Constraints on the Interactions between Dark Matter and Baryons from the X-ray
Quantum Calorimetry Experiment
Adrienne L. Erickcek1,2, Paul J. Steinhardt2,3, Dan McCammon4, and Patrick C. McGuire5
Division of Physics, Mathematics, & Astronomy,
California Institute of Technology, Mail Code 103-33, Pasadena, CA 91125, USA
Department of Physics, Princeton University, Princeton, NJ 08544, USA
Princeton Center for Theoretical Physics, Princeton University, Princeton, NJ 08544, USA
Department of Physics, University of Wisconsin, Madison, WI 53706, USA and
McDonnell Center for the Space Sciences, Washington University, St. Louis, M0 63130, USA
Although the rocket-based X-ray Quantum Calorimetry (XQC) experiment was designed for X-
ray spectroscopy, the minimal shielding of its calorimeters, its low atmospheric overburden, and its
low-threshold detectors make it among the most sensitive instruments for detecting or constraining
strong interactions between dark matter particles and baryons. We use Monte Carlo simulations to
obtain the precise limits the XQC experiment places on spin-independent interactions between dark
matter and baryons, improving upon earlier analytical estimates. We find that the XQC experiment
rules out a wide range of nucleon-scattering cross sections centered around one barn for dark matter
particles with masses between 0.01 and 105 GeV. Our analysis also provides new constraints on
cases where only a fraction of the dark matter strongly interacts with baryons.
PACS numbers: 95.35.+d, 12.60.-i, 29.40.Vj
I. INTRODUCTION
From Vera Rubin’s discovery that the rotation curves
of galaxies remain level to radii much greater than pre-
dicted by Keplerian dynamics [1] to the Wilkinson Mi-
crowave Anisotropy Probe (WMAP) measurement of
the cosmic microwave background (CMB) temperature
anisotropy power spectrum [2], observations indicate that
the luminous matter we see is only a fraction of the
mass in the Universe. The three-year WMAP CMB
anisotropy spectrum is best-fit by a cosmological model
with Ωm = 0.241 ± 0.034 and a baryon density that is
less than one fifth of the total mass density. The cold
collisionless dark matter (CCDM) model has emerged
as the predominant paradigm for discussing the missing
mass problem. The dark matter is assumed to consist
of non-relativistic, non-baryonic, weakly interacting par-
ticles, often referred to as Weakly Interacting Massive
Particles (WIMPs).
Although the CCDM model successfully predicts ob-
served features of large-scale structure at scales greater
than one megaparsec [3], there are indications that it may
fail to match observations on smaller scales. Numerical
simulations of CCDM halos [4, 5, 6, 7, 8, 9, 10, 11, 12]
imply that CCDM halos have a density profile that in-
creases sharply at small radii (ρ ∼ r−1.2 according to
Ref. [12]). These predictions conflict with lensing ob-
servations of clusters [13, 14] that indicate the presence
of constant-density cores. X-ray observations of clusters
have found cores in some clusters, although density cusps
have also been observed [15, 16, 17]. On smaller scales,
observations of dwarf and low-surface-brightness galaxies
[18, 19, 20, 21, 22, 23, 24] indicate that these dark mat-
ter halos have constant-density cores with lower densities
than predicted by numerical simulations. Observations
also indicate that cores are predominant in spiral galaxies
as well, including the Milky Way [25, 26, 27]. Numerical
simulations of CCDM halos also predict more satellite
halos than are observed in the Local Group [28, 29] and
fossil groups [30].
Astrophysical explanations for the discord between the
density profiles predicted by CCDM simulations and ob-
servations have been proposed: for instance, dynamical
friction may transform density cusps into cores in the
inner regions of clusters [31], and the triaxiality of galac-
tic halos may mask the true nature of their inner den-
sity profiles [32]. There are also models of substructure
formation that explain the observed paucity of satellite
halos [33, 34, 35, 36].
Another possible explanation for the apparent failure
of the CCDM model to describe the observed features
of dark matter halos is that dark matter particles scat-
ter strongly off one another. The discrepancies between
observations and the CCDM model are alleviated if one
introduces a dark matter self-interaction that is compa-
rable in strength to the interaction cross section between
neutral baryons [37, 38]:
= 8× 10−25 − 1× 10−23cm2 GeV−1, (1)
where σDD is the cross section for scattering between dark
matter particles and mdm is the mass of the dark mat-
ter particle. Numerical simulations have shown that in-
troducing dark matter self-interactions within this range
reduces the central slope of the halo density profile and
reduces the central densities of halo cores, in addition to
destroying the extra substructure [39, 40].
The numerical coincidence between this dark mat-
ter self-interaction cross section and the known strong-
interaction cross section for neutron-neutron or neutron-
proton scattering has reinvigorated interest in the pos-
sibility that dark matter interacts with itself and with
http://arxiv.org/abs/0704.0794v2
baryons through the strong nuclear force. We refer to
dark matter of this type as “strongly interacting dark
matter” where “strong” refers specifically to the strong
nuclear force. Strongly interacting dark matter candi-
dates include the dibaryon [41, 42], the Q-ball [43], and
O-helium [44].
Surprisingly, the possibility that the dark matter may
be strongly interacting is not ruled out. While there are
numerous experiments searching for WIMPs, they are
largely insensitive to dark matter that interacts strongly
with baryons. The reason is that WIMP searches are typ-
ically conducted at or below ground level based on the
fact that WIMPs can easily penetrate the atmosphere
or the Earth, whereas strongly interacting dark matter is
multiply scattered and thermalized by the time it reaches
ground level and its thermal kinetic energy is too small
to produce detectable collisions with baryons in WIMP
detectors. Consequently, there are few experiments ca-
pable of detecting strongly interacting dark matter di-
rectly. Starkman et al. [45] summarized the constraints
on strongly interacting dark matter from experiments
prior to 1990, and these constraints were later refined
[38, 46, 47, 48]. The strength of dark matter interactions
with baryons may also be constrained by galactic dynam-
ics [45], cosmic rays [45, 49], Big Bang nucleosynthesis
(BBN) [49], the CMB [50], and large-scale structure [50].
The X-ray Quantum Calorimetry (XQC) project
launched a rocket-mounted micro-calorimeter array in
1999 [51]. At altitudes above 165 km, the XQC detector
collected data for a little less than two minutes. Although
its primary purpose was X-ray spectroscopy, the limited
amount of shielding in front of the calorimeters and the
low atmospheric overburden makes the XQC experiment
a sensitive detector of strongly interacting dark matter.
In this article, we present a new numerical analysis
of the constraints on spin-independent interactions be-
tween dark matter particles and baryons from the XQC
experiment using Monte Carlo simulations of dark mat-
ter particles interacting with the XQC detector and the
atmosphere above it. Our work is a significant improve-
ment upon the earlier analytic estimates presented by
some of us in Refs. [38, 48] because it accurately models
the dark matter particle’s interactions with the atmo-
sphere and the XQC instrument. Our calculation here
also supersedes the analytic estimate by Zaharijas and
Farrar [52] because they only considered a small por-
tion of the XQC data and did not include multiple scat-
tering events nor the overburden of the XQC detector.
We restrict our analysis to spin-independent interactions
because the XQC calorimeters are not highly sensitive
to spin-dependent interactions. Only a small fraction of
the target nuclei in the calorimeters have non-zero spin;
consequently, the bound on spin-dependent interactions
between baryons and dark matter from the XQC experi-
ment is about four orders of magnitude weaker than the
bound on spin-independent interactions [52].
This article is organized as follows. In Section II we
summarize the specifications of the XQC detector. We
then review dark matter detection theory in Section III.
This Section includes a discussion of coherent versus in-
coherent scattering and how we account for the loss of
coherence in our analysis. A complete description of our
analysis follows in IV, and our results are presented in
Section V. Finally, in Section VI, we summarize our
findings and compare the constraints to strongly inter-
acting dark matter from the XQC experiment to those
from other experiments.
II. THE XQC EXPERIMENT
Calorimetry is the use of temperature deviations to
measure changes in the internal energy of a material. By
drastically reducing the specific heat of the absorbing
material, the use of cryogenics in calorimetry allows the
absorbing object to have a macroscopic volume and still
be sensitive to minute changes in energy. These detectors
are sensitive enough to register the energy deposited by a
single photon or particle and gave birth to the technique
of “quantum calorimetry,” the thermal measurement of
energy quanta.
The quantum calorimetry experiment [51] we use to
constrain interactions between dark matter particles and
baryons is the second rocket-born experiment in the XQC
(X-ray Quantum Calorimetry) Project, a joint under-
taking of the University of Wisconsin and the Goddard
Space Flight Center [53, 54]. It launched on March 28,
1999 and collected about 100 seconds of data at alti-
tudes between 165 and 225 km above the Earth’s surface.
The detector consisted of thirty-four quantum calorime-
ters operating at a temperature of 0.06 K; for detailed
information on the XQC detector functions, please refer
to Refs. [51, 54]. These detectors were separated from
the exterior of the rocket by five thin filter panes [51].
The small atmospheric overburden at this altitude and
the minimal amount of shielding in front of the calorime-
ters makes this experiment a promising probe of strongly
interacting dark matter.
The absorbers in the XQC calorimeters are composed
of a thin film of HgTe (0.96 µm thick) deposited on a sili-
con (Si) substrate that is 14 µm thick. The absorbers rest
on silicon spacers and silicon pixel bodies. Figs. 1 and
2 show side and top views of the detectors with the di-
mensions of each layer. Temperature changes in all four
components are measured by the calorimeter’s internal
thermometer. The calorimeters report the average tem-
perature over an integration time of 7 ms in order to
reduce the effect of random temperature fluctuations on
the measurement. Multiple scatterings by a dark matter
particle will register as a single event because the time it
takes the dark matter particle to make its way through
the calorimeter is small compared to the integration time.
The detector array consists of two rows of detec-
tors, with seventeen active calorimeters and one inactive
calorimeter in each row, and is located at the bottom of
a conical detector chamber. Within a 32-degree angle
   Si Substrate
0.5 mm x 2.0 mm
14 µm
12 µm
HgTe Absorber
0.96µm
      Si Pixel Body
0.25 mm x 1.0 mm
Si Spacer
0.245 mm x 0.245 mm
FIG. 1: A vertical cross section of an XQC calorimeter. The
relative thicknesses of the layers are drawn to scale, as are
their relative lengths, but the two scales are not the same. To
facilitate the display of the layers, the vertical dimension has
been stretched relative to the horizontal dimension.
2.0 mm
1.0 mm
Absorber Panel
  (HgTe on Si)
Pixel Body
     (Si)
   Spacer
     (Si)
  0.245 mm 
x 0.245 mm
FIG. 2: A top view of an XQC calorimeter. The absorber is
the top layer and underneath it lies the spacer, followed by
the pixel body. These dimensions are drawn to scale.
from the detector normal, the incoming particles only
pass through the aforementioned filters. The five filters
are located 2 mm, 6 mm, 9 mm, 11 mm and 28 mm above
the detectors. Each filter consists of a thin layer of alu-
minum (150 Å) supported on a parylene (CH) substrate
(1380 Å). The pressure inside the chamber is less than
10−6 Torr. At this level of evacuation, a dark matter
particle with a mass of 106 GeV and a baryon interac-
tion cross section of 106 barns, would have less than a
20% chance of colliding with an air atom in the cham-
ber. Therefore, we assume that the chamber is a perfect
vacuum in our analysis.
While the atmospheric pressure at the altitudes at
which the detector operated is about 10−8 times the at-
mospheric pressure at sea level, the atmospheric over-
burden of the XQC detector is still sufficient to scat-
ter incoming strongly interacting dark matter particles.
Simulating a dark matter particle’s path through the
atmosphere requires number-density profiles for all the
molecules in the atmosphere. These profiles were ob-
tained using the MSIS-E-90 model1 for the time (1999
1 Available at http://modelweb.gsfc.nasa.gov/models/msis.html
FIG. 3: The points depict the MSIS-E-90 density profiles for
the seven most prevalent constituents of the atmosphere above
the XQC detector, and the lines show the piecewise exponen-
tial fits used in our analysis.
March 28 9:00 UT) and location (White Sands Missile
Range, New Mexico) of the XQC rocket launch.
During the data collection period, the average altitude
of the XQC rocket was 201.747 km. At this altitude and
above, the primary constituents of the atmosphere are
molecular and atomic oxygen, molecular and atomic ni-
trogen, helium, atomic hydrogen, and argon. The MSIS-
E-90 model provides tables of the number densities of
each of these seven chemical species. In our analysis,
computational efficiency demanded that we fit analytic
functions to these data. We found exponential fits for
the density profiles in three altitude ranges: 200-300 km,
300-500 km and 500-1000 km. The error in the proba-
bility of a collision between a dark matter particle and
an element of the atmosphere introduced by using these
fits instead of the original data is 0.02%. Fig. 3 shows
the number density profiles provided by the MSIS-E-90
model and the exponential fits used to model the data.
The XQC detector collected data for a total of 150
seconds. During these 150 seconds of activity, the thirty-
four individual calorimeters were not all operational at
all times. Furthermore, events that could not be ac-
curately measured by the calorimeters and events at-
tributed to cosmic rays hitting the base of the detec-
tor array were removed from the XQC spectrum, and
these cuts also contribute to the dead time of the sys-
tem. Specifically, events that arrived too close together
for the calorimeters to accurately measure distinct ener-
gies were discarded. This criterion removed 12% of the
observed events and the resulting loss of sensitivity was
http://modelweb.gsfc.nasa.gov/models/msis.html
FIG. 4: Top panel: The XQC energy spectrum from 0 - 4
keV in 5 eV bins. This spectrum does not have non-linearity
corrections applied (see Ref. [51]), so the calibration lines at
3312 eV and 3590 eV appear slightly below their actual en-
ergies. The cluster of counts to the left of each calibration
peak result from X-rays passing through the HgTe layer and
being absorbed in the Si components where up to 12% of the
energy may then be trapped in metastable states. Bottom
panel: The XQC energy spectrum from 0 - 2.5 keV in 5 eV
bins. This spectrum, combined with the over-saturation rate
of 0.6 events per second with energies greater than 4000 eV,
was used in our analysis.
included in the dead time of the calorimeters. When a
cosmic ray penetrates the silicon base of the detector ar-
ray, the resulting temperature increase is expected to reg-
ister as multiple, nearly simultaneous, low-energy events
on nearby calorimeters. To remove these events from the
spectrum, we cut out events that were part of either a
pair of events in adjacent detectors or a trio of events
in any of the detectors that arrived within 3 ms of each
other and had energies less than 2.5 keV. This procedure
was expected to remove more than 97% of the events that
resulted from cosmic rays hitting the base of the array.
Nearly all of the events attributed to heating from cosmic
rays had energies less than 300 eV, and a high fraction
of the observed low-energy events were included in this
cut. For example, seventeen of the observed twenty-four
events with energies less than 100 eV were removed. The
expected loss of sensitivity due to events being falsely
attributed to cosmic rays was included in the calculated
dead time of the calorimeters. Once all the dead time is
accounted for, the 150 seconds of data collection is equiv-
alent to 100.7 seconds of observation with all thirty-four
calorimeters operational.
The XQC calorimeters are capable of detecting energy
deposits that exceed 20 eV, but full sensitivity is not
reached until the energy surpasses 36 eV, and for approx-
imately half of the detection time, the detector’s lower
threshold was set to 120 eV. The calorimeters cannot re-
solve energies above 4 keV, and the 2.5-4 keV spectrum is
dominated by the detector’s interior calibration source:
a ring of 2µCi 41Ca that generates Kα and Kβ lines at
3312 eV and 3590 eV, respectively. We refer the reader
to Ref. [51] for a complete discussion of the calibration of
the detector. These limitations restrict the useful portion
of the XQC spectrum to 0.03-2.5 keV. This spectrum is
shown in Fig. 4, along with the full spectrum from 0-4
keV. The XQC field of view was centered on a region of
the sky known to have an enhanced X-ray background
in the 100-300 eV range, possibly due to hot gas in the
halo, and this surge in counts can be seen in Fig. 4. In
addition to the information present in this spectrum, we
know that the XQC detector observed an average over-
saturation event rate of 0.6 per second. This corresponds
to a total of 60 events that deposited more than 4000 eV
in a calorimeter. In Section IVB, we describe how we use
the observed spectrum between 29 eV and 2500 eV and
the integrated over-saturation rate to constrain the total
cross section for elastic scattering between dark matter
particles and nucleons.
III. DETECTING DARK MATTER
A. Incidence of dark matter particles
The expected flux of dark matter particles into the de-
tector depends on the density of the dark matter halo in
the Solar System. Unfortunately, the local dark matter
density is unknown and the range of theoretical predic-
tions is wide. By constructing numerous models of our
galaxy with various dark matter density profiles and halo
characteristics, rejecting those models that contradict ob-
servations, and finding the distribution of local dark mat-
ter densities in the remaining viable models, Ref. [55] pre-
dicted that the local dark matter density is between 0.3
and 0.7 GeV cm−3 assuming that the dark matter halo
is flattened, and the predicted local density decreases as
the halo is taken to be more spherical. Another approach
[56] used numerical simulations of galaxies similar to our
own to find the dark matter density profile and then fit
the profile parameters to Galactic observations, predict-
ing a mean local dark matter density between 0.18 GeV
cm−3 and 0.30 GeV cm−3. Given that it lies in the in-
tersection of these two ranges, we use the standard value
of 0.3 GeV cm−3 for the local dark matter density in
our primary analysis. This assumption ignores the possi-
ble presence of dark matter streams or minihalos, which
do occur in numerical simulations [56] and could lead to
local deviations from the mean dark matter density.
We also assume that the velocities of the dark matter
particles with respect to the halo are isotropic and have
a bounded Maxwellian distribution: the probability that
a particle has a velocity within a differential volume in
velocity-space centered around a given velocity ~v is
P (~v) =
d3~v if v ≤ vesc,
0 if v > vesc.
where v0 is the dispersion velocity of the halo, vesc is the
Galactic escape velocity at the Sun’s position, and k is a
normalization factor [57]:
k = (πv20)
. (3)
Numerical simulations indicate that dark matter particle
velocities may not have an isotropic Maxwellian distri-
bution [56]. Ref. [58] examines how assuming a more
complicated velocity distribution would alter the flux of
dark matter particles into an Earth-based detector.
Given the flat rotation curve of the spiral disk at the
Sun’s radius and beyond and assuming a spherical halo,
the local dispersion speed v0 is the maximum rotational
velocity of the Galaxy vc [59]. Reported values for the
rotational speed include 222±20 km s−1 [60], 228±19 km
s−1 [61], 184± 8 km s−1 [62] and 230 ± 30 km s−1 [63].
Recent measurements of the Galaxy’s angular velocity
have yielded values of Ωgal = 28 ± 2 km s−1 kpc−1 [64]
and 32.8 ± 2 km s−1 kpc−1 [65]. If the Sun is located
8.0 kpc from the Galactic center, these angular velocities
correspond to tangential velocities 224± 16 km s−1 and
262 ± 16 km s−1 respectively. We adopt vc = 220 ± 30
km s−1 as a centrally conservative value for the Galaxy’s
circular velocity at the Sun’s location.
The final parameter we need to obtain the dark mat-
ter’s velocity distribution is the escape velocity in the So-
lar vicinity. The largest observed stellar velocity at the
Sun’s radius in the Milky Way is 475 km s−1, which es-
tablishes a lower bound for the local escape velocity [66].
Ref. [67] used the radial motion of Carney-Latham stars
to determine that the escape velocity is between 450 and
650 km s−1 to 90% confidence, and Ref. [68] obtained a
90% confidence interval of 498 to 608 km s−1 from ob-
servations of high-velocity stars. A kinematic derivation
of the escape velocity [59] gives
v2esc = 2v
1 + ln
, (4)
where R0 is the distance from the Sun to the center of the
Galaxy, and Rgal is radius of the Galaxy. Observations of
other galaxies suggest that our galaxy extends to about
100 kpc [59], and observations of Galactic satellites indi-
cate that the Galaxy’s flat rotation curve extends to at
least 110 kpc [63]. The commonly accepted value for the
Solar radius is R0 = 8.0 kpc [69]. Recent measurements
include R0 = 7.9± 0.3 kpc [70] and R0 = 8.01± 0.44 kpc
[71], and a compilation of measurements over the past
decade [71] yields an average value of R0 = 7.80 ± 0.33
kpc. To estimate the escape velocity, we use 100 kpc as
a conservative estimate of the Galactic radius and the
standard value R0 = 8.0 kpc. These parameters, com-
bined with vc = 220 km s
−1, predict an escape velocity
of 584 km s−1, which falls near the middle of the ranges
proposed in Refs. [67, 68].
The isotropic Maxwellian velocity distribution given
by Eq. (2) specifies the dark matter particles’ motion
relative to the halo. However, we are interested in their
motion relative to the XQC detector: ~vobserved = ~vdm −
~vdetector where the latter two velocities are measured with
respect to the halo. The velocity of the detector with
respect to the halo has three components: the velocity of
the Sun relative to halo, the velocity of the Earth with
respect to the Sun, and the velocity of the detector with
respect to the Earth.
When discussing these velocities, it is useful to de-
fine a Galactic Cartesian coordinate system. In Galactic
coordinates, the Sun is located at the origin, and the
xy-plane is defined by the Galactic disk. The x-axis
points toward the center of the Galaxy, and the y-axis
points in the direction of the Sun’s tangential velocity
as it revolves around the Galactic center. The z-axis
points toward the north Galactic pole and is antiparal-
lel to the angular momentum of the rotating disk. The
motion of the Sun through the halo has two compo-
nents. First, there is the Sun’s rotational velocity as
it orbits the Galactic center: vc in the y direction. Sec-
ond, there is the motion of the Sun relative to the spiral
disk [72]: ~v⊙ = (10.00 ± 0.36, 5.25 ± 0.62, 7.17 ± 0.38)
km s−1 in Galactic Cartesian coordinates. When the
Earth’s motion through the Solar System during its an-
nual orbit of the Sun is expressed in Galactic coor-
dinates [57], the resulting velocity at the time of the
XQC experiment (7.3 days after the vernal equinox) is
~vEarth = (29.14, 5.330,−3.597) km s−1.
The final consideration is the velocity of the detector
relative to the Earth. The maximum velocity attained by
the XQC rocket was less than 1.2 km s−1. This velocity is
insignificant compared to the motion of the Sun relative
to the halo. Moreover, the XQC detector collected data
while the rocket rose and while it fell, and the average
velocity of the rocket was only 0.104 km s−1. Therefore,
we neglect the motion of the rocket in the calculation of
the dark matter wind. Combining the motion of the Sun
and the Earth then gives the total velocity of the XQC
detector with respect to the halo during the experiment
in Galactic Cartesian coordinates: ~vdetector = (39.14 ±
0.36, 230.5 ± 30, 3.573 ± 0.38) km s−1. Subtracting the
velocity vector of the detector relative to the halo from
the velocity vector of the dark matter relative to the halo
gives the dark matter’s velocity relative to the detector in
Galactic coordinates. However, we want the dark matter
particles’ velocities in the coordinate frame defined by
the detector, where the z-axis is the field-of-view vector.
The XQC field of view was centered on l = 90◦, b = +60◦
in Galactic latitude and longitude [51], so the rotation
from Galactic coordinates to detector coordinates may
be described as a clockwise 30◦ rotation of the z-axis
around the x-axis, which is taken to be the same in both
coordinate systems.
B. Dark Matter Interactions
Calorimetry measures the kinetic energy transferred
from the dark matter to the absorbing material without
regard for the specific mechanism of the scattering or
any other interactions. Consequently, the dark matter
detection rate for a calorimeter depends only on the mass
of the dark matter particle and the total cross section for
elastic scattering between the dark matter particle and an
atomic nucleus of mass number A, which is proportional
to the cross section for dark matter interactions with a
single nucleon (σDn). The calorimeter measures the recoil
energy of the target nucleus (mass mT),
Erec =
2mTmdm
(mT +mdm)2
(1 − cos θCM), (5)
where mdm and vdm are dark matter particle’s mass and
velocity prior to the collision in rest frame of the target
nucleus and θCM is the scattering angle in the center-of-
mass frame.
If the momentum transferred to the nucleus, q2 =
2mTErec, is small enough that the corresponding de
Broglie wavelength is larger than the radius R of the
nucleus (qR ≪ ~), then the scattering is coherent. In co-
herent scattering, the scattering amplitudes for each in-
dividual component in the conglomerate body are added
prior to the calculation of the cross section, so the total
cross section is proportional to the square of the mass
number of the target nucleus. Including kinematic fac-
tors [45, 73], the cross section for coherent scattering off
a nucleus is given by
σcoh(A) = A
mred(DM,Nuc)
mred(DM, n)
σDn, (6)
where mred(DM,Nuc) is the reduced mass of the nucleus
and the dark matter particle, mred(DM, n) is the reduced
mass of a nucleon and the dark matter particle, and A is
the mass number of the nucleus. Coherent scattering is
isotropic in the center-of-mass frame of the collision.
Dark matter particles may be massive and fast-moving
enough that the scattering is not completely coherent
when the target nucleus is large [74]. When the scat-
tering is incoherent, the dark matter particle “sees” the
internal structure of the nucleus, and the cross section for
scattering is reduced by a “form factor,” which is a func-
tion of the momentum transferred to the nucleus during
the collision (q) and the nuclear radius (R):
F 2(q, R). (7)
Since q depends on the recoil energy, which in turn de-
pends on the scattering angle, incoherent scattering is
not isotropic.
In this discussion of coherence, we have neglected the
possible effects of the dark matter particle’s internal
structure by assuming that σDn is independent of recoil
energy. If the dark matter particle is not point-like then
σDn decreases as the recoil momentum increases due to a
loss of coherence within the dark matter particle. Inco-
herence within the dark matter particle has observational
consequences [75], but these effects depend on the size of
the dark matter particle. To avoid restricting ourselves
to a particular dark matter model, we assume that the
dark matter particle is small enough that nucleon scat-
tering is always coherent; when we discuss incoherence,
we are referring to the effects of the nucleus’s internal
structure.
According to the Born approximation, the form fac-
tor for nuclear scattering defined in Eq. (7) is the
Fourier transform of the nuclear ground-state mass den-
sity [57, 76]. The most common choice for the form factor
[74, 77] is F 2(q, R) = exp[−(qRrms)2/(3~2)], where Rrms
is the root-mean-square radius of the nucleus. For a solid
sphere, R2rms = (3/5)R
2, so this form factor is equivalent
to the form factor used in Ref. [52]. This form factor is
an accurate approximation of the Fourier transform of
a solid sphere for (qR)/~ ∼< 2, but it grossly underesti-
mates the reduction in σ for larger values of q [57]. The
maximum speed of a dark matter particle with respect
to the XQC detector is ∼ 800 km s−1 (escape velocity
+ detector velocity), and at that speed, the maximum
possible value of qR/~ for a collision with a Hg nucleus
(A = 200) is nearly ten for a 100 GeV dark matter parti-
cle, and the maximum possible value of qR/~ increases as
the mass of the dark matter particle increases. Clearly,
this approximation is not appropriate for a large portion
of the dark matter parameter space probed by the XQC
experiment.
Furthermore, a solid sphere is not a very realistic model
of the nucleus. A more accurate model of the nuclear
mass density is ρ(r) =
d3r′ρ0(r
′)ρ1(r− r′), where ρ0 is
constant inside a radius R20 = R
2 − 5s2 and zero beyond
that radius and ρ1 = exp[−r2/(2s2)], where s is a “skin
thickness” for the nucleus [78]. The resulting form factor
F (q, R) = 3
sin(qR/~)− (qR/~) cos(qR/~)
(qR/~)3
× exp
(qs/~)2
. (8)
We follow Ref. [57] in setting the parameters in Eq. (8):
s = 0.9 fm and
R2 = [(1.23A1/3 − 0.6)2 + 0.631π2 − 5s2] fm2, (9)
where A is the mass number of the target nucleus.
Despite its simple analytic form, the form factor given
by Eq. (8) is computationally costly to evaluate repeat-
edly. We use an approximation:
F 2 =
0.9 fm
if qR
9(0.81)
(qR/~)4
0.9 fm
if qR
The low-q approximation combines the standard approx-
imation for the solid sphere with the factor accounting
for the skin depth of the nucleus. The high-q approxi-
mation was derived from the asymptotic form of the first
spherical Bessel function and normalized so that the to-
tal cross section is as close as possible to the exact result.
The error in the total cross section due to the use of the
approximation is less than 1% for nearly all dark mat-
ter masses; the sole exception is mdm ∼ 10 − 100 GeV,
and even then the error is less than 5%. Unless other-
wise noted, we use this approximation for the form factor
throughout this analysis. We also assume that the dark
matter particle does not interact with nuclei in any way
other than elastic scattering.
IV. ANALYSIS OF XQC CONSTRAINTS
To obtain an accurate description of the XQC experi-
ment’s ability to detect strongly interacting dark matter
particles, we turned to Monte Carlo simulations. The
Monte Carlo code we wrote to analyze the XQC experi-
ment simulates a dark matter particle’s journey through
the atmosphere to the XQC detector chamber, its path
through the detector chamber to a calorimeter, and its in-
teraction with the sensitive components of the calorime-
ter. This latter portion of the code also records how much
energy the particle deposits in the calorimeter through
scattering. The results of several such simulations for the
same set of dark matter properties may be used to pre-
dict the likelihood that a given dark matter particle will
deposit a particular amount of energy into the calorime-
ter. These probabilities of various energy deposits predict
the recoil-energy spectrum the XQC detector would ob-
serve if the dark matter particles have a given mass and
nucleon-scattering cross section. This simulated spec-
trum may then be compared to the XQC data to find
which dark matter parameters are excluded by the XQC
experiment.
A. Generating Simulated Energy-Recoil Spectra
The basic subroutine in our Monte Carlo algorithm
is the step procedure. The step procedure begins with
a particle with a certain velocity vector and position in
a given material and moves the particle a certain dis-
tance in the material, returning its new position and ve-
locity. The step procedure also determines whether or
not a scattering event occurred during the particle’s trek
and updates the velocity accordingly. The number of ex-
pected collisions in a step of length l through a material
with target number density n is n× σtot × l, where σtot
is the total scattering cross section obtained by integrat-
ing Eq. (7) over the scattering angle, or equivalently, the
recoil momentum q:
σtot =
q2max
F 2(q, R) dq2, (11)
where qmax is the maximum possible recoil momentum.
The step length l is chosen so that it is at most a tenth
of the mean free path through the material, so the num-
ber of expected collisions is less than one and represents
the probability of a collision. After each step, a ran-
dom number between zero and one is generated using
the “Mersenne Twister” (MT) algorithm [79] and if that
random number is less than the probability of a collision,
the particle’s energy and trajectory are updated. First, a
recoil momentum is selected according to the probability
distribution P (q2) = F 2(q, R)σcoh/(q
maxσtot), where the
exact form factor is used for qR/~ > 2 so that the oscilla-
tory nature of the form factor is not lost. The recoil mo-
mentum determines the recoil energy and the scattering
angle in the center-of-mass frame through Eq. (5). The
scattering is axisymmetric around the scattering axis, so
the azimuthal angle is assigned a random value between
0 and 2π. The scattering angles are used to update the
particle’s trajectory, and its speed is decreased in accor-
dance with the kinetic energy transferred to the target
nucleus. The step subroutine repeats until the particle
exits the simulation, or its kinetic energy falls below 0.1
eV, or the energy deposited in the calorimeter exceeds
the saturation point of 4000 eV.
Our simulation treats the atmosphere as a 4.6×4.6 cm
square column with periodic boundary conditions, the
bottom face of which covers the top of the conical detec-
tor chamber described in Section II. This implementation
assumes that for every particle that exits one side of the
column, there is a particle that enters the column from
the opposite side with the same velocity. The infinite
extent of the atmosphere and its translational invariance
makes this assumption reasonable. The atmosphere col-
umn extends to an altitude 1000 km; increasing the at-
mosphere height beyond 1000 km has a negligible effect
on the total number of collisions in the atmosphere. The
simulation begins with a dark matter particle at the top
of the atmosphere column at a random initial position
on the 4.6×4.6 cm square. Its initial velocity with re-
spect to the dark matter halo is selected according to the
isotropic Maxwellian velocity distribution function given
by Eq. (2), and then the velocity relative to the detector
is found via the procedure described in Section IIIA.
The dark matter particle’s path from the top of the at-
mosphere to the detector is modeled using the step pro-
cedure described above. The simulation of the particle’s
interaction with the atmosphere ends if the particle’s al-
titude exceeds 1000 km or if the particle falls below the
height of the XQC rocket. We use the time-averaged alti-
tude (201.747 km) as the constant altitude of the rocket.
We made this simplification because it allows us to ig-
nore the periodic inactivity of each calorimeter and treat
the detector as thirty-four calorimeters that are active for
100.7 seconds. When the dark matter particle hits the
rocket, its path through the five filter layers is also mod-
eled using the step procedure, as is its path through the
calorimeters. In addition to being smaller than the mean
free path, the step length is chosen so that the particle’s
position relative to the boundaries of the detectors is ac-
curately modeled. The simulation ends when the dark
matter particle’s random-walk trajectory takes it out of
the detector chamber. As mentioned in Section II, the
calorimeter detects the sum of all the recoil energies if
the dark matter particle is scattered multiple times.
When the dark matter particle is unlikely to experience
more than one collision in the calorimeter, this simula-
tion is far more detailed than is required to accurately
predict the energy deposited by the dark matter parti-
cle. This is the case for the lightest (mdm ≤ 102 GeV)
and weakest-interacting (σDn ≤ 10−26 cm2) dark mat-
ter particles that the XQC calorimeters are capable of
detecting. Since the lightest dark matter particles are
also the most numerous, many Monte Carlo trials are
required to sample all the possible outcomes of a dark
matter particle’s encounter with the detector. The sim-
ulation described above is too computationally intensive
to run that many trials, so we used a faster and simpler
simulation to model the interactions of these dark matter
particles. This simulation assumes that the particle will
experience at most one collision in the atmosphere and
at most one collision in each filter layer and each layer of
the calorimeter. The simulation ends if the probability
of two scattering events in either the atmosphere or any
of the filter layers exceeds 0.1. Instead of tracking the
dark matter particle’s path through the atmosphere, the
total overburden for the atmosphere is used to determine
the probability that the dark matter particle scatters in
the atmosphere, and the particle only reaches the detec-
tor if its velocity vector points toward the detector after
the one allowed scattering event. Also, instead of the
small step lengths required to accurately model the ran-
dom walk of a strongly interacting particle, each layer is
crossed with a single step. These simplifications reduce
the runtime of the simulation by a factor of 100, making
it possible to run 1010 trials in less than one day.
B. Comparing the Simulations to the XQC Data
In order to compare the probability spectra produced
by our Monte Carlo routine to the results of the XQC
experiment, we must multiply the probabilities by the
number of dark matter particles that are encountered by
the initial surface of the Monte Carlo routine. When the
initial velocity of the dark matter particle is chosen, the
initial velocity may point toward or away from the detec-
tor; in the latter case, the trial ends immediately. Con-
FIG. 5: Simulated event spectra for dark matter particles with
masses of 1, 10 and 100 GeV and a total nucleon-scattering
cross section of 10−27.3 cm2. In addition to the events de-
picted in these spectra, the simulations predict 1300 ± 160
events with energies greater than 4000 eV when mdm = 10
GeV and 10, 000 ± 1200 such events when mdm = 100 GeV.
The histogram represents the XQC observations.
sequently, the Monte Carlo probability that the particle
deposits no energy in the calorimeter already includes the
probability that the dark matter particle does not have a
halo trajectory that takes it into the atmosphere. There-
fore, the probabilities resulting from the Monte Carlo
routine should be multiplied by the number of particles
in the volume swept out by the initial 4.6×4.6 cm2 square
surface during the 100.7f(E) seconds of observation time,
where f is the fraction of the observing time that the
XQC detector was sensitive to deposits of energy E. For
energies between 36 and 88 eV, f is 0.5083, and the value
of f increases to one over energies between 88 and 128 eV.
The detector was also slightly sensitive to lower energies:
between 29 and 35 eV, f increases from 0.3815 to 0.5083.
The normal of the initial surface points along the detec-
tor’s field of view, and the surface moves with the detec-
tor; using the detector velocity given in Section IIIA, the
number of dark matter particles encountered by the ini-
tial surface is Ndm = f × (ρdm/mdm)× [(2.5± 0.3)× 1010
cm3], where ρdm is the local dark matter density.
The simulated event spectra produced by our Monte
Carlo routine indicate that particles with masses less
than 1 GeV very rarely deposit more than 100 eV inside
the XQC calorimeters. Conversely, particles with masses
greater than 100 GeV nearly always deposit more than
4000 eV when they interact with the XQC calorimeters,
so constraints on σDn for these mdm values arise from the
FIG. 6: Simulated event spectra for 10-GeV dark mat-
ter particles with total nucleon-scattering cross sections of
10−21.6, 10−27.3 and 10−28.3 cm2. In addition to the events
depicted in these spectra, the simulations predict 140 ± 37
events with energies greater than 4000 eV when σDn = 10
−21.6
cm2, 1300 ± 160 such events when σDn = 10
−27.3 cm2, and
120±15 such events when σDn = 10
−28.3 cm2. The histogram
represents the XQC observations.
over-saturation (E ≥ 4000 eV) event rate. Fig. 5 shows
simulated spectra for three mdm values that lie between
these two extremes, along with a histogram that depicts
the XQC observations. Given an initial velocity of 300
km s−1 relative to the XQC detector, a 1-GeV particle
can only deposit up to 66 eV in a single collision with an
Si nucleus, so the spectrum for these particles is confined
to very low energies. Meanwhile, a 10-GeV particle and
a 100-GeV particle with the same initial velocity can de-
posit up to 900 eV and 44,000 eV, respectively, in a single
collision with an Hg nucleus. In fact, ignoring any loss
of coherence, all recoil energies between 0 and 44,000 eV
are equally likely during a collision between an Hg nu-
cleus and a 100-GeV dark matter particle. That’s why
the mdm = 100 GeV spectrum in Fig. 5 is flat below 2500
eV and why the simulations predict 10,000 events with
energies greater than 4000 eV for this value of mdm and
Fig. 6 shows how changing the total cross section for
elastic scattering off a nucleon affects the simulated spec-
tra generated by our Monte Carlo routine for a single
dark matter particle mass (mdm = 10 GeV). We see that
increasing σDn from 10
−28.3 cm2 to 10−27.3 cm2 increases
all of the counts by a factor of ten but leaves the ba-
sic shape of the spectrum unchanged. For much larger
values of σDn, however, the particle loses a considerable
Energy Range (eV) Counts Energy Range (eV) Counts
29 - 36 0 945 - 1100 31
36 - 128 11 1100 - 1310 30
128 - 300 129 1310 - 1500 29
300 - 540 80 1500 -1810 32
540 - 700 90 1810 - 2505 15
700 - 800 32 ≥ 4000 60
800 - 945 48
TABLE I: The binned XQC results used for comparison with
our Monte Carlo simulations.
amount of its energy while traveling through the atmo-
sphere. Consequently, high-energy recoil events become
less frequent, as shown by the spectrum for σDn = 10
−21.6
cm2. For larger values of σDn, too much energy is lost in
the atmosphere for the particle to be detectable by the
XQC experiment.
When comparing the simulated measurements to the
XQC data, we group the events into the thirteen energy
bins given in Table I. We generally use large bins be-
cause it reduces the fractional error in the probabilities
generated by our Monte Carlo routine by increasing the
probability of each bin: δpi/pi = 1/
pit, where t is the
number of trials and pi is the probability of an energy
deposit in the ith bin. Given that the number of trials is
limited by runtime constraints, increasing the bin size is
often the only way to obtain bin probabilities with δpi/pi
values much less than one. When choosing our binning
scheme, we attempted to maximize bin size while pre-
serving as many features of the observed spectrum as
possible. We also grouped all energies for which f 6= 1
into two bins; we ignore the variation in f within these
bins and set f = 0.3815 in the lowest-energy bin and
f = 0.5083 in the next-to-lowest bin.
Unfortunately, we do not know the number of X-ray
events in any of the bins listed in Table I. We considered
using a model to subtract off the X-ray background but,
given any model’s questionable accuracy, we decided not
to use it in our analysis. Our ignorance of the X-ray
background forces us to treat the number of observed
counts in each bin as an upper limit on the number of
dark matter events in that energy range. Consequently,
we define a parameter X2 that measures the extent of
the discrepancy between the simulated results for a given
mdm and σDn and the XQC observations while ignoring
bins in which the observed event count exceeds the pre-
dicted contribution from dark matter:
i=# of Bins
(Ei − Ui)2
with Ui < Ei
, (12)
where Ei = Ndm × pi is the number of counts in the ith
bin predicted by the Monte Carlo simulation and Ui is
the number of observed counts in the same bin. We use
a second Monte Carlo routine to determine how likely it
is that a set of observations would give a value of X2 as
large or larger than the one derived from the XQC data
given a mean signal described by the set of Ei derived
from the simulation Monte Carlo.
In the comparison Monte Carlo, a trial begins by gen-
erating a new set of Ei by sampling the error distribu-
tions of Ndm and pi. The distribution of Ndm values is
assumed to be Gaussian with the mean and standard de-
viation given above. The probability pi is derived from
pi× t events in the simulation Monte Carlo (recall that t
is the number of trials), so a new value for pi is generated
by sampling a Poisson distribution with a mean of pi × t
and dividing the resulting number by t. Once a new set
of Ei has been found, the routine generates a simulated
number of observed counts for each bin according to a
Poisson distribution with a mean of Ei. The value of the
X2 parameter for the new Ei and Ui is computed and
compared to the value for the original Ei and the XQC
observations, X2XQC. The number of trials needed to ac-
curately measure the probability P(X) thatX2 ≥ X2XQC
is determined by requiring that the variation in the mean
value of X2 over ten Monte Carlo simulations does not
exceed (100-C)%, where C% is the desired confidence
level and that the range P(X)±(5× the variation in P(X))
does not contain (100− C)/100.
V. RESULTS AND DISCUSSION
The XQC experiment rules out the enclosed region in
(mdm, σDn) parameter space shown in Fig. 7. The over-
burden from the atmosphere and the filtering layers as-
sures that there will be a limit to how strongly a dark
matter particle can interact with baryons and still reach
the XQC calorimeters; this overburden is responsible for
the top edge of the exclusion region. Conversely, if σDn is
too small, the dark matter particles will pass through the
calorimeters without interacting. The low-energy thresh-
old of the XQC calorimeters places a lower bound on the
excluded dark matter particle masses; ifmdm is too small,
then the recoil energies are undetectable. On the other
side of the mass range, the XQC detector is not sensitive
to mdm ∼> 105 GeV because the number density of such
massive dark matter particles is too small for the XQC
experiment to detect.
The exclusion region shown in Fig. 7 has a complicated
shape, but its features are readily explicable. As mdm in-
creases, the range of excluded σDn values shifts to lower
values and then moves up again. The downward shift for
mdm between 0.1 GeV and 100 GeV is due to the effects of
coherent nuclear scattering. Since σcoh increases with in-
creasing mdm for fixed σDn, a 100-GeV particle interacts
more strongly in the atmosphere and in the detector than
a 1-GeV particle with the same σDn. Consequently, both
the upper and lower boundaries of the excluded region
decrease with increasing mass for mdm ∼< 100 GeV. The
scattering of dark matter particles with larger masses is
incoherent, and the form factor discussed in Section III B
FIG. 7: The region of dark matter parameter space excluded
by the XQC experiment; σDn is the total cross section for scat-
tering off a nucleon and mdm is the mass of the dark matter
particle. This exclusion region follows from the assumption
that the local dark matter density is 0.3 GeV cm−3 and that
all of the dark matter shares the same value of σDn.
causes σtot to decrease as mass increases for fixed σDn.
Moreover, particles that are more massive than the target
nuclei have straighter trajectories than lighter dark mat-
ter particles due to smaller scattering angles in the de-
tector rest frame. The loss of coherence also contributes
because incoherent scattering makes small scattering an-
gles more probable. A straight trajectory is shorter than
a random walk, so the more massive particles interact
less in the atmosphere and the detector than the more
easily-deflected lighter particles. Due to both of these
effects, the upper and lower boundaries of the exclusion
region increase with increasingmdm for mdm ∼> 100 GeV.
The lower left corner of the exclusion region also has
two interesting features. First, the lower bound on the
excluded value of σDn decreases sharply as mdm increases
from 0.1 GeV to 0.5 GeV. A dark matter particle with the
maximum possible velocity with respect to the detector
(800 km s−1) must have a mass greater than 0.24 GeV
to be capable of depositing 29 eV in the calorimeter in
a single collision. Lighter particles are only detectable
if they scatter multiple times inside the calorimeter, and
multiple scatters require a higher value of σDn. Since
their analysis does not allow multiple collisions, the XQC
exclusion region found in Ref. [52] does not extend to
masses lower than 0.3 GeV for any value of σDn. Second,
there is a kink in the lower boundary at mdm = 10 GeV;
the constraint on σDn is not as strong for this mass. The
FIG. 8: The region of dark matter parameter space excluded
to 90% confidence by the XQC experiment for several values of
the local density of dark matter with a total nucleon scattering
cross section σDn and mass mdm. The four densities shown
are 0.3 GeV cm−3 (solid line), 0.15 GeV cm−3 (long dashed
line), 0.075 GeV cm−3 (short dashed line) and 0.03 GeV cm−3
(dotted line).
simulated spectra produced by our Monte Carlo routine
for mdm = 10 GeV and σDn ∼< 10−25 cm2 reveal that the
particle is most likely to deposit between 100 and 600
eV, as exemplified by the spectra depicted in Fig. 6. The
background in this energy range is very high, so the XQC
constraints are not as strict at these energies.
Altering the local density of dark matter that strongly
interacts with baryons changes the exclusion region.
Fig. 8 shows the 90%-confidence exclusion regions for
four values of the local density of dark matter particles
with strong baryon interactions: 0.3 GeV cm−3 (solid
line), 0.15 GeV cm−3 (long dashed line), 0.075 GeV cm−3
(short dashed line) and 0.03 GeV cm−3 (dotted line).
These different local densities could arise due to varia-
tions in the local dark matter density due to mini-halos
or streams. They also describe models where the dark
matter does not consist of a single particle species and
the dark matter that strongly interacts with baryons is
a fraction fd of the local dark matter. In that case, the
four exclusion regions in Fig. 8 correspond to fd =1, 0.5,
0.25, and 0.1.
Fig. 8 indicates that the top and left boundaries of the
XQC exclusion region are not highly sensitive to the dark
matter density. In particular, the upper left corner of the
exclusion region (0.01 ≤ mdm ≤ 0.1 GeV) is nearly un-
affected by lowering the dark matter density. This con-
sistency indicates our Monte Carlo-generated exclusion
region is smaller than the true exclusion region in this
corner. If the dark matter is light (mdm ∼< 0.1 GeV),
then the number of dark matter particles encountered by
the XQC detector is very large (Ndm ∼> 7 × 1010). As
previously mentioned, the upper left corner of the XQC
exclusion region results from multiple scattering events,
so the simpler version of our Monte Carlo code described
in Section IVA is not applicable. Consequently, it is not
possible to run more than 109 trials in a week, so each
scattering event in the simulation corresponds to more
than one scattering event in the detector for all the den-
sities shown in Fig. 8. Therefore, decreasing the density
does not change the result. If it were possible to run 1011
trials, then the upper left corner of the exclusion region
would expand and differences between the different den-
sity contours would emerge. Since the upper left corner
of the XQC exclusion region is already ruled out by astro-
physical constraints (see Fig. 9), we have not invested in
the computational time necessary to expand this corner.
The upper boundary of the exclusion region is also not
greatly affected by decreasing the particle density, even
when Ndm is small enough that the Monte Carlo routine
is capable of running more than Ndm trials (mdm ≥ 100
GeV). This robustness indicates that the overburden of
the XQC experiment effectively prevents all dark matter
particles with σDn values greater than the upper bound
of the exclusion region from reaching the detector, so
it does not matter how many particles are encountered.
Finally, as discussed previously, the lower portion of the
exclusion region’s left boundary (σDn ≤ 10−23 cm2) is
set by the energy threshold for detection and is therefore
independent of Ndm.
Examining the features of the excluded region allows
us to predict how the region may be expanded by a fu-
ture XQC-like experiment. Decreasing the overburden by
either increasing the rocket’s altitude or reducing the fil-
tering will push the top boundary of the excluded region
upwards. Decreasing the energy detection threshold will
extend the excluded region to lower masses. It may also
extend the exclusion region to higher values of σDn for all
masses since strongly interacting particles lose much of
their energy in the atmosphere and arrive at the calorime-
ter with too little energy to produce a detectable sig-
nal. Increasing the size or number of calorimeters would
increase the sensitivity and extend the excluded region
to lower values of σDn. Finally, increasing the observa-
tion time would increase Ndm, and that would extend the
right and bottom boundaries of the excluded region.
VI. CONCLUSION
The X-ray Quantum Calorimetry (XQC) experiment is
a powerful detector of dark matter that interacts strongly
with baryons due to its high altitude and minimal shield-
ing. The XQC measurements rule out a large range of
hitherto unconstrained dark matter masses and scatter-
FIG. 9: Plot of the scattering cross section for dark matter particles and nucleons (σDn) versus dark matter particle mass
(mdm) showing the new XQC limits along with other current experimental limits. The red XQC exclusion region is the same
as shown in Fig. 7, and the other experiments are discussed in the text. The dark gray region shows the maximal range of
dark matter self-interaction cross section consistent with the strongly self-interacting dark matter model of structure formation
[37, 38]. The square marks the value of the scattering cross section for neutron-nucleon interactions.
ing cross sections. The excluded range was first derived
in Refs. [38, 48] based on rough analytic estimates. In
this paper, we have improved upon these results using
detailed Monte Carlo simulations to predict how a dark
matter particle of a given mass and cross section for nu-
cleon scattering would interact with the XQC calorime-
ters. Unlike Ref. [52], our analysis includes the atmo-
sphere and the shielding of the detector, so our result in-
cludes the upper limit on excluded σDn values, which had
not yet been accurately determined. Our simulation also
models the internal geometry of the XQC detector and
the random walk of particles through it, which is not pos-
sible using the analytical approaches of Refs. [38, 48, 52].
The resulting exclusion region is significantly different
than its analytical predecessors. When multiple scatter-
ings are included, the XQC experiment is sensitive to
dark matter particles with masses below 0.3 GeV and
cross sections for nucleon scattering between 10−24 and
10−20 cm2. Unlike Ref. [52], we find that the XQC ex-
clusion region does not include σDn < 10
−29 cm2 for
dark matter masses less than 10 GeV. Ref. [52] obtained
a more restrictive upper bound because they assumed a
specific X-ray background while we treat all events as po-
tential dark matter interactions. At higher masses, the
lower boundary of our exclusion region is much higher
than in Refs. [38, 48] because they over-estimated the
XQC sensitivity by assuming coherent scattering. It also
appears that Refs. [38, 48] underestimated the atmo-
spheric and shielding overburden for the XQC detector
because our exclusion region does not extend to values of
σDn as large as those included in their exclusion region.
We also assume a lower local dark matter density than
Refs. [38, 48] (0.3 instead of 0.4 GeV cm−3), so some of
the shrinkage of the exclusion region may be attributed
to the reduction in the assumed number density of dark
matter particles.
Fig. 9 shows how the XQC exclusion region depicted in
Fig. 7 complements the exclusion regions from other ex-
periments that are sensitive to similar values of σDn and
mdm. For a summary of some of the other experimen-
tal constraints as of 1994, see Ref. [46]. The constraints
to σDn from Pioneer 11 [80], Skylab [81], and IMP7/8
[82] were interpreted by Refs. [38, 46, 48]. There have
been two balloon-borne searches for dark matter, the
IMAX experiment [46, 47] and the Rich, Rocchia & Spiro
(RRS) [83] experiment. Although underground detectors
are designed to detect WIMPs, DAMA [84, 85] does ex-
clude σDn values within the range of interest, and relevant
constraints may be derived from Edelweiss (EDEL) and
CDMS [86, 87].
All of the exclusion regions shown in Fig. 9 were de-
rived assuming that all the dark matter is strongly in-
teracting. A local dark matter density of 0.4 GeV cm−3
was assumed in the analysis of the exclusion regions from
Pioneer 11, Skylab and the RRS experiment, while all
the other exclusion regions were derived assuming a lo-
cal dark matter density of 0.3 GeV cm−3. Furthermore,
the derivations of all the shown exclusion regions other
than the XQC region and the EDEL+CDMS region as-
sume that the scattering between dark matter particles
and nuclei is coherent. Therefore, these exclusion regions
are likely too broad because they over-estimate the cross
section for nuclear scattering. A comparison of the XQC
exclusion region reported in Refs. [38, 48] and our exclu-
sion region indicates that assuming coherent scattering
extends the exclusion region for mdm ≥ 1000 GeV to
σDn values that are roughly A× smaller than the lower
boundary of our exclusion region, where A is the mass
number of the largest target nucleus.
Fig. 9 also shows the bound on σDn from the CMB
and large-scale structure (LSS) obtained when one as-
sumes prior knowledge of the Hubble constant H0 and
the cosmic baryon fraction (from BBN) [50]. This bound
is nominally stronger than the bound from disk stabil-
ity [45], but it is less direct in that it requires combining
different measurements and depends on the cosmologi-
cal model; consequently, we show both bounds in Fig. 9.
Measurements of primordial element abundances give an
upper limit of σDn/mdm ∼< 4 × 10−16 cm2 GeV−1 [49].
Since this upper bound lies well beyond the upper bound
from disk stability, we do not include it in Fig. 9. We
also do not display the constraints from cosmic rays [49]
because they are derived from inelastic interactions that
are model-dependent.
As shown in Fig. 9, the XQC experiment rules out
a wide region of (mdm, σDn) parameter space that was
not probed by prior dark matter searches. Of partic-
ular interest is the darkly shaded range of σDn values
that corresponds to the maximal range of dark matter
self-interaction cross sections consistent with the strongly
self-interacting dark matter model of structure formation
[37, 38]. If the dark matter consists of exotic hadrons
whose interactions with nucleons are comparable to their
self-interactions, then σDn for these particles would lie in
or near the darkly shaded region in Fig. 9. Previous esti-
mates of the XQC exclusion region [38, 48] indicated that
the XQC experiment rules out all the darkly shaded σDn
values for 1 ∼< mdm ∼< 104 GeV. Our analysis reveals that
this is not the case; portions of the darkly shaded region
for mdm ∼> 20 GeV are not excluded by the XQC ex-
periment, although they are ruled out by observations of
LSS and the CMB. The mass-σ combination correspond-
ing to nucleon-neutron scattering (the square in Fig. 9)
lies within the exclusion region of the XQC experiment,
and the only portion of the darkly shaded region that is
unconstrained corresponds to dark matter masses smaller
than 0.25 GeV.
It is important to note, however, that the cross section
for dark matter self-interactions need not be comparable
to the cross section for nucleon scattering; σDn could dif-
fer by a few orders of magnitude from the self-interaction
cross section (as is the case for Q-balls). Furthermore, no
interactions with baryons are required for self-interacting
dark matter to resolve the tension between the collision-
less dark matter model and observations of small-scale
structure.
Another XQC detector is scheduled to launch in the
upcoming year. This experiment will have twice the ob-
serving time of the XQC experiment used in this analy-
sis. As discussed in Section V, increasing the observing
time will extend the exclusion region to higher masses
and weaker interactions. The future XQC experiment
will also have a lower energy threshold (15 eV) and will
maintain sensitivity to all energies above this threshold
throughout the run. The increased sensitivity to low en-
ergies will shift the lower (σDn ≤ 10−23 cm2) left bound-
ary of the exclusion region to lower masses. A lower en-
ergy threshold of 15 eV will make the experiment sensi-
tive to single recoil events involving dark matter particles
more massive than 0.17 GeV, as discussed in Section V.
Clearly, the next-generation XQC experiment will be an
even more powerful probe of interactions between dark
matter particles and baryons than its predecessor.
Acknowledgments
A. L. E. would like to thank Robert Lupton and
Michael Ramsey-Musolf for useful discussions. D. M.
thanks the Wallops Flight Facility launch support team
and the many undergraduate and graduate students that
made this pioneering experiment possible. The authors
also thank Randy Gladstone for his assistance with the
atmosphere model. A. L. E. acknowledges the support
of an NSF Graduate Fellowship. P. J. S. is supported
in part by US Department of Energy grant DE-FG02-
91ER40671. P. C. M. acknowledges current support for
this project from a Robert M. Walker Senior Research
Fellowship in Experimental Space Science from the Mc-
Donnell Center for the Space Sciences, as well as prior
institutional support for this project from the Instituto
Nacional de Técnica Aeroespacial (INTA) in Spain, from
the University of Bielefeld in Germany, and from the Uni-
versity of Arizona.
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  Although the rocket-based X-ray Quantum Calorimetry (XQC) experiment was
designed for X-ray spectroscopy, the minimal shielding of its calorimeters, its
low atmospheric overburden, and its low-threshold detectors make it among the
most sensitive instruments for detecting or constraining strong interactions
between dark matter particles and baryons. We use Monte Carlo simulations to
obtain the precise limits the XQC experiment places on spin-independent
interactions between dark matter and baryons, improving upon earlier analytical
estimates. We find that the XQC experiment rules out a wide range of
nucleon-scattering cross sections centered around one barn for dark matter
particles with masses between 0.01 and 10^5 GeV. Our analysis also provides new
constraints on cases where only a fraction of the dark matter strongly
interacts with baryons.

Constraints on the Interactions between Dark Matter and Baryons from the X-ray
Quantum Calorimetry Experiment
Adrienne L. Erickcek1,2, Paul J. Steinhardt2,3, Dan McCammon4, and Patrick C. McGuire5
Division of Physics, Mathematics, & Astronomy,
California Institute of Technology, Mail Code 103-33, Pasadena, CA 91125, USA
Department of Physics, Princeton University, Princeton, NJ 08544, USA
Princeton Center for Theoretical Physics, Princeton University, Princeton, NJ 08544, USA
Department of Physics, University of Wisconsin, Madison, WI 53706, USA and
McDonnell Center for the Space Sciences, Washington University, St. Louis, M0 63130, USA
Although the rocket-based X-ray Quantum Calorimetry (XQC) experiment was designed for X-
ray spectroscopy, the minimal shielding of its calorimeters, its low atmospheric overburden, and its
low-threshold detectors make it among the most sensitive instruments for detecting or constraining
strong interactions between dark matter particles and baryons. We use Monte Carlo simulations to
obtain the precise limits the XQC experiment places on spin-independent interactions between dark
matter and baryons, improving upon earlier analytical estimates. We find that the XQC experiment
rules out a wide range of nucleon-scattering cross sections centered around one barn for dark matter
particles with masses between 0.01 and 105 GeV. Our analysis also provides new constraints on
cases where only a fraction of the dark matter strongly interacts with baryons.
PACS numbers: 95.35.+d, 12.60.-i, 29.40.Vj
I. INTRODUCTION
From Vera Rubin’s discovery that the rotation curves
of galaxies remain level to radii much greater than pre-
dicted by Keplerian dynamics [1] to the Wilkinson Mi-
crowave Anisotropy Probe (WMAP) measurement of
the cosmic microwave background (CMB) temperature
anisotropy power spectrum [2], observations indicate that
the luminous matter we see is only a fraction of the
mass in the Universe. The three-year WMAP CMB
anisotropy spectrum is best-fit by a cosmological model
with Ωm = 0.241 ± 0.034 and a baryon density that is
less than one fifth of the total mass density. The cold
collisionless dark matter (CCDM) model has emerged
as the predominant paradigm for discussing the missing
mass problem. The dark matter is assumed to consist
of non-relativistic, non-baryonic, weakly interacting par-
ticles, often referred to as Weakly Interacting Massive
Particles (WIMPs).
Although the CCDM model successfully predicts ob-
served features of large-scale structure at scales greater
than one megaparsec [3], there are indications that it may
fail to match observations on smaller scales. Numerical
simulations of CCDM halos [4, 5, 6, 7, 8, 9, 10, 11, 12]
imply that CCDM halos have a density profile that in-
creases sharply at small radii (ρ ∼ r−1.2 according to
Ref. [12]). These predictions conflict with lensing ob-
servations of clusters [13, 14] that indicate the presence
of constant-density cores. X-ray observations of clusters
have found cores in some clusters, although density cusps
have also been observed [15, 16, 17]. On smaller scales,
observations of dwarf and low-surface-brightness galaxies
[18, 19, 20, 21, 22, 23, 24] indicate that these dark mat-
ter halos have constant-density cores with lower densities
than predicted by numerical simulations. Observations
also indicate that cores are predominant in spiral galaxies
as well, including the Milky Way [25, 26, 27]. Numerical
simulations of CCDM halos also predict more satellite
halos than are observed in the Local Group [28, 29] and
fossil groups [30].
Astrophysical explanations for the discord between the
density profiles predicted by CCDM simulations and ob-
servations have been proposed: for instance, dynamical
friction may transform density cusps into cores in the
inner regions of clusters [31], and the triaxiality of galac-
tic halos may mask the true nature of their inner den-
sity profiles [32]. There are also models of substructure
formation that explain the observed paucity of satellite
halos [33, 34, 35, 36].
Another possible explanation for the apparent failure
of the CCDM model to describe the observed features
of dark matter halos is that dark matter particles scat-
ter strongly off one another. The discrepancies between
observations and the CCDM model are alleviated if one
introduces a dark matter self-interaction that is compa-
rable in strength to the interaction cross section between
neutral baryons [37, 38]:
= 8× 10−25 − 1× 10−23cm2 GeV−1, (1)
where σDD is the cross section for scattering between dark
matter particles and mdm is the mass of the dark mat-
ter particle. Numerical simulations have shown that in-
troducing dark matter self-interactions within this range
reduces the central slope of the halo density profile and
reduces the central densities of halo cores, in addition to
destroying the extra substructure [39, 40].
The numerical coincidence between this dark mat-
ter self-interaction cross section and the known strong-
interaction cross section for neutron-neutron or neutron-
proton scattering has reinvigorated interest in the pos-
sibility that dark matter interacts with itself and with
http://arxiv.org/abs/0704.0794v2
baryons through the strong nuclear force. We refer to
dark matter of this type as “strongly interacting dark
matter” where “strong” refers specifically to the strong
nuclear force. Strongly interacting dark matter candi-
dates include the dibaryon [41, 42], the Q-ball [43], and
O-helium [44].
Surprisingly, the possibility that the dark matter may
be strongly interacting is not ruled out. While there are
numerous experiments searching for WIMPs, they are
largely insensitive to dark matter that interacts strongly
with baryons. The reason is that WIMP searches are typ-
ically conducted at or below ground level based on the
fact that WIMPs can easily penetrate the atmosphere
or the Earth, whereas strongly interacting dark matter is
multiply scattered and thermalized by the time it reaches
ground level and its thermal kinetic energy is too small
to produce detectable collisions with baryons in WIMP
detectors. Consequently, there are few experiments ca-
pable of detecting strongly interacting dark matter di-
rectly. Starkman et al. [45] summarized the constraints
on strongly interacting dark matter from experiments
prior to 1990, and these constraints were later refined
[38, 46, 47, 48]. The strength of dark matter interactions
with baryons may also be constrained by galactic dynam-
ics [45], cosmic rays [45, 49], Big Bang nucleosynthesis
(BBN) [49], the CMB [50], and large-scale structure [50].
The X-ray Quantum Calorimetry (XQC) project
launched a rocket-mounted micro-calorimeter array in
1999 [51]. At altitudes above 165 km, the XQC detector
collected data for a little less than two minutes. Although
its primary purpose was X-ray spectroscopy, the limited
amount of shielding in front of the calorimeters and the
low atmospheric overburden makes the XQC experiment
a sensitive detector of strongly interacting dark matter.
In this article, we present a new numerical analysis
of the constraints on spin-independent interactions be-
tween dark matter particles and baryons from the XQC
experiment using Monte Carlo simulations of dark mat-
ter particles interacting with the XQC detector and the
atmosphere above it. Our work is a significant improve-
ment upon the earlier analytic estimates presented by
some of us in Refs. [38, 48] because it accurately models
the dark matter particle’s interactions with the atmo-
sphere and the XQC instrument. Our calculation here
also supersedes the analytic estimate by Zaharijas and
Farrar [52] because they only considered a small por-
tion of the XQC data and did not include multiple scat-
tering events nor the overburden of the XQC detector.
We restrict our analysis to spin-independent interactions
because the XQC calorimeters are not highly sensitive
to spin-dependent interactions. Only a small fraction of
the target nuclei in the calorimeters have non-zero spin;
consequently, the bound on spin-dependent interactions
between baryons and dark matter from the XQC experi-
ment is about four orders of magnitude weaker than the
bound on spin-independent interactions [52].
This article is organized as follows. In Section II we
summarize the specifications of the XQC detector. We
then review dark matter detection theory in Section III.
This Section includes a discussion of coherent versus in-
coherent scattering and how we account for the loss of
coherence in our analysis. A complete description of our
analysis follows in IV, and our results are presented in
Section V. Finally, in Section VI, we summarize our
findings and compare the constraints to strongly inter-
acting dark matter from the XQC experiment to those
from other experiments.
II. THE XQC EXPERIMENT
Calorimetry is the use of temperature deviations to
measure changes in the internal energy of a material. By
drastically reducing the specific heat of the absorbing
material, the use of cryogenics in calorimetry allows the
absorbing object to have a macroscopic volume and still
be sensitive to minute changes in energy. These detectors
are sensitive enough to register the energy deposited by a
single photon or particle and gave birth to the technique
of “quantum calorimetry,” the thermal measurement of
energy quanta.
The quantum calorimetry experiment [51] we use to
constrain interactions between dark matter particles and
baryons is the second rocket-born experiment in the XQC
(X-ray Quantum Calorimetry) Project, a joint under-
taking of the University of Wisconsin and the Goddard
Space Flight Center [53, 54]. It launched on March 28,
1999 and collected about 100 seconds of data at alti-
tudes between 165 and 225 km above the Earth’s surface.
The detector consisted of thirty-four quantum calorime-
ters operating at a temperature of 0.06 K; for detailed
information on the XQC detector functions, please refer
to Refs. [51, 54]. These detectors were separated from
the exterior of the rocket by five thin filter panes [51].
The small atmospheric overburden at this altitude and
the minimal amount of shielding in front of the calorime-
ters makes this experiment a promising probe of strongly
interacting dark matter.
The absorbers in the XQC calorimeters are composed
of a thin film of HgTe (0.96 µm thick) deposited on a sili-
con (Si) substrate that is 14 µm thick. The absorbers rest
on silicon spacers and silicon pixel bodies. Figs. 1 and
2 show side and top views of the detectors with the di-
mensions of each layer. Temperature changes in all four
components are measured by the calorimeter’s internal
thermometer. The calorimeters report the average tem-
perature over an integration time of 7 ms in order to
reduce the effect of random temperature fluctuations on
the measurement. Multiple scatterings by a dark matter
particle will register as a single event because the time it
takes the dark matter particle to make its way through
the calorimeter is small compared to the integration time.
The detector array consists of two rows of detec-
tors, with seventeen active calorimeters and one inactive
calorimeter in each row, and is located at the bottom of
a conical detector chamber. Within a 32-degree angle
   Si Substrate
0.5 mm x 2.0 mm
14 µm
12 µm
HgTe Absorber
0.96µm
      Si Pixel Body
0.25 mm x 1.0 mm
Si Spacer
0.245 mm x 0.245 mm
FIG. 1: A vertical cross section of an XQC calorimeter. The
relative thicknesses of the layers are drawn to scale, as are
their relative lengths, but the two scales are not the same. To
facilitate the display of the layers, the vertical dimension has
been stretched relative to the horizontal dimension.
2.0 mm
1.0 mm
Absorber Panel
  (HgTe on Si)
Pixel Body
     (Si)
   Spacer
     (Si)
  0.245 mm 
x 0.245 mm
FIG. 2: A top view of an XQC calorimeter. The absorber is
the top layer and underneath it lies the spacer, followed by
the pixel body. These dimensions are drawn to scale.
from the detector normal, the incoming particles only
pass through the aforementioned filters. The five filters
are located 2 mm, 6 mm, 9 mm, 11 mm and 28 mm above
the detectors. Each filter consists of a thin layer of alu-
minum (150 Å) supported on a parylene (CH) substrate
(1380 Å). The pressure inside the chamber is less than
10−6 Torr. At this level of evacuation, a dark matter
particle with a mass of 106 GeV and a baryon interac-
tion cross section of 106 barns, would have less than a
20% chance of colliding with an air atom in the cham-
ber. Therefore, we assume that the chamber is a perfect
vacuum in our analysis.
While the atmospheric pressure at the altitudes at
which the detector operated is about 10−8 times the at-
mospheric pressure at sea level, the atmospheric over-
burden of the XQC detector is still sufficient to scat-
ter incoming strongly interacting dark matter particles.
Simulating a dark matter particle’s path through the
atmosphere requires number-density profiles for all the
molecules in the atmosphere. These profiles were ob-
tained using the MSIS-E-90 model1 for the time (1999
1 Available at http://modelweb.gsfc.nasa.gov/models/msis.html
FIG. 3: The points depict the MSIS-E-90 density profiles for
the seven most prevalent constituents of the atmosphere above
the XQC detector, and the lines show the piecewise exponen-
tial fits used in our analysis.
March 28 9:00 UT) and location (White Sands Missile
Range, New Mexico) of the XQC rocket launch.
During the data collection period, the average altitude
of the XQC rocket was 201.747 km. At this altitude and
above, the primary constituents of the atmosphere are
molecular and atomic oxygen, molecular and atomic ni-
trogen, helium, atomic hydrogen, and argon. The MSIS-
E-90 model provides tables of the number densities of
each of these seven chemical species. In our analysis,
computational efficiency demanded that we fit analytic
functions to these data. We found exponential fits for
the density profiles in three altitude ranges: 200-300 km,
300-500 km and 500-1000 km. The error in the proba-
bility of a collision between a dark matter particle and
an element of the atmosphere introduced by using these
fits instead of the original data is 0.02%. Fig. 3 shows
the number density profiles provided by the MSIS-E-90
model and the exponential fits used to model the data.
The XQC detector collected data for a total of 150
seconds. During these 150 seconds of activity, the thirty-
four individual calorimeters were not all operational at
all times. Furthermore, events that could not be ac-
curately measured by the calorimeters and events at-
tributed to cosmic rays hitting the base of the detec-
tor array were removed from the XQC spectrum, and
these cuts also contribute to the dead time of the sys-
tem. Specifically, events that arrived too close together
for the calorimeters to accurately measure distinct ener-
gies were discarded. This criterion removed 12% of the
observed events and the resulting loss of sensitivity was
http://modelweb.gsfc.nasa.gov/models/msis.html
FIG. 4: Top panel: The XQC energy spectrum from 0 - 4
keV in 5 eV bins. This spectrum does not have non-linearity
corrections applied (see Ref. [51]), so the calibration lines at
3312 eV and 3590 eV appear slightly below their actual en-
ergies. The cluster of counts to the left of each calibration
peak result from X-rays passing through the HgTe layer and
being absorbed in the Si components where up to 12% of the
energy may then be trapped in metastable states. Bottom
panel: The XQC energy spectrum from 0 - 2.5 keV in 5 eV
bins. This spectrum, combined with the over-saturation rate
of 0.6 events per second with energies greater than 4000 eV,
was used in our analysis.
included in the dead time of the calorimeters. When a
cosmic ray penetrates the silicon base of the detector ar-
ray, the resulting temperature increase is expected to reg-
ister as multiple, nearly simultaneous, low-energy events
on nearby calorimeters. To remove these events from the
spectrum, we cut out events that were part of either a
pair of events in adjacent detectors or a trio of events
in any of the detectors that arrived within 3 ms of each
other and had energies less than 2.5 keV. This procedure
was expected to remove more than 97% of the events that
resulted from cosmic rays hitting the base of the array.
Nearly all of the events attributed to heating from cosmic
rays had energies less than 300 eV, and a high fraction
of the observed low-energy events were included in this
cut. For example, seventeen of the observed twenty-four
events with energies less than 100 eV were removed. The
expected loss of sensitivity due to events being falsely
attributed to cosmic rays was included in the calculated
dead time of the calorimeters. Once all the dead time is
accounted for, the 150 seconds of data collection is equiv-
alent to 100.7 seconds of observation with all thirty-four
calorimeters operational.
The XQC calorimeters are capable of detecting energy
deposits that exceed 20 eV, but full sensitivity is not
reached until the energy surpasses 36 eV, and for approx-
imately half of the detection time, the detector’s lower
threshold was set to 120 eV. The calorimeters cannot re-
solve energies above 4 keV, and the 2.5-4 keV spectrum is
dominated by the detector’s interior calibration source:
a ring of 2µCi 41Ca that generates Kα and Kβ lines at
3312 eV and 3590 eV, respectively. We refer the reader
to Ref. [51] for a complete discussion of the calibration of
the detector. These limitations restrict the useful portion
of the XQC spectrum to 0.03-2.5 keV. This spectrum is
shown in Fig. 4, along with the full spectrum from 0-4
keV. The XQC field of view was centered on a region of
the sky known to have an enhanced X-ray background
in the 100-300 eV range, possibly due to hot gas in the
halo, and this surge in counts can be seen in Fig. 4. In
addition to the information present in this spectrum, we
know that the XQC detector observed an average over-
saturation event rate of 0.6 per second. This corresponds
to a total of 60 events that deposited more than 4000 eV
in a calorimeter. In Section IVB, we describe how we use
the observed spectrum between 29 eV and 2500 eV and
the integrated over-saturation rate to constrain the total
cross section for elastic scattering between dark matter
particles and nucleons.
III. DETECTING DARK MATTER
A. Incidence of dark matter particles
The expected flux of dark matter particles into the de-
tector depends on the density of the dark matter halo in
the Solar System. Unfortunately, the local dark matter
density is unknown and the range of theoretical predic-
tions is wide. By constructing numerous models of our
galaxy with various dark matter density profiles and halo
characteristics, rejecting those models that contradict ob-
servations, and finding the distribution of local dark mat-
ter densities in the remaining viable models, Ref. [55] pre-
dicted that the local dark matter density is between 0.3
and 0.7 GeV cm−3 assuming that the dark matter halo
is flattened, and the predicted local density decreases as
the halo is taken to be more spherical. Another approach
[56] used numerical simulations of galaxies similar to our
own to find the dark matter density profile and then fit
the profile parameters to Galactic observations, predict-
ing a mean local dark matter density between 0.18 GeV
cm−3 and 0.30 GeV cm−3. Given that it lies in the in-
tersection of these two ranges, we use the standard value
of 0.3 GeV cm−3 for the local dark matter density in
our primary analysis. This assumption ignores the possi-
ble presence of dark matter streams or minihalos, which
do occur in numerical simulations [56] and could lead to
local deviations from the mean dark matter density.
We also assume that the velocities of the dark matter
particles with respect to the halo are isotropic and have
a bounded Maxwellian distribution: the probability that
a particle has a velocity within a differential volume in
velocity-space centered around a given velocity ~v is
P (~v) =
d3~v if v ≤ vesc,
0 if v > vesc.
where v0 is the dispersion velocity of the halo, vesc is the
Galactic escape velocity at the Sun’s position, and k is a
normalization factor [57]:
k = (πv20)
. (3)
Numerical simulations indicate that dark matter particle
velocities may not have an isotropic Maxwellian distri-
bution [56]. Ref. [58] examines how assuming a more
complicated velocity distribution would alter the flux of
dark matter particles into an Earth-based detector.
Given the flat rotation curve of the spiral disk at the
Sun’s radius and beyond and assuming a spherical halo,
the local dispersion speed v0 is the maximum rotational
velocity of the Galaxy vc [59]. Reported values for the
rotational speed include 222±20 km s−1 [60], 228±19 km
s−1 [61], 184± 8 km s−1 [62] and 230 ± 30 km s−1 [63].
Recent measurements of the Galaxy’s angular velocity
have yielded values of Ωgal = 28 ± 2 km s−1 kpc−1 [64]
and 32.8 ± 2 km s−1 kpc−1 [65]. If the Sun is located
8.0 kpc from the Galactic center, these angular velocities
correspond to tangential velocities 224± 16 km s−1 and
262 ± 16 km s−1 respectively. We adopt vc = 220 ± 30
km s−1 as a centrally conservative value for the Galaxy’s
circular velocity at the Sun’s location.
The final parameter we need to obtain the dark mat-
ter’s velocity distribution is the escape velocity in the So-
lar vicinity. The largest observed stellar velocity at the
Sun’s radius in the Milky Way is 475 km s−1, which es-
tablishes a lower bound for the local escape velocity [66].
Ref. [67] used the radial motion of Carney-Latham stars
to determine that the escape velocity is between 450 and
650 km s−1 to 90% confidence, and Ref. [68] obtained a
90% confidence interval of 498 to 608 km s−1 from ob-
servations of high-velocity stars. A kinematic derivation
of the escape velocity [59] gives
v2esc = 2v
1 + ln
, (4)
where R0 is the distance from the Sun to the center of the
Galaxy, and Rgal is radius of the Galaxy. Observations of
other galaxies suggest that our galaxy extends to about
100 kpc [59], and observations of Galactic satellites indi-
cate that the Galaxy’s flat rotation curve extends to at
least 110 kpc [63]. The commonly accepted value for the
Solar radius is R0 = 8.0 kpc [69]. Recent measurements
include R0 = 7.9± 0.3 kpc [70] and R0 = 8.01± 0.44 kpc
[71], and a compilation of measurements over the past
decade [71] yields an average value of R0 = 7.80 ± 0.33
kpc. To estimate the escape velocity, we use 100 kpc as
a conservative estimate of the Galactic radius and the
standard value R0 = 8.0 kpc. These parameters, com-
bined with vc = 220 km s
−1, predict an escape velocity
of 584 km s−1, which falls near the middle of the ranges
proposed in Refs. [67, 68].
The isotropic Maxwellian velocity distribution given
by Eq. (2) specifies the dark matter particles’ motion
relative to the halo. However, we are interested in their
motion relative to the XQC detector: ~vobserved = ~vdm −
~vdetector where the latter two velocities are measured with
respect to the halo. The velocity of the detector with
respect to the halo has three components: the velocity of
the Sun relative to halo, the velocity of the Earth with
respect to the Sun, and the velocity of the detector with
respect to the Earth.
When discussing these velocities, it is useful to de-
fine a Galactic Cartesian coordinate system. In Galactic
coordinates, the Sun is located at the origin, and the
xy-plane is defined by the Galactic disk. The x-axis
points toward the center of the Galaxy, and the y-axis
points in the direction of the Sun’s tangential velocity
as it revolves around the Galactic center. The z-axis
points toward the north Galactic pole and is antiparal-
lel to the angular momentum of the rotating disk. The
motion of the Sun through the halo has two compo-
nents. First, there is the Sun’s rotational velocity as
it orbits the Galactic center: vc in the y direction. Sec-
ond, there is the motion of the Sun relative to the spiral
disk [72]: ~v⊙ = (10.00 ± 0.36, 5.25 ± 0.62, 7.17 ± 0.38)
km s−1 in Galactic Cartesian coordinates. When the
Earth’s motion through the Solar System during its an-
nual orbit of the Sun is expressed in Galactic coor-
dinates [57], the resulting velocity at the time of the
XQC experiment (7.3 days after the vernal equinox) is
~vEarth = (29.14, 5.330,−3.597) km s−1.
The final consideration is the velocity of the detector
relative to the Earth. The maximum velocity attained by
the XQC rocket was less than 1.2 km s−1. This velocity is
insignificant compared to the motion of the Sun relative
to the halo. Moreover, the XQC detector collected data
while the rocket rose and while it fell, and the average
velocity of the rocket was only 0.104 km s−1. Therefore,
we neglect the motion of the rocket in the calculation of
the dark matter wind. Combining the motion of the Sun
and the Earth then gives the total velocity of the XQC
detector with respect to the halo during the experiment
in Galactic Cartesian coordinates: ~vdetector = (39.14 ±
0.36, 230.5 ± 30, 3.573 ± 0.38) km s−1. Subtracting the
velocity vector of the detector relative to the halo from
the velocity vector of the dark matter relative to the halo
gives the dark matter’s velocity relative to the detector in
Galactic coordinates. However, we want the dark matter
particles’ velocities in the coordinate frame defined by
the detector, where the z-axis is the field-of-view vector.
The XQC field of view was centered on l = 90◦, b = +60◦
in Galactic latitude and longitude [51], so the rotation
from Galactic coordinates to detector coordinates may
be described as a clockwise 30◦ rotation of the z-axis
around the x-axis, which is taken to be the same in both
coordinate systems.
B. Dark Matter Interactions
Calorimetry measures the kinetic energy transferred
from the dark matter to the absorbing material without
regard for the specific mechanism of the scattering or
any other interactions. Consequently, the dark matter
detection rate for a calorimeter depends only on the mass
of the dark matter particle and the total cross section for
elastic scattering between the dark matter particle and an
atomic nucleus of mass number A, which is proportional
to the cross section for dark matter interactions with a
single nucleon (σDn). The calorimeter measures the recoil
energy of the target nucleus (mass mT),
Erec =
2mTmdm
(mT +mdm)2
(1 − cos θCM), (5)
where mdm and vdm are dark matter particle’s mass and
velocity prior to the collision in rest frame of the target
nucleus and θCM is the scattering angle in the center-of-
mass frame.
If the momentum transferred to the nucleus, q2 =
2mTErec, is small enough that the corresponding de
Broglie wavelength is larger than the radius R of the
nucleus (qR ≪ ~), then the scattering is coherent. In co-
herent scattering, the scattering amplitudes for each in-
dividual component in the conglomerate body are added
prior to the calculation of the cross section, so the total
cross section is proportional to the square of the mass
number of the target nucleus. Including kinematic fac-
tors [45, 73], the cross section for coherent scattering off
a nucleus is given by
σcoh(A) = A
mred(DM,Nuc)
mred(DM, n)
σDn, (6)
where mred(DM,Nuc) is the reduced mass of the nucleus
and the dark matter particle, mred(DM, n) is the reduced
mass of a nucleon and the dark matter particle, and A is
the mass number of the nucleus. Coherent scattering is
isotropic in the center-of-mass frame of the collision.
Dark matter particles may be massive and fast-moving
enough that the scattering is not completely coherent
when the target nucleus is large [74]. When the scat-
tering is incoherent, the dark matter particle “sees” the
internal structure of the nucleus, and the cross section for
scattering is reduced by a “form factor,” which is a func-
tion of the momentum transferred to the nucleus during
the collision (q) and the nuclear radius (R):
F 2(q, R). (7)
Since q depends on the recoil energy, which in turn de-
pends on the scattering angle, incoherent scattering is
not isotropic.
In this discussion of coherence, we have neglected the
possible effects of the dark matter particle’s internal
structure by assuming that σDn is independent of recoil
energy. If the dark matter particle is not point-like then
σDn decreases as the recoil momentum increases due to a
loss of coherence within the dark matter particle. Inco-
herence within the dark matter particle has observational
consequences [75], but these effects depend on the size of
the dark matter particle. To avoid restricting ourselves
to a particular dark matter model, we assume that the
dark matter particle is small enough that nucleon scat-
tering is always coherent; when we discuss incoherence,
we are referring to the effects of the nucleus’s internal
structure.
According to the Born approximation, the form fac-
tor for nuclear scattering defined in Eq. (7) is the
Fourier transform of the nuclear ground-state mass den-
sity [57, 76]. The most common choice for the form factor
[74, 77] is F 2(q, R) = exp[−(qRrms)2/(3~2)], where Rrms
is the root-mean-square radius of the nucleus. For a solid
sphere, R2rms = (3/5)R
2, so this form factor is equivalent
to the form factor used in Ref. [52]. This form factor is
an accurate approximation of the Fourier transform of
a solid sphere for (qR)/~ ∼< 2, but it grossly underesti-
mates the reduction in σ for larger values of q [57]. The
maximum speed of a dark matter particle with respect
to the XQC detector is ∼ 800 km s−1 (escape velocity
+ detector velocity), and at that speed, the maximum
possible value of qR/~ for a collision with a Hg nucleus
(A = 200) is nearly ten for a 100 GeV dark matter parti-
cle, and the maximum possible value of qR/~ increases as
the mass of the dark matter particle increases. Clearly,
this approximation is not appropriate for a large portion
of the dark matter parameter space probed by the XQC
experiment.
Furthermore, a solid sphere is not a very realistic model
of the nucleus. A more accurate model of the nuclear
mass density is ρ(r) =
d3r′ρ0(r
′)ρ1(r− r′), where ρ0 is
constant inside a radius R20 = R
2 − 5s2 and zero beyond
that radius and ρ1 = exp[−r2/(2s2)], where s is a “skin
thickness” for the nucleus [78]. The resulting form factor
F (q, R) = 3
sin(qR/~)− (qR/~) cos(qR/~)
(qR/~)3
× exp
(qs/~)2
. (8)
We follow Ref. [57] in setting the parameters in Eq. (8):
s = 0.9 fm and
R2 = [(1.23A1/3 − 0.6)2 + 0.631π2 − 5s2] fm2, (9)
where A is the mass number of the target nucleus.
Despite its simple analytic form, the form factor given
by Eq. (8) is computationally costly to evaluate repeat-
edly. We use an approximation:
F 2 =
0.9 fm
if qR
9(0.81)
(qR/~)4
0.9 fm
if qR
The low-q approximation combines the standard approx-
imation for the solid sphere with the factor accounting
for the skin depth of the nucleus. The high-q approxi-
mation was derived from the asymptotic form of the first
spherical Bessel function and normalized so that the to-
tal cross section is as close as possible to the exact result.
The error in the total cross section due to the use of the
approximation is less than 1% for nearly all dark mat-
ter masses; the sole exception is mdm ∼ 10 − 100 GeV,
and even then the error is less than 5%. Unless other-
wise noted, we use this approximation for the form factor
throughout this analysis. We also assume that the dark
matter particle does not interact with nuclei in any way
other than elastic scattering.
IV. ANALYSIS OF XQC CONSTRAINTS
To obtain an accurate description of the XQC experi-
ment’s ability to detect strongly interacting dark matter
particles, we turned to Monte Carlo simulations. The
Monte Carlo code we wrote to analyze the XQC experi-
ment simulates a dark matter particle’s journey through
the atmosphere to the XQC detector chamber, its path
through the detector chamber to a calorimeter, and its in-
teraction with the sensitive components of the calorime-
ter. This latter portion of the code also records how much
energy the particle deposits in the calorimeter through
scattering. The results of several such simulations for the
same set of dark matter properties may be used to pre-
dict the likelihood that a given dark matter particle will
deposit a particular amount of energy into the calorime-
ter. These probabilities of various energy deposits predict
the recoil-energy spectrum the XQC detector would ob-
serve if the dark matter particles have a given mass and
nucleon-scattering cross section. This simulated spec-
trum may then be compared to the XQC data to find
which dark matter parameters are excluded by the XQC
experiment.
A. Generating Simulated Energy-Recoil Spectra
The basic subroutine in our Monte Carlo algorithm
is the step procedure. The step procedure begins with
a particle with a certain velocity vector and position in
a given material and moves the particle a certain dis-
tance in the material, returning its new position and ve-
locity. The step procedure also determines whether or
not a scattering event occurred during the particle’s trek
and updates the velocity accordingly. The number of ex-
pected collisions in a step of length l through a material
with target number density n is n× σtot × l, where σtot
is the total scattering cross section obtained by integrat-
ing Eq. (7) over the scattering angle, or equivalently, the
recoil momentum q:
σtot =
q2max
F 2(q, R) dq2, (11)
where qmax is the maximum possible recoil momentum.
The step length l is chosen so that it is at most a tenth
of the mean free path through the material, so the num-
ber of expected collisions is less than one and represents
the probability of a collision. After each step, a ran-
dom number between zero and one is generated using
the “Mersenne Twister” (MT) algorithm [79] and if that
random number is less than the probability of a collision,
the particle’s energy and trajectory are updated. First, a
recoil momentum is selected according to the probability
distribution P (q2) = F 2(q, R)σcoh/(q
maxσtot), where the
exact form factor is used for qR/~ > 2 so that the oscilla-
tory nature of the form factor is not lost. The recoil mo-
mentum determines the recoil energy and the scattering
angle in the center-of-mass frame through Eq. (5). The
scattering is axisymmetric around the scattering axis, so
the azimuthal angle is assigned a random value between
0 and 2π. The scattering angles are used to update the
particle’s trajectory, and its speed is decreased in accor-
dance with the kinetic energy transferred to the target
nucleus. The step subroutine repeats until the particle
exits the simulation, or its kinetic energy falls below 0.1
eV, or the energy deposited in the calorimeter exceeds
the saturation point of 4000 eV.
Our simulation treats the atmosphere as a 4.6×4.6 cm
square column with periodic boundary conditions, the
bottom face of which covers the top of the conical detec-
tor chamber described in Section II. This implementation
assumes that for every particle that exits one side of the
column, there is a particle that enters the column from
the opposite side with the same velocity. The infinite
extent of the atmosphere and its translational invariance
makes this assumption reasonable. The atmosphere col-
umn extends to an altitude 1000 km; increasing the at-
mosphere height beyond 1000 km has a negligible effect
on the total number of collisions in the atmosphere. The
simulation begins with a dark matter particle at the top
of the atmosphere column at a random initial position
on the 4.6×4.6 cm square. Its initial velocity with re-
spect to the dark matter halo is selected according to the
isotropic Maxwellian velocity distribution function given
by Eq. (2), and then the velocity relative to the detector
is found via the procedure described in Section IIIA.
The dark matter particle’s path from the top of the at-
mosphere to the detector is modeled using the step pro-
cedure described above. The simulation of the particle’s
interaction with the atmosphere ends if the particle’s al-
titude exceeds 1000 km or if the particle falls below the
height of the XQC rocket. We use the time-averaged alti-
tude (201.747 km) as the constant altitude of the rocket.
We made this simplification because it allows us to ig-
nore the periodic inactivity of each calorimeter and treat
the detector as thirty-four calorimeters that are active for
100.7 seconds. When the dark matter particle hits the
rocket, its path through the five filter layers is also mod-
eled using the step procedure, as is its path through the
calorimeters. In addition to being smaller than the mean
free path, the step length is chosen so that the particle’s
position relative to the boundaries of the detectors is ac-
curately modeled. The simulation ends when the dark
matter particle’s random-walk trajectory takes it out of
the detector chamber. As mentioned in Section II, the
calorimeter detects the sum of all the recoil energies if
the dark matter particle is scattered multiple times.
When the dark matter particle is unlikely to experience
more than one collision in the calorimeter, this simula-
tion is far more detailed than is required to accurately
predict the energy deposited by the dark matter parti-
cle. This is the case for the lightest (mdm ≤ 102 GeV)
and weakest-interacting (σDn ≤ 10−26 cm2) dark mat-
ter particles that the XQC calorimeters are capable of
detecting. Since the lightest dark matter particles are
also the most numerous, many Monte Carlo trials are
required to sample all the possible outcomes of a dark
matter particle’s encounter with the detector. The sim-
ulation described above is too computationally intensive
to run that many trials, so we used a faster and simpler
simulation to model the interactions of these dark matter
particles. This simulation assumes that the particle will
experience at most one collision in the atmosphere and
at most one collision in each filter layer and each layer of
the calorimeter. The simulation ends if the probability
of two scattering events in either the atmosphere or any
of the filter layers exceeds 0.1. Instead of tracking the
dark matter particle’s path through the atmosphere, the
total overburden for the atmosphere is used to determine
the probability that the dark matter particle scatters in
the atmosphere, and the particle only reaches the detec-
tor if its velocity vector points toward the detector after
the one allowed scattering event. Also, instead of the
small step lengths required to accurately model the ran-
dom walk of a strongly interacting particle, each layer is
crossed with a single step. These simplifications reduce
the runtime of the simulation by a factor of 100, making
it possible to run 1010 trials in less than one day.
B. Comparing the Simulations to the XQC Data
In order to compare the probability spectra produced
by our Monte Carlo routine to the results of the XQC
experiment, we must multiply the probabilities by the
number of dark matter particles that are encountered by
the initial surface of the Monte Carlo routine. When the
initial velocity of the dark matter particle is chosen, the
initial velocity may point toward or away from the detec-
tor; in the latter case, the trial ends immediately. Con-
FIG. 5: Simulated event spectra for dark matter particles with
masses of 1, 10 and 100 GeV and a total nucleon-scattering
cross section of 10−27.3 cm2. In addition to the events de-
picted in these spectra, the simulations predict 1300 ± 160
events with energies greater than 4000 eV when mdm = 10
GeV and 10, 000 ± 1200 such events when mdm = 100 GeV.
The histogram represents the XQC observations.
sequently, the Monte Carlo probability that the particle
deposits no energy in the calorimeter already includes the
probability that the dark matter particle does not have a
halo trajectory that takes it into the atmosphere. There-
fore, the probabilities resulting from the Monte Carlo
routine should be multiplied by the number of particles
in the volume swept out by the initial 4.6×4.6 cm2 square
surface during the 100.7f(E) seconds of observation time,
where f is the fraction of the observing time that the
XQC detector was sensitive to deposits of energy E. For
energies between 36 and 88 eV, f is 0.5083, and the value
of f increases to one over energies between 88 and 128 eV.
The detector was also slightly sensitive to lower energies:
between 29 and 35 eV, f increases from 0.3815 to 0.5083.
The normal of the initial surface points along the detec-
tor’s field of view, and the surface moves with the detec-
tor; using the detector velocity given in Section IIIA, the
number of dark matter particles encountered by the ini-
tial surface is Ndm = f × (ρdm/mdm)× [(2.5± 0.3)× 1010
cm3], where ρdm is the local dark matter density.
The simulated event spectra produced by our Monte
Carlo routine indicate that particles with masses less
than 1 GeV very rarely deposit more than 100 eV inside
the XQC calorimeters. Conversely, particles with masses
greater than 100 GeV nearly always deposit more than
4000 eV when they interact with the XQC calorimeters,
so constraints on σDn for these mdm values arise from the
FIG. 6: Simulated event spectra for 10-GeV dark mat-
ter particles with total nucleon-scattering cross sections of
10−21.6, 10−27.3 and 10−28.3 cm2. In addition to the events
depicted in these spectra, the simulations predict 140 ± 37
events with energies greater than 4000 eV when σDn = 10
−21.6
cm2, 1300 ± 160 such events when σDn = 10
−27.3 cm2, and
120±15 such events when σDn = 10
−28.3 cm2. The histogram
represents the XQC observations.
over-saturation (E ≥ 4000 eV) event rate. Fig. 5 shows
simulated spectra for three mdm values that lie between
these two extremes, along with a histogram that depicts
the XQC observations. Given an initial velocity of 300
km s−1 relative to the XQC detector, a 1-GeV particle
can only deposit up to 66 eV in a single collision with an
Si nucleus, so the spectrum for these particles is confined
to very low energies. Meanwhile, a 10-GeV particle and
a 100-GeV particle with the same initial velocity can de-
posit up to 900 eV and 44,000 eV, respectively, in a single
collision with an Hg nucleus. In fact, ignoring any loss
of coherence, all recoil energies between 0 and 44,000 eV
are equally likely during a collision between an Hg nu-
cleus and a 100-GeV dark matter particle. That’s why
the mdm = 100 GeV spectrum in Fig. 5 is flat below 2500
eV and why the simulations predict 10,000 events with
energies greater than 4000 eV for this value of mdm and
Fig. 6 shows how changing the total cross section for
elastic scattering off a nucleon affects the simulated spec-
tra generated by our Monte Carlo routine for a single
dark matter particle mass (mdm = 10 GeV). We see that
increasing σDn from 10
−28.3 cm2 to 10−27.3 cm2 increases
all of the counts by a factor of ten but leaves the ba-
sic shape of the spectrum unchanged. For much larger
values of σDn, however, the particle loses a considerable
Energy Range (eV) Counts Energy Range (eV) Counts
29 - 36 0 945 - 1100 31
36 - 128 11 1100 - 1310 30
128 - 300 129 1310 - 1500 29
300 - 540 80 1500 -1810 32
540 - 700 90 1810 - 2505 15
700 - 800 32 ≥ 4000 60
800 - 945 48
TABLE I: The binned XQC results used for comparison with
our Monte Carlo simulations.
amount of its energy while traveling through the atmo-
sphere. Consequently, high-energy recoil events become
less frequent, as shown by the spectrum for σDn = 10
−21.6
cm2. For larger values of σDn, too much energy is lost in
the atmosphere for the particle to be detectable by the
XQC experiment.
When comparing the simulated measurements to the
XQC data, we group the events into the thirteen energy
bins given in Table I. We generally use large bins be-
cause it reduces the fractional error in the probabilities
generated by our Monte Carlo routine by increasing the
probability of each bin: δpi/pi = 1/
pit, where t is the
number of trials and pi is the probability of an energy
deposit in the ith bin. Given that the number of trials is
limited by runtime constraints, increasing the bin size is
often the only way to obtain bin probabilities with δpi/pi
values much less than one. When choosing our binning
scheme, we attempted to maximize bin size while pre-
serving as many features of the observed spectrum as
possible. We also grouped all energies for which f 6= 1
into two bins; we ignore the variation in f within these
bins and set f = 0.3815 in the lowest-energy bin and
f = 0.5083 in the next-to-lowest bin.
Unfortunately, we do not know the number of X-ray
events in any of the bins listed in Table I. We considered
using a model to subtract off the X-ray background but,
given any model’s questionable accuracy, we decided not
to use it in our analysis. Our ignorance of the X-ray
background forces us to treat the number of observed
counts in each bin as an upper limit on the number of
dark matter events in that energy range. Consequently,
we define a parameter X2 that measures the extent of
the discrepancy between the simulated results for a given
mdm and σDn and the XQC observations while ignoring
bins in which the observed event count exceeds the pre-
dicted contribution from dark matter:
i=# of Bins
(Ei − Ui)2
with Ui < Ei
, (12)
where Ei = Ndm × pi is the number of counts in the ith
bin predicted by the Monte Carlo simulation and Ui is
the number of observed counts in the same bin. We use
a second Monte Carlo routine to determine how likely it
is that a set of observations would give a value of X2 as
large or larger than the one derived from the XQC data
given a mean signal described by the set of Ei derived
from the simulation Monte Carlo.
In the comparison Monte Carlo, a trial begins by gen-
erating a new set of Ei by sampling the error distribu-
tions of Ndm and pi. The distribution of Ndm values is
assumed to be Gaussian with the mean and standard de-
viation given above. The probability pi is derived from
pi× t events in the simulation Monte Carlo (recall that t
is the number of trials), so a new value for pi is generated
by sampling a Poisson distribution with a mean of pi × t
and dividing the resulting number by t. Once a new set
of Ei has been found, the routine generates a simulated
number of observed counts for each bin according to a
Poisson distribution with a mean of Ei. The value of the
X2 parameter for the new Ei and Ui is computed and
compared to the value for the original Ei and the XQC
observations, X2XQC. The number of trials needed to ac-
curately measure the probability P(X) thatX2 ≥ X2XQC
is determined by requiring that the variation in the mean
value of X2 over ten Monte Carlo simulations does not
exceed (100-C)%, where C% is the desired confidence
level and that the range P(X)±(5× the variation in P(X))
does not contain (100− C)/100.
V. RESULTS AND DISCUSSION
The XQC experiment rules out the enclosed region in
(mdm, σDn) parameter space shown in Fig. 7. The over-
burden from the atmosphere and the filtering layers as-
sures that there will be a limit to how strongly a dark
matter particle can interact with baryons and still reach
the XQC calorimeters; this overburden is responsible for
the top edge of the exclusion region. Conversely, if σDn is
too small, the dark matter particles will pass through the
calorimeters without interacting. The low-energy thresh-
old of the XQC calorimeters places a lower bound on the
excluded dark matter particle masses; ifmdm is too small,
then the recoil energies are undetectable. On the other
side of the mass range, the XQC detector is not sensitive
to mdm ∼> 105 GeV because the number density of such
massive dark matter particles is too small for the XQC
experiment to detect.
The exclusion region shown in Fig. 7 has a complicated
shape, but its features are readily explicable. As mdm in-
creases, the range of excluded σDn values shifts to lower
values and then moves up again. The downward shift for
mdm between 0.1 GeV and 100 GeV is due to the effects of
coherent nuclear scattering. Since σcoh increases with in-
creasing mdm for fixed σDn, a 100-GeV particle interacts
more strongly in the atmosphere and in the detector than
a 1-GeV particle with the same σDn. Consequently, both
the upper and lower boundaries of the excluded region
decrease with increasing mass for mdm ∼< 100 GeV. The
scattering of dark matter particles with larger masses is
incoherent, and the form factor discussed in Section III B
FIG. 7: The region of dark matter parameter space excluded
by the XQC experiment; σDn is the total cross section for scat-
tering off a nucleon and mdm is the mass of the dark matter
particle. This exclusion region follows from the assumption
that the local dark matter density is 0.3 GeV cm−3 and that
all of the dark matter shares the same value of σDn.
causes σtot to decrease as mass increases for fixed σDn.
Moreover, particles that are more massive than the target
nuclei have straighter trajectories than lighter dark mat-
ter particles due to smaller scattering angles in the de-
tector rest frame. The loss of coherence also contributes
because incoherent scattering makes small scattering an-
gles more probable. A straight trajectory is shorter than
a random walk, so the more massive particles interact
less in the atmosphere and the detector than the more
easily-deflected lighter particles. Due to both of these
effects, the upper and lower boundaries of the exclusion
region increase with increasingmdm for mdm ∼> 100 GeV.
The lower left corner of the exclusion region also has
two interesting features. First, the lower bound on the
excluded value of σDn decreases sharply as mdm increases
from 0.1 GeV to 0.5 GeV. A dark matter particle with the
maximum possible velocity with respect to the detector
(800 km s−1) must have a mass greater than 0.24 GeV
to be capable of depositing 29 eV in the calorimeter in
a single collision. Lighter particles are only detectable
if they scatter multiple times inside the calorimeter, and
multiple scatters require a higher value of σDn. Since
their analysis does not allow multiple collisions, the XQC
exclusion region found in Ref. [52] does not extend to
masses lower than 0.3 GeV for any value of σDn. Second,
there is a kink in the lower boundary at mdm = 10 GeV;
the constraint on σDn is not as strong for this mass. The
FIG. 8: The region of dark matter parameter space excluded
to 90% confidence by the XQC experiment for several values of
the local density of dark matter with a total nucleon scattering
cross section σDn and mass mdm. The four densities shown
are 0.3 GeV cm−3 (solid line), 0.15 GeV cm−3 (long dashed
line), 0.075 GeV cm−3 (short dashed line) and 0.03 GeV cm−3
(dotted line).
simulated spectra produced by our Monte Carlo routine
for mdm = 10 GeV and σDn ∼< 10−25 cm2 reveal that the
particle is most likely to deposit between 100 and 600
eV, as exemplified by the spectra depicted in Fig. 6. The
background in this energy range is very high, so the XQC
constraints are not as strict at these energies.
Altering the local density of dark matter that strongly
interacts with baryons changes the exclusion region.
Fig. 8 shows the 90%-confidence exclusion regions for
four values of the local density of dark matter particles
with strong baryon interactions: 0.3 GeV cm−3 (solid
line), 0.15 GeV cm−3 (long dashed line), 0.075 GeV cm−3
(short dashed line) and 0.03 GeV cm−3 (dotted line).
These different local densities could arise due to varia-
tions in the local dark matter density due to mini-halos
or streams. They also describe models where the dark
matter does not consist of a single particle species and
the dark matter that strongly interacts with baryons is
a fraction fd of the local dark matter. In that case, the
four exclusion regions in Fig. 8 correspond to fd =1, 0.5,
0.25, and 0.1.
Fig. 8 indicates that the top and left boundaries of the
XQC exclusion region are not highly sensitive to the dark
matter density. In particular, the upper left corner of the
exclusion region (0.01 ≤ mdm ≤ 0.1 GeV) is nearly un-
affected by lowering the dark matter density. This con-
sistency indicates our Monte Carlo-generated exclusion
region is smaller than the true exclusion region in this
corner. If the dark matter is light (mdm ∼< 0.1 GeV),
then the number of dark matter particles encountered by
the XQC detector is very large (Ndm ∼> 7 × 1010). As
previously mentioned, the upper left corner of the XQC
exclusion region results from multiple scattering events,
so the simpler version of our Monte Carlo code described
in Section IVA is not applicable. Consequently, it is not
possible to run more than 109 trials in a week, so each
scattering event in the simulation corresponds to more
than one scattering event in the detector for all the den-
sities shown in Fig. 8. Therefore, decreasing the density
does not change the result. If it were possible to run 1011
trials, then the upper left corner of the exclusion region
would expand and differences between the different den-
sity contours would emerge. Since the upper left corner
of the XQC exclusion region is already ruled out by astro-
physical constraints (see Fig. 9), we have not invested in
the computational time necessary to expand this corner.
The upper boundary of the exclusion region is also not
greatly affected by decreasing the particle density, even
when Ndm is small enough that the Monte Carlo routine
is capable of running more than Ndm trials (mdm ≥ 100
GeV). This robustness indicates that the overburden of
the XQC experiment effectively prevents all dark matter
particles with σDn values greater than the upper bound
of the exclusion region from reaching the detector, so
it does not matter how many particles are encountered.
Finally, as discussed previously, the lower portion of the
exclusion region’s left boundary (σDn ≤ 10−23 cm2) is
set by the energy threshold for detection and is therefore
independent of Ndm.
Examining the features of the excluded region allows
us to predict how the region may be expanded by a fu-
ture XQC-like experiment. Decreasing the overburden by
either increasing the rocket’s altitude or reducing the fil-
tering will push the top boundary of the excluded region
upwards. Decreasing the energy detection threshold will
extend the excluded region to lower masses. It may also
extend the exclusion region to higher values of σDn for all
masses since strongly interacting particles lose much of
their energy in the atmosphere and arrive at the calorime-
ter with too little energy to produce a detectable sig-
nal. Increasing the size or number of calorimeters would
increase the sensitivity and extend the excluded region
to lower values of σDn. Finally, increasing the observa-
tion time would increase Ndm, and that would extend the
right and bottom boundaries of the excluded region.
VI. CONCLUSION
The X-ray Quantum Calorimetry (XQC) experiment is
a powerful detector of dark matter that interacts strongly
with baryons due to its high altitude and minimal shield-
ing. The XQC measurements rule out a large range of
hitherto unconstrained dark matter masses and scatter-
FIG. 9: Plot of the scattering cross section for dark matter particles and nucleons (σDn) versus dark matter particle mass
(mdm) showing the new XQC limits along with other current experimental limits. The red XQC exclusion region is the same
as shown in Fig. 7, and the other experiments are discussed in the text. The dark gray region shows the maximal range of
dark matter self-interaction cross section consistent with the strongly self-interacting dark matter model of structure formation
[37, 38]. The square marks the value of the scattering cross section for neutron-nucleon interactions.
ing cross sections. The excluded range was first derived
in Refs. [38, 48] based on rough analytic estimates. In
this paper, we have improved upon these results using
detailed Monte Carlo simulations to predict how a dark
matter particle of a given mass and cross section for nu-
cleon scattering would interact with the XQC calorime-
ters. Unlike Ref. [52], our analysis includes the atmo-
sphere and the shielding of the detector, so our result in-
cludes the upper limit on excluded σDn values, which had
not yet been accurately determined. Our simulation also
models the internal geometry of the XQC detector and
the random walk of particles through it, which is not pos-
sible using the analytical approaches of Refs. [38, 48, 52].
The resulting exclusion region is significantly different
than its analytical predecessors. When multiple scatter-
ings are included, the XQC experiment is sensitive to
dark matter particles with masses below 0.3 GeV and
cross sections for nucleon scattering between 10−24 and
10−20 cm2. Unlike Ref. [52], we find that the XQC ex-
clusion region does not include σDn < 10
−29 cm2 for
dark matter masses less than 10 GeV. Ref. [52] obtained
a more restrictive upper bound because they assumed a
specific X-ray background while we treat all events as po-
tential dark matter interactions. At higher masses, the
lower boundary of our exclusion region is much higher
than in Refs. [38, 48] because they over-estimated the
XQC sensitivity by assuming coherent scattering. It also
appears that Refs. [38, 48] underestimated the atmo-
spheric and shielding overburden for the XQC detector
because our exclusion region does not extend to values of
σDn as large as those included in their exclusion region.
We also assume a lower local dark matter density than
Refs. [38, 48] (0.3 instead of 0.4 GeV cm−3), so some of
the shrinkage of the exclusion region may be attributed
to the reduction in the assumed number density of dark
matter particles.
Fig. 9 shows how the XQC exclusion region depicted in
Fig. 7 complements the exclusion regions from other ex-
periments that are sensitive to similar values of σDn and
mdm. For a summary of some of the other experimen-
tal constraints as of 1994, see Ref. [46]. The constraints
to σDn from Pioneer 11 [80], Skylab [81], and IMP7/8
[82] were interpreted by Refs. [38, 46, 48]. There have
been two balloon-borne searches for dark matter, the
IMAX experiment [46, 47] and the Rich, Rocchia & Spiro
(RRS) [83] experiment. Although underground detectors
are designed to detect WIMPs, DAMA [84, 85] does ex-
clude σDn values within the range of interest, and relevant
constraints may be derived from Edelweiss (EDEL) and
CDMS [86, 87].
All of the exclusion regions shown in Fig. 9 were de-
rived assuming that all the dark matter is strongly in-
teracting. A local dark matter density of 0.4 GeV cm−3
was assumed in the analysis of the exclusion regions from
Pioneer 11, Skylab and the RRS experiment, while all
the other exclusion regions were derived assuming a lo-
cal dark matter density of 0.3 GeV cm−3. Furthermore,
the derivations of all the shown exclusion regions other
than the XQC region and the EDEL+CDMS region as-
sume that the scattering between dark matter particles
and nuclei is coherent. Therefore, these exclusion regions
are likely too broad because they over-estimate the cross
section for nuclear scattering. A comparison of the XQC
exclusion region reported in Refs. [38, 48] and our exclu-
sion region indicates that assuming coherent scattering
extends the exclusion region for mdm ≥ 1000 GeV to
σDn values that are roughly A× smaller than the lower
boundary of our exclusion region, where A is the mass
number of the largest target nucleus.
Fig. 9 also shows the bound on σDn from the CMB
and large-scale structure (LSS) obtained when one as-
sumes prior knowledge of the Hubble constant H0 and
the cosmic baryon fraction (from BBN) [50]. This bound
is nominally stronger than the bound from disk stabil-
ity [45], but it is less direct in that it requires combining
different measurements and depends on the cosmologi-
cal model; consequently, we show both bounds in Fig. 9.
Measurements of primordial element abundances give an
upper limit of σDn/mdm ∼< 4 × 10−16 cm2 GeV−1 [49].
Since this upper bound lies well beyond the upper bound
from disk stability, we do not include it in Fig. 9. We
also do not display the constraints from cosmic rays [49]
because they are derived from inelastic interactions that
are model-dependent.
As shown in Fig. 9, the XQC experiment rules out
a wide region of (mdm, σDn) parameter space that was
not probed by prior dark matter searches. Of partic-
ular interest is the darkly shaded range of σDn values
that corresponds to the maximal range of dark matter
self-interaction cross sections consistent with the strongly
self-interacting dark matter model of structure formation
[37, 38]. If the dark matter consists of exotic hadrons
whose interactions with nucleons are comparable to their
self-interactions, then σDn for these particles would lie in
or near the darkly shaded region in Fig. 9. Previous esti-
mates of the XQC exclusion region [38, 48] indicated that
the XQC experiment rules out all the darkly shaded σDn
values for 1 ∼< mdm ∼< 104 GeV. Our analysis reveals that
this is not the case; portions of the darkly shaded region
for mdm ∼> 20 GeV are not excluded by the XQC ex-
periment, although they are ruled out by observations of
LSS and the CMB. The mass-σ combination correspond-
ing to nucleon-neutron scattering (the square in Fig. 9)
lies within the exclusion region of the XQC experiment,
and the only portion of the darkly shaded region that is
unconstrained corresponds to dark matter masses smaller
than 0.25 GeV.
It is important to note, however, that the cross section
for dark matter self-interactions need not be comparable
to the cross section for nucleon scattering; σDn could dif-
fer by a few orders of magnitude from the self-interaction
cross section (as is the case for Q-balls). Furthermore, no
interactions with baryons are required for self-interacting
dark matter to resolve the tension between the collision-
less dark matter model and observations of small-scale
structure.
Another XQC detector is scheduled to launch in the
upcoming year. This experiment will have twice the ob-
serving time of the XQC experiment used in this analy-
sis. As discussed in Section V, increasing the observing
time will extend the exclusion region to higher masses
and weaker interactions. The future XQC experiment
will also have a lower energy threshold (15 eV) and will
maintain sensitivity to all energies above this threshold
throughout the run. The increased sensitivity to low en-
ergies will shift the lower (σDn ≤ 10−23 cm2) left bound-
ary of the exclusion region to lower masses. A lower en-
ergy threshold of 15 eV will make the experiment sensi-
tive to single recoil events involving dark matter particles
more massive than 0.17 GeV, as discussed in Section V.
Clearly, the next-generation XQC experiment will be an
even more powerful probe of interactions between dark
matter particles and baryons than its predecessor.
Acknowledgments
A. L. E. would like to thank Robert Lupton and
Michael Ramsey-Musolf for useful discussions. D. M.
thanks the Wallops Flight Facility launch support team
and the many undergraduate and graduate students that
made this pioneering experiment possible. The authors
also thank Randy Gladstone for his assistance with the
atmosphere model. A. L. E. acknowledges the support
of an NSF Graduate Fellowship. P. J. S. is supported
in part by US Department of Energy grant DE-FG02-
91ER40671. P. C. M. acknowledges current support for
this project from a Robert M. Walker Senior Research
Fellowship in Experimental Space Science from the Mc-
Donnell Center for the Space Sciences, as well as prior
institutional support for this project from the Instituto
Nacional de Técnica Aeroespacial (INTA) in Spain, from
the University of Bielefeld in Germany, and from the Uni-
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