# Model-independent comparison of annual modulation and total rate with direct
detection experiments

Felix Kahlhoefer  Florian Reindl  Karoline Schäffner
Kai Schmidt-Hoberg  and Sebastian Wild

###### Abstract

The relative sensitivity of different direct detection experiments depends
sensitively on the astrophysical distribution and particle physics nature of
dark matter, prohibiting a model-independent comparison. The situation changes
fundamentally if two experiments employ the same target material. We show that
in this case one can compare measurements of an annual modulation and
exclusion bounds on the total rate while making no assumptions on astrophysics
and no (or only very general) assumptions on particle physics. In particular,
we show that the dark matter interpretation of the DAMA/LIBRA signal can be
conclusively tested with COSINUS, a future experiment employing the same
target material. We find that if COSINUS excludes a dark matter scattering
rate of about $0.01\,\text{kg}^{-1}\,\text{days}^{-1}$ with an energy
threshold of $1.8\,$keV and resolution of $0.2\,$keV, it will rule out all
explanations of DAMA/LIBRA in terms of dark matter scattering off sodium
and/or iodine.

##  1 Introduction

When lamenting the absence of conclusive evidence for particle dark matter (or
in fact any kind of new physics beyond the Standard Model), it is often
forgotten that we have been observing for many years a statistically highly
significant signal compatible with predictions for the scattering of dark
matter (DM) particles from the Galactic halo. The combined data from the
DAMA/NaI and the DAMA/LIBRA experiments show evidence for an annual modulation
of the event rate at the 9.3$\sigma$ level [1], which agrees with the DM
hypothesis in both period and phase.111One of the few unexpected features in
the DAMA data is a slight time-dependence of the modulation amplitude [2].
Even though the NaI detectors employed by the DAMA collaboration do not allow
to distinguish nuclear and electron recoils, it has not been possible to
identify a plausible background that could account for the observed signal
while satisfying all cross-checks performed by the DAMA collaboration (see
refs. [3, 4, 5, 6, 7, 8, 9] for recent discussions).

The great trouble with the DAMA signal is its incompatibility with exclusion
limits obtained from other direct detection experiments. For the simplest
assumptions, i.e. spin-independent or spin-dependent interactions between DM
and nuclei, a number of experiments exclude the parameter region preferred by
DAMA by several orders of magnitude [10, 11, 12, 13, 14, 15, 16, 17, 18, 19].
But even more exotic hypotheses, such as leptophilic DM [20], inelastic DM
[21] or isospin-violating DM [22], are strongly disfavoured by recent searches
(see e.g. ref. [23]).

While it is very challenging to construct DM models that bring all
experimental results into agreement (see refs. [24, 25, 26, 27, 28, 29] for
some recent attempts), it is even more difficult to generally exclude a DM
explanation of the DAMA signal. The reason is that any comparison between
different direct detection experiments requires a number of assumptions on
both the astrophysical distribution and the particle physics properties of DM
and the conclusions may depend sensitively on these assumptions. A number of
halo-independent methods have been developed to deal with the former issue,
allowing for a statistically meaningful comparison of different experiments
without the need to assume a specific DM density or velocity distribution [30,
31, 32, 33, 34, 35, 36].

The best way to reduce the need to make specific assumptions on the particle
physics properties of DM would be an independent test of the DAMA result with
NaI-based detectors. Indeed, a number of experimental collaborations are
currently pursuing this strategy. Most of these, namely SABRE [37], ANAIS
[38], PICO-LON [39] and COSINE [40] (the latter being a joint effort of the
DM-Ice [41] and KIMS [42] collaborations), aim to search for an annual
modulation in their event rates in much the same way as the original DAMA
experiment. The COSINUS collaboration [43], in contrast, has the more
ambitious goal to develop a cryogenic NaI detector that can distinguish
nuclear and electron recoils. This approach will allow to significantly
suppress background levels, making it possible for the first time to test the
DAMA signal using the absolute rate of nuclear recoils in NaI detectors. At
the same time, the possibility to measure phonon signals will allow COSINUS to
significantly lower the nuclear recoil energy threshold compared to DAMA,
further increasing its sensitivity.

Nevertheless, the comparison of a total rate and a modulation amplitude
appears to require specific assumptions on the DM velocity distribution, which
determines the modulation fraction of the signal. Existing halo-independent
methods deal with this issue by making very conservative assumptions on the
modulation fraction, typically allowing modulation fractions up to 100%.222A
notable exception are refs. [44, 45, 46, 47, 48], which derive halo-
independent constraints on the modulation amplitude from constraints on the
total rate. While very similar in spirit, the difference between these works
and ours is that we explicitly construct the velocity distribution that
maximizes the modulation amplitude rather than deriving conservative upper
bounds. In the present work we show that for the case of several experiments
employing the same detector material, significantly stronger consistency
requirements can be made, leading to potentially much tighter constraints.
Moreover, we will show that halo-independent methods can be generalised to
consider not only arbitrary DM velocity distributions but also very general
classes of interactions between DM and nuclei. This approach is therefore
ideally suited for comparing future results from COSINUS with the DAMA signal.

Figure 1: COSINUS exclusion power, defined as the bound on the total rate (or
equivalently the total exposure with zero observed events) that COSINUS must
achieve for excluding DAMA in a halo-independent way, as a function of the
assumed threshold in COSINUS for different DM masses. More general assumptions
correspond to weaker exclusion power, meaning that stronger bounds are
necessary to achieve an exclusion. The most general bound, valid for arbitrary
DM-nucleus scattering, is derived in section 3. The assumption of falling
recoil spectra, which is more restrictive but still covers many cases of
interest, is discussed in section 4. The specific example of halo-independent
constraints on spin-independent scattering is considered in section 5. The red
dot in each panel indicates the design sensitivity of COSINUS,
$E_{\text{thres}}=1\,\text{keV}$ and
$R_{\text{COSINUS}}^{\text{bound}}=0.1\,\mathrm{kg^{-1}\,days^{-1}}$.

A particular emphasis is placed on the question how to optimise the design of
the COSINUS detector. As in all direct detection experiments, the detector
development faces two competing goals: lowering the energy threshold and
improving the bound on the event rate above threshold (e.g. by increasing the
detector mass). We show that for a halo-independent test of the DAMA signal,
it is highly desirable to achieve an energy threshold of about
$1.8\,\mathrm{keV}$ (based on conservative assumptions on the quenching
factors in DAMA and the energy resolution in COSINUS), while a further
reduction of the threshold is less advantageous than e.g. an increase in
exposure. These conclusions are illustrated in figure 1, which shows the bound
on the total event rate that COSINUS must achieve in order to exclude the DAMA
signal in a halo-independent way as a function of the COSINUS threshold. The
different lines correspond to different assumptions on the interactions of DM,
which will be explained in the following chapters.

This paper is structured as follows: In section 2 we review the DAMA
experiment and the COSINUS proposal, and summarise the assumptions made for
our specific implementation of these experiments. In section 3 we present the
simplest – but also most conservative – way to compare the two experiments by
comparing the respective efficiencies. Section 4 discusses a more constraining
approach, valid for rather general classes of recoil spectra. We compare the
results obtained from this approach with the ones obtained for more specific
assumptions on the interactions of DM in section 5. Our conclusions are
presented in section 6, while additional technical details can be found in
appendix A.

##  2 Direct detection with DAMA and COSINUS

To introduce the relevant notation and conventions, let us briefly review the
formalism for DM direct detection. The differential event rate with respect to
recoil energy is given by

$\frac{\text{d}R^{\text{T}}}{\text{d}E_{\text{R}}}=\frac{\xi_{\mathrm{T}}\rho}{m_{\mathrm{T}}\,m_{\chi}}\int_{v_{\text{min}}}^{\infty}vf(\boldsymbol{v}+\boldsymbol{v}_{\text{E}}(t))\frac{\text{d}\sigma^{\text{T}}}{\mathrm{d}E_{\text{R}}}\text{d}^{3}v\;,$
(2.1)

where $m_{\chi}$ and $m_{\mathrm{T}}$ are the mass of the DM particle and the
target nucleus T, $\xi_{\text{T}}$ is the mass fraction of T relative to the
total detector mass, $\rho$ and $f(\mathbf{v})$ are the local DM density and
velocity distribution in the rest frame of the Sun,
$\boldsymbol{v}_{\text{E}}(t)$ is the velocity of the Earth and
$v=|\boldsymbol{v}|$.333It is conventional to define both $f(\mathbf{v})$ and
$\boldsymbol{v}_{\text{E}}(t)$ in the Galactic rest frame, where the velocity
distribution often takes a very simple form. However, in the context of halo-
independent methods it will be more convenient to specify $f(\mathbf{v})$ in
the rest frame of the Sun. In order to transfer an energy $E_{\text{R}}$ in an
elastic collision, the incoming DM particle must have a _minimum velocity_ of

$v_{\text{min}}(E_{\text{R}})=\sqrt{\frac{m_{\mathrm{T}}E_{\text{R}}}{2\,\mu_{\mathrm{T}\chi}^{2}}}\;,$
(2.2)

where
$\mu_{\mathrm{T}\chi}=m_{\mathrm{T}}\,m_{\chi}/(m_{\mathrm{T}}+m_{\chi})$ is
the reduced mass of the DM-nucleus system.

For reasons that will become apparent below, we will focus on cut-and-count
experiments considering a single signal region in energy, i.e. disregarding
the spectral information of the signal. Such an experiment can be fully
characterised by the total exposure $\mathcal{E}$ and the efficiency functions
$\epsilon^{\text{T}}(E_{\text{R}})$ that quantify the probability for a
nuclear recoil of the target nucleus T with energy $E_{\text{R}}$ to lead to
an event in the signal region. In terms of these quantities, the number of
predicted events is given by

$N=\mathcal{E}\sum_{\text{T}}\int\frac{\text{d}R^{\text{T}}}{\text{d}E_{\text{R}}}\,\epsilon^{\text{T}}(E_{\text{R}})\,\text{d}E_{\text{R}}\;.$
(2.3)

The total event rate in the signal region is given by

$R=\frac{N}{\mathcal{E}}=\sum_{\text{T}}\int\frac{\text{d}R^{\text{T}}}{\text{d}E_{\text{R}}}\,\epsilon^{\text{T}}(E_{\text{R}})\,\text{d}E_{\text{R}}\;.$
(2.4)

A null result allows an experiment to place an upper bound on $N$ for a given
exposure or, equivalently, an upper bound on $R$. For example, in the case of
zero observed events, it is possible to exclude $N>2.71$ at 90% confidence
level (CL), leading to the upper bound $R<2.71/\mathcal{E}$.

In general, both $N$ and $R$ depend on time because of the time-dependence of
$\mathbf{v}_{\text{E}}(t)$. If the DM velocity distribution is isotropic in
the Galactic rest frame, one expects an approximately sinusoidal annual
modulation that peaks around the 1st of June [49, 50]. This prediction is in
good agreement with the time-dependence of the DAMA signal. We therefore
define the mean rate444Note that for approximately sinusoidal modulation with
period of one year, our definition of $\bar{R}$ agrees with the time-averaged
rate $\langle R\rangle=\tfrac{1}{t}\int R(t^{\prime})\,\mathrm{d}t^{\prime}$.

$\displaystyle\bar{R}=\frac{1}{2}\left(R(t=\text{1st of June})+R(t=\text{1st
of December})\right)\;,$ (2.5)

and the modulation amplitude

$\displaystyle S=\frac{1}{2}\left(R(t=\text{1st of June})-R(t=\text{1st of
December})\right)\;.$ (2.6)

The results of the DAMA experiment can then be summarised in terms of the
modulation amplitude observed in different energy bins. The greatest evidence
for an annual modulation is seen in the energy range
$E^{\text{obs}}_{\mathrm{R}}\in\left[2.5\,\mathrm{keVee},3.5\,\mathrm{keVee}\right]$,
where $E^{\text{obs}}_{\mathrm{R}}$ denotes observed recoil energy after
quenching (see below). In this energy range the combined data from DAMA/NaI
and DAMA/LIBRA imply [1]

$S=(2.34\pm 0.28)\cdot 10^{-2}\,\mathrm{kg^{-1}\,days^{-1}}\;.$ (2.7)

In the following we will focus on this energy range and study whether COSINUS
can test the hypothesis that the signal observed there is due to interactions
between DM and nuclei. If COSINUS can exclude this hypothesis, it will be
impossible to find a DM model that provides a satisfactory explanation of the
DAMA signal. We have checked explicitly that our conclusions do not depend
sensitively on the energy range that we consider.

In order to predict the modulation amplitude in the DAMA signal region for a
given differential event rate, one must first of all convert the physical
recoil energy $E_{\mathrm{R}}$ into a quenched energy
$E^{\prime}_{\mathrm{R}}=Q\,E_{\mathrm{R}}$, where $Q$ denotes the quenching
factor.555To distinguish between physical recoil energy and reconstructed
(quenched) recoil energy, we use the unit keVee (ee = electron equivalent)
when referring to the latter. For the present analysis we adopt the values
$Q=0.3$ and 0.09 for sodium and iodine scatters, respectively [51]. Some
recent measurements indicate even smaller quenching factors [52], which would
further strengthen the sensitivity of COSINUS with respect to the DAMA data as
the COSINUS detection efficiency is a monotonically increasing function of
energy. To calculate the efficiency functions
$\epsilon^{\text{T}}(E_{\mathrm{R}})$, we follow refs. [53, 54] and assume a
Gaussian energy resolution with
$\sigma(E^{\prime}_{\mathrm{R}})=(0.448\,\text{keVee})\sqrt{E^{\prime}_{\mathrm{R}}/\text{keVee}}+0.0091E^{\prime}_{\mathrm{R}}$.
We only consider fluctuations of up to four standard deviations, in order to
avoid an unphysical extrapolation.

In contrast to DAMA, COSINUS aims to use NaI within a cryogenic detector,
simultaneously acquiring a phonon (heat) and a light signal from a particle
interaction. The phonon channel yields a precise and unquenched, thus,
particle-independent measurement of the energy deposited in the crystal; the
scintillation light provides particle discrimination on an event-by-event
basis. This technology has been shown to be extraordinarily sensitive and
allowed the CRESST collaboration to lead the field of direct dark matter
detection on the low DM mass frontier [55]. In the past two years COSINUS
successfully proved for the first time that cryogenic operation of NaI is
possible in ref. [56], leading to the commissioning of the final detector
design [57].666In parallel to detector optimisation the COSINUS collaboration
is also preparing measurements of the quenching factors for scatterings off
sodium and iodine at cryogenic temperatures by irradiating a COSINUS detector
to neutrons from AmBe sources and/or a neutron beam [58]. The light channel
already exceeds the final performance goals while further improvements in the
phonon channel are necessary in order to further decrease the threshold.

Our implementation of the COSINUS experiment largely follows ref. [43]. We
assume that physical recoil energy $E_{\mathrm{R}}$ and observed recoil energy
$E_{\mathrm{R}}^{\text{obs}}$ are related via a Gaussian energy resolution
with $\sigma_{E}=0.2\,\,\mathrm{keV}$, where once again we only take into
account fluctuations of up to $4\sigma_{E}$. The probability for detecting a
nuclear recoil with observed energy $E_{\mathrm{R}}^{\text{obs}}$ is given by

$\displaystyle\epsilon_{\text{det}}(E_{\mathrm{R}}^{\text{obs}})=\Theta(E_{\mathrm{R}}^{\text{obs}}-E_{\text{thres}})\times\begin{cases}0&\mbox{for
}E_{\mathrm{R}}^{\text{obs}}<1\,\text{keV}\,,\\\
0.3\cdot\frac{E_{\mathrm{R}}^{\text{obs}}}{\text{keV}}-0.1&\mbox{for
}1\,\text{keV}\leq E_{\mathrm{R}}^{\text{obs}}\leq 2\,\text{keV}\,,\\\
0.5&\mbox{for }E_{\mathrm{R}}^{\text{obs}}>2\,\text{keV}\,,\end{cases}$ (2.8)

where $\Theta(x)$ is the Heaviside step function and $E_{\text{thres}}$
denotes the low-energy threshold of the signal region considered by COSINUS.
Unless explicitly stated otherwise, we adopt the value
$E_{\text{thres}}=1\,\text{keV}$.

A crucial part of the COSINUS strategy to reduce backgrounds is to reject
events with large light yield, as these are much more likely to arise from
electron recoils than from nuclear recoils. The signal region is therefore
restricted to events with a light yield below the mean of the sodium nuclear
recoil band (see figure 2 in ref. [43]). The final efficiency function is then
given by
$\epsilon(E_{\mathrm{R}}^{\text{obs}})=\epsilon_{\text{det}}(E_{\mathrm{R}}^{\text{obs}})\times\epsilon_{\text{ly}}(E_{\mathrm{R}}^{\text{obs}})$,
where the latter factor quantifies the probability that an event with
reconstructed energy $E_{\mathrm{R}}^{\text{obs}}$ has sufficiently low light
yield to fall into the signal region. For sodium recoils, one has by
construction $\epsilon_{\text{ly}}=0.5$, whereas a fit of the iodine nuclear
recoil band gives

$\displaystyle\epsilon_{\text{ly}}(E_{\mathrm{R}}^{\text{obs}})\approx
1-0.42\exp(-0.89E_{\mathrm{R}}^{\text{obs}}/\text{keV})$ (2.9)

for iodine recoils.

It is important to emphasise that the ability of the COSINUS detector to
discriminate between electron recoils and nuclear recoils is reduced at low
energies. In other words, a lowering of $E_{\text{thres}}$ may lead to an
increase in the number of observed background events and hence to a weaker
bound on $R$. One of the central questions of the subsequent sections will be
to determine under which conditions it will be more desirable for COSINUS to
lower $E_{\text{thres}}$ or to strive for a more stringent bound on $R$.

Figure 2: Comparison of the projected sensitivity of COSINUS (black lines) and
the parameter regions favoured by DAMA (shaded grey) for spin-independent
interactions and a Maxwell-Boltzmann velocity distribution. The three
different lines for COSINUS correspond to different assumptions on
$E_{\text{thres}}$. For comparison we show the COSINUS expected sensitivity
for $E_{\text{thres}}=1\,\text{keV}$ derived in ref. [43] using an optimum
interval analysis (violet line). The orange shaded regions, which are taken
from fig. 2 of ref. [59], indicate the parameter regions favoured by DAMA when
including 8 separate bins (90% CL, rescaled to
$\rho=0.3\,\text{GeV}\,\text{cm}^{-3}$).

To conclude this section, we compare in figure 2 the projected sensitivity of
the COSINUS experiment with the parameter region favoured by DAMA for standard
assumptions on the properties of DM [60]. Specifically we assume spin-
independent interactions with

$\frac{\text{d}\sigma^{\text{T}}}{\text{d}E_{\text{R}}}=A^{2}\,F^{2}(E_{\text{R}})\frac{m_{\mathrm{T}}\,\sigma_{p}}{2\,\mu_{p\chi}^{2}\,v^{2}}\;,$
(2.10)

where $A$ and $F(E_{\mathrm{R}})$ denote the mass number and form factor of
the target nucleus, and $\sigma_{p}$ and $\mu_{p\chi}$ denote the DM-proton
scattering cross section and reduced mass, respectively. We set
$\rho=0.3\,\mathrm{GeV\,cm^{-3}}$ and adopt a Maxwell-Boltzmann distribution
with most probable speed $v_{0}=220\,\text{km}\,\text{s}^{-1}$ and a cut-off
at $v_{\text{esc}}=544\,\text{km}\,\text{s}^{-1}$ for $f(v)$ in the Galactic
rest frame. To obtain the corresponding velocity distribution in the rest
frame of the Sun, we take the velocity of the Sun in the Galactic rest frame
to be $v_{\text{sun}}=230\,\mathrm{km\,s^{-1}}$.

We assume that COSINUS can set an upper bound of
$R<0.1\,\mathrm{kg^{-1}\,days^{-1}}$ (corresponding e.g. to a 95% CL upper
limit for an exposure of $105\,\mathrm{kg\,days}$ and 5 observed events) and
illustrate the impact of changing the low-energy threshold by considering
three different values of $E_{\text{thres}}$. Our estimated sensitivity for
$E_{\text{thres}}=1\,$keV agrees quite well with the one from ref. [43], which
makes use of the optimum interval method [61] to derive constraints. Since we
implement only a single energy range for DAMA, we do not find two separate
signal regions (corresponding respectively to scattering dominantly off sodium
and dominantly off iodine) but instead obtain a band of allowed parameters. In
particular, our implementation of DAMA is conservative, because it leads to a
significantly larger allowed parameter region.

The central conclusion from figure 2 is that for the assumptions of spin-
independent interactions and a Maxwell-Boltzmann velocity distribution COSINUS
should be able to exclude the DAMA region by about two orders of magnitude in
cross section. However, as pointed out in section 1, the DAMA hypothesis is
already robustly excluded for standard assumptions by a number of experiments.
We will therefore now turn to the much more interesting question how the
sensitivity of COSINUS compares to DAMA for less restrictive assumptions.

##  3 Halo- and model-independent comparison

We begin by discussing the most general – and hence most conservative – way of
comparing DAMA and COSINUS. This approach is based on the very simple
observation that the modulation amplitude in a given experiment cannot exceed
the mean rate [32]:

$S\leq\bar{R}\;,$ (3.1)

which directly follows from eqs. (2.5) and (2.6). We now define
$S_{\text{DAMA}}^{\text{bound}}\equiv 1.78\cdot
10^{-2}\,\mathrm{kg^{-1}\,days^{-1}}$ as the smallest value of the modulation
amplitude compatible with the DAMA measurement at the $95\%$ CL, see eq.
(2.7), and $R_{\text{COSINUS}}^{\text{bound}}$ as the 95% CL upper bound on
the mean total event rate in COSINUS. Using eq. (3.1), the modulation
amplitude $S_{\text{DAMA}}$ in DAMA must then satisfy

$\displaystyle S_{\text{DAMA}}^{\text{bound}}\leq
S_{\text{DAMA}}\leq\bar{R}_{\text{DAMA}}=\sum_{\text{T}}\int\frac{\text{d}\bar{R}^{\text{T}}}{\text{d}E_{\text{R}}}\,\epsilon_{\text{DAMA}}^{\text{T}}(E_{\text{R}})\,\text{d}E_{\text{R}}\,,$
(3.2)

where we define $\text{d}\bar{R}^{\text{T}}/\text{d}E_{\text{R}}$ analogously
to eq. (2.5) as the mean differential recoil rate during the year. On the
other hand, the upper bound from COSINUS implies

$\displaystyle
R_{\text{COSINUS}}^{\text{bound}}\geq\bar{R}_{\text{COSINUS}}=\sum_{\text{T}}\int\frac{\text{d}\bar{R}^{\text{T}}}{\text{d}E_{\text{R}}}\,\epsilon_{\text{COSINUS}}^{\text{T}}(E_{\text{R}})\,\text{d}E_{\text{R}}\,.$
(3.3)

These two equations can be combined to give

$\sum_{\text{T}}\int\frac{\text{d}\bar{R}^{\text{T}}}{\text{d}E_{\text{R}}}\frac{\epsilon_{\text{DAMA}}^{\text{T}}(E_{\text{R}})}{S_{\text{DAMA}}^{\text{bound}}}\,\text{d}E_{\text{R}}\geq\sum_{\text{T}}\int\frac{\text{d}\bar{R}^{\text{T}}}{\text{d}E_{\text{R}}}\frac{\epsilon_{\text{COSINUS}}^{\text{T}}(E_{\text{R}})}{R_{\text{COSINUS}}^{\text{bound}}}\,\text{d}E_{\text{R}}\;.$
(3.4)

Crucially, the mean differential event rate
$\mathrm{d}\bar{R}^{\text{T}}/\mathrm{d}E_{\mathrm{R}}$ is the same for both
DAMA and COSINUS, irrespective of the type of interaction, because they employ
exactly the same detector material. This means that eq. (3.4) can only hold if
$\epsilon_{\text{DAMA}}^{\text{T}}(E_{\text{R}})/S_{\text{DAMA}}^{\text{bound}}>\epsilon_{\text{COSINUS}}^{\text{T}}(E_{\text{R}})/R_{\text{COSINUS}}^{\text{bound}}$
for at least one target nucleus T and some value of $E_{\text{R}}$.
Consequently, the COSINUS measurement excludes an explanation of the DAMA
signal in terms of DM-nucleus scattering for _arbitrary differential event
rates_ at more than 95% CL if

$\dfrac{\epsilon_{\text{COSINUS}}^{\text{T}}(E_{\mathrm{R}})}{R_{\text{COSINUS}}^{\text{bound}}}>\dfrac{\epsilon_{\text{DAMA}}^{\text{T}}(E_{\mathrm{R}})}{S_{\text{DAMA}}^{\text{bound}}}\quad\text{
for all }E_{\mathrm{R}}\text{ and all }\text{T}\;.$ (3.5)

Figure 3: Exposure-weighted efficiencies for DAMA (orange) and COSINUS
(violet), corresponding to the two sides of eq. (3.5). The different violet
lines reflect different assumptions on $R_{\text{COSINUS}}^{\text{bound}}$,
i.e. the bound on the total rate that COSINUS can achieve. If
$R_{\text{COSINUS}}^{\text{bound}}$ is sufficiently small such that the
COSINUS line lies above the DAMA line for all values of $E_{\mathrm{R}}$, the
DAMA signal will be excluded for arbitrary differential event rates. This is
the case for
$R_{\text{COSINUS}}^{\text{bound}}<10^{-2}\,\mathrm{kg^{-1}\,days^{-1}}$ for
scattering off sodium and $R_{\text{COSINUS}}^{\text{bound}}<2\cdot
10^{-2}\,\mathrm{kg^{-1}\,days^{-1}}$ for scattering off iodine.

In figure 3 we show a comparison of the left-hand side and right-hand side of
eq. (3.5) for different assumptions on the value of
$R_{\text{COSINUS}}^{\text{bound}}$ that can be achieved. The left panel
corresponds to scattering off sodium, the right to scattering off iodine. As
expected, the DAMA sensitivity peaks around $E_{\mathrm{R}}\approx
10\,\mathrm{keV}$ for sodium and around $E_{\mathrm{R}}\approx
33\,\mathrm{keV}$ for iodine, both of which correspond to
$E^{\prime}_{\mathrm{R}}\approx 3\,\mathrm{keVee}$ when taking into account
the appropriate quenching factors. The sensitivity of COSINUS, on the other
hand, covers a much wider range of energies and in particular extends to
significantly lower values of $E_{\mathrm{R}}$. In order to exclude the DAMA
signal for sodium scatters, COSINUS thus needs to place a bound on the total
event rate of $R_{\text{COSINUS}}^{\text{bound}}\approx
10^{-2}\,\text{kg}^{-1}\,\text{days}^{-1}$. A slightly weaker bound is
sufficient for the case of iodine scatters, because the corresponding
efficiency function in COSINUS is slightly larger. Since it is however not
known what fraction of the DAMA signal is due to iodine scatters, a fully
model-independent exclusion is only possible if COSINUS can exclude the DAMA
signal for the case of scattering exclusively on sodium.

Eq. (3.5) constitutes the most general method for comparing DAMA and COSINUS
(or more generally any experiment observing an annual modulation with another
experiment constraining the total rate for the same detector material). This
approach is valid as long as the signal in both experiments is due to nuclear
recoils. In particular it applies to all elastic and inelastic DM-nucleus
scattering processes and to arbitrary velocity distributions. The sensitivity
that COSINUS must achieve is however quite ambitious, about an order of
magnitude beyond what we assumed for figure 2. We will therefore now explore
whether stronger constraints can be obtained for more restrictive assumptions
on the differential event rate.

##  4 Halo-independent comparison for classes of cross sections

Let us begin by considering a fixed DM mass $m_{\chi}$ and a differential
cross section with fixed shape but free normalisation, which is controlled by
a parameter $\sigma$.777For the moment we assume that the differential event
rate is dominated by either scattering off sodium or scattering off iodine.
Below we will generalise to the case where both elements contribute at
comparable level. For a given DM velocity distribution $f(\mathbf{v})$ in the
rest frame of the Sun, we define $R^{f}_{\text{COSINUS}}(\sigma)$ and
$S^{f}_{\text{DAMA}}(\sigma)$ as the predicted rate and modulation amplitude
in COSINUS and DAMA, respectively. With these conventions we define

$S_{\text{DAMA}}^{\text{max}}\equiv\max_{f}\big{[}S_{\text{DAMA}}^{f}(\sigma)\text{
with }\sigma\text{ given by
}R_{\text{COSINUS}}^{f}(\sigma)=R_{\text{COSINUS}}^{\text{bound}}\big{]}\;.$
(4.1)

In other words, $S_{\text{DAMA}}^{\text{max}}$ is the largest modulation
amplitude compatible with the COSINUS bound that can be obtained in DAMA from
the assumed cross section for _any DM velocity distribution_. If
$S_{\text{DAMA}}^{\text{max}}<S_{\text{DAMA}}^{\text{bound}}$, we can
therefore conclude that COSINUS excludes DAMA for the assumed cross section in
a halo-independent way.

At first sight, calculating $S_{\text{DAMA}}^{\text{max}}$ seems rather
challenging, given that it requires varying the three-dimensional function
$f(\mathbf{v})$. To simplify the problem, we make the important observation
that eq. (2.1) is linear in $f(\mathbf{v})$, meaning that we can write

$\displaystyle R^{f}_{\text{COSINUS}}(\sigma)$
$\displaystyle=\int\mathrm{d}^{3}\,v_{0}\,f(\mathbf{v}_{0})\,\mathcal{R}_{\text{COSINUS}}(\mathbf{v}_{0},\sigma)\;.$
(4.2)

Here $\mathcal{R}_{\text{COSINUS}}(\mathbf{v}_{0},\sigma)$ is the rate
calculated from the assumed differential cross section for a velocity
distribution of the form
$f(\mathbf{v})=\delta^{3}(\mathbf{v}-\mathbf{v}_{0})$, corresponding to a DM
stream with a single velocity $\mathbf{v}_{0}$ [62]. In a fully analogous way,
$S^{f}_{\text{DAMA}}$ can be expressed as a superposition of functions
$\mathcal{S}_{\text{DAMA}}(\mathbf{v}_{0},\sigma)$.

As shown explicitly in appendix A, it turns out that in order to calculate
$S_{\text{DAMA}}^{\text{max}}$, it is sufficient to consider velocity
distributions of the form
$f(\mathbf{v})=\delta^{3}(\mathbf{v}-\mathbf{v}_{0})$. In other words, the
optimum velocity distribution that maximises $S_{\text{DAMA}}^{f}$ always
corresponds to a single DM stream:888Note that for very small DM masses the
optimal velocity can be significantly larger than the values typically assumed
for the Galactic escape velocity $v_{\text{esc}}$. Restricting velocities to
be smaller than $v_{\text{esc}}$ would lead to even stronger bounds.

$S_{\text{DAMA}}^{\text{max}}\equiv\max_{\mathbf{v}_{0}}\big{[}\mathcal{S}_{\text{DAMA}}(\mathbf{v}_{0},\sigma)\text{
with }\sigma\text{ given by
}\mathcal{R}_{\text{COSINUS}}(\mathbf{v}_{0},\sigma)=R_{\text{COSINUS}}^{\text{bound}}\big{]}\;.$
(4.3)

At this point it is essential that we consider only a single energy range for
both DAMA and COSINUS. The conclusion that a single DM stream is optimal no
longer holds for experiments with several signal regions or experiments
performing an unbinned analysis [63, 35]. In fact, the problem can be
simplified further, because
$\mathcal{R}_{\text{COSINUS}}(\mathbf{v}_{0},\sigma)$ depends only on
$v_{0}=|\mathbf{v}_{0}|$, while
$\mathcal{S}_{\text{DAMA}}(\mathbf{v}_{0},\sigma)$ is maximized for streams
that are anti-aligned with the velocity of the Earth on the 1st of June, given
by $\mathbf{v}_{\mathrm{E}}\equiv v_{\mathrm{E}}\,\mathbf{e}_{\text{summer}}$
with $v_{\text{E}}=30\,\text{km}\,\text{s}^{-1}$. It is therefore sufficient
to consider only streams with
$\mathbf{v}_{0}=v_{0}\,\mathbf{e}_{\text{summer}}$.999We note that
gravitational focusing may modify the phase of the modulation signal [64],
such that streams of a slightly different direction are needed to give the
best fit to the DAMA signal.

We conclude that, for a given cross section, COSINUS excludes DAMA in a halo-
independent way if

$\frac{\mathcal{S}_{\text{DAMA}}(v_{0},\sigma)}{S_{\text{DAMA}}^{\text{bound}}}<\frac{\mathcal{R}_{\text{COSINUS}}(v_{0},\sigma)}{R_{\text{COSINUS}}^{\text{bound}}}$
(4.4)

for all $v_{0}$. Note that, since the parameter $\sigma$ leads to the same
rescaling on both sides, it is sufficient to check eq. (4.4) for one specific
value of $\sigma$. The quantities $\mathcal{R}_{\text{COSINUS}}$ and
$\mathcal{S}_{\text{DAMA}}$ are easily calculated, allowing to determine very
quickly whether DAMA and COSINUS can be compatible for a specific particle
physics hypothesis. Alternatively, eq. (4.4) can be rearranged to

$R_{\text{COSINUS}}^{\text{bound}}<\min_{v_{0}}\left[\frac{\mathcal{R}_{\text{COSINUS}}(v_{0})}{\mathcal{S}_{\text{DAMA}}(v_{0})}\right]S_{\text{DAMA}}^{\text{bound}}\;,$
(4.5)

i.e. we can calculate the bound on the total rate that COSINUS must achieve in
order to exclude a specific hypothesis in a halo-independent way. We shall
refer to this number as the _exclusion power_ of COSINUS.

The aim of this section is however to compare COSINUS and DAMA for whole
classes of cross sections at once. The requirement derived above is
insufficient for this purpose, as $\mathcal{R}_{\text{COSINUS}}$ and
$\mathcal{S}_{\text{DAMA}}$ need to be recalculated as soon as the form of the
differential cross section changes. However, we can make use of the same trick
used to consider arbitrary velocity distributions to also consider broad
classes of cross sections. Let us therefore assume that the differential cross
sections of interest can be written as a linear combination of a set of basis
functions

$\frac{\mathrm{d}\sigma^{\mathrm{T}}}{\mathrm{d}E_{\mathrm{R}}}=\sum_{i}\alpha^{\mathrm{T}}_{i}\frac{\mathrm{d}\sigma_{i}^{\text{T}}}{\mathrm{d}E_{\mathrm{R}}}\;,$
(4.6)

where $\mathrm{T}=\text{Na},\text{I}$ denotes the two different target
elements and we require $\alpha^{\mathrm{T}}_{i}\geq 0$. For each of these
basis functions we can then calculate the total rate in COSINUS and the
modulation amplitude in DAMA, called
$\mathcal{R}^{\mathrm{T}}_{i,\text{COSINUS}}$ and
$\mathcal{S}^{\mathrm{T}}_{i,\text{DAMA}}$, respectively. Note in particular
that this calculation needs to be done separately for the two target elements.
Following the same reasoning as above, one can immediately show that COSINUS
excludes DAMA for _any cross section of the assumed form_ and _any DM velocity
distribution_ if

$\frac{\mathcal{S}^{\mathrm{T}}_{i,\text{DAMA}}(v_{0})}{S_{\text{DAMA}}^{\text{bound}}}<\frac{\mathcal{R}^{\mathrm{T}}_{i,\text{COSINUS}}(v_{0})}{R_{\text{COSINUS}}^{\text{bound}}}$
(4.7)

for all $v_{0}$, $i$ and $\mathrm{T}$. Clearly, this approach requires more
calculational effort, but potentially allows to make very general statements.

###  4.1 Monotonically decreasing cross sections

Let us now turn to a particularly interesting example and consider the class
of models for which the differential cross section is a monotonically
decreasing function of recoil energy. More specifically, we assume that the
differential cross section can be written as

$\frac{\text{d}\sigma^{\mathrm{T}}}{\text{d}E_{\mathrm{R}}}=\frac{\sigma_{0}}{m_{\text{T}}v^{2}}\,\kappa^{\mathrm{T}}(E_{\mathrm{R}})\,\Theta(E^{\mathrm{T}}_{\text{max}}(v)-E_{\mathrm{R}})$
(4.8)

with an arbitrary reference cross section $\sigma_{0}$ and dimensionless
monotonically decreasing functions $\kappa^{\mathrm{T}}(E_{\mathrm{R}})$. In
this expression the factor $1/v^{2}$ corresponds to the velocity dependence
obtained in the usual case that the matrix element for the scattering does not
depend on velocity and the final factor (with
$E_{\text{max}}^{\text{T}}(v)=2\mu_{\mathrm{T}}^{2}v^{2}/m_{\mathrm{T}}$)
enforces energy and momentum conservation for elastic scattering. Examples for
cross sections that can be written in this way are spin-independent and spin-
dependent scattering (see section 5).101010Strictly speaking, the relevant
form factors only decrease monotonically for sufficiently small momentum
transfer, but this covers the energy range relevant for COSINUS and DAMA. Now
we make use of the fact that any monotonically decreasing function can be
written as a (potentially infinite) sum of step functions:

$\displaystyle\kappa^{\mathrm{T}}(E_{\mathrm{R}})=\int_{0}^{\infty}\alpha^{\mathrm{T}}(E_{0})\,\Theta(E_{0}-E_{\mathrm{R}})\mathrm{d}E_{0}\;.$
(4.9)

We can therefore write any differential cross section of the form given in eq.
(4.8) as

$\frac{\text{d}\sigma^{\mathrm{T}}}{\text{d}E_{\mathrm{R}}}=\int_{0}^{\infty}\alpha^{\mathrm{T}}(E_{0})\frac{\text{d}\sigma^{\mathrm{T}}}{\text{d}E_{\mathrm{R}}}(E_{0})\,\mathrm{d}E_{0}$
(4.10)

in terms of the basis functions

$\frac{\text{d}\sigma^{\mathrm{T}}}{\text{d}E_{\mathrm{R}}}(E_{0})=\frac{\sigma_{0}}{m_{\text{T}}v^{2}}\,\Theta(E_{0}-E_{\mathrm{R}})\,\Theta(E^{\mathrm{T}}_{\text{max}}(v)-E_{\mathrm{R}})\;.$
(4.11)

Here the continuous parameter $\alpha^{\mathrm{T}}(E_{0})$ takes the place of
the discrete parameter $\alpha^{\mathrm{T}}_{i}$ introduced above.

In complete analogy to the previous results, we conclude that COSINUS excludes
DAMA for _any monotonically decreasing differential cross section_ and _any DM
velocity distribution_ if

$\frac{\mathcal{S}^{\mathrm{T}}_{\text{DAMA}}(v_{0},E_{0})}{S_{\text{DAMA}}^{\text{bound}}}<\frac{\mathcal{R}^{\mathrm{T}}_{\text{COSINUS}}(v_{0},E_{0})}{R_{\text{COSINUS}}^{\text{bound}}}$
(4.12)

for all $v_{0}$, $E_{0}$ and $\mathrm{T}$. The great virtue of this expression
is that $\mathcal{R}^{\mathrm{T}}(v_{0},E_{0})$ and
$\mathcal{S}^{\mathrm{T}}(v_{0},E_{0})$ are in fact very easily calculated.
Resubstituting the various definitions, we obtain the quite simple expressions

$\mathcal{R}^{\mathrm{T}}(v_{0},E_{0})=\frac{\xi_{\text{T}}\,\rho\,\sigma_{0}}{m_{\text{T}}^{2}\,m_{\chi}\,v_{0}}\int\epsilon^{\mathrm{T}}(E_{\mathrm{R}})\,\Theta(E_{0}-E_{\mathrm{R}})\,\Theta(v_{0}-v^{\mathrm{T}}_{\text{min}}(E_{\mathrm{R}}))\,\mathrm{d}E_{\mathrm{R}}$
(4.13)

and

$\mathcal{S}^{\mathrm{T}}(v_{0},E_{0})\equiv\frac{1}{2}\left[\mathcal{R}^{\mathrm{T}}(v_{0}^{+},E_{0})-\mathcal{R}^{\mathrm{T}}(v_{0}^{-},E_{0})\right]$
(4.14)

with

$v_{0}^{+}=v_{0}+v_{\mathrm{E}}\,,\qquad v_{0}^{-}=|v_{0}-v_{\mathrm{E}}|\;.$
(4.15)

Note that $\mathcal{R}^{\mathrm{T}}(v_{0},E_{0})$ and
$\mathcal{S}^{\mathrm{T}}(v_{0},E_{0})$ depend on the DM mass via
$v^{\mathrm{T}}_{\text{min}}(E_{\mathrm{R}})$ and hence eq. (4.12) must be
checked separately for each value of $m_{\chi}$. The local DM density $\rho$
as well as the reference cross section $\sigma_{0}$, on the other hand, cancel
out in the condition (4.12).

Clearly, for many values of $v_{0}$ and $E_{0}$ the modulation amplitude in
DAMA will be zero or extremely small, either because the corresponding recoil
energies are too small to be observable or because they are so large that DAMA
is predicted to observe essentially the same rate in summer and winter. A
large modulation fraction is obtained only if $v_{0}$ and $E_{0}$ are such
that typical nuclear recoils are mostly below threshold in winter but slightly
above threshold in summer. The configuration that maximises the modulation
amplitude in DAMA relative to the total rate in COSINUS (for scattering off
sodium and a DM mass of $m_{\chi}=10\,$GeV) is illustrated in the left panel
of figure 4. The dashed (dotted) line indicates the differential event rate in
summer (winter). Note that for small recoil energies the rate in winter is
larger than in summer by a factor $v_{0}^{+}/v_{0}^{-}$, see eq. (4.13).

Figure 4: Left: Differential recoil rate for scattering off sodium for the
optimum monotonically decreasing scattering cross section, i.e. the scattering
cross section satisfying eq. (4.8) that maximises the modulation amplitude in
DAMA while satisfying the (assumed) bound from COSINUS. The dashed (dotted)
line corresponds to the scattering in summer (winter). Note that we have not
included energy resolution and detector acceptance. Right: the exclusion power
of COSINUS as defined in eq. (4.5) for the case of monotonically decreasing
scattering cross sections. The blue (red) line corresponds to scattering off
sodium (iodine).

The blue shaded region in the left panel of figure 4 corresponds to the energy
window probed by DAMA in the absence of fluctuations. For perfect energy
resolution, one would therefore expect a modulation fraction in DAMA of
$100\%$. The fact that the energy resolution in DAMA is however not perfect
(and indeed pretty poor) means that some fraction of events will fluctuate
into the signal region even in winter, so that the modulation fraction ends up
being only about 46%. To achieve larger modulation fractions, one would have
to decrease $v_{0}$ further below the DAMA threshold. At this point it becomes
crucial that COSINUS has a much lower threshold than DAMA, because reducing
$v_{0}$ further would reduce the modulation amplitude in DAMA more rapidly
than the total rate predicted for COSINUS, leading to a less than optimal
configuration.

The fact that the optimum choice of $v_{0}$ and $E_{0}$ for monotonically
decreasing cross sections does not correspond to 100% modulation fraction in
DAMA means that we expect COSINUS to have more exclusion power than in the
fully model-independent case discussed in section 3. In other words, weaker
bounds on the total rate in COSINUS will be sufficient to exclude the DAMA
signal. The values of $R_{\text{COSINUS}}^{\text{bound}}$ that COSINUS must
achieve for this purpose are shown in the right panel of figure 4 as a
function of $m_{\chi}$. Blue (red) solid lines correspond to scattering off
sodium (iodine). The dashed lines show for comparison the values of
$R_{\text{COSINUS}}^{\text{bound}}$ required for a fully model-independent
exclusion (see figure 3).

We observe that the assumption of a monotonically decreasing cross section
enhances the exclusion power of COSINUS by at least a factor of about three,
meaning that three times weaker bounds on the rate (or three times less
exposure) are sufficient to exclude the DAMA signal. For sufficiently small DM
masses, the exclusion power becomes even greater owing to a reduced modulation
fraction in DAMA. The nominal sensitivity of COSINUS
$R_{\text{COSINUS}}^{\text{bound}}=0.1\mathrm{kg^{-1}\,days^{-1}}$ will be
sufficient to exclude the DAMA signal for $m_{\chi}<6\,\text{GeV}$ for
scattering off sodium and for $m_{\chi}<90\,\mathrm{GeV}$ for scattering off
iodine.

We emphasise that our approach is only suitable for addressing the question
whether COSINUS can exclude the DAMA signal. If there is a combination of
$v_{0}$ and $E_{0}$ such that

$\frac{\mathcal{S}^{\mathrm{T}}_{\text{DAMA}}(v_{0},E_{0})}{S_{\text{DAMA}}^{\text{bound}}}>\frac{\mathcal{R}^{\mathrm{T}}_{\text{COSINUS}}(v_{0},E_{0})}{R_{\text{COSINUS}}^{\text{bound}}}$
(4.16)

this does not necessarily imply that a good fit to the DAMA data can be
obtained. The reason is that we consider only the modulation amplitude from
DAMA in one specific bin and do not include the full information of the energy
and time dependence of the DAMA signal.

Figure 5: Differential modulation amplitude
$\mathrm{d}S/\mathrm{d}E_{\text{R}}^{\text{obs}}$ as a function of the
_observed_ recoil energy $E_{\text{R}}^{\text{obs}}$ in DAMA for the optimum
solution shown in figure 4. The substantial difference between the two figures
is due to the rather poor energy resolution of DAMA. Note that the data points
in this plot (which are taken from the official DAMA analysis [1]) are shown
only for illustration and have not been used to determine the optimum
configuration, which is determined only from the total modulation amplitude in
the bin $\left[2.5\,\text{keVee},3.5\,\text{keVee}\right]$. Nevertheless, both
the energy dependence and the time dependence (see inset, which shows the
energy-integrated modulation amplitude in units
$\text{kg}^{-1}\,\text{days}^{-1}\,\text{keVee}^{-1}$) of the DAMA signal can
be approximately reproduced, confirming that our treatment is neither too
crude nor too conservative.

Nevertheless, we make the interesting observation that the step function shown
in the left panel of figure 4 actually gives a surprisingly good fit to the
full DAMA data. The reason is once again that the poor energy resolution of
the DAMA detector washes out the sharp feature in the differential event rate.
This is illustrated in figure 5, which shows the differential modulation
amplitude (with respect to observed energy $E_{\mathrm{R}}^{\text{obs}}$) as
well as the time dependence of the modulation amplitude integrated over the
energy interval
$E_{R}^{\text{obs}}\in\left[2\,\text{keVee},6\,\text{keVee}\right]$ in
comparison to the actual data points from DAMA. In fact, the quality of the
fit is comparable to the one obtained for standard assumptions on the DM
interactions and velocity distribution. This leads to the reassuring
conclusion that the approach of decomposing the velocity distribution and the
differential cross section into sets of basis functions is not exceedingly
conservative.

###  4.2 Momentum-dependent cross sections

To conclude this section, let us consider possible alternatives to the
assumption of monotonically decreasing cross sections. In principle, one could
of course consider arbitrary cross sections, i.e. one could decompose the
differential cross section into $\delta$-functions rather than step functions.
It is easy to see, however, that in this case one can always arrange for 100%
modulation fraction in DAMA, so that one would recover the model-independent
bounds obtained in section 3.

A more interesting assumption is that the differential cross section can grow
no faster than some power $n$ of the recoil energy. In other words, we assume
that the differential cross section can be written as

$\frac{\text{d}\sigma^{\mathrm{T}}}{\text{d}E_{\mathrm{R}}}=\left(\frac{E_{\mathrm{R}}}{m_{\mathrm{T}}}\right)^{n}\frac{\sigma_{0}}{m_{\text{T}}v^{2}}\,\kappa^{\mathrm{T}}(E_{\mathrm{R}})\,\Theta(E^{\mathrm{T}}_{\text{max}}(v)-E_{\mathrm{R}})$
(4.17)

with monotonically decreasing functions $\kappa^{\mathrm{T}}(E_{\mathrm{R}})$.
For $n=0$ this corresponds to the case considered above, but for $n>0$ one
obtains less restrictive assumptions. As we will see in section 5, this
enables us to consider certain types of DM interactions that are not captured
by the assumption of monotonically decreasing cross sections. The
corresponding basis functions are given by

$\frac{\text{d}\sigma^{\mathrm{T}}_{n}}{\text{d}E_{\mathrm{R}}}(E_{0})=\left(\frac{E_{\mathrm{R}}}{m_{\mathrm{T}}}\right)^{n}\frac{\sigma_{0}}{m_{\text{T}}v^{2}}\,\Theta(E_{0}-E_{\mathrm{R}})\,\Theta(E^{\mathrm{T}}_{\text{max}}(v)-E_{\mathrm{R}})\;.$
(4.18)

Figure 6: Exclusion power of COSINUS for DM models in which the DM-nucleus
scattering cross section can be written as a power $n$ of the recoil energy
times a monotonically decreasing function (see eq. (4.17)) for different
values of $n$. Blue (red) lines correspond to scattering off sodium (iodine).
The case $n=0$ is identical to the case of a monotonically decreasing cross
section considered in the right panel of figure 4. For $n>0$ the exclusion
power of COSINUS is reduced due to the smaller benefit from the lower
threshold.

The values of $R_{\text{COSINUS}}^{\text{bound}}$ required to exclude DAMA for
different values of $n$ are shown in figure 6. As expected, larger values of
$n$, corresponding to less restrictive assumptions on the differential cross
section, imply weaker exclusion power from COSINUS, meaning that stronger
bounds on the rate (or larger exposure) are needed to exclude the DAMA signal.
In fact, there is also a more intuitive way to understand figure 6: As
discussed above, much of the exclusion power of COSINUS comes from the fact
that it has a lower threshold than DAMA. If the differential cross section
grows with energy, the benefit of the lower threshold, and hence the exclusion
power of COSINUS, is reduced.

##  5 Halo-independent comparison for specific interactions

In this section we compare the halo-independent bounds for different classes
of recoil spectra derived above to the halo-independent bounds obtained for
specific models that fall into these general categories. Specifically, we will
consider spin-independent and spin-dependent scattering as examples for models
with monotonically falling recoil spectra and $CP$-odd interactions as
examples for models with momentum-dependent scattering. Towards the end of
this section we will discuss inelastic DM as an example of a model that does
not fall into one of the categories discussed in section 4.

The crucial difference between the more general treatment above and the more
specific treatment in this section is that by considering a specific type of
interaction we impose certain relations between the scattering rates for
different elements. Doing so implies in particular that DAMA faces
overwhelming constraints from other direct detection experiments. There is
consequently less motivation for performing a comparison between DAMA and
COSINUS in the context of these models. Such a comparison is nevertheless
instructive in order to explore how the general bounds derived above compare
to the bounds obtained within specific models.

As we will see below, as long as the event rate in both DAMA and COSINUS is
dominated by the scattering off the same target element, the general bounds
derived above are only slightly weaker than the ones obtained within a
specific model. However, much stronger exclusion limits are obtained in those
regions of parameter space where scattering in DAMA is dominated by a single
target element, while both sodium and iodine contribute to the event rate in
COSINUS. In this case the performance of COSINUS turns out to strongly depend
on its low-energy threshold.

###  5.1 Spin-independent scattering

We begin by discussing in detail the case of spin-independent scattering,
which was also briefly considered in section 2 (see eq. (2.10) and figure 2).
Since the corresponding differential cross section is of the form of eq.
(4.8), we can immediately read off a conservative estimate of the exclusion
power of COSINUS from figure 4. A stronger bound can however be obtained by
explicitly calculating $\mathcal{R}_{\text{COSINUS}}(v_{0})$ and
$\mathcal{S}_{\text{DAMA}}(v_{0})$ for stream-like velocity distributions and
making use of eq. (4.5).

Figure 7: Differential recoil rate (without energy resolution and detector
acceptance) for spin-independent scattering with the optimum value of $v_{0}$.
Blue (red) lines correspond to scattering off sodium (iodine), dashed (dotted)
lines indicate the rates in summer (winter). The shaded regions indicate the
approximate energy ranges that can contribute to the observed modulation in
DAMA, the black arrow indicates the energy range that is relevant for COSINUS.
The different rows (columns) correspond to different values of $m_{\chi}$
($E_{\text{thres}}$).

Figure 7 shows the recoil spectrum (in terms of the true nuclear recoil energy
$E_{\mathrm{R}}$) for the optimal value of $v_{0}$, i.e. the one that imposes
the strongest requirement on $R_{\text{COSINUS}}^{\text{bound}}$. The two
different rows (columns) correspond to different values of $m_{\chi}$
($E_{\text{thres}}$). As in the left panel of figure 4 the dashed (dotted)
curves show the rates in summer (winter). The crucial difference to figure 4
is that for each case there are now two curves, corresponding to sodium (blue)
and iodine (red).

We make a number of pertinent observations: First of all, we note that the
rate from scattering off iodine is several orders of magnitude larger than the
rate from scattering off sodium, owing to the well-known coherent enhancement
for heavy target nuclei. Furthermore, as a result of the form factor
suppression, the differential event rates are no longer constant below the
cut-off. Finally, we note that for $m_{\chi}=100\,\text{GeV}$ the differential
event rate for iodine extends to larger recoil energies, while for
$m_{\chi}=5\,\text{GeV}$ the situation is reversed, due to the different
ratios of DM particle mass to target nucleus mass.

For $m_{\chi}=100\,\text{GeV}$ the situation is clear: Both the total rate and
the modulation amplitude are completely dominated by scattering off iodine,
and the contribution from sodium can effectively be neglected.111111We note
that scenarios in which the modulation amplitude in DAMA is dominated by
scattering off iodine are already constrained by existing direct detection
experiments based on target nuclei also involving iodine, such as COUPP [65],
KIMS [66] and PICO-60 [67]. The only difference between spin-independent
scattering and the general case considered in figure 4 is the form factor
suppression, which substantially suppresses the event rate in the energy range
probed by DAMA, but is less relevant for the lower energies probed by COSINUS.
We therefore expect COSINUS to have somewhat higher exclusion power (by a
factor of a few) for spin-independent interactions than for general
monotonically decreasing scattering. Quantitatively, this is confirmed by
comparing the orange and cyan curves in the lower left panel of figure 1.

For $m_{\chi}=5\,\text{GeV}$ the situation becomes more complicated, as the
optimum value of $v_{0}$ depends on $E_{\text{thres}}$. The reason is that the
value of $v_{0}$ that maximises the modulation fraction corresponds to
scattering off sodium, such that scatters off iodine are completely below
threshold for DAMA. However, this configuration is of advantage for DAMA only
if iodine scatters are also below threshold in COSINUS. If the COSINUS
threshold is low enough to be able to observe iodine scatters even if the DAMA
signal is dominated by sodium scatters, the resulting bound from COSINUS will
be overwhelmingly strong due to the coherent enhancement factor. Thus, for
$E_{\text{thres}}=1\,\text{keV}$ the optimum value of $v_{0}$ still
corresponds to the case that scattering off iodine dominates both for DAMA and
COSINUS. Scattering off sodium becomes advantageous for DAMA only if the
threshold of COSINUS is rather high, starting at about $E_{\text{thres}}\simeq
2\,\text{keV}$.121212We note that even for $m_{\chi}=5\,\text{GeV}$ and
$E_{\text{thres}}=2\,\text{keV}$ the differential event rate due to scattering
off iodine is only barely below threshold in COSINUS. As a result, the optimum
configuration looks quite different from the one in figure 4, with only upward
fluctuations contributing to the DAMA signal. Only for even smaller DM masses
and/or even higher thresholds the optimum configuration from figure 4 is
recovered.

###  5.2 Spin-dependent and momentum-dependent scattering

Having discussed in detail the case of spin-independent scattering, we now
consider a number of alternative possibilities. In each case there is a well-
defined relation between the scattering rate off iodine and the one off
sodium, meaning that we expect COSINUS to have greater exclusion power than
for model-independent assumptions.

The elastic scattering of Galactic DM particles off nuclei can be fully
characterised in terms of 18 non-relativistic effective operators, which can
depend on the spin of the DM particle and the nucleon, $\mathbf{s}_{\chi}$ and
$\mathbf{s}_{N}$, the relative velocity $\mathbf{v}$ and the momentum transfer
$\mathbf{q}$ [68].131313In addition, the cross section may be multiplied by an
overall factor that depends on the momentum transfer in a target-independent
way. For example, a factor of $q_{\text{ref}}^{4}/q^{4}$ is present for the
case of long-range interactions arising from the exchange of a very light
mediator [69]. Within this nomenclature, standard spin-independent scattering
corresponds to $\mathcal{O}_{1}$, while standard spin-dependent scattering
corresponds to $\mathcal{O}_{4}$. Without reviewing the formalism in detail,
we note that several of these operators predict cross sections that grow with
increasing momentum transfer, corresponding to recoil spectra that do not
decrease monotonically. Particularly well-known examples are the operators
generated by the exchange of a spin-0 mediator with general $CP$ phase [70].
These are

$\displaystyle\mathcal{O}_{10}^{N}$
$\displaystyle=i\,\mathbf{s}_{N}\cdot\mathbf{q}/m_{N}\,,$ (5.1)
$\displaystyle\mathcal{O}_{11}^{N}$
$\displaystyle=i\,\mathbf{s}_{\chi}\cdot\mathbf{q}/m_{N}\,,$ (5.2)
$\displaystyle\mathcal{O}_{6}^{N}$
$\displaystyle=(\mathbf{s}_{N}\cdot\mathbf{q})(\mathbf{s}_{\chi}\cdot\mathbf{q})/m_{N}^{2}\,,$
(5.3)

with $m_{N}$ the nucleon mass. The differential event rate resulting from each
of these operators can be written as some power $n$ of the nuclear recoil
energy times a monotonically decreasing function (see eq. (4.17)).
Specifically, since $E_{\mathrm{R}}\propto q^{2}$, one finds $n=1$ for
$\mathcal{O}_{10,11}$ and $n=2$ for $\mathcal{O}_{6}$. In the following we
assume that the corresponding spin-0 mediator couples to each Standard Model
quark proportionally to its mass, as required by minimal flavour violation.
Using the quark spin contents from ref. [71] and quark masses from ref. [72],
the neutron-to-proton coupling ratios of the operators are given by $\simeq 1$
for $\mathcal{O}_{11}^{N}$ and $-0.26$ for $\mathcal{O}_{6}^{N}$ and
$\mathcal{O}_{10}^{N}$.

The COSINUS exclusion power for each of these operators can be calculated in
complete analogy to the case of standard spin-independent scattering discussed
above. On the one hand, this enables us to determine which particle physics
explanations of the DAMA signal can be excluded by COSINUS in a halo-
independent way. On the other hand, we can use the results to determine
whether the more model-independent approach from section 4 is overly
conservative. Our results are summarized in figure 8. The orange curves
correspond to the ones already shown in figure 6 (solid for sodium, dashed for
iodine), while the other curves correspond to specific particle physics
assumptions (i.e. to specific non-relativistic operators). The first three
panels correspond to $n=0,1,2$, the final panel will be discussed in the
following sub-section.

Figure 8: COSINUS exclusion power for different assumptions regarding the
underlying particle physics of DM-nucleus interactions. The first three panels
consider interactions for which the cross section can be written as a power
$n$ of the recoil energy times a monotonically decreasing function. In each
panel the black lines show the fully model-independent constraint derived in
section 3 (solid for scattering off sodium, dashed for scattering off iodine),
the orange lines show the bounds for classes of recoil spectra obtained from
eq. (4.17), see also figure 6, and the remaining lines correspond to specific
interactions. The final panel considers the case of inelastic DM, which cannot
be written in the form of eq. (4.17) and can therefore only be compared to the
fully model-independent constraint.

In the first panel, we show in addition to spin-independent scattering also
the case of spin-dependent scattering off protons (denoted SDp). Due to the
lack of coherent enhancement for scattering off iodine, the best-fit
configuration for small DM masses corresponds to scattering dominantly off
sodium. The transition from scattering off iodine to scattering off sodium
leads to a visible kink in the exclusion power. We observe that in both
regimes the exclusion power of COSINUS is quite close to the respective bounds
derived in a model-independent way in section 4 (orange lines). Nevertheless,
we conclude that the design sensitivity of COSINUS will be sufficient to
exclude both spin-independent and spin-dependent scattering as an explanation
of the DAMA signal in a halo-independent way, independently of the DM mass. We
also show the case of spin-independent long-range interactions containing an
additional factor of $1/q^{4}$ (denoted $\text{SI}\times 1/q^{4}$). Due to the
steeply falling recoil spectrum, the exclusion power of COSINUS is very large,
and it should be easy for COSINUS to rule out any interpretation of DAMA in
terms of long-range interactions (as proposed e.g. in ref. [69]).

In the second and third panel we see that, as expected, an explanation of the
DAMA signal in terms of momentum-suppressed scattering is more difficult to
exclude with COSINUS, as its lower threshold is less beneficial. Nevertheless,
for the design sensitivity
$R_{\text{COSINUS}}^{\text{bound}}=0.1\,\mathrm{kg^{-1}\,days^{-1}}$, COSINUS
will be able to rule out also these scenarios in a halo-independent way for DM
masses below $\simeq 20\,$GeV, $\simeq 40\,$GeV and $\simeq 75\,$GeV for
scattering mediated by the non-relativistic operators $\mathcal{O}_{6}$,
$\mathcal{O}_{10}$ and $\mathcal{O}_{11}$, respectively. The qualitative
difference between the behaviour of $\mathcal{O}_{11}$ and the one of
$\mathcal{O}_{6}$ and $\mathcal{O}_{10}$ is once again due to the absence of
coherent enhancement for the latter two.

###  5.3 Inelastic scattering

It is well known that if DM-nucleus scattering is inelastic, i.e. if it
requires the transition between two DM states of slightly different mass, the
tension between the DAMA signal and other direct detection experiments is
reduced [21]. For an inelasticity parameter of $\delta$ (corresponding to the
mass difference of the two DM states), the minimum velocity required for a
recoil of energy $E_{\mathrm{R}}$ is given by

$v_{\text{min}}=\left|\delta+\frac{m_{\mathrm{T}}\,E_{\mathrm{R}}}{\mu}\right|\frac{1}{\sqrt{2\,E_{\mathrm{R}}\,m_{\mathrm{T}}}}\;.$
(5.4)

The requirement that $v_{\text{min}}$ is smaller than the Galactic escape
velocity places a lower bound on $E_{\mathrm{R}}$, meaning that there are no
low-energy nuclear recoils and as a result the modulation fraction can be
large.

The scattering rate for inelastic DM cannot be written in the form of eq.
(4.17), providing a prime example for a scenario in which the rather general
constraints derived in section 4 do not apply. Nevertheless, the fully model-
independent bound derived in section 3 does still apply, so we can immediately
obtain a conservative estimate of the exclusion power of COSINUS. In the final
panel of figure 8 we study how this conservative estimate compares to the
sensitivity that COSINUS must achieve to exclude a specific model of inelastic
scattering as an explanation of the DAMA signal in a halo-independent way. For
the latter, we choose for illustration the case of spin-independent scattering
with $f_{n}/f_{p}\simeq-0.72$ and a mass splitting $\delta=50\,$keV. Similar
scenarios have been studied e.g. in refs. [73, 74] in order to suppress the
scattering rate in xenon- and/or germanium-based experiments.141414The idea of
inelastic DM was recently revisited in ref. [29] for a scenario with proto-
philic spin-dependent coupling to nucleons. We find that such a scenario can
be probed by COSINUS with much less exposure, as in contrast to spin-
independent interactions with $f_{n}/f_{p}\simeq-0.72$ there is no suppression
for scatterings off iodine recoils.

The mass dependence of the rate required in COSINUS for excluding a DM
interpretation of DAMA can be understood as follows. For $m_{\chi}\lesssim
5\,$GeV, the suppression of iodine scatters due to the choice
$f_{n}/f_{p}\simeq-0.72$ leads to a situation in which both the rate in
COSINUS and DAMA are dominated by sodium recoils. For these DM masses, the
range of kinematically allowed energies corresponding to the optimal choice of
the stream velocity $v_{0}$ gets more and more narrow with increasing
$m_{\chi}$. At $m_{\chi}\simeq 5\,$GeV, the situation is already nearly
equivalent to adopting a single recoil energy $E_{\text{R}}=E_{0}$ for
scattering off sodium, and hence the fully model-independent bound shown as a
solid black curve is almost saturated. For $5\,\text{GeV}\lesssim
m_{\chi}\lesssim 10\,\text{GeV}$, iodine recoils start to become relevant in
COSINUS but not in DAMA, due the larger threshold of the latter experiment;
consequently, a less stringent bound on the rate in COSINUS is necessary for
excluding DAMA. Finally, for $m_{\chi}\gtrsim 10\,$GeV iodine recoils dominate
over sodium both in COSINUS and DAMA (see also the lower panel in figure 7).
The initial decrease of the required rate for these DM masses is again due to
the smaller range of kinematically allowed energies, favouring DAMA over
COSINUS. For $m_{\chi}\gtrsim 50\,$GeV, the situation is again nearly
identical to the case of recoils with a single energy, such that in this case
the fully model-independent bound for scattering off iodine is nearly
saturated.

##  6 Conclusions

While the interpretation of the DAMA annual modulation in terms of DM is
strongly challenged by null results from other direct detection experiments,
it is extremely difficult to refute this interpretation in a model-independent
way. In particular, the different target elements as well as techniques
employed by the various experiments, while offering important complementarity,
also imply larger uncertainties due to the unknown local velocity distribution
as well as the particle physics properties of DM.

In this article we have discussed how the DM interpretation of the DAMA signal
in terms of nuclear scatterings can be tested independently of astrophysical
and particle physics unknowns with COSINUS, a future experiment which employs
the same target material and is thus sensitive to exactly the same recoil
spectrum as DAMA (before taking into account detection efficiencies). Compared
to other planned experiments also based on a NaI target, a great advantage of
the cryogenic detector developed by the COSINUS collaboration lies in its
ability to measure the total DM induced scattering rate with a significantly
lower threshold than DAMA.

Our main findings are summarised in figure 1, where we show the halo-
independent exclusion power of COSINUS, i.e. the bound on the total rate that
COSINUS must achieve for excluding DAMA in a halo-independent way, as a
function of the assumed threshold in COSINUS. The vertical axis on the right
of each panel indicates the required exposure assuming zero observed events.
The four panels correspond to different values of the DM mass $m_{\chi}$, the
various lines in each panel result from different assumptions on the
differential event rate. We make the following observations:

  1. 1.

The black shaded regions in figure 1 show which combinations of the bound on
the total rate and the threshold in COSINUS suffice to exclude a DM
interpretation of DAMA, _fully independently_ of the assumed nuclear recoil
spectrum. This is only possible if the COSINUS threshold is significantly
lower than the one of DAMA, $E_{\text{thres}}\lesssim 1.8\,$keV. While this is
compatible with the design goal $E_{\text{thres}}=1\,\text{keV}$, the required
bound on the total rate is about an order of magnitude stronger than the
current sensitivity estimate
$R_{\text{COSINUS}}^{\text{bound}}=0.1\mathrm{\,kg^{-1}\,days^{-1}}$ for
COSINUS.

  2. 2.

More restrictive assumptions on the differential scattering cross section of
DM lead to greater exclusion power, meaning that a weaker bound on the total
rate (or equivalently a smaller exposure) is sufficient to exclude the DAMA
signal. In particular, the orange curves in figure 1 show the required bound
for the still very general class of differential scattering cross sections
that are monotonically decreasing in energy and have the standard $1/v^{2}$
dependence on velocity. For the design sensitivity of COSINUS, such scenarios
can be robustly excluded for DM masses $m_{\chi}\lesssim 6\,\text{GeV}$. If
the COSINUS sensitivity is further increased by a factor of $\simeq 3$, this
class of scattering cross sections can be excluded for all relevant DM masses.
Interestingly, it follows from figure 1 that this conclusion would still hold
if the threshold in COSINUS is raised to significantly larger values; this is
a prime example of how our results can be employed in order to optimise the
detector layout of COSINUS for testing the DAMA hypothesis.

  3. 3.

Finally, we show in figure 1 the required bound on the rate in COSINUS when
assuming a given particle physics scenario for the scattering process of DM
with nuclei (in this case standard spin-independent interactions). In contrast
to the other curves, this fixes the relation between the scattering rate off
sodium and iodine. In particular, depending on the COSINUS threshold and the
DM mass, it can happen that the modulation signal in DAMA is dominated by
sodium recoils, while COSINUS can also observe the corresponding signal from
iodine, thanks to its lower threshold. This further lowers the required
sensitivity of COSINUS compared to e.g. the more general results based on
monotonically decreasing cross sections. Similar results for various other
interaction types are shown in figure 8.

In summary, we have shown that if COSINUS is able to set a bound on the
scattering rate of DM of $\simeq(0.01-0.1)\,\text{kg}^{-1}\,\text{days}^{-1}$
with an energy threshold of $\lesssim 1.8\,$keV, it can exclude very general
classes of interactions in which the signal is dominated by DM scattering off
sodium or iodine in a halo-independent way. As other explanations of the DAMA
data such as DM-electron scattering [75], scattering off thallium [76] or
scattering off OH impurities [77] are already strongly constrained by other
experiments [78, 55, 17], this would strongly point towards a so far unknown
source of background being responsible for the modulation seen by the DAMA
collaboration.

To conclude, let us mention that for the last few years DAMA has been taking
data with improved photo-multiplier tubes, which promise a significant
lowering of the threshold [79]. Measurements of the modulation amplitude in
this new energy range will provide important clues to the origin of the DAMA
signal [59]. To make full use this new information it will be crucial to
extend the methods developed here to also include spectral information when
calculating the model-independent exclusion power of COSINUS. Clearly, the
fact that COSINUS can achieve an even lower threshold and a much better energy
resolution than DAMA will then become particularly important.

## Note added

Figure 9: Best-fit recoil spectra in DAMA for low-mass DM (left),
corresponding to scattering dominantly on sodium, and high-mass DM (right),
corresponding to scattering dominantly on iodine. In both cases we have
included the first twelve bins from the combined data sets of DAMA/NaI,
DAMA/LIBRA-phase1 and DAMA/LIBRA-phase2.

After the completion of this work, DAMA presented new data with a lower
threshold of $1\;\mathrm{keVee}$ [80]. While the measurements above
$2\,\mathrm{keVee}$ are in agreement with previous results, two new data
points at lower energies provide very interesting additional information,
because they exhibit neither the steep rise expected for scattering of low-
mass DM, nor the turn-over towards anti-modulation expected for high-mass DM
[59]. We have performed a combination of the two data sets, assuming that the
energy resolution has not changed significantly during the detector upgrade.
We find that it is no longer possible to obtain a good fit for DM scattering
with the standard assumptions discussed in section 2. The best-fit spectra for
spin-independent scattering on sodium and spin-independent scattering on
iodine are shown in figure 9. The goodness of fit of the best-fit point for
scattering on sodium is unacceptable ($p\sim 10^{-8}$ when including the first
12 bins), while the hypothesis of scattering on iodine is only marginally
compatible with data ($p\approx 0.003$). This observation implies that any
interpretation of the DAMA signal in terms of DM requires non-standard
interactions or non-standard astrophysical distributions (or both),
independently of (but already implied by) the exclusion bounds from other
experiments. New experiments based on NaI, like COSINUS, and model-independent
methods, like the one presented in this work, will therefore be essential to
further investigate the nature of the DAMA signal.

## Acknowledgments

We thank Suchita Kulkarni, Josef Pradler and Thomas Schwetz for discussion and
Vanessa Zema for helpful comments on the manuscript. This work is supported by
the German Science Foundation (DFG) under the Collaborative Research Center
(SFB) 676 “Particles, Strings and the Early Universe” and the Emmy Noether
Grant No. KA 4662/1-1 as well as the ERC Starting Grant ‘NewAve’ (638528).

##  Appendix A Best-fit velocity distribution

In this appendix we prove that a single stream is enough to determine
$S_{\text{DAMA}}^{\text{max}}$ as defined in eq. (4.1). First of all, we
rewrite this equation as

$\displaystyle
S_{\text{DAMA}}^{\text{max}}=\max_{f}\frac{S^{f}_{\text{DAMA}}(\sigma)}{R_{\text{COSINUS}}^{f}(\sigma)/R_{\text{COSINUS}}^{\text{bound}}}\;.$
(A.1)

The right-hand side of this equation is in fact independent of $\sigma$, as
the cross section cancels out in the ratio. It is therefore sufficient to
calculate $S_{\text{DAMA}}^{\text{max}}$ for an arbitrary reference cross
section, so that we will simply drop the dependence on $\sigma$ for the rest
of this appendix.

Let us now define

$\displaystyle
S_{\text{DAMA}}^{\text{stream}}\equiv\max_{\mathbf{v}_{0}}\frac{S^{\mathbf{v}_{0}}_{\text{DAMA}}}{R_{\text{COSINUS}}^{\mathbf{v}_{0}}/R_{\text{COSINUS}}^{\text{bound}}}$
(A.2)

in complete analogy to eq. (A.1), but considering only DM velocity
distributions of the form $f(\mathbf{v})=\delta(\mathbf{v}-\mathbf{v}_{0})$.
We now want to show that

$\displaystyle S_{\text{DAMA}}^{\text{max}}\equiv
S_{\text{DAMA}}^{\text{stream}}\;.$ (A.3)

Since it is obviously true that $S_{\text{DAMA}}^{\text{max}}\geq
S_{\text{DAMA}}^{\text{stream}}$, it remains to be shown that
$S_{\text{DAMA}}^{\text{max}}\leq S_{\text{DAMA}}^{\text{stream}}$. To do so,
we denote the velocity that maximises the right-hand side of eq. (A.2) as
$\mathbf{w}_{0}$, such that

$\displaystyle
S_{\text{DAMA}}^{\text{stream}}=\frac{S^{\mathbf{w}_{0}}_{\text{DAMA}}}{R_{\text{COSINUS}}^{\mathbf{w}_{0}}/R_{\text{COSINUS}}^{\text{bound}}}\;.$
(A.4)

For a given velocity distribution $f(\mathbf{v})$ we can then consider the
following expression:

$\displaystyle
R_{\text{COSINUS}}^{f}\,\,-\,\,\frac{S_{\text{DAMA}}^{f}}{S_{\text{DAMA}}^{\mathbf{w}_{0}}/R_{\text{COSINUS}}^{\mathbf{w}_{0}}}=\int\text{d}^{3}v_{0}\,f(\mathbf{v}_{0})\left(R_{\text{COSINUS}}^{\mathbf{v}_{0}}\,\,-\,\,\frac{S_{\text{DAMA}}^{\mathbf{v}_{0}}}{S_{\text{DAMA}}^{\mathbf{w}_{0}}/R_{\text{COSINUS}}^{\mathbf{w}_{0}}}\right)\;,$
(A.5)

where we have made use of the fact that the velocity distribution enters
linearly into the calculation of $R_{\text{COSINUS}}$ and $S_{\text{DAMA}}$
(see eq. (4.2)). By definition of $\mathbf{w}_{0}$ we have

$\frac{S_{\text{DAMA}}^{\mathbf{w}_{0}}}{R_{\text{COSINUS}}^{\mathbf{w}_{0}}}\,\geq\,\frac{S_{\text{DAMA}}^{\mathbf{v}_{0}}}{R_{\text{COSINUS}}^{\mathbf{v}_{0}}}$
(A.6)

for all $\mathbf{v}_{0}$, and hence the integrand in the right-hand side of
eq. (A.5) is always positive. It therefore follows immediately that

$R_{\text{COSINUS}}^{f}\,\,-\,\,\frac{S_{\text{DAMA}}^{f}}{S_{\text{DAMA}}^{\mathbf{w}_{0}}/R_{\text{COSINUS}}^{\mathbf{w}_{0}}}\geq
0\;,$ (A.7)

which can be rearranged to

$\displaystyle\frac{S_{\text{DAMA}}^{f}}{R_{\text{COSINUS}}^{f}/R_{\text{COSINUS}}^{\text{bound}}}\leq\frac{S_{\text{DAMA}}^{\mathbf{w}_{0}}}{R_{\text{COSINUS}}^{\mathbf{w}_{0}}/R_{\text{COSINUS}}^{\text{bound}}}=S_{\text{DAMA}}^{\text{stream}}\;.$
(A.8)

Since this inequality holds for arbitrary velocity distributions $f$, it must
hold in particular for the velocity distribution that maximises the left-hand
side, thus completing the proof.

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