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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.
2 THE 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
3
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
2.1 The 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).
2.3Halo 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.
2.4 Radial 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 =
3.2Fitting 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
Page 8
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.
3.5Self-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).
3.6The 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 and 6 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)
4.4 The 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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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
[km/s]
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[km/s]
Aq-A-1
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1.250×103
1.000×104
3.585×104
2.868×105
2.294×106
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87
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179.41
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179.36
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1.343×1012
1.345×1012
1.341×1012
1.342×1012
1.357×1012
1074.06
134.47
37.39
4.68
0.59
208.75
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209.22
209.24
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20.69
20.54
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20.58
20.84
117.47
117.13
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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.
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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
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