Page 1
Complementarity of Dark Matter Direct Detection Targets
Miguel Pato,1,2,3, ∗Laura Baudis,4Gianfranco Bertone,1,2
Roberto Ruiz de Austri,5Louis E. Strigari,6and Roberto Trotta7
1Institute for Theoretical Physics, Univ. of Z¨ urich, Winterthurerst. 190, 8057 Z¨ urich CH
2Institut d’Astrophysique de Paris, UMR 7095-CNRS,
Univ. Pierre & Marie Curie, 98bis Bd Arago 75014 Paris, France
3Dipartimento di Fisica, Universit` a degli Studi di Padova, via Marzolo 8, I-35131, Padova, Italy
4Physics Institute, Univ. of Z¨ urich, Winterthurerst. 190, 8057 Z¨ urich CH
5Instituto de F´ ısica Corpuscular, IFIC-UV/CSIC, Valencia, Spain
6Kavli Institue for Particle Astrophysics & Cosmology, Stanford University, Stanford, CA, 94305
7Astrophysics Group, Imperial College London
Blackett Laboratory, Prince Consort Road, London SW7 2AZ, UK
(Dated: December 17, 2010)
We investigate the reconstruction capabilities of Dark Matter mass and spin-independent cross-
section from future ton-scale direct detection experiments using germanium, xenon or argon as
targets. Adopting realistic values for the exposure, energy threshold and resolution of Dark Matter
experiments which will come online within 5 to 10 years, the degree of complementarity between
different targets is quantified. We investigate how the uncertainty in the astrophysical parameters
controlling the local Dark Matter density and velocity distribution affects the reconstruction. For a
50 GeV WIMP, astrophysical uncertainties degrade the accuracy in the mass reconstruction by up
to a factor of ∼ 4 for xenon and germanium, compared to the case when astrophysical quantities are
fixed. However, combination of argon, germanium and xenon data increases the constraining power
by a factor of ∼ 2 compared to germanium or xenon alone. We show that future direct detection
experiments can achieve self-calibration of some astrophysical parameters, and they will be able to
constrain the WIMP mass with only very weak external astrophysical constraints.
I.INTRODUCTION
Many experiments are currently searching for Dark
Matter (DM) in the form of Weakly Interacting Mas-
sive Particles (WIMPs), by looking for rare scattering
events off nuclei in the detectors, and many others are
planned for the next decade [1–6]. This direct DM de-
tection strategy has brought over the last year several
interesting observations and upper limits. The results of
the DAMA/LIBRA [7] and, more recently, the CoGeNT
[8] collaborations have been tentatively interpreted as
due to DM particles. It appears however that these re-
sults cannot be fully reconciled with other experimen-
tal findings, in particular with the null searches from
XENON100 [9–11] or CDMS [12], and are also in tension
with ZEPLIN-III [13]. In this context, the next genera-
tion of low-background, underground detectors is eagerly
awaited and will hopefully confirm or rule out a DM in-
terpretation.
If convincing evidence is obtained for DM particles
with direct detection experiments, the obvious next step
will be to attempt a reconstruction of the physical param-
eters of the DM particle, namely its mass and scattering
cross-section (see e.g. Refs. [14, 15]). This is a non-trivial
task, hindered by the different uncertainties associated
with the computation of WIMP-induced recoil spectra.
In particular, Galactic model uncertainties – i.e. uncer-
∗Electronic address: pato@iap.fr
tainties pertaining to the density and velocity distribu-
tion of WIMPs in our neighbourhood – play a crucial role.
In attempting reconstruction, the simplest assumption to
make is a fixed local DM density ρ0= 0.3 GeV/cm3and a
“standard halo model”, i.e. an isotropic isothermal sphere
density profile and a Maxwell-Boltzmann distribution of
velocities with a given galactic escape velocity vescand
one-dimensional dispersion σ2≡ v2
ing the most probable velocity and vlsrthe local circular
velocity, see below). However, the Galactic model param-
eters are only estimated to varying degrees of accuracy,
so that the true local population of DM likely deviates
from the highly idealised standard halo model.
0/2 = v2
lsr/2 (v0 be-
Several attempts have been made to improve on the
standard approach [15–18].
signal at one experiment, recent analyses have studied
how complementary detectors can extract dark matter
properties, independent of our knowledge of the Galac-
tic model [19]. Certain properties of dark matter may
also be extracted under assumptions about the nature
of the nuclear recoil events [20]. Furthermore, eventual
multiple signals at different targets have been shown to
be useful in constraining both dark matter and astro-
physical properties [21] and in extracting spin-dependent
and spin-independent couplings [22, 23]. Here, using a
Bayesian approach, we study how uncertainties on Galac-
tic model parameters affect the determination of the DM
mass mχand spin-independent WIMP-proton scattering
cross-section σp
SI. In particular we focus on realistic ex-
perimental capabilities for the future generation of ton-
scale detectors – to be reached within the next 10 years
In the case of a detected
arXiv:1012.3458v1 [astro-ph.CO] 15 Dec 2010
Page 2
2
– with noble liquids (argon, xenon) and cryogenic (ger-
manium) technologies.
The main focus of this paper is the complementarity
between different detection targets. It is well-known (see
e.g. [2]) that different targets are sensitive to different
directions in the mχ− σp
to achieve improved reconstruction capabilities – or more
stringent bounds in the case of null results. This problem
has often been addressed without taking proper account
of Galactic model uncertainties. Using xenon (Xe), argon
(Ar) and germanium (Ge) as case-studies, we ascertain
to what extent unknowns in Galactic model parameters
limit target complementarity. A thorough understand-
ing of complementarity will be crucial in the near future
since it provides us with a sound handle to compare ex-
periments and, if needed, decide upon the best target to
bet on future detectors. Our results also have important
consequences for the combination of collider observables
and direct detection results (for a recent work see [24]).
Besides degrading the extraction of physical proper-
ties like mχand σp
will challenge our ability to distinguish between different
particle physics frameworks in case of a positive signal.
Other relevant unknowns are hadronic uncertainties, re-
lated essentially to the content of nucleons [25]. Here, we
undertake a model-independent approach without speci-
fying an underlying WIMP theory and using mχand σp
as our phenomenological parameters – for this reason we
shall not address hadronic uncertainties (hidden in σp
A comprehensive work complementary to ours and done
in the supersymmetric framework has been presented re-
cently [26, 27].
The paper is organised as follows. In the next section,
we give some basic formulae for WIMP-nucleus recoil
rates in direct detection experiments. In Section III the
upcoming experimental capabilities are detailed, while
Section IV describes our Bayesian approach. We outline
the relevant Galactic model uncertainties and our mod-
elling of the velocity distribution function in Section V
and present our results in Section VI before concluding
in Section VII.
SIplane, which is very useful
SI, uncertainties in the Galactic model
SI
SI).
II.BASICS OF DIRECT DARK MATTER
DETECTION
Several thorough reviews on direct dark matter
searches exist in the literature [1–6]. In this section, we
simply recall the relevant formulae, emphasizing the im-
pact of target properties and unknown quantities.
The elastic recoil spectrum produced by WIMPs of
mass mχand local density ρ0on target nuclei N(A,Z)
of mass mN is
?
dR
dER(ER) =
ρ0
mχmN
V
d3? v vf (? v + ? ve)dσχ−N
dER
(v,ER),
(1)
where ? v is the WIMP velocity in the detector rest frame,
? veis the Earth velocity in the Galactic rest frame, f(? w) is
the WIMP velocity distribution in the galactic rest frame
and σχ−N is the WIMP-nucleus cross-section. The inte-
gral is performed over V : v > vmin(ER), where vmin
is the minimum WIMP velocity that produces a nuclear
recoil of energy ER. Eq. (1) simply states that the recoil
rate is the flux of WIMPs ρ0v/mχ, averaged over the ve-
locity distribution f(? w), times the probability of interac-
tion with one target nucleus σχ−N. Anticipating the scale
of future detectors, we will think of measuring dR/dERin
units of counts/ton/yr/keV. For non-relativistic (elastic)
collisions – as appropriate for halo WIMPs, presenting
v/c ∼ 10−3– the kinematics fixes the recoil energy
ER(mχ,v,A,θ?) =µ2
Nv2(1 − cosθ?)
mN
,
and the minimum velocity
vmin(mχ,ER,A) =
?
mNER
2µ2
N
in which θ?is the scattering angle in the centre of mass
and µN=
In principle, all WIMP-nucleus couplings enter in the
cross-section σχ−N. However, we shall focus solely on
spin-independent (SI) scalar interactions so that
mχmN
mχ+mNis the WIMP-nucleus reduced mass.
dσχ−N
dER
=
mN
2µ2
Nv2σN
SIF2(A,ER),
where σN
nucleus spin-independent cross-section at zero momen-
tum transfer and F(A,ER) is the so-called form factor
that accounts for the exchange of momentum. Assuming
that the WIMP couplings to protons and neutrons are
similar, fp∼ fn, and defining σp
WIMP-proton reduced mass), one gets
SI =
4µ2
π[Zfp+ (A − Z)fn]2is the WIMP-
N
SI≡
4µ2
πf2
p
p(µpbeing the
dσχ−N
dER
=
mN
2µ2
pv2σp
SIA2F2(A,ER). (2)
For the form factor, we use the parameterisation in [2]
appropriate for spin-independent couplings, namely
F(A,ER) = 3sin(qrn) − (qrn)cos(qrn)
(qrn)3
exp(−(qs)2/2) ,
with qr = 6.92 × 10−3A1/2(E/keV)1/2r/fm, s ? 0.9 fm,
r2
fm.
As noticed above, in Eq. (1) ? veis the Earth velocity
with respect to the galactic rest frame and amounts to
? ve= ? vlsr+? vpec+? vorb, where vlsr∼ O(250) km/s is the
local circular velocity, vpec∼ O(10) km/s is the peculiar
velocity of the Sun (with respect to ? vlsr) and vorb ∼
O(30) km/s is the Earth velocity with respect to the Sun
(i.e. the Earth orbit).Here, we are not interested in
the annual modulation signal nor directional signatures
n= c2+7
3π2a2−5s2, c/fm = 1.23A1/3−0.6 and a ? 0.52
Page 3
3
but rather in the average recoil rate – therefore we shall
neglect ? vpecand ? vorband take ? ve? ? vlsr= const.
Under these assumptions, Eq. (1) may be recast in a
very convenient way:
dR
dER(ER) =
ρ0σp
2µ2
pmχ
F (vmin(mχ,ER,A),? ve;v0,vesc) , (3)
SI
× A2F2(A,ER) ×
where we have used Eq. (2), defined
F ≡
?
V
d3? vf (? v + ? ve)
v
(4)
and made explicit the dependence of F on the velocity
distribution parameters v0 and vesc. Below we discuss
in more detail the connection between the parameters v0
and vlsr. The distribution of DM is encoded in the fac-
tor F (and ρ0), whereas the detector-related quantities
appear in A2F2(A,ER) (and vmin). The apparent de-
generacy along the direction ρ0σp
broken by using different recoil energies and/or different
targets since F is sensitive to a non-trivial combination
of mχ, ERand A. Nevertheless, for very massive WIMPs
mχ? mN∼ O(100) GeV ? mp, the minimum velocity
becomes independent of mχ, vmin?
the degeneracy ρ0σp
on the target being used, this usually happens for WIMP
masses above a few hundred GeV.
Ultimately, the observable we will be interested in is
the number of recoil events in a given energy bin E1<
ER< E2:
?E2
SI/mχ= const may be
?ER/(2mN), and
SI/mχcannot be broken. Depending
NR(E1,E2) =
E1
dER?eff
d˜R
dER
,(5)
?eff being the effective exposure (usually expressed in
ton×yr) and d˜R/dERthe recoil rate smeared according
to the energy resolution of the detector σ(E),
d˜R
dER
=
?
dE?dR
dER(E?)
1
√2πσ(E?)exp
?
−(E − E?)2
2σ2(E?)
?
.
Three fiducial WIMP models will be used to assess
the capabilities of future direct detection experiments:
mχ=25, 50 and 250 GeV, all with σp
models are representative of well-motivated candidates
such as neutralinos in supersymmetric theories [28].
SI= 10−9pb. These
III.UPCOMING EXPERIMENTAL
CAPABILITIES
Currently, the most stringent constraints on the SI
WIMP-nucleon coupling are those obtained by the
CDMS [29] and XENON [9] collaborations.
XENON100 should probe the cross-section region down
to 5 × 10−45cm2with data already in hand, the
While
XENON1T [30] detector, whose construction is sched-
uled to start by mid 2011, is expected to reach another
order of magnitude in sensitivity improvement. To test
the σp
a new generation of detectors with larger WIMP target
masses and ultra-low backgrounds is needed. Since we
are interested in the prospects for detection in the next 5
to 10 years, we discuss new projects that can realistically
be built on this time scale, adopting the most promising
detection techniques, namely noble liquid time projec-
tion chambers (TPCs) and cryogenic detectors operated
at mK temperatures.
In Europe, two large consortia, DARWIN [31] and EU-
RECA [32], gathering the expertise of several groups
working on existing DM experiments are funded for R&D
and design studies to push noble liquid and cryogenic ex-
periments to the multi-ton and ton scale, respectively.
DARWIN is devoted to noble liquids, having as main
goal the construction of a multi-ton liquid Xe (LXe)
and/or liquid Ar (LAr) instrument [33], with data tak-
ing to start around 2016.
WARP collaborations participate actively in the DAR-
WIN project. EURECA is a design study dedicated to
cryogenic dark matter detectors operated at mK tem-
peratures. The proposed roadmap is to improve upon
CRESST [34] and EDELWEISS [35] technologies and
build a ton-scale detector by 2018, with a SI sensitivity of
about 10−46cm2≡ 10−10pb. The complementarity be-
tween DARWIN and EURECA is of utmost importance
for dark matter direct searches since a solid, uncontro-
versial discovery requires signals in distinct targets and
preferentially distinct technologies. In an international
context, two engineering studies (MAX [36] and LZS [37])
are funded in the US for ton to multi-ton scale LXe and
LAr TPCs and the SuperCDMS/GEODM collaboration
[38] plans to operate an 1.5ton Ge cryogenic experiment
at DUSEL [39]. In Japan, the XMASS experiment [40],
using a total of 800kg of liquid xenon in a single-phase
detector, is under commissioning at the Kamioka under-
ground laboratory [41], while a large single-phase liquid
argon detector, DEAP-3600 [42], using 3.6tons of LAr is
under construction at SNOLab [43].
Given these developments, we will focus on the three
most promising targets: Xe and Ar as examples of noble
liquid detectors, and Ge as a case-study for the cryogenic
technique.In the case of a Ge target, we assume an
1.5 ton detector (1ton as fiducial target mass), 3 years
of operation, an energy threshold for nuclear recoils of
Ethr,Ge= 10 keV and an energy resolution given by
SIregion down to 10−47cm2≡ 10−11pb and below,
The XENON, ArDM and
σGe(E) =
?
(0.3)2+ (0.06)2E/keV keV.(6)
For a liquid Xe detector, we assume a total mass of
8tons (5tons in the fiducial region), 1 year of operation,
an energy threshold for nuclear recoils of Ethr,Xe = 10
keV and an energy resolution of
σXe(E) = 0.6 keV
?
E/keV.(7)
Page 4
4
target ? [ton×yr] ηcut ANR ?eff [ton×yr] Ethr [keV] σ(E) [keV] background events/?eff
Xe5.0 0.80.5 2.00
Ge
3.00.8 0.92.16
Ar10.0 0.8 0.86.40
10
10
30
Eq. (7)
Eq. (6)
Eq. (8)
< 1
< 1
< 1
TABLE I: Characteristics of future direct dark matter experiments using xenon, germanium and argon as target nuclei. In all
cases the level of background in the fiducial mass region is negligible for the corresponding effective exposure. See Section III
for further details.
Finally, for a liquid Ar detector, we assume a total
mass of 20tons (10tons in the fiducial region), 1 year
of operation, an energy threshold for nuclear recoils of
Ethr,Ar= 30 keV and an energy resolution of [44]
σAr(E) = 0.7 keV
?
E/keV. (8)
To calculate realistic exposures, we make the following
assumptions: nuclear recoils acceptances ANR of 90%,
80% and 50% for Ge, Ar and Xe, respectively, and an
additional, overall cut efficiency ηcutof 80% in all cases,
which for simplicity we consider to be constant in energy.
We hypothesise less than one background event per given
effective exposure ?eff, which amounts to 2.16 ton×yr in
Ge, 6.4 ton×yr in Ar and 2 ton×yr in Xe, after allow-
ing for all cuts. Such an ultra-low background will be
achieved by a combination of background rejection using
the ratio of charge-to-light in Ar and Xe, and charge-to-
phonon in Ge, the timing characteristics of raw signals,
the self-shielding properties and extreme radio-purity of
detector materials, as well as minimisation of exposure
to cosmic rays above ground.
The described characteristics are summarised in Table
I. We note that in the following we shall consider recoil
energies below 100 keV only; to increase this maximal
value may add some information but the effect is likely
small given the exponential nature of WIMP-induced re-
coiling spectra.
IV.STATISTICAL METHODOLOGY
We take a Bayesian approach to parameter inference.
We begin by briefly summarizing the basics, and we refer
the reader to [45] for further details. Bayesian inference
rests on Bayes theorem, which reads
p(Θ|d) =p(d|Θ)p(Θ)
p(d)
, (9)
where p(Θ|d) is the posterior probability density func-
tion (pdf) for the parameters of interest, Θ, given data
d, p(d|Θ) = L(Θ) is the likelihood function (when viewed
as a function of Θ for fixed data d) and p(Θ) is the prior.
Bayes theorem thus updates our prior knowledge about
the parameters to the posterior by accounting for the in-
formation contained in the likelihood. The normalization
constant on the r.h.s. of Eq. (9) is the Bayesian evidence
and it is given by the average likelihood under the prior:
?
p(d) =dΘp(d|Θ)p(Θ).(10)
The evidence is the central quantity for Bayesian model
comparison [46], but it is just a normalisation constant
in the context of the present paper.
The parameter set Θ contains the DM quantities we
are interested in (mass and scattering cross-section), and
also the Galactic model parameters, which we regard as
nuisance parameters, entering the calculation of direct
detection signals, namely ρ0, v0, vesc, k, see Eq. (3) and
Section V. We further need to define priors p(Θ) for all of
our parameters. For the DM parameters, we adopt flat
priors on the log of the mass and cross-section, reflecting
ignorance on their scale. For the Galactic model param-
eters, we choose priors that reflect our state of knowl-
edge about their plausible values, as specified in the next
section. Those priors are informed by available observa-
tional constraints as well as plausible estimations of un-
derlying systematical errors, for example for ρ0. Finally,
the likelihood function for each of the direct detection ex-
periments is given by a product of independent Poisson
likelihoods over the energy bins:
L(Θ) =
?
b
N
ˆ Nb!
ˆ
Nb
R
exp(−NR),(11)
whereˆ Nbis the number of counts in each bin (generated
from the true model with no shot noise, as explained be-
low) and NR= NR(Emin
b
,Emax
b
in the b-th bin (in the energy range Emin
when the parameters take on the value Θ, and it is given
by Eq. (5). We use 10 bins for each experiment, uniformly
spaced on a linear scale between the threshold energy and
100 keV. We have checked that our results are robust if
we double the number of assumed energy bins. Using the
experimental capabilities outlined in Section III, we com-
pute the counts NRthat the benchmark WIMPs would
generate, and include no background events since the ex-
pected background level in the fiducial mass region is
negligible (cf. Table I). The mock counts are generated
from the true model, i.e. without Poisson scatter. This
is because we want to test the reconstruction capabilities
without having to worry about realization noise (such a
data set has been called “Asimov data” in the particle
physics context [47]).
To sample the posterior distribution we employ the
MultiNest code [48–50], an extremely efficient sampler
of the posterior distribution even for likelihood functions
defined over a parameter space of large dimensionality
with a very complex structure. In our case, the likeli-
hood function is unimodal and well-behaved, so Monte
) is the number of counts
≤ E ≤ Emax
bb
)
Page 5
5
Parameter
log10(mχ/GeV) (0.1,3.0)
log10(σp
ρ0/(GeV/cm3)
v0/(km/s)
vesc/(km/s)
k
Prior range Prior constraint
Uniform prior
(−10,−6)
(0.001,0.9) Gaussian: 0.4 ± 0.1
(80,380)Gaussian: 230 ± 30
(379,709) Gaussian: 544 ± 33
(0.5,3.5) Uniform prior
SI/pb)
Uniform prior
TABLE II: Parameters used in our analysis, with their prior range (middle column) and the prior constraint adopted (rightmost
column). See Section IV and V for further details.
Carlo Markov Chain (MCMC) techniques would be suf-
ficient to explore it. However, MultiNest also computes
the Bayesian evidence (which MCMC methods do not re-
turn), as it is an implementation of the nested sampling
algorithm [51]. In this work, we run MultiNest with 2000
live points, an efficiency parameter of 1.0 and a tolerance
of 0.8 (see [48, 49] for details).
V.
GALACTIC MODEL PARAMETERS
VELOCITY DISTRIBUTION AND
We now move onto discussing our modeling of the ve-
locity distribution function and the Galactic model pa-
rameters that are input for Eq. (3).
the smooth component of the velocity distribution –
recent results from numerical simulations indicate that
the velocity distribution component arising from lo-
calised streams and substructures is likely sub-dominant
in the calculation of direct dark matter detection sig-
nals [52, 53].
We model the velocity distribution function as spheri-
cal and isotropic, and parameterise it as [54],
We model only
f(w) =
?
1
Nf
0
?
exp
?v2
esc−w2
kv2
0
?
− 1
?k
if w ≤ vesc
if w > vesc
. (12)
This velocity distribution function was found to be flex-
ible enough to describe the range of dark matter halo
profiles found in cosmological simulations [54]. Boosting
into the rest frame of the Earth implies the transforma-
tion w2= v2+v2
e+2vvecosθ, where θ is the angle between
? v and ? ve∼ ? vlsr. The shape parameter that determines
the power law tail of the velocity distribution is k, the
escape velocity is vesc, while v0is a fit parameter that we
discuss in detail below, and Nf is the appropriate nor-
malisation constant. The special case k = 1 represents
the standard halo model with a truncated Maxwellian
distribution, and the corresponding expressions for Nf
and F have been derived analytically in the literature –
see for instance [17]. Note as well that, for any value
of k, this distribution matches a Maxwellian distribution
for sufficiently small velocities w and if vesc> v0.
The high-velocity tail of the distributions found in nu-
merical simulations of pure dark matter galactic halos are
well modelled by 1.5 < k < 3.5 [54]. In our analysis we
will expand this range to also include models that behave
similar to pure Maxwellian distributions near the tail of
the distribution, so that in our analysis we vary k in the
range
k = 0.5 − 3.5(flat). (13)
We adopt an uniform (i.e., flat) prior within the above
range for k.
The range we take for the vescis motivated by the re-
sults of Ref. [55], where a sample of high-velocity stars is
used to derive a median likelihood local escape velocity
of ¯ vesc= 544 km/s and a 90% confidence level interval
498 km/s < vesc< 608 km/s. Assuming Gaussian errors
this translates into a 1σ uncertainty of 33 km/s. It is im-
portant to note that this constraint on the escape velocity
is derived assuming a range in the power law tail for the
distribution of stars in the local neighbourhood, which
is then related to the power law tail in the dark matter
distribution [55]. Motivated by obtaining conservative
limits on the reconstructed mass and cross-section of the
dark matter, in our modelling we will not include such
correlations between the escape velocity and the power
law index k, so that in the end we take a Gaussian prior
on vescwith mean and standard deviation given by
vesc= 544 ± 33 km/s (1σ).(14)
Having specified ranges for vesc and k, it remains to
consider a range for v0in Eq. (12). As defined in that
equation, the quantity v0 does not directly correspond
to the local circular velocity, vlsr, but rather is primarily
set by vlsrand the dark matter profile. Following a pro-
cedure similar to that discussed in Ref. [54], we find the
range of values v0compatible with a given a dark matter
halo profile, ρ0and a range for vlsr. For the above range
in vlsrand the values ρ0in Eq. (16) below, we find that
the parameter v0can take values in the range 200 − 300
km/s for pure Navarro-Frenk-White (NFW) dark matter
halos with outer density slopes ρ ∝ r−3. Larger values of
v0are allowed for steeper outer density slopes, though the
range is found to not expand significantly if we restrict
ourselves to models with outer slopes similar to the NFW
case. With these caveats in mind regarding the mapping
between v0and vlsrfor steeper outer slopes, for simplic-
ity and transparency in our analysis, we will consider a
similar range for v0as for the local circular velocity, so
we take v0= vlsr(that holds in the case of the standard
halo model).
Page 6
6
[GeV]
χ
m
[pb]
p
σ
SI
2
10
3
10
-9
10
50
Xe
Ge
Ar
DM benchmarks
=230 km/s, k=1
0
=544 km/s, v
esc
, v
3
=0.4 GeV/cm
0
ρ
[GeV]
χ
m
[pb]
p
σ
SI
2
10
3
10
-9
10
50
Xe
Xe+Ge
Xe+Ge+Ar
DM benchmarks
=230 km/s, k=1
0
=544 km/s, v
esc
, v
3
=0.4 GeV/cm
0
ρ
FIG. 1: The joint 68% and 95% posterior probability contours in the mχ − σp
(mχ = 25,50,250 GeV) with fixed Galactic model, i.e. fixed astrophysical parameters. In the left frame we show the re-
construction capabilities of Xe, Ge and Ar configurations separately, whereas in the right frame the combined data sets Xe+Ge
and Xe+Ge+Ar are shown.
SIplane for the three DM benchmarks
For the local circular velocity and its uncertainty, a va-
riety of measurements presents a broad range of central
values and uncertainties [56]. To again remain conserva-
tive we use an interval bracketing recent determinations:
v0= vlsr= 230 ± 30 km/s(1σ),(15)
where we take a Gaussian prior with the above mean and
standard deviation. To account for the variation of the
local density of dark matter in our modeling, we will take
a mean value and error given by [57, 58]
ρ0= 0.4 ± 0.1 GeV/cm3
(1σ),(16)
There are several other recent results that determine ρ0,
both consistent [59] and somewhat discrepant [60] with
our adopted value. Even in light of these uncertainties,
we take Eq. (16) to represent a conservative range for the
purposes of our study.
For completeness Table II summarises the information
on the parameters used in our analysis.
VI. RESULTS
A. Complementarity of targets
We start by assuming the three dark matter bench-
mark models described in Section II (mχ = 25,50,250
GeV with σp
parameters to their fiducial values, ρ0= 0.4 GeV/cm3,
v0= 230 km/s, vesc= 544 km/s, k = 1. With the exper-
imental capabilities outlined in Section III, we generate
SI= 10−9pb) and fix the Galactic model
mock data that in turn are used to reconstruct the poste-
rior for the DM parameters mχand σp
of Fig. 1 presents the results for the three benchmarks
and for Xe, Ge and Ar separately. Contours in the figure
delimit regions of joint 68% and 95% posterior probabil-
ity. Several comments are in order here. First, it is ev-
ident that the Ar configuration is less constraining than
Xe or Ge ones, which can be traced back to its smaller A
and larger Ethr. Moreover, it is also apparent that, while
Ge is the most effective target for the benchmarks with
mχ= 25,250 GeV, Xe appears the best for a WIMP with
mχ= 50 GeV (see below for a detailed discussion). Let
us stress as well that the 250 GeV WIMP proves very
difficult to constrain in terms of mass and cross-section
due to the high-mass degeneracy explained in Section II.
Taking into account the differences in adopted values and
procedures, our results are in qualitative agreement with
Ref. [26], where a study on the supersymmetrical frame-
work was performed. However, it is worth noticing that
the contours in Ref. [26] do not extend to high masses
as ours for the 250 GeV benchmark – this is likely be-
cause the volume at high masses in a supersymmetrical
parameter space is small.
SI. The left frame
In the right frame of Fig. 1 we show the reconstruction
capabilities attained if one combines Xe and Ge data,
or Xe, Ge and Ar together, again for when the Galac-
tic model parameters are kept fixed.
mχ= 25,50 GeV, the configuration Xe+Ar+Ge allows
the extraction of the correct mass to better than O(10)
GeV accuracy. For reference, the (marginalised) mass
accuracy for different mock data sets is listed in Table
III. For mχ = 250 GeV, it is only possible to obtain a
lower limit on mχ.
In this case, for
Page 7
7
[GeV]
χ
m
[pb]
p
σ
SI
2
10
-9
10
50
Xe
fixed astrophysics
0.1 GeV/cm
±
=0.4
0
30 km/s
±
=230
all
DM benchmark
3
ρ
0v
[GeV]
χ
m
[pb]
p
σ
SI
2
10
3
10
-9
10
50
Xe
Xe+Ge
Xe+Ge+Ar
DM benchmarks
30 km/s, k=0.5-3.5
±
=230
0
33 km/s, v
±
=544
esc
, v
3
0.1 GeV/cm
±
=0.4
0
ρ
FIG. 2: The joint 68% and 95% posterior probability contours in the mχ − σp
uncertainties are taken into account. In the left frame, the effect of marginalising over ρ0, v0 and all four (ρ0, v0, vesc, k)
astrophysical parameters is displayed for a Xe detector and the 50 GeV benchmark WIMP. In the right frame, the combined
data sets Xe+Ge and Xe+Ge+Ar are used for the three DM benchmarks (mχ = 25,50,250 GeV).
SIplane for the case in which astrophysical
Percent 1σ accuracy
mχ = 25 GeV mχ = 50 GeV
6.5% (14.3%)
5.5% (16.0%)
12.3% (23.4%) 14.7% (86.5%)
3.9% (10.9%)
3.6% (9.0%)
Xe
Ge
Ar
8.1% (20.4%)
7.0% (29.6%)
Xe+Ge
Xe+Ge+Ar
5.2% (15.2%)
4.5% (10.7%)
TABLE III: Marginalised percent 1σ accuracy of the DM mass reconstruction for the benchmarks mχ = 25,50 GeV. Figures
between brackets refer to scans where the astrophysical parameters were marginalised over (with priors as in Table II), while
the other figures refer to scans with the fiducial astrophysical setup.
Fig. 2 shows the results of a more realistic analysis,
that keeps into account the large uncertainties associated
with Galactic model parameters, as discussed in Section
V. The left frame of Fig. 2 shows the effect of varying
only ρ0(dashed lines, blue surfaces), only v0(solid lines,
red surfaces) and all Galactic model parameters (dotted
lines, yellow surfaces) for Xe and mχ = 50 GeV. The
Galactic model uncertainties are dominated by ρ0 and
v0, and, once marginalised over, they blow up the con-
straints obtained with fixed Galactic model parameters.
This amounts to a very significant degradation of mass
(cf. Table III) and scattering cross-section reconstruction.
Inevitably, the complementarity between different targets
is affected – see the right frame of Fig. 2. Still, for the
50 GeV benchmark, combining Xe, Ge and Ar data im-
proves the mass reconstruction accuracy with respect to
the Xe only case, essentially by constraining the high-
mass tail.
In order to be more quantitative in assessing the use-
fulness of different targets and their complementarity, we
use as figure of merit the inverse area enclosed by the
95% marginalised contour in the log10(mχ) − log10(σp
plane.Fig. 3 displays this figure of merit for several
SI)
cases, where we have normalised to the Ar target at
mχ = 250 GeV with fixed Galactic model.
with fixed Galactic model parameters are represented by
empty bars, while the cases where all Galactic model pa-
rameters are marginalised over with priors as in Table II
are represented by filled bars. Firstly, one can see that all
three targets perform better for WIMP masses around 50
GeV than 25 or 250 GeV if the Galactic model is fixed.
When astrophysical uncertainties are marginalised over,
the constraining power of the experiments becomes very
similar for benchmark WIMP masses of 25 and 50 GeV.
Secondly, Fig. 3 also confirms what was already appar-
ent from Fig. 1: Ge is the best target for mχ= 25,250
GeV (although by a narrow margin), whereas Xe appears
the most effective for a 50 GeV WIMP (again, by a nar-
row margin). Furthermore, the inclusion of uncertainties
drastically reduces the amount of information one can
extract from the data: the filled bars are systematically
below the empty ones. Now, astrophysical uncertainties
affect the complementarity between different targets in a
non-trivial way. To understand this point, let us focus
on the two rightmost bars for each benchmark in Fig. 3,
corresponding to the data sets Xe+Ge and Xe+Ge+Ar.
Analyses
Page 8
8
Figure of merit
1
10
=25 GeV
χ
m=50 GeV
χ
m=250 GeV
χ
m
Empty: fixed astrophysicsFilled: incl. astrophysical uncertainties
Ar
Ge
XeXe+Ge
Xe+Ge+Ar
Ar
Ge
XeXe+Ge
Xe+Ge+Ar
Ar
Ge
Xe
Xe+Ge
Xe+Ge+Ar
FIG. 3: Figure of merit quantifying the relative information
gain on Dark Matter parameters for different targets and com-
binations thereof. The values of the figure of merit are nor-
malised to the Ar case at mχ = 250 GeV with fixed astrophys-
ical parameters. Empty (filled) bars are for fixed astrophysical
parameters (including astrophysical uncertainties).
For instance, in the case of a 250 GeV WIMP, astrophysi-
cal uncertainties seem to reduce target complementarity:
adding Ar to Xe+Ge leads to a significant increase in
the figure of merit for analyses with fixed astrophysics
(empty bars) but has a negligible effect for analyses with
varying astrophysical parameters (filled bars). For low
mass benchmarks, the effect of combining two (Xe+Ge)
or three targets (Xe+Ge+Ar) is to increase the figure of
merit by about a factor of 2 compared to Xe alone or Ge
alone, almost independently of whether the astrophysical
parameters are fixed or marginalised over. However, the
overall information gain on the Dark Matter parameters
(for light WIMPs) is reduced by a factor ∼ 10 if astro-
physical uncertainties are taken into account, compared
to the case where the Galactic model is fixed.
B.Reduction in uncertainties and self-calibration
The uncertainties used thus far and outlined in Section
V are a reasonable representation of the current knowl-
edge. For illustration it is also interesting to consider
the effect of tighter constraints on Galactic model pa-
rameters in the reconstruction of WIMP properties. We
start by computing the correlation coefficient between
the parameters (mχ, σp
are constrained by the combined data set Xe+Ge+Ar
– see Table IV. Clearly, for all benchmark models, σp
and ρ0as well as mχand v0are strongly anti-correlated.
The anti-correlation between σp
dR/dER ∝ σp
and v0, it is easy to verify that, for vmin? ve∼ v0?
SI, ρ0, v0, vesc, k) when they
SI
SIand ρ0is obvious since
SIρ0. As for the degeneracy between mχ
[GeV]
χ
m
[pb]
p
σ
SI
2
10
3
10
-9
10
50
Xe+Ge+Ar
fixed astrophysics
full uncertainties
reduced uncertainties
DM benchmarks
FIG. 4: The effect of reducing the uncertainty on the astro-
physical parameters ρ0 and v0. The red surfaces refer to the
scan using the fiducial astrophysical setup; the yellow sur-
faces (and dotted lines) indicate the effect of marginalising
over the uncertainties in Table II; the blue surfaces (and solid
lines) correspond to the reduced uncertainties ρ0 = 0.4 ±
0.028 GeV/cm3, v0 = 230±9.76 km/s, vesc = 544±33 km/s,
k = 0.5 − 3.5.
vesc, F defined in Eq. (4) goes approximately as 1/v0and
thus dR/dER∝ 1/(mχv0). Table IV also shows a small
(anti-)correlation between σp
tions are negligible. Therefore, ρ0and v0are the domi-
nant sources of uncertainty and their more accurate de-
termination will lead to a significant improvement on the
reconstruction of mχand σp
follow [57] and apply a 7% (4.2%) uncertainty on ρ0(v0),
while maintaining the same central values as before, thus
reducing the realistic error bars used above by a factor
∼ 3.0−3.5 for both parameters. The results are shown in
Fig. 4 where we consider the combination Xe+Ge+Ar. A
future more constrained astrophysical setup may indeed
lead to a better reconstruction of the WIMP mass and
scattering cross-section.
To this point we have studied the impact of Galac-
tic model uncertainties on the extraction of DM proper-
ties from direct detection data. However, once a positive
signal is well-established, it may be used to determine
some of the Galactic parameters directly from direct de-
tection data (see e.g. [21]), without relying on external
priors. This would amount to achieving a self-calibration
of the astrophysical uncertainties affecting direct detec-
tion rates. In order to explore such possibility we re-ran
our analysis but dropping the Gaussian priors on ρ0, v0
and vesc described in Section V. Instead, we used uni-
form, non-informative priors on ρ0, v0, vescand k in the
ranges indicated in the middle column of Table II. We
focus on the 50 GeV benchmark and use the data sets
Xe, Xe+Ge and Xe+Ge+Ar. With this large freedom on
SIand v0; all other correla-
SI. To illustrate this point we
Page 9
9
mχ= 25 GeV
ρ0
mχ= 50 GeV
ρ0
mχ= 250 GeV
ρ0
mχ
− 0.039 -0.006 -0.850 -0.238 -0.002
−
−
−
−
σp
SI
v0
vesc
kmχ
− 0.098 -0.006 -0.870 -0.079 -0.004
−
−
−
−
σp
SI
v0
vesc
kmχ
− 0.874 -0.011 -0.615 -0.027 0.022
−
−
−
−
σp
SI
v0
vesc
k
mχ
σp
SI
ρ0
v0
vesc
−
−
−
−
-0.887 -0.237 0.116 0.010
−
−
−
−
−
−
−
-0.957 -0.175 0.026 -0.031
−
−
−
−
−
−
−
-0.452 -0.525 -0.024 0.015
−
−
−
0.013 -0.005 0.005
−
−
0.014 -0.010 0.030
−
−
0.002 0.015 0.010
−
−
-0.087 -0.004
−
-0.151 0.011
−
-0.049 -0.008
−
0.000-0.0090.001
TABLE IV: The correlation factors r(X,Y ) = cov(X,Y )/(σ(X)σ(Y )) for the posteriors obtained from the combined data set
Xe+Ge+Ar and including the astrophysical uncertainties with priors as in Table II.
[GeV]
χ
m
0
0.2
0.4
0.6
0.8
1
2
10
3
10
50
Xe
Xe+Ge
Xe+Ge+Ar
=50 GeV, flat astrophysics
χ
m
[km/s]
0v
50100 150 200 250300 350400450
0
0.2
0.4
0.6
0.8
1
Xe
Xe+Ge
Xe+Ge+Ar
=50 GeV, flat astrophysics
χ
m
FIG. 5: The marginalised posterior distribution function for mχ (left frame) and v0 (right frame) with the data sets Xe, Xe+Ge
and Xe+Ge+Ar for the 50 GeV benchmark. The parameters ρ0, v0, vesc and k were varied in the ranges indicated in the
middle column of Table II with a uniform prior and no constraint on astrophysics was applied. The probability distributions
are therefore a result of the constraining power of direct detection data only, which have the potential to achieve self-calibration
of the circular velocity.
the astrophysical side, it turns out that direct detection
data alone leave ρ0, vescand k unconstrained within their
ranges while σp
SIis pinpointed within approximately one
order of magnitude. Only the DM mass mχand the cir-
cular velocity v0can be constrained by direct detection,
as shown in Fig. 5. This figure stresses two interest-
ing results. First, if mχ = 50 GeV (and σp
pb), the next generation of experiments will be able to
determine the WIMP mass within a few tens of GeV
(percent 1σ accuracy of 11.8%) even with very loose as-
sumptions on the local DM distribution.
right frame in Fig. 5 shows that the combination of Xe,
Ge and Ar targets is very powerful in constraining v0on
its own without external priors. In particular, the data
set Xe+Ge+Ar (solid blue line) is sufficient to infer at
1σ v0= 238 ± 22 km/s (compared to the top-hat prior
in the range 80−380 km/s). This represents already a
smaller uncertainty than the present-day constraint that
we have taken, v0= 230±30 km/s – in case of a positive
signal, a combination of direct detection experiments will
probe in an effective way the local circular velocity. Re-
peating the same exercise for the 25 GeV benchmark we
find good mass reconstruction but a weaker constraint:
SI= 10−9
Second, the
v0= 253±39 km/s. Again, we stress that the quoted v0
uncertainties in this paragraph do not take into account
possible systematic deviations from the parameterisation
in Eq. (12).
VII.CONCLUSIONS
We have discussed the reconstruction of the key phe-
nomenological parameters of WIMPs, namely mass and
scattering cross-section off nuclei, in case of positive de-
tection with one or more direct DM experiments planned
for the next decade. We have in particular studied the
complementarity of ton scale experiments with Xe, Ar
and Ge targets, adopting experimental configurations
that may realistically become available over this time
scale.
To quantify the degree of complementarity of different
targets we have introduced a figure of merit measuring
the inverse of the area enclosed by the 95% marginalised
contours in the plane log10(mχ)−log10(σp
high degree of complementarity of different targets: for
our benchmark with mχ= 50 GeV and our fiducial set of
SI). There is a
Page 10
10
Galactic model parameters, the relative error on the re-
constructed mass goes from 8.1% for an analysis based on
a xenon experiment only, to 5.2% for a combined analysis
with germanium, to 4.5% adding also argon. Allowing
the parameters to vary within the observational uncer-
tainties significantly degrades the reconstruction of the
mass, increasing the relative error by up to a factor of ∼4
for xenon and germanium, especially due to the uncer-
tainty on ρ0and v0. However, we found that combining
data from Ar, Ge and Xe should allow to reconstruct a 50
GeV WIMP mass to 11.8% accuracy even under weaker
astrophysical constraints than currently available.
Although the mass reconstruction accuracy may ap-
pear modest, any improvement of this reconstruction is
important, in particular in view of the possible measure-
ment of the same quantity at the Large Hadron Collider
at CERN. The existence of a particle with a mass com-
patible, within the respective uncertainties, with that de-
duced from direct detection experiments would provide a
convincing proof that the particles produced in acceler-
ators are stable over cosmological time scales. Although
this is not sufficient to claim discovery of DM [24], it
would certainly be reassuring.
Despite the strong dependence of direct detection ex-
periments on the Galactic model degrades the reconstruc-
tion of DM properties, it does open up the possibility to
potentially constrain the local distribution of DM, in case
of detection with multiple targets. For example in the
case of a low mass 50 GeV WIMP, we have shown that
the local circular velocity can be determined from direct
detection data alone more accurately than it is presently
measured using the local distribution of stars and gas
clouds. Additionally, directly detecting DM provides the
most realistic way of measuring the local DM velocity
distribution. This will in principle provide invaluable in-
formation on the structure and formation of the Milky
Way halo.
Acknowledgements: G.B., R.T. and M.P. would like to
thank the organisers of the workshop “Dark Matter all
around” for a stimulating meeting. We wish to thank the
authors of the paper [26] for providing their preliminary
results, as well as Henrique Araujo and Alastair Cur-
rie for useful discussions. We also acknowledge support
from the SNF grant 20AS21-29329 and the University of
Zurich. M.P. is supported by Funda¸ c˜ ao para a Ciˆ encia
e Tecnologia (Minist´ erio da Ciˆ encia, Tecnologia e Ensino
Superior).
[1] Particle
Searches, ed. G. Bertone, Cambridge University Press
(2010)
[2] J. D. Lewin and P. F. Smith, Astropart. Phys. 6, 87
(1996).
[3] L. Bergstrom,Rept. Prog. Phys. 63,
[arXiv:hep-ph/0002126].
[4] C. Munoz, Int. J. Mod. Phys. A 19, 3093 (2004)
[arXiv:hep-ph/0309346].
[5] G. Bertone, D. Hooper and J. Silk, Phys. Rept. 405, 279
(2005) [arXiv:hep-ph/0404175].
[6] D. G. Cerdeno and A. M. Green, arXiv:1002.1912 [astro-
ph.CO].
[7] R. Bernabei et al., Eur. Phys. J. C 67 (2010) 39
[arXiv:1002.1028 [astro-ph.GA]].
[8] C. E.Aalsethetal.
arXiv:1002.4703 [astro-ph.CO].
[9] E. Aprile et al. [XENON100 Collaboration], Phys.
Rev. Lett. 105, 131302 (2010) [arXiv:1005.0380 [astro-
ph.CO]].
[10] C. Savage, G. Gelmini, P. Gondolo and K. Freese,
arXiv:1006.0972 [astro-ph.CO].
[11] F.Bezrukov, F.Kahlhoefer
arXiv:1011.3990 [astro-ph.IM].
[12] Z.Ahmedet al.
arXiv:1011.2482 [astro-ph.CO].
[13] V. N. Lebedenko et al., Phys. Rev. D 80 (2009) 052010
[arXiv:0812.1150 [astro-ph]].
[14] A. M. Green, JCAP 0807, 005 (2008) [arXiv:0805.1704
[hep-ph]].
[15] L. E. Strigari and R. Trotta, JCAP 0911, 019 (2009)
[arXiv:0906.5361 [astro-ph.HE]].
DarkMatter:Observations,Models and
793 (2000)
[CoGeNT collaboration],
andM.Lindner,
[CDMS-IICollaboration],
[16] F. S. Ling, E. Nezri, E. Athanassoula and R. Teyssier,
JCAP 1002, 012 (2010) [arXiv:0909.2028 [astro-ph.GA]].
[17] C.McCabe,Phys.Rev.
[arXiv:1005.0579 [hep-ph]].
[18] A. M. Green, JCAP 1010, 034 (2010) [arXiv:1009.0916
[astro-ph.CO]].
[19] P. J. Fox, J. Liu and N. Weiner, arXiv:1011.1915 [hep-
ph].
[20] P. J. Fox, G. D. Kribs and T. M. P. Tait, arXiv:1011.1910
[hep-ph].
[21] A. H. G. Peter, Phys. Rev. D 81 (2010) 087301
[arXiv:0910.4765 [astro-ph.CO]].
[22] M. Cannoni,J. D. Vergados and M. E. Gomez,
arXiv:1011.6108 [hep-ph].
[23] G. Bertone, D. G. Cerdeno, J. I. Collar and B. C. Odom,
Phys. Rev. Lett. 99 (2007) 151301 [arXiv:0705.2502
[astro-ph]].
[24] G. Bertone, D. G. Cerdeno, M. Fornasa, R. R. de Aus-
tri and R. Trotta, Phys. Rev. D 82, 055008 (2010)
[arXiv:1005.4280 [hep-ph]].
[25] J. R. Ellis, K. A. Olive and C. Savage, Phys. Rev. D 77,
065026 (2008) [arXiv:0801.3656 [hep-ph]].
[26] Y. Akrami, C. Savage, P. Scott, J. Conrad and J. Edsjo,
arXiv:1011.4318 [astro-ph.CO].
[27] Y. Akrami, C. Savage, P. Scott, J. Conrad and J. Edsjo,
arXiv:1011.4297 [hep-ph].
[28] R. R. de Austri, R. Trotta and L. Roszkowski, JHEP
0605, 002 (2006) [arXiv:hep-ph/0602028].
[29] Z. Ahmed et al. [The CDMS-II Collaboration], Science
327, 1619 (2010) [arXiv:0912.3592 [astro-ph.CO]].
[30] E. Aprile et al. (XENON1T Collaboration), XENON1T
at LNGS, Proposal, April (2010) and Technical Design
D82,023530 (2010)
Page 11
11
Report, October (2010).
[31] http://darwin.physik.uzh.ch/
[32] http://www.eureca.ox.ac.uk/
[33] L. Baudis (for the DARWIN Consortium), in Proceedings
of Science, PoS(IDM2010)122 (2010).
[34] http://www.cresst.de/
[35] http://edelweiss.in2p3.fr/
[36] http://www.fnal.gov/pub/max/index.html
[37] D. N. McKinsey, Journal of Physics, Conference Series
203 (2010) 012026
[38] P. Brink, Talk at DM2010, Marina del Rey (2010)
http://www.physics.ucla.edu/hep/dm10/talks/brink.pdf
[39] http://www.dusel.org/
[40] Hiroyuki Sekiya (for the XMASS collaboration), in pro-
ceedings of the 1st International Workshop towards
the Giant Liquid Argon Charge Imaging Experiment,
arXiv:1006.1473 (2010)
[41] http://www-sk.icrr.u-tokyo.ac.jp/index-e.html
[42] http://deapclean.org/
[43] http://www.snolab.ca/
[44] Christian Regenfus, private communication.
[45] R. Trotta, Bayes in the sky: Bayesian inference and
model selection in cosmology, Contemp. Phys. 49 (2008)
71–104.
[46] R. Trotta, Mon. Not. Roy. Astron. Soc., 378, 72 (2007).
[47] G. Cowan, K. Cranmer, E. Gross and O. Vitells,
arXiv:1007.1727 [physics.data-an].
[48] F. Feroz and M. P. Hobson, Multimodal nested sampling:
an efficient and robust alternative to MCMC methods for
astronomical data analysis, Mon. Not. Roy. Astron. Soc.
384 (2008) 449–463, 0704.3704.
[49] F. Feroz, M. P. Hobson, and M. Bridges, Mon. Not. Roy.
Astron. Soc. 398 (2009) 1601–1614, 0809.3437.
[50] R. Trotta, F. Feroz, M. Hobson, L. Roszkowski, and
R. Ruiz de Austri, Journal of High Energy Physics 12
(Dec., 2008) 24, 0809.3792.
[51] J. Skilling, Nested Sampling for Bayesian Computations,
Proc. Valencia/ISBA 8thWorld Meeting on Bayesian
Statistics (2006).
[52] M. Vogelsberger et al., Mon. Not. Roy. Astron. Soc. 395,
797 (2009) [arXiv:0812.0362 [astro-ph]].
[53] M. Kuhlen, N. Weiner, J. Diemand et al., JCAP 1002,
030 (2010). [arXiv:0912.2358 [astro-ph.GA]].
[54] M. Lisanti, L. E. Strigari,
[arXiv:1010.4300 [astro-ph.CO]].
[55] M. C. Smith et al., Mon. Not. Roy. Astron. Soc. 379
(2007) 755 [arXiv:astro-ph/0611671].
[56] X. X. Xue et al. [SDSS Collaboration], Astrophys. J.
684 (2008) 1143 [arXiv:0801.1232 [astro-ph]]; Y. So-
fue, arXiv:0811.0860 [astro-ph]; M. J. Reid et al., As-
trophys. J. 700, 137 (2009) [arXiv:0902.3913 [astro-
ph.GA]]; J. Bovy, D. W. Hogg and H. W. Rix, Astrophys.
J. 704 (2009) 1704 [arXiv:0907.5423 [astro-ph.GA]];
P. J. McMillan and J. J. Binney, arXiv:0907.4685 [astro-
ph.GA].
[57] R. Catena and P. Ullio,
[arXiv:0907.0018 [astro-ph.CO]].
[58] M. Pato,O. Agertz, G. Bertone,
R.Teyssier,Phys.Rev.
[arXiv:1006.1322 [astro-ph.HE]].
[59] P. Salucci, F. Nesti, G. Gentile and C. F. Martins,
arXiv:1003.3101 [astro-ph.GA].
[60] M. Weber, W. de Boer, Astron. Astrophys. 509, A25
(2010). [arXiv:0910.4272 [astro-ph.CO]].
J. G. Wacker et al.,
JCAP 1008 (2010) 004
B. Moore and
023531D 82, (2010)
Download full-text