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arXiv:0810.1522v2 [astro-ph] 11 Jun 2009

Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 11 June 2009 (MN LATEX style file v1.4)

The Diversity and Similarity of Simulated Cold Dark

Matter Halos

Julio F. Navarro1,5, Aaron Ludlow1, Volker Springel2, Jie Wang2,

Mark Vogelsberger2, Simon D.M. White2, Adrian Jenkins3, Carlos S. Frenk3, and

Amina Helmi4,

1Dept. of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 5C2, Canada

2Max-Planck-Institut f¨ ur Astrophysik, Karl-Schwarzschild-Straße 1, 85740 Garching bei M¨ unchen, Germany

3Institute for Computational Cosmology, Dep. of Physics, Univ. of Durham, South Road, Durham DH1 3LE, UK

4Kapteyn Astronomical Institute, Univ. of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands

5Department of Astronomy, University of Massachusetts, Amherst, MA 01003-9305, USA

11 June 2009

ABSTRACT

We study the mass, velocity dispersion, and anisotropy profiles of ΛCDM halos using

a suite of N-body simulations of unprecedented numerical resolution. The Aquarius

Project follows the formation of 6 different galaxy-sized halos simulated several times

at varying numerical resolution, allowing numerical convergence to be assessed directly.

The highest resolution simulation represents a single dark matter halo using 4.4 billion

particles, of which 1.1 billion end up within the virial radius. Our analysis confirms a

number of results claimed by earlier work, and clarifies a few issues where conflicting

claims may be found in the recent literature. The mass profile of ΛCDM halos deviates

slightly but systematically from the form proposed by Navarro, Frenk & White. The

spherically-averaged density profile becomes progressively shallower inwards and, at

the innermost resolved radius, the logarithmic slope is γ ≡ −dlnρ/dlnr<

totic inner slopes as steep as the recently claimed ρ ∝ r−1.2are clearly ruled out.

The radial dependence of γ is well approximated by a power-law, γ ∝ rα(the Einasto

profile). The shape parameter, α, varies slightly but significantly from halo to halo,

implying that the mass profiles of ΛCDM halos are not strictly universal: different

halos cannot, in general, be rescaled to look identical. Departures from similarity are

also seen in velocity dispersion profiles and correlate with those in density profiles

so as to preserve a power-law form for the spherically averaged pseudo-phase-space

density, ρ/σ3∝ r−1.875. The index here is identical to that of Bertschinger’s sim-

ilarity solution for self-similar infall onto a point mass from an otherwise uniform

Einstein-de Sitter Universe. The origin of this striking behaviour is unclear, but its

robustness suggests that it reflects a fundamental structural property of ΛCDM halos.

Our conclusions are reliable down to radii below 0.4% of the virial radius, providing

well-defined predictions for halo structure when baryonic effects are neglected, and

thus an instructive theoretical template against which the modifications induced by

the baryonic components of real galaxies can be judged.

∼1. Asymp-

Key words: cosmology: dark matter – methods: numerical

1INTRODUCTION

A couple of decades of steady progress in the simulation of

non-linear structures in a cold dark matter (CDM) domi-

nated universe have resulted in significant advances in our

⋆E-mail: jfn@uvic.ca

understanding of the clustering of dark matter on the scale

of galactic halos. There is now widespread consensus that

the hierarchical assembly of CDM halos yields: (1) mass pro-

files that are approximately “universal” (i.e., independent of

mass and cosmological parameters aside from simple physi-

cal scalings (Navarro et al., 1996, 1997, hereafter NFW), (2)

strongly triaxial shapes, with a slight preference for nearly

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Navarro et al..

prolate systems (e.g., Frenk et al., 1988; Jing & Suto, 2002;

Allgood et al., 2006; Hayashi et al., 2007), (3) abundant,

but non-dominant, substructure (Klypin et al., 1999; Moore

et al., 1999a; Ghigna et al., 2000; Gao et al., 2004), and

(4) “cuspy” inner mass profiles, where the central density

increases systematically as the numerical resolution of the

calculation is improved (see, e.g., NFW, Moore et al., 1999b;

Fukushige & Makino, 2001; Navarro et al., 2004; Diemand

et al., 2005).

Despite this consensus, there are a number of issues

where conflicting claims may be found in the recent liter-

ature, hindering the design and interpretation of observa-

tional tests aimed at validating or ruling out various aspects

of the CDM theory on these scales. One contentious issue

concerns the statistics, spatial distribution, and structure of

substructure, and their consequences for the discovery and

interpretation of possible signals of dark matter annihilation

in the gamma-ray sky (Stoehr et al., 2003; Diemand et al.,

2007; Kuhlen et al., 2008; Springel et al., 2008a, and refer-

ences therein). The controversy extends to the structure of

the inner cusps both of the main halo and of substructure

halos, where some recent work has claimed a well-defined

central slope of ρ ∝ r−1.2(Diemand et al., 2004, 2005, 2008)

whereas others have argued that no compelling evidence for

such power-law behaviour is apparent (Navarro et al., 2004;

Graham et al., 2006).

Considerable debate also surrounds whether the struc-

ture of CDM halos is truly “universal”. This is indeed the

case if halos have mass profiles that are well-described by

two-parameter formulae, such as the NFW profile or some

of its modifications; see, for example, Moore et al. (1999b,

hereafter M99). These profiles have two scaling parameters

(mass and size) but fixed shape, so that two different halos

can, in principle, be rescaled to be indistinguishable from

each other.

On the other hand, recent work suggests that at least

three parameters may be needed to describe halo mass pro-

files accurately. An example is the Einasto formula (Einasto,

1965), shown by Navarro et al. (2004) to improve signif-

icantly the accuracy of the fits to the inner density pro-

files of simulated halos. It is unclear from that work, how-

ever, whether the improvement is due to the fact that the

Einasto formula has a different asymptotic inner behaviour

than NFW or to the extra shape parameter it introduces.

Merritt et al. (2005, 2006) explored this further and argued

that the third parameter is indeed needed to account faith-

fully for the curvature in the shape of the density profile.

Merritt et al.’s conclusions have received support from the

work of Gao et al. (2008) and Hayashi & White (2008), who

have stacked density profiles of many halos of similar mass to

show that mean profile shape, and, in particular, the Einasto

shape parameter α (see eq. 4 below), depends systematically

on halo mass. This implies that the mass profile of ΛCDM

halos is not strictly universal; no simple scaling of the av-

erage profile of cluster halos will provide an accurate fit to

the average profile of galaxy halos.

Many of these controversies and uncertainties may be

traced to the fact that earlier work has lacked the numerical

resolution and the representative halo sample needed to set-

tle the debate. For example, the dark matter annihilation

flux observable from Earth depends crucially on resolving

not only substructures but also the nested “substructure

within substructure” expected from the hierarchical assem-

bly of CDM halos. Only the most recent simulations have

been able to begin addressing this issue (see, e.g., Diemand

et al., 2008; Springel et al., 2008a,b).

A similar comment applies to the structure of the inner

cusp, where pinning down the asymptotic inner behaviour of

the dark matter density profile depends crucially on under-

standing the limitations introduced by, for example, finite

particle number, gravitational softening, and time-stepping

technique.

We have shown in earlier work (Power et al., 2003, here-

after P03) that, when suitable choices of the numerical pa-

rameters are made, the main factor determining the inner-

most radius where the mass profile may be measured reli-

ably is the total number of particles used in the simulation.

Empirically, the boundary of the region where numerical

convergence is achieved roughly corresponds to the radius

where the two-body relaxation time, trelax, exceeds the age

of the Universe. Since trelax scales roughly like the enclosed

number of particles times the local orbital timescale, and

the latter drops sharply toward the centre, extending the

resolved region inwards even modestly requires a dramatic

increase in the total number of particles.

These difficulties, coupled to the significant halo-to-halo

scatter already seen in early work, imply that substantive

progress on these issues requires a concerted numerical ef-

fort where several different halos are simulated with varying

numerical resolution, so that cosmic variance and numerical

convergence may be assessed directly.

These are the aims of The Aquarius Project, a recently

completed suite of numerical simulations of the formation of

galaxy-sized halos in the ΛCDM cosmogony. The series in-

cludes re-simulations of six different ∼ 1012M⊙ halos where

the number of particles is systematically varied. In one case,

the same halo is simulated 5 times, increasing gradually the

number of particles in the halo from about one million to

∼ 1.1 billion within the virial radius. The highest resolution

simulations of the other 5 halos have roughly 100-200 million

particles each within the virialized region.

The simulation series has been presented recently by

Springel et al. (2008a,b), where the interested reader may

find relevant details. Our first paper (Springel et al., 2008a)

deals with predictions of the annihilation signal whereas the

second (Springel et al., 2008b) addresses the statistics, spa-

tial distribution, and structure of dark matter substructures.

Here we deal with the structure of the main halo, with spe-

cial emphasis on the structure of the inner cusp. The plan

of the paper is as follows. Sec. 2 summarizes briefly the nu-

merical parameters of our simulations; Sec. 3 and 4 present

our main results. We conclude with a brief discussion and

summary in Sec. 5.

2THE NUMERICAL SIMULATIONS

We present here for completeness a brief summary of the

numerical simulations, and refer the reader to Springel et

al. (2008a,b) for further details.

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The Diversity and Similarity of Simulated Cold Dark Matter Halos

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Figure 1. Spherically-averaged density (left) and circular velocity (right) profiles for the Aq-A halo simulation series. Different colours

correspond to different resolution runs, as labeled in the figure. The density profile is multiplied by r2in order to emphasize small

deviations. The bumps in the outer regions may be traced to the presence of substructure and unrelaxed tidal debris. Profiles are shown

from ∼ 3r200 down to the “convergence radius”, r(1)

conv, corresponding to the radius where the relaxation time, trelax, is of the order of

the age of the Universe. The thick portion of each profile indicates the region r > r(7)

universe and where stricter convergence is achieved. Outside r(7)

convcircular velocity estimates converge to better than 2.5% (see Fig. 2).

The dot-dashed line shows an Einasto profile with α = 0.17 matched at (r−2,ρ−2), the peak in the r2ρ profile. This provides an excellent

fit to the structure of the inner regions of the halo, as shown by the residuals plotted in the bottom panels. Arrows indicate the softening

length hs of each simulation.

convwhere trelaxis more than 7 times the age of the

2.1The Cosmological Parameters

All our simulations assume a ΛCDM cosmogony with the

following parameters: Ωm = 0.25, ΩΛ = 0.75, σ8 = 0.9,

ns = 1, and Hubble constant H0 = 100hkms−1Mpc−1=

73kms−1Mpc−1. These cosmological parameters are the

same adopted in previous numerical work by our group,

such as the Millennium Simulation of Springel et al. (2005),

and are consistent, within their uncertainties, with con-

straints derived from the WMAP 1- and 5-year data analy-

ses (Spergel et al., 2003; Komatsu et al., 2008) and with the

recent cluster abundance analysis of Henry et al. (2008).

2.2The Code

The simulations were carried out with a new version of

the GADGET (Springel et al., 2001; Springel, 2005) parallel

cosmological code. This version, which we call GADGET-3,

has been especially developed for this project, and imple-

ments a novel domain decomposition technique in order to

achieve unprecedented dynamic range in massively-parallel

computer systems without sacrificing load balancing or nu-

merical accuracy. Time stepping is carried out with a kick-

drift-kick leap-frog integrator where the timesteps are based

on the local gravitational acceleration, together with a con-

servatively chosen maximum timestep for all particles.

Pairwise particle interactions are softened with a spline

of scalelength hs, so that they are strictly Newtonian for par-

ticles separated by more than hs. The resulting softening is

roughly equivalent to a traditional Plummer-softening with

scalelength ǫG ∼ hs/2.8. The gravitational softening length

is kept fixed in comoving coordinates throughout the evo-

lution of all our halos. The dynamics is then governed by a

Hamiltonian and the phase-space density of the discretized

particle system should be strictly conserved as a function of

time (Springel, 2005).

2.3 Halo Selection

All halos in the Aquarius suite were identified for resimu-

lation in a 9003-particle parent simulation of a 100h−1Mpc

box. The identification technique selects all ∼ 1012M⊙halos

in the box and chooses, at random, a few of them that sat-

isfy a mild isolation criterion (no neighbour exceeding half

its mass within 1h−1Mpc). This criterion is only imposed

in order to remove halos in the vicinity of massive groups

and clusters, which may have evolved differently from the

average.

Each halo is then resimulated at various resolutions,

making sure that each resimulation shares the same power

spectrum and phases at all resolved spatial frequencies. Ini-

tial displacements are imprinted using the Zeldovich ap-

proximation and a ‘glass-like’ uniform particle load (White,

1996). The 100h−1Mpc simulation box is divided into

a “high-resolution” region, which corresponds to the La-

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Navarro et al..

Figure 2. Top panel: Fractional deviations in the circular velocity

profile of the Aq-A convergence series versus the (enclosed) relax-

ation time, trelax, expressed in units of the circular orbit period at

the virial radius, tcirc(r200). Deviations are measured relative to

the highest resolution halo, Aq-A-1. Note that departures from

convergence for all simulations are similar when expressed this

way, indicating that trelax is the main parameter determining

convergence. Solid circles mark the location of the convergence

criterion proposed by P03. Note that Vc estimates converge there

to about 10%. A stricter convergence criterion, e.g., 2.5% conver-

gence in Vc, is achieved at larger radii, where trelax∼ 7tcirc(r200)

(right vertical line). Bottom panel: Relaxation time versus radius

for all five Aq-A simulations. Arrows indicate hs = 2.8ǫG, the

lengthscale where pairwise interactions become Newtonian.

grangian region surrounding the target halo, and a low-

resolution region (the rest of the box), which is represented

with a smaller number of particles with mass increasing with

distance to the target halo. We have carefully designed the

geometry of the high-resolution region in order to avoid con-

tamination of the halo by massive low-resolution particles.

Typically, about 30% of particles in the high-resolution re-

gion end up in the virialized region of the final halo, and no

higher mass particles end up within the virial radius of the

final halo.

Table 1 lists some basic information about each simu-

lation. This includes a symbolic simulation name, the parti-

cle mass in the high-resolution region, mp, the gravitational

softening length, ǫG, the virial radius⋆, r200, as well as the

total mass, M200 and the total number of particles, N200,

enclosed within r200. Other structural parameters of inter-

est include the location of the peak in the circular velocity

profile, specified by rmax and Vmax, as well as that of the ve-

locity dispersion profile (σmax and r(σmax)). σhost indicates

the 1D rms velocity of the main halo within r200 (excluding

substructures).

Table 1 lists only information on the halos used in this

paper. A more complete list of numerical parameters may

be found in Springel et al. (2008b). One of our halos, la-

beled Aq-A, has been resimulated 5 times, spanning a fac-

tor of ∼ 2000 in particle mass. Our naming convention uses

the tags “Aq-A” through “Aq-F” to refer to each of the six

Aquarius halos. An additional suffix “1” to “5” denotes the

resolution level. “Aq-A-1” is our highest resolution calcula-

tion: it follows the surroundings of Aq-A with ∼ 4.4 billion

particles, ∼ 1.1 billion of which end up within r200. We have

level-2 simulations of all 6 halos, corresponding to between

100 and 200 million particles per halo (within r200). The

softening parameters of each simulation adopt the “optimal”

softening recommendation of P03, which aims to balance the

number of timesteps required for accurate integration whilst

minimizing the loss of spatial resolution.

2.4Radial Profiles

Our analysis uses spherically-averaged profiles of the ba-

sic dynamical properties describing the structure of ΛCDM

halos: the density, circular velocity, velocity dispersion,

and anisotropy profiles. Typically, these are computed in

50 spherical shells equally spaced in log10r (where r is

the distance to the halo center), and spanning the range

1.5 × 10−4< r/r200 < 3. (When different choices for ei-

ther the number of bins or the radial range are made, this

is stated explicitly in the analysis below.) These concentric

⋆We define the virial mass of a halo, M200, as that contained

within a sphere of mean density 200 × ρcrit. The virial mass de-

fines implicitly the virial radius, r200, and virial velocity, V200=

(GM200/r200)1/2, of a halo, respectively. We note that other defi-

nitions of “virial radius” have been used in the literature; the most

popular of the alternatives adopts a density contrast (relative to

critical) of ∆ ≈ 178Ω0.45

m

∼ 100 (for our adopted cosmological

parameters, see Eke et al. (1996)). We shall refer to these alter-

native choices, where appropriate, with a subscript indicating the

value of ∆; i.e., r50would be the virial radius obtained assuming

∆ = 50, and so an enclosed density 200 times the mean cosmic

value.

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The Diversity and Similarity of Simulated Cold Dark Matter Halos

5

Figure 3. Left: Spherically-averaged density profiles of all level-2 Aquarius halos. Density estimates have been multiplied by r2in order

to emphasize details in the comparison. Radii have been scaled to r−2, the radius where the logarithmic slope has the “isothermal” value,

−2. Thick lines show the profiles from r(7)

profiles, which are fixed in these scaled units. This scaling makes clear that the inner profiles curve inward more gradually than NFW,

and are substantially shallower than predicted by M99. The bottom panels show residuals from the best fits (i.e., with the radial scaling

free) to the profiles using various fitting formulae (Sec. 3.2). Note that the Einasto formula fits all profiles well, especially in the inner

regions. The shape parameter, α, varies significantly from halo to halo, indicating that the profiles are not strictly self-similar: no simple

physical rescaling can match one halo onto another. The NFW formula is also able to reproduce the inner profiles quite well, although

the slight mismatch in profile shapes leads to deviations that increase inward and are maximal at the innermost resolved point. The

steeply-cusped Moore profile gives the poorest fits. Right: Same as left, but for the circular velocity profiles, scaled to match the peak of

each profile. This cumulative measure removes the bumps and wiggles induced by substructures and confirms the lack of self-similarity

apparent in the left panel.

convoutward; thin lines extend inward to r(1)

conv. For comparison, we also show the NFW and M99

shells are centered at the location of the particle identified by

the SUBFIND algorithm (Springel et al., 2001) as having the

minimum gravitational potential. Extensive tests show that

this procedure identifies the region where the local density of

the main subsystem of each halo peaks, and is coincident in

most cases (except perhaps major ongoing mergers between

comparable-mass halos) with the results of other methods,

such as the “shrinking sphere” method discussed by P03.

The mass density in each radial bin is estimated as the

dark mass in the bin divided by its volume, and assigned to

a radius corresponding to the bin center. Circular velocities

are computed by adding up the mass of each bin plus all

interior ones, and assigned to the radius corresponding to the

outer edge of the bin. The construction of velocity dispersion

and anisotropy profiles is described in detail in Sec. 4.1.

When differentiation is necessary, such as when computing

the logarithmic slopes shown in Figs. 5 and 6, we use a simple

3-point Lagrangian interpolation to perform the numerical

differentiation (as implemented by the DERIV subroutine of

the IDL software package).

3MASS PROFILES

3.1Numerical Convergence

We begin our study of the mass profile by using our series

of re-simulations of the Aq-A halo in order to assess the ra-

dial range where numerical convergence is achieved. Figure 1

shows the mass profile of the five Aq-A resimulations; the left

panels show the spherically-averaged density profile (multi-

plied by r2in order to emphasize small departures); the right

panels the corresponding circular velocity profile. Lines of

different colours correspond to different resimulations, as la-

beled. Arrows indicate hs = 2.8ǫG, the lengthscale where

softened pairwise interactions become fully Newtonian.

This figure demonstrates the striking numerical conver-

gence achieved in our re-simulations. Outside some char-

acteristic radius (which we discuss below), all the profiles

are essentially indistinguishable from each other, even down

to details such as “bumps” in the outer regions caused by

the presence of substructure. As discussed by Springel et al.

(2008b), this reflects the high quality of the numerical in-

tegration of GADGET-3 and the careful approach we have

taken to building our initial conditions; indeed, the Aq-A

resimulations not only reproduce faithfully the properties

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Navarro et al..

of the main halo, but even the mass, location and internal

structure of most major substructures.

Inevitably, near the centre the mass profiles diverge as a

consequence of numerical limitations. Each profile is plotted

down to the “convergence radius” proposed by P03. These

authors demonstrate that deviations from convergence de-

pend (for appropriate choices of other numerical parame-

ters) solely on the number of particles, and scale roughly

with the collisional “relaxation” time, trelax. Expressed in

units of the circular orbit timescale at r200 (which is of the

order of the age of the Universe), κ = trelax/tcirc(r200), the

relaxation time may be written as:

√200

8

κ(r) =

N

8lnN

r/Vc

r200/V200

=

N(r)

lnN(r)

?ρ(r)

ρcrit

?−1/2

,(1)

where N = N(r) is the enclosed number of particles and

ρ(r) is the mean enclosed density within r.

According to P03, deviations of roughly 10% are ex-

pected in the Vc profile where κ ≈ 1, and they adopted this

condition to define the convergence radius, rconv. Stricter

convergence demands larger values of κ, and we shall use

a superscript on rconv to denote the value of κ adopted for

its definition. For instance, r(1)

κ = 1.

Profiles in Fig. 1 are thus plotted in the range [r(1)

3r200]. As shown in the bottom right panel, this inner radius

indeed corresponds to the point where systematic deviations

in Vc(r) reach ∼ 10%. It is also clear from this figure that

convergence in the local density profile is always much easier

to achieve, so concentrating our analysis on the enclosed

mass profile, or on the circular velocity, is a conservative

approach.

Although each halo converges over a different radial

range, the departures from convergence are all similar when

expressed in terms of κ. This is shown in the top panel in

Fig. 2, where differences in Vc from our highest-resolution

halo, Aq-A-1, are shown as a function of κ for the other Aq-

A resimulations. Deviations of ∼ 10% are typical at κ = 1;

convergence to better than ∼ 2.5%, on the other hand, re-

quires κ ≈ 7 (indicated by the right dashed vertical line).

We may use these results to estimate convergence radii

for our highest resolution run, Aq-A-1: its Vc profile con-

verges to better than 10% for radii r > r(1)

pc; 2.5% convergence or better is expected for r > r(7)

253h−1pc (see bottom panel of Fig. 2). Convergence radii

for various values of κ are listed in Table 2 for each simulated

halo.

conv = r(κ=1)

conv

corresponds to

conv,

conv = 112h−1

conv =

3.2 Fitting formulae

The fitting formulae we have used to describe the mass pro-

file of our simulated halos are the following: (i) The NFW

profile, given by

ρs

(r/rs)(1 + r/rs)2,

ρ(r) =

(2)

(ii) the modification to the NFW profile proposed by M99,

ρM

(r/rM)1.5[1 + (r/rM)1.5],

ρ(r) =

(3)

and (iii) the Einasto profile,

ln(ρ(r)/ρ−2) = (−2/α)[(r/r−2)α− 1]. (4)

Figure 4. Minimum-Q values as a function of the Einasto param-

eter α for best fits to all level-2 halo profiles in the radial range

0.01 < r/r−2 < 5. Colors identify different halos, and line types

the number of bins chosen for the profile. The minimum-Q val-

ues obtained for NFW and M99 best fits are also shown, and are

plotted at arbitrary values of α for clarity. Note that Einasto fits

are consistently better than NFW which are consistently better

than M99, and that a significant improvement in Q is obtained

when letting α vary in the Einasto formula. Q is approximately

independent of the number of bins used in the profile, and is min-

imized for different values of α for each individual halo. See text

for further details.

Because each of these formulae defines the character-

istic parameters in a slightly different way, we choose to

reparametrise them in terms of r−2and ρ−2 ≡ ρ(r−2), which

identify the “peak” of the r2ρ profile shown in the left panel

of Fig. 1. This marks the radius where the logarithmic slope

of the profile, γ(r) = −dlnρ/dlnr, equals the isothermal

value, γ = 2.

The characteristic radius, r−2, is a well-defined scale-

length which is relatively easy to identify in each halo with-

out resorting to any particular fitting formula. In practice,

we determine r−2 by computing the logarithmic slope pro-

file, γ(r), and identifying where a low-order polynomial fit

to it intersects the isothermal value. Each r2ρ profile is then

visually inspected in order to ensure that r−2 corresponds

to the main peak of the profile, and that it is not unduly

influenced by secondary peaks that arise as the result of sub-

structure. (See the left panel of Fig. 1.) Table 2 lists r−2 and

ρ−2 for all our simulated halos. Note that for the NFW pro-

file, r−2 = rs and ρ−2 = ρs/4, while for the Moore profile,

ρ−2 = (4/3)ρM and r−2 = 2−2/3rM.

We note that, unlike NFW or M99, when α is al-

lowed to vary freely the Einasto profile is a 3-parameter

fitting formula. This is not, of course, the only possible

extension of NFW-like profiles which allows for a variable

shape with the aid of an extra free parameter. For example,

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The Diversity and Similarity of Simulated Cold Dark Matter Halos

7

Figure 5. Logarithmic slope of the density profile as a function

of radius for our Aq-A convergence series. As in other plots, thick

lines show results for r > r(7)

conv, thin lines extend the profiles down

to the less strict convergence radius r(1)

excellent numerical convergence for the slope is achieved down

to a radius intermediate between these two convergence radii.

Applied to the highest-resolution Aq-A-1 simulation, this implies

that the slope is shallower than the asymptotic value of the NFW

profile (r−1) in the inner regions. We see no sign of convergence to

an asymptotic inner power-law. Instead, the profiles get shallower

toward the centre as predicted by the Einasto formula (a straight

line in this plot). The “critical solution” of Taylor & Navarro

(2001) (which has a r−0.75asymptotic inner cusp) does better

than NFW but not as well as Einasto in reproducing the inner

profile of the halo.

conv. Comparison shows that

Merritt et al. (2006) compared N-body halos with the 3-

parameter Einasto formula, as well as with the anisotropic

model of Dehnen & McLaughlin (2005) and with the de-

projected Sersic (1968) model of Prugniel & Simien (1997).

Merritt et al. conclude that, overall, Einasto’s formula per-

forms best. Therefore, we adopt it here for the rest of our

analysis, although we do not exclude the possibility that

other 3-parameter formulae may perform at least as well as

Einasto’s. A full exploration of this issue is beyond the scope

of this paper.

3.3Fitting procedure

Best-fit parameters are found by minimizing the deviation

between model and simulation across all bins in a specified

radial range. In the case of the density profile, the best fit

is found by minimizing the figure-of-merit function, Q2, de-

fined by

Q2=

1

Nbins

Nbins

?

i=1

(lnρi− lnρmodel

i

)2. (5)

Figure 6. As Fig. 5, but for all level-2 resolution Aquarius halos,

after scaling radii to r−2.

This function provides an intuitively simple measure

of the level of disagreement between simulated and model

profiles. It is dimensionless; it weights different radii loga-

rithmically; and, for given radial range, Q2is approximately

independent of the number of bins used in the profile. Thus,

minimizing Q2yields for each halo well-defined estimates of

a model’s best-fit parameters. Note that when Q is small,

it is just the rms fractional deviation of the data from the

model.

It is less clear how to define a goodness-of-fit mea-

sure associated with Q2and, consequently, how to assign

statistically-meaningful confidence intervals to the best-fit

parameter values. This difficulty arises because, at the very

high resolution of the simulations analyzed here, discreteness

noise in the binned density estimates is negligible. The figure

of merit of a fit therefore depends not only on how faithfully

a model approximates a halo but also on the presence of indi-

vidual halo features that no simple fitting formula can hope

to reproduce. These distinct features are present on small

scales (substructure) and large scales (such as streams, as-

phericity, and other relics of each halo’s specific assembly

history; see, e.g., Vogelsberger et al., 2009). As a result, bin-

to-bin residuals are distinctly non-Gaussian and highly cor-

related, precluding the use of simple statistical tools such as

the χ2distribution in order to assess goodness of fit.

Assessing the acceptability of various Q values would

require the definition of a detailed statistical model in order

to measure reliably the departures of individual halos from

a smooth profile whose average shape (and scatter) could be

obtained directly by averaging various numerical realizations

of halos of the same mass. Unfortunately, such procedure is

unlikely to be robust with only 6 halos in our sample.

Therefore, we limit our analysis to comparing the

minimum-Q values obtained with various formulae, and to

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8

Navarro et al..

discussing how Q changes as the fitting parameters are var-

ied. The actual value of Q is, after all, a reliable and objec-

tive measure of the average per-bin deviation from a par-

ticular model. As we discuss below, this is, in many cases,

enough to prefer unequivocally one fitting formula over an-

other and to make a compelling case for the need of an extra

parameter in the fit.

3.4Einasto vs NFW vs M99

The left panel of Fig. 3 compares the density profiles of

all six level-2 Aquarius halos, after scaling radii to r−2 and

densities to ρ−2. The right-hand panel shows the circular

velocity profiles, scaled in an analogous manner to match

the peak of the profile, identified by rmax and Vmax. In these

scaled units, the fitting formulae introduced in Sec. 3.2 are

curves of fixed shape and normalization, as shown by the

thin solid, dashed, and dot-dashed curves in Fig. 3. (The

Einasto curve adopts α = 0.159 in this figure.)

Comparison with the simulations (thick curves) indi-

cates that there is a clear mismatch between the shape of

the halo profiles and those of the NFW and M99 fitting

formulae. This is not just a result of enforcing the r−2-ρ−2

scaling. We illustrate this by showing, in the two bottom

panels of Fig. 3, residuals from best fits obtained by adjust-

ing both fit parameters of the NFW and M99 profiles (r−2

and ρ−2) in order to minimize Q2. (The radial range chosen

for these fits is r(1)

in the residuals, which are largest (and increasing) at the

innermost radius of the profile. Because of the shape mis-

match, extrapolating either the NFW or M99 fits further

inwards, to regions less well resolved numerically, is almost

guaranteed to incur substantial error.

The large-scale radial trend of the residuals from the

best Einasto fits (middle panels of Fig. 3), on the other hand,

is rather weak, suggesting that the shape of the simulated

halo profiles are much better accommodated by this formula.

This is not just a result of the extra shape parameter in

the Einasto formula: even when keeping α fixed to a single

value, residuals are smaller and have less radial structure

than those from either NFW or M99.

We show this in Fig. 4, where we plot the minimum-

Q (Qmin) values of the best Einasto fits for all six level-2

Aquarius halos, as a function of the shape parameter α. For

given value of α the remaining two free parameters of the

Einasto formula are allowed to vary in order to minimize

Q2. Different line types correspond to different numbers of

bins used to construct the profile (from 20 to 50), chosen to

span in all cases the same radial range, 0.01 < r/r−2 < 5,

a factor of 500 in radius. Minimum-Q values are computed

using a similar procedure for the NFW and M99 formulae,

and are shown, for each halo, with symbols of corresponding

colour.

In terms of Qmin, Einasto fits are consistently superior

to NFW or M99, whether or not the α parameter is adjusted

freely. For example, for fixed α = 0.15, all Einasto best fits

have minimum-Q values below ∼ 0.03. For comparison, best

NFW and M99 fits have an average ?Qmin? ∼ 0.06 and 0.095,

respectively. These numbers correspond to Nbins = 20, but

they are rather insensitive to Nbins, as may be judged from

conv < r < 0.5r200.) Note the “S” shape

the small difference between the various lines corresponding

to each halo in Fig. 4.

We emphasize that, although the improvement obtained

with Einasto’s formula is significant, NFW fits are still ex-

cellent, with a typical rms deviation of just ∼ 6% over a

range of 500 in radius. The use of the NFW formula may

thus be justified for applications where this level of accuracy

is sufficient over this radial range.

When α is adjusted as a free parameter, ?Qmin? ∼ 0.018

for Einasto fits. Furthermore, there is, for each halo, a well

defined value of α that yields an absolute minimum in Q.

The Q-dependence on α about this minimum is roughly sym-

metric and, as expected, nearly independent of the number

of bins used in the profile. The minimum in Q is sharp;

a shift of just 0.015 in α typically leads to an increase of

∼ 50% in Q around the minimum. Given that the value of α

that minimizes Q varies from 0.130 for Aq-E-2 to 0.173 for

Aq-B-2, we conclude that the improvement obtained when

allowing α to vary is significant. We quote nominal “error

bars” for α in Table 2 that bracket the interval where Q

deviates by less than 50% from the absolute minimum in

Fig. 4.

3.5 Self-similarity?

The need for a variable α discussed above illustrates one

of our main findings: namely, that the mass profiles of our

Aquarius halos are not strictly self similar. The shapes of the

profiles are subtly but significantly different from each other,

and no rescaling can match one exactly to another. Halo

Aq-E-2 provides the most striking example, deviating from

halo Aq-D-2, for example, by almost a factor of 2 in density

at ∼ 0.03r−2. The same differences in mass profile shape

are also easily appreciated in the scaled circular velocity

profiles, which indicate that the departures from similarity

are genuine and not just caused by inaccuracies in the scaling

or by the “bumps and wiggles” caused by unrelaxed tidal

debris and substructure.

We have verified this further by performing the same

analysis after removing bound substructure clumps iden-

tified by SUBFIND: the same conclusion applies to the

“cleaned” profiles of the main smooth halo. With hindsight,

this is perhaps not too surprising. Bound substructures do

not amount to more than ∼ 10% of the halo mass (Springel

et al., 2008b), and therefore cannot alter the results dis-

cussed above.

We have also checked that the differences in α are not

caused by transient departures from equilibrium or numeri-

cal resolution: the same qualitative trends, and indeed very

similar α values, are seen at earlier times and in runs with

fewer particles. There also seems to be little correlation be-

tween α and the overall triaxiality of the system; however,

we shall only deal here with spherically-averaged profiles,

and defer a detailed study of departures from sphericity to

a later paper.

Although the departures from similarity appear signif-

icant, we must also emphasize that they are rather subtle,

and are only clearly evident because of the large radial range

resolved by our simulations, about three decades in radius

within the virialized region of a halo. Simulations with more

limited numerical resolution have hinted at this but had dif-

ficulty making such a compelling case for non-similarity (see,

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The Diversity and Similarity of Simulated Cold Dark Matter Halos

9

Figure 7. Maximum value of the asymptotic inner slope of the

density cusp, as a function of radius for our Aq-A convergence se-

ries. Excellent numerical convergence is achieved at radii compa-

rable to r(7)

conv(the inner limit of the thick lines; thin lines extend

down to r(1)

conv). This shows that there is not enough mass near

the centre of Aq-A to sustain a cusp steeper than ρ ∝ r−0.9±0.1.

Arrows are as in Fig. 1.

e.g., Navarro et al., 2004; Merritt et al., 2005; Stoehr, 2006;

Merritt et al., 2006).

3.6 The Cusp

It is clear from the residuals in the bottom panels of Fig. 3

that, near the centre, the M99 profile approximates the sim-

ulated halos more poorly than either NFW or Einasto. The

weak performance of the M99 formula may be traced to

its steep asymptotic inner slope, ρ ∝ r−1.5. Indeed, all six

Aquarius halos have measured slopes in the inner regions

that are substantially shallower than −1.5. This is shown

in Figs. 5 and 6, where the thick portion of each curve cor-

responds to r > r(7)

r(1)

side r(7)

beyond r(1)

Interestingly, the slope of the Aq-A-1profile at r = r(7)

is exactly −1, and becomes shallower inward, so it is clear

that at least for this halo we are able to resolve a region

where the dark matter profile has become shallower than

−1, the asymptotic value of the NFW profile. Fig. 6 shows

the radial dependence of the logarithmic slope for all six

level-2 halos and confirms the general applicability of the

Aq-A results: the measured slopes of all halos approach −1

(and are certainly shallower than −1.5) at the innermost

resolved point.

Figs. 5 and6 also make clear that there is no sign

that the profiles are approaching power-law behaviour near

conv and the innermost point plotted to

conv. In all cases, the logarithmic slopes converge well in-

conv, and only minor deviations may be seen at radii

conv.

conv

Figure 8. As Fig. 7, but for our six level-2 Aquarius halos. Re-

sults are similar in all cases and rule out cusps steeper than r−1

for ΛCDM halos.

the centre: they keep getting shallower to the innermost re-

solved radius. This behaviour is well captured by the Einasto

model, where the logarithmic slope is simply a power-law of

radius, dlnρ/dlnr ∝ rα. Our results thus rule out recent

claims of cusps as steep as r−1.2in typical ΛCDM halos

(Diemand et al., 2004, 2005, 2008).

This conclusion is unlikely to depend on the details of

our profile construction and/or fitting procedures. Indeed, as

we show in the next subsection, there is actually not enough

mass within the innermost resolved radius to allow for a cusp

as steep as r−1.2. Recent work by Stadel et al. (2008), also

based on very high-resolution simulations, agrees with our

present conclusions, and argues for asymptotic inner slopes

shallower than −1, as previously suggested by Navarro et al.

(2004).

3.7The Asymptotic Inner Slope

The results presented above do not preclude the possibility

that a shallow power-law cusp may be present in the inner-

most regions which are still unresolved in our simulations. It

is therefore interesting to estimate the maximum value that

the slope of such a cusp may take. This is constrained, at

any radius, by the total enclosed mass and the local value

of the spherically averaged density: slopes steeper than γmax

require more mass than is available within that radius. This

constraint assumes only that the logarithmic slope is mono-

tonic with radius and that the halo is not hollow. It is then

straightforward to show that the maximum possible inner

asymptotic slope is γmax = 3(1 − ρ(r)/¯ ρ(r)), where ¯ ρ(r) is

the mean density enclosed within r. Evaluated at the inner-

most radius where both local density and enclosed mass (or,

equivalently, circular velocity) have converged, this quan-

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Navarro et al..

tity provides an important constraint on the density profile

at radii that remain unresolved even in our best simulations.

We show this parameter as a function of radius for our

Aq-A convergence series in Fig. 7. This figure shows that

γmax converges to better than 0.1 for r > r(7)

most point of the thick portion of the profiles). Our data for

Aq-A thus indicates that there is not enough mass in the

unresolved region to support a cusp steeper than r−0.9±0.1.

Fig. 8 shows that the results for Aq-A are not exceptional:

all our level-2 Aquarius halos suggest maximum possible

asymptotic slopes of about −1.

conv (the inner-

4DYNAMICAL PROFILES

4.1Velocity Dispersion Structure

Fig. 9 shows velocity dispersion and anisotropy profiles for

our Aq-A series and demonstrates that the excellent numer-

ical convergence of our simulations extends to their veloc-

ity dispersion structure. The velocity dispersion (squared)

is computed simply as twice the specific kinetic energy in

each spherical shell and the anisotropy as β = 1−σ2

where σ2

tangential and radial motions, respectively. Besides numer-

ical convergence, the panels in this figure illustrate two im-

portant points. The first concerns the shape of the velocity

dispersion profiles (left panel in Fig. 9), which is remarkably

similar to that of the r2ρ profiles shown in Fig. 1. This co-

incidence suggests an intimate connection between density

and velocity dispersion, which we explore in more detail in

Sec. 4.4. The second point concerns the anisotropy profile,

which is clearly non-monotonic. It is nearly isotropic at the

centre, becomes radially anisotropic at intermediate radii,

but the dominance of radial motions decreases again near

the virial radius. As shown in Fig. 10, these properties ap-

pear to be rather general, since all six Aquarius halos have

non-monotonic anisotropy profiles and similar velocity dis-

persion profile shapes.

t/(2σ2

r),

t and σ2

r are the (squared) velocity dispersion in

4.2Self-similarity?

Fig. 10 also demonstrates a clear lack of self-similarity in the

structure of the simulated halos. We have chosen to empha-

size this by rescaling all profiles so as to match the peak of

the σ(r) curve, which occurs at r(σmax). This scaling demon-

strates that, as with the density profiles, the shape of the

σ(r) profiles differs subtly but significantly amongst halos.

We have checked that these differences in shape are not due

to bound subhalos; removing all the subhalos identified by

our SUBFIND algorithm and recalculating the dispersion

and anisotropy profiles results in only rather minor changes

The most striking case is again that of halo Aq-E-2 (blue

curve), whose σ(r) profile is much broader than the others.

Recall that this halo also stands out in Fig. 3 as having an

unusually broad r2ρ profile. Halo Aq-E-2 also has an unusual

velocity anisotropy profile, with less predominance of radial

motions than the rest of the series. The departures from sim-

ilarity in mass and velocity structure therefore seem closely

linked, suggesting that these halos may share a common

property that combines density and velocity dispersion. We

explore this in Sec. 4.4 below.

4.3Anisotropy-slope relation

We may use the results of the previous subsection to assess

recent claims by Hansen & Moore (2006) of a general con-

nection between the local values of logarithmic slope, γ, and

the velocity anisotropy, β. We show this in Fig. 11, where

we plot β vs γ for all level-2 Aquarius halos. Open circles

correspond to the inner regions of the halo (r(1)

whereas filled circles correspond to the outer regions (r−2 <

r < r200). As in other figures, different colours correspond

to the different Aquarius halos. The relation proposed by

Hansen & Moore is shown by a dashed line and accounts

reasonably well (albeit not perfectly) for our data in the in-

ner regions where both the anisotropy and the logarithmic

slope are monotonic functions of r.

However, there are large departures from this relation in

the outer regions, where the density profile steepens further

but the velocity ellipsoid tends to become less anisotropic.

The failure of the Hansen & Moore relation in the outer

regions is not unexpected since γ, unlike β, is monotonic

with radius. We conclude that, if a simple relation links

anisotropy and slope, it can only hold in the inner regions

of halos.

conv< r < r−2)

4.4The Phase-Space Density Profile

The similarity in shape between the σ2and r2ρ profiles high-

lighted above suggests that there may be a simple scaling

between densities and velocity dispersions in halos. This is

best appreciated by considering the quantity ρ/σ3, which,

for dimensional reasons, we shall call the pseudo-phase-space

density, although it is important to realise that it is not

the true coarse-grained phase-space density at the resolu-

tion of our simulations, or even the average of this quantity

in spherical shells. For consistency with the rest of our anal-

ysis, we calculate ρ/σ3directly from the estimates of ρ and

σ computed in concentric spherical shells.

Fig. 12 shows the ρ/σ3profile for our Aq-A conver-

gence series. As noted by Taylor & Navarro (2001), the pro-

file of this quantity is remarkably well approximated by a

power-law. More remarkable still is the fact that the power

law is indistinguishable from that predicted by the similar-

ity solution of Bertschinger (1985) for infall onto a point

mass in an otherwise unperturbed Einstein-de Sitter uni-

verse, ρ/σ3∝ r−1.875(dot-dashed line in Fig. 12). This so-

lution is spherically symmetric, involves purely radial mo-

tions, and is violently dynamically unstable, so its relevance

to ΛCDM halos is far from clear. The residuals in the bot-

tom panel of Fig. 12 are deviations from a Bertschinger law

matched within the characteristic radius r−2, where sub-

structure bumps and wiggles are minimal.

Note that, although there is only one free parameter in

this fit (the vertical scaling), the residuals do not exceed

∼ 20% anywhere within the virial radius, even though sub-

structures add significant noise to the dynamical measure-

ments in the outskirts of the halo. Interestingly, the residuals

increase when σr, the velocity dispersion in radial motions,

is used in place of the full 3D rms velocity, σ,to estimate the

“phase-space density”. Thus, the r−1.875behaviour seems to

concern the full kinetic energy content of each shell rather

than just radial or tangential motions.

Fig. 13 shows that similar conclusions apply to the rest

Page 11

The Diversity and Similarity of Simulated Cold Dark Matter Halos

11

Figure 9. Left panel: Velocity dispersion profiles for our Aq-A convergence series. Arrows, line-types and colours are as in Fig. 1. Note

the excellent numerical convergence. The shape of the velocity dispersion profile is remarkably similar to that of the r2ρ profile shown in

Fig. 1, highlighting the intimate connection between the density and velocity dispersion profiles which is responsible for the power-law

behaviour of the pseudo-phase-space density profile discussed in Sec. 4.4. Right panel: Anisotropy profiles for the Aq-A convergence

series. Note the non-monotonic variation with radius: the halo is nearly isotropic near the centre, is dominated by radial motions at

intermediate radii, but becomes markedly less anisotropic near the virial radius.

Figure 10. As Fig. 9, but for all six level-2 resolution Aquarius halos, scaled to match at the peak of the profile, identified by σmax

and r(σmax). This scaling highlights small but significant departures from similarity in the velocity dispersion structure of ΛCDM halos.

Note the correspondence in shape between the velocity dispersion and r2ρ profiles shown in Fig. 1, which reflects the “universal” pseudo-

phase-space density profile of the halos (Fig. 13). Note also that the non-monotonic behaviour of the anisotropy highlighted in Fig. 9 is

common to all six halos.

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12

Navarro et al..

of the Aquarius halos. Residuals from the Bertschinger law

are small for all halos, and are typically larger when the

radial velocity dispersion is used. Note that there is some

“curvature” in the residual profiles, suggesting that a power-

law is a good, but perhaps not perfect, description of the

radial dependence of ρ/σ3. We are currently investigating

the origin of this curvature and plan to report on it in a

future paper (Ludlow et al., in preparation).

A power-law radial dependence is approximately pre-

served when σr is used, but the best fitting value of the

exponent differs systematically from −1.875. This may be

seen in the bottom panels of Fig. 13, which show the resid-

uals from the best fitting ρ/σ3∝ rχlaw. The values of the

best-fit exponent for both ρ/σ3and ρ/σ3

tively) are listed in Table 2.

r(χ and χr, respec-

Perhaps the most important result from Fig. 13 is that

there seems to be very little scatter between halos when

considering their ρ/σ3profiles. Take, for example, the case of

halo Aq-E-2, which was a clear outlier in the density, velocity

dispersion, and anisotropy profiles. When considering ρ/σ3

this halo is unremarkable, and follows the Bertschinger law

as closely as the others.

This shows that there is a sense in which ΛCDM halos

are nearly universal, but that universality does not extend

to their density or velocity dispersion profiles separately, but

rather only to their pseudo-phase-space density profile. This

may appear a bold statement, and it certainly needs to be

corroborated by future work, but it offers an intriguing per-

spective into the origin of the near-universal density profiles

of halos, the meaning of the Einasto shape parameter, α,

and the provenance of their velocity dispersion structure.

These issues deserve further investigation.

We end by noting that, although it is still not clear

what leads to the power-law stratification of ρ/σ3, these re-

sults may be used to place constraints on the structure of

the central cusp, under the plausible (but admittedly un-

proven) assumption that the power-law behaviour of the

phase-space density continues all the way to the centre. For

example, Taylor & Navarro (2001) used this assumption to

show that, for isotropic systems, a power-law pseudo-phase-

space density implies an inner density cusp with ρ ∝ r−0.75.

This is certainly consistent with the results shown in Fig. 7,

which only exclude cusps steeper than r−0.9±0.1. However,

as we show in Fig 5, the detailed profile which they derive for

an isotropic halo with Bertschinger’s power-law ρ/σ3profile

is a significantly worse fit to our numerical data than the

Einasto profile.

The power-law behaviour of the pseudo phase-space

density has been confirmed by a number of authors, and

seems to be present even at early redshift (Vass et al., 2008).

Interestingly, the average power-law exponent to the ρ/σ3

profile is ?χr? ≈ 1.97, close to the “critical” 1.94 required by

Dehnen & McLaughlin (2005) to have a dynamical model

that is well behaved at all radii. Simulations of even larger

dynamic range seem required in order to explore the true

asymptotic inner behaviour of the dynamical profile of a

halo, if indeed there is any such asymptote.

r

Figure 11. Local values of the logarithmic slope of the density

profile plotted versus velocity anisotropy. The relation proposed

by Hansen & Moore (2006) is shown as a dashed line. Because

the density profile steepens gradually from the centre outwards

whereas the velocity anisotropy is non-monotonic, no simple re-

lation between these two quantities is valid throughout the halos.

The Hansen & Moore formula approximates our results quite well

in the inner regions, but large deviations may be seen outside r−2,

particularly at the largest radii where our halos are approximately

isotropic but their density profiles are steepest. Open circles cor-

respond to r(1)

conv < r < r−2, filled circles to r−2 < r < r200.

Colors are as in Fig. 3.

5 SUMMARY

We have analysed density, velocity dispersion, anisotropy

and pseudo-phase-space density profiles at redshift zero for

simulated halos from the Aquarius Project. This is a set of

six galaxy-sized halos whose formation and evolution have

been simulated at a variety of resolutions in their proper

ΛCDM context. The set includes the largest simulation of

this kind reported so far; a ∼ 4.4 billion particle simulation

in which the final halo has 1.1 billion particles within its

virial radius, r200. The set also includes simulations of all

six halos with 100 – 200 million particles within the virial

radius, as well as a comprehensive numerical convergence

study for the largest system. Our main conclusions are as

follows.

• Density profiles deviate slightly but significantly from

the NFW model, and are approximated well by a fitting

formula where the logarithmic slope is a power-law of ra-

dius: the Einasto profile (eq. 4). The steeply-cusped profile

of Moore et al. (1999) is a poor fit to the structure of our

six halos.

• We find convincing evidence that the shape parame-

ter of the Einasto formula varies from halo to halo at given

mass (see Table 2). This complements the earlier conclusion

of Merritt et al. (2006), Gao et al. (2008) and Hayashi &

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The Diversity and Similarity of Simulated Cold Dark Matter Halos

13

Figure 12. Pseudo-phase-space density profiles for our Aq-A convergence series, estimated as ρ/σ3, computed in concentric spherical

shells. Arrows, line-types, and colours are as in Fig. 1. Note the remarkable power-law behaviour of this quantity, a result already noted by

Taylor & Navarro (2001). The dot-dashed line is not a fit to the data, but rather the prediction of the similarity solution of Bertschinger

(1985) for infall onto a point mass in an otherwise unperturbed Einstein-de Sitter universe, ρ/σ3∝ r−1.875. This has been scaled to

match Aq-A at r < r−2. Residuals from the Bertschinger solution are shown in the bottom panels. Note that this power-law behaviour

is most evident when the full 3D velocity dispersion is used (left panels). When only the radial velocity dispersion is used (right panels)

deviations from the Bertschinger solution are considerably larger.

White (2008) that its mean value varies systematically with

halo mass. Together these results imply that the density

profiles of ΛCDM halos are not strictly self-similar: different

halos cannot be rescaled to look alike. This lack of similar-

ity extends to the kinematic structure, as measured by the

velocity dispersion and anisotropy profiles.

• Intriguingly, departures from similarity are minimized

when analyzing a pseudo-phase-space density profile defined

as ρ/σ3. This suggests a limited sense in which ΛCDM halos

are indeed nearly “universal”. The pseudo-phase-space den-

sity profiles are very well approximated by ρ/σ3∝ r−1.875,

the power law predicted by Bertschinger’s similarity solution

for infall onto a point mass in an otherwise unperturbed

Einstein-de Sitter universe. This simple law has only one

scaling parameter and no shape parameters, yet it approxi-

mates, for over six decades, the ρ/σ3profiles to better than

20-30%, all the way from the innermost resolved point to

the virial radius. The power-law description is, however, not

perfect, and further work designed to understand better its

origin and limitations seems warranted.

• Density profiles become monotonically shallower in-

wards, down to the innermost resolved point, with no indi-

cation that they approach power-law behaviour. The inner-

most slope we measure is slightly shallower than −1, a result

supported by estimates of the maximum possible asymptotic

inner slope.

• These results convincingly rule out recent claims that

typical ΛCDM halos may have asymptotic central cusps as

steep as r−1.2(Diemand et al., 2004, 2005, 2008). Shallower

cusps, such as the asymptotic r−0.75behaviour predicted

by the model of Taylor & Navarro (2001), cannot yet be

excluded. These results should discourage further work as-

suming CDM cusps steeper than r−1except possibly around

central black holes.

• Velocity anisotropy does not depend monotonically on

radius beyond r−2. Halos are roughly isotropic near the cen-

tre, are dominated by radial motions at intermediate radii,

but become more isotropic again as the virial radius is ap-

proached. This behaviour does not appear to be driven by

the presence of substructure. Given that the slope of the

density profile does increase monotonically with radius, this

implies that no simple relation between anisotropy and slope

can hold throughout a halo. The relation recently proposed

by Hansen & Moore (2006) works reasonably well in the

inner regions (r < r−2), but fails at larger radii.

The main aim of the Aquarius Project is to provide reli-

able theoretical predictions for the structure and formation

history of dark matter halos like that surrounding the Milky

Way down to radii of order 100 pc. This permits direct com-

parisons with a number of observations with minimal extrap-

olation, and it helps to design new observational strategies

aimed at testing the cold dark matter paradigm on these

very non-linear scales.

We recognize, however, that many of these tests and

predictions will apply to regions where baryons play an im-

portant dynamical role. Our numerical work provides robust

results for the limiting but unrealistic case of pure dark mat-

ter halos, and these will undoubtedly be modified in non-

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Navarro et al..

Figure 13. Pseudo-phase-space density profiles of all six level-2 Aquarius halos. Radii have been scaled to r−2, and the pseudo-phase-

space densities to maximise agreement within r−2. Note that for all six halos these profiles are very well approximated by power laws

with an exponent very close to that of the Bertschinger solution. All halos, including those that were outliers in the density, velocity

dispersion, and anisotropy profiles, are almost indistinguishable in this plot. Deviations from the Bertschinger law are typically more

pronounced when radial velocity dispersion is used instead of the full 3D velocity dispersion. Residuals from the best-fit power-laws,

ρ/σ3∝ rχ, are shown in the bottom panels. The values of χ are listed for each halo in Table 2.

trivial ways by the presence of baryons. Providing a full

account of the coupled structure of the cold dark matter

and baryonic components in galaxies like our own is clearly

the next major computational challenge, and it is likely to

exercise us for some time to come.

ACKNOWLEDGMENTS

The simulations for the Aquarius Project were carried out at

the Leibniz Computing Center, Garching, Germany, at the

Computing Centre of the Max-Planck-Society in Garching,

at the Institute for Computational Cosmology in Durham,

and on the ‘STELLA’ supercomputer of the LOFAR ex-

periment at the University of Groningen. This work was

supported in part by an STFC rolling grant to the ICC.

CSF acknowledges a Royal Society Wolfson Research Merit

award. AH acknowledges financial support from NOVA and

NWO.

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Page 15

The Diversity and Similarity of Simulated Cold Dark Matter Halos

15

Halomp

[M⊙/h]

ǫG

[pc/h]

r200

[kpc/h]

M200

[M⊙/h]

N200

[106]

Vmax

[km/s]

rmax

[kpc/h]

σhost

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Aq-A-2

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87

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179.41

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1.342×1012

1.357×1012

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208.75

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209.24

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20.69

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20.58

20.84

117.47

117.13

117.31

117.23

116.61

261.70

261.88

262.80

262.29

260.59

Aq-B-2

Aq-C-2

Aq-D-2

Aq-E-2

Aq-F-2

4.706×103

1.021×104

1.020×104

7.002×103

4.946×103

48

48

48

48

48

137.02

177.26

177.28

154.96

152.72

5.982×1011

1.295×1012

1.295×1012

8.652×1011

8.282×1011

127.09

126.77

126.98

123.56

167.45

157.68

222.40

203.20

179.00

169.08

29.31

23.70

39.48

40.52

31.15

89.59

124.08

113.15

101.73

96.78

190.74

270.50

254.28

215.14

204.53

Table 1. Basic parameters of the Aquarius simulations. We have simulated 6 different halos, each at several different numerical resolutions.

The leftmost column gives the simulation name, encoding the halo (A to F), and the resolution level (1 to 5; 1 is our highest resolution,

5 the lowest). mp is the particle mass in the high-resolution region, ǫGis the Plummer-equivalent gravitational softening length, r200 is

the virial radius, defined as the radius enclosing a mean overdensity 200 times the critical value for closure, M200is the mass within the

virial radius, N200 is the total number of particles within r200. Other characteristic properties of the halos listed are the position (rmax)

of the peak (Vmax) of the circular velocity profile, as well as the 1D velocity dispersion of the main halo (σhost), and the peak (σmax) of

the velocity dispersion profile.

Halor(1)

conv

[kpc/h]

r(7)

conv

[kpc/h]

ρ−2

[1010h2M⊙/Mpc3]

r−2

[kpc/h]

αχχr

γmax

Aq-A-1

Aq-A-2

Aq-A-3

Aq-A-4

Aq-A-5

0.113

0.250

0.417

0.952

2.206

0.253

0.575

0.966

2.277

5.530

7.462×105

7.322×105

7.456×105

6.501×105

7.534×105

11.05

11.15

11.09

11.90

11.02

0.170 ± 0.0259

0.163 ± 0.0249

0.174 ± 0.0266

0.160 ± 0.0248

0.165 ± 0.0268

-1.898

-1.917

-1.926

-1.991

-2.015

-1.948

-1.976

-1.995

-2.061

-2.111

0.894

1.051

1.128

1.321

1.493

Aq-B-2

Aq-C-2

Aq-D-2

Aq-E-2

Aq-F-2

0.219

0.248

0.281

0.223

0.209

0.507

0.573

0.652

0.516

0.486

1.830×105

4.973×105

2.075×105

2.058×105

1.673×105

16.79

14.37

20.30

17.88

18.84

0.173 ± 0.0123

0.159 ± 0.0125

0.170 ± 0.0124

0.130 ± 0.0200

0.145 ± 0.0167

-1.868

-1.948

-1.862

-1.912

-1.911

-1.938

-2.010

-1.942

-1.947

-1.980

1.039

1.077

1.070

1.084

1.298

Table 2. Fit parameters of Aquarius halos. The first column labels each halo, as in Table 1, the second and third list the convergence

radii obtained for κ = 1 and κ = 7. These radii, r(1)

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