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# Light Dark Matter Accumulating in Terrestrial Planets: Nuclear Scattering

Joseph Bramante  The Arthur B. McDonald Canadian Astroparticle Physics
Research Institute,
Department of Physics, Engineering Physics, and Astronomy,
Queen’s University, Kingston, Ontario, K7L 2S8, Canada Perimeter Institute for
Theoretical Physics, Waterloo, ON N2J 2W9, Canada Jason Kumar  Department of
Physics and Astronomy, University of Hawai’i, Honolulu, HI 96822, USA Gopolang
Mohlabeng  The Arthur B. McDonald Canadian Astroparticle Physics Research
Institute,
Department of Physics, Engineering Physics, and Astronomy,
Queen’s University, Kingston, Ontario, K7L 2S8, Canada Perimeter Institute for
Theoretical Physics, Waterloo, ON N2J 2W9, Canada Department of Physics and
Astronomy, University of California, Irvine, CA 92697, USA Nirmal Raj  TRIUMF,
4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada Ningqiang Song  Department
of Mathematical Sciences,
University of Liverpool, Liverpool, L69 7ZL, United Kingdom The Arthur B.
McDonald Canadian Astroparticle Physics Research Institute,
Department of Physics, Engineering Physics, and Astronomy,
Queen’s University, Kingston, Ontario, K7L 2S8, Canada

###### Abstract

The direct capture and accumulation of Galactic dark matter in astrophysical
bodies can occur as a result of its scattering with nuclei. In this work we
investigate the detailed capture and evaporation of dark matter in terrestrial
planets, taking Earth as an example. We focus on the strongly interacting case
in which Earth may be opaque to dark matter, referred to as the “optically
thick” limit. We investigate light dark matter in particular, addressing
important dynamical processes such as the “ping-pong effect” during dark
matter capture and the heating of Earth by dark matter annihilation. We do
this using Monte-Carlo simulations as well as detailed analytical
computations, and obtain improved bounds on dark matter direct detection for
both spin-dependent and spin-independent scattering, and also allowing for the
interacting species to make up a sub-component of the dark matter density.

††preprint:

##  I Introduction

The ongoing search for non-gravitational interactions of dark matter with the
standard model continues to ramp up. Terrestrial experiments and astrophysical
probes are placing more and more stringent constraints on the various parts of
the dark matter parameter space. This has not only prompted the emergence of
exciting new ideas to search for this elusive matter, but also the need to
revisit already established concepts and reconsider previous assumptions in
order to better understand the dark matter parameter space.

Among many such ideas is the notion of using celestial bodies as laboratories
to search for dark matter. This is well motivated due to the ability of the
astrophysical objects to capture dark matter particles and enhance the flux of
their annihilation products. The standard framework for studying dark matter
capture and evaporation was largely established, for the Sun and the Earth, by
Refs. Gould:1987ju ; Gould:1987ir ; Gould:1987ww . Subsequently, a substantial
body of work then studied dark matter capture by and evaporation from; the Sun
Zentner:2009is ; Kong:2014mia ; Alhazmi:2016qcs ; Feng:2016ijc ;
Garani:2017jcj , Earth Feng:2015hja ; Baum:2016oow ; Catena:2016kro  and other
stellar and planetary systems Adler:2008ky ; Iocco:2008xb ; Guver:2012ba ;
Baryakhtar:2017dbj ; Raj:2017wrv ; Joglekar:2019vzy ; Acevedo:2019agu ;
Joglekar:2020liw ; Leane:2020wob ; Dasgupta:2020dik ; Garani:2020wge ;
Leane:2021tjj ; Leane:2021ihh ; Lopes:2021jcy ; Garani:2021feo ; Bell:2021fye
; Banks:2021sba ; Bramante:2021dyx ; Coffey:2022eav , in part because the
annihilation of dark matter in these systems can yield signals of weakly
coupled particles which can be observed by large volume experiments on Earth
and various other space-based detectors. In much of the previous literature,
the standard framework for studying the dark matter capture and evaporation is
in the so-called “optically thin” limit, in which dark matter typically
scatters at most once as it traverses the celestial body. In the optically
thin limit, there is an “evaporation mass” for the dark matter particle, below
which dark matter will be rapidly evaporated from the body, see Ref.
Garani:2021feo  for a detailed study of the evaporation mass from various
astrophysical bodies.

Using this approach, A. Gould considered the study of dark matter capture and
evaporation, from the Sun Gould:1989tu  and subsequently the Earth
Gould:1991va , in the “optically thick limit”. In this limit, the dark matter
gets captured by interacting many times with nuclei during transit. Over the
years, many subsequent works have refined these calculations. Much of the
focus of these works has been on the Sun, largely because the regions of
parameter space for which searches for the annihilation of WIMP-like dark
matter in the Earth would be most sensitive ($m_{\chi}\sim{\cal
O}(100\leavevmode\nobreak\ {\rm GeV})$ Gould:1987ir ) are already ruled out by
direct detection experiments LZ:2022ufs . More recently various other
astrophysical systems have been considered, in which the multi-scatter capture
of dark matter with masses in the range $m_{\chi}\sim$ (10 MeV - 1 PeV) can be
probed Bramante:2017xlb ; Dasgupta:2019juq ; Ilie:2021iyh ; Leane:2022hkk .

The focus of this work is on revisiting dark matter capture and evaporation in
the Earth, with a focus on the optically thick regime. In this regime, dark
matter may be shielded from deep underground direct detection experiments by
the overburden, yielding open parameter space at large coupling, even for
relatively small masses. This motivates a more detailed study, focusing on
less explored regions of parameter space. There has already been recent work
on this topic, for the purpose of determining if dark matter models can be
excluded through the excess heating of the Earth which would be caused by dark
matter annihilation to visible matter Mack:2007xj ; Bramante:2019fhi . Our
work will address some subtleties in the treatment of capture and evaporation
in the optically thick scenario, especially for low-mass dark matter. One of
our main improvements will be the use of a full Monte Carlo simulation of the
dark matter capture process, rather than an analytic treatment. We will see
that a full numerical treatment reveals some important features which can have
a significant effect on the result.

The rest of the article is organized as follows: In Section II we describe the
evolution of the number density of dark matter throughout the Earth. Section
III is devoted to our treatment of the capture process, comparing with the
standard analytical approach. Subsequently, in Sections IV and V, we present
the evaporation and annihilation rates respectively. In Section VI we show the
experimental limits and finally conclude in Section VII.

##  II Dark Matter Number Density in the Earth

We assume that the processes which affect the distribution of dark matter in
the Earth can be summarized as follows:

  * •

Dark matter particles are gravitationally captured at a rate $C_{\oplus}$.
Capture occurs when a dark matter particle, which has scattered one or more
times against nuclei which are nearly at rest, drops below the Earth’s escape
velocity.

  * •

Captured dark matter particles strongly interact with SM matter and
thermalize, yielding a distribution of dark matter which is approximately in
local thermal equilibrium (LTE). The total number of dark matter particles in
LTE in the Earth is denoted by $N_{C}$.

  * •

Dark matter particles, in LTE, can annihilate with each other. The
annihilation rate is $(1/2)A_{\oplus}N_{C}^{2}$.

  * •

Dark matter particles in LTE can also evaporate as a result of scattering
against SM particles at the tail of the thermal distribution, if the DM
particles gain enough energy to escape the Earth, accounting for the
possibility of dark matter scattering again before leaving the Earth. The
total evaporation rate from the Earth is denoted by $E_{\oplus}$.

Note, we have assumed that the thermalization time for captured dark matter
particles is very short compared with the characteristic time scale for a dark
matter particle to either annihilate or evaporate away.

The rate of change of the number of dark matter particles captured within the
Earth is then expressed as

$\displaystyle\frac{dN_{C}}{dt}$ $\displaystyle=$ $\displaystyle
C_{\oplus}-\left(\frac{E_{\oplus}}{N_{C}}\right)N_{C}-\frac{1}{2}A_{\oplus}N_{C}^{2}\,,$
(1)

(assuming the dark matter particle is its own anti-particle) where
$C_{\oplus}$, $E_{\oplus}/N_{C}$ and $A_{\oplus}$ will be solved for. In the
equilibrium limit, we find $dN_{C}/dt=0$. But more generally, we can solve ths
equation for $N_{C}(t)$, and evaluate the total annihilation rate,

$\Gamma_{\oplus}(t)=(1/2)A_{\oplus}N_{C}(t)^{2}\,,$ (2)

at present times.

##  III Dark Matter Capture

###  III.1 Analytical Approach

In the analytical approach, one calculates the probability for a dark matter
particle in the halo to scatter once in the Earth and get captured. We will
review this approach here, and highlight aspects of the capture process which
are not fully described in this approach, and are more completely revealed by
numerical simulation. For this we primarily follow the approach outlined in
Garani:2017jcj  (dubbed Model 1, or M1), and briefly mention the approach in
Baum:2016oow  (dubbed Model 2, or M2).

If we assume that dark matter follows a Maxwell-Boltzmann distribution in the
rest frame of the Galactic Center, with dark matter velocity dispersion
$v_{d}=270$ km/s, then in the rest frame of Earth, the dark matter velocity
distribution is given by Garani:2017jcj

$f(u_{\chi})=\sqrt{\dfrac{3}{2\pi}}\dfrac{u_{\chi}}{v_{\oplus}v_{d}}\left(\mathrm{exp}\left[-\frac{3(u_{\chi}-v_{\oplus})^{2}}{2v_{d}^{2}}\right]-\mathrm{exp}\left[-\frac{3(u_{\chi}+v_{\oplus})^{2}}{2v_{d}^{2}}\right]\right)\,,$
(3)

where we take the Earth velocity $v_{\oplus}=220$ km/s. A dark matter particle
in the halo will be accelerated by the Sun’s and Earth’s gravitational
potential. If an infalling dark matter particle has speed $u_{\chi}$ when far
from the Sun, we assume that it will have speed
$w(r)=\sqrt{u_{\chi}^{2}+v_{s}^{2}+v_{e}(r)^{2}}$ at a distance $r$ from the
Earth center, where $v_{s}=42.2$ km/s is the the escape velocity from the Sun
at 1 AU and $v_{e}(r)$ is the escape velocity of the Earth. We neglect the
subtleties in the solar velocity boost in the frame of the Earth, see for
example Griest:1987vc . We have also neglected the effect of the Galactic
escape velocity of dark matter at the position of the solar system, as the
Maxwellian distribution above the escape velocity only makes up a small
fraction of dark matter, and the capture of dark matter is dominated by the
low velocity part.

A dark matter particle may scatter with the nuclei in the Earth, losing part
of its kinetic energy. If the particle falls below the Earth’s escape
velocity, it will be gravitationally captured. If the dark matter-nucleus
interaction is strong enough that all dark matter particles that bombard the
Earth are captured, the capture rate saturates at the geometric capture rate

$C_{\oplus}^{\mathrm{geom}}=\pi R_{\oplus,\rm
atm}^{2}\sqrt{\dfrac{8}{3\pi}}\dfrac{\rho_{\chi}v_{d}}{m_{\chi}}\left(1+\dfrac{3v_{e}^{2}(R_{\oplus,\rm
atm})}{v_{d}^{2}}\right)\xi\,,$ (4)

where

$\xi=\left[v_{d}^{2}e^{-\frac{3v_{\oplus}^{2}}{2v_{d}^{2}}}+\sqrt{\dfrac{3\pi}{2}}\dfrac{v_{d}}{v_{\oplus}}\left(v_{e}^{2}(R_{\oplus,\rm
atm})+v_{\oplus}^{2}+\dfrac{v_{d}^{2}}{3}\right)\mathrm{Erf}\left(\sqrt{\dfrac{3}{2}}\dfrac{v_{\oplus}}{v_{d}}\right)\right]\left(2v_{d}^{2}+3v_{e}^{2}(R_{\oplus,\rm
atm})\right)^{-1}\,.$ (5)

Essentially, the geometric capture rate is the product of the Earth’s
geometric cross section ($\pi R_{\oplus,\rm atm}^{2}$), the dark matter flux
($\sim\rho_{\chi}v_{d}/m_{\chi}$) and a factor which accounts for the
gravitational version of Sommerfeld-enhancement. We take into consideration
the effect of the atmosphere on the capture, evaporation and annihilation of
dark matter. The atmosphere is assumed to be 100 km thick, beyond which the
number density of atmospheric particles is negligibly small. As a consequence,
the geometric size of the Earth is $R_{\oplus,\rm atm}=6471$ km. We take
$\rho_{\chi}=0.3$ GeV/cm3 for the local dark matter density.

If the interaction between dark matter and nuclei is weak, a dark matter
particle will likely scatter at most once as it crosses the Earth. The
corresponding capture rate is Gould:1987ir

$C_{\oplus}^{\mathrm{weak}}=\sum\limits_{j}\dfrac{\rho_{\chi}}{m_{\chi}}\int_{0}^{{R_{\oplus}}}4\pi
r^{2}\int_{0}^{\infty}du_{\chi}f(u_{\chi})\dfrac{w(r)}{u_{\chi}}\int_{0}^{v_{e}(r)}R_{j}^{-}(w\rightarrow
v)|F_{j}(q)|^{2}dv\,.$ (6)

The sum runs over all nuclear targets $j$ in the Earth and
$R_{j}^{-}(w\rightarrow v)$ describes the rate for a dark matter particle of
velocity $w(r)$ to slow down to $v<v_{e}(r)$ by scattering with nucleus $j$
(see Garani:2017jcj  for the explicit expressions of $R_{j}^{\pm}$). The
$R_{j}^{\pm}$ are proportional to the dark matter-nucleus scattering cross
section. We consider two types of interactions in this paper, spin-independent
(SI) and spin-dependent (SD) nuclear scattering (both taken to be velocity-
independent). In the former case, the nuclear scattering cross section at zero
momentum transfer is given by

$\sigma_{j,0}^{\rm
SI}=\left(\dfrac{\mu_{A_{j}}}{\mu_{N}}\right)^{2}A_{j}^{2}\sigma^{\rm
SI}_{\chi N}\,,$ (7)

where $A_{j}$ is the mass number of a Standard Model nucleus, $\mu_{A_{j}}$
($\mu_{N}$) is the reduced mass between dark matter and the nucleus (nucleon),
and $\sigma^{\rm SI}_{\chi N}$ is the dark matter nucleon spin-independent
scattering cross section. For the latter, we have instead

$\sigma_{j,0}^{\rm
SD}=\left(\dfrac{\mu_{A_{j}}}{\mu_{N}}\right)^{2}\dfrac{4(J_{j}+1)}{3J_{j}}\left(a_{p}\langle
S_{p}\rangle+a_{n}\langle S_{n}\rangle\right)^{2}\sigma^{\rm SD}_{\chi N}\,,$
(8)

where $J_{j}$ is the total nuclear spin, $\sigma^{\rm SD}_{\chi N}$ is the
dark matter nucleon spin-dependent scattering cross section, and $\langle
S_{p}\rangle$ and $\langle S_{n}\rangle$ represent the average spin of protons
and neutrons in a nucleus, respectively. We also explore three scenarios,
isospin-independent scattering $a_{p}=a_{n}=1$, proton-only scattering
$a_{p}=1$, $a_{n}=0$ and neutron-only scattering $a_{p}=0$, $a_{n}=1$. As we
have checked that including the nuclear form factor will not change the
results significantly, we always set the momentum-dependent form factor to
unity to speed up the computation, unless explicitly stated. We use the Earth
composition listed in Table 1 of Ref. Bramante:2019fhi . For the atmosphere,
we use the NRLMSISE-00 model NRLMSISE-00  with the average oxygen, nitrogen,
argon, helium mass fractions of 23.18%, 75.6%, 1.2% and $7.25\times 10^{-7}$
respectively. We also list the isotope abundance and spin information relevant
for the SD calculation in Table 1 in Appendix A.

When the dark matter-nucleus scattering cross section is small, the capture
rate is given by $C_{\oplus}^{\mathrm{weak}}$. But no matter how large the
scattering cross section is, the capture rate can never exceed
$C_{\oplus}^{\mathrm{geom}}$, which is the rate at which dark matter is
incident on the Earth. The capture rate can thus be approximated by
Bernal:2012qh ; Garani:2017jcj

$C_{\oplus}=C_{\oplus}^{\mathrm{weak}}(1-\exp(-C_{\oplus}^{\mathrm{geom}}/C_{\oplus}^{\mathrm{weak}}))\,.$
(9)

The capture rates computed from Eq. (6) and Eq. (9) are shown in Figure 1
(dashed lines, M1). The capture is maximized when the dark matter mass matches
the mass of the target nucleus, and the typical recoil energy is maximized,
resulting in the peaks around 10-100 GeV. For spin-independent scattering,
from left to right we can identify the the peaks caused by scattering with O,
with Mg, and with Si and Fe. For spin-dependent scattering, there are instead
two peaks arising from scattering with ${}^{29}\mbox{Si}$, ${}^{27}\mbox{Al}$,
${}^{25}\mbox{Mg}$, as well as ${}^{57}\mbox{Fe}$. At large cross sections,
the peaks are smeared out as the capture saturates the geometric rate.

Figure 1: Total dark matter capture rates in the Earth in the optically thick
regime for various dark matter-nucleon scattering cross sections. The dashed,
dash-dotted and solid lines show the capture rates from Garani:2017jcj  (M1),
Baum:2016oow  (M2) and from Monte Carlo (MC) simulations using DaMaSCUS-
EarthCapture, modified from the DaMaSCUS code Emken:2017qmp ;
emken2017damascus , respectively. Solar acceleration of halo dark matter is
included. The capture fractions from simulations are extrapolated below
$10^{-5}$ GeV. See text for details. The dashed gray line depicts the Earth
geometric dark matter capture rate when all dark matter particles that
encounter the Earth get captured. Left: Results for spin-independent
scattering. Right: Results for spin-dependent scattering with $a_{p}=a_{n}=1$.

For comparison we also show in Figure 1 the results extrapolated from
Baum:2016oow  (dash-dotted lines, M2), where the capture rate is computed
using Eq. (9), but with $C_{\oplus}^{\mathrm{weak}}$ given by Baum:2016oow

$C_{\oplus}^{\mathrm{weak}}=\sum\limits_{j}\dfrac{\rho_{\chi}}{m_{\chi}}\sigma_{j,0}\int_{0}^{{R_{\oplus}}}4\pi
r^{2}v_{e}^{2}(r)n_{j}(r)\int_{0}^{u_{\chi,\max}}du_{\chi}\dfrac{f(u_{\chi})}{u_{\chi}}\left(1-\dfrac{u_{\chi}^{2}+v_{s}^{2}}{u_{\chi,\max}^{2}+v_{s}^{2}}\right)\,,$
(10)

with $n_{j}$ the number density of an isotope $j$ in the Earth. Note that we
have included the solar gravitational acceleration explicitly but discarded
the momentum transfer dependence of the form factors. In a single scattering,
the maximum momentum transfer is $q_{\max}=2\mu_{A_{j}}w$, causing the maximum
kinetic energy loss $E_{R,\max}=2\mu_{A_{j}}^{2}w^{2}/m_{A_{j}}$. If the dark
matter particle is captured, the dark matter kinetic energy after scatter must
satisfy the relation
$E_{\chi,f}=\frac{1}{2}m_{\chi}w^{2}-E_{R,\max}\leq\frac{1}{2}m_{\chi}v_{e}^{2}(r)$.
This caps the dark matter initial velocity at

$w\leq\left\lvert\dfrac{m_{\chi}+m_{A_{j}}}{m_{\chi}-m_{A_{j}}}\right\rvert
v_{e}(r)\,.$ (11)

With the relation $w(r)=\sqrt{u_{\chi}^{2}+v_{s}^{2}+v_{e}(r)^{2}}$, this
translates to the maximum dark matter halo speed such that single-scatter
capture is possible:

$u_{\chi,\max}=\sqrt{\dfrac{4m_{\chi}m_{A_{j}}}{(m_{\chi}-m_{A_{j}})^{2}}v_{e}^{2}(r)-v_{s}^{2}}\,.$
(12)

If the dark matter speed (at infinity) is greater than $u_{\chi,\max}$ in the
halo, its speed after in-fall to radius $r$ will be such that, even if scatter
results in the maximum momentum transfer, the dark matter still will not have
dropped below the escape velocity. Hence, gravitational capture with a single
scatter is highly improbable. This is intuitive as the dark matter velocity
change in scattering is proportional to the momentum transfer $q$, which
decreases with smaller dark matter mass. It is also understood that the square
root in Eq. (12) has to be real, which determines the minimum captured dark
matter mass in single scattering

$m_{\chi,\min}\simeq
m_{A,\min}\left(1-2\dfrac{v_{e}}{v_{s}}+2\dfrac{v_{e}^{2}}{v_{s}^{2}}-\dfrac{v_{e}^{3}}{v_{s}^{3}}\right)\simeq
0.59\leavevmode\nobreak\ m_{A,\min}\,,$ (13)

where we have dropped higher powers of $v_{e}/v_{s}$. In the second equation
we have used $v_{e}=11.2$ km/s, the escape speed at the surface of the Earth.
If scattering occurred at the center of the Earth, we would instead find
$m_{\chi,\min}\sim 0.5\leavevmode\nobreak\ m_{A,\min}$. Using
${}^{14}\mbox{N}$ puts the minimum dark matter mass at 7.7 GeV, while He might
facilitate the capture of dark matter as light as 2.2 GeV, despite the fact
that the He abundance in the atmosphere is extremely small. Similarly, there
is a maximum dark matter mass such that capture is possible with a single
scatter, given by

$m_{\chi,\max}\simeq
m_{A,\max}\left(1+2\dfrac{v_{e}}{v_{s}}+2\dfrac{v_{e}^{2}}{v_{s}^{2}}+\dfrac{v_{e}^{3}}{v_{s}^{3}}\right)\simeq
1.64\leavevmode\nobreak\ m_{A,\max}\,.$ (14)

The main difference between the M1 and M2 analyses is that the M1 analysis
incorporates the thermal motion of nuclei in the scattering process. In both
formalisms, the weak capture rate is proportional to the scattering cross
section. Although the M2 analysis does not account for the thermal motion of
Standard Model nuclei, it agrees with M1 remarkably well at all dark matter
masses once the effects of the solar gravitational potential are included. It
is important to note that the upper and lower limits on the dark matter mass
at which single-scatter capture is possible only appear because the Earth-DM
system is in the external gravitational potential of the Sun. In the absence
of this external gravitational potential, single-scatter capture would be
possible for any dark matter mass, for some choice of the dark matter speed
far from the Earth. The Sun’s gravitational potential has a major impact on
the behavior of the capture rate at large scattering cross section. Given the
approximation in Eq. (9), we see that if single-scatter capture of any non-
zero fraction of the incident dark matter flux is kinematically possible, then
at sufficiently large scattering cross section, the capture rate will saturate
to the geometric capture rate. But if the dark matter mass is such that
single-scatter capture is kinematically impossible, then the capture rate in
Eq. (9) will vanish, no matter how strong the coupling is.

###  III.2 Monte Carlo Simulations

Multiple scattering processes are especially important if the dark matter is
either significantly heavier or lighter than the nuclei it scatters against,
in which case the fraction of the kinetic energy lost in a typical scatter is
small, and multiple scatters may be needed before the dark matter falls below
the escape velocity. The analytical treatment we have described above only
treats multiple scattering in an approximate manner, by interpolating between
the limit in which dark matter scatters at most once and the limit in which
the capture rate asymptotes to the geometric rate. Essentially, this treatment
accounts for the attenuation of the dark matter flux as it passes through the
Earth (or Sun) due to single-scatter capture, avoiding double-counting. But it
does not take account of the change in the dark matter velocity distribution
due to scatters which did not lead to capture. In particular, if single-
scatter capture is not kinematically allowed, then this approach will yield a
vanishing capture rate, regardless of how many times dark matter may scatter
in the target.

If dark matter is of intermediate mass, such that single-scatter capture is
possible, then this treatment, although failing at intermediate scattering
cross sections, reproduces the saturation of the capture rate at large
scattering cross section. But if the dark matter mass is either small enough
or large enough that single-scatter capture is very rare or non-existent, then
this treatment will fail. For these scenarios, dark matter may drop below the
escape velocity only after multiple scatters. For heavy dark matter, this
implies that, for a sufficiently large scattering cross section, the capture
rate will saturate the geometric capture rate. On the other hand, for low-mass
dark matter, it is not possible to define a cross section large enough so that
all the low-mass dark matter that encounters the Earth will be captured. The
key distinction, is that for low-mass dark matter, even at arbitrarily high
cross section, a large fraction of the light dark matter is reflected in
multiple scattering processes, whereas the high-mass dark matter follows a
nearly linear trajectory as it crosses through the Earth, so all of the high-
mass dark matter is captured for a sufficiently large cross section.

If the dark matter is much heavier than nuclei, then a scatter results in a
loss of energy, but a small change in direction. In the limit of many
scatters, the dark matter is essentially guaranteed to be captured, leading to
the geometric capture rate at strong coupling. But if dark matter is much
lighter than the nuclei it scatters against, then one would expect the dark
matter particle’s direction to be essentially randomized after every scatter,
much like a ping-pong ball scattering off a bowling ball. In that case, even
if dark matter couples arbitrarily strongly to nuclei, it need not scatter
more than once, as it may be reflected away at the first scatter. This would
lead to a sharp suppression in the capture rate.

To study multiple scattering in dark matter capture, we use the DaMaSCUS-
EarthCapture code, developed from the DaMaSCUS code Emken:2017qmp ;
emken2017damascus . We improve the code in several ways: 1) We add the crust
layer and the atmosphere of the Earth and their chemical compositions,
allowing dark matter to scatter and stop in the crust and the atmosphere. 2)
We draw the dark matter velocity distribution from Eq. (3). Upon arriving at
the Earth111For simplicity, we ignore the motion of the Earth around the Sun.
In a more precise treatment, the dark matter velocity distribution would be
boosted to the frame of the Sun, retaining angular information. Drawing
particles from this distribution with a randomly drawn impact parameter
determines the angular momentum $\vec{L}$ and speed $u_{\chi}$ of a dark
matter particle with respect to the Sun, when far from the Sun. Each particle
is then assigned a speed $u^{\prime}_{\chi}=\sqrt{u_{\chi}^{2}+v_{s}^{2}}$
when it reaches a distance of 1 AU from the Sun. It is then boosted to speed
$u_{\chi}^{\prime\prime}$ in the frame of the Earth, given the velocity which
the Earth would have with respect to the Sun at the point where the dark
matter particle intersects the Earth’s orbit. The particle is finally assigned
a speed $w=\sqrt{u_{\chi}^{{}^{\prime\prime}2}+v_{e}^{2}(R_{\oplus,\rm atm})}$
at the top of the Earth’s atmosphere., dark matter is assigned a velocity
$w=\sqrt{u_{\chi}^{2}+v_{s}^{2}+v_{e}^{2}(R_{\oplus,\rm atm})}$. 3) If dark
matter leaves the Earth with a velocity $v\leq v_{e}(R_{\oplus,\rm atm})$, it
reenters the Earth at the exit point with the opposite velocity, as these dark
matter particles will follow an elliptical path and reenter the Earth at some
point, due to gravity. We neglect the thermal motion of Earth nuclei, and
simulate the trajectories of dark matter passing through the Earth, either
streaming freely or scattering with the nuclei. The probability of dark matter
scattering after traveling freely over a length $L$ is
$P=1-\exp(-\int_{0}^{L}dx/\lambda(x))$, with $\lambda$ the mean free path of
dark matter in Earth. The scattering angle is randomized between 0 and $\pi$
in the center-of-mass frame for velocity and momentum transfer independent
scattering, which is then translated to the Earth frame. A dark matter
particle is considered lost when it leaves the Earth with a velocity
$v>v_{e}(R_{\oplus,\rm atm})$, and captured when it reaches a velocity
$v<v_{e}(R_{\oplus,\rm atm})$ inside the Earth. We do not consider the
indirect capture of particles which first become gravitationally bound to the
Sun after scattering in the Sun, the Earth, or Jupiter, and are then
subsequently captured by the Earth. Our estimate is, in that sense,
conservative. We then determine the capture fraction, $f_{C}$, which is
defined as the fraction of dark matter particles reaching the surface of the
Earth and subsequently captured.

Figure 2: Dark matter capture in the Earth for spin-independent interactions
from Monte Carlo simulation using DaMaSCUS-EarthCapture. Left: The fraction of
dark matter particles that are captured among those that impinge on the Earth
as a function of dark matter mass, for the various cross sections indicated.
Right: The capture fraction of dark matter as a function of dark matter-
nucleon scattering cross section, for the various dark matter masses
indicated.

We show the capture fraction of dark matter with spin-independent interaction
in Figure 2. For relatively small cross sections $\sigma_{\chi N}^{\rm
SI}\lesssim 10^{-33}$ cm2, the capture fraction becomes suppressed when
$m_{\chi}\ll m_{A}$. In this limit, dark matter scatters rarely as it crosses
the Earth, and because of the small mass, each scatter results in only a small
loss in kinetic energy ($\Delta E_{k}/E_{k}\propto m_{\chi}/m_{A}\ll 1$),
making it unlikely that a dark matter particle is captured. This suppression
tends to get milder as the cross section increases, since dark matter may
scatter more often as it crosses the Earth. At $\sigma_{\chi N}^{\rm
SI}\gtrsim 10^{-30}$ cm2, the capture saturates down to keV dark matter mass,
i.e. increasing the cross section further contributes little to the capture
fraction. However, note that for $m_{\chi}\ll m_{A}$, the $f_{C}$ saturates to
a value which is $\ll 1$. A numerical fit suggests that the saturation capture
fraction $f_{C}\propto\sqrt{m_{\chi}}$. The reason is that for very low-mass
dark matter, each scatter roughly randomizes the direction of the dark matter
particle. There is a roughly $50\%$ chance that, after the first scatter, the
dark matter particle will reflect back away from the Earth and escape. If the
DM particle does not escape after this first scatter, it will then random-walk
through the Earth, with the number of steps before capture roughly given by
$\frac{m_{A}}{m_{\chi}}\ln\frac{w_{i}}{w_{f}}$, where the initial dark matter
velocity $w_{i}\sim 200$ km/s and the final velocity $w_{f}\sim 10$ km/s (see
Appendix B). The number of steps needed for capture increases as $m_{\chi}$
decreases. If, in this random walk, the dark matter ever leaves the Earth, it
will escape. As a result, the maximum fraction of dark matter which can be
captured may be very small, and gets smaller as $m_{\chi}$ decreases.

On the other hand, the capture fraction is close to 100% for $m_{\chi}\gtrsim
100\leavevmode\nobreak\ {\rm GeV}$ when $\sigma_{\chi N}^{\rm SI}\gtrsim
10^{-35}$ cm2. In this scenario, since the dark matter is much heavier than
the typical nuclei it scatters against, one finds that although dark matter
scatters isotropically in the center-of-mass frame, it is typically forward-
directed in the frame of the target nucleus. As a result, the dark matter
particle’s path through the Earth is not a random walk, but a nearly straight
line which ends inside the Earth when the dark matter drops below the escape
velocity. We also notice that the capture fraction of light dark matter
($m_{\chi}\ll m_{A}$) is peaked near $\sigma_{\chi N}^{\rm SI}\gtrsim
10^{-26}$ cm2. This is due to the summation of possible dark matter
trajectories, as we explain in Appendix B.

In Figure 1 we also compare the analytical capture rates obtained from Eq. (9)
with Monte Carlo (MC) simulations. The MC capture rates are computed with
$C^{\rm MC}_{\oplus}=f_{C}C_{\oplus}^{\rm geom}$. Due to the limitation of the
number of dark matter samples in the simulations, we do not have reliable data
below $f_{C}=10^{-5}$. We therefore extrapolate the capture fraction to low
dark matter masses and compute the corresponding capture fraction for
$\sigma_{0}\leq 10^{-32}$ cm2. The extrapolation does not affect our dark
matter bounds since we are primarily interested in $m_{\chi}\gtrsim$ GeV and
large cross sections, where the capture fraction is always larger than
$10^{-5}$.

We find that using Eq. (9) to extrapolate the capture rate beyond the weak
scattering limit tends to underestimate the capture rate, unless this
approximation reaches the saturation limit. As a sanity check, we also show in
Appendix C that, in the limit where cross section is small and multiple
scattering is less important, the MC capture rates match the analytical
results well.

For completeness, we also study the effect of solar gravitational potential on
the capture process, and display in Figure 3 the capture rates with and
without including solar acceleration. The latter is simply achieved by setting
$v_{s}=0$ in Eqs. (6) and (10). The agreement between M1 and M2 analytical
results remain robust in the high mass limit above 10 GeV. However, in the low
mass end, the capture rate in M2 tends to be flat, while the M1 rate scales as
$1/m_{\chi}$ when solar acceleration is ignored. In stark contrast to the
analytical results, the capture rates in MC simulations change marginally with
or without including $v_{s}$. As shown in Appendix B, the number of collisions
of dark matter before capture in the optically thick limit is only
logarithmically sensitive to the dark matter initial velocity. As the capture
probability of low mass dark matter is related to the number of collisions, it
is not surprising that increasing the lowest dark matter velocity has little
effect on the results. In the following sections, we always use the MC capture
rate with $v_{s}$ to compute the dark matter annihilation rate.

Interestingly, we also find the analytical results yield a result closer to
the Monte Carlo result at high masses if one fails to take into account the
solar gravitational potential than if one correctly accounts for it. If one
does not account for the solar gravitational potential, then single-scatter
capture would be possible, which is extrapolated to a capture rate that
saturates the geometric rate at sufficiently large cross section. However, the
absence of the solar gravitational potential implies that multi-scatter
capture is not correctly accounted for. On the other hand, at low masses, the
extrapolation in Eq. (9) does not match the Monte Carlo result regardless of
whether or not solar acceleration is included, because it does not account for
the reflection of dark matter from the Earth.

Figure 3: Total dark matter capture rates in the Earth in the optically thick
regime. The dashed, dash-dotted and solid lines show the capture rates from
Garani:2017jcj  (M1), Baum:2016oow  (M2) and from Monte Carlo simulations
using DaMaSCUS-EarthCapture, respectively. We assume the spin-independent
cross section $\sigma_{\chi N}^{\rm SI}=10^{-34}$ cm2. The red curves include
the effect of solar gravitational acceleration, while the grey curves not. See
text for details.

##  IV Dark Matter Evaporation

###  IV.1 Dark matter number density distribution

In the optically thick regime where the cross section is large (for SI (SD)
this corresponds to $\sigma_{\chi N}^{\rm SI}\gtrsim 10^{-36}$ cm2
($\sigma_{\chi N}^{\rm SD}\gtrsim 10^{-32}$ cm2)) where the Knudsen number
$K\gtrsim 1$, dark matter particles may thermalize with their local ambient
environment due to frequent scattering momentum exchange. In this case, we may
take the dark matter to be in local thermal eqilibrium (LTE); essentially, we
may assume that, in every small volume element, dark matter is an ideal gas in
thermal equilibrium with SM matter at temperature $T(r)$, and in diffusive
equilibrium with dark matter in the surrounding volume. Then, the dark matter
radial distribution follows

$\displaystyle
n_{\chi}(r)=n_{\chi}(0)\left(\dfrac{T_{\oplus}(r)}{T_{\oplus}(0)}\right)^{3/2}\exp\left(-\int_{0}^{r}\left[\alpha(r^{\prime})\frac{dT_{\oplus}(r^{\prime})}{dr^{\prime}}+m_{\chi}\frac{d\phi(r^{\prime})}{dr^{\prime}}\right]T_{\oplus}^{-1}dr^{\prime}\right)\,,$
(15)

where $n_{\chi}(0)$ and $T_{\oplus}(0)$ are the dark matter density and Earth
temperature, respectively, at $r=0$, and $\phi(r)$ is the external
gravitational potential
($\phi(r)=\int_{0}^{r^{\prime}}GM_{\oplus}(r^{\prime})/r^{\prime
2}dr^{\prime}$). The thermal diffusivity $\alpha(r)$ is related to the mean
free path of dark matter in the Earth (see Garani:2017jcj  for its explicit
form). For velocity-independent interaction, $\alpha$ is independent of cross
section and mildly dependent on the nature of the interaction. A self-
consistent study was recently carried out in Ref. Leane:2022hkk  including the
effects of diffusion and gravity, which can result in a floating distribution
of dark matter on the Earth’s surface. We have checked that this ‘bouyant’
dark matter distribution does not change the overall distribution for masses
above 1 GeV significantly and our results remain robust.

Figure 4: Left: Normalized dark matter number density distribution as a
function of the distance from the center of the Earth including 100 km
atmosphere, for a benchmark spin-independent cross-section of $10^{-30}\rm
cm^{2}$. Line styles correspond to dark matter masses of 1 keV, 1 MeV, 100
MeV, 1 GeV, 100 GeV, separately, from top to bottom. Right: The normalized
evaporation rate per captured particle $E_{\oplus}/N_{C}$ as a function of
dark matter mass, for the various spin-independent dark matter-nucleon cross
sections indicated.

The radial distribution of dark matter is depicted in the left panel of Figure
4. For relatively light dark matter $m_{\chi}\lesssim 0.1$ GeV, captured dark
matter dwells towards the surface of the Earth due to the temperature gradient
in the upper mantle and the crust. Due to the temperature profile in the
atmosphere, the density distribution fluctuates at a radius
${R_{\oplus}}<r\leq R_{\oplus,\rm atm}$. Heavy dark matter particles tend to
sink down. In particular, heavier dark matter $m_{\chi}\gtrsim 10$ GeV is more
clustered in the Earth core.

###  IV.2 Evaporation rate

Assuming local thermal equilibrium, the velocity of captured dark matter
follows a Maxwell-Boltzmann distribution (at zeroth order in the temperature
gradient), with a cutoff at the local escape velocity

$f_{\oplus}(\vec{u}_{\chi},r)=\dfrac{1}{(\sqrt{\pi}v_{\chi}(r))^{3}}e^{-u_{\chi}^{2}/v_{\chi}^{2}(r)}\left({\rm
Erf}(x)-\frac{2}{\sqrt{\pi}}xe^{-x^{2}}\right)^{-1}\Theta(v_{e}(r)-u_{\chi})\,,$
(16)

where again $v_{e}(r)$ is the escape velocity at radius $r$. The dark matter
thermal velocity is $v_{\chi}=\sqrt{2T_{\oplus}(r)/m_{\chi}}$ and we define
$x\equiv v_{e}/v_{\chi}$. Due to the thermal motion of Earth nuclei, dark
matter may scatter with a nucleus and acquires a high enough velocity to
escape from the Earth. The corresponding evaporation rate is described as
Garani:2017jcj

$E_{\oplus}=\sum\limits_{j}\int_{0}^{R_{\oplus}}4\pi
r^{2}n_{\chi}(r)s(r)dr\int_{0}^{v_{e}(r)}4\pi
u_{\chi}^{2}f_{\oplus}(u_{\chi},r)du_{\chi}\int_{v_{e}(r)}^{\infty}R_{j}^{+}(u_{\chi}\rightarrow
v)dv\,.$ (17)

Note that in the optically thick regime, dark matter particles with a velocity
above the escape velocity may not actually evaporate, as they may scatter with
the Earth matter several times before making their way out. This effect is
encapsulated in the $s(r)$ factor defined as Garani:2017jcj

$s(r)=\eta_{\rm ang}\eta_{\rm mult}e^{-\tau(r)}\,.$ (18)

$\eta_{\rm ang}$ and $\eta_{\rm mult}$ take into account the angular
trajectory and multiple scattering respectively, and $\tau(r)$ is the optical
depth. Note that the only dark matter properties which $E_{\oplus}$ depends on
are the dark matter mass, scattering cross section and the constant
$n_{\chi}(0)$. Since the total evaporation rate depends on the total number of
captured dark matter $N_{C}=\int_{0}^{r^{\prime}}4\pi
n_{\chi}(r^{\prime})r^{\prime 2}dr^{\prime}$ which is yet to be solved for, we
define the evaporation rate per particle ${E_{\oplus}}\equiv
E_{\oplus}/N_{C}$, which now depends only on the dark matter mass and
scattering cross section.

The dark matter evaporation rates as obtained from Eq. (17) for spin-
independent interaction are displayed in the right panel of Figure 4. As the
mass increases, the evaporation rate is exponentially suppressed by the
thermal velocity of dark matter. Dark matter with mass $m_{\chi}\gtrsim 10$
GeV can hardly escape from the Earth regardless of the cross section.
Evaporation is enhanced when $\sigma_{\chi N}^{\rm SI}$ goes above $10^{-36}$
cm2, thanks to the rise of the scattering rate between dark matter and nuclei.
However, further increasing the cross section to $10^{-34}$ cm2 or higher will
not facilitate more evaporation, as the $s(r)$ factor becomes important and
dark matter from the inner layer could hardly find a way out of the Earth. In
this scenario, the evaporation is more and more sourced from the dark matter
particles near the Earth crust or even the atmosphere. The combination of the
effects of more scattering and narrower evaporation region makes the
evaporation of light dark matter rather insensitive to the cross section when
$\sigma_{\chi N}\gtrsim 10^{-34}$ cm2. Similar to the capture analysis, Eq.
(17) might overestimate the evaporation rate for $m_{\chi}\ll m_{A}$ while the
direction of the dark matter particle is effectively randomized at every
scatter when dark matter makes its way out. A dedicated Monte Carlo study is
required to obtain the proper evaporation rate at the low mass regime, which
we leave for future work. Our analysis using Eq. (17) is conservative.

##  V Dark Matter Annihilation and Earth Heating

The dark matter annihilation rate, normalized by the total number of dark
matter particles, is given by

$A_{\oplus}=\langle\sigma
v\rangle_{\chi\chi}\int_{0}^{{R_{\oplus}}}n_{\chi}^{2}4\pi
r^{2}dr\left(\int_{0}^{{R_{\oplus}}}n_{\chi}4\pi r^{2}dr\right)^{-2}\,,$ (19)

where we assume the thermal averaged s-wave annihilation cross section
$\langle\sigma v\rangle_{\chi\chi}\simeq 3\times 10^{-26}$ cm3/s. Note that
the total annihilation rate, $(1/2)A_{\oplus}N_{C}^{2}$, only depends mildly
on the annihilation cross section. In the limit in which evaporation is
negligible, equilibrium is achieved when the total annihilation rate is equal
to the capture rate, regardless of what the annihilation cross section is. On
the other hand, if evaporation is dominant, then the annihilation rate will be
small. Thus, the total annihilation rate depends on the annihilation cross
section only in the regime in which the annihilation and evaporation rates are
comparable, which is a small part of parameter space.

When evaporation is not significant, the total annihilation rate will scale
with the scattering cross section, which in turn determines the capture rate.
But the capture rate largely determines the normalization of the dark matter
density distribution (that is, $N_{C}$), not the radial profile. Consequently,
the normalized annihilation rate, $A_{\oplus}$, depends only on the radial
profile of the dark matter distribution, not the normalization, and thus is
largely independent of the scattering cross section.

Bounds on Earth heating by dark matter are obtained by requiring that dark
matter annihilating inside the Earth cannot contribute more thermal energy
than that observed flowing out of the Earth’s surface. It is well established
that the heat presently flowing from the Earth is less than $\sim$44
terrawatts williams1974 ; lister1990 ; JAUPART2015223 ; Davies1980 ;
Davies1980_2 ; sclater1980 ; pollack1993 ; davies2010 , where a substantial
portion of this heat flow is attributable to the decay of potassium and
uranium arevalo2009 . Since the temperature profile of the Earth should be
mildly different at early geological times, we evaluate the number of dark
matter trapped in the Earth by solving Eq. (1) after $t=10^{9}$ years. This is
conservative in the sense that additional heating of the Earth over geological
timescales should only increase the predicted present-day heat flowing from
the Earth’s surface. The total annihilation rate is obtained using Eq. (2),
which, multiplied by $2m_{\chi}$, yields the heat flow powered by dark matter
annihilation. We always assume dark matter annihilates to visible final
states, and the annihilation deposits 100% of its mass-energy in the form of
heat.

In Figure 5 right (left) panel we plot the total (normalized) dark matter
annihilation rate as a function of $m_{\chi}$ for various choices of the spin-
independent dark matter-nucleon scattering cross section $\sigma_{\chi N}^{\rm
SI}$. The total annihilation rate decreases at larger dark matter mass, mainly
due to the drop in the geometric dark matter capture rate. The annihilation
rate also drops rapidly below around a few GeV, where evaporation becomes very
efficient. The mass when the annihilation rate crumbles depends on the cross
section. Larger scattering cross section tends to shift annihilation towards
lower masses, whereas increasing it further to $\sigma_{\chi N}^{\rm
SI}\gtrsim 10^{-31}$ cm2 has little impact on the overall annihilation rate.

Figure 5: Left: Normalized annihilation rate as a function of dark matter mass
as defined in Eq. (19) assuming local thermal equilibrium and assuming
$\langle\sigma v\rangle_{\chi\chi}=3\times 10^{-26}{\rm cm}^{3}/{\rm s}$. The
rate is insensitive to the scattering cross section and the nature of the
scattering. Right: The total annihilation rate of dark matter trapped in the
Earth as a function of dark matter mass for spin-independent scattering with
the cross sections indicated.

##  VI Experimental Limits and Earth Heating Constraints

We now are ready to set bounds on light dark matter captured and annihilating
inside the Earth. In Figure 6, we show bounds obtained by requiring that the
power produced by dark matter annihilation inside the Earth not exceed 44
terrawatts beardsmore_cull_2001 , assuming dark matter annihilates to visible
SM final states. The left panel shows the limits on the spin-independent dark
matter-nucleon scattering cross section from Earth heating assuming that
$\chi$ is all of the cosmological dark matter. To determine the maximum cross-
section for which the Earth heating bound is applicable (the upper edge of the
Earth heating bound shown in Figure 6), it is necessary to consider the
maximum cross-section for which dark matter annihilation occurs predominantly
_below_ the surface of the Earth. For a large enough dark matter-nucleon
scattering cross-section, dark matter can drift slowly enough through the
Earth’s atmosphere that it annihilates predominantly within the atmosphere
before reaching the Earth’s surface. The time for the dark matter to drift and
reach the Earth’s surface is Acevedo:2020gro ,

$t_{\rm drift}(R_{\oplus},R_{\oplus,\rm
atm})=\frac{1}{Gm_{\chi}}\Bigg{[}\sum_{j}\sigma_{j}\int_{R_{\oplus}}^{R_{\oplus,\rm
atm}}\frac{n_{j}(r)\sqrt{3m_{A_{j}}T(r)}}{M(r)}r^{2}\textrm{d}r\bigg{]}\,$
(20)

where a sum is taken over nuclei present in the Earth’s atmosphere, and
$\sigma_{j}$ is the dark matter nuclear scattering cross section which we
approximate by $\sigma_{j,0}$ in Eq. (7) or Eq. (8) as appropriate. We obtain
a conservative upper limit on the Earth heating bound cross-section by
requiring that no more than 10% of the dark matter captured annihilates on its
way to the Earth’s surface $\Gamma_{\oplus,\rm atm}<C_{\oplus}/10$. To
determine this condition, we sum over annihilation in atmospheric shells of
one kilometer thickness,

$\Gamma_{\oplus,\rm atm}=\sum_{r_{i}=R_{\oplus}}^{R_{\oplus,\rm
atm}}\frac{\langle\sigma
v\rangle_{\chi\chi}\Big{(}f_{\chi}(r_{i})C_{\oplus}t_{\rm
drift}(R_{\oplus},R_{\oplus,\rm atm})\Big{)}^{2}}{V_{\rm shell}},$ (21)

where the volume of each shell is simply $V_{\rm
shell}=\frac{4\pi}{3}(r_{i+1}^{3}-r_{i}^{3})$, and the fraction of captured,
drifting dark matter in each shell is

$f_{\chi}(r_{i})=\frac{t_{\rm drift}(r_{i},r_{i+1})}{t_{\rm
drift}(R_{\oplus},R_{\oplus,\rm atm})}.$ (22)

We also show constraints from CMB Dvorkin:2013cea ; Gluscevic:2017ywp , XQC
Erickcek:2007jv , RRS rich1987search , CRESST 2017 surface run CRESST:2017ues
, CDMS-I CDMS:2002moo , CRESST-III CRESST:2019jnq  and XENON1T XENON:2018voc .
For the CRESST surface run we use the upper limit from Refs. Kavanagh:2017cru
; Davis:2017noy  and the lower limit from Ref. CRESST:2017ues . For CDMS-I we
adopt the upper limit from Kavanagh:2017cru  and the lower limit from
CDMS:2002moo . As RRS placed a constraint on the dark matter-silicon
scattering cross section, we translate that to a dark matter-nucleon
scattering cross section using Eq. (8). To determine bounds in this parameter
space for for CRESST-III and XENON1T, we derive our limits by assuming that
dark matter particles all arrive from the average zenith angle of
$54^{\circ}$, and do not change their direction of motion significantly as
they pass through the Earth. The speed change of dark matter per unit distance
is given by Bhoonah:2020dzs ; Kavanagh:2017cru

$\dfrac{du_{\chi}}{dD}=-\dfrac{u_{\chi}}{m_{\chi}}\sum\limits_{j}\dfrac{\mu_{A_{j}}^{2}n_{j}}{m_{A_{j}}}\sigma_{j}\,,$
(23)

where $m_{A_{j}}$, $\mu_{A_{j}}$ and $n_{j}$ are the nucleus mass, dark
matter-nucleus reduced mass, and number density in the Earth, respectively, of
the $j$th isotope. The final one-dimensional dark matter velocity at the
detector is connected to the halo dark matter velocity distribution222For this
purpose, we can ignore the effects of gravitational infall. Particles which
are slow enough that gravitational effects are important will in any case not
deposit enough energy in the detector to exceed threshold. via

$f(u_{f})=f(u_{\chi})\dfrac{du_{\chi}}{du_{f}}=f(u_{\chi})\exp\left(\int
dD\sum\limits_{j}\dfrac{\mu_{A_{j}}^{2}n_{j}}{m_{\chi}m_{A_{j}}}\sigma_{j}\right)\,.$
(24)

For simplicity we do not include form factors in the overburden calculation
using Eq. (24), i.e. $\sigma_{j}\simeq\sigma_{j,0}$. The expected number of
events in an experiment is333Note that we do not include the momentum
transfer-dependent form factors in overburden calculations for simplicity, but
we always include form factors in the terrestrial experiments to produce as
accurate experimental limits as possible.

$N_{\rm exp}=\sum\limits_{i}N_{i}T\dfrac{\rho_{\chi}}{m_{\chi}}\int
u_{f}f(u_{f})du_{f}\int\dfrac{d\sigma_{i}}{dE_{R}}\epsilon(E_{R})dE_{R}\,,$
(25)

where

$\dfrac{d\sigma_{i}}{dE_{R}}=\dfrac{\sigma_{i,0}m_{A_{i}}}{2\mu_{A_{i}}^{2}u_{f}^{2}}F_{i}^{2}(E_{R})\,,$
(26)

$N_{i}$ is the number of target nuclei, and $T$ is the exposure time. We also
include the efficiency factor $\epsilon(E_{R})$ from the respective
experiments. For spin-independent interactions, $F_{i}$ is the Helm form
factor. We use Eq. (25) to find the cross section limits for CRESST-III and
XENON1T. 441 dark matter candidate events were identified with the exposure of
3.64 kg$\cdot$days using CaWO4 crystal CRESST:2019jnq , while for XENON1T we
adopt 3.7 events at $90\%$ upper limit with 0.9 tonne reference mass
XENON:2018voc .

In the right panel of Figure 6, we show limits when $\chi$ constitutes $5\%$
dark matter. The Earth heating limit is obtained by setting the capture rate
$C_{\oplus}=0.05f_{C}C_{\oplus}^{\rm geom}$ while using the conditions
$2m_{\chi}\Gamma_{\oplus}>44$ TW and $\Gamma_{\oplus,\rm atm}<C_{\oplus}/10$.
To produce these limits, we scale up the lower limit of XQC and RRS by a
factor of 20. We also scale up the CMB limit by the same factor, as the
collision terms enters the Boltzmann equations in the form of
$\rho_{\chi}\sigma_{\chi p}$ Gluscevic:2017ywp . We assume the overburden line
remains robust with reduced dark matter flux for XQC and RRS. We use the verne
verne ; Kavanagh:2017cru  code to compute the dark matter limits from CRESST
surface run and CDMS-I. We again use Eq. (25) to find the constraints from
CRESST-III and XENON1T, when $N_{\exp}$ matches the observation limit with a
reduced dark matter density.

Figure 6: Earth heat limit from dark matter annihilation for spin-independent
nuclear scattering interactions. Also shown are results from CMB
Dvorkin:2013cea ; Gluscevic:2017ywp , XQC Erickcek:2007jv , RRS rich1987search
, CRESST 2017 surface run CRESST:2017ues , CDMS-I CDMS:2002moo , CRESST-III
CRESST:2019jnq  and XENON1T XENON:2018voc . See text for details. The region
enclosed by the dashed gray lines is the Earth heat limit from Mack et. al.
Mack:2007xj . Left: $\chi$ constitutes 100% dark matter. Right: $\chi$
constitutes 5% dark matter.

We also derived Earth heating limits on spin-dependent nuclear scattering.
Figure 7, Figure 8, and Figure 9 show the constraints on isospin-independent,
proton-only and neutron-only scattering, respectively, assuming that $\chi$ is
either 100% (left panels) or 5% (right panels) of cosmological dark matter.
The Earth heating limits are obtained with the corresponding capture rates for
spin-dependent scattering shown in Appendix C, using the dark matter nuclear
scattering cross section in Eq. (8). We also follow Eqs. 23 to 26 to derive
the constraints from XENON1T and CDMSlite SuperCDMS:2017nns . The spin-
dependent form factor $F^{2}_{\rm
SD}=(a_{p}S_{p}(E_{R})+a_{n}S_{n}(E_{R}))/S_{T}(0)$, where

$S_{T}(0)=\dfrac{(2J+1)(J+1)}{\pi J}\left(a_{p}\langle
S_{p}\rangle+a_{n}\langle S_{n}\rangle\right)^{2}\,.$ (27)

We employ the updated nuclear structure factors and average spins in
Klos:2013rwa , which are also listed in Appendix A. We use the first two
energy bins in CDMSlite Run 2, which well reproduces the results in
SuperCDMS:2017nns . We again use verne to compute the limits from CRESST
surface run and CDMS-I, neglecting the form factors in the overburden, but not
in the scattering rates in the detector, except for ${}^{17}\mbox{O}$ whose
momentum dependent structure factor is still missing. The XQC and RRS limits
are translated using Eqs. 7 and 8. For the overburden of these two experiments
we assume ${}^{14}\mbox{N}$ is the main source of dark matter deceleration in
both spin-dependent and spin-independent scattering, and translate the
constraints accordingly. For the case in which $\chi$ is 5% of cosmological
dark matter we follow the same treatment as in the spin-independent scattering
case.

For spin-independent interaction, the Earth heating places constraints on the
dark matter scattering cross section for $m_{\chi}>0.84$ GeV (6.2 GeV)
assuming 100% (5%) dark matter in the parameter space where our analysis is
valid. We constrain the cross section $\sigma_{\chi N}^{\rm SI}>10^{-36}$ cm2,
where local thermal equilibrium is justified, and $\sigma_{\chi N}^{\rm
SI}\lesssim 10^{-20}$ cm2, where dark matter dominantly drifts down below the
surface of the Earth and annihilates there. This effectively cuts the
exclusion region of Mack et al. Mack:2007xj  at lower masses and extends the
upper and lower cross section reach. For spin-dependent interactions of all
types, the lower mass limits shift to 0.93 GeV ($\sim 5.5$ GeV) assuming 100%
(5%) dark matter. The excluded cross section is above $10^{-32}$ cm2 to
satisfy the local thermal equilibrium distribution, and below about $10^{-17}$
cm2 for dark matter to drift down. The Earth heating limits exclude some
parameter space that is not covered by direct detection experiments. For spin-
dependent scattering Earth heating closes the gap between CMB and CREEST in
the 0.9 GeV and 2.4 GeV mass window, and the gap between CMB, XQC and RRS
above about 10 GeV, particularly for proton-only interaction.

Figure 7: Earth heat limit from dark matter annihilation for spin-dependent
interaction, with isospin-independent nuclear response $a_{p}=a_{n}=1$ (black
shaded region). Overlaid are results from CMB Dvorkin:2013cea ;
Gluscevic:2017ywp , XQC Erickcek:2007jv , RRS rich1987search , CRESST 2017
surface run CRESST:2017ues , CDMS-I CDMS:2002moo , CDMSlite SuperCDMS:2017nns
and XENON1T XENON:2018voc . See text for details. The region enclosed by the
dashed gray lines is the Earth heat limit from Mack et. al. Bramante:2019fhi .
Left: $\chi$ constitutes 100% dark matter. Right: $\chi$ constitutes 5% dark
matter.

Figure 8: Same as Figure 7 but for spin-dependent proton-only interaction.

Figure 9: Same as Figure 7 but for spin-dependent neutron-only interaction.

##  VII Discussion and Conclusions

We have considered in detail the distribution of dark matter gravitationally
captured by the Earth, including the processes of evaporation. We have made
several improvements over previous results in the literature, chiefly
involving a detailed treatment through Monte Carlo simulation of the limit in
which multiple scattering is important, and a more thorough treatment of the
effect of the density and temperature profile of the Earth on evaporation of
captured dark matter.

Among the main differences we have found with previous results is the effect
of multiple scattering in the intermediate regime, where dark matter scatters
more than once in the Earth, but not so often that all dark matter incident on
the Earth is captured. We have found that, compared to previous
approximations, high mass dark matter tends to saturate the geometric capture
rate more quickly. But for lower mass dark matter, we find that even if the
scattering cross section is large enough that the capture rate saturates, this
saturation capture rate is much smaller than the rate at which dark matter is
incident upon Earth. Essentially, this is because a light dark matter particle
can effectively bounce off the Earth during its first few scatters and escape.

We have focused on the distribution of dark matter in the Earth for the case
in which dark matter can annihilate to visible matter, leading to anomalous
heating of the Earth that may be bounded by data. But dark matter annihilation
in the Earth can also produce a flux of neutrinos that may be observed at
neutrino detectors. Thus our analysis framework is equally useful for
constraining this neutrino flux, which would be an interesting topic of future
work. Our analysis can also be easily generalized to other astrophysical
bodies, including the Sun.

## Acknowledgements

We thank Rebecca Leane for useful discussions. NS would like to thank the UK
Science and Technology Facilities Council (STFC) for funding this work through
support for the Quantum Sensors for the Hidden Sector (QSHS) collaboration
under grants ST/T006102/1, ST/T006242/1, ST/T006145/1, ST/T006277/1,
ST/T006625/1, ST/T006811/1, ST/T006102/1 and ST/T006099/1. Computations were
performed at the Queen’s Centre for Advanced Computing, supported in part by
the Canada Foundation for Innovation, and at Barkla HPC cluster housed at the
University of Liverpool. JK is supported in part by DOE grant DE-SC0010504. JB
is supported by the Natural Sciences and Engineering Research Council of
Canada (NSERC). GM acknowledges support from the Arthur B. McDonald Canadian
Astroparticle Physics Research Institute and from the UC office of the
President via the UCI Chancellor’s Advanced Postdoctoral Fellowship. NR is
supported in part by the Natural Sciences and Engineering Research Council
(NSERC) of Canada. Research at Perimeter Institute is supported in part by the
Government of Canada through the Department of Innovation, Science and
Economic Development Canada and by the Province of Ontario through the
Ministry of Colleges and Universities. TRIUMF receives federal funding via a
contribution agreement with the National Research Council (NRC) of Canada.

##  Appendix A List of spin information for different isotopes

Isotope | Abundance[%] | $J$ | $\langle S_{p}\rangle$ | $\langle S_{n}\rangle$
---|---|---|---|---
${}^{14}\mbox{N}$ | 99.6 | 1 | 0.5 | 0.5
${}^{15}\mbox{N}$ | 0.4 | 1/2 | -0.145 | 0.037
${}^{17}\mbox{O}$ | 0.04 | 5/2 | -0.036 | 0.508
${}^{29}\mbox{Si}$ | 4.7 | 1/2 | 0.016 | 0.156
${}^{27}\mbox{Al}$ | 100 | 5/2 | 0.326 | 0.038
${}^{57}\mbox{Fe}$ | 2.12 | 1/2 | 0 | 0.5
${}^{43}\mbox{Ca}$ | 0.135 | 7/2 | 0 | 0.5
${}^{23}\mbox{Na}$ | 100 | 3/2 | 0.224 | 0.024
${}^{39}\mbox{K}$ | 100 | 3/2 | -0.196 | 0.055
${}^{25}\mbox{Mg}$ | 10 | 5/2 | 0.040 | 0.376
${}^{47}\mbox{Ti}$ | 7.44 | 5/2 | 0 | 0.21
${}^{49}\mbox{Ti}$ | 5.44 | 7/2 | 0 | 0.29
${}^{61}\mbox{Ni}$ | 1.14 | 3/2 | 0 | -0.357
${}^{59}\mbox{Co}$ | 100 | 7/2 | 0.5 | 0
${}^{31}\mbox{P}$ | 100 | 1/2 | 0.181 | 0.032
${}^{33}\mbox{S}$ | 75 | 3/2 | 0 | -0.3
${}^{73}\mbox{Ge}$ | 7.76 | 9/2 | 0.031 | 0.439
${}^{129}\mbox{Xe}$ | 26.4 | 1/2 | 0.010 | 0.329
${}^{131}\mbox{Xe}$ | 21.2 | 3/2 | -0.009 | -0.272
Table 1: The fractional isotope abundance, nuclear spin $J$, average proton
spin $\langle S_{p}\rangle$, and average neutron spin $\langle S_{n}\rangle$
for isotopes with non-zero spin in the Earth. The average spins for
${}^{29}\mbox{Si}$, ${}^{27}\mbox{Al}$ ${}^{23}\mbox{Na}$, ${}^{73}\mbox{Ge}$,
${}^{129}\mbox{Xe}$ and ${}^{131}\mbox{Xe}$ are taken from Ref. Klos:2013rwa ,
and the rest from Ref. Bednyakov:2004xq  (EOGM $g_{A}/g_{V}=1$). The spin
information of Ge and Xe isotopes is used to produce the spin-dependent
results for CDMSlite and XENON1T.

##  Appendix B Resonant light dark matter capture

As shown in Figure 2, for spin-independent scattering the capture fraction is
peaked near $10^{-26}$ cm2 for dark matter mass $m_{\chi}\lesssim 1$ GeV. To
understand this behavior, we show the trajectories of dark matter captured in
the Earth or Earth’s atmosphere in Figure 10 and Figure 11. Assuming dark
matter with a velocity $w$ scatters with a nucleus at rest, the momentum
transfer $q\sim\mu_{\chi A}w\sim m_{\chi}w$ for light dark matter $m_{\chi}\ll
m_{A}$. The dark matter kinetic energy loss in each scattering is just the
recoil energy of the nucleus $E_{R}\sim q^{2}/(2m_{A})$. Therefore, the speed
change of dark matter in a typical scattering is

$\dfrac{\Delta w^{2}}{w^{2}}\sim-\dfrac{m_{\chi}}{m_{A}}\,,$ (28)

i.e. the speed change is smaller for lighter dark matter, where more
scattering is necessary to bring the dark matter velocity down below the
escape velocity. The number of scatters required in the capture process is
therefore

$N_{\rm scatt}\simeq 2\dfrac{m_{A}}{m_{\chi}}\ln\dfrac{w_{i}}{w_{f}}\,.$ (29)

If dark matter scatters with nitrogen in the atmosphere, with $w_{i}\sim 200$
km/s and $w_{f}\sim 10$ km/s, then we find $N_{\rm scatt}\sim 8\times 10^{4}$,
consistent with the number of steps in Figure 10.

The dark matter velocity after one scattering is

$\vec{w}^{\prime}=\dfrac{m_{A}w\hat{n}+m_{\chi}\vec{w}}{m_{\chi}+m_{A}}\,,$
(30)

where $\hat{n}$ is the direction vector of dark matter in the center of mass
frame. Defining $\cos\alpha=\hat{n}\cdot\hat{w}$, $\cos\alpha$ is uniformly
distributed between -1 and 1 in the absence of the form factor. In the lab
frame, we have the scattering angle

$\cos\alpha^{\prime}=\dfrac{m_{A}\cos\alpha+m_{\chi}}{\sqrt{m_{A}^{2}+m_{\chi}^{2}+2m_{A}m_{\chi}\cos\alpha}}\,.$
(31)

In the light dark matter limit $m_{\chi}\ll m_{A}$,
$\cos\alpha^{\prime}\simeq\cos\alpha$, and dark matter is deflected randomly
between 0 and $\pi$. As we can see from Figure 10 and Figure 11, light dark
matter scatters multiple times before captured. Amid these scatterings, for
large enough cross section $\sigma_{\chi N}\sim 10^{-24}$ cm2, light dark
matter is likely to be reflected in the Earth’s atmosphere and escape, while
for small enough cross section $\sigma_{\chi N}\lesssim 10^{-28}$ cm2, dark
matter is reflected in the Earth’s crust or mantle before they leave. For
cross sections in between, dark matter reflected in the Earth’s crust might be
deflected back in the atmosphere, causing dark matter to be finally captured
after bouncing back and force. The capture probability is therefore the sum of
all possible trajectories (or paths) along which dark matter is captured. The
path integral is apparently larger for intermediate cross section, where dark
matter’s trajectories cross different media, than for very high or very low
cross section, where the trajectories mostly cross one medium. The regime
where path integral maximizes corresponds to the peak around $10^{-26}$ cm2.

It is also worth noting that more intense scattering is demanded for smaller
dark matter mass, and the peak in Figure 2 moves to higher cross section. The
peak is also more pronounced for lighter dark matter, as more scatterings
before capture facilitate more viable trajectories, which enhances the overall
path integral compared with other cross sections. The peak is also missing for
spin-dependent interactions, where the cross section up to $10^{-24}$ cm2 is
not high enough to raise the peak.

Figure 10: Example trajectories of MeV mass dark matter captured inside the
Earth or the Earth’s atmosphere from DaMaSCUS-EarthCapture simulations, for
various per-nucleon spin-independent scattering cross sections. Dark matter
starts from the top of the atmosphere with $r=6471$ km. At each step, dark
matter scatters with a Standard Model nucleus at a specific radius and finally
stops at a radius $r<6471$ km.

Figure 11: Example trajectories of MeV mass dark matter NOT captured inside
the Earth or the Earth’s atmosphere from DaMaSCUS-EarthCapture simulations,
for various per-nucleon spin-independent scattering cross sections. Dark
matter starts from the top of the atmosphere with $r=6471$ km. At each step,
dark matter scatters with Standard Model nucleus at a specific radius and
finally exits at a radius $r=6471$ km.

##  Appendix C Capture fractions for different types of interaction

To give a complete picture of dark matter capture, we show the dark matter
capture fraction for isospin-independent, proton-only and neutron-only spin
dependent scatterings in Figure 12 to Figure 14. It is clearly seen from the
left panel of Figure 12 that in the optically thin limit where single scatter
dominates the capture ($\sigma_{\chi N}^{\rm SD}=10^{-36}$ cm2), the capture
fraction peaks between 10 GeV and 100 GeV, where the dark matter mass
kinematically matches the masses of ${}^{29}\mbox{Si}$, ${}^{27}\mbox{Al}$,
${}^{25}\mbox{Mg}$, as well as ${}^{57}\mbox{Fe}$, and the dark matter kinetic
energy loss is also maximized. A comparison between MC simulation and
analytical results is also shown in Figure 15. The simulation matches the
analytical results well, apart from small difference attributable to limited
statistics. However, at higher cross section, multiple scattering becomes
important and the peaks are smeared out.

Figure 12: Dark matter capture in the Earth for spin-independent interactions
from Monte Carlo simulation using DaMaSCUS-EarthCapture, assuming isospin-
independent nuclear response $a_{p}=a_{n}=1$. Left: The fraction of dark
matter particles that are captured among those impinge on the Earth as a
function of dark matter mass. The orange, magenta, purple, green, light blue
and yellow lines correspond to the cross section of $10^{-34}$, $10^{-32}$,
$10^{-30}$, $10^{-28}$, $10^{-26}$, $10^{-24}$ cm2, respectively. Right: The
capture fraction of dark matter as a function of dark matter-nucleon
scattering cross section. From bottom to top, different line types delineate
dark matter masses of $m_{\chi}=10$ GeV, 1 GeV, 100 MeV, 1 MeV, and 1 keV
respectively.

Figure 13: Same as Figure 12 but for spin-dependent proton-only interaction.

Figure 14: Same as Figure 12 but for spin-dependent neutron-only interaction.
Figure 15: Dark matter capture rate in the Earth for spin-dependent scattering
cross section $\sigma_{\chi N}^{\rm SD}=10^{-36}$ cm2, $a_{p}=a_{n}=1$. The
dashed, dash-dotted and solid lines show the capture rates from M1
Garani:2017jcj , M2 Baum:2016oow  and from Monte Carlo simulations using
DaMaSCUS-EarthCapture, respectively.

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