11institutetext: Department of Physics, Lancaster University, Lancaster, LA1
4YB, UK. 22institutetext: Department of Physics, Royal Holloway University of
London, Egham, Surrey, TW20 0EX, UK. 33institutetext: RIKEN Center for Quantum
Computing, RIKEN, Wako, 351-0198, Japan. 44institutetext: Rudolf Peierls
Centre for Theoretical Physics, University of Oxford 1 Keble Road, Oxford, OX1
3NP, UK.

# QUEST-DMC superfluid 3He detector for sub-GeV dark matter

QUEST-DMC collaboration: S. Autti<EMAIL_ADDRESS>A. Casey 22 N. Eng
22 N. Darvishi 22 P. Franchini<EMAIL_ADDRESS>R. P. Haley 11
P.J. Heikkinen 22 A. Jennings 33 A. Kemp 22 E. Leason
<EMAIL_ADDRESS>L.V. Levitin 22 J. Monroe 22 J. March-Russel 44
M. T. Noble 11 J. R. Prance 11 X. Rojas 22 T. Salmon 11 J. Saunders 22 R.
Smith 22 M. D. Thompson 11 V. Tsepelin 11 S. M. West 22 L. Whitehead 11 V. V.
Zavjalov 11 D. E. Zmeev 11

(Received: date / Revised version: date)

###### Abstract

The focus of dark matter searches to date has been on Weakly Interacting
Massive Particles (WIMPs) in the GeV/$c^{2}$-TeV/$c^{2}$ mass range. The
direct, indirect and collider searches in this mass range have been extensive
but ultimately unsuccessful, providing a strong motivation for widening the
search outside this range. Here we describe a new concept for a dark matter
experiment, employing superfluid 3He as a detector for dark matter that is
close to the mass of the proton, of order 1 GeV/$c^{2}$. The QUEST-DMC
detector concept is based on quasiparticle detection in a bolometer cell by a
nanomechanical resonator. In this paper we develop the energy measurement
methodology and detector response model, simulate candidate dark matter
signals and expected background interactions, and calculate the sensitivity of
such a detector. We project that such a detector can reach sub-eV nuclear
recoil energy threshold, opening up new windows on the parameter space of both
spin-dependent and spin-independent interactions of light dark matter
candidates.

##  1 Introduction

Dark matter plays a vital role in the evolution of the universe, for example,
it played a central role in the formation of structure in the early universe
and today plays a key role in stopping galaxies flying apart. The focus of
dark matter studies and searches to date has been on Weakly Interacting
Massive Particles (WIMPs) whose predicted mass range is broadly speaking
between 10-1000 times that of the proton. The direct, indirect and collider
searches for this dark matter candidate to date have been extensive but so far
unsuccessful. There is a strong motivation to widen the search.

Dark matter with a mass of order or below the mass of the proton, commonly
referred to as the “sub-GeV/$c^{2}$” mass region has received significant
recent attention driven by the widening exploration of possible connections
between the visible and hidden sectors. Coupled with the idea that hidden
sectors may contain more than just the dark matter state leads to a rich set
of new dark matter-Standard Model interactions with novel dark matter genesis
mechanisms moving beyond the vanilla WIMP picture. A comprehensive list of
sub-GeV/$c^{2}$ models with a review of the growing international landscape of
small detectors for low-mass dark matter searches can be found in Essig  _et
al._ (2022).

Helium is an attractive target for scattering searches as its relatively light
mass is well-matched, kinematically, for sensitivity to dark matter
elastically scattering off nuclei in the GeV mass range. As a noble element,
helium has the ease of purification Dmitriev  _et al._ (2005), high light
yield, and transparency to its own scintillation produced in excimer decay.

Helium-4 based dark matter detection, to search for sub-GeV/$c^{2}$ dark
matter candidates with spin-independent interactions, is being intensively
explored, e.g. by the HeRALD and DELight collaborations Hertel  _et al._
(2019); von Krosigk  _et al._ (2022), who both employ the noble gas as their
target media. The QUEST-DMC programme proposes to build and demonstrate the
capability of a superfluid helium-3 detector to improve on the sensitivity to
spin-dependent dark matter in a unique and complimentary search in the same
well-motivated sub-GeV/$c^{2}$ dark matter mass range. The use of superfluid
3He as a particle detector was first proposed Pickett (1988) in 1988. Later,
MACHe3 project led to the ULTIMA programme Mayet  _et al._ (2000); Winkelmann
_et al._ (2007), exploring a 100 GeV/$c^{2}$ WIMP dark matter detector. The
second quantum revolution has led to very significant improvements in
achievable energy resolution since then, and QUEST-DMC aims to exploit these
to reach world-leading sensitivity to spin-dependent interactions of sub-
GeV/$c^{2}$ mass dark matter.

Figure 1:  Schematic illustration of the QUEST-DMC experiment: The mixing
chamber of a 3He-4He dilution refrigerator provides a stable temperature of
$2$ mK. This is used to pre-cool a single-shot magnetic cooldown stage made
from copper, reaching sub-$100\text{\,}\mathrm{\SIUnitSymbolMicro K}$
temperatures. The superfluid 3He dark matter target is contained in a
transparent bolometer volume (above the red demagnetisation stage), surrounded
by a secondary superfluid volume that is cooled down by the copper coolant.
The bolometer is connected to the rest of the superfluid container via an
orifice of $\sim 1\,\mathrm{mm}^{2}$ surface area. Scintillation in the
bolometer can monitored by a photon detector (purple shell), but that
possibility is not included in the sensitivity analysis presented in this
Article. The front half of the whole cylindrical apparatus shown here is cut
out and the size of the bolometer is exaggerated for illustrational purposes.
The bolometer is instrumented with two resonator wires (not shown here). The
magnetic field is vertical.

The QUEST-DMC detector concept is as follows. In a dark matter-3He scattering
event, energy transferred to the struck 3He atom will be deposited in the
detector volume as heat and ionization energy loss, leading to the formation
of thermal excitations (broken Cooper pairs, called quasiparticles) and
excimers respectively. The superfluid 3He target is enclosed in a
$\sim\\!$1\text{\,}\mathrm{c}\mathrm{m}^{3}$$ bolometer box, instrumented with
a nanomechanical resonator (NEMS) sensitive to thermal quasiparticle
production in the 3He. The NEMS records the temperature in the box as a
function of time Pickett (1988); Bradley  _et al._ (1995); Hayes  _et al._
(1996). A sketch of a sample container suitable for such purposes is shown in
Figure 1. To achieve the desired performance, the bolometer is operated at the
lowest achievable temperatures, around $100\text{\,}\mathrm{\SIUnitSymbolMicro
K}$; the NEMS is constructed from nanowires with diameters reaching well below
the $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ scale previously achieved; and,
we implement SQUID readout of NEMS at these temperatures for the first time.

In this paper we study the sensitivity of QUEST-DMC using a complete
simulation model of the detector and its surroundings. Combining a simple
model of the superfluid physics, measured NEMS characteristics, radio-assay
data of the materials involved, and a comprehensive description of the
collision physics using the particle physics simulation library GEANT4 Allison
_et al._ (2016, 2006); Agostinelli  _et al._ (2003), we show that the QUEST-
DMC detector will provide recoil energy sensitivity down to the
$\text{\,}\mathrm{eV}$ scale. Provided realistic cosmic ray background
rejection, this energy sensitivity enables the exploration of the sub-
GeV/$c^{2}$ dark matter mass range with a significant projected improvement in
sensitivity to the spin-dependent elastic scattering cross section. For
example at a dark matter mass of 1 GeV/$c^{2}$, the leading constraint on the
spin-dependent dark matter-neutron cross section stands at $\sim 10^{-34}$ cm2
from CRESST III (LiAlO2) Angloher and others. (2022); the projected
sensitivity of a 6-month run of QUEST-DMC with a 4.9 g day exposure in a
facility located at the planet’s surface reaches down to $\sim 10^{-36}$ cm2.
The details of the profile likelihood ratio sensitivity analysis and
background model are described in Sections 4 and 3.

##  2 Detection Principle

Dark matter can interact with the 3He target material via scattering. This may
be with the target nucleus, transferring a small fraction of the dark matter
kinetic energy, or with the electron cloud, in which case the recoil energy
can be as large as the dark matter particle mass. In this paper we focus on
nuclear scattering, considering both spin-dependent and spin-independent dark
matter interactions. Backgrounds for these dark matter signals include cosmic
rays and cosmogenics, radiogenic sources and solar neutrinos. Figure 2
summarises the expected background spectra along with the differential signal
event rate for spin-dependent dark matter nuclear scattering of a 1
GeV/$c^{2}$ mass dark matter particle. Details on signal generation and
background models can be found in Sections 2.1 and 3. For illustration, the
maximum recoil energy resulting from interaction of a 1 GeV/$c^{2}$ mass dark
matter particle is at the scale of 1 keV. This drives the detector design of
QUEST-DMC to reach the energy threshold of 10 eV and below.

Figure 2: Energy spectra showing the results of simulated backgrounds incident
on the detector target volume, before the detector response has been applied.
As outlined in the Background section, these consist of cosmic rays (including
cosmogenics), radiogenic decays of isotopes within the detector materials and
surroundings and solar neutrinos - which can result in electron recoil (ER) or
nuclear recoil (NR) interactions. The first run of the experiment will take
place on the planetary surface with negligible shielding from cosmic rays. A
cosmic muon veto system is planned for which a 90% tagging efficiency is
assumed. Here, the nominal radiogenic background for a typical cryostat setup
is shown, however this could be further improved using shielding and more
radiopure materials. The energy spectrum for spin-dependent dark matter
nuclear scattering is shown for comparison using a dashed line, for a dark
matter mass of 1 GeV/$c^{2}$ and cross section of $10^{-36}\rm cm^{2}$.

###  2.1 Distribution of collision energy

The energy deposited by a collision is divided between two observable
channels, scintillation photons and heat. Heat is ultimately released and
detected as superfluid quasiparticles. We calculate the division of energy by
first splitting the initial collision energy into elastic scattering,
ionization and excitation. Then each of these is divided into the observable
channels and the contributions are summed.

The initial deposited energy, $E$, is split between electronic stopping,
$\eta$, and nuclear stopping, $\nu$, with: $E=\eta(\epsilon)+\nu(\epsilon)$.
For electron recoil (ER) interactions there is little energy transfer to the
nucleus and all energy goes into electronic stopping. For nuclear recoil (NR)
interactions the fraction of energy in electronic stopping $f$, the nuclear
quenching, is calculated using the Lindhard model Lindhard  _et al._ (1963):

$f=\frac{\eta(\epsilon)}{E}=\frac{kg(\epsilon)}{1+kg(\epsilon)}.$ (1)

This depends on the atomic number, $Z$, and mass, $A$, through reduced energy,
$\epsilon=11.5Z^{-7/3}E/\rm keV$ and $k=0.133Z^{2/3}A^{-1/2}$. The function
$g(\epsilon)=3\epsilon^{0.15}+0.7\epsilon^{0.6}+\epsilon$ is determined from
fits to data Mei  _et al._ (2008). At very low recoil energies the Lindhard
model is acknowledged to be uncertain, due to the assumed potential and lack
of atomic binding energy, this is included in the energy scale systematic
described in Section 4

All of the energy in nuclear stopping goes into the elastic scattering
channel, but the energy in electronic stopping must be split further into
ionization and excitation. This is done using the ratio of measured cross
sections for ionization and excitation. For nuclear recoils we use He–He
impact cross sections for ionization and excitation (summed over all states)
Ito and Seidel (2013). No data exists below 1 keV, so the ratio is
extrapolated down to the ionization threshold. For electron recoils we use
electron impact cross sections for ionization and excitation to both singlet
and triplet states Ralchenko  _et al._ (2008).

The next step is to map the energy deposited in elastic scattering, ionization
and excitation, to the quasiparticles and photons seen in the detector. For
the elastic scattering all energy goes into quasiparticle excitations. For
ionization and excitation channels energy is split between singlet, triplet
and infrared photons and quasiparticles, with the ratio depending on both
recoil energy and interaction type.

Following ionization, the ejected electron forms a ‘bubble’, repelling nearby
He atoms, and the ion forms a ‘snowball’, attracting nearby He atoms. This
phenomenon slows the movement of the charges and at small or zero applied
field almost all recombine forming excited He${}_{2}^{*}$ dimers. These
subsequently relax, via IR cascades, to the first excited state - singlets
($A^{1}\Sigma u$) or triplets ($a^{3}\Sigma u$). Both excimers decay emitting
UV photons, however the singlet decays on a short (ns) timescale whilst the
triplet decay should have a much longer (s) timescale. This should prompt and
delay UV scintillation photons, correspondingly. At least one experiment has
shown that the triplet lifetime is significantly shorter Zmeev  _et al._
(2013) in the presence of 3He atoms. The lifetime of triplets in superfluid
3He, therefore, remains an open question.

In the case of excitation of a helium atom in the nuclear recoil interaction,
the excited helium atom will form an excited dimer. Again, this subsequently
relaxes via an IR emission cascade to singlet and triplet states, which emit
UV photons as above.

The number of singlet/triplet states depends on the interaction type. For
nuclear recoil interactions ionization results in a 0.25:0.75 singlet:triplet
ratio from recombination due to the availability of states. Excitation results
in a 0.86:0.14 singlet:triplet ratio from transition probabilities Ito and
Seidel (2013). For electron recoil interactions ionization is assumed to give
0.5:0.5 singlet:triplet ratio from geminate recombination, whilst the
singlet:triplet ratio for excitation can be determined from cross section
measurements.

A further process that must be accounted for is Penning quenching Ito and
Seidel (2013), which results in non radiative destruction of two excimers
producing helium atoms, an ion and an electron. When the ion and electron
recombine a new single excimer is formed, giving a net reduction in the number
of excimers and UV scintillation photons. The differential equation for the
rate of change of excited states around an interaction site can be solved to
give the Penning quenching factor Ito  _et al._ (2012). This modifies the
fraction of singlet energy going into UV photons and quasiparticles, with the
fraction of singlets lost due to bimolecular processes ranging from $\sim$20%
below 10 keV to $\sim$40% for few MeV interactions.

The energy going into each channel is summed for each step with quasiparticle
energies of 8 eV from subthreshold electrons, 2 eV from dimerisation and 4 eV
from ground state dissociation Ito and Seidel (2013); Seidel (2019). The mean
energies going into the IR channel are 4 eV from ionization and 0.5 eV from
excitation and mean energies for the UV scintillation photons are 16 eV for
singlets and triplets. Production of quanta are random processes, which can be
modelled using Poisson fluctuations. For each energy, samples are randomly
drawn from a Poisson distribution to give a smeared distribution of the number
of quanta and the quanta can then be converted back to energy, as shown in
Figure 3.

Figure 3: Partitioning of the deposited energy by (a) nuclear recoil or (b)
electron recoil interactions into signal channels. Energy is split between
quasiparticles, infrared photons, singlet and triplet UV scintillation
photons. Production of quanta is a Poisson process - dashed lines show the
mean fraction of energy in each channel and points show Poisson smearing of
the discrete quanta produced.

The different fractions of energy in photons and quasiparticles for electron
and nuclear recoil events can potentially be used to discriminate between
these two interaction types. Below the ionization energy of 19.7 eV all the
energy will go into quasiparticles and for a search below this energy, the
photons can be used as a veto for higher energy background events. The
sensitivity projection reported in this paper does not assume any background
reduction associated with particle identification based on energy partition,
or timing within the ionization partition; however, we note the promise of
pulse-shape discrimination in future.

###  2.2 Detecting the deposited heat

Figure 4:  Simulation of bolometer readout: (a) If the detector is operated at
$0.12\,T_{\mathrm{c}}$ ($T_{\mathrm{c}}\sim 1$ mK is the superfluid transition
temperature), the conventional readout of the thermometer wire (blue line,
equivalent amplifier noise $\approx 0.1\,\mathrm{nV}/\sqrt{\mathrm{Hz}}$)
reaches 10% uncertainty at deposited energy $Q=100$ eV. The SQUID readout
circuit has equivalent readout noise $\approx
0.1\,\mathrm{pV}/\sqrt{\mathrm{Hz}}$, yielding sufficient sensitivity down to
sub-eV deposited energies (red dashed line). In both cases the readout error
is inversely proportional to the deposited energy $Q$. The shot noise (dotted
green line) is shown separately assuming no other noise sources are present.
(b) The most important optimisation parameter of the bolometer is the
operation temperature. The simulated readout error for fixed deposited energy
(here $Q=10$ eV) is proportional to the thermal quasiparticle density. The
black dotted line shows the temperature dependence of the thermal
quasiparticle density $\propto\exp(-\frac{\Delta}{k_{\mathrm{B}}T})$. Both
panels show calculations for a 2 mm-long, 400 nm-thick detector wire, operated
at saturated vapour pressure; the conventional and SQUID readouts are
simulated, respectively, for a 100 mT and 0.4 mT magnetic field. Dependencies
on wire dimensions and operation pressure are weaker than those shown here and
are shown in Figure 7 in Appendix B.

The detector concept we consider consists of a bolometer cell with a volume of
1 cm3, filled with superfluid helium-3 in the B phase (3He-B) Vollhardt and
Wölfle (2013). In order to maximise the sensitivity of the detector to changes
in the quasiparticle density, the ambient quasiparticle density needs to be
made as low as possible. This is achieved by cooling the superfluid in the
bolometer to $\approx$$100\text{\,}\mathrm{\SIUnitSymbolMicro K}$. In this
temperature regime, thermal quasiparticles propagate ballistically with a mean
free path of several kilometres.

The bolometer is surrounded by large superfluid volume, refrigerated by a
nuclear demagnetization refrigerator as shown in Figure 1. The two volumes are
connected via a pinhole in the bolometer wall (diameter $\sim 0.5$ mm), and
any deposited heat flows out as carried by the ballistic quasiparticles. This
process is described by the bolometer time constant $\tau_{\mathrm{b}}\sim 5$
s. The long time constant allows measuring the deposited heat once the
collision energy has been thermalised between the quasiparticles in the
bolometer (see Appendix A). Note that based on the simulations described in
the next sections, the expected integrated event rate in the bolometer is
$<2.5$ events/minute (without a veto), which is slow as compared with
$\tau_{\mathrm{b}}$. Below we have assumed the bolometer readout cannot be
much slower than $\sim 10$ s.

Each quasiparticle carries an energy equal to the superfluid gap
$\Delta\approx 10^{-7}$ eV so that at a recoil energy of 10 eV, 108
quasiparticles are created. In comparison, the ionization energy threshold to
liberate one electron is $\sim$20 eV.

The density of quasiparticles is measured using a nanomechanical wire
resonator made from superconducting metal. The moving wire, driven by an AC
current in a magnetic field, experiences a drag force proportional to the
quasiparticle density. The Full Width at Half Maximum (labelled $\Delta f$) of
the mechanical resonance is proportional to the drag force, and the measured
signal amplitude to the peak velocity of the probe. In 3He-B, this force is
orders of magnitude larger than for an equivalent ideal gas Cousins  _et al._
(1997) because (i) all quasiparticles carry the Fermi momentum instead of a
thermal momentum distribution and (ii) quasiparticles approaching the moving
probe from behind are Andreev reflected, while those inbound, with a direction
of motion opposite to that of the probe, are reflected normally Fisher  _et
al._ (1989); Bradley  _et al._ (2017). Andreev reflection is a quasiparticle
reflection process that occurs in quantum condensates that reverses the
direction of motion of the quasiparticle but does not transmit momentum
(conversion from “particle” to “hole”). Thus, net momentum transfer from the
collisions is dramatically amplified.

The quasiparticle drag force measurement is done in two steps:

  1. 1.

In a sweep measurement the resonator response is measured as a function of
drive frequency to characterise the resonator. The resonance is parametrised
by the resonance frequency, the resonance width $\Delta f$, and the amplitude
of the response on resonance. These quantities are extracted from fits of a
Lorentzian function to the in-phase and out-of-phase part of the voltage.

  2. 2.

If the resonator is driven on resonance, a rapid increase in the quasiparticle
drag force produces a decrease in the amplitude of the resonator motion and
thus the measured voltage. Once the resonator parameters are extracted using a
frequency sweep, driving the resonator on resonance allows extracting such
changes as a function of time. This mode of operation is termed resonance
tracking.

A particle depositing energy in the bolometer produces an increase in the
measured width Winkelmann  _et al._ (2007)

$\displaystyle\Delta f(t)$ $\displaystyle=$ $\displaystyle\Delta
f_{\mathrm{base}}+$ (2) $\displaystyle+$ $\displaystyle\Delta(\Delta
f){\left(\frac{\tau_{\mathrm{b}}}{\tau_{\mathrm{w}}}\right)}^{\tau_{\mathrm{w}}/(\tau_{\mathrm{b}}-\tau_{\mathrm{w}})}\times$
$\displaystyle\times$
$\displaystyle\frac{\tau_{\mathrm{b}}}{\tau_{\mathrm{b}}-\tau_{\mathrm{w}}}\left(e^{-t/\tau_{\mathrm{b}}}-e^{-t/\tau_{\mathrm{w}}}\right).$

Here the energy deposition takes place at time $t=0$ and the peak amplitude
$\Delta(\Delta f)$ is proportional to the energy deposited for energies up to
several MeV. The peak shape is determined by the bolometer time constant
$\tau_{\mathrm{b}}$ and the wire time constant $\tau_{\mathrm{w}}=1/(\pi\Delta
f_{\mathrm{base}})$, where the latter dependence is valid assuming the change
in the width is small as compared with the total width ($\Delta(\Delta
f)\ll\Delta f_{\mathrm{base}}$). Further detail on the underlying physics can
be found in Appendix A.

We simulate the bolometer performance by generating bolometer and probe wire
responses from Eq. (2). Two techniques of readout have been considered: a
passive cold amplifier with amplification factor 100 ("conventional readout"),
and a SQUID readout circuit. In both cases, the signal is eventually recorded
using a room-temperature lock-in amplifier (see Appendices B and C). For the
conventional readout the simulated signals are combined with readout noise in
the room temperature amplifier. In the case of the SQUID readout circuit the
room temperature amplifier noise is negligible, and the noise considered is
fundamental noise in the resonator and the cold readout circuit. We fit the
obtained noisy signals with Eq. (2), and repeat the process to gather
statistics on the error in the fitted outcome. This allows us to calculate the
average measurement error as a function of energy deposited by the collision,
detector temperature, pressure and other relevant physical parameters.

The resonance tracking signal measured from the nanowire oscillator is much
larger than the noise. Thus, signal to noise ratio in detecting bolometer
events is directly given by the ratio of $\Delta(\Delta f)$ (in voltage units)
and noise. The obtained uncertainty in the recoil energy detection is
inversely proportional to the deposited energy (Figure 4a). It is also
proportional to the quasiparticle density in the bolometer before the
collision, which decreases exponentially with decreasing temperature (Figure
4b). We conclude that if only readout noise is taken into account, decreasing
the temperature in the bolometer is the most effective way of improving the
detection sensitivity.

Statistical fluctuations in the quasiparticle collisions with the detector
wire result in fluctuations in the measured force proportional to the square
root of the measured force (see Appendix D). We term this noise quasiparticle
shot noise. This noise contribution is shown separately in Figure 4. For the
detector wire studied here, this noise contribution cannot be observed using
traditional readout, but we predict it will be the dominant noise source in a
SQUID readout system. Taking the shot noise into account (details in Appendix
A and B), we predict that a 10% readout error can be achieved at deposited
energy $Q=10$ eV using the SQUID circuit up to and possibly even above
operation temperature $0.15\,T_{\mathrm{c}}$.

Figure 5:  Superconducting wire resonators for superfluid thermometry: (a) A
fake-colour micrograph of a 390 nm-thick superconducting wire resonator from
above (red string), spanning approximately 1 mm between two copper posts
(copper-coloured rods). The white scale bar corresponds to
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (b) The blue circles show the
Full Width at Half Maximum (FWHM) of thermometer wire resonances for different
wire diameters. The data is measured in superfluid 3He-B at
$0.24T_{\mathrm{c}}$ (220 $\mu$K) and saturated vapour pressure (0 bar). The
red dashed line is the theoretical expression detailed in Eq. (11) in Appendix
A. Approaching the coherence length (vertical black dash line), the measured
resonance width is expected to remain smaller than the theoretical line owing
to qualitative changes in the quasiparticle scattering process. The grey
shaded area shows where the resonator response time becomes longer than 10 s
if temperature is lowered to $0.13T_{\mathrm{c}}$
($120\text{\,}\mathrm{\SIUnitSymbolMicro K}$ at 0 bar pressure) and the blue
area shows where the response time becomes longer than 10 s if temperature is
lowered to $0.11T_{\mathrm{c}}$ ($100\text{\,}\mathrm{\SIUnitSymbolMicro K}$).

At the lowest temperatures, the temperature measurement can become too slow to
distinguish subsequent collisions. As a crude estimate of the readout speed
needed to avoid pile-up, we assume that the detector needs to be able to cope
with background events occurring every $\sim 10$ s. The expected background
spectra are discussed in detail in the next section. The mechanical resonator
adjusts to changes in the quasiparticle density with the time constant
$\tau_{\mathrm{w}}=1/(\pi\Delta f)$, which increases as

$\Delta f\propto\exp\left(-\frac{\varDelta}{k_{\mathrm{B}}T}\right)$ (3)

with decreasing temperature. Mechanical resonators also have intrinsic
dissipation that does not depend on quasiparticle collisions and thus sets a
minimum for $\Delta f$. Lighter resonators (thinner wires) react faster to
changes in the quasiparticle density and become saturated at the intrinsic
width at a lower temperature. Using sub-$\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ resonators is therefore essential for the operation of the superfluid
bolometers.

We have developed a technique of manufacturing superconducting niobium-
titanium (Nb-Ti) resonator wires down to 400 nm thickness. In brief, a copper
matrix only containing a few filaments of the desired size is etched away
across the planned resonator length, and the filaments are manually removed
down to one using tweezers. The one remaining cylindrical superconducting
filament remains attached to the copper legs. An example of such resonator is
shown in the Scanning Electron Microscope picture in the inset of Figure 5.
Details of the manufacturing process will be published separately.

Figure 5 shows resonance widths of wires with diameters down to 400 nm. The
shaded areas in Figure 5 indicate an extrapolation to lower temperatures using
Eq. (3), showing which probes become slower than $\tau_{\mathrm{w}}\approx 10$
s as temperature is lowered to $100\text{\,}\mathrm{\SIUnitSymbolMicro K}$.
That is, probe wires thinner than a micrometre are fast enough to operate in
this temperature regime, allowing us to avoid pile-up even if the eV-scale
backgrounds turn out to be more frequent than our simulation model predicts.
Note that the smallest wires have resonance widths significantly smaller than
expected based on theory. This may be either because of impurities attached to
the resonator wire in the assembly process or because the wire diameter is
approaching the coherence length of the superfluid where Andreev reflection is
expected to gradually switch off. For our purposes the obtained widths are
large enough for $100\text{\,}\mathrm{\SIUnitSymbolMicro K}$ operation.

We conclude that the desired energy sensitivity of 10 eV can be achieved if
the $\mathrm{cm}^{3}$-volume bolometer is operated at
$100\text{\,}\mathrm{\SIUnitSymbolMicro K}$ temperature and the readout of a 2
mm-long thermometer wire is instrumented with a SQUID readout circuit.
Achieving sufficient readout speed at this operating temperature to deal with
background events requires using a sub-$\text{\,}\mathrm{\SIUnitSymbolMicro
m}$-thick thermometer wire.

The bolometer will be energy calibrated using two techniques. Following Ref.
Fisher  _et al._ (1992), the first technique is to inject heat into the
bolometer volume using a second superconducting wire resonator in the
bolometer. Intrinsic losses in such a heater are negligible, and therefore the
measured driving power is directly dissipated by heating the quasiparticle
gas. The steady-state increase in the bolometer temperature is measured as a
function of power injected. In this state the power going out via the orifice
equals that injected by the heater. The total energy that flows out via the
orifice during an event described by Eq. (3) can be obtained by integrating
the corresponding bolometer temperature evolution Winkelmann  _et al._ (2007).
The second calibration technique is to place radioactive sources, with fixed
energy lines, as near to the active bolometer volume as possible. This
calibration gives a direct relation to the deposited energy in the bolometer,
from the radioactive decay, to the measured wire response.

##  3 Background

Beyond reaching low recoil energy sensitivity, the key challenge in dark
matter searches is to achieve sufficiently low background event rates in the
dark matter search energy region of interest. The 3He target itself is
intrinsically radiopure. That is, the only other atomic species that remains
liquid at microkelvin temperatures is 4He. At
$100\text{\,}\mathrm{\SIUnitSymbolMicro K}$, the solubility of 4He in 3He is
so low that not a single atom is expected to be contained in a bolometer
volume of any size. Note also that the rare 4He atoms produced by neutron
capture will be adsorbed on the container walls after ballistic propagation
and that 4He atoms in the bulk of the superfluid are not expected to disturb
the operation of the bolometer. All other impurities are solidified on filling
line surfaces and heat exchangers well before entering the bolometer volume.
This extreme purity means that the superfluid has absolutely no intrinsic
radioactivity or other background that could impede a dark matter search. As a
result, we consider only background sources arising from the materials the
detector is made of, as well as external particles incident on the
detector–neutrinos and cosmic ray-induced activity.

###  3.1 Radiogenic Background Sources

We calculate estimated rates for radiogenic backgrounds observed in the
superfluid detector using a detailed GEANT4 model of the detector geometry and
surrounding materials. The model includes the major features of the cryostat
and laboratory, and it is based on a combination of dedicated gamma ray
spectroscopy assay at Boulby Underground laboratory Scovell  _et al._ (2018)
for all the materials involved that we have direct access to, as well as
material radioassay results from the SNOLAB radiopurity database Loach  _et
al._ (2016). The specific activities of the main detector materials are
summarized in Table 1.

Table 1: Activity of materials comprising the bolometer, cryostat and surrounding area: Assay values in mBq kg-1 from gamma ray spectroscopy screening Scovell  _et al._ (2018) of materials used in the QUEST-DMC detector, cryostat and laboratory separated into relevant isotopes. * denotes values from dedicated QUEST-DMC screening. Estimate values for materials with no direct access (concrete and insulation of the cryostat’s dewar), or isotopic abundances below the sensitivity of dedicated screening, are taken from the SNOLAB radiopurity database Loach  _et al._ (2016). If no value for 235U could be found the theoretical ratio of natural abundance 235U/238U = 0.007257 is used Liu  _et al._ (2011). 137Cs measurement for Araldite epoxy is also used for similar epoxies (Stycast).  Material | Up. 238U | Lo. 238U |  210Pb | Up. 232Th | Lo. 232Th |  235U |  137Cs |  40K |  60Co |  54Mn
---|---|---|---|---|---|---|---|---|---|---
Concrete | $<1.60\times 10^{5}$ | 1.50$\times 10^{4}$ | $1.00\times 10^{7}$ | 7.57$\times 10^{3}$ | 7.57$\times 10^{3}$ | < 7.20$\times 10^{3}$ | 800 | 4.20$\times 10^{4}$ | < 700 | 0.00
Aluminium | 8.33$\times 10^{3}$* | 15.3* | 70.7* | 356* | 334* | 60.5 | < 0.940* | < 3.12* | < 1.10* | 0.00*
Insulation | 679 | < 200 | < 3.90$\times 10^{3}$ | 200 | 200 | 4.93 | 0.00 | 3.50$\times 10^{3}$ | 400 | 0.00
Stainless Steel | 16* | 2.5* | 82.2* | 3.1* | 3.90* | 0.120 | 2.00 | < 6.20* | < 5.20* | 1.70
Steel | < 12.4 | 12 | 1.20$\times 10^{4}$ | 4.88 | 4.88 | 3.00 | 2.00 | 34.1 | 30.0 | 1.00
Araldite | < 3.60* | < 4.80* | 14.5* | < 3.40* | < 2.20* | 0.0260 | 2.00 | 25.5* | 8.00* | 0.00*
Stycast | < 10.5* | < 9.50* | < 14.9* | < 12.8* | < 6.20* | 0.0762* | 2.00 | 122* | 10.0* | 0.00*

The main contributors to radioactive backgrounds in rare-event experiments
tend to come from the decay chains of 238U and 232Th which have a high
abundance in nature, and can produce electrons, alpha particles, photons from
de-excitation, or radiogenic neutrons. Secular equilibrium is assumed for
isotopes in the Thorium chain unless known otherwise from particular assay
results (with equilibrium breaking at 228Th) and for isotopes in the upper and
lower parts of the Uranium decay chain with equilibrium breaking at 226Ra and
210Pb. Radon emanation effects are not simulated in initial studies. Other
isotopes analysed include 235U, 60Co, 40K, 137Cs and 54Mn. If assay results
are not able to detect trace amounts of 235U, the theoretical ratio of natural
abundance 235U/238U = 0.007257 is used Liu  _et al._ (2011).

Radioactive decays from relevant isotopes, shown in Table 1, are simulated
originating from each separate volume comprising the GEANT4 model. For each
volume between 105 and 1010 primary decays are simulated per isotope,
dependent on the distance between the volume and the detector. The normalised
deposited energy spectra from each decay are summed over all volumes, taking
into account the activity and geometric acceptance of the decay products, to
provide an estimate rate of “true” deposited energy within the 3He. The
normalised background recoil energy spectra associated with these
probabilities can be seen in Figure 2. The full contributions from radiogenic
backgrounds are summarized in Table 2.

Radiogenic backgrounds could be further reduced by the addition of optimized
passive shielding. For example, we have simulated the addition of 10 cm thick
external lead shielding around the cryostat and 2 cm of radiopure copper
within the cryostat itself in GEANT4, which gives $\sim$65% reduction in the
simulated radiogenic background in the Region Of Interest (ROI) (0-10 keV),
compared to the current background model used to assess the sensitivity reach
here.

###  3.2 Solar Neutrino Backgrounds

Neutrinos produced in nuclear processes in the Sun reach the Earth and undergo
nuclear recoil or electron recoil interactions in the target. Coherent
neutrino-nucleus scattering mediated by $Z$ exchange produces NRs that can
populate the dark matter search energy region of interest, whilst $Z$ and $W$
boson-mediated neutrino-electron elastic scattering produces ERs.

The incoming neutrino flux depends on the solar model. We use the recommended
normalisation values from Baxter  _et al._ (2021). Experimental values are
used for 8B (SNO Aharmim  _et al._ (2013)) and 7Be (Borexino Agostini  _et
al._ (2019)), whilst theoretical predictions from the high metallicity B16
solar model Vinyoles  _et al._ (2017) are used for all other sources.

The background contribution from solar neutrinos is summarized in Table 2.
Most solar neutrinos are produced in the initial proton-proton fusion step,
which dominates both backgrounds at low energies. Neutrinos produced in the
later stages by 7Be electron capture and 8B positron emission extend to higher
energies.

###  3.3 Cosmic Ray-induced Backgrounds

While low background experiments are usually hosted in underground facilities
to provide a natural shielding against cosmic radiation, QUEST-DMC will be
initially operated at ground level in the ultra-low temperature cryostats at
the Lancaster physics department. In these conditions, the majority of the
background will be caused by cosmic rays, and cosmic ray-induced radiation.

The flux of cosmic radiation at ground level has been estimated using the
Cosmic-ray Shower Library (CRY) cry  particle generator, together with the
GEANT4 detector simulation used for the radiogenic background studies. GEANT4
propagates the primary and secondary particles (including cosmogenics
production) inside the cryostat materials, accounting for the total deposited
energy inside the 3He cell. We do not distinguish ER and NR, and we identify
as cosmic ray-induced background anything that produces energy deposition in
the ROI; below 19.7 eV there is no ionization partition so there is no
particle identification information. To mitigate the rate of events coming
from cosmic rays, we consider using a tagging setup of plastic scintillator
planes at the top and at the bottom of the superfluid sample container cell.
The light could be carried to the external readout electronic by a light
guide. This tagging scheme has an estimated efficiency of 90% that would
permit to reject most of the particles crossing the cell. The background
contribution from cosmic ray-induced backgrounds, with a tagger of 90%
efficiency, is summarized in Table 2.

The assumed cosmic ray flux in the background model at the surface is
0.017/cm2/s. This would be greatly suppressed if operated underground as is
typical for rare event search experiments; i.e. at Boulby Underground
Laboratory (2805 metres of water equivalent depth) the muon flux has been
measured as (4.09$\pm 0.15$)$\times 10^{-8}$/cm2/s Robinson  _et al._ (2003).

Table 2:  Background event rates: Expected mean counts for each background component in the 0–10 keV energy range in units of events per kg per day and per cell per day. The interaction types are indicated by ER for electron recoil and NR for nuclear recoil. The associated uncertainties on the number of counts arise from material screening, flux models and Monte Carlo statistics.  Component | Expected counts [0-10 keV] | Uncertainty
---|---|---
| /kg/day | /cell/day |
Cosmic ray | $1.05\times 10^{5}$ | $3.31$ | 11 %
Radiogenic ER | $8.31\times 10^{4}$ | $2.61$ | 14 %
Solar $\nu$ ER | $1.51\times 10^{-2}$ | $4.76\times 10^{-7}$ | 2 %
Solar $\nu$ NR | $6.37\times 10^{-4}$ | $2.01\times 10^{-9}$ | 2 %
TOTAL | $1.88\times 10^{5}$ | 5.92 |

##  4 Sensitivity analysis

This section summarises the signal model; more details are provided in
Appendix E. The differential event rate for a dark matter particle of mass
$m_{\chi}$ scattering with a target nucleus of mass, $m_{N}$, is given by

$\frac{dR}{dE_{\rm NR}}=\frac{\rho_{\chi}}{m_{\chi}m_{N}}\int^{\infty}_{v_{\rm
min}}\frac{d\sigma}{dE_{\rm NR}}\;v\;f(\vec{v})\;d^{3}v,$ (4)

where $\rho_{\chi}$ is the local dark matter density and $f(\vec{v})$ is the
local dark matter velocity distribution in the rest frame of the detector
Lewin and Smith (1996); Savage  _et al._ (2006) and $v_{\rm min}$ is the
minimum dark matter velocity needed to impart a recoil energy of $E_{\rm NR}$
in the detector. The Standard Halo Model is assumed with the dark matter
distributed as an isothermal sphere with an isotropic Maxwell Boltzmann
velocity distribution truncated at the escape velocity, $v_{\rm esc}$. We use
the Halo parameters recommended in Baxter  _et al._ (2021).

We consider spin-dependent and spin-independent scattering separately. At zero
momentum transfer the differential cross sections for spin-dependent and spin-
independent scattering can be written, respectively, as

$\displaystyle\frac{d\sigma^{\rm SD}}{dE_{\rm NR}}$ $\displaystyle=\sigma^{\rm
SD}_{\rm\chi n}\frac{m_{N}}{2\mu^{2}_{\rm\chi n}v^{2}},$ (5)
$\displaystyle\frac{d\sigma^{\rm SI}}{dE_{\rm NR}}$ $\displaystyle=\sigma^{\rm
SI}_{\rm\chi p}\frac{A^{2}m_{N}}{2\mu^{2}_{\rm\chi p}v^{2}},$ (6)

where $A=3$ for 3He, $\sigma^{\rm SD}_{\rm\chi n}$ and $\sigma^{\rm
SI}_{\rm\chi p}$ are the spin-dependent dark matter-neutron and spin-
independent dark matter-nucleon cross sections respectively and where
$\mu_{\rm\chi n}$ and $\mu_{\rm\chi p}$ are the reduced dark matter-neutron
and dark matter-nucleon masses respectively.

The differential event rates for spin-dependent and spin-independent
scattering are then

$\displaystyle\frac{dR^{\rm SD}}{dE_{\rm NR}}=\frac{\rho_{\chi}\,\sigma^{\rm
SD}_{\rm\chi n}}{2m_{\chi}\,\mu^{2}_{\rm\chi n}}\int^{\infty}_{v_{\rm
min}}\frac{1}{v}\;f(\vec{v})\;d^{3}v,$ (7) $\displaystyle\frac{dR^{\rm
SI}}{dE_{\rm NR}}=\frac{9\rho_{\chi}\sigma^{\rm SI}_{\rm\chi
p}}{2m_{\chi}\,\mu^{2}_{\rm\chi p}}\int^{\infty}_{v_{\rm
min}}\frac{1}{v}\;f(\vec{v})\;d^{3}v.$ (8)

Detector response effects are applied to the predicted differential event
rates with energy from the readout error, shot noise and intrinsic fluctuation
outlined above and a lower limit determined by the threshold energy of the
detector.

The energy threshold of the bolometer is calculated as the energy that can be
statistically determined to be non-zero at 95% confidence level. For a range
of deposited energy values, the mean number of quasiparticles is calculated as
described in Section 2.1. A smeared distribution about this mean is made using
the readout error and intrinsic fluctuations calculated as described in
Appendix D. Then, for each true number of quasiparticles, a Gaussian is fitted
to the smeared distribution. The fraction of this Gaussian which falls below
zero is calculated and the true number of quasiparticles corresponding to 5%
below zero is determined. This number of quasiparticles is then converted into
an energy, which defines the threshold.

Using this method an energy threshold of 39 eV is found for the conventional
readout and a threshold of 0.71 eV for the SQUID readout, at a temperature of
$0.12\,T_{\mathrm{c}}$. These threshold energies are used as energy cuts on
the background and signal distributions that go into the sensitivity
projection.

For the sensitivity projections that follow, an exposure of 4.9 g day is
assumed, corresponding to 5 $\times$ 0.03 g cells with 6 months live data
taking time.

A profile likelihood ratio analysis is used to evaluate the sensitivity to
dark matter interactions Cowan  _et al._ (2011); Baxter  _et al._ (2021). An
unbinned likelihood function is defined as the product of Poisson
probabilities for the signal plus background rate in all bins. The parameter
of interest is defined as the dark matter interaction rate. Other unknown
parameters in the model are nuisance parameters, which will be profiled out in
the analysis. These include the rates of different background components in
the background model, the galactic escape velocity of dark matter in the
signal model $v_{\textrm{esc}}$ and the energy scale calibration.

Using the RooStats package Moneta  _et al._ (2010) a test statistic is
constructed using the ratio of the conditional to global maximum likelihood.
This is evaluated for different values of the parameter of interest, floating
all nuisance parameters within Gaussian constraints. The resulting test
statistic distribution is used to determine two sided 90% confidence limit on
the parameter of interest, which is converted to a limit on the dark matter
interaction cross section.

The probability density functions used in the likelihood function are
distributions of the dark matter signal and expected backgrounds in a
reconstructed energy variable. The reconstructed energy probability
distribution functions are made by applying the energy threshold cutoff and
Gaussian energy resolution to “true” energy probability distribution functions
which have been simulated. Background rates and uncertainties used are shown
in Table 2.

The likelihood is extended to include constrain terms on the nuisance
parameters accounting for systematics in the signal and background models. For
each background component a Gaussian constraint term with width equal to the
rate uncertainty is added - arising from solar neutrino flux uncertainty,
radiogenic material screening and Monte Carlo simulation statistics.

Uncertainty in the signal model can arise from the astrophysical parameters,
which determine the dark matter velocity distribution. Changes in galactic
escape velocity can significantly affect the endpoint of this distribution,
which is important at low dark matter masses for low threshold experiments.
The recommended value of 544 km/s Baxter  _et al._ (2021) was measured by the
RAVE survey Smith  _et al._ (2007) with a 90% confidence limit of (498, 608)
km/s. The value of $v_{\mathrm{esc}}$ is allowed to float in the analysis,
with a Gaussian constraint term corresponding to the 1 $\sigma$ width.

The final source of uncertainty considered in this analysis is in the energy
measured in the bolometer - accounted for using an energy scale parameter.
This arises from uncertainty in the expected quasiparticle energy, from the
deposited energy distribution, transport and cell thermalisation, and from
uncertainty in the energy calibration and measurements. The calibration
uncertainty is expected to dominate with $\sim$10%, as seen in the measured
calibration coefficient in Fisher  _et al._ (1992), and a corresponding
Gaussian constraint term is included.

The resulting 90% confidence limit sensitivity projection for spin-dependent
dark matter-neutron scattering is shown in Figure 6 contrasted with existing
limits from Xenon 1T S2-only MIGD Aprile  _et al._ (2019), CRESST III (LiAlO2)
Angloher and others. (2022), LUX (Xe) Akerib  _et al._ (2016), CDMSlite (Ge)
Agnese  _et al._ (2018) and PandaX-II Xia  _et al._ (2019). The projected
limits from this work would improve significantly on the existing limits of
Xenon 1T Aprile  _et al._ (2019) and CRESST III (LiAlO2) Angloher and others.
(2022) in the mass range $\sim(0.025-4)$ GeV/$c^{2}$ for the QUEST-DMC SQUID
readout.

The 90% confidence limit sensitivity projection for spin-independent dark
matter-nucleon scattering is also shown in the lower plot of Figure 6 along
with existing limits from DarkSide-50 Agnes  _et al._ (2023) and XENON1T
Aprile  _et al._ (2018). Even though 3He has a significantly lower atomic mass
number compared with xenon and argon, the low energy threshold coupled with
the scattering kinematics of the 3He with dark matter allows QUEST-DMC to
project stronger limits than DarkSide-50 and XENON1T for masses below
$\sim$350 MeV/$c^{2}$, with the SQUID readout projection providing sensitivity
down to $\sim$25 MeV/$c^{2}$.

There are a number of sensitivity projections in the literature related to
planned upgrades of existing direct detection experiments and to new proposed
experiments that are in various stages of development. For spin-dependent dark
matter-neutron scattering DarkSPHERE Balogh  _et al._ (2023) has projections
for a helium-isobutane gas target over the mass range $\sim(0.05-10)$
GeV/$c^{2}$ reaching as low as $\sim 5\times 10^{-38}$ cm2 at a dark matter
mass of $\sim 550$ MeV/$c^{2}$. The QUEST-DMC SQUID readout limit is the most
sensitive projection below $\sim 150$ MeV/$c^{2}$.

For spin-independent dark matter-nucleon scattering, the landscape of
projections is more populated for the sub-GeV/$c^{2}$ mass range. For example,
at a dark matter mass of $200$ MeV/$c^{2}$ DarkSPHERE Balogh  _et al._ (2023)
projects a limit of $\sim 5\times 10^{-42}$cm2. Projections from He-based dark
matter detection experiments (HeRALD (kg-day exposure) Hertel  _et al._ (2019)
and DELight von Krosigk  _et al._ (2022)) are useful comparisons, with
projected limits at $200$ MeV/$c^{2}$ of $\sim 4\times 10^{-39}$cm2 and $\sim
5\times 10^{-40}$cm2 respectively, although we note that the DELight
projection is background free. We also note that HeRALD Hertel  _et al._
(2019) include projections for 100 kg year with a 1 meV energy threshold
giving sensitivity to masses $\sim$MeV/$c^{2}$ with a cross section of $\sim
10^{-45}$ cm2.

Figure 6: Projected 90% exclusion limit sensitivity to dark matter nucleon
interactions, assuming 4.9 g day exposure and the background rates outlined in
Table 2. Calculated for the conventional (cold transformer plus lock-in
amplifier) and SQUID readout schemes, which are expected to achieve energy
thresholds of 39 and 0.71 eV respectively. (a) Cross section limit for spin-
dependent dark matter-nucleon interactions including existing limits from
Xenon 1T S2-only MIGD Aprile  _et al._ (2019), CRESST III (LiAlO2) Angloher
and others. (2022), LUX (Xe) Akerib  _et al._ (2016), CDMSlite (Ge) Agnese
_et al._ (2018) and PandaX-II Xia  _et al._ (2019). (b) Cross section limit
for spin-independent dark matter-nucleon interactions including existing
limits from DarkSide-50 Agnes  _et al._ (2023) and XENON1T Aprile  _et al._
(2018).

At low dark matter masses and relatively high cross sections it is important
to consider the effect of dark matter interactions with particles in the Earth
and atmosphere Collar and Avignone (1992, 1993); Hasenbalg  _et al._ (1997);
Kouvaris and Shoemaker (2014); Kouvaris (2016); Bernabei  _et al._ (2015);
Kavanagh  _et al._ (2017). Scattering of incoming dark matter will alter the
number density and velocity distribution at the detector location. This effect
causes the signal spectra and resulting sensitivity to be modified or even
lost for sufficiently large dark matter scattering cross sections. In this
paper we focus only on the limit at low cross sections, where this effect is
minimised, and leave the inclusion of these effects to future work.

##  5 Discussion

In this paper we have described a new concept for a dark matter experiment,
employing superfluid 3He as a detector for interactions of dark matter with
mass of order 1 $\text{\,}\mathrm{G}\mathrm{eV}\mathrm{/}\mathrm{c}^{2}$ or
below. The detector concept is based on quasiparticle detection in a bolometer
cell by a nanomechanical resonator. We have developed the energy measurement
methodology and detector response model, simulated candidate dark matter
signals and expected background interactions, and calculated the sensitivity
of such a detector. We project that such a detector can reach sub-eV recoil
energy threshold, opening up new windows on the parameter space of spin-
independent interactions of light dark matter candidates. Simultaneously, this
complete description of the system allows for future investigations of
fundamental superfluid physics, such as understanding the elusive homogeneous
first order phase transitions between the superfluid A and B phases where
external radiation may trigger the transition instead of thermal fluctuations
Schiffer and Osheroff (1995); Halperin  _et al._ (2021).

The projected energy threshold of the QUEST-DMC detector is limited by
quasiparticle shot noise in measuring the energy deposited in the detector.
Equipping the bolometer with $N$ independent nanowire oscillators Noble  _et
al._ (2022); Ahlstrom  _et al._ (2014) would enhance the projected energy
sensitivity by a factor $\sqrt{N}$ Fleischmann  _et al._ (2020).
Alternatively, turning the nanowire into a paddle by attaching a light thin
membrane such as a graphene flake or PMMA film would increase the number of
quasiparticle collisions without making the resonator significantly heavier,
thus reducing the shot noise. In principle, one could also use the shot noise
measurement as a direct means of thermometry Spietz  _et al._ (2003), or
switch to non-local thermometry, such as measuring the relaxation rate of a
magnon BEC occupying the bolometer volume Zavjalov  _et al._ (2015); Heikkinen
_et al._ (2014a, b); Fisher  _et al._ (2000); Autti  _et al._ (2018).
Implementing one or more of these improvements will allow pushing the
sensitivity to sub-eV energies.

## Data availability

The data used in this study are available at
https://doi.org/10.17637/rh.23574687.

## Code availability

The simulation codes are available at https://doi.org/10.17637/rh.23261534.

## Acknowledgements

This work was funded by UKRI EPSRC and STFC (Grants ST/T006773/1,
EP/P024203/1, EP/W015730/1 and EP/W028417/1), as well the European Union’s
Horizon 2020 Research and Innovation Programme under Grant Agreement no 824109
(European Microkelvin Platform). We thank George Pickett for the materials
used to create Figure 1. We acknowledge M.G. Ward and A. Stokes for their
excellent technical support. S.A. acknowledges financial support from the
Jenny and Antti Wihuri Foundation. We thank Boulby Underground Laboratory for
radio-assaying the materials. M.D.T acknowledges financial support from the
Royal Academy of Engineering (RF\201819\18\2).

## Author contributions

The manuscript was written by S.A., P.F., E.L., J.M., S.M.W., and R.S. with
contributions from all authors. The nanowire fabrication and characterisation
experiments were carried out by S.A., A.J., J.R.P., T.S., M.D.T., L.W., V.V.Z.
and D.E.Z. The radio assay of the cryostat components was carried out by S.A.,
N.D., P.F., A.K., E.L., J.M., R.S. and D.E.Z. The SQUID readout scheme was
designed by S.A., A.C., P.J.H., L.V.L, J.S. and D.E.Z. The numerical
simulation model was constructed and used by S.A., N.D., P.F., A.K., E.L.,
J.M., R.S., S.M.W. and D.E.Z. The project was conceived and supervised by
S.A., A.C., R.P.H., M.H., S.J.H., J.M., J.M-R., J.S., V.T., S.M.W. and D.E.Z.

## Competing interests

The authors declare no competing interests.

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## Appendices

##  Appendix A Superfluid 3He as a bolometer

The density of thermal quasiparticles in 3He-B decreases exponentially with
decreasing temperature. Below $0.25T_{\mathrm{c}}$ ($T_{\mathrm{c}}\sim 1$ mK
is the superfluid transition temperature), the density is so low that the
quasiparticles do not interact with each other. That is, the mean free path
exceeds the sample container dimensions by orders of magnitude. Here, the heat
capacity of the superfluid contained by the bolometer can be calculated from
first principles Vollhardt and Wölfle (2013),

$C_{V}(T)=2(2\pi)^{1/2}k_{B}N_{0}\varDelta\left(\frac{\varDelta}{k_{B}T}\right)\exp\left(-\frac{\varDelta}{k_{B}T}\right)$
(9)

where $k_{B}$ is the Boltzmann constant, $N_{0}$ is BCS density of states in
the normal phase at Fermi energy for one spin component, and $\varDelta$ is
the superfluid energy gap.

The density of quasiparticles can be measured using a mechanical resonator
immersed in the superfluid. Here we consider a cylindrical superconducting
resonator wire, driven by an AC current in a magnetic field. The velocity of
the wire can be measured by recording the induced voltage. The resonator wire
experiences a drag force due to collisions with quasiparticles. For a
cylindrical resonator wire, the drag force per unit length from quasiparticle
collisions follows Enrico (1995); Enrico  _et al._ (1995)

$F=vd\frac{\pi}{4}p^{2}_{\mathrm{F}}v_{\mathrm{F}}N_{0}\exp{\left(-\frac{\varDelta}{k_{B}T}\right)}.$
(10)

Here $v$ is the velocity of the wire, $d$ is its diameter, $p_{\mathrm{F}}$ is
the Fermi momentum and $v_{\mathrm{F}}$ is the Fermi velocity. This linear
expression holds up to
$v\approx$$1\text{\,}\mathrm{m}\mathrm{m}\,\mathrm{s}^{-1}$. In this paper we
have therefore assumed that the resonator is operated so that the peak
velocity is $v=$$1\text{\,}\mathrm{m}\mathrm{m}\,\mathrm{s}^{-1}$ for
superfluid at 0 bar pressure. Increasing the velocity much beyond this not
only makes the drag force nonlinear in velocity but also eventually enables
direct vortex production and emission of surface-bound quasiparticles Autti
_et al._ (2020, 2023).

The resonance line shape in the linear regime is Lorentzian, characterised by
the central frequency $f_{0}\sim$1 kHz and the Full Width at Half Maximum

$\Delta f=\frac{2F}{\pi^{2}\rho d^{2}v},$ (11)

where $\rho$ is the mass density of the wire. The resonance width can be
measured directly by sweeping the AC drive frequency across $f_{0}$ and
recording the induced voltage. The resonance width is, therefore, a direct
measure of the superfluid’s temperature.

The induced voltage depends on the geometry of the resonator, but to a good
approximation the on-resonance voltage follows

$V_{0}\approx lBv$ (12)

for both semi-loop-shaped resonator wires and straight wires. Here $B$ is the
applied magnetic field and $l$ is the length of the wire projected to the
plane perpendicular to $B$ (length of the wire for a straight wire, leg
spacing for a semi-loop). For the state of the art fabrication of the thinnest
wires $l\approx 2$ mm (longer wires result in greater induced Faraday voltages
and, therefore, more sensitive measurements).

To reach the lowest possible temperature, the bolometer needs to be placed as
close to the demagnetisation stage as possible, as shown in Figure 1. Thus,
the largest magnetic field available for operating the resonator wires is the
typical end-of-demagnetisation field $B=100$ mT. We have used this field for
calculating the sensitivity of the conventional readout. The SQUID readout can
possibly be impedance-matched to work in such field, but would ideally work at
a significantly lower magnetic field. We have assumed it is operated at 0.4 mT
as detailed in the following Section.

For a fixed drive current, Eqs. (11, 12) show the product $\Delta f\,V_{0}$ is
conserved. It is therefore possible to operate the resonator in a secondary
readout mode by constantly driving it at $f_{0}$. Changes in the measured
$V_{0}$ can be directly converted into corresponding $\Delta f$. In this mode,
the temperature readout time is limited by the intrinsic response time of the
resonator, $\tau_{\mathrm{w}}=1/(\pi\Delta f)$.

The response of a bolometer operated at temperature $T$, corresponding to
thermometer wire resonance width $\Delta f_{\mathrm{base}}$, is modelled as
follows. The heat deposited by a collision in the bolometer volume is denoted
$Q$. After the collision, the bolometer reaches a new temperature $\Delta
f=\Delta f_{\mathrm{base}}+\Delta(\Delta f)$. The increase in the resonator
width $\Delta(\Delta f)\propto Q$. In practical experiments, the bolometer
leaks energy to the surrounding superfluid bath via a ‘weak link’ — a small
opening in the bolometer wall — which allows the temperature to slowly return
to the value before the collision. A typical hole diameter used is $\sim 0.5$
mm yielding a bolometer time constant $\tau_{\mathrm{b}}\approx 5$ s. The
superfluid is well-decoupled from the container walls by the Kapitza
resistance, that is, the thermalisation time into the bolometer walls is $\gg
10^{5}$ s. On the other hand, localised heat release up to MeV energies is
distributed into the superfluid state within a few microseconds Ruutu  _et
al._ (1996) and while the quasiparticle thermalisation time in the bolometer
is not known precisely, it is of the same order as the time measured in Ruutu
_et al._ (1996) and thus much shorter than $\tau_{\mathrm{b}}$. This can be
confirmed by measuring the Andreev-reflection-based deviation from Eq. (11) at
resonator peak velocities $v>$$1\text{\,}\mathrm{m}\mathrm{m}\mathrm{s}^{-1}$
as described by Eq. (17) in Fisher  _et al._ (1991) (see also Enrico (1995)).
This deviation arises from the thermal velocity distribution of the bulk
quasiparticles. Therefore, the heat deposited by a collision in the bolometer
volume, $Q$, can be measured slowly by recording the resulting thermometer
response Winkelmann  _et al._ (2007):

$\displaystyle\Delta f(t)$ $\displaystyle=$ $\displaystyle\Delta
f_{\mathrm{base}}+$ (13) $\displaystyle+$ $\displaystyle\Delta(\Delta
f){\left(\frac{\tau_{\mathrm{b}}}{\tau_{\mathrm{w}}}\right)}^{\tau_{\mathrm{w}}/(\tau_{\mathrm{b}}-\tau_{\mathrm{w}})}\times$
$\displaystyle\times$
$\displaystyle\frac{\tau_{\mathrm{b}}}{\tau_{\mathrm{b}}-\tau_{\mathrm{w}}}\left(e^{-t/\tau_{\mathrm{b}}}-e^{-t/\tau_{\mathrm{w}}}\right).$

Note that the peak value of this function is independent of
$\tau_{\mathrm{b}}$ and $\tau_{\mathrm{w}}$, and thus the readout sensitivity
does not depend strongly on either time constant.

The dark matter detection concept presented here relies on detecting changes
in thermal energy in the bolometer smaller than any existing experiment is
sensitive to. While the heat capacity and Andreev reflection responses have
not been measured in this energy range, the theoretical description works well
because of the lack of quasiparticle interactions, and so no deterioration of
the agreement with theory is expected at even lower quasiparticle densities.
This justifies extrapolation to lower-than-yet-unexplored energy scales.

Figure 7:  Simulated bolometer sensitivity as a function of pressure and wire
diameter (a) The magnitude of the bolometer response as measured by the
thermometer wire only weakly depends on the wire diameter. Thus, the error of
the conventional readout (blue line) does not depend on wire diameter. Noise
in the SQUID circuit (red dashed line) increases as the wire diameter
decreases due to larger thermal motion of the wire, and therefore the readout
error increases. The relative shot noise (dotted green line) decreases as the
wire size increases due to the increase in the total number of quasiparticles
hitting the wire. The pressure in this panel is 0 bar. (b) The highest
sensitivity at a fixed relative temperature ($T/T_{\mathrm{c}}$) is achieved
at zero pressure. Note that the superfluid decouples from heat exchangers at
approximately the same $T/T_{\mathrm{c}}$ regardless of pressure. This panel
shows calculations for a 400 nm resonator wire; the conventional and SQUID
readouts are simulated, respectively, for a 100 mT and 0.4 mT magnetic field.
In both panels the temperature is $T/T_{\mathrm{c}}=0.12$ and the the
collision energy deposited $Q=10$ eV.

##  Appendix B Simulation of bolometer readout

We consider two circuits for the readout of induced voltage $V_{0}$: a
‘conventional’ four-point measurement with the induced voltage read out via
passive pre-amplification by a cold transformer (amplification $\times 100$),
and a SQUID-based preamplifier circuit (Figure 9). The latter amplifies the
signal orders of magnitude more than the passive transformer, as described
below. Both readout schemes use a room-temperature lock-in amplifier to record
the signal, operated with time constant 100 ms. The lock-in input noise is
white with the RMS amplitude $V_{\mathrm{RMS}}$=7.9 nV. This is the dominant
noise source in the passive pre-amplification circuit, while in the SQUID
circuit the lock-in noise can be neglected. Noise in the SQUID circuit is
derived in the next section.

The resolution of the energy measurement determines the recoil energy
threshold in the detector. In order to simulate an experiment, we add noise
onto a signal generated using Eq. (13). The obtained noisy signal is then
fitted with the same equation to obtain a fitted deposited energy, and the
outcome is compared with the known actual $Q$. The energy deposition is always
considered to take place at $t=0$ so the time alignment is not part of the
fitting procedure; this is justified by the fact that, given the current
estimated rate of events, any relevant pile up of events is not expected
(which would require an additional peak finding procedure). An example signal
with added conventional readout noise is fitted with Eq. (13) in Figure 8.
Repeating this process yields a Monte Carlo set of pseudo-experiments. The
outcome is a Gaussian distribution centred at $Q$ with a standard deviation
$\sigma$. The uncertainty shown in Figure 4 and 7 corresponds to $\sigma/Q$.

Figure 8:  A fit to a simulated bolometer response: At $t<0$ the bolometer is
at a steady temperature $T/T_{\mathrm{c}}$ = 0.11. With a 400 nm readout wire
this corresponds to $\Delta f\approx$ 300 mHz (black dots). When a sudden heat
release of Q = 1 keV takes place at $t=0$, the bolometer heats up and $\Delta
f$ increases as determined by the time constants $\tau_{\mathrm{w}}$ and
$\tau_{\mathrm{b}}$. The initial temperature rise in the bolometer is assumed
to be instantaneous. The blue line shows a fit to Eq. (13). The bolometer
temperature, extracted from the fit, returns to the value before $t=0$
exponentially as determined by $\tau_{\mathrm{b}}$ (red dash line). The green
dashed line shows the base width in the fit. Pressure in this simulation is
$P=0$ bar and the noise in the simulated signal corresponds to that in the
conventional readout circuit.

With the conventional readout scheme, the recoil energy can be measured with
$<$100% uncertainty above deposited energy of 100 eV, whilst with SQUID
readout smallest detectable energy with this criterion is approximately 0.1
eV. The conventional readout sensitivity does not depend on the wire diameter
($\tau_{\mathrm{w}}\propto 1/d$) as seen in Figure 7a. However, the SQUID
readout sensitivity does depend on $d$ because this scheme is sensitive to the
thermal motion of the wire as detailed below, and that depends on the
thickness $d$. Regardless of the readout scheme, the highest sensitivity is
reached at saturated vapour pressure, which is essentially zero at microkelvin
temperatures (Figure 7b). We note that helium density increases $\sim$40% from
zero pressure to 30 bar, and that $T_{\mathrm{c}}$ is higher at higher
pressures so a given relative temperature can be held for longer in practical
experiments. These factors need to be taken into account when optimising the
detector performance and maximising the useful exposure, but this remains a
task for the future.

##  Appendix C SQUID readout

Figure 9:  Readout schemes for a vibrating wire moving in a magnetic field
$B$: (Top) Conventional readout scheme. The vibrating wire with impedance
$Z(f)$ is driven with current $I_{\mathrm{x}}$. The voltage across the wire is
detected with a room-temperature lock-in amplifier (LIA), coupled via a cold
transformer. (Bottom) SQUID readout scheme. The voltage excitation is applied
via a transformer with mutual inductance $M_{\mathrm{x}}$. SQUID current
sensor detects the current $I_{\mathrm{i}}$ flowing through the wire, contact
resistance $R$, and SQUID input coil $L_{\mathrm{i}}$. The SQUID is connected
to LIA via room temperature flux-locked-loop (FLL) electronics. In both
circuits all noise sources can be represented by equivalent voltage
$V_{\mathrm{n}}$ in the wire. In our simulation the two readouts are operated
at different optimal magnetic fields.

To improve the energy resolution, we consider nanowire resonator readout based
on a superconducting quantum interference device (SQUID) current sensor Drung
_et al._ (2007). The circuit diagram is shown in Figure 9. The nanowire is
excited with a voltage $V_{\mathrm{x}}=2i\pi fM_{\mathrm{x}}I_{\mathrm{x}}$
applied by driving a current $I_{\mathrm{x}}$ via the transformer with mutual
inductance $M_{\mathrm{x}}$. The impedance of the resonator

$Z(f)=V_{\mathrm{x}}\big{/}I_{\mathrm{i}}-R-2i\pi fL_{\mathrm{i}}$ (14)

is inferred from the current $I_{\mathrm{i}}$ measured by the SQUID, after
subtracting the contact resistance $R$ (the only resistive element of the
circuit, we consider an upper bound $R=1\,\mathrm{\Omega}$) and impedance of
the SQUID input coil $L_{\mathrm{i}}=1.5\,\mu$H, which are connected in series
with the nanowire. The self-inductance of the secondary of the drive
transformer is included in $L_{\mathrm{i}}$. On resonance the wire impedance
$Z_{0}=Z(f_{0})=lB^{2}\big{/}\big{(}2\pi m\,\Delta f\big{)}$ is directly
related to the linewidth $\Delta f$. Here $m$ is the wire mass per unit
length.

We now turn to the sensitivity of the on-resonance readout mode. We assume the
measurement bandwidth $\delta f$ determined by the lock-in time constant to be
greater than $\Delta f$. It is convenient to represent all noise sources in
terms of equivalent voltage in the wire, shown by $V_{n}$ in Figure 9.

The intrinsic SQUID flux noise has typical power spectral density
$S_{\phi}\sim 10^{-13}\Phi_{0}^{2}/\mathrm{Hz}$, where $\Phi_{0}=h/(2e)\approx
2\,$\text{\,}\mathrm{fWb}$$ represents the superconducting magnetic flux
quantum. This noise corresponds to the input current variance
$I_{\mathrm{i}}^{2}=S_{\phi}\,\delta f/M_{\mathrm{i}}^{2}$, where
$M_{\mathrm{i}}=10$ nH is the mutual inductance between the input coil and the
SQUID. The equivalent voltage noise is

$V_{\text{SQUID}}^{2}=\left|Z+R+i2\pi
f_{0}L_{\mathrm{i}}\right|^{2}S_{\phi}\,\delta f\big{/}M_{\mathrm{i}}^{2}.$
(15)

The back action from the SQUID Tesche and Clarke (1977) adds a prefactor
Levitin  _et al._ (2007) $1-\alpha^{2}\eta\sim 0.6$ of order unity in front of
$L_{\mathrm{i}}$ in Eq. (15), that has no significant effect. The Johnson-
Nyquist noise in the wire contact resistance is

$V_{\mathrm{JN}}^{2}=4k_{B}TR\,\delta f.$ (16)

The thermal motion of the wire (resonant phonons) leads to

$V_{\text{wire}}^{2}=k_{B}TlB^{2}\big{/}m.$ (17)

Thus, the total RMS noise in the voltage measurement is

$V_{\mathrm{n}}=\small\sqrt{V_{\text{SQUID}}^{2}+V_{\mathrm{JN}}^{2}+V_{\text{wire}}^{2}}.$
(18)

Here we assume that the resonant phonons in the wire and electrons in the
contacts are at the same temperature $T$ as the surrounding superfluid.

The optimum readout magnetic field is found by comparing $V_{\mathrm{n}}$ to
the voltage $V_{0}$ associated with the motion of the wire, For the $l=2$ mm
wire resonating at $f_{0}=0.8$ kHz with $\Delta f=60$ mHz, considered in our
simulations, the best sensitivity is achieved at $B=0.4$ mT, where the
impedance $Z_{0}$ of the wire matches the rest of the circuit $R+2i\pi
fL_{\mathrm{i}}$. Integrating the SQUID readout into the experimental setup
shown in Figure 1 may require tailoring the magnetic field profile or
increasing $L_{\mathrm{i}}$.

##  Appendix D Quasiparticle shot noise

The three-dimensional calculation that yields the drag force from
quasiparticle collisions, Eq. (10), considers all possible quasiparticle
scattering trajectories including those where the quasiparticle is scattered
multiple times before leaving the vicinity of the wire. The contribution from
escape trajectories requiring multiple scatterings is not negligible Enrico
(1995). To estimate the fluctuations in the force, Eq. (10), we nevertheless
neglect these scattering trajectories and simply calculate the quasiparticle
flux from the quasiparticle number density.

The temperature-dependent quasiparticle number density $n$ is

$n=\frac{1}{\pi^{2}}\left(\frac{2m^{*}}{\hbar^{2}}\right)^{3/2}\sqrt{E_{F}}k_{B}T\exp\left({-\frac{\varDelta}{k_{B}T}}\right)\\\
$ (19)

where $E_{F}=p_{F}^{2}/(2m^{*})$. As a two-dimensional approximation, a flux
$J=\frac{1}{2}nv_{F}lr$ of quasiparticles of effective mass $m^{*}$ impacts on
a wire of length $l$ and radius $r$, from one direction. The QPs hitting the
wire follow a Poisson distribution in time. That is, the error (input shot
noise, or spectral noise) corresponds to the standard deviation
$\sqrt{J}/\sqrt{\mathrm{Hz}}$ in a bandwidth $\delta f$ is expected to be
Bradley and Hayes (2000)

$\displaystyle\sigma_{N}$
$\displaystyle=\sqrt{J}=\sqrt{\frac{nv_{F}lr}{2\delta f}}.$ (20)

Therefore the relative error (or fractional noise) per $\sqrt{\mathrm{Hz}}$ is

$\sqrt{J}/J=\sqrt{\frac{2}{nv_{F}lr}}.$ (21)

The relative shot noise becomes larger at low temperatures.

We note that the magnitude of the quasiparticle shot noise in 3He has not been
verified experimentally, and it may deviate from the above estimate due to the
complicated three-dimensional scattering processes involved. If necessary, the
shot noise can be reduced significantly by modifying the geometry of the
oscillator from a solid-metal cylinder to a paddle-like structure. This can be
done by attaching a graphene flake or PMMA film to the superconducting
nanowire. This way the flux will significantly increase without making the
oscillator much heavier. This modification will change the flow around the
sensor and might result in reduction of the critical velocity. Further
experiments required to verify this geometry in superfluid 3He are underway,
however they are outside the scope of this paper.

##  Appendix E Signal model

We are interested in the sensitivity to both spin-dependent and spin-
independent scattering. In this work we consider interactions either arising
from $\mathcal{O}_{4}$ (spin-dependent) or $\mathcal{O}_{1}$ (spin-
independent) in the conventions of Fitzpatrick  _et al._ (2013).

The differential event rate for a dark matter particle of mass $m_{\chi}$
scattering with a target nucleus of mass $m_{N}$ is given by

$\frac{dR}{dE_{\rm NR}}=\frac{\rho_{\chi}}{m_{\chi}m_{N}}\int_{v_{\rm
min}}^{\infty}\frac{d\sigma}{dE_{\rm R}}\;v\;f(\vec{v})\;d^{3}v,$ (22)

where $\rho_{\chi}$ is the local dark matter density, $f(\vec{v})$ is the dark
matter velocity distribution evaluated in the frame of the detector, truncated
at $v=v_{\rm esc}$ with a lower limit $v_{\rm min}=\sqrt{m_{N}E_{\rm
NR}/2\mu^{2}_{\chi N}}$, which is the minimum velocity required to produce a
recoil energy $E_{\rm NR}$ in the detector, where $\mu^{2}_{\chi
N}=m_{\chi}m_{\rm N}/(m_{\chi}+m_{\rm N})$ is the reduced mass of the dark
matter-target nucleus system.

The differential cross section, $\frac{d\sigma}{dE_{\rm NR}}$, has in general
contributions from both spin-dependent and spin-independent processes of the
form

$\frac{d\sigma}{dE_{\rm NR}}=\frac{d\sigma^{\rm SD}}{dE_{\rm
NR}}+\frac{d\sigma^{\rm SI}}{dE_{\rm NR}}.$ (23)

In order to set useful limits that can be compared across experiments and
against theory it is assumed that only one type of process is present.

Focusing first on the spin-dependent only case. The differential cross section
can be written as Engel  _et al._ (1992)

$\frac{d\sigma^{\rm SD}}{dq^{2}}=\frac{8G_{F}^{2}}{(2J+1)v^{2}}\;S_{A}(q),$
(24)

where $q^{2}=2E_{\rm NR}m_{N}$ and the spin structure function, $S_{A}(q)$, is
given by

$S_{A}(q)=a_{0}^{2}S_{00}+a_{0}a_{1}S_{01}+a_{1}^{2}S_{11},$ (25)

where the $S_{ij}$s are the sum of all contributions to the spin-dependent
interactions. The parameters $a_{0},a_{1}$ encode the model dependent
isoscalar and isovector effective dark matter-nucleon couplings respectively.

In the limit of zero momentum transfer ($q=0$), which is a very good limit for
dark matter scattering with Helium-3, the spin-structure function can be
written as,

$\displaystyle S_{A}(0)=\frac{(2J+1)(J+1)}{\pi J}\left(a_{p}\langle{\bf
S_{p}}\rangle+a_{n}\langle{\bf S_{n}}\rangle\right)^{2},$ (26)

where $\langle{\bf S_{p}}\rangle$ and $\langle{\bf S_{n}}\rangle$ are the mean
spins of the proton and neutron in the target nucleus, $J$ is the total spin
of the target nucleus and $a_{p},a_{n}$ are the effective dark matter-proton
and dark matter-neutron couplings, defined by the relations by
$a_{0}=a_{p}+a_{n}$ and $a_{1}=a_{p}-a_{n}$. For 3He, $\langle{\bf
S_{p}}\rangle=0$ leaving only dark matter-neutron spin-dependent scattering
Catena and Schwabe (2015).

The differential cross section can be written in terms of the spin-dependent
dark matter-neutron scattering cross section, $\sigma^{\rm SD}_{\rm\chi n}$,
as

$\frac{d\sigma^{\rm SD}}{dq^{2}}=\frac{\sigma^{\rm SD}_{\rm\chi
n}}{3\mu^{2}_{\chi n}v^{2}}\frac{J+1}{J}\langle{\bf S_{n}}\rangle^{2}.$ (27)

Setting $J=1/2$, $\langle{\bf S_{n}}\rangle=1/2$ and converting back to use
recoil energy in the differential cross section,

$\frac{d\sigma^{\rm SD}}{dE_{\rm NR}}=\sigma^{\rm SD}_{\rm\chi
n}\frac{m_{N}}{2\mu^{2}_{\chi n}v^{2}}.$ (28)

Substituting this into the differential event rate we arrive at

$\frac{dR^{\rm SD}}{dE_{\rm NR}}=\frac{\rho_{\chi}\,\sigma^{\rm SD}_{\rm\chi
n}}{2m_{\chi}\,\mu^{2}_{\rm\chi n}}\int^{\infty}_{v_{\rm
min}}\frac{1}{v}\;f(\vec{v})\;d^{3}v.$ (29)

For the spin-independent case, the differential cross section is

$\frac{d\sigma^{\rm SI}}{dE_{\rm NR}}=\frac{m_{N}\sigma_{0}F^{2}(E_{\rm
NR})}{2v^{2}\mu_{\rm\chi N}^{2}},$ (30)

where $F^{2}(E_{\rm NR})$ is a nuclear form factor accounting for the loss of
coherence for large momentum transfer, $\mu_{\chi N}$ is the dark matter-
nucleus reduced mass. The parameter $\sigma_{0}$ contains the dark matter
interaction details and can be written as Kopp  _et al._ (2010)

$\sigma_{0}=\frac{\left(Zf_{p}+(A-Z)f_{n}\right)^{2}}{f_{p}^{2}}\frac{\mu^{2}_{\chi
N}}{\mu^{2}_{\chi p}}\sigma_{\rm\chi p}^{\rm SI},$ (31)

where the parameters $f_{p,n}$ are the dark matter-proton/neutron couplings
and $\sigma_{\rm\chi p}^{\rm SI}$ is the spin-independent dark matter-proton
cross section. We assume $f_{p}=f_{n}$ and in the limit of zero momentum
transfer, where $F^{2}(E_{\rm NR})\rightarrow 1$, the differential cross
section simplifies to

$\frac{d\sigma^{\rm SI}}{dE_{\rm NR}}=\frac{m_{N}A^{2}}{2v^{2}\mu_{\chi
p}^{2}}\sigma_{\rm\chi p}^{\rm SI}.$ (32)

Substituting this into the differential event rate and setting $A=3$ gives

$\frac{dR^{\rm SI}}{dE_{\rm NR}}=\frac{9\rho_{\chi}\sigma^{\rm SI}_{\rm\chi
p}}{2m_{\chi}\,\mu^{2}_{\rm\chi p}}\int^{\infty}_{v_{\rm
min}}\frac{1}{v}\;f(\vec{v})\;d^{3}v.$ (33)