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Scalelengths in Dark Matter Halos
Eric I. Barnes, Liliya L. R. WilliamsDepartment of Astronomy, University of Minnesota, Minneapolis, MN 55455
Arif Babul1
Department of Physics and Astronomy, University of Victoria, BC, Canada
Julianne J. Dalcanton2
Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195
ABSTRACT
We investigate a hypothesis regarding the origin of the scalelength in halos formed in cosmo-logical N-body simulations. This hypothesis can be viewed as an extension of an earlier idea putforth by Merritt and Aguilar. Our findings suggest that a phenomenon related to the radial orbitinstability is present in such halos and is responsible for density profile shapes. This instabilitysets a scalelength at which the velocity dispersion distribution changes rapidly from isotropicto radially anisotropic. This scalelength is reflected in the density distribution as the radius atwhich the density profile changes slope. We have tested the idea that radially dependent veloc-ity dispersion anisotropy leads to a break in density profile shape by manipulating the input ofa semi-analytic model to imitate the velocity structure imposed by the radial orbit instability.Without such manipulation, halos formed are approximated by single power-law density profilesand isotropic velocity distributions. Halos formed with altered inputs display density distribu-tions featuring scalelengths and anisotropy profiles similar to those seen in cosmological N-bodysimulations.
Subject headings: dark matter —galaxies:formation, evolution
1. Introduction
Some of the earliest studies to lay the foun-dation of galaxy formation have relied on ana-lytical methods to follow the collisionless evolu-tion of dark matter (e.g., Gunn & Gott 1972;Gott 1975; Gunn 1977) and lead to pure power-law density distributions. Several other studies
1Leverhulme Visiting Professor, Universities of Oxfordand Durham
2Alfred P. Sloan Foundation Fellow
have built on this early analytical framework. Fill-more & Goldreich (1983) and Bertschinger (1984)have found similarity solutions for the sphericalcollapse of cosmological perturbations that pro-duce nearly pure power-law density profiles. Thehalos in these studies are populated solely by ra-dial orbits. Ryden & Gunn (1987) have intro-duced nonradial motion, but the resulting den-sity profiles are still well-approximated by a singlepower-law over at least two orders of magnitude inradius, with noticeable deviations only on small
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[log (r/rvir) . −2] and large [log (r/rvir) & 0]scales (rvir is the virial radius).
As computing power has increased, another ap-proach has been adopted to investigate cosmo-logical structure formation, N-body simulations.These studies do not suffer from the same limita-tions of adopted approximations that are inherentin the analytically-based work, however such simu-lations do rely on other approximations, like grav-itational softening. Given appropriate initial con-ditions, cosmological N-body simulations (CNS)can follow structure formation through non-lineardevelopment and investigate the impact of hierar-chical merging (e.g., Dubinski & Carlberg 1991;Navarro et al. 1996, 1997; Moore et al. 1998, 1999;Bullock et al. 2001; Power et al. 2003). These stud-ies agree that the equilibrium dark matter halosthat form in CNS have nearly universal densityprofiles that are nothing like single power-laws.These halos generically have logarithmic slopesγ = −d log ρ/d log r that become larger with in-creasing radius; γ ≈ 1 at one-hundredth of a virialradius and γ ≈ 3 near the virial radius. There re-main, however, disagreements regarding the exactvalues and behavior of γ, especially at small radii.
The fitting function that has been widely usedto characterize the density profiles of CNS halos,the NFW profile (Navarro et al. 1996), has an ex-plicit scalelength that divides the inner and outerbehaviors. The NFW profile γ asymptotes to 1 asr → 0 and 3 as r → ∞. Recently, Navarro et al.
(2004) have discussed another fitting function todescribe the density profiles of higher resolutionCNS halos. Unlike the NFW profile, γ of the newfunction does not approach asymptotic values atsmall and large radii and does not have the samewell-defined scalelength as the NFW profile. In-stead, γ of this new profile changes continuouslywith radius, implying that there are no regionsin which the density profile behaves like a power-law. At the same time, another recent study (Die-mand et al. 2005) finds that the r → 0 asymptoticpower-law behavior persists in their high resolu-tion CNS. Given this disagreement, it is unclearwhether or not CNS halos actually have preferredscalelengths. However, the NFW profile does pro-vide a reasonable description of density distribu-tions for galaxy and cluster scale CNS halos (Mer-ritt et al. 2005). For the sake of brevity, we willrefer to the radius at which γ = 2 as the scale-
length.
Despite these fairly minor disagreements re-garding the behavior of γ, CNS halos (as a class)appear qualitatively similar. This similarity doesnot extend to the analytically-based (AB) halosdescribed above. Like previous studies that havesought to understand the differences between CNSand AB halos (e.g., Avila-Reese, Firmani, & Her-nandez 1998; Lokas 2000; Nusser 2001), we are in-vestigating how the input physics differs betweenthe methods and whether or not such physics canexplain the differences. We find evidence that aphenomenon related to the radial orbit instabil-ity (ROI) can account for the differences as wellas shed some light on the universality and scale-lengths of CNS halos.
A brief introduction to the ROI is given in §2.We lay out the hypothesis, drawing links betweenthe ROI, CNS halos, and AB halos, in §3. Thedemonstration of the hypothesis relies on evolvingdark matter halos semi-analytically and is basedon the method presented in Ryden & Gunn (1987).Our modifications to this method have been fullydiscussed in Williams et al. (2004). Briefly, thismethod relies on spherical symmetry and ignorescomplications due to merging. This simplicity pro-vides us with a degree of control not present inCNS. At the same time, inclusion of secondaryperturbations allows us to approximate the correctevolution of an isolated dark matter halo. Detailsof the method and results of testing the hypoth-esis are presented in §4. We discuss the link be-tween the ROI and the shapes of CNS halo densityprofiles in §5. The final section summarizes andpresents the conclusions drawn from this work.
2. Overview of the Radial Orbit Instabil-
ity
Early analytical and N-body studies have foundthat equilibrium spherical systems composed ofpurely radial orbits are unstable to forming barsor triaxial systems (Polyachenko 1981; Fridman &Polyachenko 1984; Binney & Tremaine 1987, §5.2).Although there is no general theory explaining theROI, Palmer & Papaloizou (1987) argue that if aspherical system develops a small m = 2 distor-tion, precession of radial orbits can draw them toreinforce the distortion, thereby increasing the dis-tortion and leading to bar formation. Whatever
2
the exact mechanism is, it is clear that nonradialforces play a key role, as we will discuss in the nextsection.
Merritt & Aguilar (1985) have further inves-tigated the ROI by creating equilibrium systemswith initial anisotropy (β = 1 − σ2
φ/σ2r) profiles
described by,
β(r) =r2
r2a + r2
, (1)
where ra is referred to as the anisotropy radius(Merritt 1985). These profiles are completelyisotropic (β = 0) for ra/r1/2 = ∞, where r1/2
is the initial half-mass radius. For smaller ra/r1/2
values, the anisotropy increases from the centeroutwards. At the limit ra/r1/2 = 0, the en-tire model is composed of radial orbits. Mer-ritt & Aguilar (1985) list several fundamental re-sults of the ROI. First, an initially spherical sys-tem that undergoes the ROI transforms into anearly prolate bar with long-to-short axis ratio≈ 2 − 2.5. Second, for the anisotropy profileof Equation 1 there appears to be a fairly dis-tinct border between systems that form a sig-nificant bar (ra/r1/2 . 0.2) and those that re-main relatively spherical (ra/r1/2 & 0.3); however,other anisotropy profiles produce different stabil-ity criteria. Third, the global anisotropy measure2Tr/Tt (where Tr and Tt are the kinetic energiesassociated with radial and tangential motions, re-spectively) is not a universal arbiter of whether ornot a system is unstable; larger values of this ratioindicate instability, but there does not appear tobe a definitive demarcation value (Polyachenko &Shukhman 1981; Barnes, Goodman, & Hut 1986).The Merritt & Aguilar (1985) work also highlightsthat the stability of any equilibrium system de-pends (at a minimum) on the anisotropy distribu-tion, with centrally isotropic systems being morestable than those with radially anisotropic cores.In brief, collisionless systems will undergo the ROIif a sufficient fraction of orbits are predominantlyradial.
While we have so far discussed the ROI in equi-librium systems, collisionless collapses also appearto relate to the ROI. The link with the ROI is thatthese cold collapses produce mostly radial motion,a condition favorable to the onset of the ROI. Wecompare the results of noncosmological and cos-mological collapses and discuss the signature of
the ROI in the next section.
3. Collapses & the Radial Orbit Instability
Studies of cold collisionless collapses in non-cosmological settings (van Albada 1982; McGlynn1984; Aguilar & Merritt 1990; Trenti, Bertin, &van Albada 2005) report that the end resultsof collapses are good descriptions of observedsurface brightness profiles of elliptical galaxies.For example, the collapses presented in van Al-bada (1982) result in mildly nonspherical systemswith two-dimensional projected density profilesthat are well-approximated by the Sersic func-tion ln (Σ/Σe) = −bn[(R/Re)
1/n − 1] (Sersic1968), with n ≈ 4 like a de Vaucouleurs profile.These collapse products also display characteristicanisotropy profiles; the inner regions are isotropicand transitions to radial anisotropy occur at largerradii. Merritt & Aguilar (1985) also point outthat such collapses are less centrally concentratedthan purely radial collapses. We extend and fo-cus these thoughts by specifically investigating thelink between density scalelengths and monotonicvariations in velocity anisotropy.
While some choices of initial conditions in thepreviously mentioned studies have an eye towardsdescribing cosmological situations, their maingoals relate to understanding elliptical galaxies.Two studies directly deal with the differences be-tween cosmological and noncosmological collapses.Katz (1991) argues that the ROI does not occur incosmological settings because there is no correla-tion between final shape and initial 2T/W value.However, this study also reports that the finalstates of collapses have density profiles that arewell-approximated by a Jaffe distribution (Jaffe1983), which is similar to a de Vaucouleurs (Sersicn = 4) profile when projected to two dimen-sions. More importantly, it is stated that theequilibria are supported by anisotropic velocitydispersions, like the results of the noncosmologi-cal studies. Carpintero & Muzzio (1995) find thatrealistic cosmological collapses do not succumb tothe bar instability, but, like Katz (1991), the fi-nal density profiles are approximated by a Jaffedistribution (again, similar to an n = 4 Sersicprofile). Unfortunately, this study focuses solelyon the final shapes of the collapses and does notpresent any kinematic information to allow a com-
3
parison of anisotropy profiles. Beyond these twostudies, it is well-reported that halos formed inCNS tend to appear similar to the products ofnoncosmological collapse simulations; i.e., theyappear prolate spheroidal/triaxial (e.g., Cole &Lacey 1996; Springel, White, & Hernquist 2004)and have monotonically increasing anisotropy dis-tributions (Cole & Lacey 1996; Huss et al. 1999).All of these points lead us to conclude that cosmo-logical collapse simulations result in systems thathave the same qualitative density and anisotropyprofiles as those from noncosmological studies.
The final states of these collapses are similar,but have the systems undergone the ROI? Fromthe literature, it appears that most researchersconsider bar formation to be the signpost of theinstability. The Katz (1991) and Carpintero &Muzzio (1995) studies suggest that the ROI is ab-sent in cosmological situations due to the lack ofbars. Aguilar & Merritt (1990) refer to the merelyslightly nonspherical results in van Albada (1982)as stable.
We propose that bar formation may not be theonly sign of the ROI, but possibly only the mostflagrant. The end results of collapses like those invan Albada (1982), where no strong bar is formed,display a less drastic result of the ROI. The equi-librium states resulting from collisionless collapseshave similar anisotropy profiles; anisotropy in-creases (becomes more radial) with radius. Thisis in line with the discussion in Merritt & Aguilar(1985) which regards systems with isotropic cen-tral regions as more stable than those with radi-ally anisotropic central regions. From this pointof view, the ROI changes the mostly radial initialvelocity distribution into one that has an isotropiccore surrounded by a radial “mantle”. As we con-tinue this discussion, we will use the term “mROI”(mild aspect of the ROI) to include the onset ofthe anisotropy profile and possible mild triaxial-ity and to distinguish it from the usual (and moreextreme) bar-formation criterion.
Figure 1 shows anisotropy profiles that can beconstructed from information in several previousstudies. Panel a) shows the results of van Albada(1982) (plus symbols) and the analytical expres-sion utilized by Merritt & Aguilar (1985), Equa-tion 1. In the figure, we have set ra = re, wherere is the effective radius of the corresponding deVaucouleurs profile. Note that the analytical ex-
pression is not a fit to the data points, it merelyillustrates the initial anisotropy profile utilized inthe ROI study of Merritt & Aguilar (1985). Panelb) shows the results from the “n = −2” (here,n refers to the power spectrum P (k) ∝ kn, notthe Sersic profile) simulation of Cole & Lacey(1996) (asterisks) as well as the analytical expres-sion adopted by Carlberg et al. (1997),
β = βm4r/r178
4 + (r/r178)2, (2)
where r178 is the virial radius and we have cho-sen βm = 0.3. This expression is intended to fitthe data points from equilibrium systems. Thelast panel shows the results of the simulations ofHuss et al. (1999). Admittedly, the amplitude ofthe anisotropy in the Cole & Lacey (1996) result(panel b) is much smaller than the other cases,but the monotonically increasing nature is stillpresent. The mROI anisotropy profile is foundgenerically in collapses.
We find support for a link between the mROIand the presence of scalelengths in CNS haloslooking closely at the information from Huss et
al. (1999). This work has investigated 5 modelsof isolated dark matter halo formation. Four ofthese have simple initial conditions that were cho-sen to highlight the impact of differing amountsof initial random motion. The fifth represents astandard NFW type halo. Their Model I is es-pecially interesting since it has no initial tangen-tial motions and tangential forces are not allowed,essentially forcing radial collapse. The resultinghalo is described by a constant radial anisotropy(β = 1, diamonds in Figure 1c) and a single power-law density profile ρ ∝ r−2. Once this restrictionon tangential motion is lifted (Models II-IV haveincreasing amounts of initial random motion andallow nonradial forces), the resulting anisotropyprofiles show basically isotropic cores with β in-creasing with radius (Figure 1c), evidence that themROI is present in these cases. The correspond-ing density profiles resemble NFW profiles, i.e., ascalelength is introduced. While generic equilib-ria do not have a unique relation between β andγ, Hansen & Moore (2004) point out that a widevariety of CNS halos appear to have a specific linkbetween the anisotropy distribution and γ. In §5,we further discuss this connection and the specificforms of the density and anisotropy profiles.
4
These previous studies provide evidence oftwo things. One, the mROI, and its consequentanisotropy profile that has a distinct scalelengthdividing an isotropic core from a radial mantle, isubiquitous in collapses1. Two, this scalelength islinked to (if not the same as) the scalelength ofthe resulting density distributions. These pointsare the basis of the hypothesis that the mROI a)is the key physics missing from previous AB stud-ies of dark matter halo formation and b) leads todensity profiles with scalelengths. We will nowproceed to describe and discuss our testing of thishypothesis.
4. Demonstrating the Link Between Anisotropy
& Density Profiles
We are not interested in modeling the mROI.What we want to do is mimic the key result ofthe instability, the anisotropy profile, by suitablealterations of inputs. This approach will let ustest the root of the hypothesis, the impact of theanisotropy distribution on the resulting densitydistribution.
To create the halos we will investigate, we followthe procedure outlined in Ryden & Gunn (1987)and Williams et al. (2004). Initially thin spheri-cal shells expand with the Hubble flow until theyreach their turn-around radius, at which time theycollapse onto the mass interior. At the moment ashell begins to collapse, the integrated effects ofsecondary perturbations are introduced: a shell,which can be thought of as a mass of particles,is given a velocity dispersion. This can also bepictured as giving the shell a thickness, with apo-and pericentric distances. At the same time, theperturbation velocity is oriented randomly, intro-ducing angular momentum for that shell. The per-turbation velocities calculated per Ryden & Gunn(1987) are actually RMS values. Therefore, wefollow Williams et al. (2004) and randomly choosethe magnitude of the velocity from the Maxwellspeed distribution (truncated at 4 standard devi-ations) with the RMS value from Ryden & Gunn(1987). Since our method relies on choosing ran-
1We note that this evidence cannot be considered conclusiveas other studies have found that different mechanisms, mostnotably density inhomogeneities, can also lead to similaroutcomes (e.g., van Albada 1982; Londrillo, Messina, &Stiavelli 1991).
dom magnitudes and directions, for the rest of thispaper we will average over some number (usually20) of halos and refer to the average quantities asbelonging to a single halo.
The benefit of the Williams et al. (2004) for-malism is that it allows us to impose an arbitraryvelocity dispersion profile and then study the halosthat result from the subsequent collapse. However,before we explore variations in the anisotropy pro-file, we first establish a “standard” halo to use asa benchmark. By construction, it will be similarto the halos that have been created in Ryden &Gunn (1987) and Williams et al. (2004). The spa-tial density and phase-space density proxy ρ/σ3
profiles of the standard halo are shown in Figure 2(as usual, r200 is the radius at which the halo den-sity is 200ρcrit). The top panel highlights the de-viations from ρ ∝ r−2. Over a substantial radialrange (−2.5 . log (r/r200) . 0.0), the density iswell described by a single power-law ρ ∝ r−1.8.This agrees well with the halo described in Ry-den & Gunn (1987, see their Figure 11). We showthe ρ/σ3 distribution for the standard halo in thebottom panel. This distribution is nicely approxi-mated by a power-law with exponent ≈ −2 over awide radial range −2.5 < log (r/r200) < 0 (Austinet al. 2005). We also present the velocity distribu-tions in Figure 3. The radial, one-dimensional tan-gential, and total velocity dispersion profiles alongwith the anisotropy distribution are shown. Notethat the velocities are basically isotropic over theentire range (the upturn outside r200 is from in-falling material). Not surprisingly, given its scale-free power-law properties, we find that the stan-dard halo is a poor match to those from cosmologi-cal N-body simulations, represented in Figure 2 bya NFW profile of concentration 10 (dash-dottedline).
We now turn our attention to altering the pa-rameters of the standard halo by varying the ve-locities caused by secondary perturbations. Weintroduce a factor ν, which is a function of initialcomoving radius x, to effect the desired changes.Since we are interested in mimicking the end re-sult of the mROI, we introduce changes that pro-duce an anisotropy profile that is isotropic nearthe center and becomes radially anisotropic withincreasing radius.
The prescription for doing this in Williams et
al. (2004) is a simple one; halve the perturba-
5
tion velocities in a specified comoving radial range(xlo, xhi) giving an inverted top-hat ν distribution,
ν(x) =
1 : x < xlo
0.5 : xlo < x < xhi
1 : x > xhi.(3)
In this prescription, the perturbation velocities aresimply multiplied by ν. The impact of this pre-scription is to decrease the angular momentumimparted by perturbations in the specified radialrange. This prescription also alters the total per-turbation velocity, thereby changing the energyimparted to the halo by the perturbations. Theend result is a halo that loses the power-law prop-erties of the standard halo and instead resemblesan NFW profile (see Figure 3 in Williams et al.
2004).
A representative ν(x) distribution used in thisstudy is shown in Figure 4. The distance measureof the x-axis is the initial comoving radius of ashell in Mpc. The algebraic expression is,
ν(x) = ν∗(x) − [ν∗(x) − d]e−(x−xh)n/2ω2
, (4)
where n and ω control the shape and width of thetrough, d is the depth of the trough, xh is themid-point of the trough,
ν∗(x) = 0.5(ν0−ν∞)(1−tanh[2s(x−xh)/(ν0−ν∞)])+ν∞,
2s/(ν0 − ν∞) controls the slope of ν∗ at xh, andν0 and ν∞ are the asymptotic values. These pa-rameters give us considerable flexibility to explorethe impact of secondary perturbations on the finalhalos. After some experimentation, we have foundthat the interesting parameters to vary are ν0, ν∞,and xh. We fix the remaining parameters; n = 4,ω = 0.4, d = 0.5, s = 20. In general, ν0 > 1 andν∞ = 1. If we simply multiply the perturbationvelocities by ν, then near the center (ν > 1) we in-troduce more angular momentum than in the stan-dard halo. Conversely, ν < 1 saps angular momen-tum relative to the standard case. Through vari-ous trials, this change from ν > 1 to ν < 1 appearsessential to reproducing the mROI anisotropy pro-file, and we point out that ν is simply a means ofachieving this end. We have not determined theexact link between the behavior of ν, which is im-posed only at a shell’s initial turn-around point,and the physical process of the mROI. It is possi-ble that the RMS perturbations calculated accord-ing to Ryden & Gunn (1987) are larger than those
in CNS because of the one-dimensional nature ofthis method. If this is the case, then ν < 1 is sim-ply reducing the effects of exaggerated secondaryperturbations. While further work may elucidatethe relationship between ν and the physics of themROI, for the purposes of this study, it sufficesthat this ν distribution gives rise to an mROI-typeanisotropy profile.
As it seems unlikely that the mROI changes theenergies of particles, we turn to a new prescrip-tion that leaves shell energies unchanged and in-stead affects the orientations of the perturbationvelocities. The prescription presented here (andshown in Figure 4) uses the following ν distribu-tion parameters; ν0 = 1.6, ν∞ = 1.0, xh = 0.8Mpc. The magnitude of the perturbation veloc-ity remains unchanged from its value derived fromthe Maxwell speed distribution, but ν > 1 in-creases the probability that the velocity is ori-ented tangentially and vice versa. We reiteratethat this prescription is intended solely to producean mROI-type anisotropy profile. There is no di-rect link between ν and the physics of the mROI,and so the correspondence between the final pro-files is bound to be inexact.
Figure 5 illustrates the resulting spatial den-sity and ρ/σ3 profiles. We can see that adoptingthis new prescription has indeed provided a scale-length for the halo. While we have not set outto exactly recreate NFW halos, the similarity isclear. Some of the differences between the averageand NFW profiles is due to the fact that many in-dividual halos are good approximations to NFWprofiles with different concentrations (5 ≤ c ≤ 9).Averaging these together tends to smear the peaksinto a broader profile. Looking at the distributionin panel (b), we see a profile that is a decent ap-proximation to a power-law over a couple ordersof magnitude. It also appears to be a bit shallower(slope ≈ −1.8) than the standard halo’s ρ/σ3 pro-file and more in line with the results of Taylor& Navarro (2001) who find a slope of −1.875 forthree CNS halos. Figure 6 contains the same ve-locity information as Figure 3. The break in thepower-law that is present in the density has ananalogous presence in the velocity distributions.Also, the anisotropy profile is familiar from thepreceding discussion of the mROI; i.e.,the veloci-ties are isotropic near the center with a fairly sharptransition to anisotropy. This result is in qualita-
6
tive agreement with the discussion of collapses inMerritt & Aguilar (1985); the ROI reduces thecentral density relative to purely radial collapses.
To this point, the discussion and figures pre-sented in this section are all based on a single setof parameters that determine the ν distribution.We have done some parameter space explorationand have found that the ν0 parameter can havean important impact on the final profiles. This isnot very surprising, since it primarily determinesthe impact of the isotropic core. As ν0 is reducedfrom 1.6 to 1, the inner γ of the density profilechanges from ≈ 1 to ≈ 1.3. Also, while we are nottrying to exactly reproduce NFW halos, we havenoted that by eliminating the introduction of ran-dom perturbation velocity magnitudes (i.e., justusing the RMS values) we can make the agree-ment between our halos (not shown) and NFWhalos better. We speculate that this occurs be-cause it provides “colder” initial conditions (in thesense discussed regarding the ROI). We also findthat the anisotropy profile is hardly affected if ν∞is varied between 1.0 and d (the effect of changingν∞ is to lower the ν value from 1.0 to d = 0.5 overthe range 1.5 . x ≤ 2.5, see Figure 4).
A more thorough investigation has been maderegarding the importance of xh, the mid-point ofthe ν trough. This parameter directly impactsthe extent of the isotropic core present in the fi-nal anisotropy profile. In Figure 7, we show den-sity profiles (left column) and the correspondinganisotropy profiles for 7 values of xh surroundingthe canonical value; from top to bottom xh={0.6,0.7, 0.8, 0.9, 1.0, 1.1, 1.2}. We note that allthe density profiles have scalelengths, but theagreement with the NFW form is within the er-ror bars only for xh = {0.7, 0.8, 0.9}. More im-portantly, these models highlight the correspon-dence between the density scalelength rd and theanisotropy scalelength rb, defined here to be theradius at which β = 0.5, as in Equation 1. Fig-ure 8 displays this correlation (solid line) and alsoshows how ri, the largest radius at which β = 0,changes with rd (dash-dotted line). The densityand anisotropy scalelengths are roughly the same(within ≈ 20%) in these halos, further support-ing the hypothesis that mROI induced anisotropyprofiles lead to scalelengths in density profiles.
5. Relating the mROI and Shapes of N-
body Halo Density Profiles
In this section, we take up a discussion of the“true” shape of N-body halos. The preceding dis-cussion has purposely ignored the differences be-tween the “canonical” NFW profile and other CNShalo fitting functions (e.g., Navarro et al. 2004).One must realize that our lack of detailed un-derstanding of the physics of halo formation hasforced us to approach halo structure from a veryreactive posture; halos are formed and then func-tions are chosen that resemble the resulting den-sity profiles. Unfortunately, these functions aresomewhat arbitrary and do not, by themselves,lead to a deeper understanding of halo dynamics.The central idea of this study, that the mROI leadsto an anisotropy distribution that is linked to thedensity distribution, can potentially provide a wayaround this arbitrariness. A full understanding ofthis physical mechanism can provide us with anexpected density profile for collapse equilibria. Inreality, the ROI is not fully understood and so noab initio prediction can be made. At present, weare left to continue the passive role of finding thefunction(s) that best explain the simulation resultsand we now discuss the density profile specifically.
The density profile that has long been found as-sociated with both noncosmological (e.g., van Al-bada 1982; Londrillo, Messina, & Stiavelli 1991)and cosmological (e.g., Katz 1991; Carpintero &Muzzio 1995) N-body simulations is the de Vau-couleurs (Sersic n = 4) profile. However, Sersicprofiles are not very similar to the NFW profile,the dominant density fitting function over the pastdecade. While NFW profiles do fit CNS halo den-sity profiles well, recent work by Navarro et al.
(2004) suggests that the asymptotic behavior ofthe NFW profile as r → 0 and r → ∞ does notfit density profiles as well as a function similar tothe Sersic form (however, for a dissenting view seeDiemand et al. 2005). Additionally, Dalcanton &Hogan (2001) and Merritt et al. (2005) have pre-sented evidence that the Sersic form is closer tothe true shapes of halo density profiles. As thedensity profiles of the various ROI studies listedin this paper also find Sersic (specifically de Vau-couleurs) profiles, the N-body findings mesh nicelywith the hypothesis that the ROI is a key part offorming collapse equilibria.
7
Taking the link between the mROI and equi-librium density distribution as established, wenow speculate about the causal relationship be-tween them. Kazantzidis et al. (2004) point outthat there are a wide variety of equilibria avail-able to halos with density profiles like those fromCNS. This variety arises from different velocitydistributions, but we have shown that CNS ha-los display mROI-type anisotropy profiles exclu-sively (see also Hansen & Moore 2004). We pro-pose that this choice between the equilibria is dy-namically set by the mROI, which brings aboutthe anisotropy distribution that is associated witha Sersic density profile. As mentioned before, wecannot currently demonstrate this association ana-lytically, but Trenti, Bertin, & van Albada (2005),following the work of Stiavelli & Bertin (1987) andMerritt, Tremaine, & Johnstone (1989), discusshow the Sersic density profile goes hand-in-handwith the isotropic core and radially anisotropicmantle associated with the mROI. They presenta distribution function meant to describe sys-tems that have experienced incomplete violent re-laxation. When this distribution function is in-tegrated over velocity space, the density profileis very similar to the Sersic form. Integratingthis function over spatial coordinates, one derivesan anisotropy profile that has the characteristicmROI shape. In the context of this idea, the linkbetween anisotropy and density profiles is presentregardless of initial conditions (i.e., cosmologicalor noncosmological), but we speculate that suchdifferent initial conditions most likely lead to dif-fering Sersic n values. Returning to the “true”shape of N-body halos, we conclude that Sersic,rather than NFW, profiles are better descriptionsof the density profiles of CNS halos and that afuller understanding of the ROI will provide in-sight to the actual density profiles.
6. Summary & Conclusions
There is an important difference between mod-els of collisionless dark matter halos created fromanalytically-based methods and those resultingfrom cosmological N-body simulations. Unless theinput physics (Avila-Reese, Firmani, & Hernandez1998; Lokas 2000; Nusser 2001) or parameters arevaried (as in §4), the halos that result from ana-lytical calculations tend to be best described by asingle power-law density profile and approximately
constant isotropic velocity dispersion anisotropyprofiles. Cosmological N-body simulation halostend to have density distributions that steepenand anisotropy profiles that become more radialwith increasing radius.
We have explored the hypothesis, akin to thediscussion in Merritt & Aguilar (1985, §6), thatthe presence of scalelengths in cosmological N-body simulations can be attributed to a mild as-pect of the radial orbit instability, which producesequilibrium halos with isotropic cores surroundedby regions of radial anisotropy. The hypothesis isthat it is this fundamental change in the charac-ter of the orbits supporting the halo that leads toscalelengths. Cosmological N-body simulation ha-los incorporate the physics of the mild aspect ofthe radial orbit instability and present such scale-lengths. Analytically-based halos do not have thisinstability “built-in” and therefore lack the conse-quent scalelengths.
Utilizing an extension of the analytically-basedmethod of Ryden & Gunn (1987), we have inves-tigated whether or not a preferred scalelength canbe introduced into a halo that would naturallyhave a single power-law density distribution. Tomimic the impact of the mild aspect of the radialorbit instability, we have artificially altered the ve-locities introduced by secondary perturbations tobe more tangential near the center and more ra-dial in the outer parts of a halo. This is an ap-proximate way to include the effect of the instabil-ity in a technique that does not explicitly includesuch physics. In our prescription, the magnitudesof the perturbation velocities are left unchanged,only the directions are altered. This leaves theenergy budget of the halo unchanged, which weargue is an acceptable approximation to the insta-bility. The important result is that the final ha-los have an anisotropy distribution reminiscent ofthose from cosmological N-body simulation halosand a density profile akin to an NFW profile. Anexploration of the parameter space of our modelsshows that the presence of a scalelength is insensi-tive to the relative sizes of the isotropic and radi-ally anisotropic regions, and the anisotropy radiiof halos are approximately equal to their densityscalelengths.
If cosmological N-body simulations are environ-ments in which the mild aspect of the radial orbitinstability plays a significant role (and they ap-
8
pear to be so), two central features of such simula-tions have a ready explanation. One is the seeminguniversality of halo density profiles which appearsin both noncosmological and cosmological N-bodysimulations as Sersic density profiles. A physicalmechanism, like the mild aspect of the radial or-bit instability, that generically acts in collapses,even when mergers are absent, provides a simplegenerator for such universality. The outcome ofthe instability is a variable anisotropy distribu-tion with isotropic orbits in central regions andradially anisotropic orbits in outer regions. Thisanisotropy distribution gives rise to density pro-files with scalelengths, the second defining featureof halos formed in cosmological N-body simula-tions.
We thank an anonymous referee for a carefulreading and insightful comments that improvedthis paper. This work has been supported by NSFgrant AST-0307604. Research support for ABcomes from the Natural Sciences and EngineeringResearch Council (Canada) through the Discoverygrant program. AB would also like to acknowledgesupport from the Leverhulme Trust (UK) in theform of the Leverhulme Visiting Professorship atthe Universities of Oxford and Durham. JJD waspartially supported through the Alfred P. SloanFoundation.
REFERENCES
Aguilar, L., Merritt, D. 1990, ApJ, 354, 33
Austin, C., Williams, L. L. R., Barnes, E. I.,Babul, A., Dalcanton, J. J. 2005, submitted toApJ
Avila-Reese, V., Firmani, C., Hernandez, X. 1998,ApJ, 505, 37
Barnes, J., Goodman, J, Hut, P. 1986, ApJ, 300,112
Bertschinger, E. 1984, ApJS, 58, 39
Binney, J., Tremaine, S. 1987, Galactic Dynamics,(Princeton:Princeton Univ. Press)
Bullock, J. S., Dekel, A., Kilatt, T. S., Kravtsov,A. V., Klypin, A. A., Porciani, C., Primack, J.R. 2001, ApJ, 555, 240
Carlberg, R. G., Yee, H. K. C., Ellingson, E. 1997,ApJ, 485, L13
Carpintero, D. D., Muzzio, J. C. 1995, ApJ, 440,5
Cole, S., Lacey, C. 1996, MNRAS, 281, 716
Dalcanton, J. J., Hogan, C. J. 2001, ApJ, 561, 35
Diemand, J., Zemp, M., Moore, B., Stadel, J, Car-ollo, M. 2005, astro-ph/0504215
Dubinski, J., Carlberg, R. G. 1991, ApJ, 369, 13
Fillmore, J. A., Goldreich, P. 1983, ApJ, 281, 1
Fridman, A. M., Polyachenko, V. L. 1984, Physicsof Gravitating Systems (New York:Springer)
Gott, J. R. 1975, ApJ, 201, 296
Gunn, J. E., Gott, J.R. 1972, ApJ, 176, 1
Gunn, J. E. 1977, ApJ, 218, 592
Hansen, S. H., Moore, B. 2004, astro-ph/0411473
Huss, A., Jain, B., Steinmetz, M. 1999, ApJ, 517,64
Jaffe, W. 1983, MNRAS, 202, 995
Katz, N. 1991, ApJ, 368, 325
Kazantzidis, S., Magorrian, J., Moore, B. 2004,ApJ, 601, 37
Lokas, E. L., 2000, MNRAS, 311, 423
Londrillo, P., Messina, A., Stiavelli, M. 1991, MN-RAS, 250, 54
McGlynn, T. A. 1984, ApJ, 281, 13
Merritt, D. 1985, MNRAS, 214, 25P
Merritt, D., Aguilar, L. 1985, MNRAS, 217, 787
Merritt, D., Tremaine, S., Johnstone, D. 1989,MNRAS, 236, 829
Merritt, D., Navarro, J. F., Ludlow, A., Jenkins,A. 2005, ApJ, 624, 85L
Moore, B., Governato, F., Quinn, T., Stadel, J.,Lake, G. 1998, ApJ, 499, L5
9
Moore, B., Quinn, T., Governato, F., Stadel, J.,Lake, G. 1999, ApJ, 310, 1147
Navarro, J. F., Frenk, C. S., White, S. D. M. 1996,ApJ, 462, 563
Navarro, J. F., Frenk, C. S., White, S. D. M. 1997,ApJ, 490, 493
Navarro, J. F., Hayashi, E., Power, C., Jenkins, A.R., Frenk, C. S., White, S. D. M., Springel, V.,Stadel, J., Quinn, T. R. 2004, MNRAS, 349,1039
Nusser, A., 2001, MNRAS, 325, 1397
Palmer, P. L., Papaloizou, J. 1987, MNRAS, 224,1043
Polyachenko, V. L. 1981, Pis’ma Astr. Zh., 7, 142(translated in Soviet Astr. Lett., 7, 79)
Polyachenko, V. L., Shukhman, I. G. 1981, Astr.
Zh., 58, 933 (translated in Soviet Astr., 25, 533)
Power, C., Navarro, J. F., Jenkins, A., Frenk, C.S., White, S. D. M., Springel, V., Stadel, J.,Quinn, T. 2003, MNRAS, 338, 14
Ryden, B. S., Gunn, J. E. 1987, ApJ, 318, 15
Sersic, J. L. 1968, Atlas de Galaxies Australes(Cordoba: Obs. Astron., Univ. Nac. Cordoba)
Springel, V., White, S. D. M., Hernquist, L. 2004,IAUS, 220, 421
Stiavelli, M., Bertin, G. 1987, MNRAS, 229, 61
Taylor, J. E., Navarro, J. F. 2001, ApJ, 563, 483
Trenti, M., Bertin, G., van Albada, T. S. 2005,A&A, 433, 57
van Albada, T. S. 1982, MNRAS, 201, 939
Williams, L. L. R., Babul, A., Dalcanton, J. J.2004, ApJ, 604, 18
This 2-column preprint was prepared with the AAS LATEXmacros v5.2.
Fig. 1.— Plots showing the behavior of theanisotropy parameter β for various simulations. a)The plus symbols denote the results of van Albada(1982). The line is the initial anisotropy distri-bution utilized in Merritt & Aguilar (1985) (seeEquation 1). b) The asterisks mark the values re-sulting in the n = −2 simulation of Cole & Lacey(1996). This line is the distribution suggested inCarlberg et al. (1997). c) The following symbolsrepresent the various models described in Huss et
al. (1999); diamonds – Model I (purely radial), tri-angles – Model II, squares – Model III, crosses –Model IV, filled circles – Model V.
Fig. 2.— Standard halo spatial and phase-spacedensity proxy distributions. a) Multiplying thespatial density by r2 enhances the deviations fromthat power-law. The dash-dotted line is an NFWprofile with a concentration of 10. b) The phase-space density proxy distribution. The 1-σ errorbars are shown and the normalization is arbitraryin both panels.
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Fig. 3.— Standard halo velocity distributions. a)The radial velocity dispersion profile. b) The one-dimensional tangential velocity dispersion profile.c) The total velocity dispersion profile. d) Theanisotropy profile. The dash-dotted line shows theanisotropy of purely radial orbits. The solid line isfor isotropic orbits. The 1-σ error bars are shownin all panels.
Fig. 4.— The distribution of ν values versus initialcomoving radius in Mpc. See §4 for the utility ofν.
Fig. 5.— The spatial (a) and ρ/σ3 (b) densitydistributions resulting from the prescription de-scribed in §4. The dash-dotted line in (a) is anNFW profile with a concentration of 8. The 1-σ error bars are shown and the normalization isarbitrary in both panels.
Fig. 6.— The velocity distributions resulting fromthe prescription described in §4. The panels arethe same as in Figure 3.
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Fig. 7.— Density (left column) and anisotropy(right column) profiles for different values of xh.From top to bottom, xh={0.6, 0.7, 0.8, 0.9, 1.0,1.1, 1.2}. The dash-dotted line in the density pro-files is an NFW profile with a concentration of8. The ordinate values listed are the same foreach panel in their respective columns. The solidand dash-dotted lines in the anisotropy profilesmark isotropy (β = 0) and total radial anisotropy(β = 1), respectively. Each model is the averageof 10 individual halos.
Fig. 8.— The correlations between anisotropy ra-dius rb (solid line) and maximum isotropic radiusri (dash-dotted line) and the density scalelengthrd. The dashed line represents a one-to-one corre-spondence and the numbers are the correspondingxh values.
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