) and the subscript0denotes the present-day values of the parameters.However,the time and the scale factor a are related by the equationt=12(u)d u(11) Setting t=t c in the above equation and solving for a one finds the scale factor a c at the epoch of collapse.If we call δi,c(t)the initial overdensity required for the spherical region to collapse at that time t and take into account the linear theory for the evolution of the matter density contrastδ=δρ/ρ,we haveδ∝1a a0X−3/2(u)d u=D(a),(12) (Peebles1980),thenδc(t)is givenδc(t)=δi,c(t)D(t0)D(t i)D(t0)D(t)(14) whereδcrit(t)is the linear extrapolation of the initial over-density up to the time t of its collapse.In an Einstein-de Sitter universe(Ωm=1,ΩΛ=0)this value is independent on the time of collapse and isδcrit≈1.686.In other cos-mologies it has a weak dependence on the time of collapse (e.g.Eke,Cole&Frenk(1996)).In aflat universe it can beapproximated by the formulaδcrit(t)≈1.686Ω0.0055m,0(t).The PS mass function agrees relatively well with the results of N-body simulations(e.g.Efstathiou,Frenk& White(1985),Efstathiou&Rees(1988);White,Efstathiou &Frenk(1993),Lacey&Cole(1994);Gelb&Bertschinger (1994);Bond&Myers(1996))while it deviates in detail at both the high and low masses.Recent improvements(Sheth &Tormen(1999);Sheth,Mo&Tormen(2001),see also Jenkins et al.(2001))allow a better approximation involving some more parameters.The application of the above approx-imation to the model studied in this paper is a subject of future research.Lacey&Cole(1993)extended the PS theory using the idea of a Brownian random walk,and were able to calcu-late analytically tractable expressions for the mass function,c 2003RAS,MNRAS000,1–9Extended Press-Schechter theory and the density profiles of dark matter haloes3merger rates,and other properties.They show that the in-stantaneous transition rate at t from haloes with mass M to haloes with mass between M ′,M ′+d M ′is given byr (M →M ′,t )d M ′=2d t1d M ′1−σ2(M ′)21σ2(M )d M ′(15)This provides the fraction of the total number of haloeswith mass M at t ,which give rise per unit time to haloes with mass in the range M ′,M ′+d M ′through instantaneous mergers of any amplitude.An interesting modification of the extended PS theory is the distinction between minor and major mergers (Manrique &Salvador-Sol´e (1996);Kitayama &Suto (1996);Salvador-Sol´e et al.(1998);Percival,Miller &Peacock (2000),Cohn et al.(2001)).Mergers that produce a fractional increase below a given threshold ∆m are regarded as minor.This kind of mergers corresponds to an accretion.Consequently,the rate at which haloes increase their mass due to minor mergers is the in-stantaneous mass accretion rate and is given by the relationr a m (M,t )=M (1+∆m )M(M ′−M )r (M →M ′,t )d M ′(16)Thus the rate of the increase of halo’s mass due to the ac-cretion is d M (t )ρb (a )=1a i3(1+∆i )(18)where c f is the collapse factor of the sphere defined as the ratio of its final radius to its turnaround -hav et al.(1991)applied the virial theorem to the viri-alized final sphere assuming a flat overdensity and found the collapse factor to be c f ≈(1−n/2)/(2−n/2)wheren =(Λr 3ta )/(3GM ).For an Einstein-de Sitter Universe ∆vir (a )≈18π2at any time.For flat models with cosmolog-ical constant,significantly good analytical approximations of ∆vir exist.Bryan &Norman (1998)proposed for ∆vir the following approximation∆vir (a )≈(18π2−82x −39x 2)/Ωm (a )(19)where x ≡1−Ωm (a ).MRSSS considered the following picture of the forma-tion of a halo:At time t i an halo of virial mass M i and virial radius R i is formed and at later times it accretes mass ac-cording to the Eq.(17).Assuming that the accreted mass is deposited in an outer spherical shell without changing the density profile inside its current radius,then M vir (t )−M i =R vir (t )R i4πr 2ρ(r )d r(20)The current radius R vir contains a mass with mean den-sity ∆vir (a )times the mean density of the Universe ρb (t ).Therefore,R vir (t )=3M vir (t )r a m [M vir (t ),t ]d[ln[ρb (t )∆vir (t )]]8πG∆vir (a )×1+M vir (a )a −d ln[∆vir (a )]d a=H −10X −1/2(a )r am [M vir (t ),t ].(24)Integrating Eq.17and using Eqs.21and 23,we obtain the growth of virial mass and virial radius and,in a parametric form,the density profile of haloes.3DENSITY PROFILES OF DARK MATTERHALOESThe results described in this section are derived for a flat universe with Ωm,0=0.3and ΩΛ,0=0.7.We used two forms of power spectrum.The first one -named spect1-is the one proposed by Efstathiou,Bond &White (1992).It is based on the results of the COBE DMR experiment and is given by the relation:P spect 1(k )=Bk[1+a 1k 1/2+a 2k +a 3k 3/2+a 4k 2]b(26)The values for the parameters are:n =1,a 1=−1.5598,a 2=47.986,a 3=117.77,a 4=321.92and b =1.8606.We used the top-hat window function that has a Fourier transform given by:ˆW(kR )=3(sin(kR )−kR cos(kR ))4Nicos Hiotelis-4-22log (M/M unit )0.51l o g (M /M u n i t )σFigure 1.rms mass fluctuation as a function of mass.Solid line:for spect1,dotted line:for spect2are shown.It must be noted that we use a system of units with M unit =1012h −1M ⊙,R unit =h −1Mpc and t unit =1.515×107h −1years.In this system of units H 0/H unit =1.5276.3.1Present day structuresIn the approximation used in this paper,for given models of the Universe and power spectrum there is only one free pa-rameter,that is the value of the threshold ∆m (see Eq.16).We found that the resulting density profiles are sensitive to the value of ∆m .As an example,the density profiles of two systems with the same present-day mass 1012h −1M ⊙and different values of ∆m are plotted in Fig.2.Both den-sity profiles are derived from spect2.The solid line -shown in Fig.2-corresponds to the system derived for ∆m =0.21while the dotted line to the system for ∆m =0.5.The den-sity profile for smaller ∆m is steeper at both the inner and the outer regions.Additionally,for different ∆m the concen-trations of the haloes are different too (a detailed description of the way the concentration is calculated is given below).The system that results for ∆m =0.21has c vir =15.2,while the one for ∆m =0.5has c vir =8.7.In order to calculate the density profiles (that will be presented below),we used as a basic criterion the concentrations of the present-day struc-tures.In fact,we have chosen the values of ∆m =0.23and ∆m =0.21for spect1and spect2respectively,because the concentrations resulting from these values are close to the results of the toy-model of BKPD.This model is constructed by BKPD to reproduce the results of their N-body simula-tions and it is able to give the concentration c vir of a virial mass M vir at any scale factor a .First,the scale factor a c at the epoch of collapse is calculated,solving the following equation-2.5-2-1.5-1log R1234l o g ρFigure 2.For spect2:Density profiles for two haloes having the same mass 1012h −1M ⊙.Solid line:∆m =0.21.Dotted line:∆m =0.5σ[M ∗(a c )]=σ[F M vir (a )](28)where F =0.01and M ∗is the typical collapsing mass.Then,the concentration is calculated using the formulac vir ,BKPD [M vir (a ),a ]=Kaλ+µν−n1/µr s (31)This formula gives the radius r n at which the logarithmic slope equals to n .According to the model presented in this paper,haloes grow inside-out.Thus,the value of c vir repre-sents the way of halo growth.In Fig.3,the concentration isc2003RAS,MNRAS 000,1–9Extended Press-Schechter theory and the density profiles of dark matter haloes510101102M vir /M unit681012141618c Figure 3.Concentration as a function of present-day virial mass.From the top,the first pair of curves are for spect1and the second for spect2.Solid curves:our results.Dotted curves:BKPD toy-model results.plotted as a function of the present-day virial mass.From the top of the figure,the first pair of curves (solid and dot-ted)correspond to spect1and the second pair to spect2.Solid curves show our results while dotted curves depict the results of the toy-model of BKPD.A very good agreement between the values of the concentration is shown.In partic-ular,concentrations resulting from spect2are in agreement with those obtained for the model of BKPD for the whole range of mass presented.On the other hand,small differ-ences appear for very small and very large masses in the case of spect1.In Fig.4we present the density profiles of the re-sulting structures with present-day masses in the range of 0.2×1011h −1M ⊙to 8×1014h −1M ⊙.The left-hand side fig-ures (a1,b1,c1,d1,e1)have been produced using spect1,while the right-hand ones using spect2.Figures (a1)and (a2)correspond to mass 0.2×1011h −1M ⊙,(b1)and (b2)have mass 1012h −1M ⊙,(c1)and (c2)to mass 1013h −1M ⊙,(d1)and (d2)to mass 1014h −1M ⊙and (e1)and (e2)cor-respond to mass 8×1014h −1M ⊙.Solid lines represent the resulting density profiles while dotted lines are the fits using the general formula of Eq.1.It is shown that the fits using the general formula of Eq.1are exact.We also fit every halo density profile using the analytical models that have been proposed in the literature (H90,NFW,MGQSL,JS)and are described in Section 1.The best fit of these models to our resulting profiles is shown in Fig.4(circles).This best fit is found by the minimizing procedure described above,for λ,µand νconstants and equal to the proposed values,while ρc and r s are the only fitting parameters.Best fit for the result-ing density profile in (a1)is the H90model,in (a2)and (b2)the NFW model,in (b1),(c1)and (c2)the MGQSL model24l o g ()ρJS(d1)5JS(d2)-1.5-0.5log (R /R vir )024JS(e1)-1.5-0.5log (R /R vir )24JS(e2)-1.5-0.5024MGQSL(c2)24MGQSL(c1)024l o g ()MGQSLρ(b1)24NFW(b2)24H90(a1)24NFW(a2)Figure 4.Density profiles as a function of radius.Solid curves:re-sulting density profiles.Dotted curves:fits of the resulting densityprofiles using the formula of Eq.1.Circles:best bit to our results using models proposed in the literature (H90,NFW,MGQSL,JS).Left-hand side:spect1.Right-hand side:spect2and in (d1),(d2),(e1)and (e2)the JS model.Additionally,haloes of different mass are fitted well by different analytical models.This is due to the different inner and outer slopes of the density profiles.Inner slope,(defined as that at ra-dial distance r =10−2R vir ),is an increasing function of the virial mass of the halo.For example,in the case of spect2the inner slope varies from 1.43for M =1012h −1M ⊙to 1.65for M =8×1014h −1M ⊙.Additionally,outer slope -at r =R vir -is a decreasing function of the virial mass and it varies from 3.67to 2.64for the above range of masses.Although density profiles resulting in simulations seem to be similar,systematic trends that relate them with the power spectrum have been reported.For example,Subra-manian,Cen &Ostriker (2000)found in the results of their N-body simulations the following:for power spectra of the form P (k )∝k n the density profiles have steeper cores for larger n .Therefore,a dependence of the density profile on the power spectrum is expected.This dependence is shown in our results comparing the profiles of haloes with the samec2003RAS,MNRAS 000,1–96Nicos Hiotelis200400600800log (M vir /M unit )11.5λFigure 5.Exponent λas a function of present-day virial mass for both power spectra.It is shown that λis an increasing function of the virial mass.Solid curve:spect1.Dashed curve:spect2present day mass.It should be noted that the method stud-ied in this manuscript is applicable for the era of slow accre-tion when the infalling matter is in the form of small haloes that have mass less than ∆m times the mass of the parent halo.This kind of accretion occurs at the late stages of for-mation and thus determines the profile of the outer regions of the halo under study.However,the values of the inner slopes may be questionable.Real haloes have followed dif-ferent mass growth histories and thus their properties show a significant scatter about a mean value.Unfortunately,the method studied in this manuscript results one profile for a halo of given mass.Thus,its purpose is just to approximate the mean density profile of a large number of mass growth histories.Since the mass growth history resulting from the method is in good agreement with the mean growth his-tory resulting from N-body simulation -as it will be shown below-then the values of the inner slopes could be close to the ones of N-body simulations.A Monte Carlo analytic ap-proach based on the construction of a large number of mass accretion histories is under study.This study could answer to some of the above problems.In Fig.5the exponent λis plotted,that gives the asymptotic slope at R →0,derived by the general fit as a function of present-day virial mass for both power spec-tra.It is shown that the exponent λis an increasing function of virial mass.This trend of the inner density profile is also found in the results of recent N-body simulations (Ricotti (2002)).3.2Time evolutionIn Fig.6we plot mass growth curves.The curves show M vir (a )as a function of a in a logarithmic slope.The solidlines show our resulting structures and the dotted lines show the mass growth curves of the model proposed by Wech-sler et al.(2002).The curves of the left panel correspond to spect1while those of the right panel to spect2.From the top to the bottom,the curves correspond to masses 2×1011h −1M ⊙,1012h −1M ⊙,1013h −1M ⊙,1014h −1M ⊙and 8×1014h −1M ⊙respectively.It is obvious that massive haloes show substantial increase of their mass up to late times while the growth curves of less massive haloes tend to flatten out earlier.This behaviour of mass growth curves characterizes the hierarchical clustering scenario where small haloes are formed earlier than more massive ones.Addition-ally,it helps to define the term ”formation time”by a mea-surable way.Wechsler et al.(2002)define as formation scale factor ˜a c the scale factor when the logarithmic slope of mass growth,(d lnM (a )/d lna ),falls below some specified value,S .They use the value S =2.It should be noted that this def-inition of formation scale factor differs from a c ,defined by BKPD,since a c is the value of the scale factor at the epoch the typical collapsing mass is F times the virial mass of the halo.We found that the values of ˜a c and a c for F =0.01and S =2are different.This is also noticed in Wechsler et al.(2002)since they state that ˜a c and a c have similar values for S =2but for F =0.015.However,the use of the value F =0.015in the toy-model of BKPD changes the resulting concentrations and so our basic criterion for the choice of the threshold ∆m is not satisfied.Therefore,it is preferable to choose a different value of S for the definition of ˜a c ,that of S =1.5.In Fig.6,the dotted lines show the mass growth curves of the model proposed by Wechsler et al.(2002).In this model the mass growth is calculated using the relation:M vir (a )=M vir,0exp[−˜a c S (1/a −1)](32)where M vir,0is the present-day virial mass and the formation scale factor ˜a c is defined by the condition d lnM (a )/d lna =S with S =1.5.In Fig.6,a very satis-factory agreement is shown,particularly for the less massive haloes.We have to note that our model haloes grow inside-out.Therefore,in early enough times -when the slope of the density is smaller that 2all the way from the centre up to the current radius-it is meaningless to define c vir .Once the building of the halo has proceeded beyond the point with slope 2,the evolution of c vir is due to the growth of the virial radius and is given by c vir [M (a ),a ]=c vir (M 0)R vir [M (a )]Extended Press-Schechter theory and the density profiles of dark matter haloes 7-1-0.8-0.6-0.4-0.20log(a)-1-0.8-0.6-0.4-0.2l o g (M v i r /M 0)-1-0.8-0.6-0.4-0.20log(a)Figure 6.Mass growth curves as a function of scale factor a .Left panel,right panel:spect1,spect2respectively.From top to the bottom the curves correspond to masses 0.2×1011h −1M ⊙,1012h −1M ⊙,1013h −1M ⊙,1014h −1M ⊙and 8×1014h −1M ⊙respectively.0.50.60.70.80.91a024*********.50.60.70.80.91a2468101214161820c [M (a )]Figure 7.Concentration as a function of the scale factor a .Left panel:spect1.Right panel:spect2.Solid lines:our results.Dotted lines:the results of toy-model of BKPD.From top to the bottom the lines correspond to masses 0.2×1011h −1M ⊙,1012h −1M ⊙,1013h −1M ⊙,1014h −1M ⊙and 8×1014h −1M ⊙respectively.struction of analytical models requires a number of crucial assumptions.The model studied in this paper was proposed by MRSSS and assumes that(i)The rate of mass accretion is defined by the rate ofminor mergers(ii)Haloes grow inside-out.The accreted mass is de-posited at the outer shells without changing the density pro-file of the halo inside its current virial radiusc2003RAS,MNRAS 000,1–98Nicos HiotelisThefirst assumption indicates that structures presented in this paper formed by a gentle accretion of mass.The phys-ical process implied by the second assumption is that the infalling matter does not penetrate the current virial radius. This process requires an amount of non-radial motion.This amount has to be large enough so that the pericenter of the accreted mass is larger than the current virial radius.It should be noted that a density profile that results from a radial collapse has inner slope steeper than2.It is the pres-ence of non-radial motion during the collapse that leads to inner slopes shallower than2.(e.g.Nusser(2001),Hiotelis (2002),Subramanian,Cen&Ostriker(2000)).Non-radial motions are always present in the structures formed in N-body simulations.Despite the above assumptions,the results of the model studied in this paper are in good agreement with the results of N-body simulations.The summary of these results is as follows:(i)Density profiles of haloes are close to the analytical models proposed in the literature as goodfits to the results of N-body simulations.A trend of the inner slope of the density profile as an increasing function of the mass of the halo is also found,in agreement with recent results of N-body simulations.(ii)Concentration is a decreasing function of virial mass. Its values are in agreement with the results of numerical methods.(iii)Massive haloes increase their mass substantially up to late times.Growth curves of less massive haloes tend to flatten out earlier.The concentrations of less massive haloes evolve more rapidly while those of more massive haloes evolve slowly.Taking into account the number of assumptions and approx-imations used to build the model presented in this paper,we can conclude that the agreement with the results of N-body simulations is very good.Consequently,this model provides a very promising method to deal with the process of struc-ture formation.Further improvements to this model could help to understand better the physical picture during this process.5ACKNOWLEDGEMENTSI would like to thank the Empirikion Foundation for itsfi-nancial support.REFERENCESAvila-Reese V.,Firmani C.,Hern´a dez X.,1998,ApJ,505, 37Bond J.R.,Cole S.,Efstathiou G.,Kaiser N.,1991,ApJ, 379,440Bond J.R.,Myers S.,1996,ApJS,103,41Bower R.J.,1991,MNRAS,248,332Bryan G.,Norman M.,1998,ApJ,495,80.Bullock J.S.,Kolatt A.,Primack J.R.,Dekel A.,2001,MN-RAS,321,559(BKPD)Cohn J.D.,Bagla J.S.,White M.,2001MNRAS,325,1053 Cole S.,Lacey C.,1996,MNRAS,281,716Crone M.M.,Evrard A.E.,Richstone D.O.,1994,ApJ,434, 402Dubinski J.,Calberg R.,1991,ApJ,378,496.Efstathiou G.,Frenk C.S.,White S.D.M.,1985,ApJ,292, 371Efstathiou G.,Rees M.,1988,MNRAS,230,5 Efstathiou G.,Bond J.R.,White S.D.M.,1992,MNRAS, 258,1Eke V.R,Cole S.,Frenk C.S,1996,MNRAS,282,263 Frenk C.S,White S.D.M.,Davis M.,Efstathiou G.,1988, ApJ,327,507Fukushige T.,Makino J.,1997,ApJ,477,L9Gelb J.,Bertschinger E.,1994,ApJ,436,467Henriksen R.N.,Widrow L.M.,1999,MNRAS,302,321 Hernquist L.,1990,ApJ,356,359Hiotelis N.,2002,A&A,382,84Huss A.,Jain B.,Steinmetz M.,1999,MNRAS,308,1011 Jenkins A.,Frenk C.S.,White S.D.M.,Colberg J.M.,Cole S.,Evrard A.E.,Couchman H.M.P.,Yoshida N.,2001,MN-RAS,321,372Jing Y.P.,Suto Y.,2000,ApJ,529,L69(JS)Kitayama T.,Suto Y.,1996,MNRAS,280,638Kravtsov A.V.,Klypin A.A.,Bullock J.S.,Primack J.R., 1998,ApJ,502,48Klypin A.A.,Kravtsov A.V.,Bullock J.S.,Primack J.R., 2001,ApJ,554,903Kull A.,1999,ApJ,516,L5Lacey C.,Cole S.,1993,MNRAS,262,627Lacey C.,Cole S.,1994,MNRAS,271,676Lahav O.,Lilje P.B.,Primack J.R.,Rees M.J.,1991,MN-RAS,251,128Lokas E.L.,2000,MNRAS,311,423Manrique A.,Salvador-Sol´e E.,1996,ApJ,467,504 Manrique A.,Raig A.,Salvador-Sol´e E.,Sanchis T.,Solanes J.M.,2003,preprint(astro-ph/0304378)(MRSSS)Moore B.,Governato F.,Quinn T.,Stadel ke G.,1998, ApJ,499,L5(MGQSL)Navarro J.F.,Frenk C.S.White S.D.M.,1997,ApJ,490, 493(NFW)Nusser A.,Sheth R.,1999,MNRAS,303,685Nusser A.,2001,MNRAS,325,1397Peebles P.J.E.,1980,The Large-Scale Structure of the Uni-verse,Princeton Univ.Press,Princeton,NJPercival W.J.,Miller L.,Peacock J.A.,2000,MNRAS,318, 273Press W.H.,Schechter P.,1974,ApJ,187,425.Quinn P.J.,Salmon J.K.,Zurek W.H.,1986,Nature,322, 329Raig A.,Gonz´a lez-Casado G.,Salvador-Sol´e,1998,ApJ, 508,L129Ricotti M.,2002,preprint(astro-ph/0212146)Salvador-Sol´e E.,Solanes J.M.,Manrique A.,1998,ApJ, 499,542Sheth R.K.,Tormen G.,1999,MNRAS,308,119Sheth R.K.,Mo H.J.,Tormen G.,2001,MNRAS,323,1 Smith C.C.,Klypin A.,Gross M.A.K.,Primack J.R., Holtzman J.,1998,MNRAS,297,910Subramanian K.,Cen R.Y.,Ostriker J.P.,2000,ApJ,538, 528Syer D.,White S.D.M.,1988,MNRAS,293,337 Wechsler R.H.,Bullock J.S.,Primack J.R.,Kratsov A.V., Dekel A.,2002,ApJ,568,52c 2003RAS,MNRAS000,1–9Extended Press-Schechter theory and the density profiles of dark matter haloes9 White S.D.M.,Efstathiou G.,Frenk C.,1993,MNRAS,262,1023This paper has been typeset from a T E X/L A T E Xfile preparedby the author.c 2003RAS,MNRAS000,1–9。

a r X i v :a s t r o -p h /0504557v 2 25 J u l 2005D RAFT VERSION F EBRUARY 2,2008Preprint typeset using L A T E X style emulateapj v.6/22/04IMPACT OF DARK MATTER SUBSTRUCTURE ON THE MATTER AND WEAK LENSING POWER SPECTRAB RADLEY H AGAN 1,C HUNG -P EI M A 2,A NDREY V.K RAVTSOV 3Draft version February 2,2008ABSTRACTWe explore the effect of substructure in dark matter halos on the power spectrum and bispectrum of matter fluctuations and weak lensing shear.By experimenting with substructure in a cosmological N =5123simu-lation,we find that when a larger fraction of the host halo mass is in subhalos,the resulting power spectrum has less power at 1 k 100h Mpc −1and more power at k 100h Mpc −1.We explain this effect using an analytic halo model including subhalos,which shows that the 1 k 100h Mpc −1regime depends sensitively on the radial distribution of subhalo centers while the interior structure of subhalos is important at k 100h Mpc −1.The corresponding effect due to substructures on the weak lensing power spectrum is up to ∼11%at angular scale l 104.Predicting the nonlinear power spectrum to a few percent accuracy for future surveys would therefore require large cosmological simulations that also have exquisite numerical resolution to model accurately the survivals of dark matter subhalos in the tidal fields of their hosts.Subject headings:cosmology:theory —dark matter —large-scale structure of the universe1.INTRODUCTIONOne phenomenon to emerge from N -body simulations of increasingly higher resolution is the existence of substructure (or subhalos)in dark matter halos (e.g.,Tormen et al.1998;Klypin et al.1999b;Moore et al.1999;Ghigna et al.2000).These small subhalos,relics of hierarchical structure forma-tion,have accreted onto larger host halos and survived tidal forces.Depending on their mass,density structure,orbit,and accretion time,the subhalos with high central densities can avoid complete tidal destruction although many lose a large fraction of their initial mass.These small and dense dark matter substructures,however,are prone to numerical arti-facts and can be disrupted due to insufficient force and mass resolution.Disentangling these numerical effects from the actual subhalo dynamics is an essential step towards under-standing the composition and formation of structure.Quan-tifying the effects due to dark matter substructure is also im-portant for interpreting weak lensing surveys,which are sen-sitive to the clustering statistics of the overall density field.The level of precision for which surveys such as SNAP are striving (Massey et al.2004)suggests that theoretical predic-tions for the weak lensing convergence power spectrum need to be accurate to within a few percent over a wide of range of scales (e.g.Huterer &Takada 2005).At this level,subhalos may contribute significantly to the nonlinear power spectrum because they typically constitute about 10%of the host mass.In the sections to follow,we examine the effects of sub-structure on the matter and weak lensing power spectra with two methods.In §2we use the result of a high resolution N -body simulation and quantify the changes in the power spec-tra when we smooth out increasing amounts of substructures.Our other approach,detailed in §3,is to incorporate substruc-ture into the analytic halo model.The results are dependent on the parameters used in the model,but they provide useful physical insight into the results from N -body simulations.We summarize and discuss the results in §4.1Department of Physics,University of California,Berkeley,CA 947202Department of Astronomy,University of California,Berkeley,CA 947203Dept.of Astronomy and Astrophysics,Enrico Fermi Institute,Kavli Institute for Cosmological Physics,The University of Chicago,Chicago,IL 606372.SUBSTRUCTURE IN SIMULATIONSWe use the outputs of a cosmological dark-matter-only simulation that contains a significant amount of substruc-ture.This simulation is a concordance,flat ΛCDM model:Ωm =1−ΩΛ=0.3,h =0.7and σ8=0.9.The box size is 120h −1Mpc,the number of particles is 5123,and the particle mass is 1.07×109h −1M ⊙.The simulation uses the Adaptive Refinement Tree N -body code (ART;Kravtsov et al.1997;Kravtsov 1999)to achieve high force resolution in dense re-gions.In this particular run the volume is initially resolved with a 10243grid,and the smallest grid cell found at the end of the simulation is 1.8h −1kpc.The actual resolution is about twice this value (Kravtsov et al.1997).More details about the simulation can be found in Tasitsiomi et al.(2004).To quantify the effects of subhalos on the matter and weak lensing power spectra,we first identify the simulation parti-cles that comprise subhalos within each halo.This is achieved using a version of the Bound Density Maxima algorithm (Klypin et al.1999a),which identifies all local density peaks and therefore finds both halos and subhalos.It identifies the particles that make up each of the peaks and removes those not bound to the corresponding halo.As a controlled exper-iment,we then smooth out the subhalos within the virial ra-dius of each host halo by redistributing these subhalo parti-cles back in the smooth component of the host halo according to a spherically-symmetric NFW profile (Navarro,Frenk,&White 1996).For the concentration parameter c of the profile,we do not use the fitting formulae (e.g.,Bullock et al.2001;Dolag et al.2004)but instead fit each host halo individually to take into account the significant halo-to-halo scatter in c .We therefore smooth over the subhalos and increase the normal-ization but not the shape of the spherically averaged profile of the smooth component to accommodate the mass from the subhalo component.This smoothing procedure also serves as a simple model for the effects of resolution on the abundance of subhalos in sim-ulations,in which the lack of sufficient resolution will cause an incoming small halo to be disrupted quickly and lose most of its particles over its short-lived orbit.We quantify this ef-fect by experimenting with different cut-offs on the subhalo mass:subhalos with masses below the cut-off are removed2F IG . 1.—Effects of dark matter substructure on the matter fluctuationpower spectrum (top)and the equilateral bispectrum (bottom)of a N =5123cosmological simulation.Plotted is the ratio of the original spectrum to that with a subset of the substructures smoothed out.The curves (from top down)correspond to increasing subhalo mass cut-offs,below which the mass in the subhalos is redistributed smoothly back into the host halo.The bottom panel is plotted to a lower k because B (k )becomes too noisy.The effects due to substructures are up to ∼12%in ∆2and 24%in B .and have their particles spread over the host halo;subhalos with masses above the cut-off are left alone.Increasing mass cut-offs should roughly mimic increasingly lower resolution simulations because only higher mass subhalos will stay in-tact in the halo environment.We then calculate the matter fluctuation power spectrum P (k ),the Fourier transform of the 2-point correlation function,and the matter bispectrum B (k 1,k 2,k 3),the Fourier transform of the 3-point correlation function.In order to compute P and B at large k without using an enormous amount of mem-ory,we subdivide the simulation cube into smaller cubes and stack these on top of each other (called "chaining the power"in Smith et al.2003).A typical stacking level used is 8,mean-ing that we subdivide the box into 83cubes and stack these.We use stacked spectra for the high-k regime and unstacked spectra for low-k .Finally,we subtract shot noise (∝1/N )from the outputted spectra to eliminate discreteness effects.Fig.1shows the effects of substructure on the dimension-less power spectrum ∆2(k )≡k 3P (k )/(2π2)and the equilat-eral bispectrum B (k 1)(k 1=k 2=k 3).Plotted is the ratio of the spectrum from the raw ART output divided by the spec-trum from the altered data.The altered data have no subha-los with masses below the labeled mass cutoff.The mass in the removed subhalos has been redistributed smoothly into thehost halo as described above.The deviation from the original spectrum becomes larger as the cutoff is increased because more subhalos have been smoothed out.For a given cutoff,the figure shows that a simulation with dark matter substruc-tures (such as the raw ART output)has more power at k >100h Mpc −1and less power at 1 k 100h Mpc −1than a simu-lation with smoother halos.We believe these opposite behav-iors reflect the two competing factors present in our numeri-cal experiments:removal of mass within subhalos,which af-fects scales comparable to or below subhalo radii (and hence k 100h Mpc −1),and addition of this mass back into the smooth component of the halo,which affects the larger scales of 1 k 100h Mpc −1.The ratio approaches unity for k 1h Mpc −1simply because the mass distribution on scales above individual host halos is unaltered.We will examine these ef-fects further in the context of the halo model in §3.For a given curve in Fig.1,we have also calculated the contributions from subhalos in host halos of varying masses to quantify the relative importance of cluster versus galactic host halos.For the 1012.5h −1M ⊙curve,e.g.,we find that smooth-ing over the subhalos in host halos above 1014and 1013M ⊙account for 5%and 10%in the total 12%dip seen in Fig.1,respectively.For the 1011.5h −1M ⊙cutoff,the numbers are 2%and 5%of the total 6%dip.The halos found in N -body simulations are generally triax-ial.When we redistribute the subhalo particles,however,we assume for simplicity a spherical distribution.This assump-tion makes the altered halos slightly rounder.One can esti-mate how this effect changes the power spectrum by using the halo model without substructure.Smith &Watts (2005)in-corporated a distribution of halo shapes found by Jing &Suto (2002)from cosmological simulations into the halo model (ig-noring the substructure contribution).Compared with the case where all the halos are spherical,they observed a peak decre-ment in the power spectrum of about 4%for k ≈1Mpc −1.The corresponding effect in our calculations would be much smaller since we redistribute only the subset of particles that belong to subhalos into the rounder shape (e.g.,about 10%of all particles in the case where the cutoff was 1012.5h −1M ⊙).Thus,by extending the results of Smith &Watts,we expect the spurious rounder halos in our study to account for less than 0.5%of the total 12%drop.Fig.2shows the weak lensing convergence power spectrum ∆2κ(l )corresponding to the matter power spectrum ∆2(k )in Fig.1.It is calculated from ∆2(k )using Limber’s approxima-tion and assumption of a flat universe:∆2κ(ℓ)=9πc 4 χmaxχ3d χW 2(χ)3F IG.2.—Effects of substructure on the weak lensing convergence powerspectrum for the same subhalo mass cutoffs as in Fig.1.The sources areassumed to be at redshift1.The gray band is the1-σstatistical error assumingGaussianfields.We take f sky=0.25,γrms=0.2,¯n=100/arcmin2,and a bandof widthℓ/10.surements assuming Gaussian densityfields(Kaiser1998):σ(∆2κ)22π¯n∆2κ ,(2)where f sky is the fraction of the sky surveyed,γrms is the rmsellipticity of galaxies,and¯n is the number density of galaxieson the sky.The error is dominated by the sample varianceon large scales(first term in eq.[2])and the"shape noise"onsmall scales.Our assumption of Gaussianity is not applicablefor the angular scales shown in the plot because the scalesplotted are near or below the size of individual halos,but theerrors shown should be a useful reference and have been usedin previous studies.A reliable estimate of the error wouldpresumably require a ray tracing calculation which is beyondthe scope of this paper.3.SUBSTRUCTURE IN THE HALO MODELTo gain a deeper understanding of the simulation results inFigs.1and2,we use the semi-analytic halo model to buildup the nonlinear power spectrum from different kinds of pairsof mass elements that may occur in halos(e.g.Ma&Fry2000;Peacock&Smith2000;Seljak2000;Scoccimarro etal2001).The original halo model assumes that all mass re-sides in virialized,spherical halos without substructures.Onecan then build the matter power spectrum from the differentkinds of pairs of particles that contribute to the2-point clus-tering statistics by writing P(k)=P1h(k)+P2h(k),where the1-halo term P1h contains contributions from particle pairs whereboth particles reside in the same halo,and the2-halo term P2his from pairs where the two particles reside in different halos.The1-halo term is a mass-weighted average of single haloprofiles and dominates on the scales of interest(k 1h Mpc−1)in Fig.1because close pairs of particles are more likely to befound in the same halo.The2-halo term is closely related tothe linear power spectrum and is important only at large sepa-ration(i.e.small k)where a pair of particles is more likely tobe found in two distinct halos.Similarly,the bispectrum canbe constructed from the different classes of triplets of particles(see,e.g.,Ma&Fry2000).The original halo model can be readily extended to take intoaccount a clumpy subhalo component in an otherwise smoothhost halo.Sheth&Jain(2003),e.g.,decompose the original1-halo term into P1h=P ss+P sc+P1c+P2c,where"s"denotessmooth and"c"denotes clump.The smooth-smooth term,P ss,arises from pairs of particles that both belong to the smoothcomponent of the same host halo.This term is identical to theoriginal1-halo term except for an overall decrease in ampli-tude by the factor(1−f)2,where f is the fraction of the totalhalo mass that resides in subhalos.The smooth-clump term,P sc,is due to having one particle in a subhalo(clump)and theother in the host halo(smooth).The1-and2-clump terms,P1cand P2c,come from having both particles in the same subhaloand in two different subhalos,respectively.Explicitly,P ss(k)=(1−f)2¯ρ2 dM N(M)MU(k,M)U c(k,M)× dm n(m,M)mu(k,m)(4)P1c(k)=1¯ρ2 dMN(M)U2c(k,M)× dmn(m,M)mu(k,m) 2(6)where U(k,M),u(k,m),and U c(k,M)are the Fourier trans-forms of the host halo radial density profile,the subhalo radialdensity profile,and the radial distribution of subhalo centers,respectively.N(M)dM gives the number density of host ha-los with mass M,and n(m,M)dm gives the number density ofsubhalos of mass m inside a host halo of mass M.A similarexpression can be written down for the2-halo term P2h,whichis also included in our calculations.Fig.3illustrates the contributions from the individual termsin the halo model.We use the NFW profile(truncated atthe virial radius)for the input host halo U(k,M)and subhalou(k,m),and the concentration c(M)=c0(M/1014M⊙)−0.1withc0=11that wefind to approximate the ART host halos andis identical to Dolag et al(2004)except for a15%increasein amplitude.We use c sub=3for the subhalos to take intoaccount tidal stripping but also compare different values inFig.4below.For the distribution of subhalo centers,U c(k),we compare the profile of NFW with that of Gao et al.(2004),whofind the number of subhalos within a host halo’s virial ra-dius r v to beN(<x)1+acxα,x=r/r v(7)where a=0.244,α=2,β=2.75,c=r v/r s,and N tot is the to-tal number of subhalos in the host.Since this distribution atsmall r is shallower(∝r−0.25)than the inner part of the NFWprofile(∝r−1),its Fourier transform U c(k)at high k is abouta factor of10lower than that of the NFW profile.This decre-ment results in a much lower∆sc and∆2c as shown in Fig.3(dashed vs dotted curves).Wefind the subhalo centers in theART simulation to follow approximately the distribution ofGao et al.although there is a large scatter.4F IG . 3.—Comparison of individual subhalo terms in the halo model.The smooth-clump ∆2sc and 2-clump ∆22c terms depend on the distributionof subhalo centers U c within a host halo,having a much lower amplitude at k >1h Mpc −1when U c has the cored isothermal profile of Gao et al.(2004)compared with the cuspy NFW profile.The smooth-smooth ∆2ss and 1-clump ∆21c terms are independent of U c .See text for parameters used in the model.We use the mass function of Sheth &Tormen (1999)for the host halos N (M )and a power law n (m ,M )∝m −1.9that well approximates the subhalo mass function in the ART sim-ulation.The latter is normalized so that the total mass of subhalos in a host halo adds up to f times the host mass M (f =0.14in Fig.3).To compare the halo model with simula-tions,we set the lower limit on the P ss integral to 1010h −1M ⊙,which is the smallest halo present in the simulation (about ten times the simulation particle mass).The lower limit on the outer integrals of P sc ,P 1c ,and P 2c corresponds to the smallest halo that contains substructure,which we set to the small-est halo that we considered for erasing substructure in §2:2×1012h −1M ⊙.Similarly,the lower limit on the inner inte-grals of these terms is set to the smallest subhalo that can be resolved (i.e.1010h −1M ⊙).Fig.4compares the sum of all the terms in thehalo model with the simulation result from Fig.1.As in Fig.1,we il-lustrate the effects due to substructures by dividing out the power spectrum from the original (smooth)halo model,i.e.,∆2smooth =k 3P ss /[(1−f )2(2π2)],where P ss is given before in eq.(4).Fig.4shows that the halo model is able to reproduce qualitatively the simulation results when the subhalo centers in the halo model are assigned the shallower distribution ofGao et al.The feature of ∆2sub /∆2smooth <1at 1 k 100h Mpc −1is mainly caused by the drop in the smooth-clump term relative to the smooth-smooth term at k 1h Mpc −1shown inFig.3.The ratio ∆2sub /∆2smooth becomes >1only at k 100h Mpc −1when the 1-clump term finally takes over.Fitting the halo model to actual simulation results is clearly not exact in part due to the large scatters in the properties of simulated ha-los,e.g.,the concentration (for both hosts and subhalos),the subhalo mass fraction f ,and the maximum subhalo mass in each host halo.The halo model allows us to study the depen-dence of clustering statistics on these parameters (see Fig.4).In addition,a number of effects are neglected in the current halo model,e.g.,tidal effects are likely to reduce the number of subhalos (modeled by U c of Gao et al.here)as well as theirF IG . 4.—Same ratio of ∆2as in Fig.1but comparing simulation(symbols;same 1012.5h −1M ⊙curve in Fig.1)with halo model predictions (plain curves).The two agree qualitatively when the shallower distribu-tion of Gao et al.for subhalo centers U c (k )is used (bottom 3curves)in the halo model (but not for an NFW U c (k );dotted).The detailed model prediction depends on halo parameters:the solid curve uses the same pa-rameters as in Fig.3;the dashed shows how a larger subhalo concentration(c sub =c sub 0(M /1014M ⊙)−0.1;c sub 0=11vs 3)steepens the curve at high k ;the dash-dotted shows how a smaller subhalo mass fraction f (0.1vs.0.14)raises the dip.outer radii (not modeled here)towards host halo centers;a larger amount of stripped subhalo mass may also be deposited to the inner parts of the hosts,resulting in a radius-dependent subhalo mass fraction f within the host.Fig.4also shows that the halo model predicts the oppo-site effect due to substructure (i.e.∆2sub /∆2smooth >1at all k )if the subhalo centers U c (k )are assumed to follow the NFW distribution like the underlying dark matter.The sign of this effect is consistent with the previous subhalo model study of Dolney et al.(2004),which assumed the same NFW profile for the subhalo centers and the hosts and obtained a matter power spectrum that had a higher amplitude for all k when the substructure terms were included.Their results differ slightly from ours because of different integration limits.Subhalos in recent simulations like that of Gao et al.,however,show a much shallower radial distribution in the central regions of the host halos,and inclusion of gas dynamics appears to have little effect on the survivability of subhalos (Nagai &Kravtsov 2005).The shallow distribution is apparently due to tidal disruptions,even though the precise shape of the distri-bution is still a matter of debate (e.g.,Zentner et al.2005).We have also experimented with a third distribution U c (k )that has the NFW form but is less concentrated.We are able tobring ∆2sub /∆2smooth below unity only when the concentration is reduced by a factor of more than 2.5,and only when this reduction factor is increased to ∼100would we get a com-parable dip as the curves for ART simulation and Gao et al.in Fig.4.It is interesting to see if we can mimic the behav-ior of the ART simulation without using subhalos in the halo model.We try replacing the one-halo term,P 1h ,by one that is a simple superposition of a Gao et al.profile and the usual NFW profile.This accounts for the fact that ∼90%of the mass is in a smooth NFW profile and that ∼10%is in sub-halos,which follow a flatter profile.One would not expect the high-k regime to agree as it is dominated by the subha-5los(the1-clump term,specifically).The intermediate range, 1 k 100h Mpc−1,is dominated by the host halo itself,but wefind no similarity in this range either.Subhalos are there-fore needed if the halo model is to recreate the ART results.4.DISCUSSIONThe purpose of this work is to provide a physical under-standing of the effects of substructures on clustering statistics. By experimenting with dark matter substructures in a cosmo-logical simulation with5123particles,we have shown that the power spectra of matterfluctuations and weak lensing shear can change by up to∼12%(and up to∼24%in the bispec-trum)if a significant amount of substructures is not resolved in a simulation.When a larger mass fraction of the host halos is in the form of lumpy subhalos,wefind the effect is to lower the amplitude of the matter and weak lensing power spectra at the observationally relevant ranges of k∼1to100h Mpc−1 and l 105,and to raise the amplitude on smaller scales.A similar drop in power is also seen in our analytic halo-subhalo model when the subhalo centers U c within a host halo are dis-tributed with a shallower radial profile than the underlying dark matter(as expected due to tidal effects).A way to un-derstand the drop involves looking at where the dense regions are.When U c has an NFW form the subhalos basically trace the smooth background.Thus,there is never a decrease in power when the smooth-smooth and smooth-clump terms are added because dense regions are in nearly the same relative positions.When we use a shallower profile for U c,the sub-halos are not as numerous in the denser inner regions of the background halo.This decrease in the overlap between dense clump regions and the dense inner regions causes the drop in power.We have quantified the effects of substructures on clustering statistics by erasing substructures in an N=5123simulation. An important related question is whether N=5123,single-mass resolution simulations such as the one used in our study has sufficient resolution to measure the power spectrum to the few-percent accuracy required by future surveys.Note that at least hundreds of particles and force resolution of∼kilo-parsec are required to ensure subhalo survival against tidal forces,placing stringent requirements on the dynamic range of simulations.Multi-mass resolution simulations designed for subhalo studies,on the other hand,do not give reliable predictions for P(k)on quasi-linear scales due to compro-mised resolution outside highly clustered regions.The fact that the curves in Figs.1and2continue to change at the few-percent level each time the mass threshold is lowered by 0.5dex from1012.5to1010.5h−1M⊙suggests that subhalos of M 1010.5h−1M⊙may still be affecting the power spectra at a comparable level and that N>5123would be required.We alsofind>3%changes in P(k)at k∼10h Mpc−1in the halo model as the minimum subhalo mass in the integration limit of eq.(3)is lowered to107h−1M⊙(although the exact predic-tions are sensitive to the slope of the subhalo mass function, which is assumed to be−1.9here.)Careful convergence stud-ies with higher resolution aided by insight from this study and detailed semi-analytic models for halo substructure will likely be needed to determine N.There are other challenges to predicting accurately the weak lensing signal on single halo scales.The effect of neutrino clustering could cause a rise in weak lensing convergence of ∼1%atℓ∼2000(Abazajian et al.2005).Two recent groups have investigated different aspects of baryon effects.White (2004)found that baryonic contraction and its subsequent im-pact on the dark matter distribution is capable of causing an increase in the weak lensing convergence power of a few per-cent atℓ 3000.Zhan&Knox(2004),on the other hand,use the fact that the hot intracluster medium does not follow the dark matter precisely and predict an opposite effect:a sup-pression of weak lensing power of a few percent atℓ 1000. Unlike the effects of substructure and neutrino clustering,the baryon effects cause departures from the pure dark matter weak lensing signal that only get larger with increasingℓ. We thank M.Boylan-Kolchin,W.Hu,D.Huterer,C.Vale, and P.Schneider for useful discussions.BH is supported by an NSF Graduate Student Fellowship.CPM is supported in part by NSF grant AST0407351and NASA grant NAG5-12173.A VK is supported by NSF grants AST-0206216and 0239759,NASA grant NAG5-13274,and the Kavli Institute for Cosmological Physics at the University of Chicago.REFERENCESAbazajian,K.,Switzer,E.,R.,Dodelson,S.,Heitmann,K.,Habib,S.2005, Phys.Rev.D,71,043507Bartelmann M.&Schneider P.2001,Phys.Rep.,340,291Bullock,J.S.et al.2001,MNRAS,321,559Dolag,K.,Bartelmann,M.,Perrotta,F.,Baccigalupi,C.,Moscardini,L., Meneghetti,M.,Tormen,G.2004,A&A,416,853Dolney,D.,Jain,B.,&Takada,M.2004,MNRAS,352,1019Gao,L.,White,S.D.M.,Jenkins,A.,Stoehr,F.,Springel,V.2004,MNRAS, 355,819Ghigna,S.,Moore,B.,Governato,F.,Lake,G.,Quinn,T.,Stadel,J.2000, ApJ,544,616Huterer,D.,&Takada,M.2005,astro-ph/0412142Jing,Y.P.,Suto,Y.2002,574,538Kaiser,N.1998,ApJ,498,26Klypin,A.,Gottloeber,S.,Kravtsov,A.V.,Khokhlov,M.1999a,ApJ,516, 530Klypin,A.,Kravtsov,A.V.,Valenzuela,O.,&Prada,F.1999b,ApJ,522,82 Kravtsov,A.V.,Klypin,A.A.,&Khokhlov,A.M.1997,ApJS,111,73 Kravtsov,A.V.&Klypin,A.A.1999,ApJ,520,437Ma,C.-P.&Fry,J.2000,ApJ,543,503Massey,R.et al.2004,AJ,127,3089Moore,B.et al.1999,ApJ,524,L19Nagai,D.,Kravtsov,A.V.2005,ApJ,618,557Navarro,J.,Frenk,C.,White,S.D.M.1996,ApJ,462,563Peacock,J.,A.,Smith,R.,E.2000,MNRAS,318,1144Scoccimarro,R.,Sheth,R.,Hui,L.,Jain,B.2001,ApJ,546,20Seljak,U.2000,MNRAS,318,203Sheth,R.&Jain,B.2003,MNRAS,345,529Sheth,R.&Tormen,G.1999,MNRAS,308,119Smith,R.E.et al.2003,MNRAS,341,1311Smith,R.E.,Watts,P.I.R.2005,MNRAS,360,203Tasitsiomi,A.,Kravtsov,A.V.,Wechsler,R.H.,Primack,J.R.2004,ApJ, 614,533Tormen,G.,Diaferio,A.,Syer,D.1998,MNRAS,299,728White,M.2004,Astroparticle Phys.,22,211Zentner,A.et al.2005,ApJ,in press,astro-ph/0411586Zhan,H.&Knox,L.2004,ApJ,616,L75。

暗晕的并合董符煜;汪洋【期刊名称】《天文学进展》【年(卷),期】2015(033)004【摘要】The article makes a basic introduction to the halo formation theory. The mass growth of dark matter halos includes two componenets: halo-halo merger and continuous accretion.Halo-halo merger can be associated with galaxy-galaxy merger, so a proper defi-nition to it is important. However there're a lot of influencing factors preventing us from geeting a reliable fitting form.The difficulties mainly comes from definitions: the definitions of halo-halo merger, the definition of constructting a merger tree, the definition of the halo mass and even the time step. We make some comparisons between different definitions and obtain the halo merger rates corresponding to them. In addition, we find the environment of haloes can also lead to different behaviors in merger rate. There fitting to the halo-halo merger rate with simulations can be carried out two folded: fitting to different redshift or fitting to the halo mergering histories. Usurally, the latter gives a universal fitting formula to haloes with different mass.%系统介绍暗晕-暗晕并合在理论方面的形成与发展,并简要介绍了其在数值模拟检验(冷暗物质模型)及拟合方面的进展.暗晕-暗晕并合率问题可与星系-星系并合率联系,正确地定义暗晕-暗晕并合率很重要,影响并合率的因素却很多,从数值模拟所得到的拟合公式往往存在较大差异,这些差异主要来自于定义:对并合的定义(并合树的建立),对暗晕质量的定义以及并合暗晕前身质量比的定义.数值模拟输出的时间步长也会对此有所影响,此外暗晕并合率也与环境强烈相关.对这些因素所导致的差异做了简单比较.此外,对于暗晕并合率的统计可分为对不同红移处的拟合和对z=0时刻对其并合历史的拟合,其中后者对不同质量的暗晕往往能给出一个较统一的拟合公式.【总页数】21页(P432-452)【作者】董符煜;汪洋【作者单位】中国科学院上海天文台星系与宇宙学重点实验室,上海 200030;中国科学院上海天文台星系与宇宙学重点实验室,上海 200030【正文语种】中文【中图分类】P159【相关文献】1.暗晕次结构的演化 [J], 甘建铃2.用强引力透镜研究暗晕中心的物质分布 [J], 王琳;陈大明3.暗晕子结构的数值搜寻与统计性质 [J], 李昭洲;韩家信4.宇宙速度场两点相关的暗晕模型描述 [J], 曹亮5.滴线原子核中的晕现象——中子晕、超子晕、巨晕和形变晕 [J], 孟杰;吕洪凤;张双全;周善贵因版权原因，仅展示原文概要，查看原文内容请购买。

a r Xi v:as tr o-p h /16380v121J un21SHAPES OF DARK MATTER HALOS J.S.BULLOCK Department of Astronomy,The Ohio State University ,140W.18th Ave,Columbus,OH 43210E-maIl:james@ I present an analysis of the density shapes of dark matter halos in ΛCDM and ΛWDM cosmologies.The main results are derived from a statistical sample of galaxy-mass halos drawn from a high resolution ΛCDM N-body simulation.Halo shapes show significant trends with mass and redshift:low-mass halos are rounder than high mass halos,and,for a fixed mass,halos are rounder at low z .Contrary to previous expectations,which were based on cluster-mass halos and non-COBE normalized simulations,ΛCDM galaxy-mass halos at z =0are not strongly flat-tened,with short to long axis ratios of s =0.70±0.17.I go on to study how the shapes of individual halos change when going from a ΛCDM simulation to a simulation with a warm dark matter power spectrum (ΛWDM).Four halos were compared,and,on average,the WDM halos are more spherical than their CDM counterparts (s ≃0.77compared to s ≃0.71).A larger sample of objects will be needed to test whether the trend is significant.1Introduction A variety of observational indicators suggest that galaxies are embedded within massive,extended dark matter halos,lending support to the idea that a hi-erarchical,cold dark matter (CDM)based cosmology may provide a real de-scription of the universe,especially on large scales (see,e.g.,the review by Primack 11and references therein).A useful small-scale test of CDM and variant theories may come from observations aimed at inferring the density structure of dark halos.Specifically,the quest to measure dark halo shapes has developed into a rich and complex subfield of its own.4,12Predictions for shapes of dark matter halos formed by dissipationless gravi-tational collapse are most reliably derived using numerical studies.6,5,14Thesepast investigations focused on galaxy-sized halos formed from power-law or pre-COBE CDM power spectra,and found that halos were typically flattened triaxial structures,with short-to-long axis ratios of s ≃0.5±0.15.Examina-tions based on currently favored cosmologies have been done,but they studied only cluster mass halos,8,13,9and also found significantly flattened objects,s ∼0.4−0.5.Interestingly,Thomas and collaborators 13saw an indication that low matter density models (with early structure formation)tended to produce more spherical clusters than high density models.This result was qualitatively consistent with indications from Warren et al.14that high-mass halos are more flattened than (early-forming)low-mass halos.In light of these1hints that formation history plays a role in setting halo shapes,it is impor-tant to reexamine the question for a currently standard cosmological model. Specifically,this work aims at characterizing haloflattening as a function of mass and redshift for the popularflatΛCDM cosmology.I also explore how shapes of halos are affected when going to aΛWDM model,in which the power spectrum is damped on small scales.2Simulations and shape measurementThe simulations were performed using the Adaptive Refinement Tree(ART) code.7The main results are derived from aΛCDM simulation withΩm= 0.3,ΩΛ=0.7,h=0.7,andσ8=1.0,which followed2563particles of mass1.1×109h−1M⊙within a periodic box of comoving length60h−1Mpc,obtaining a formal force resolution of1.8h−1kpc.A second simulation was run until z=1.8with the same particle number,half the box size,and thus eight times the mass resolution;it was used to check resolution issues.Dark halos were identified using a(spherical)bound density maxima method,2and masses were determined by counting particles within the spherical virial radius R v, inside which the mean overdensity has dropped to a value∆v≃(18π2+82p−39p2)/(1+p),where p≡Ωm(z)−1.This sample contains∼800halos that span the mass range3×1011−5×1014.A second pair of simulations1were used to explore how shapes of halos are affected by damping the power spectrum.We compare four halos,each simulated from the same initial conditions,for aΛCDM andΛWDM model with the multiple-mass ART code and the same cosmological parameters and effective mass per particle discussed above.Thefilter mass scale for theΛWDM run was1.7×1014h−1M⊙,and the four halos we compare have masses(2−8)×1013h−1M⊙.a Simulation results were kindly supplied by P.Col´ın.Halo axis ratios are determined using the moments of the particle dis-tributions within the virial radius R v.The short-to-long axis ratio,s,and intermediate-to-long axis ratio q are calculated by iteratively diagonalizing the tensor5M ij=Σx i x jx2+y2s2,(1)where q2≡M yy/M xx and s2≡M zz/M xx.Figure1:(Left)The average short-to-long axis ratio,s,as a function of halo mass at z=0, 1,and3.Error bars reflect the Poisson uncertainty associated with the number of halos in each mass bin and not the scatter about the relation.(Right)Distribution of measured s values for∼1012h−1M⊙halos as a function of z3ResultsThe left panel in Figure1shows the average value of s as a function of halo mass for three redshifts,z=0,1,and3.Low mass halos,on average,are rounder than high mass halos,as are halos offixed mass at low z.The relation is well-approximated by s≃0.7(M/1012h−1M⊙)−0.05(1+z)−0.2over the mass and redshift ranges explored.The right panel of Figure1shows the distribution of s parameters for galaxy-mass halos as a function of z.The average and rms dispersion in these distributions are s=0.70±0.17,0.61±0.17,and 0.55±0.15,for z=0,1,and3,respectively.Note that the distributions are quite non-Gaussian,with a significant tail of highlyflattened halos.b How do the shapes of halos change with radius?The axial ratios presented in Figure1were obtained using particles within R v.The left panel of Figure2 shows this average“virial”flattening measurement as a function of halo mass (at z=0)compared with the average s measured within a sphere of radius 30h−1kpc for each halo.Although the difference is quite small for the low mass halos(since30h−1kpc contains much of the halo mass),generally halosFigure2:(Left)s as a function of mass measured within halo virial radii R v(solid)and within30h−1kpc spheres from halo centers.(Right)q and s parameters for halos simulated usingΛCDM(solid symbols)andΛWDM(open symbols)cosmologies.Individual halos, identified by mass and location between the two runs maintain the same symbol shape. are rounder at small radii.Interestingly,within thisfixed central radius,halos typically have the sameflattening,s∼0.7,independent of the halo mass.Finally in the right panel of Figure2,we compare the measured s and q values of four halos,simulated inΛCDM andΛWDM.There is a tendency for the halos to be rounder(approaching the upper right corner)in WDM, although one halo(designated by the triangles)does become slightlyflatter in WDM.The averageflattening shifts from s≃0.71for CDM to0.77for WDM, but it is difficult to make strong conclusions based on four halos.Indeed, Moore9has simulated a single halo using CDM and WDM power spectra and finds a very similar shape for each.A larger sample of objects will be needed to test for systematic trends.4ConclusionsWefind thatΛCDM galaxy-mass halos at z=0are more spherical than pre-viously believed,with s=0.70±0.17.High mass halos show more substantial flattening,as do halos offixed mass at high redshift:s∝(1+z)−0.2M−0.05. Halos are also more spherical in their centers,and tend to become moreflat-tened near the virial radius.These trends suggest collapsed structures become more spherical with time,perhaps because they have had more time to phase mix and to obtain isotropic orbit distributions.It is also possible that the4accretion history itself plays a role.Halos formed within aΛWDM simulation show a slight indication of being lessflattened than theirΛCDM counterparts. 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