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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

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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)

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4

target ? [ton×yr] ηcut ANR ?eff [ton×yr] Ethr [keV] σ(E) [keV] background events/?eff

Xe 5.00.8 0.5 2.00

Ge

3.0 0.8 0.92.16

Ar10.00.8 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

)

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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).

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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 150200250 300350 400450

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).

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