aainstitutetext: Stanford Institute for Theoretical Physics, Stanford
University, Stanford, CA 94305, USAbbinstitutetext: The Oskar Klein Centre for
Cosmoparticle Physics, Department of Physics, Stockholm University, Alba Nova,
10691 Stockholm, Swedenccinstitutetext: Department of Physics, University of
Texas, Austin, TX 78722, USAddinstitutetext: Nordita, KTH Royal Institute of
Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm,
Swedeneeinstitutetext: Scuola Internazionale Superiore di Studi Avanzati
(SISSA), via Bonomea 265, 34136 Trieste, Italyffinstitutetext: INFN, Sezione
di Trieste, via Valerio 2, 34127 Trieste, Italygginstitutetext: Institute for
Fundamental Physics of the Universe (IFPU), via Beirut 2, 34151 Trieste, Italy

# New Projections for Dark Matter Searches with Paleo-Detectors

Sebastian Baum b Thomas D. P. Edwards c,d,b Katherine Freese e,f,g Patrick
Stengel<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Paleo-detectors are a proposed experimental technique to search for dark
matter (DM). In lieu of the conventional approach of operating a tonne-scale
real-time detector to search for DM-induced nuclear recoils, paleo-detectors
take advantage of small samples of naturally occurring rocks on Earth that
have been deep underground ($\gtrsim 5\,$km), accumulating nuclear damage
tracks from recoiling nuclei for $\mathcal{O}(1)\,$Gyr. Modern microscopy
techniques promise the capability to read out nuclear damage tracks with
nanometer resolution in macroscopic samples. Thanks to their
$\mathcal{O}(1)\,$Gyr integration times, paleo-detectors could constitute
nuclear recoil detectors with keV recoil energy thresholds and 100 kilotonne-
yr exposures. This combination would allow paleo-detectors to probe DM-nucleon
cross sections orders of magnitude below existing upper limits from
conventional direct detection experiments. In this article, we use improved
background modeling and a new spectral analysis technique to update the
sensitivity forecast for paleo-detectors. We demonstrate the robustness of the
sensitivity forecast to the (lack of) ancillary measurements of the age of the
samples and the parameters controlling the backgrounds, systematic mismodeling
of the spectral shape of the backgrounds, and the radiopurity of the mineral
samples. Specifically, we demonstrate that even if the uranium concentration
in paleo-detector samples is $10^{-8}$ (per weight), many orders of magnitude
larger than what we expect in the most radiopure samples obtained from ultra
basic rock or marine evaporite deposits, paleo-detectors could still probe DM-
nucleon cross sections below current limits. For DM masses $\lesssim
10\,$GeV/$c^{2}$, the sensitivity of paleo-detectors could still reach down
all the way to the conventional neutrino floor in a Xe-based direct detection
experiment.

##  1 Introduction

More than a century after evidence for the existence of Dark Matter (DM) in
our Universe started to emerge Bertone:2016nfn , we still do not know what DM
is made out of. Weakly Interacting Massive Particles (WIMPs) remain one of the
best-motivated DM candidates to date Jungman:1995df ; Arcadi:2017kky ;
Roszkowski:2017nbc . Among the most sensitive probes of the WIMP DM paradigm
are direct detection experiments Drukier:1983gj ; Goodman:1984dc ;
Drukier:1986tm ; Spergel:1987kx ; Collar:1992qc , where one searches for
interactions of DM with atomic nuclei. Over the past few decades, direct
detection experiments have made impressive progress: liquid noble gas
detectors have reached $\mathcal{O}(\mbox{tonne-yr})$ exposures with nuclear
recoil energy thresholds as small as a few keV Akerib:2016vxi ; Agnes:2018fwg
; Aprile:2018dbl ; Wang:2020coa , while cryogenic bolometric detectors have
pushed the nuclear recoil energy threshold below 100 eV, albeit as yet only
with detectors accumulating $\mathcal{O}(\mbox{kg\,days})$ of exposure
Angloher:2015ewa ; Armengaud:2016cvl ; Agnese:2017jvy ; Petricca:2017zdp ;
Armengaud:2019kfj . A large experimental program is underway to extend the
sensitivity of direct detection experiments both to lower WIMP masses and
smaller WIMP-nucleon cross sections Aprile:2015uzo ; Mount:2017qzi ;
Aalbers:2016jon ; Aalseth:2017fik ; Amaudruz:2017ibl , as well as to develop
directional detectors Daw:2011wq ; Cappella:2013rua ; Battat:2014van ;
Riffard:2013psa ; Santos:2013hpa ; Monroe:2012qma ; Leyton:2016nit ;
Miuchi:2010hn ; Nakamura:2015iza ; Battat:2016xxe ; Vahsen:2020pzb . However,
these detectors are becoming ever more costly and more challenging to operate.

Paleo-detectors Baum:2018tfw ; Drukier:2018pdy ; Edwards:2018hcf  are a
proposed alternative technique to search for interactions of DM with nuclei:
instead of operating a nuclear recoil detector in real time, as in a
conventional direct detection experiment, one would search for the persistent
traces of WIMP-nucleon interactions recorded in natural minerals over
timescales as long as $\mathcal{O}(1)\,$Gyr. Many minerals are excellent solid
state nuclear track detectors Fleischer:1964 ; Fleischer383 ; Fleischer:1965yv
; GUO2012233  and retain these traces (i.e. damage tracks) caused by a
recoiling atomic nucleus for timescales which can exceed the age of the Earth
by many orders of magnitude (i.e. $\gg 4.5\,\mathrm{Gyr}$). Natural minerals
commonly found on Earth are up to $\mathcal{O}(1)\,$Gyr old and can be dated
to a few-percent accuracy GTS2012 ; Gallagher:1998 ; vandenHaute:1998 . Modern
microscopy technologies, including electron microscopy Fleischer:1965yv ;
Toulemonde:2006 , Helium Ion Beam Microscopy (HIBM) HILL201265 ;
VANGASTEL20122104 , Small Angle X-ray scattering (SAXs) RODRIGUEZ2014150 ;
SAXS3d ; SAXSres , and optical microscopy GUO2012233 ; BARTZ2013273 ;
Kouwenberg:2018  potentially allow for the readout of such nuclear damage
tracks in macroscopic samples of target material. For example, HIBM may allow
one to read out $\mathcal{O}(10)\,$mg of material with $\mathcal{O}(1)\,$nm
precision when combined with pulsed-laser and fast-ion-beam ablation
techniques bioHIB ; ECHLIN20151 ; PFEIFENBERGER2017109 ; Randolph:2018 . On
the other hand, SAXs promises spatial resolutions of $\mathcal{O}(15)\,$nm in
$\mathcal{O}(100)\,$g of target material SAXS3d ; SAXSres  (see the discussion
in Refs. Baum:2018tfw ; Drukier:2018pdy ). These advanced microscopy
techniques enable paleo-detectors to reach much larger exposures with better
track length resolutions than previous ideas to probe rare events using
natural minerals; see Refs. Goto:1958 ; Goto:1963zz ; Fleischer:1969mj ;
Fleischer:1970zy ; Fleischer:1970vm ; Alvarez:1970zu ; Kolm:1971xb ;
Eberhard:1971re ; Ross:1973it ; Price:1983ax ; Kovalik:1986zz ; Price:1986ky ;
Ghosh:1990ki ; Jeon:1995rf ; Collar:1999md  and, in particular, the work by
Snowden-Ifft et al. searching for damage tracks from WIMP-nucleus interactions
in ancient phlogopite mica samples SnowdenIfft:1995ke ; Collar:1994mj ;
Engel:1995gw ; SnowdenIfft:1997hd . While not the focus of this article,
paleo-detectors have also been proposed as a probe of astrophysical neutrino
fluxes from Galactic supernovae Baum:2019fqm , cosmic ray interactions in
Earth’s atmosphere Jordan:2020gxx , and from our Sun Arellano:2021jul . There
has been recent related work on using crystal-defects created in certain
minerals by WIMP-nucleus interactions to search for light DM Rajendran:2017ynw
; Essig:2016crl ; Budnik:2017sbu  or to monitor nuclear reactors
Cogswell:2021qlq , and to use very long damage tracks in ancient quartz
samples to search for ultra-heavy DM Ebadi:2021cte .

Using natural minerals found on Earth to record DM-nucleus interactions over
$\mathcal{O}(100)\,{\rm Myr}-\mathcal{O}(1)\,$Gyr timescales provides paleo-
detectors with two crucial advantages over conventional direct detection
experiments. First, the enormous integration times promise exposures many
orders of magnitude larger than what is feasible in laboratory experiments.
Reading out 100 g of material that has been recording nuclear damage tracks
for 1 Gyr yields an exposure of $\varepsilon=100\,$g Gyr, or, in units more
appropriate for a real-time experiment, $\varepsilon=100\,$kilotonne yr. The
$\mathcal{O}(15)\,$nm spatial resolution achievable with SAXs readout
corresponds to a nuclear recoil energy threshold of $\mathcal{O}(1)\,$keV.
Thus, paleo-detectors would combine the recoil energy threshold achievable in
direct detection experiments utilizing liquid noble gases as target materials
with the exposures achievable in large (planned) neutrino detectors such as
Super-/Hyper-Kamiokande or DUNE. Second, while conventional direct detection
experiments measure events in real time, paleo-detectors integrate over
timescales up to $\mathcal{O}(1)\,$Gyr, comparable to the age of the Solar
System, $t_{\odot}\sim 4.5\,$Gyr, and the orbital period of the Solar System
around the Milky Way, $T_{\odot}\sim 250\,$Myr. Using a series of samples with
different ages, paleo-detectors offer a unique pathway to gaining information
about a signal’s time-evolution over Gyr timescales Baum:2019fqm ;
Jordan:2020gxx ; Arellano:2021jul ; SUBSTRUCTURE .

Of course, paleo-detectors must pay a price for these advantages: compared to
a conventional detector operated in a controlled laboratory environment,
background mitigation might be much more challenging. The background sources
for DM searches in paleo-detectors are qualitatively the same as in
conventional direct detection experiments since both approaches are, at their
heart, nuclear recoil searches. Extensive background studies for paleo-
detectors have been performed in previous work Baum:2018tfw ; Drukier:2018pdy
; Baum:2019fqm  and we will discuss the different sources further below. As we
will see, the most important background sources are cosmic rays, solar
neutrinos, and radioactivity. Regarding the first, the fundamental strategy
for paleo-detectors is to use natural minerals extracted from deep within the
Earth where the rock-overburden has shielded them from cosmogenic backgrounds.
Conventional direct detection experiments are operated in deep underground
laboratories for the same reason. However, paleo-detectors require only (at
most) a few kg of target material compared to actively operating a tonne-scale
experiment; samples of such modest sizes can be extracted from much greater
depths than those of existing underground laboratories (for example from
existing boreholes drilled for geological R&D and oil exploration). For the
backgrounds from solar neutrinos and radioactivity it is important to note
that, due to their large exposures, paleo-detectors would contain a sizable
number of background events. A rare event search, such as for DM-induced
nuclear recoils, would thus not proceed as in conventional direct detection
experiments where one typically attempts to find a background-free signal
region. Instead, a paleo-detector analysis is more similar to a collider
physics search: one would model the distribution of background events (for
example in track-length space) and search for deviations of the observed data
from the background-only prediction using a signal template. One important
consequence of the large number of background (and possible signal) events is
that one can characterize many aspects of the backgrounds from the data
themselves. Just as in collider searches, paleo-detectors can use control
regions, where the backgrounds dominate, to better characterize the background
contribution in the signal region of the search.

The remainder of this article is organized as follows: in Section 2 we discuss
the calculation of DM signals in paleo-detectors followed by the most
important background sources in Section 3. In Section 4 we show projections
for the reach of paleo-detectors in the conventional WIMP mass–cross section
plane. Compared to previous estimates of the sensitivity of paleo-detectors to
DM signals Baum:2018tfw ; Drukier:2018pdy ; Edwards:2018hcf , in this work we
make use of both improved background modeling and new statistical techniques.
In this sense, the results presented here supersede the sensitivity
projections shown in Refs. Baum:2018tfw ; Drukier:2018pdy ; Edwards:2018hcf .
In Section 4 we also explore the robustness of the sensitivity projections to
changes in the assumptions we make in our analysis; for example, we remove
external constraints on the age of the minerals and the parameters controlling
the background normalizations, and explore the effect of systematic
mismodeling of the spectral shape of the backgrounds. Furthermore, we show how
the projected sensitivity changes with the radiopurity of the samples, and
demonstrate that, even if the uranium concentration in paleo-detector samples
is $10^{-8}$ (per weight, denoted as “g/g” throughout), many orders of
magnitude larger than what we expect in the most radiopure samples obtained
from ultra basic rock or marine evaporite deposits, paleo-detectors could
still probe DM-nucleon cross sections below current limits. Specifically, for
WIMP masses $m_{\chi}\lesssim 10\,$GeV/$c^{2}$, the sensitivity of paleo-
detectors could still reach down all the way to the conventional neutrino
floor in a Xe-based direct detection experiment. We summarize and conclude in
Section 5. We make the code used in this work available: paleoSpec PaleoSpec
for the computation of the signal and background spectra, and paleoSens
PaleoSens  for the sensitivity forecasts.

##  2 Dark Matter Signals in Paleo-Detectors

The signature of WIMP DM in a paleo-detector is the spectrum of damage tracks
left by nuclei in the mineral sample after receiving a “kick” from WIMP-
nucleus scattering. In this section, we discuss how this signal is computed.
The differential rate of nuclear recoils per unit target mass for a WIMP with
mass $m_{\chi}$ scattering off target nuclei $N$ with recoil energy $E_{R}$ is
given by Engel:1991wq ; Engel:1992bf

$\left(\frac{dR}{dE_{R}}\right)_{N}=\frac{A_{N}^{2}\sigma_{p}^{\rm
SI}F_{N}^{2}(E_{R})}{2\mu_{p}^{2}m_{\chi}}\rho_{\chi}\eta_{\chi}(v_{\rm
min})\;,$ (1)

where $A_{N}$ is the atomic mass number of $N$,
$\mu_{p}=m_{p}m_{\chi}/(m_{p}+m_{\chi})$is the reduced mass of the WIMP-proton
system with the proton mass $m_{p}$, and we have assumed isospin-conserving
spin-independent WIMP-nucleon interactions with the (zero momentum transfer)
WIMP-proton cross section $\sigma_{p}^{\rm SI}$. Expressions analogous to Eq.
(1) for isospin-violating SI interaction and spin-dependent WIMP-nucleon
interactions can be found in Refs. Engel:1991wq ; Engel:1992bf ;
Ressell:1993qm ; Bednyakov:2004xq ; Bednyakov:2006ux , and for more general
WIMP-nucleus interactions in Refs. Fan:2010gt ; Fitzpatrick:2012ix  . Note
that while we focus on spin-independent interactions in this work, paleo-
detectors could also probe spin-dependent WIMP-nucleus interactions, see Ref.
Drukier:2018pdy .

In Eq. (1), the nuclear form factor $F_{N}(E_{R})$ accounts for the finite
size of the nucleus; we will use the Helm parameterization Helm:1956zz ;
Lewin:1995rx ; Duda:2006uk  in our numerical calculations.111Note that more
refined calculations of the form factors are available, although only for a
few isotopes, see e.g. Refs. Vietze:2014vsa ; Gazda:2016mrp ; Korber:2017ery ;
Hoferichter:2018acd . The distribution of WIMPs is described by the local WIMP
(mass) density, $\rho_{\chi}$, and the mean inverse speed,

$\eta_{\chi}(v_{\rm min})=\int_{v\geq v_{\rm
min}}d^{3}{\bm{v}}\;\frac{f({\bm{v}})}{v}\;,$ (2)

where $f({\bm{v}})$ is the velocity distribution of WIMPs with ${\bm{v}}$
($v=\left|{\bm{v}}\right|$) the velocity (speed) of the WIMP relative to the
detector. In Eq. (2), $v_{\rm min}=\sqrt{m_{N}E_{R}/2\mu_{N}^{2}}$ accounts
for the scattering kinematics; $v_{\rm min}$ is the minimal speed, with
respect to a nucleus of mass $m_{N}$, that a WIMP must have to give rise to a
recoil with energy $E_{R}$ (where $\mu_{N}$ is the reduced mass of the WIMP-
nucleus system). In our numerical calculations we set
$\rho_{\chi}=0.3\,$GeV/cm3 while for $f({\bm{v}})$ we use a Maxwell-Boltzmann
velocity distribution in the Galactic rest frame with velocity dispersion
$\sigma_{v}=166\,$km/s Koposov:2009hn , truncated at the escape speed $v_{\rm
esc}=550\,$km/s Piffl:2013mla , and boosted to the Solar System frame by
$v_{\odot}=248\,$km/s Bovy:2012ba  as in the Standard Halo Model (SHM)
Drukier:1986tm ; Lewin:1995rx ; Freese:2012xd .

Figure 1: Left: Range $x_{N}$ of different ions with initial energy $E_{R}$ in
gypsum [Ca(SO${}_{4})\\!\cdot\\!2$(H2O)]. Right: Differential rate of tracks
per track length and unit target mass in gypsum from a $5\,$GeV/$c^{2}$ and a
$500\,$GeV/$c^{2}$ WIMP compared to background spectra induced by neutrinos
($\nu$), radiogenic neutrons ($n$), and ${{}^{238}{\rm U}}\to{{}^{234}{\rm
Th}}+\alpha$ recoils (${{}^{234}{\rm Th}}$ ), see the discussion in Section 3.
For the normalization of the WIMP DM spectra we set the spin-independent WIMP-
nucleon cross sections to $\sigma_{p}^{\rm SI}=10^{-43}\,{\rm cm}^{2}$ for the
5 GeV/$c^{2}$ case and $\sigma_{p}^{\rm SI}=10^{-46}\,{\rm cm}^{2}$ for the
500 GeV/$c^{2}$ case, compatible with current upper bounds Aprile:2018dbl ;
Aprile:2019xxb .

In order to obtain the differential track length spectrum (per unit target
mass), $dR/dx$, we use the range of a nucleus $N$ with energy $E_{R}$,

$x_{N}(E_{R})=\int_{0}^{E_{R}}dE\;\left|\frac{dE}{dx}\right|_{N}^{-1}\;,$ (3)

where $\left(dE/dx\right)_{N}$ is the stopping power of $N$ in the target
material. The true length of a damage track in a given target may differ from
the range if, for example, the nucleus creates a lasting damage track only
along some portion of the length it travels through the material, or if the
nucleus’ trajectory deviates significantly from a straight line. Our previous
studies suggest that such effects should be small (see the discussion in Ref.
Drukier:2018pdy ), making Eq. (3) a good proxy for the track length.
Furthermore, the effects of thermal annealing could potentially be significant
over geological timescales; fortunately, any associated modifications to the
track lengths would be similar for both the signal and background recoils.
While detailed experimental studies for each target material will be required
to account for these possible corrections, for the purposes of this work we
will use Eq. (3) for the track length and SRIM Ziegler:1985 ; Ziegler:2010  to
calculate stopping powers. The left panel of Figure 1 shows the range of the
different constituent nuclei222Note that we do not show the range of H in
Figure 1 since we do not expect ions with charge $Z\leq 2$ (in particular, H
and He) to produce lasting damage tracks in most materials, see the discussion
in Drukier:2018pdy . in gypsum [Ca(SO${}_{4})\\!\cdot\\!2$(H2O)] as an
example; we see that a 1 keV recoil gives rise to a few-nm long track, while a
100 keV recoil produces a track with a length of a few hundred nm.

In a composite target material composed of different isotopes with mass
fraction $\xi_{N}$, the differential rate at which tracks are produced (per
unit target mass) is then given by

$\frac{dR}{dx}=\sum_{N}\xi_{N}\left(\frac{dR}{dE_{R}}\right)_{N}\left(\frac{dE}{dx}\right)_{N}\;,$
(4)

with $dR/dE_{R}$ given in Eq. (1). In the right panel of Figure 1 we show
$dR/dx$ for a 5 GeV/$c^{2}$ (with cross section $\sigma_{p}^{\rm
SI}=10^{-43}\,{\rm cm}^{2}$) and a 500 GeV/$c^{2}$ (with $\sigma_{p}^{\rm
SI}=10^{-46}\,{\rm cm}^{2}$) WIMP in gypsum as an example together with
various background spectra which we will discuss in Section 3. Note that for
the $5\,$GeV/$c^{2}$ case, the largest contribution to the signal is at track
lengths of a few nm, while heavier WIMPs, such as the 500 GeV/$c^{2}$ case
shown in Figure 1, give rise to longer tracks with typical lengths of order
$100\,$nm.

The nuclear recoil tracks in a paleo-detector sample can be read out with a
variety of readout techniques as briefly discussed in the introduction (see
Ref. Drukier:2018pdy  for a more detailed discussion). Here, we will consider
two scenarios:

  * •

High-resolution readout scenario: We assume that 10 mg of material can be read
out with a track length resolution of $\sigma_{x}=1\,$nm. This scenario may be
achievable with Helium Ion Beam Microscopy (HIBM) combined with pulsed-laser
and fast-ion-beam ablation techniques HILL201265 ; VANGASTEL20122104 ; bioHIB
; ECHLIN20151 ; PFEIFENBERGER2017109 ; Randolph:2018 . As we will see, this
scenario is advantageous for low-mass ($m_{\chi}\lesssim 10\,$GeV$/c^{2}$)
WIMP searches.

  * •

High-exposure readout scenario: We assume that 100 g of material can be read
out with $\sigma_{x}=15\,$nm. This scenario could be realized via Small Angle
X-ray scattering (SAXs) tomography at a synchrotron facility RODRIGUEZ2014150
; SAXS3d ; SAXSres ; it is better suited for heavier ($m_{\chi}\gtrsim
10\,$GeV$/c^{2}$) WIMP searches.

These are the same readout scenarios as what we have used in previous paleo-
detector studies Baum:2018tfw ; Drukier:2018pdy ; Edwards:2018hcf ;
Baum:2019fqm ; Jordan:2020gxx . We note that these scenarios represent
challenging applications of the respective techniques; demonstrating their
feasibility and especially their scalability to the proposed sample masses
would be an important step towards the realization of a paleo-detector
experiment. There are ongoing efforts to demonstrate the readout of few-keV
recoils using a variety of microscopy techniques at JAMSTEC Hirose  and at
Rensselaer Polytechnic Institute RPI .

In order to assess the sensitivity of paleo-detectors, we will consider the
binned track length spectrum as the experimental observable. Accounting for
the finite resolution of the readout method, the number of tracks in the
$i$-th bin with reconstructed length $x\in\left[x_{i}^{min},x_{i}^{\rm
max}\right]$ in a sample of mass $M$ which has been recording tracks for a
time $t_{\rm age}$ is

$N_{i}=M\times t_{\rm age}\int dx^{\prime}\;W(x^{\prime};x_{i}^{\rm
min},x_{i}^{\rm max})\frac{dR}{dx}(x^{\prime})\;,$ (5)

where $dR/dx$ is given by Eq. (4) and we have assumed that $dR/dx$ is
independent of time.333For time-dependent signals, replace $t_{\rm
age}\to\int_{0}^{t_{\rm age}}dt$ in Eq. (5). The window function, $W$,
accounts for the finite resolution effects of the readout. In our numerical
calculations, we will assume that the probability of measuring a track length
$x$ from a track with true length $x^{\prime}$ is Gaussian-distributed with
variance $\sigma_{x}^{2}$. The corresponding window function is

$W(x^{\prime};x_{i}^{\rm min},x_{i}^{\rm max})=\frac{1}{2}\left[{\rm
erf}\left(\frac{x^{\prime}-x_{i}^{\rm min}}{\sqrt{2}\sigma_{x}}\right)-{\rm
erf}\left(\frac{x^{\prime}-x_{i}^{\rm
max}}{\sqrt{2}\sigma_{x}}\right)\right]\;.$ (6)

Note that in order to avoid artificial sensitivity to tracks which are much
shorter than the spatial resolution of the readout, we remove all tracks with
true track length $x^{\prime}<\sigma_{x}/2$ from the spectra ($dR/dx$) before
applying Eq. (5). For all numerical results in this work, we use 100 log-
spaced bins between $\sigma_{x}/2$ and $1000\,$nm.

Figure 2: Track length spectra for a 5 GeV/$c^{2}$ and a 500 GeV/$c^{2}$ WIMP
compared to the background spectra induced by neutrinos ($\nu$), radiogenic
neutrons ($n$), and ${{}^{238}{\rm U}}\to{{}^{234}{\rm Th}}+\alpha$ recoils
(${{}^{234}{\rm Th}}$ ) after taking finite resolution effects into account.
The left panels are for our high resolution scenario ($M=10\,$mg of sample
material read out with track length resolution of $\sigma_{x}=1\,$nm) while
the right panels are for the high exposure scenario ($M=100\,$g,
$\sigma_{x}=15\,$nm). We assume that the samples have been recording tracks
for $t_{\rm age}=1\,$Gyr. In all panels, we are using 100 logarithmically
spaced bins between $\sigma_{x}/2\leq x\leq 10^{3}\,$nm; note that these bins
and the scales of the axes differ between the left and the right panels. The
upper panels show the number of tracks per bin from the respective sources.
For the backgrounds, the shaded bands around the respective spectra show the
associated error (systematic and statistical added in quadrature). The bottom
panels show the ratio of the number of signal ($S_{i}$) to the (summed) number
of background ($B_{i}$) events per bin, and the sand-colored shade shows the
relative uncertainty of the total number of background events per bin. The
signal-to-noise ratio per bin can be read off from the lower panels by
dividing $S_{i}/B_{i}$ for the respective signal with the relative uncertainty
of the background. For the normalization of the WIMP DM spectra we set the
spin-independent WIMP-nucleon cross sections to $\sigma_{p}^{\rm
SI}=10^{-43}\,{\rm cm}^{2}$ for the 5 GeV/$c^{2}$ case and $\sigma_{p}^{\rm
SI}=10^{-46}\,{\rm cm}^{2}$ for the 500 GeV/$c^{2}$ case, as in Figure 1.

Figure 2 shows the same 5 GeV/$c^{2}$ and 500 GeV/$c^{2}$ WIMP DM spectra as
the right panel of Figure 1 together with the various background sources after
taking into account finite resolution effects. The left panel is for the high-
resolution scenario, while the right panel is for the high-exposure scenario.
We will discuss this figure further at the end of Section 3 after introducing
the relevant background sources. Let us here only note that the high-
resolution scenario is better suited for relatively light WIMPs with masses
$m_{\chi}\lesssim 10\,$GeV/$c^{2}$, while the high-exposure scenario is aimed
at heavier WIMPs, $m_{\chi}\gtrsim 10\,$GeV/$c^{2}$. This is for two reasons:
First, lighter WIMPs give rise to softer nuclear recoils, resulting in (on
average) shorter signal tracks, making excellent track length resolution
crucial to resolving the details of the signal and background spectra in the
region where the ratio of signal to background events is largest for such
light WIMPs. Second, current bounds from direct detection experiments rule out
much smaller cross sections for heavier WIMPs with $m_{\chi}\gtrsim
10\,$GeV/$c^{2}$, hence, searches for WIMPs with $m_{\chi}\gtrsim
10\,$GeV/$c^{2}$ and cross sections smaller than current upper limits require
very large exposures.

##  3 Backgrounds

The background sources for DM searches with paleo-detectors are the same as
for conventional direct detection experiments. However, the relative
importance of the respective sources is different because, compared to
conventional direct detection experiments, paleo-detectors use much smaller
target masses ($\lesssim 1\,$kg) and integrate over much longer timescales
($t_{\rm age}\sim 0.1-1\,$Gyr) without information about the timing of
individual signal or background events. The experimental observables in paleo-
detectors are nuclear damage tracks with, for all practical purposes, perfect
rejection of electronic recoils. Furthermore, natural defects in minerals are
either single-site or span across the entire (mono)-crystalline volume and do
not resemble nuclear damage tracks. Hence, the only relevant backgrounds are
nuclear recoils. In the remainder of this section we will give a brief
overview of the most important background sources; we refer the reader to
Refs. Drukier:2018pdy ; Baum:2019fqm  for more detailed discussions.

  * •

Cosmogenics: In order to mitigate backgrounds from cosmic rays, minerals to be
used as paleo-detectors must have been shielded by a large overburden for as
long as they have been recording nuclear damage tracks (just as conventional
direct detection experiments are operated in deep underground laboratories).
Paleo-detectors require only (at most) a few kg of target material rather than
actively operating a tonne-scale experiment; such modest amounts of materials
can be sourced from even greater depths than those of existing underground
laboratories. One promising source of samples are (existing) boreholes drilled
for geological R&D or oil exploration. For example, for an overburden of 5 km
rock, the cosmogenic-muon-induced neutron flux is $\mathcal{O}(10^{2})\,{\rm
cm}^{-2}\,{\rm Gyr}^{-1}$ Mei:2005gm , leading to negligible backgrounds for
the purposes of a paleo-detector. We note that at depths $\gtrsim 6\,$km rock,
backgrounds from atmospheric neutrinos producing neutrons in the vicinity of
the target become comparable to the backgrounds from cosmogenic muons
Aharmim:2009zm . We also note that paleo-detector samples can be stored near
the surface after extraction from deep in the Earth before readout. For
example, the cosmogenic-muon-induced neutron flux in a $\sim$50 m deep storage
facility is $\lesssim 0.2\,{\rm cm}^{-2}\,{\rm yr}^{-1}$.

  * •

Astrophysical Neutrinos: Neutrinos scattering off nuclei in a paleo-detector
sample give rise to nuclear damage tracks. We include neutrinos from the Sun,
supernovae, and the interactions of cosmic rays with the Earth’s atmosphere in
our background modeling. We take the solar and atmospheric neutrino fluxes
from Ref. OHare:2020lva . Because of the long integration times, paleo-
detectors are sensitive not only to neutrinos from supernovae in far-away
galaxies throughout our Universe (the Diffuse Supernova Neutrino Background,
DSNB), but also to neutrinos from local supernovae - the supernova rate in the
Milky Way is estimated to be 2–3 per 100 years Cappellaro:2003eg ;
Diehl:2006cf ; Strumia:2006db ; Li:2011 ; Botticella:2011nd ; Adams:2013ana .
We compute the DSNB spectrum and the contribution from Galactic supernovae as
in Ref. Baum:2019fqm , see also Refs. Beacom:2010kk ; Adams:2013ana ;
Billard:2013qya ; Madau:2014bja ; Strolger:2015kra . The neutrino-induced
background is dominated by solar neutrinos at track lengths below $\sim$100
nm, with supernova neutrinos giving the largest contributions for tracks of a
few 100 nm, before atmospheric neutrinos dominate the neutrino-induced
backgrounds at even longer track lengths. Note that while neutrino-induced
nuclear recoils are a background for DM searches, they can also be an
interesting signal for paleo-detectors, see Refs. Baum:2019fqm ;
Jordan:2020gxx ; Arellano:2021jul .

  * •

Radiogenics: Any natural mineral used as a paleo-detector will contain trace
amounts of radioactive materials. In order to mitigate radiogenic backgrounds,
it is crucial to use minerals with as low a concentration of radioactive
elements as possible. The most important radioactive isotope for paleo-
detectors is ${{}^{238}{\rm U}}$ . Typical minerals formed in the Earth’s
crust have ${{}^{238}{\rm U}}$ concentrations of order $C^{238}\sim
10^{-6}\,$g/g, leading to prohibitively large backgrounds for DM searches.
However, minerals which constitute so-called Ultra Basic Rocks (UBRs), formed
from the material of the Earth’s mantle, and Marine Evaporites (MEs), formed
from evaporated sea water, have typical ${{}^{238}{\rm U}}$ concentrations
orders of magnitude lower Thomson:1954 ; Condie:1957 ; Adams:1959 ; Seitz:1974
; Dean:1978 ; Yui:1998 ; Sanford:2013 , (see also Ref. Drukier:2018pdy  and
especially the discussion in the appendix of Ref. Baum:2019fqm ). As in
previous works on paleo-detectors Baum:2018tfw ; Drukier:2018pdy ;
Edwards:2018hcf ; Baum:2019fqm ; Jordan:2020gxx  we will assume benchmark
${{}^{238}{\rm U}}$ concentrations of $C^{238}=10^{-10}\,$g/g for UBRs and
$C^{238}=10^{-11}\,$g/g for MEs.

Since paleo-detectors record only nuclear recoils, the most relevant
radiogenic background arises from $\alpha$-decays and spontaneous fission.
Note that it is not the $\alpha$-particles themselves or the fission fragments
which lead to backgrounds for DM searches: $\alpha$-particles (i.e. He nuclei)
do not give rise to lasting damage tracks in most materials and, even if they
would, the range of the $\alpha$’s is of order $10\,\mu$m in natural minerals,
orders of magnitude larger than the tracks from DM-induced nuclear recoils.
Similarly, the fission fragments from spontaneous fission of heavy elements
have typical energies of tens of MeV, producing much longer tracks than DM-
induced recoils. However, in an $\alpha$-decay, the decaying nucleus recoils
with typical energies of some tens of keV against the $\alpha$-particle. Due
to the long integration time of paleo-detectors, almost all of the
${{}^{238}{\rm U}}$ nuclei in the mineral which undergo the first
(${{}^{238}{\rm U}}\to{{}^{234}{\rm Th}}+\alpha$) decay in the ${{}^{238}{\rm
U}}$ decay chain will have continued on to decay to the stable 206Pb. There
are eight $\alpha$-decays in the ${{}^{238}{\rm U}}$ decay chain, leading to a
characteristic pattern of connected nuclear recoil tracks. We expect such
patterns of connected tracks to be easily distinguishable from the isolated
recoil tracks DM-nucleus interactions would cause and, hence, do not consider
them a relevant background for DM searches. However, the second $\alpha$-decay
in the ${{}^{238}{\rm U}}$ decay chain (${}^{234}{\rm U}\to{{}^{230}{\rm
Th}}+\alpha$) has a relatively long half-life, $T_{1/2}(^{234}{\rm
U})=0.25\,$Myr. This leads to a population of isolated tracks from
${{}^{238}{\rm U}}$ atoms which have undergone the initial (${{}^{238}{\rm
U}}\to{{}^{234}{\rm Th}}+\alpha$) decay, but not the second $\alpha$-decay
(${}^{234}{\rm U}\to{{}^{230}{\rm Th}}+\alpha$) in the uranium series
Collar:1995aw ; SnowdenIfft:1996zz . The ${{}^{234}{\rm Th}}$ nucleus receives
$72\,$keV of recoil energy in the (${{}^{238}{\rm U}}\to{{}^{234}{\rm
Th}}+\alpha$) decay, leading to a population of monochromatic isolated tracks
with approximate number density $n({{}^{234}{\rm Th}})\simeq 10^{6}\,{\rm
g}^{-1}\times\left(C^{238}/10^{-11}\,{\rm g/g}\right)$ Drukier:2018pdy .

The second important source of radiogenic backgrounds are neutrons:
spontaneous fission of heavy isotopes as well as ($\alpha,n$)-reactions
between the $\alpha$-particles produced in $\alpha$-decays and nuclei
comprising the target material give rise to neutrons with MeV energies. These
neutrons in turn scatter off the nuclei in the paleo-detector sample, giving
rise to a broad spectrum of nuclear recoil tracks. In order to model this
background, we use SOURCES-4A sources4a:1999  to compute the neutron spectra
from spontaneous fission and $(\alpha,n)$-reactions of $\alpha$’s from all
$\alpha$-decaying nuclei in the ${{}^{238}{\rm U}}$ decay chain; note that the
contributions from spontaneous fission and ($\alpha,n$)-reactions to the
neutron flux are typically of the same order of magnitude, the precise balance
depends on the chemical composition of the target. From the neutron spectra,
we calculate the induced nuclear recoil spectra using TENDL-2017
Koning:2012zqy ; Rochman:2016 ; Sublet:2015 ; Fleming:2015  neutron-nucleus
cross sections tabulated in the JANIS4.0 Soppera:2014zsj  database.444We take
only elastic neutron-nucleus scattering into account; this yields a
conservative estimate of the background because neutrons typically lose a
larger fraction of their energies through inelastic processes than in elastic
scattering. Note also that our Monte Carlo simulation of the nuclear recoils
induced by radiogenic neutrons has been validated with a calculation of the
nuclear recoils induced by the same neutron spectra with FLUKA Ferrari:2005zk
; Bohlen:2014buj ; NUNDIS  in Ref. Jordan:2020gxx  for the particular case of
a halite target. The neutron-induced backgrounds are strongly suppressed in
target materials containing hydrogen: since neutrons and H nuclei have
approximately the same mass, neutrons lose a large fraction of their energy in
a single interaction with hydrogen, leading to efficient moderation of the
radiogenic neutrons even if hydrogen comprises only a small fraction of the
atoms in the target material Baum:2018tfw ; Drukier:2018pdy .

In Figures 1 and 2 we show the track-length spectra produced by these various
background sources together with two benchmark DM signals in gypsum
[Ca(SO${}_{4})\\!\cdot\\!2$(H2O)]. At short track lengths, the background
budget is dominated by (solar) neutrino induced nuclear recoils. At
intermediate ranges between a few tens and a hundred nm (the precise values
depend on the particular mineral), the isolated ${{}^{234}{\rm Th}}$ tracks
from ${{}^{238}{\rm U}}\to{{}^{234}{\rm Th}}+\alpha$ decays contribute, before
the background budget becomes dominated by radiogenic-neutron-induced
backgrounds at larger track lengths. Comparing to the benchmark WIMP spectra,
we see that solar neutrinos are the most relevant background source for WIMPs
with masses in the few GeV/$c^{2}$ range, while radiogenic neutrons are the
limiting background in searches for heavier WIMPs with masses larger than
$\sim$10 GeV/$c^{2}$.

In Figure 2 we show the (binned) number of tracks one would observe in the
high-resolution (left panel) and high-exposure (right panel) scenarios after
accounting for the effects of finite readout resolution. Because of the large
exposure of paleo-detectors, a sizable number of background events will be
present. A DM search with paleo-detectors would thus proceed quite differently
from a traditional direct detection experiment, in which one typically
attempts to construct a signal region with very few (or, ideally, zero)
background events, and then searches for an $\mathcal{O}(1)$ number of DM-
induced events. Given the large number of background events, in a paleo-
detector analysis one would model the distribution of background events (for
example, in track-length space) and search for deviations of the observed data
from the background-only prediction. Thus, the uncertainty in the background
prediction is much more important than the absolute number of background
events.

For illustration, we include error bands around the respective background
components in Figure 2. These bands include both the (statistical) Poisson
noise and a systematic error in the background prediction. For the latter, we
make well-motivated assumptions in order to illustrate how the sensitivity of
paleo-detectors to signals from DM of various masses would depend on different
background components in a cut-and-count analysis, see previous studies of
paleo-detectors Baum:2018tfw ; Drukier:2018pdy ; Edwards:2018hcf ;
Baum:2019fqm ; Jordan:2020gxx . However, we note that in the spectral analysis
discussed in Section 4, the sensitivity of paleo-detectors to a WIMP DM signal
is virtually independent of any prior knowledge of the background
normalizations. Let us briefly describe our choices for the systematic errors
in Figure 2. Neutrino-induced backgrounds are controlled by sources which are
not associated with the environment of the paleo-detector target sample: the
fluxes of solar, supernova, or atmospheric neutrinos. These fluxes may vary by
an $\mathcal{O}(1)$ factor over the $\sim\,$Gyr integration time of paleo-
detectors Baum:2019fqm ; Jordan:2020gxx ; Arellano:2021jul , hence, we assign
a (relative) systematic uncertainty of $\pm 100\,\%$ to the neutrino-induced
backgrounds.555In Figure 2 we plotted all neutrino-induced backgrounds
together to avoid excessive clutter. In our sensitivity projections we treat
the solar, DSNB, Galactic supernova, and atmospheric neutrino induced
backgrounds as four independent background contributions. The radiogenic
backgrounds, on the other hand, are controlled by the ${{}^{238}{\rm U}}$
concentrations which do not change over time. Furthermore, the shape of the
track length spectra induced by radiogenic backgrounds can be calibrated by
studying the track length distributions in mineral samples with large
concentrations of ${{}^{238}{\rm U}}$ . Finally, the ${{}^{238}{\rm U}}$
concentration in a given sample can be measured using, for example, mass
spectrometry or gamma ray spectroscopy Povinec:2018 ; Povinec:2018wgd . Hence,
we expect the uncertainty on the radiogenic background predictions to be much
smaller and assign them a (relative) systematic error of $\pm 1\,\%$.

From the bottom panels in Figure 2 we can directly read off the signal-to-
noise ratio (per bin), i.e. the ratio of the number of signal events to the
uncertainty of the background prediction: in these panels, the lines show the
ratio of the number of signal to the (total) number of background events
($S/B$), while the sand-colored shaded region shows the relative uncertainty
of the background prediction. For a given signal, the ratio of $S/B$ to the
(relative) background uncertainty directly corresponds to the significance one
would obtain in a simple cut-and-count analysis. As discussed in Refs.
Baum:2018tfw ; Drukier:2018pdy , any signal for which $S/B$ is larger than the
shaded region indicating the background uncertainty in the bottom panels of
Figure 2 in at least one bin could be probed with a simple cut-and-count
analysis.

Given the large number of events expected in a paleo-detector, one can go
beyond the simple cut-and-count analysis and perform a spectral analysis in
track-length space: Armed with predictions for the spectral shape of the
various background components and of a possible signal, one can fit all of
these spectral templates to the data and ask if the data is better described
by either the backgrounds-only or the backgrounds+signal hypothesis.
Heuristically, such a search uses control regions, where some background
dominates, to measure that background component, and uses this information to
reduce the error of the background prediction in the signal region, where the
signal-to-background ratio is largest. For example, short track lengths can
serve as a control region for the solar-neutrino-induced backgrounds (as well
as the ${{}^{234}{\rm Th}}$ tracks in the high-exposure scenario), while track
lengths longer than $\sim$500 nm can provide an excellent control region for
the neutron-induced backgrounds. We have first used such a spectral analysis
in Ref. Edwards:2018hcf  and will discuss an alternative approach for the
spectral analysis in the next section.

##  4 Sensitivity Forecasts

In Section 2 we described how to compute the DM signal in paleo-detectors. In
particular, we showed how to obtain the track length spectrum one would
observe in a paleo-detector from the differential recoil rate per recoil
energy and how to account for finite track-length resolution effects. In
Section 3 we discussed the most relevant background sources and briefly
sketched how one would perform a DM search with a paleo-detector. Due to the
potentially large exposure, a sizable number of background (and, possibly,
signal) events will be present in any paleo-detector sample. In order to
search for DM, one would perform a spectral analysis of the observed recoil
spectrum in track-length space and ask if that spectrum is better fit by a
background-only model or by a background+signal hypothesis. In Refs.
Baum:2018tfw ; Drukier:2018pdy  we have used a much simpler cut-and-count
analysis to project the sensitivity of paleo-detectors to DM, while in Ref.
Edwards:2018hcf  we have estimated the sensitivity using a spectral analysis
based on the Fisher-matrix formalism using the swordfish package
Edwards:2017mnf ; Edwards:2017kqw ; swordfish . In the remainder of this
section, we describe a profile likelihood ratio based approach to calculate
the projected sensitivity of paleo-detectors to a DM signal with a spectral
analysis. As we will see, one advantage of using this statistical approach is
that we can easily incorporate the effect of background mismodeling into our
sensitivity forecast. We show results for the sensitivity forecasts in Figures
3 – 5.

The predicted track length spectrum in a paleo-detector is

$\begin{split}{\bm{N}}({\bm{\theta}};\sigma_{p}^{\rm
SI},m_{\chi})&={\bm{N}}_{\nu}^{\rm sol}(\Phi_{\nu}^{\rm
sol})+{\bm{N}}_{\nu}^{\rm DSNB}(\Phi_{\nu}^{\rm DSNB})+{\bm{N}}_{\nu}^{\rm
GSNB}(\Phi_{\nu}^{\rm GSNB})+{\bm{N}}_{\nu}^{\rm atm}(\Phi_{\nu}^{\rm atm})\\\
&\quad+{\bm{N}}_{\rm rad}^{{{}^{234}{\rm Th}}}(C^{238})+{\bm{N}}_{\rm
rad}^{n}(C^{238})\\\ &\quad+{\bm{N}}^{\rm DM}(\sigma_{p}^{\rm
SI};m_{\chi})\;,\end{split}$ (7)

where the ${\bm{N}}$ are the track length spectra described in Eq. (5). Each
${\bm{N}}$ is computed from the appropriate recoil energy distribution, given
by Eq. (1) for the DM signal and the corresponding expressions for the
different background components discussed in Section 3. In Eq. (7), the terms
in the first line describe the contributions induced by solar neutrinos
(${\bm{N}}_{\nu}^{\rm sol}$), the DSNB (${\bm{N}}_{\nu}^{\rm DSNB}$), the
Galactic Supernova Neutrino Background (${\bm{N}}_{\nu}^{\rm GSNB}$), and
atmospheric neutrinos (${\bm{N}}_{\nu}^{\rm atm}$). The terms in the second
line describe the radiogenic backgrounds, separated into: the isolated
${{}^{234}{\rm Th}}$ decays from ${{}^{238}{\rm U}}$ nuclei, which have
undergone only the inital $\alpha$-decay during the time the sample has been
recording nuclear damage tracks (${\bm{N}}_{\rm rad}^{{{}^{234}{\rm Th}}}$),
and the background induced by radiogenic neutrons (${\bm{N}}_{\rm rad}^{n}$).
Finally, the term in the third line is the DM signal. For each $\bm{N}$, we
have indicated in parantheses the parameter controlling its normalization: the
flux ($\Phi_{\nu}^{i}$) for each neutrino contribution, the ${{}^{238}{\rm
U}}$ concentration in the sample ($C^{238}$) for the radiogenic backgrounds,
and the WIMP-nucleon cross section ($\sigma_{p}^{\rm SI}$) for the DM signal.
Note that all contributions are also proportional to the mass of the sample,
$M$, and all contributions except for ${\bm{N}}_{\rm rad}^{{{}^{234}{\rm
Th}}}$ are proportional to the length of time the sample has been recording
damage tracks, $t_{\rm age}$; we have suppressed these dependencies in Eq. (7)
for compactness. Finally, the DM signal also depends on the mass of the WIMP,
$m_{\chi}$.

To easily compare to existing upper limits and the projected sensitivities of
conventional direct detection experiments, we calculate the projected upper
(exclusion) limit on $\sigma_{p}^{\rm SI}$ at 90 % confidence level as a
function of $m_{\chi}$ for a paleo-detector. The set of nuisance parameters in
Eq. (7) is then

${\bm{\theta}}=\left\\{M,t_{\rm age},\Phi_{\nu}^{\rm sol},\Phi_{\nu}^{\rm
DSNB},\Phi_{\nu}^{\rm GSNB},\Phi_{\nu}^{\rm atm},C^{238}\right\\}\;.$ (8)

Since paleo-detectors are ultimately a counting experiment, we expect the data
to be Poisson-distributed. The appropriate log-likelihood to observe the data
${\bm{D}}$ given the parameters $\left\\{{\bm{\theta}};\sigma_{p}^{\rm
SI},m_{\chi}\right\\}$ is thus666Here and in the following, we drop constant
factors in the expression of the likelihood which cancel in the likelihood
ratio we are ultimately interested in.

$\ln{\mathcal{L}}_{\rm
Poisson}\left({\bm{D}}\middle|{\bm{\theta}};\sigma_{p}^{\rm
SI},m_{\chi}\right)=\sum_{i}\left[D_{i}\ln N_{i}({\bm{\theta}};\sigma_{p}^{\rm
SI},m_{\chi})-N_{i}({\bm{\theta}};\sigma_{p}^{\rm SI},m_{\chi})\right]\;.$ (9)

The nuisance parameters will be constrained by other measurements. For
example, the neutrino fluxes will be constrained by existing neutrino
experiments and theoretical modeling. The mass, age, and uranium concentration
of the sample will be constrained by direct measurements of the sample
mineral. In a frequentist analysis, these constraints can be taken into
account by jointly analyzing these ancillary measurements with the track
length spectrum data. To mimick such a joint analysis, we include a set of
external (Gaussian) constraints on the nuisance parameters,

$\ln{\mathcal{L}}_{\rm
ext.~{}const.}({\bm{\theta}})=\sum_{j}\left[-\frac{1}{2}\left(\frac{\theta_{j}-\overline{\theta}_{j}}{c_{j}\overline{\theta}_{j}}\right)^{2}\right]\;.$
(10)

Here, $\overline{\theta}_{j}$ is the central value for the $j$-th nuisance
parameter supplied by the ancillary measurement(s), and $c_{j}$ is the
relative uncertainty of the measurement(s). The likelihood is then

$\ln{\mathcal{L}}\left({\bm{D}}\middle|{\bm{\theta}};\sigma_{p}^{\rm
SI},m_{\chi}\right)=\ln{\mathcal{L}}_{\rm
Poisson}\left({\bm{D}}\middle|{\bm{\theta}};\sigma_{p}^{\rm
SI},m_{\chi}\right)+\ln{\mathcal{L}}_{\rm ext.~{}const.}({\bm{\theta}})\;.$
(11)

To calculate the projected sensitivity of paleo-detectors we consider the
maximum log-likelihood ratio Cowan:2010js ; Billard:2012 ; Conrad:2014nna

$q(\sigma_{p}^{\rm
SI};m_{\chi})=-2\ln\left[\frac{\mathcal{L}\left({\bm{D}}\middle|\hat{\hat{\bm{\theta}}};\sigma_{p}^{\rm
SI},m_{\chi}\right)}{\mathcal{L}\left({\bm{D}}\middle|\hat{\bm{\theta}};\hat{\sigma}_{p}^{\rm
SI},m_{\chi}\right)}\right]\;.$ (12)

In the numerator, $\hat{\hat{{\bm{\theta}}}}$ is the set of nuisance
parameters that maximizes $\mathcal{L}$ for given values of $\sigma_{p}^{\rm
SI}$ (and $m_{\chi}$), while in the denominator, $\mathcal{L}$ is maximized
over both ${\bm{\theta}}$ and $\sigma_{p}^{\rm SI}$ (for a given value of
$m_{\chi}$), with best-fit values $\hat{\bm{\theta}}$ and
$\hat{\sigma}_{p}^{\rm SI}$. In order to calculate the projected exclusion
limit, we use the Asimov data set Cowan:2010js  under the assumption of no
signal,

${\bm{D}}={\bm{N}}(\overline{\bm{\theta}};\sigma_{p}^{\rm SI}=0)\;,$ (13)

where $\overline{\bm{\theta}}$ is a set of fiducial choices for the nuisance
parameters. The 90 % confidence level exclusion limit for a given value of
$m_{\chi}$ is obtained by finding the smallest value of $\sigma_{p}^{\rm SI}$
for which $q(\sigma_{p}^{\rm SI};m_{\chi})\geq q_{\rm crit}=2.71$.777By Wilks’
theorem Wilks:1938dza , the distribution of $q$ is (asymptotically) $\chi^{2}$
distributed, and for a one-dimensional $\chi^{2}$-distribution,
$\chi^{2}=2.71$ for a $p$-value of 0.1.

We describe the the fiducial values for the nuisance parameters listed in Eq.
(8) and the relative uncertainty of the external constraints on those
parameters, $c_{j}$, defined by Eq. (10). For the high-resolution (high-
exposure) scenario, we choose a fiducial value of $\overline{M}=10\,$mg
($\overline{M}=100\,$g) for the sample mass and, for both scenarios,
$\overline{t}_{\rm age}=1\,$Gyr for the time the sample has been recording
damage tracks. We assume that these parameters can be constrained by ancillary
measurements to $c_{M}=0.01\,\%$ and $c_{t_{\rm age}}=5\,\%$; while measuring
the mass of a target sample to high precision is trivial, its age can be
inferred with few-percent precision using standard geological dating
techniques GTS2012 ; Gallagher:1998 ; vandenHaute:1998 . For the normalization
of the neutrino fluxes, $\overline{\Phi}_{\nu}^{i}$, and the ${{}^{238}{\rm
U}}$ concentrations in our target minerals, $\overline{C}^{238}$, we take
fiducial values described in Section 3 (the latter are also summarized in
Table 1). As a benchmark, we assume that the associated nuisance parameters
can be constrained to $c_{\Phi_{\nu}^{i}}=100\,\%$ and $c_{C^{238}}=1\,\%$;
these choices are equivalent to the respective systematic uncertainties of the
backgrounds assumed in previous studies Baum:2018tfw ; Drukier:2018pdy ;
Edwards:2018hcf ; Baum:2019fqm ; Jordan:2020gxx . However, as we will discuss
below, the projected sensitivity of paleo-detectors to a DM signal is largely
independent of external constraints on the nuisance parameters.

Mineral | Composition | Fiducial 238U concentration [g/g]
---|---|---
Halite | NaCl | $10^{-11}$
Gypsum | Ca(SO${}_{4})\\!\cdot\\!2$(H2O) | $10^{-11}$
Sinjarite | CaCl${}_{2}\\!\cdot\\!2($H2O) | $10^{-11}$
Olivine | Mg1.6Fe${}^{2+}_{0.4}$(SiO4) | $10^{-10}$
Phlogopite | KMg3AlSi3O10F(OH) | $10^{-10}$
Nchwaningite | Mn${}^{2+}_{2}$SiO3(OH)${}_{2}\\!\cdot\\!($H2O) | $10^{-10}$
Table 1: List of minerals considered in this work with their chemical
composition and our fiducial assumption for the ${{}^{238}{\rm U}}$
concentration ($C^{238}$) in radiopure samples of these minerals.

In Figure 3 we show the projected 90 % confidence level upper limit under
these fiducial assumptions in both the high-resolution (sample mass
$M=10\,$mg, track-length resolution $\sigma_{x}=1\,$nm; left panel) and the
high-exposure ($M=100\,$g, $\sigma_{x}=15\,$nm; right panel) readout scenario.
We show results for a selection of minerals, see Table 1: three examples of
Ultra-Basic Rocks (UBRs), the very common olivine and two less common UBRs
which contain hydrogen, phlogopite mica and nchwaningite. We also show the
projected sensitivity for three examples of Marine Evaporites (MEs): halite,
gypsum, and sinjarite where the latter two contain hydrogen. As discussed
above, the high-resolution scenario has better sensitivity for WIMP masses
$m_{\chi}\lesssim 10\,$GeV/$c^{2}$ where the dominant background stems from
solar neutrinos; here, excellent track length resolution is crucial to
resolving the differences between the shapes of the signal and background
track length spectra. For $m_{\chi}\gtrsim 10\,$GeV/$c^{2}$, the high-exposure
scenario has better sensitivity since for such WIMP masses radiogenic
backgrounds are dominant and maximizing the exposure is crucial in order to
achieve the best sensitivity. Comparing the sensitivity for the different
target materials we can note that materials containing hydrogen generally have
better sensitivity than those without hydrogen. This is due to the effective
suppression of the radiogenic neutron backgrounds by hydrogen, see the
discussion in Section 3. Furthermore, we note that MEs typically have better
sensitivity than UBRs; this is because we assume a lower ${{}^{238}{\rm U}}$
concentration ($C^{238}=10^{-11}\,$g/g) for MEs than for UBRs
($C^{238}=10^{-10}\,$g/g). Out of the minerals considered in Figure 3,
throughout the shown WIMP mass range and for both the high-resolution and the
high-exposure readout scenarios, sinjarite promises the best sensitivity.
Primarily, this is due to sinjarite being an ME containing hydrogen, and hence
having relatively small ${{}^{238}{\rm U}}$ -induced backgrounds. However, the
sensitivity is also enhanced by its particular chemical composition: the other
example of an ME containing hydrogen shown in Figure 3, gypsum, is very
oxygen-rich, and hence, both the DM-induced and the background spectra are
dominated by the contribution of oxygen. Sinjarite on the other hand has one
calcium, two oxygen, and two chlorine atoms; each of these elements can
therefore contribute comparably to the recoil spectra. This adds additional
structure to the track length spectra which is advantageous in a spectral
analysis.

Figure 3: Projected 90 % confidence level upper limits in the WIMP mass
($m_{\chi}$) – spin-independent WIMP-nucleus scattering cross section
($\sigma_{p}^{\rm SI}$) plane in the high-resolution (sample mass $M=10\,$mg,
track length resolution $\sigma_{x}=1\,$nm; left panel) and high-exposure
($M=100\,$g, $\sigma_{x}=15\,$nm; right panel) readout scenarios. The
different lines are for different target materials as indicated in the legend,
see Table 1. The gray-shaded region of parameter space is disfavored by
current upper limits from direct detection experiments Angloher:2015ewa ;
Petricca:2017zdp ; Agnes:2018ves ; Aprile:2018dbl ; Aprile:2019xxb , while the
sand-colored region indicates the neutrino floor for a Xe-based experiment
Ruppin:2014bra . Colors and linestyles are the same in both panels.

In Figure 3 we also show existing limits from conventional direct detection
experiments Angloher:2015ewa ; Petricca:2017zdp ; Agnes:2018ves ;
Aprile:2018dbl ; Aprile:2019xxb  as well as the neutrino floor for a Xe-based
experiment Ruppin:2014bra  with the shaded regions. At small WIMP masses,
$m_{\chi}\lesssim 10\,$GeV/$c^{2}$, paleo-detectors could probe WIMPs with
WIMP-nucleon cross sections many orders of magnitude smaller than those
currently probed by conventional experiments; for example, at
$m_{\chi}=1\,$GeV/$c^{2}$ the best existing upper limit is $\sigma_{p}^{\rm
SI}\lesssim 10^{-38}\,{\rm cm}^{2}$ Angloher:2015ewa ; Petricca:2017zdp .
Moreover, paleo-detectors offer sensitivity deep into the conventional (solar)
neutrino floor in the $m_{\chi}\lesssim 10\,$GeV/$c^{2}$ mass range. Due to
the large exposures, a paleo-detector would observe $\mathcal{O}(100-1000)$
recoil tracks from solar neutrinos in the high-resolution readout scenario,
see Figure 2. In a spectral analysis, this large number of neutrino-induced
events combined with the excellent track-length (and, in turn, recoil energy)
resolution would allow one to characterize the solar neutrino background in
the “signal region” and hence probe WIMP-nucleon cross sections deep into the
conventional neutrino floor. For heavier WIMPs ($m_{\chi}\gtrsim
10\,$GeV/$c^{2}$), we see that paleo-detectors could probe WIMP-nucleon cross
sections roughly one and a half orders of magnitude below current upper
limits, but somewhat larger than the conventional neutrino floor in a Xe-based
direct detection experiment. In this WIMP mass region, the sensitivity of
paleo-detectors is limited by the radiogenic neutron backgrounds, as we will
discuss further below.

In the remainder of this section, we will explore the robustness of the
sensitivity projections shown in Figure 3 to the assumptions we have made. Let
us first note that the projected exclusion limits are virtually independent of
the ancillary measurements. For fixed normalization of the backgrounds in the
Asimov data, removing the Gaussian constraints [defined by Eq. (10)] on the
parameters controlling the normalization of the neutrino-induced backgrounds
($\Phi_{\nu}^{i}$) and the radiogenic backgrounds ($C^{238}$) as well as
removing the external constraint on the age of the samples ($t_{\rm age}$)
leads to a depreciation of the projected upper limits by less than a factor
two across the WIMP mass range shown in Figure 3. We find the most pronounced
depreciation of the upper limits in the $m_{\chi}\lesssim 1\,$GeV/$c^{2}$ WIMP
range when the constraints on the $\Phi_{\nu}^{i}$ are removed. Hence, the
data from the track-length spectrum alone suffices to search for DM with
paleo-detectors, even in the absence of any ancillary information about the
normalization of the backgrounds.

Figure 4: Projected 90 % confidence level exclusion limits in a gypsum
[Ca(SO${}_{4})\\!\cdot\\!2$(H2O)] paleo-detector in the high-resolution (left)
and high-exposure (right) readout scenarios. The black solid lines show the
projected limit using the Poisson log-likelihood, see Eq. (9), in our maximum
likelihood ratio test, Eq. (12), while the dashed lines show the projected
upper limit if we instead use a $\chi^{2}$ distribution, Eq. (14), for the
log-likelihood. The differently colored dashed lines are for different
assumptions on the relative systematic error of the background modeling
parameterized by $\Delta_{\rm Bkg}^{\rm sys}$, see Eq. (15). The gray and
sand-colored shaded areas indicate current upper limits and the conventional
neutrino floor in a Xe-based direct detection experiment, respectively, see
Figure 3. Colors and linestyles are the same in both panels. Note that in the
left panel, the $\mathcal{L}_{\rm Poisson}$, the
$\mathcal{L}_{\chi^{2}}(\Delta_{\rm Bkg}^{\rm sys}=0)$, and the
$\mathcal{L}_{\chi^{2}}(\Delta_{\rm Bkg}^{\rm sys}=0.01)$ lines are
practically laying on top of each other.

In order to explore how robust the sensitivity forecasts are with respect to
mismodeling of the shape of each background component, we replace the Poisson
log-likelihood, Eq. (9), with a $\chi^{2}$ distribution,

$\ln{\mathcal{L}}_{\chi^{2}}\left({\bm{D}}\middle|{\bm{\theta}};\sigma_{p}^{\rm
SI},m_{\chi}\right)=\sum_{i}\left[-\frac{1}{2}\left(\frac{D_{i}-N_{i}({\bm{\theta}};\sigma_{p}^{\rm
SI},m_{\chi})}{\sigma_{i}}\right)^{2}\right]\;,$ (14)

where $\sigma_{i}^{2}$ is the variance in the $i$-th bin. If we include only
the Poisson error of the data, $\sigma_{i}^{2}\to D_{i}$, the Poisson log-
likelihood and the $\chi^{2}$ likelihood become asymptotically equivalent in
the large-$N_{i}$ limit. Given the number of signal events one would observe
in a paleo-detector, see Figure 2, we thus expect to find the same sensitivity
for both forms of the likelihood. The $\chi^{2}$ distribution allows us to
include additional uncertainties, which we model as

$\sigma_{i}^{2}=D_{i}+\left(\Delta_{\rm Bkg}^{\rm sys}D_{i}\right)^{2}\;,$
(15)

where $\Delta_{\rm Bkg}^{\rm sys}$ parameterizes the (relative) systematic
modeling error. We note that it is straightforward to extend this to
$\sigma_{i}^{2}=D_{i}+\sum_{j}\left(\Delta_{i}^{j}D_{i}^{j}\right)^{2}$,
allowing for the assignment of a systematic error to the $j$-th contribution
to the data in the $i$-th bin. In this work, we will only employ a simple
overall systematic error $\Delta$ to explore the robustness of the sensitivity
projections.

In Figure 4 we show the projected upper limit for a gypsum paleo-detector
under different assumptions for the background-shape mismodeling. First, we
note that for $\Delta_{\rm Bkg}^{\rm sys}=0$, i.e. accounting only for the
Poisson error in the variance, $\sigma_{i}^{2}=D_{i}$, both the Poisson log-
likelihood and the $\chi^{2}$ distribution yield the same projected limit. In
the high-resolution scenario (left panel of Figure 4) we find that if we
include an error of $\Delta_{\rm Bkg}^{\rm sys}=0.1$, corresponding to 10 %
bin-to-bin mismodeling of the shape of the backgrounds, the sensitivity does
not depreciate significantly. For $\Delta_{\rm Bkg}^{\rm sys}=1$, the
projected upper limit is up to an order of magnitude worse than in the
$\Delta_{\rm Bkg}^{\rm sys}=0$ case for $m_{\chi}\lesssim 30\,$GeV/$c^{2}$,
with smaller changes at larger WIMP masses. In the high-exposure scenario
(right panel of Figure 4), the sensitivity is not quite as robust against
background-shape mismodeling. For small systematic errors, $\Delta_{\rm
Bkg}^{\rm sys}=0.01$, we find that the sensitivity becomes only slightly worse
than for $\Delta_{\rm Bkg}^{\rm sys}=0$, while for $\Delta_{\rm Bkg}^{\rm
sys}=0.1$ the projected upper limit is a factor of $\sim 5$ larger than for
$\Delta_{\rm Bkg}^{\rm sys}=0$. The high-exposure-readout scenario is most
relevant for the mass range $m_{\chi}\gtrsim 10\,$GeV/$c^{2}$, where the
dominant background stems from radiogenic neutrons. The corresponding track
length spectrum is straightforward to calibrate in paleo-detectors by
measuring the tracks in samples of the same mineral but with larger
${{}^{238}{\rm U}}$ concentration. Thus, it seems unlikely that the systematic
(bin-to-bin) shape errors would be as large as $\Delta_{\rm Bkg}^{\rm
sys}=0.1$ in an actual paleo-detector experiment. However, we note from Figure
4 that even for errors as large as $\Delta_{\rm Bkg}^{\rm sys}=0.1$, a gypsum
paleo-detector could still probe WIMP-nucleon cross sections almost an order
of magnitude below current experimental limits for $m_{\chi}\gtrsim
100\,$GeV/$c^{2}$.

Figure 5: Projected 90 % confidence level exclusion limits in a gypsum
[Ca(SO${}_{4})\\!\cdot\\!2$(H2O)] paleo-detector in the high-resolution (left)
and high-exposure (right) readout scenarios for different assumptions on the
${{}^{238}{\rm U}}$ concentration $C^{238}$ controlling the radiogenic
backgrounds. The solid black line shows the sensitivity with our fiducial
value for marine evaporites, $C^{238}=10^{-11}\,$g/g, while the differently
colored dashed lines show the projected upper limit for both larger and
smaller $C^{238}$. The gray and sand-colored shaded areas indicate current
upper limits and the conventional neutrino floor in a Xe-based direct
detection experiment, respectively, see Figure 3. Colors and linestyles are
the same in both panels.

In Figure 5 we show the effect of assuming different ${{}^{238}{\rm U}}$
concentrations on the projected upper limits in a gypsum paleo-detector.
Trivially, we find that paleo-detectors could probe smaller WIMP cross
sections if the samples are more radiopure. For most of the range of
${{}^{238}{\rm U}}$ concentrations, $10^{-12}\,{\rm g/g}\leq C^{238}\leq
10^{-7}\,$g/g shown in Figure 5 and most of the shown WIMP mass range, the
projected upper limit scales like $\sigma_{p}^{\rm SI}\propto\sqrt{C^{238}}$.
This scaling arises from the uncertainty of the radiogenic backgrounds in the
“signal region” being given by the associated Poisson error, with the
normalization of the radiogenic backgrounds inferred from the relevant
“control regions”. The $\propto\sqrt{C^{238}}$ scaling of the sensitivity is
broken if other backgrounds provide significant contributions in the “signal
region.” For example, the radiogenic background contribution can be
subdominant to that of solar neutrinos for $m_{\chi}\lesssim 10\,$GeV/$c^{2}$
or, for very small ${{}^{238}{\rm U}}$ concentrations, atmospheric neutrinos
for $m_{\chi}\gtrsim 100\,$GeV/$c^{2}$. Note that in the high-resolution
scenario, paleo-detectors could probe WIMP-nucleon cross sections down to the
conventional (solar) neutrino floor even for ${{}^{238}{\rm U}}$
concentrations as large as $C^{238}=10^{-8}\,$g/g in the $m_{\chi}\lesssim
10\,$GeV/$c^{2}$ mass range. For heavier WIMPs, we find that a gypsum paleo-
detector in the high-exposure readout scenario could still probe WIMP-nucleus
cross sections below current experimental limits even if the ${{}^{238}{\rm
U}}$ concentration is orders of magnitude larger than our fiducial assumption
of $C^{238}=10^{-11}\,$g/g.

Before moving on, let us note that the statistical framework used here to
project the sensitivity of paleo-detectors can easily be adapted to answer
different statistical questions or to incorporate additional uncertainties.
For example, by using the Asimov data as defined above (backgrounds only) but
also including a DM signal contribution, one can calculate the projected
discovery reach rather than the exclusion limit. Using the $\chi^{2}$
likelihood, it is straightforward to include arbitrarily complicated
uncertainties; another approach to including (modeling) uncertainties would be
to use random realizations of noisy mock data instead of the Asimov data set.
It is also straightforward to extend the framework to simultaneously analyze a
series of paleo-detectors. Samples of different ages can be used to explore
the sensitivity of paleo-detectors to time-dependent signals. For instance,
the Solar System could have traversed some denser region of the DM halo during
its past few revolutions around our Galaxy SUBSTRUCTURE . Time-dependent
neutrino signals may also arise from the evolution of our Sun Arellano:2021jul
, the Galactic supernova rate Baum:2019fqm , or the flux of cosmic rays
hitting Earth’s atmosphere Jordan:2020gxx . The code used in this work is
available here PaleoSens . Some of the options mentioned in this paragraph are
already implemented in the code, and we invite the reader to add the
capabilities in order to answer their own analysis questions.

##  5 Summary and Conclusions

This work surveys the prospects of searching for DM with paleo-detectors,
natural minerals which can record and retain nuclear damage tracks for
geological timescales. The exposure times of paleo-detectors can be of the
order of a billion years, and modern microscopy techniques may allow for the
readout of these damage tracks with nanometer spatial resolution in
macroscopic samples. Thus, paleo-detectors could constitute nuclear recoil
detectors with $\sim\,$keV recoil energy thresholds but with exposures
rivaling those of large (planned) neutrino detectors such as Super/Hyper-
Kamiokande or DUNE. Building on our previous work Baum:2018tfw ;
Drukier:2018pdy ; Edwards:2018hcf ; Baum:2019fqm ; Jordan:2020gxx , we have
discussed the relevant backgrounds for DM searches with paleo-detectors and
presented updated sensitivity forecasts using a full spectral analysis. For
the first time, the background induced by neutrinos from Galactic supernovae
is included in our projections and, in this sense, the results presented here
supersede the forecasts in Refs. Baum:2018tfw ; Drukier:2018pdy ;
Edwards:2018hcf . We have also put particular emphasis on exploring the
robustness of our sensitivity forecasts: we have demonstrated that DM-searches
with paleo-detectors are possible even in the absence of any external
information about the age of the samples, or the neutrino fluxes and the
${{}^{238}{\rm U}}$ concentration, which control the normalization of the
dominant backgrounds in paleo-detectors; the track length spectra alone
contain sufficient information to infer these parameters. Furthermore, we have
shown that the projected sensitivity is robust against uncertainties in the
modeling of the spectral shape of the backgrounds. Finally, we have
demonstrated that even if the uranium concentration in paleo-detector samples
is $C^{238}=10^{-8}\,$g/g, many orders of magnitude larger than what we expect
in the most radiopure samples obtained from ultra basic rock or marine
evaporite deposits, paleo-detectors could still probe WIMP-nucleon cross
sections below current limits. In particular, for WIMP masses
$m_{\chi}\lesssim 10\,$GeV/$c^{2}$, the sensitivity of paleo-detectors could
still reach down all the way to the conventional neutrino floor in a Xe-based
direct detection experiment.

We hope that this article serves as further motivation for experimental
efforts towards paleo-detectors. Ongoing efforts to demonstrate the
feasibility of reading out tracks from keV-scale nuclear recoils Hirose ; RPI
are exciting first steps in this direction. Conventional direct detection
experiments have been making impressive progress in their sensitivity to DM
during the last few decades. However, despite detectors becoming ever larger
and more sophisticated, they have failed to deliver (conclusive) evidence of
WIMP-nucleon interactions to date. Future experiments are becoming
increasingly expensive and challenging to operate. Thus, paleo-detectors are a
timely proposal to probe the remaining WIMP parameter space. Moreover, paleo-
detectors have applications to physics beyond DM searches, such as measuring
solar, supernova, or atmospheric neutrinos, and offer a unique tool to explore
the time-dependence of any signal over gigayear timescales.

###### Acknowledgements.

We thank our collaborators on previous and ongoing paleo-detector projects,
Ethan Brown, Francesco Capozzi, William DeRocco, Andrzej Drukier, Alfredo
Ferrari, Maciej Górski, Shigenobu Hirose, Shunsaku Horiuchi, Jonathon Jordan,
Saarik Kalia, Bradley Kavanagh, Kirsten McMichael, Maria Morone, Kelly Odgers,
Paola Sala, Morgan Schaller, Joshua Spitz, and Christoph Weniger. We are
deeply grateful to Frank Avignone, John Beacom, Juan Collar, Rodney Ewing, Ben
Feldman, Alfredo Ferella, Ariel Goobar, Peter Graham, Chris Kelso, Rafael
Lang, Dan Snowden-Ifft, Kai Sun and Martin Winkler for discussion about
various aspects of paleo-detectors. The code used to produce the results in
this work is available at: paleoSpec PaleoSpec  for the computation of the
signal and background spectra, and paleoSens PaleoSens  for the sensitivity
forecast. SB is supported in part by NSF Grant PHY-1720397, DOE HEP QuantISED
award #100495, and the Gordon and Betty Moore Foundation Grant GBMF7946. TE
and KF acknowledge support by the Vetenskapsrådet (Swedish Research Council)
through contract No. 638-2013-8993 and the Oskar Klein Centre for
Cosmoparticle Physics. KF is the Jeff & Gail Kodosky Endowed Chair of Physics
at the University of Texas in Austin and is grateful for support. PS is
partially supported by the research grant “The Dark Universe: A Synergetic
Multi-messenger Approach” number 2017X7X85K under the program PRIN 2017 funded
by the Ministero dell’Istruzione, Università e della Ricerca (MIUR), and by
the “Hidden” European ITN project (H2020-MSCA-ITN-2019//860881-HIDDeN).

## References

  * (1) G. Bertone and D. Hooper, _History of dark matter_ , _Rev. Mod. Phys._ 90 (2018) 045002, [1605.04909].
  * (2) G. Jungman, M. Kamionkowski and K. Griest, _Supersymmetric dark matter_ , _Phys. Rept._ 267 (1996) 195–373, [hep-ph/9506380].
  * (3) G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, M. Pierre et al., _The waning of the WIMP? A review of models, searches, and constraints_ , _Eur. Phys. J. C_ 78 (2018) 203, [1703.07364].
  * (4) L. Roszkowski, E. M. Sessolo and S. Trojanowski, _WIMP dark matter candidates and searches—current status and future prospects_ , _Rept. Prog. Phys._ 81 (2018) 066201, [1707.06277].
  * (5) A. Drukier and L. Stodolsky, _Principles and Applications of a Neutral Current Detector for Neutrino Physics and Astronomy_ , _Phys. Rev._ D30 (1984) 2295.
  * (6) M. W. Goodman and E. Witten, _Detectability of Certain Dark Matter Candidates_ , _Phys. Rev._ D31 (1985) 3059.
  * (7) A. K. Drukier, K. Freese and D. N. Spergel, _Detecting Cold Dark Matter Candidates_ , _Phys. Rev._ D33 (1986) 3495–3508.
  * (8) D. N. Spergel, _The Motion of the Earth and the Detection of Wimps_ , _Phys. Rev._ D37 (1988) 1353.
  * (9) J. I. Collar and F. T. Avignone, _Diurnal modulation effects in cold dark matter experiments_ , _Phys. Lett._ B275 (1992) 181–185.
  * (10) LUX collaboration, D. S. Akerib et al., _Results from a search for dark matter in the complete LUX exposure_ , _Phys. Rev. Lett._ 118 (2017) 021303, [1608.07648].
  * (11) DarkSide collaboration, P. Agnes et al., _DarkSide-50 532-day Dark Matter Search with Low-Radioactivity Argon_ , _Phys. Rev. D_ 98 (2018) 102006, [1802.07198].
  * (12) XENON collaboration, E. Aprile et al., _Dark Matter Search Results from a One Ton-Year Exposure of XENON1T_ , _Phys. Rev. Lett._ 121 (2018) 111302, [1805.12562].
  * (13) PandaX-II collaboration, Q. Wang et al., _Results of dark matter search using the full PandaX-II exposure_ , _Chin. Phys. C_ 44 (2020) 125001, [2007.15469].
  * (14) CRESST collaboration, G. Angloher et al., _Results on light dark matter particles with a low-threshold CRESST-II detector_ , _Eur. Phys. J. C_ 76 (2016) 25, [1509.01515].
  * (15) EDELWEISS collaboration, E. Armengaud et al., _Constraints on low-mass WIMPs from the EDELWEISS-III dark matter search_ , _JCAP_ 05 (2016) 019, [1603.05120].
  * (16) SuperCDMS collaboration, R. Agnese et al., _Low-mass dark matter search with CDMSlite_ , _Phys. Rev. D_ 97 (2018) 022002, [1707.01632].
  * (17) CRESST collaboration, F. Petricca et al., _First results on low-mass dark matter from the CRESST-III experiment_ , _J. Phys. Conf. Ser._ 1342 (2020) 012076, [1711.07692].
  * (18) EDELWEISS collaboration, E. Armengaud et al., _Searching for low-mass dark matter particles with a massive Ge bolometer operated above-ground_ , _Phys. Rev. D_ 99 (2019) 082003, [1901.03588].
  * (19) XENON collaboration, E. Aprile et al., _Physics reach of the XENON1T dark matter experiment_ , _JCAP_ 1604 (2016) 027, [1512.07501].
  * (20) B. J. Mount et al., _LUX-ZEPLIN (LZ) Technical Design Report_ , 1703.09144.
  * (21) DARWIN collaboration, J. Aalbers et al., _DARWIN: towards the ultimate dark matter detector_ , _JCAP_ 1611 (2016) 017, [1606.07001].
  * (22) C. E. Aalseth et al., _DarkSide-20k: A 20 tonne two-phase LAr TPC for direct dark matter detection at LNGS_ , _Eur. Phys. J. Plus_ 133 (2018) 131, [1707.08145].
  * (23) DEAP-3600 collaboration, P. A. Amaudruz et al., _Design and Construction of the DEAP-3600 Dark Matter Detector_ , _Astropart. Phys._ 108 (2019) 1–23, [1712.01982].
  * (24) E. Daw et al., _The DRIFT Directional Dark Matter Experiments_ , _EAS Publ. Ser._ 53 (2012) 11–18, [1110.0222].
  * (25) F. Cappella et al., _On the potentiality of the $ZnWO_{4}$ anisotropic detectors to measure the directionality of Dark Matter_, _Eur. Phys. J. C_ 73 (2013) 2276.
  * (26) DRIFT collaboration, J. B. R. Battat et al., _First background-free limit from a directional dark matter experiment: results from a fully fiducialised DRIFT detector_ , _Phys. Dark Univ._ 9-10 (2015) 1–7, [1410.7821].
  * (27) Q. Riffard et al., _Dark Matter directional detection with MIMAC_ , in _Proceedings, 48th Rencontres de Moriond on Very High Energy Phenomena in the Universe: La Thuile, Italy, March 9-16, 2013_ , pp. 227–230, 2013, 1306.4173, http://inspirehep.net/record/1238915/files/arXiv:1306.4173.pdf.
  * (28) D. Santos et al., _MIMAC: MIcro-tpc MAtrix of Chambers for dark matter directional detection_ , _J. Phys. Conf. Ser._ 469 (2013) 012002, [1311.0616].
  * (29) DMTPC collaboration, J. Monroe, _Status and Prospects of the DMTPC Directional Dark Matter Experiment_ , _EAS Publ. Ser._ 53 (2012) 19–24.
  * (30) DMTPC collaboration, M. Leyton, _Directional dark matter detection with the DMTPC m 3 experiment_, _J. Phys. Conf. Ser._ 718 (2016) 042035.
  * (31) K. Miuchi et al., _First underground results with NEWAGE-0.3a direction-sensitive dark matter detector_ , _Phys. Lett._ B686 (2010) 11–17, [1002.1794].
  * (32) K. Nakamura et al., _Direction-sensitive dark matter search with gaseous tracking detector NEWAGE-0.3b’_ , _PTEP_ 2015 (2015) 043F01.
  * (33) DRIFT collaboration, J. B. R. Battat et al., _Low Threshold Results and Limits from the DRIFT Directional Dark Matter Detector_ , _Astropart. Phys._ 91 (2017) 65–74, [1701.00171].
  * (34) S. E. Vahsen et al., _CYGNUS: Feasibility of a nuclear recoil observatory with directional sensitivity to dark matter and neutrinos_ , 2008.12587.
  * (35) S. Baum, A. K. Drukier, K. Freese, M. Górski and P. Stengel, _Searching for Dark Matter with Paleo-Detectors_ , _Phys. Lett. B_ 803 (2020) 135325, [1806.05991].
  * (36) A. K. Drukier, S. Baum, K. Freese, M. Górski and P. Stengel, _Paleo-detectors: Searching for Dark Matter with Ancient Minerals_ , _Phys. Rev. D_ 99 (2019) 043014, [1811.06844].
  * (37) T. D. P. Edwards, B. J. Kavanagh, C. Weniger, S. Baum, A. K. Drukier, K. Freese et al., _Digging for dark matter: Spectral analysis and discovery potential of paleo-detectors_ , _Phys. Rev. D_ 99 (2019) 043541, [1811.10549].
  * (38) R. L. Fleischer, P. B. Price, R. M. Walker and E. L. Hubbard, _Track registration in various solid-state nuclear track detectors_ , _Phys. Rev._ 133 (Mar, 1964) A1443–A1449.
  * (39) R. L. Fleischer, P. B. Price and R. M. Walker, _Tracks of charged particles in solids_ , _Science_ 149 (1965) 383–393.
  * (40) R. L. Fleischer, P. B. Price and R. M. Walker, _Solid-state track detectors: applications to nuclear science and geophysics_ , _Ann. Rev. Nucl. Part. Sci._ 15 (1965) 1–28.
  * (41) S.-L. Guo, B.-L. Chen and S. Durrani, _Chapter 4 - solid-state nuclear track detectors_ , in _Handbook of Radioactivity Analysis (Third Edition)_ (M. F. L’Annunziata, ed.), pp. 233 – 298.  Academic Press, Amsterdam, third edition ed., 2012.  DOI.
  * (42) M. D. S. Felix M. Gradstein, James G. Ogg and G. M. Ogg, eds., _The Geologic Time Scale_.  Elsevier, 2012, 10.1016/c2011-1-08249-8.
  * (43) K. Gallagher, R. Brown and C. Johnson, _Fission track analysis and its applications to geological problems_ , _Annual Review of Earth and Planetary Sciences_ 26 (1998) 519–572, [https://doi.org/10.1146/annurev.earth.26.1.519].
  * (44) P. van den Haute and F. de Corte, eds., _Advances in Fission-Track Geochronology_.  Springer, 1998, 10.1007/978-94-015-9133-1.
  * (45) M. Toulemonde, W. Assmann, C. Dufour, A. Meftah, F. Studer and C. Trautmann, _Experimental Phenomena and Thermal Spike Model Description of Ion Tracks in Amorphisable Inorganic Insulators_ , in _Ion Beam Science: Solved and Unsolved problems, Mat. Fys. Medd. Dan. Vid. Selsk._ (P. Sigmund, ed.), vol. 52, (Copenhagen), p. 263, Det Kongelige Danske Videnskabernes Selskab, 2006.
  * (46) R. Hill, J. A. Notte and L. Scipioni, _Chapter 2 - scanning helium ion microscopy_ , in _Advances in Imaging and Electron Physics_ (P. W. Hawkes, ed.), vol. 170 of _Advances in Imaging and Electron Physics_ , pp. 65 – 148.  Elsevier, 2012.  DOI.
  * (47) R. van Gastel, G. Hlawacek, H. J. Zandvliet and B. Poelsema, _Subsurface analysis of semiconductor structures with helium ion microscopy_ , _Microelectronics Reliability_ 52 (2012) 2104 – 2109.
  * (48) M. Rodriguez, W. Li, F. Chen, C. Trautmann, T. Bierschenk, B. Afra et al., _Saxs and tem investigation of ion tracks in neodymium-doped yttrium aluminium garnet_ , _Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms_ 326 (2014) 150 – 153.
  * (49) F. Schaff, M. Bech, P. Zaslansky, C. Jud, M. Liebi, M. Guizar-Sicairos et al., _Six-dimensional real and reciprocal space small-angle x-ray scattering tomography_ , _Nature_ 527 (11, 2015) 353.
  * (50) M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen et al., _X-ray ptychographic computed tomography at 16 nm isotropic 3d resolution_ , _Scientific Reports_ 4 (01, 2014) 3857.
  * (51) J. Bartz, C. Zeissler, V. Fomenko and M. Akselrod, _An imaging spectrometer based on high resolution microscopy of fluorescent aluminum oxide crystal detectors_ , _Radiation Measurements_ 56 (2013) 273 – 276.
  * (52) J. Kouwenberg, G. Kremers, J. Slotman, H. Woltenbeek, A. Houtsmuller, A. Denkova et al., _Alpha particle spectroscopy using fntd and sim super-resolution microscopy_ , _Journal of Microscopy_ 270 326–334.
  * (53) M. S. Joens, C. Huynh, J. M. Kasuboski, D. Ferranti, Y. J. Sigal, F. Zeitvogel et al., _Helium ion microscopy (him) for the imaging of biological samples at sub-nanometer resolution_ , _Sci. Rep._ 3 (12, 2013) 3514.
  * (54) M. P. Echlin, M. Straw, S. Randolph, J. Filevich and T. M. Pollock, _The tribeam system: Femtosecond laser ablation in situ sem_ , _Materials Characterization_ 100 (2015) 1 – 12.
  * (55) M. J. Pfeifenberger, M. Mangang, S. Wurster, J. Reiser, A. Hohenwarter, W. Pfleging et al., _The use of femtosecond laser ablation as a novel tool for rapid micro-mechanical sample preparation_ , _Materials & Design_ 121 (2017) 109 – 118.
  * (56) S. J. Randolph, J. Filevich, A. Botman, R. Gannon, C. Rue and M. Straw, _In situ femtosecond pulse laser ablation for large volume 3D analysis in scanning electron microscope systems_ , _Journal of Vacuum Science & Technology B_ 36 (2018) 06JB01.
  * (57) E. Goto, _On the Observation of Magnetic Monopoles_ , _J. Phys. Soc. Japan_ 13 (1958) 1413–1418.
  * (58) E. Goto, H. H. Kolm and K. W. Ford, _Search for Ferromagnetically Trapped Magnetic Monopoles of Cosmic-Ray Origin_ , _Phys. Rev._ 132 (1963) 387.
  * (59) R. L. Fleischer, I. S. Jacobs, W. M. Schwarz, P. B. Price and H. G. Goodell, _Search for multiply charged dirac magnetic poles_ , _Phys. Rev._ 177 (1969) 2029–2035.
  * (60) R. L. Fleischer, H. R. Hart, I. S. Jacobs, P. B. Price, W. M. Schwarz and F. Aumento, _Search for magnetic monopoles in deep ocean deposits_ , _Phys. Rev._ 184 (1969) 1393–1397.
  * (61) R. L. Fleischer, P. B. Price and R. T. Woods, _Search for tracks of massive, multiply charged magnetic poles_ , _Phys. Rev._ 184 (1969) 1398–1401.
  * (62) L. W. Alvarez, P. H. Eberhard, R. R. Ross and R. D. Watt, _Search for Magnetic Monopoles in the Lunar Sample_ , _Science_ 167 (1970) 701–703.
  * (63) H. H. Kolm, F. Villa and A. Odian, _SEARCH FOR MAGNETIC MONOPOLES_ , _Phys. Rev._ D4 (1971) 1285.
  * (64) P. H. Eberhard, R. R. Ross, L. W. Alvarez and R. D. Watt, _SEARCH FOR MAGNETIC MONOPOLES IN LUNAR MATERIAL_ , _Phys. Rev._ D4 (1971) 3260.
  * (65) R. R. Ross, P. H. Eberhard, L. W. Alvarez and R. D. Watt, _SEARCH FOR MAGNETIC MONOPOLES IN LUNAR MATERIAL USING AN ELECTROMAGNETIC DETECTOR_ , _Phys. Rev._ D8 (1973) 698.
  * (66) P. B. Price, S.-l. Guo, S. P. Ahlen and R. L. Fleischer, _Search for GUT Magnetic Monopoles at a Flux Level Below the Parker Limit_ , _Phys. Rev. Lett._ 52 (1984) 1265.
  * (67) J. M. Kovalik and J. L. Kirschvink, _New Superconducting Quantum Interface Device Based Constraints on the Abundance of Magnetic Monopoles Trapped in Matter: An Investigation of Deeply Buried Rocks_ , _Phys. Rev._ A33 (1986) 1183–1187.
  * (68) P. B. Price and M. H. Salamon, _Search for Supermassive Magnetic Monopoles Using Mica Crystals_ , _Phys. Rev. Lett._ 56 (1986) 1226–1229.
  * (69) D. Ghosh and S. Chatterjea, _Supermassive magnetic monopoles flux from the oldest mica samples_ , _Europhys. Lett._ 12 (1990) 25–28.
  * (70) H. Jeon and M. J. Longo, _Search for magnetic monopoles trapped in matter_ , _Phys. Rev. Lett._ 75 (1995) 1443–1446, [hep-ex/9508003].
  * (71) J. I. Collar and K. Zioutas, _Limits on exotic heavily ionizing particles from the geological abundance of fullerenes_ , _Phys. Rev. Lett._ 83 (1999) 3097, [astro-ph/9902310].
  * (72) D. P. Snowden-Ifft, E. S. Freeman and P. B. Price, _Limits on dark matter using ancient mica_ , _Phys. Rev. Lett._ 74 (1995) 4133–4136.
  * (73) J. I. Collar and F. T. Avignone, III, _Nuclear tracks from cold dark matter interactions in mineral crystals: A Computational study_ , _Nucl. Instrum. Meth._ B95 (1995) 349, [astro-ph/9505055].
  * (74) J. Engel, M. T. Ressell, I. S. Towner and W. E. Ormand, _Response of mica to weakly interacting massive particles_ , _Phys. Rev._ C52 (1995) 2216–2221, [hep-ph/9504322].
  * (75) D. P. Snowden-Ifft and A. J. Westphal, _Unique signature of dark matter in ancient mica_ , _Phys. Rev. Lett._ 78 (1997) 1628–1631, [astro-ph/9701215].
  * (76) S. Baum, T. D. P. Edwards, B. J. Kavanagh, P. Stengel, A. K. Drukier, K. Freese et al., _Paleodetectors for Galactic supernova neutrinos_ , _Phys. Rev. D_ 101 (2020) 103017, [1906.05800].
  * (77) J. R. Jordan, S. Baum, P. Stengel, A. Ferrari, M. C. Morone, P. Sala et al., _Measuring Changes in the Atmospheric Neutrino Rate Over Gigayear Timescales_ , _Phys. Rev. Lett._ 125 (2020) 231802, [2004.08394].
  * (78) N. T. Arellano and S. Horiuchi, _Measuring solar neutrinos over Gigayear timescales with Paleo Detectors_ , 2102.01755.
  * (79) S. Rajendran, N. Zobrist, A. O. Sushkov, R. Walsworth and M. Lukin, _A method for directional detection of dark matter using spectroscopy of crystal defects_ , _Phys. Rev. D_ 96 (2017) 035009, [1705.09760].
  * (80) R. Essig, J. Mardon, O. Slone and T. Volansky, _Detection of sub-GeV Dark Matter and Solar Neutrinos via Chemical-Bond Breaking_ , _Phys. Rev. D_ 95 (2017) 056011, [1608.02940].
  * (81) R. Budnik, O. Chesnovsky, O. Slone and T. Volansky, _Direct Detection of Light Dark Matter and Solar Neutrinos via Color Center Production in Crystals_ , _Phys. Lett. B_ 782 (2018) 242–250, [1705.03016].
  * (82) B. K. Cogswell, A. Goel and P. Huber, _Passive low-energy nuclear recoil detection with color centers_ , 2104.13926.
  * (83) R. Ebadi et al., _Ultra-Heavy Dark Matter Search with Electron Microscopy of Geological Quartz_ , 2105.03998.
  * (84) S. Baum, W. DeRocco, T. D. P. Edwards and S. Kalia, _Throwing rocks at dinosaurs: Time-varying dark matter signals in paleo-detectors_ , .
  * (85) https://github.com/sbaum90/paleoSpec.
  * (86) https://github.com/sbaum90/paleoSens.
  * (87) J. Engel, _Nuclear form-factors for the scattering of weakly interacting massive particles_ , _Phys. Lett._ B264 (1991) 114–119.
  * (88) J. Engel, S. Pittel and P. Vogel, _Nuclear physics of dark matter detection_ , _Int. J. Mod. Phys._ E1 (1992) 1–37.
  * (89) M. T. Ressell, M. B. Aufderheide, S. D. Bloom, K. Griest, G. J. Mathews and D. A. Resler, _Nuclear shell model calculations of neutralino - nucleus cross-sections for Si-29 and Ge-73_ , _Phys. Rev._ D48 (1993) 5519–5535, [hep-ph/9307228].
  * (90) V. A. Bednyakov and F. Simkovic, _Nuclear spin structure in dark matter search: The Zero momentum transfer limit_ , _Phys. Part. Nucl._ 36 (2005) 131–152, [hep-ph/0406218].
  * (91) V. A. Bednyakov and F. Simkovic, _Nuclear spin structure in dark matter search: The Finite momentum transfer limit_ , _Phys. Part. Nucl._ 37 (2006) S106–S128, [hep-ph/0608097].
  * (92) J. Fan, M. Reece and L.-T. Wang, _Non-relativistic effective theory of dark matter direct detection_ , _JCAP_ 1011 (2010) 042, [1008.1591].
  * (93) A. L. Fitzpatrick, W. Haxton, E. Katz, N. Lubbers and Y. Xu, _The Effective Field Theory of Dark Matter Direct Detection_ , _JCAP_ 1302 (2013) 004, [1203.3542].
  * (94) R. H. Helm, _Inelastic and Elastic Scattering of 187-Mev Electrons from Selected Even-Even Nuclei_ , _Phys. Rev._ 104 (1956) 1466–1475.
  * (95) J. D. Lewin and P. F. Smith, _Review of mathematics, numerical factors, and corrections for dark matter experiments based on elastic nuclear recoil_ , _Astropart. Phys._ 6 (1996) 87–112.
  * (96) G. Duda, A. Kemper and P. Gondolo, _Model Independent Form Factors for Spin Independent Neutralino-Nucleon Scattering from Elastic Electron Scattering Data_ , _JCAP_ 0704 (2007) 012, [hep-ph/0608035].
  * (97) L. Vietze, P. Klos, J. Menéndez, W. C. Haxton and A. Schwenk, _Nuclear structure aspects of spin-independent WIMP scattering off xenon_ , _Phys. Rev._ D91 (2015) 043520, [1412.6091].
  * (98) D. Gazda, R. Catena and C. Forssén, _Ab initio nuclear response functions for dark matter searches_ , _Phys. Rev._ D95 (2017) 103011, [1612.09165].
  * (99) C. Körber, A. Nogga and J. de Vries, _First-principle calculations of Dark Matter scattering off light nuclei_ , _Phys. Rev._ C96 (2017) 035805, [1704.01150].
  * (100) M. Hoferichter, P. Klos, J. Menéndez and A. Schwenk, _Nuclear structure factors for general spin-independent WIMP-nucleus scattering_ , _Phys. Rev. D_ 99 (2019) 055031, [1812.05617].
  * (101) S. E. Koposov, H.-W. Rix and D. W. Hogg, _Constraining the Milky Way potential with a 6-D phase-space map of the GD-1 stellar stream_ , _Astrophys. J._ 712 (2010) 260–273, [0907.1085].
  * (102) T. Piffl et al., _The RAVE survey: the Galactic escape speed and the mass of the Milky Way_ , _Astron. Astrophys._ 562 (2014) A91, [1309.4293].
  * (103) J. Bovy et al., _The Milky Way’s circular velocity curve between 4 and 14 kpc from APOGEE data_ , _Astrophys. J._ 759 (2012) 131, [1209.0759].
  * (104) K. Freese, M. Lisanti and C. Savage, _Colloquium: Annual modulation of dark matter_ , _Rev. Mod. Phys._ 85 (2013) 1561–1581, [1209.3339].
  * (105) XENON collaboration, E. Aprile et al., _Light Dark Matter Search with Ionization Signals in XENON1T_ , _Phys. Rev. Lett._ 123 (2019) 251801, [1907.11485].
  * (106) J. F. Ziegler, J. P. Biersack and U. Littmark, _The Stopping and Range of Ions in Solids_.  Pergamon Press, New York, 01, 1985.
  * (107) J. F. Ziegler, M. D. Ziegler and J. P. Biersack, _SRIM – The stopping and range of ions in matter (2010)_ , _Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms_ 268 (2010) 1818 – 1823.
  * (108) S. Hirose. private communication.
  * (109) E. Brown. private communication.
  * (110) D. Mei and A. Hime, _Muon-induced background study for underground laboratories_ , _Phys. Rev._ D73 (2006) 053004, [astro-ph/0512125].
  * (111) SNO collaboration, B. Aharmim et al., _Measurement of the Cosmic Ray and Neutrino-Induced Muon Flux at the Sudbury Neutrino Observatory_ , _Phys. Rev. D_ 80 (2009) 012001, [0902.2776].
  * (112) C. A. J. O’Hare, _Can we overcome the neutrino floor at high masses?_ , _Phys. Rev. D_ 102 (2020) 063024, [2002.07499].
  * (113) E. Cappellaro, R. Barbon and M. Turatto, _Supernova statistics_ , _Springer Proc. Phys._ 99 (2005) 347–354, [astro-ph/0310859].
  * (114) R. Diehl et al., _Radioactive Al-26 and massive stars in the galaxy_ , _Nature_ 439 (2006) 45–47, [astro-ph/0601015].
  * (115) A. Strumia and F. Vissani, _Neutrino masses and mixings and…_ , hep-ph/0606054.
  * (116) W. Li, R. Chornock, J. Leaman, A. V. Filippenko, D. Poznanski, X. Wang et al., _Nearby supernova rates from the Lick Observatory Supernova Search - III. The rate-size relation, and the rates as a function of galaxy Hubble type and colour_ , _Mon. Not. Roy. Astron. Soc._ 412 (2011) 1473–1507, [1006.4613].
  * (117) M. T. Botticella, S. J. Smartt, R. C. Kennicutt, Jr., E. Cappellaro, M. Sereno and J. C. Lee, _A comparison between star formation rate diagnostics and rate of core collapse supernovae within 11 Mpc_ , _Astron. Astrophys._ 537 (2012) A132, [1111.1692].
  * (118) S. M. Adams, C. S. Kochanek, J. F. Beacom, M. R. Vagins and K. Z. Stanek, _Observing the Next Galactic Supernova_ , _Astrophys. J._ 778 (2013) 164, [1306.0559].
  * (119) J. F. Beacom, _The Diffuse Supernova Neutrino Background_ , _Ann. Rev. Nucl. Part. Sci._ 60 (2010) 439–462, [1004.3311].
  * (120) J. Billard, L. Strigari and E. Figueroa-Feliciano, _Implication of neutrino backgrounds on the reach of next generation dark matter direct detection experiments_ , _Phys. Rev._ D89 (2014) 023524, [1307.5458].
  * (121) P. Madau and M. Dickinson, _Cosmic Star Formation History_ , _Ann. Rev. Astron. Astrophys._ 52 (2014) 415–486, [1403.0007].
  * (122) L.-G. Strolger, T. Dahlen, S. A. Rodney, O. Graur, A. G. Riess, C. McCully et al., _The Rate of Core Collapse Supernovae to Redshift 2.5 From The CANDELS and CLASH Supernova Surveys_ , _Astrophys. J._ 813 (2015) 93, [1509.06574].
  * (123) S. Thomson and G. Wardle, _Coloured natural rocksalts: a study of their helium contents, colours and impurities_ , _Geochimica et Cosmochimica Acta_ 5 (1954) 169 – 184.
  * (124) K. C. Condie, C. S. Kuo, R. M. Walker and V. R. Murthy, _Uranium Distribution in Separated Clinopyroxenes from Four Eclogites_ , _Science_ 165 (1969) 57–59.
  * (125) J. A. S. Adams, J. K. Osmond and J. J. W. Rogers, _The geochemistry of thorium and uranium_ , _Phys. Chem. Earth_ 3 (1959) 298–348.
  * (126) M. Seitz and S. Hart, _Uranium and boron distributions in some oceanic ultramafic rocks_ , _Earth and Planetary Science Letters_ 21 (1973) 97 – 107.
  * (127) W. E. Dean, _Section 5 trace and minor elements in evaporites_ , in _Marine Evaporites_.  SEPM Society for Sedimentary Geology, 1987.  DOI.
  * (128) M. Yui, Y. Kikawada, T. Oi, T. Honda and T. Nozaki, _Abundance of Uranium and Thorium in Rock Salts_ , _Radioisotopes_ 47 (1998) 488–492.
  * (129) W. Sanford, M. Doughten, T. Coplen, A. Hunt and T. Bullen, _Evidence for high salinity of Early Cretaceous sea water from the Chesapeake Bay crater_ , _Nature_ 503 (2013) 252–256.
  * (130) J. I. Collar, _Comments on ’limits on dark matter using ancient mica’_ , _Phys. Rev. Lett._ 76 (1996) 331, [astro-ph/9511055].
  * (131) D. P. Snowden-Ifft, E. S. Freeman and P. B. Price, _Snowden-Ifft, Freemen, and Price Reply (A Reply to the Comment by Juan I Collar)_ , _Phys. Rev. Lett._ 76 (1996) 332.
  * (132) _SOURCES 4A: A Code for Calculating (alpha,n), Spontaneous Fission, and Delayed Neutron Sources and Spectra_ , Tech. Rep. LA-13639-MS, Los Alamos National Lab, 1999.  10.2172/15215.
  * (133) A. J. Koning and D. Rochman, _Modern Nuclear Data Evaluation with the TALYS Code System_ , _Nucl. Data Sheets_ 113 (2012) 2841–2934.
  * (134) D. Rochman, A. J. Koning, J. C. Sublet, M. Fleming et al., _The TENDL library: hope, reality and future_ , in _Proceedings of the International Conference on Nuclear Data for Science and Technology, September 11-16, 2016, Bruges, Belgium_ , 2016, https://tendl.web.psi.ch/bib_rochman/tendl.nd2016.pdf.
  * (135) J. C. Sublet, A. J. Koning, D. Rochman, M. Fleming and M. Gilbert, _TENDL-2015: Delivering Both Completeness and Robustness_ , in _Advances in Nuclear Nonproliferation Technology and Policy Conference, Sept. 25-30, Santa Fe, NM, USA_ , https://tendl.web.psi.ch/bib_rochman/ANTPC_TENDL.pdf.
  * (136) M. Fleming, J. C. Sublet, J. Kopecky, D. Rochman and A. J. Koning, _Probing experimental and systematic trends of the neutron-induced TENDL-2014 nuclear data library_ , in _CCFE report UKAEA-R(15)29, October 2015_ , 2015, https://tendl.web.psi.ch/bib_rochman/UKAEA-R(15)29_30.9.15.pdf.
  * (137) N. Soppera, M. Bossant and E. Dupont, _JANIS 4: An Improved Version of the NEA Java-based Nuclear Data Information System_ , _Nucl. Data Sheets_ 120 (2014) 294–296.
  * (138) A. Ferrari, P. R. Sala, A. Fasso and J. Ranft, _FLUKA: A multi-particle transport code (Program version 2005)_ , .
  * (139) T. T. Böhlen, F. Cerutti, M. P. W. Chin, A. Fassò, A. Ferrari, P. G. Ortega et al., _The FLUKA Code: Developments and Challenges for High Energy and Medical Applications_ , _Nucl. Data Sheets_ 120 (2014) 211–214.
  * (140) G. Battistoni, A. Ferrari, M. Lantz, P. R. Sala and G. I. Smirnov, _A neutrino-nucleon interaction generator for the fluka monte carlo code_ , in _CERN-Proceedings-2010-001_ , pp. 387–394, 2010.
  * (141) P. P. Povinec, _New ultra-sensitive radioanalytical technologies for new science_ , _J. Radioanal. Nucl. Chem._ 316 (2018) 893–931.
  * (142) P. P. Povinec et al., _Ultra-sensitive radioanalytical technologies for underground physics experiments_ , _J. Radioanal. Nucl. Chem._ 318 (2018) 677–684.
  * (143) T. D. P. Edwards and C. Weniger, _A Fresh Approach to Forecasting in Astroparticle Physics and Dark Matter Searches_ , _JCAP_ 1802 (2018) 021, [1704.05458].
  * (144) T. D. P. Edwards and C. Weniger, _swordfish: Efficient Forecasting of New Physics Searches without Monte Carlo_ , 1712.05401.
  * (145) https://github.com/cweniger/swordfish.
  * (146) G. Cowan, K. Cranmer, E. Gross and O. Vitells, _Asymptotic formulae for likelihood-based tests of new physics_ , _Eur. Phys. J. C_ 71 (2011) 1554, [1007.1727].
  * (147) J. Billard, F. Mayet and D. Santos, _Assessing the discovery potential of directional detection of dark matter_ , _Phys. Rev. D_ 85 (2012) 035006, [1110.6079].
  * (148) J. Conrad, _Statistical Issues in Astrophysical Searches for Particle Dark Matter_ , _Astropart. Phys._ 62 (2015) 165–177, [1407.6617].
  * (149) S. S. Wilks, _The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses_ , _Annals Math. Statist._ 9 (1938) 60–62.
  * (150) DarkSide collaboration, P. Agnes et al., _Low-Mass Dark Matter Search with the DarkSide-50 Experiment_ , _Phys. Rev. Lett._ 121 (2018) 081307, [1802.06994].
  * (151) F. Ruppin, J. Billard, E. Figueroa-Feliciano and L. Strigari, _Complementarity of dark matter detectors in light of the neutrino background_ , _Phys. Rev._ D90 (2014) 083510, [1408.3581].