Halo-to-Halo Similarity and Scatter in the Velocity Distribution of Dark Matter Page: 2 of 6
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Mao et al.
scatter from the uncertain position of the Earth within a
given halo. We further identify the largest uncertainties
that currently exist in our understanding of the VDF
at the location of the Earth in our Galaxy, and quan-
tify their relevance for inferences from direct detection
2. UNIVERSAL VELOCITY DISTRIBUTION IN
To identify the relevant physical quantities which af-
fect the VDF and to quantify scatter in the distribu-
tions among different halos in cosmological simulations,
we must examine a large number of halos across a wide
range of mass. We also need high resolution to reduce
sampling error and distinguish differences in VDFs for
In this study, we use halos from the RHAPSODY and
BOLSHOI simulations; state-of-the-art dark-matter-only
simulations with high mass resolution. RHAPSODY con-
sists of re-simulations of 96 massive cluster-size halos
with Mvir - 1014.80.05Mh--. The particle mass is
1.3 x 108M h-1, resulting in ~ 5 x 106 particles in each
halo. This simulation set currently comprises the largest
number of halos simulated with this many particles in a
narrow mass bin (Fig. 1 of Wu et al. 2012). BOLSHOI is
a full cosmological simulation, with similar mass resolu-
tion, 1.3 x 108M h--1. For detailed descriptions of the
RHAPSODY and BOLSHOI simulations, refer to Wu et al.
(2012) and Klypin et al. (2011) respectively.
We use the phase-space halo finder ROCK-
STAR (Behroozi et al. 2011) to identify host halos
at z = 0. The masses and radii of the halos are
defined by the spherical overdensity of virialization,
M(< rvir) 3 rirAvirpc, where Avir 94 and pc is
the critical density. We examine the VDFs at a range
of radii. A VDF at radius r uses all particles within a
spherical shell centered at the halo center with the inner
and outer radii of 10 065r, so that the ratio of the shell
width to the radius is fixed. In each shell, we assign
the escape velocity (vesc) as the spherically-averaged
vesc of all particles in the shell. We have verified
that vesc determined from this method is consistent
with the same quantity deduced from the best-fitting
spherically-averaged smooth density profile.
We fit each halo with an NFW density profile,
P(r) (r/rs)(1 + r/rs)2, (2)
where r, is the scale radius at which the log log slope is
-2. The fit uses maximum-likelihood estimation based
on particles within rvir. The halo concentration is defined
as c = rvir/rs.
Fig. 1 shows the VDF at different values of r/r. The
value of r/r, affects the shape of VDF dramatically. The
peak of the distribution is a strong function of r/r. If
instead the velocity is normalized by the circular veloc-
ity at each radius rather than the escape velocity, this
trend will be slightly weakened but still significant. This
trend in r/r, is not surprising because the VDF heav-
ily depends on the gravitational potential. If the density
profiles of simulated halos can be described by the NFW
profile, which is a function of r/r only (up to a nor-
malization constant), the VDF should mostly depend on
0.2 0.4 0.6 0.8 1.0
Figure 1. Solid colored lines show the stacked velocity distribu-
tion for 96 halos in RHAPSODY, at different values of r/rs: (from left
to right) 0.15 (blue), 0.3 (red), 0.6 (green), 1.2 (magenta). Bands
show the 68% halo-to-halo scatter in those VDFs. Dashed and dot-
ted colored lines indicate the same values of r/rs in BOLSHOI with
halos of Mvir ~ 1012 and 1013Moh-1 respectively. The VDFs of
low-mass halos are cut at the head and tail due to limited particle
number, and their scatter is not shown. The SHM (vo = 220 km/s
and vesc = 544 km/s) is shown for comparison (black).
r/r, until the isolated NFW potential breaks down at
The above trend is robust for halo masses down to
~ 1012 M, as shown by the BOLSHOI simulation in
Fig. 1. The scatter of the VDFs in the low-mass halos
considered is somewhat larger due to resolution. How-
ever, when the high-mass halos are downsampled to have
the same particle number, the spreads in the stacked
VDF are comparable to the low-mass halos. We further
investigated the impact of a variety of parameters char-
acterizing the halo on the shape of the VDF, and found
that for a fixed value of r/r, the halo-to-halo scatter in
the VDFs is not significantly reduced when binning on
concentration, shape, or formation history. A detailed
discussion on this halo-to-halo scatter is in Section 4.
3. MODELS OF THE VELOCITY DISTRIBUTION
The dark matter velocity distribution in halos is set
by a sequence of mergers and accretion. The process of
violent relaxation (Lynden-Bell 1967) may be responsi-
ble for the resulting near-equilibrium distributions ob-
served in dark matter halos and in galaxies. These near-
equilibrium distributions explain why existing VDF mod-
els (see e.g. Frandsen et al. 2012), including the Stan-
dard Halo Model (SHM), King model, the double power-
law model, and the Tsallis model, are all variants of the
Maxwell Boltzmann distribution. Recent studies have
shown that the widely-used SHM, which is a Maxwell
Boltzmann distribution with a cut-off put in by hand, is
inconsistent with the VDF found in a handful of individ-
ual simulations (Stiff & Widrow 2003; Vogelsberger et al.
2009; Kuhlen et al. 2010; Purcell et al. 2012) and in the
study of rotation curve data (Bhattacharjee et al. 2012).
The double power-law model was proposed to suppress
the tail of the distribution, by raising the SHM to the
power of a parameter k (Lisanti et al. 2011). The Tsal-
lis model replaces the Gaussian in Maxwell Boltzmann
distribution with a q-Gaussian, which approaches to a
Gaussian as q -> 1 (Vergados et al. 2008). It was argued
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Mao, Yao-Yuan; Strigari, Louis E.; Wechsler, Risa H.; Wu, Hao-Yi & Hahn, Oliver. Halo-to-Halo Similarity and Scatter in the Velocity Distribution of Dark Matter, article, December 14, 2012; United States. (digital.library.unt.edu/ark:/67531/metadc842242/m1/2/: accessed March 18, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.