††thanks: present address: Mechatronics Research, Samsung Electronics,
Hwaseong, 18448, South Korea††thanks: present address: Université de Franche-
Comté, SupMicroTech-ENSMM, UMR 6174 CNRS, institut FEMTO-ST, 25000 Besançon,
France††thanks: present address: Department of Physics, University of
Colorado, Boulder Colorado 80309, USA ††thanks: present address: Department of
Physics, Harvard University, Cambridge, Massachusetts, 02138, USA††thanks:
present address: Department of Physics and Astronomy, University of Rochester,
Rochester, New York 14627, USA††thanks: present address: Department of
Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801,
USA

# What can a GNOME do?

Search targets for the Global Network of
Optical Magnetometers for Exotic physics searches

S. Afach  Helmholtz-Institut, GSI Helmholtzzentrum fur Schwerionenforschung,
Mainz 55128, Germany  Johannes Gutenberg University, Mainz 55128, Germany  D.
Aybas Tumturk  Department of Physics, University of California at Berkeley,
Berkeley, California 94720-7300, USA  H. Bekker  Helmholtz-Institut, GSI
Helmholtzzentrum fur Schwerionenforschung, Mainz 55128, Germany  Johannes
Gutenberg University, Mainz 55128, Germany  B. C. Buchler  Research School of
Physics, Australian National University, Canberra ACT 2601, Australia  D.
Budker  Johannes Gutenberg University, Mainz 55128, Germany  Helmholtz-
Institut, GSI Helmholtzzentrum fur Schwerionenforschung, Mainz 55128, Germany
Department of Physics, University of California at Berkeley, Berkeley,
California 94720-7300, USA  K. Cervantes  Department of Physics, California
State University – East Bay, Hayward, California 94542-3084, USA  A.
Derevianko  Department of Physics, University of Nevada, Reno, Nevada 89557,
USA  J. Eby  Kavli Institute for the Physics and Mathematics of the Universe
(WPI), The University of Tokyo Institutes for Advanced Study, The University
of Tokyo, Kashiwa, Chiba 277-8583, Japan  N. L. Figueroa  Johannes Gutenberg
University, Mainz 55128, Germany  Helmholtz-Institut, GSI Helmholtzzentrum fur
Schwerionenforschung, Mainz 55128, Germany  R. Folman  Department of Physics,
Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel  D. Gavilán
Martín  Johannes Gutenberg University, Mainz 55128, Germany  Helmholtz-
Institut, GSI Helmholtzzentrum fur Schwerionenforschung, Mainz 55128, Germany
M. Givon  Department of Physics, Ben-Gurion University of the Negev, Be’er
Sheva 84105, Israel  Z. D. Grujić  Physics Department, University of Fribourg,
CH-1700 Fribourg, Switzerland  Institute of Physics Belgrade, Pregrecica 118,
Belgrade 11080, Serbia  H. Guo  State Key Laboratory of Advanced Optical
Communication Systems and Networks, Department of Electronics, and Center for
Quantum Information Technology, Peking University, Beijing 100871, China  P.
Hamilton  Department of Physics & Astronomy, University of California at Los
Angeles, Los Angeles, California, 90095-1547, USA  M. P. Hedges  Research
School of Physics, Australian National University, Canberra ACT 2601,
Australia  D. F. Jackson Kimball<EMAIL_ADDRESS>Department of Physics, California State University – East Bay, Hayward,
California 94542-3084, USA  S. Khamis  Department of Physics & Astronomy,
University of California at Los Angeles, Los Angeles, California, 90095-1547,
USA  D. Kim  Department of Physics, KAIST, Daejeon 34141, South Korea  Center
for Axion and Precision Physics Research, IBS, Daejeon, 34051, South Korea  E.
Klinger  Helmholtz-Institut, GSI Helmholtzzentrum fur Schwerionenforschung,
Mainz 55128, Germany  Johannes Gutenberg University, Mainz 55128, Germany  A.
Kryemadhi  Dept. Computing,Math & Physics, Messiah University, Mechanicsburg
PA 17055, USA  X. Liu  State Key Laboratory of Advanced Optical Communication
Systems and Networks, Department of Electronics, and Center for Quantum
Information Technology, Peking University, Beijing 100871, China  G.
Łukasiewicz  Marian Smoluchowski Institute of Physics, Jagiellonian
University, Łojasiewicza 11, 30-348, Kraków, Poland  H. Masia-Roig  Johannes
Gutenberg University, Mainz 55128, Germany  Helmholtz-Institut, GSI
Helmholtzzentrum fur Schwerionenforschung, Mainz 55128, Germany  M. Padniuk
Marian Smoluchowski Institute of Physics, Jagiellonian University,
Łojasiewicza 11, 30-348, Kraków, Poland  C. A. Palm  Department of Physics,
California State University – East Bay, Hayward, California 94542-3084, USA
S. Y. Park  Department of Physics & Astronomy, Oberlin College, Oberlin, Ohio
44074, USA  H. R. Pearson  Department of Physics & Astronomy, Oberlin College,
Oberlin, Ohio 44074, USA  X. Peng  State Key Laboratory of Advanced Optical
Communication Systems and Networks, Department of Electronics, and Center for
Quantum Information Technology, Peking University, Beijing 100871, China  M.
Pospelov  School of Physics and Astronomy, University of Minnesota,
Minneapolis, MN 55455, USA  William I. Fine Theoretical Physics Institute,
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN
55455, USA  S. Pustelny  Marian Smoluchowski Institute of Physics,
Jagiellonian University, Łojasiewicza 11, 30-348, Kraków, Poland  Y.
Rosenzweig  Department of Physics, Ben-Gurion University of the Negev, Be’er
Sheva 84105, Israel  O. M. Ruimi  Racah Institute of Physics, Hebrew
University of Jerusalem, Jerusalem 9190401, Israel  T. Scholtes  Physics
Department, University of Fribourg, CH-1700 Fribourg, Switzerland  Leibniz
Institute of Photonic Technology, Albert-Einstein-Strasse 9, D-07745 Jena,
Germany  P. C. Segura  Department of Physics & Astronomy, Oberlin College,
Oberlin, Ohio 44074, USA  Y. K. Semertzidis  Center for Axion and Precision
Physics Research, IBS, Daejeon, 34051, South Korea  Department of Physics,
KAIST, Daejeon 34141, South Korea  Y. C. Shin  Center for Axion and Precision
Physics Research, IBS, Daejeon, 34051, South Korea  J. A. Smiga  Johannes
Gutenberg University, Mainz 55128, Germany  Helmholtz-Institut, GSI
Helmholtzzentrum fur Schwerionenforschung, Mainz 55128, Germany  Y. V. Stadnik
School of Physics, University of Sydney, Sydney, NSW 2006, Australia  J. E.
Stalnaker  Department of Physics & Astronomy, Oberlin College, Oberlin, Ohio
44074, USA  I. A. Sulai  Department of Physics & Astronomy, Bucknell
University, Lewisburg, Pennsylvania 17837, USA  D. Tandon  Department of
Physics & Astronomy, Oberlin College, Oberlin, Ohio 44074, USA  K. Vu
Department of Physics, California State University – East Bay, Hayward,
California 94542-3084, USA  A. Weis  Physics Department, University of
Fribourg, CH-1700 Fribourg, Switzerland  A. Wickenbrock  Johannes Gutenberg
University, Mainz 55128, Germany  Helmholtz-Institut, GSI Helmholtzzentrum fur
Schwerionenforschung, Mainz 55128, Germany  T. Z. Wilson  Department of
Physics, California State University – East Bay, Hayward, California
94542-3084, USA  T. Wu  State Key Laboratory of Advanced Optical Communication
Systems and Networks, Department of Electronics, and Center for Quantum
Information Technology, Peking University, Beijing 100871, China  W. Xiao
State Key Laboratory of Advanced Optical Communication Systems and Networks,
Department of Electronics, and Center for Quantum Information Technology,
Peking University, Beijing 100871, China  Y. Yang  State Key Laboratory of
Advanced Optical Communication Systems and Networks, Department of
Electronics, and Center for Quantum Information Technology, Peking University,
Beijing 100871, China  Beijing Institute of Applied Meteorology, Beijing
100029, China  D. Yu  State Key Laboratory of Advanced Optical Communication
Systems and Networks, Department of Electronics, and Center for Quantum
Information Technology, Peking University, Beijing 100871, China  F. Yu
PRISMA+ Cluster of Excellence & Mainz Institute for Theoretical Physics,
Johannes Gutenberg University, Mainz 55128, Germany  J. Zhang  State Key
Laboratory of Advanced Optical Communication Systems and Networks, Department
of Electronics, and Center for Quantum Information Technology, Peking
University, Beijing 100871, China  Y. Zhao  State Key Laboratory of Advanced
Optical Communication Systems and Networks, Department of Electronics, and
Center for Quantum Information Technology, Peking University, Beijing 100871,
China

###### Abstract

Numerous observations suggest that there exist undiscovered beyond-the-
Standard-Model particles and fields. Because of their unknown nature, these
exotic particles and fields could interact with Standard Model particles in
many different ways and assume a variety of possible configurations. Here we
present an overview of the Global Network of Optical Magnetometers for Exotic
physics searches (GNOME), our ongoing experimental program designed to test a
wide range of exotic physics scenarios. The GNOME experiment utilizes a
worldwide network of shielded atomic magnetometers (and, more recently,
comagnetometers) to search for spatially and temporally correlated signals due
to torques on atomic spins from exotic fields of astrophysical origin. We
survey the temporal characteristics of a variety of possible signals currently
under investigation such as those from topological defect dark matter (axion-
like particle domain walls), axion-like particle stars, solitons of complex-
valued scalar fields (Q-balls), stochastic fluctuations of bosonic dark matter
fields, a solar axion-like particle halo, and bursts of ultralight bosonic
fields produced by cataclysmic astrophysical events such as binary black hole
mergers.

##  I Introduction

There are widespread hints from nature suggesting there exist exotic,
heretofore undiscovered particles. Perhaps the most prominent hint is the
accumulated evidence for dark matter. A leading hypotheses to explain dark
matter is that it consists of ultralight bosons such as axions or axion-like
particles (ALPs) with masses $m_{a}\ll 1\,{\rm{eV}}$ [1, 2]. Such ultralight
bosonic dark matter (UBDM) can couple to Standard Model particles through a
variety of “portals” [3, 4], one of which is the direct interaction of the
UBDM field with atomic spins [5, 6]. If such an interaction exists, a UBDM
field would generate a spin-dependent energy shift similar to that caused by
the Zeeman effect due to an external magnetic field. This opens the
possibility of using atomic-spin-based magnetometers [7, 8] to search for
UBDM.

Several experiments use atomic magnetometers and nuclear magnetic resonance
(NMR) techniques to search for the interaction of UBDM fields with spins [9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The results of these experiments
are interpreted using models that assume that the UBDM is a virialized
ensemble of non-interacting bosons described by the standard halo model (SHM)
[21, 22, 23]. These isotropic SHM UBDM models typically ignore any small-scale
structure in the dark matter halo, beyond the stochastic fluctuations of the
UBDM due to its finite coherence time [24, 25, 26, 27]. Thus the sensors in
these experiments are assumed to be quasi-continuously bathed in the UBDM
field.

The Global Network of Optical Magnetometers for Exotic physics searches
(GNOME) [28, 29, 30] tests a different hypothesis. Perhaps the energy density
of the UBDM field is concentrated in large composite structures. In this case,
most of the time the Earth would be in a region of space where there is little
or no dark matter [31]. In this case, the Earth would only occasionally and
briefly pass through dark matter, leading to rare and short-lived signals in
dark matter detectors. In principle, a single sensor could detect such
transient events. However, it would be exceedingly difficult to confidently
distinguish a signal generated by an encounter with a composite UBDM structure
from “false positives”. Such false positives can be induced by occasional
abrupt changes of sensor operational conditions (such as those due to
electronic noise spikes, laser mode hops, or vibrations). The GNOME is a time-
synchronized array of atomic magnetometers, widely distributed geographically
(Fig. 1). The design of GNOME enables vetoing of false positive events,
suppression of uncorrelated noise, and confident identification of transient
signals attributable to exotic, beyond-the-Standard-Model physics.

Figure 1: Map and list of locations of GNOME stations. Note that the Fribourg
station was moved to Jena in 2018.

In this paper, we review a variety of theoretical models describing sources of
transient signals potentially detectable with GNOME. We focus in particular on
phenomenological descriptions of the temporal characteristics of the signals
that would manifest in the GNOME sensors, which informs our data-analysis
strategies.

##  II Interactions between atomic spins and ultralight bosonic fields

The optical atomic magnetometers (OAMs) [7, 8] comprising GNOME are sensitive
to interactions of atomic spins $\boldsymbol{F}$ with hypothetical exotic
fields $\boldsymbol{\Upsilon}$, as well as magnetic fields $\boldsymbol{B}$,
where $\boldsymbol{F}$ is the total atomic angular momentum. For the exotic
spin-dependent interactions considered here, the Hamiltonian has the form:

$\displaystyle\mathcal{H}_{\Upsilon}=-\sum_{i=e,p,n}{g_{\Upsilon
i}\frac{\boldsymbol{S}_{i}}{\left|S_{i}\right|}\cdot\boldsymbol{\Upsilon}}=-\sum_{i}{g_{\Upsilon
i}\sigma_{i}\frac{\boldsymbol{F}}{\left|F\right|}\cdot\boldsymbol{\Upsilon}}\,,$
(1)

where $g_{\Upsilon i}$ is the coupling constant characterizing the interaction
of $\boldsymbol{\Upsilon}$ with the fermion spin [where the fermions
considered include electrons ($i=e$), protons ($i=p$), and neutrons ($i=n$)],
$\sigma_{i}$ is the fractional fermion spin polarization for a given atom (see
the Supplemental Material in Ref. [30]), $\boldsymbol{F}$ is the total atomic
angular momentum of the atomic state probed, and $\left|S_{i}\right|=1/2$ and
$\left|F\right|$ are the maximum spin projections. The exotic-field
Hamiltonian $\mathcal{H}_{\Upsilon}$ can be compared to the Zeeman Hamiltonian
describing the interaction of atomic spins with a magnetic field:

$\displaystyle\mathcal{H}_{B}=g_{F}\mu_{B}\boldsymbol{F}\cdot\boldsymbol{B}\,,$
(2)

where $g_{F}$ is the Landé g-factor, $\mu_{B}$ is the Bohr magneton, and
$\boldsymbol{B}$ is the external magnetic field experienced by the atom.
Comparing Eqs. (1) and (2), it is evident that the physical manifestations of
the ordinary magnetic field coupling and the coupling of exotic fields to
spins are analogous. Furthermore, by measuring OAM response to
$\boldsymbol{B}$, the response to an exotic field $\boldsymbol{\Upsilon}$ can
be inferred [32]. Therefore, we can consider $\boldsymbol{\Upsilon}$ to be a
“pseudo-magnetic” field: a field that shifts Zeeman energy levels and
generates torques on atomic spins, but does not couple proportionally to spins
of different species according to their gyromagnetic ratios (i.e., comparing
across different atomic species, $\sum_{i}g_{\Upsilon
i}\sigma_{i}/\left|F\right|$ is not proportional to $g_{F}\mu_{B}$).
Furthermore, unlike a magnetic field, it may be the case that
$\boldsymbol{\Upsilon}$ has nonzero divergence (see, for example, Ref. [33]).

In order to reduce environmental noise from ambient magnetic fields, the
atomic vapor cells that contain the gases at the heart of GNOME’s OAMs are
placed inside multilayer magnetic shields composed of soft ferrimagnetic or
ferromagnetic materials (such as mu-metal) [34]. As noted in Ref. [35], if the
exotic field $\boldsymbol{\Upsilon}$ interacts primarily with electron spins,
there is an approximate cancellation of the effect of the field
$\boldsymbol{\Upsilon}$ on electron spins within the magnetic shield. This is
due to the fact that it is the electron spins within shielding materials such
as mu-metal that respond to the external magnetic field, so they similarly
respond to an electron-coupled field $\boldsymbol{\Upsilon}$. This response
generates a corresponding magnetic field approximately cancelling the
electron-spin-dependent energy shift within the shield [35].111Note that
cancellation of the effects of an electron-spin-coupled
$\boldsymbol{\Upsilon}$ is not exact within the ferromagnetic shielding due to
imperfect shielding and nonzero nuclear magnetic moments. If the techniques of
noble-gas-alkali-metal comagnetometry are employed (see Sec. III.5), electron-
spin-coupled fields $\boldsymbol{\Upsilon}$ are measurable inside the shields
because the nuclear spins of the noble gas respond to the
$\boldsymbol{\Upsilon}$-induced magnetic field from the shielding material.
Consequently, GNOME OAMs are primarily sensitive to exotic field couplings to
nuclear spins. At present, GNOME OAMs use alkali atoms such as rubidium (Rb)
and cesium (Cs) whose nuclei have valence protons, and thus GNOME
predominantly measures interactions of exotic fields with the proton spin
$\boldsymbol{S}_{p}$ [36, 37].

The principal hypothesis GNOME has sought to test is that dark matter is
composed of ultralight spin-0 bosons known as axions or axion-like particles
(ALPs). Such exotic spin-0 bosons are ubiquitous features of the theoretical
landscape beyond the Standard Model. The axion originally emerged from a
proposed solution to the strong-CP problem [38, 39, 40, 41], the mystery of
why nucleon EDMs and CP-violating nuclear electromagnetic moments are many
orders of magnitude smaller than nominally predicted by quantum chromodynamics
(QCD). Since then, a variety of other beyond-the-Standard-Model theories have
emerged predicting similar spin-0 bosons known as ALPs [1, 42, 43, 44, 45].
Axions and ALPs are commonly thought to be ultralight (masses $m_{a}\ll
1\,{\rm eV}$). They can be copiously produced in the early universe [46, 47,
48, 49, 50, 51, 52] and have all the requisite characteristics to be the dark
matter [2, 53, 52, 54].

The most commonly considered manifestation of a coupling between an ALP field
$\varphi$ and atomic spins is given by the Lagrangian [55]

$\displaystyle\mathcal{L}_{l}=\frac{{\left(\hbar
c\right)}^{3/2}}{f_{l}}J^{\mu}\partial_{\mu}\varphi\,,$ (3)

where $f_{l}$ is the characteristic energy scale associated with the spin
“portal” between ALPs and fermions (the subscript $l$ denoting that the
interaction is linear in the ALP field $\varphi$) and $J^{\mu}$ is the axial-
vector current for fermions $\psi$,

$\displaystyle J^{\mu}=\bar{\psi}\gamma^{\mu}\gamma_{5}\psi\,,$ (4)

where $\gamma^{\mu}$ and $\gamma_{5}$ are Dirac matrices. The corresponding
Hamiltonian $\mathcal{H}_{l}$ can be derived from the Euler-Lagrange equations
(see, for example, Refs. [1, 56]):

$\displaystyle\mathcal{H}_{l}\psi=-\frac{{\left(\hbar
c\right)}^{3/2}}{f_{l}}\gamma_{0}\gamma^{\mu}\gamma_{5}{\left(\partial_{\mu}\varphi\right)}\psi~{}.$
(5)

In the nonrelativistic limit, where the spacelike component of the derivative
of $\varphi$ is much larger than the timelike component,

$\displaystyle\mathcal{H}_{li}=-\frac{{\left(\hbar
c\right)}^{3/2}}{f_{li}}\frac{\boldsymbol{S}_{i}}{\left|S_{i}\right|}\cdot\boldsymbol{\nabla}\varphi~{},$
(6)

where the subscript $i$ specifies the interaction with fermion $i=e,p,n$.
Comparing Eq. (6) to Eq. (1), we see that the coupling constant for ALPs in
the above parameterization is given by $g_{\Upsilon i}={\left(\hbar
c\right)}^{3/2}/f_{li}$ and the exotic pseudo-magnetic field is described by
$\boldsymbol{\Upsilon}_{l}=\boldsymbol{\nabla}\varphi$. Note that not only
does $\mathcal{H}_{l}$ generate an interaction between spins and the spatial
gradient of $\varphi$, but $\mathcal{H}_{l}$ also generates an interaction
between spins that move with respect to the ALPs, since the momentum is
related to the gradient operator via
$\boldsymbol{p}=-i\hbar\boldsymbol{\nabla}$. The latter interaction is often
referred to as the “axion wind” [57, 55, 5, 58].

We also consider an alternative “quadratic” coupling between spins and the
gradient of the intensity of the ALP field, $\boldsymbol{\nabla}\varphi^{2}$
[31]. Whereas the QCD axion associated with the Peccei-Quinn solution to the
strong-CP problem [38, 39] generally possesses the linear gradient interaction
described by Eqs. (3) and (6) [55], the quadratic gradient interaction can
arise in effective field theories predicting ALPs not associated with the QCD
sector [59]. It is possible that under certain circumstances (see, for
example, Refs. [60, 61, 62]), linear-in-$\varphi$ interactions may be
suppressed, and the interaction of photons, electrons and nuclei with scalar
fields starts at the quadratic order, $\varphi^{2}$. A very important
consequence of such a modification is the relaxation of the most stringent
astrophysics bounds compared to the linear case [59], opening up a parameter
space for the direct searches of $\varphi^{2}$ coupling to spins. Also note
that quadratic-in-$\varphi$ interactions are required for complex-valued
$\varphi$, as is the case for the Q-ball scenario discussed in Sec. IV.3.
Furthermore, there are novel experimental signatures and modalities that can
be employed to search for the quadratic-in-$\varphi$ interactions [63].

The Lagrangian describing the quadratic gradient coupling between an ALP field
and atomic spins is given by [31]

$\displaystyle\mathcal{L}_{q}=\frac{\hbar^{2}c^{2}}{f_{q}^{2}}J^{\mu}\partial_{\mu}\varphi^{2}~{}.$
(7)

The corresponding Hamiltonian is described by

$\displaystyle\mathcal{H}_{q}\psi=-\frac{\hbar^{2}c^{2}}{f_{q}^{2}}\gamma_{0}\gamma^{\mu}\gamma_{5}{\left(\partial_{\mu}\varphi^{2}\right)}\psi\,,$
(8)

and in the nonrelativistic limit the Hamiltonian describing the interaction of
$\varphi$ with the spin of fermion $i$ is approximately

$\displaystyle\mathcal{H}_{qi}=-\frac{\hbar^{2}c^{2}}{f_{qi}^{2}}\frac{\boldsymbol{S}_{i}}{\left|S_{i}\right|}\cdot\boldsymbol{\nabla}\varphi^{2}~{}.$
(9)

Comparing Eq. (9) to Eq. (1), we see that the coupling constant for ALPs in
the above parameterization is given by $g_{\Upsilon i}={\left(\hbar
c\right)}^{2}/f_{qi}^{2}$ and the exotic pseudo-magnetic field is described by
$\boldsymbol{\Upsilon}_{q}=\boldsymbol{\nabla}\varphi^{2}$.222Note that, as
defined here, the units of
$\boldsymbol{\Upsilon}_{q}=\boldsymbol{\nabla}\varphi^{2}$ differ from those
of $\boldsymbol{\Upsilon}_{l}=\boldsymbol{\nabla}\varphi$. The correct units
for the associated Hamiltonian, described by Eq. (1), are obtained through the
respective coupling constants $g_{\Upsilon}$ also having different units for
the quadratic and linear ALP gradient interactions.

It is of interest to note the discrete symmetry properties of the interactions
described by Eqs. (6) and (9). Consider as a reference the discrete symmetry
properties of the standard Zeeman interaction of Eq. (2). The atomic angular
momentum $\boldsymbol{F}$ is even under parity (P-even), since it is an axial-
or pseudo-vector, and odd under time-reversal (T-odd), since angular momentum
reverses its sign when time runs backward. The magnetic field
$\boldsymbol{B}$, generated by current flow, is also P-even and T-odd, and
thus $\mathcal{H}_{B}$ is P- and T-even. For the linear gradient interaction
of Eq. (6), the pseudoscalar ALP field $\varphi$ is P-odd and T-odd and the
gradient $\boldsymbol{\nabla}$ is P-odd and T-even, and thus
$\boldsymbol{\nabla}\varphi$ is a P-even, T-odd quantity, matching the
discrete symmetry properties of the magnetic field. Consequently,
$\mathcal{H}_{l}\propto\boldsymbol{S}\cdot\boldsymbol{\nabla}\varphi$ is P-
and T-even. On the other hand, for the quadratic gradient interaction of Eq.
(9), $\boldsymbol{\nabla}\varphi^{2}$ is P-odd and T-even because of the extra
factor of the ALP field $\varphi$, which means that the quantity
$\boldsymbol{S}\cdot\boldsymbol{\nabla}\varphi^{2}$ is P- and T-odd. Based on
CPT invariance (where C is the charge conjugation symmetry), it follows that
$\mathcal{H}_{q}$ describes a CP-violating interaction, and as a result could
play a role in baryogenesis [64].

##  III GNOME overview

The idea of the GNOME experiment is to carry out synchronous measurements of
spin-dependent interactions using OAMs operating within magnetically-shielded
environments in distant locations. In this section we review the basic feature
of the GNOME network, give an overview of the data collected so far, and
describe ongoing improvements to the GNOME sensors that will significantly
enhance the network sensitivity.

###  III.1 GNOME magnetometers

As mentioned above, OAMs utilize the interaction of atomic spins with external
magnetic fields [7]. Typically, alkali metal vapors, contained in glass cells,
are used for the measurements. To prevent spin-depolarizing collisions with
cell walls, which can limit the OAM sensitivity, either the walls are coated
with special (e.g., paraffin) layer or the cells are filled with an additional
inert gas (e.g., noble gas) to slow down diffusion. In OAMs, the atoms are
optically polarized, resulting in optical anisotropy of the medium. As the
spins of the polarized atoms precess due to a nonzero magnetic field (or,
perhaps, due to an exotic field coupled to atomic spins), detection of the
corresponding change in optical properties of the medium provides quantitative
information about the field. The signals of the GNOME magnetometers are
recorded using a custom data acquisition (DAQ) system [65], providing accurate
timing from the Global Positioning System (GPS).

In the first incarnation of the GNOME, various OAMs employed different
elements (Rb and Cs) and were based on different techniques; spin-exchange-
relaxation-free (SERF) [66], Mx [67], and nonlinear-magneto-optical-rotation
(NMOR) [68] magnetometers were used for the measurements. On the one hand,
this diversity offers flexibility, opens a greater range of theoretical
parameter space for exploration, and improves the robustness of the network.
On the other hand, it results in sensors having different sensitivities and
bandwidths, which complicates data analysis. In general, however,
magnetometers used in the GNOME have an operational sensitivity better than 1
pT/$\sqrt{\text{Hz}}$, corresponding to a sensitivity to Zeeman energy shifts
below 10-17 eV/$\sqrt{\text{Hz}}$, and bandwidths up to 100 Hz [29].

Although OAMs enable searches for non-magnetic spin couplings, the devices are
highly sensitive to magnetic fields. Therefore, despite shielding from the
external environment, uncontrollable magnetic disturbances are a significant
source of noise. In order to reduce magnetic noise, the next generation of the
GNOME experiment (Advanced GNOME) is using comagnetometers [69, 70]: sensors
with limited sensitivity to magnetic fields that still maintain sensitivity to
non-magnetic spin couplings. Comagnetometers are briefly described in Sec.
III.5.

Figure 2: One-day rolling average of the GNOME noise level according to Eq.
(10) over the course of the first five Science Runs. The standard deviation at
each magnetometer is calculated using the data for one second. Then this
information is averaged for one hour. The color of the line indicates the
number of active stations as indicated on the color map legend at the upper
right.

###  III.2 Monitoring of glitches

While by its nature the GNOME can suppress uncorrelated noise and false
positive events, additional measures are implemented in the network to further
increase the data quality and trustworthiness of each station. In order to
veto signatures in the data which might have been produced by technical issues
or changing experimental conditions (e.g., mechanical shocks, magnetic or
electric pulses from neighboring technical devices), each GNOME station
features a tailored automated system to continuously check for environmental
perturbations. The system is based on the Arduino microcontroller board MEGA
2560 additionally equipped with the Arduino 9 Axes motion shield
(magnetometer, accelerometer and gyroscope, three axes each) in a separate
“sensor box”. The sensor box is mounted on the optical table of the GNOME
station near the magnetic shielding. In this way, it can check for mechanical
shocks or vibration of the optical setup. In addition, the system features
additional analog voltage inputs, which are used to monitor the operational
status of the station (e.g., magnetometer signal amplitude, error signal of
the magnetometer feedback and/or the laser lock, readings of temperature
sensors, and signals of a photo diode monitoring laser or ambient light
power). For each GNOME station, depending on the specific setup, the system is
set to monitor its critical parameters.

A dedicated Python-based software allows the system to display, save, and
define “sane” ranges for all monitored parameters. If one of the parameters
falls out of the “sane” range, the system will output a signal to the DAQ
system indicating the data recorded meanwhile should be rejected in data
analysis. The software writes a log file allowing one to trace back the event
to the individual monitored parameter. It is also possible to set alarms to
notify station operators about irregularities over email and/or the Telegram
app.

###  III.3 Calibration pulses

A possible concern with the continuous operation of the magnetometers over the
course of several months is variation in the calibration and bandwidth of the
detectors. Such variations could result from drifts in laser power, laser
frequency, or temperature of the vapor cells. To monitor this, a series of
oscillating magnetic fields are periodically applied to each magnetometer
station via coils inside the magnetic shields. The frequency of the applied
magnetic field is stepped from 1 to 180 Hz over the course of 9 s using a
programmable function generator.333The pulse sequence used in a recent Science
Run 5 was 1 Hz for 4 s, 10 Hz for 2 s, 35 Hz for 1 s, 55 Hz for 0.6 s, 70 Hz
for 0.4 s, 80 Hz for 0.2 s, 90 Hz for 0.2 s, 110 Hz for 0.2 s, 130 Hz for 0.2
s, 160 Hz for 0.1 s, and 180 Hz for 0.1 s. During the most recent experimental
campaign (Science Run 5), the pulse sequence was applied hourly. The response
of the magnetometers at the different frequencies provides a convenient check
on the operation of the magnetometers as well as a method for measuring the
frequency response and bandwidth of the detectors. The pulses also provide a
test of the timing of the stations. The pulses are triggered by the GPS pulse-
per-second (pps) signal of the GNOME DAQ system or a time synchronized
computer clock. These tests indicate that the stations are synchronized at a
level of better than the sample period, $T_{\textrm{samp}}=1/512$ s.444The GPS
DAQ system provides timing with a precision better than 100 ns.

###  III.4 The GNOME experiment so far

To date, the GNOME collaboration has completed five “Science Runs” as well as
a number of test and calibration runs. According to Eqs. (6) and (9) and the
surrounding discussion in Sec. II, GNOME sensors seek to measure
characteristic global patterns of pseudo-magnetic fields
$\boldsymbol{\Upsilon}$. Typically, GNOME magnetometers are sensitive to the
projection of $\boldsymbol{\Upsilon}$ along a particular sensitive axis. Thus
the amplitudes of signals from exotic fields $\boldsymbol{\Upsilon}$ scale
proportionally to $\hat{\boldsymbol{m}}\cdot\boldsymbol{\Upsilon}$, where
$\hat{\boldsymbol{m}}$ is a unit vector pointing along the sensitive axis of a
magnetometer. Since the direction of the field $\boldsymbol{\Upsilon}$ is
essentially unknown, in order to assess the sensitivity of the GNOME, this
directional sensitivity must be taken into account. Additionally, the various
parameters of individual sensors must be considered along with the
relationship between the sensor response and the underlying physical theory
(which must account for atomic and nuclear structure [36, 37]). These issues
are discussed in more detail in Refs. [71, 30, 72].

A summary of the network performance for the five GNOME Science Runs is shown
in Fig. 2. Since the GNOME data can be utilized in different ways to test
different exotic physics hypotheses, discussed in Sec. IV, for simplicity we
adopt for the summary a relatively simple, model-independent evaluation. The
plot in Fig. 2 shows the one-hour-average noise level as defined by

$\displaystyle\sigma{{}_{\mbox{\scriptsize
network}}}\equiv\sqrt{\frac{1}{\sum_{j}\sigma_{j}^{-2}}}\,,$ (10)

where $\sigma_{j}$ is the variance of magnetometer $j$, calculated using the
standard deviation for each second of data. As can be seen from Fig. 2, the
GNOME experiment has accumulated over a year of data sensitive to pseudo-
magnetic fields with equivalent magnitudes $\lesssim\textrm{pT}$ that can be
searched for a variety of exotic physics signals.

###  III.5 Advanced GNOME: noble gas comagnetometers

The further development of the GNOME experiment focuses on the diversification
and upgrade of the sensors implemented in the network. There are three main
directions for the improvement of sensors: enhancing their sensitivity,
increasing their bandwidth, and expanding the types of couplings probed.
Although a number of various experimental techniques could be used for GNOME
sensors (e.g., spin-based amplifiers [73], noble gas masers [74, 75], dual-
species nuclear-spin comagnetometers [76, 77], alkali comagnetometers [78, 79,
*kimball2023GDMerratum, 81], liquid-state NMR comagnetometers operating in the
zero-to-ultralow field (ZULF) regime [82, 83], etc.), efforts are presently
focused on developing self-compensating noble-gas-alkali-metal comagnetometers
[69, 70] for implementation in the “Advanced GNOME” experiment [32, 84].

In addition to the coupling of exotic fields to proton spins, which was the
only coupling probed at a competitive level by the first-generation GNOME
[30], self-compensating noble-gas-alkali-metal comagnetometers can also probe
both neutron and electron spin couplings. The ability to measure neutron spin
couplings to $\boldsymbol{\Upsilon}$ comes from the fact that the noble gases
(such as 3He) employed in these sensors have nuclei with valence neutrons
[36].555Note that 3He-alkali-metal comagnetometers retain sensitivity to
proton couplings via a reasonably well-measured and understood proton-spin
polarization in the 3He nucleus and the proton-spin polarization of the alkali
metal nucleus [36]. However, the reason for the ability of these sensors to
measure electron spin couplings is somewhat more subtle. If we assume that
$\boldsymbol{\Upsilon}$ couples primarily to electron spins, then the
ferromagnetic (or ferrimagnetic) shielding responds to $\boldsymbol{\Upsilon}$
by creating an induced magnetic field that cancels the effect of
$\boldsymbol{\Upsilon}$ on electrons within the shield where the vapor is
located [35]. However, if we assume no coupling of $\boldsymbol{\Upsilon}$ to
neutron spins, then neutrons will respond to the induced magnetic field from
the shields and thus a detectable effect is generated. This further highlights
the advantages of comagnetometry for exotic physics searches with the GNOME.
At sub-Hz frequencies, the sensitivity of the comagnetometer is also
significantly improved by the suppression of the magnetic field response due
to the self-compensation regime in which the comagnetometer is operated.

Initial tests of Advanced GNOME sensors demonstrate a sensitivity at the level
of 10-21 eV/$\sqrt{\mathrm{Hz}}$ (at 1 Hz) for exotic fields coupling to
neutron spins, and about 10-19 eV/$\sqrt{\mathrm{Hz}}$ (at 1 Hz) for the
proton spin coupling (surpassing that of GNOME magnetometers by a factor of
100), see Fig. 3. For the electron spin couplings, the expected sensitivity is
comparable to the sensitivity of the first-generation GNOME magnetometers to
the proton exotic spin couplings. The noble-gas-alkali-metal comagnetometers
have an optimal sensitivity to the nuclear spin couplings for frequencies
below a few hertz. However, enhanced sensitivity to exotic fields (as compared
to the first-generation of GNOME magnetometers) is expected over the whole
bandwidth of the ordinary magnetometers, even at frequencies for which the
comagnetometer performance is sub-optimal.

The Advanced GNOME sensors also bring another significant qualitative
advantage for exotic physics searches. The difference in the response to the
magnetic and non-magnetic spin couplings can be used to discriminate events of
magnetic origin from those driven by non-magnetic spin couplings, just from a
single sensor readout [32]. Although ultimate verification of a global exotic
physics event will rely on the correlation between signals observed in
multiple GNOME stations, more effective discrimination between magnetic and
non-magnetic signals will enable efficient suppression of the “false positive”
rate and therefore improve the overall sensitivity of the network.

Figure 3: Sensitivities of Advanced GNOME stations in Mainz (left) and Kraków
(right) obtained during a Test Run in January 2023, compared with the typical
sensitivity of the first-generation GNOME stations during Science Run 5 (see
Fig. 2). Red curves: sensitivities of the comagnetometers to the neutron
exotic spin couplings; blue curves: sensitivity of the comagnetometers to the
proton spin couplings; black curves: magnetometer sensitivities to exotic
proton couplings. The sensitivities are presented in terms of Zeeman-like
energy splitting generated by the considered perturbation. The presented data
sets were undersampled for illustration purposes.

##  IV Search targets and their signals

By analyzing correlations between signals from the geographically separated
magnetometers and comagnetometers comprising GNOME, it is possible to probe a
wide variety of exotic beyond-the-Standard-Model hypotheses. In this section
we survey some exotic physics scenarios that can be searched for using GNOME
data and highlight examples of their particular temporal signatures.

###  IV.1 Axion domain walls

Figure 4: Upper diagram: schematic depiction of a GNOME sensor passing through
an ALP domain wall (thickness $\approx R_{E}/10$), indicating with the long
black arrow the trajectory of the sensor along $\boldsymbol{v}$ (parallel to
$\boldsymbol{k}$, along $\hat{\boldsymbol{x}}$, $v\approx 10^{-3}c$) and the
sensitive axis of a GNOME sensor (blue arrow, tilted at $45^{\circ}$ to
$\hat{\boldsymbol{x}}$). The plots below show example signals due to the
linear spin coupling (middle plot) and quadratic spin coupling (lower plot)
from an ALP-domain-wall encounter corresponding to the schematic at top.

The first proposed search targets for GNOME were axion or ALP domain walls
[31, 28]. Domain walls are topological defects that form between regions of
space in which the ALP field possesses different, but energy-degenerate,
vacuum states [85, 86]. Such a scenario arises due to the non-trivial vacuum
topology that ALP fields typically possess: there can be multiple local field
energy minima (and corresponding vacuum states) that, in the abstract space
describing the field, are not “simply connected” in a topological sense [87].
When spontaneous symmetry breaking occurs in the early universe, different
regions of space (domains) acquire different vacuum states. The domain walls
are the field configurations at the boundaries between these domains. In order
for the ALP field to transition from one energy minimum to another across
these boundaries, the ALP field necessarily acquires values above those
corresponding to the minimum energy. These non-vaccuum states are associated
with considerable potential energy, and the change of the field over space
means that the ALP field has a nonzero gradient.

####  IV.1.1 Theoretical description

ALP domain walls are macroscopic field configurations that could be stable and
long lived, and continue to exist today [86]. They can be of astrophysical
extent, potentially much larger than the size of Earth. If the ALP field
primarily manifests in the form of domain walls, the associated energy density
would be concentrated into compact spatial regions. While in most theoretical
models such domain walls are unstable [85, 88], in some other models [31, 89,
86] domain walls can compose a significant fraction of dark matter. These
objects can be considered virialized in the galaxy according to the standard
halo model (SHM). Based on this assumption, the rate and duration of the ALP
domain wall encounters can be estimated. A region of theoretically plausible
parameter space is expected to have an average encounter rate with Earth of a
year or less [31, 89]. This opens the possibility to directly search for such
encounters. Atomic magnetometers and comagnetometers are sensitive to domain
walls, since the ALP-field gradient can interact with the spin of electrons,
neutrons and protons [see Eqs. (6) and (9)].

####  IV.1.2 Signal model

Consider an ALP domain wall in the $yz$-plane ($x=0$) separating two
degenerate ALP vacuum states as pictured in the upper diagram in Fig. 4. The
solution of the field equations for $\varphi$ yields [90, 31, 30]

$\displaystyle\varphi(x)=\varphi_{0}\arcsin{\left[\tanh{\left(x/\lambda_{c}\right)}\right]}\,,$
(11)

where $\varphi_{0}$ is a constant proportional to the spontaneous symmetry-
breaking scale associated with the ALP and
$\lambda_{c}=\hbar/{\left(m_{a}c\right)}$ is the Compton wavelength of the ALP
of mass $m_{a}$. Atomic spins can interact with the ALP field through the
linear and quadratic gradient interactions of Eqs. (6) and (9), which can be
described by the pseudo-magnetic fields

$\displaystyle\boldsymbol{\Upsilon}_{l}=\boldsymbol{\nabla}\varphi(x)=\frac{\varphi_{0}}{\lambda_{c}}\operatorname{sech}{\left(x/\lambda_{c}\right)}\,\hat{\boldsymbol{x}}$
(12)

and

$\displaystyle\boldsymbol{\Upsilon}_{q}=\boldsymbol{\nabla}\varphi^{2}(x)=\frac{2\varphi_{0}^{2}}{\lambda_{c}}\arcsin{\left[\tanh{\left(x/\lambda_{c}\right)}\right]}\operatorname{sech}{\left(x/\lambda_{c}\right)}\,\hat{\boldsymbol{x}}\,,$
(13)

respectively. The signal measured by a GNOME sensor is
$\propto\hat{\boldsymbol{m}}\cdot\boldsymbol{\Upsilon}$, where
$\hat{\boldsymbol{m}}$ is directed along the sensitive axis of the
magnetometer. The expected time-dependent lineshapes for the linear and
quadratic couplings are shown in the lower plots of Fig. 4. While all the
GNOME sensors experience a common transient pseudo-magnetic field
$\boldsymbol{\Upsilon}$, because of the varying directions of their sensitive
axes $\hat{\boldsymbol{m}}$ (as well as differences in the magnetometers
employed), the signals in different sensors would have varying amplitudes and
signs. For a given relative velocity between the ALP domain wall and Earth,
there is a characteristic signal timing and amplitude pattern that can be used
to distinguish true domain-wall-crossing events from spurious noise [71, 91,
30]. Once an event is detected, one can relate the signal properties (width
and the amplitude of the signal, as well as the inferred expected distribution
of domain walls) to ALP parameters, including the mass of the ALP and the
interaction and symmetry-breaking scales [30].

The GNOME collaboration has developed data analysis algorithms to search for
ALP domain walls [71, 91], and has completed a full analysis of the data from
Science Run 2 carried out in 2017 [30]. Our searches did not find any
statistically significant signals above background that could point to the
existence of ALP domain walls. Consequently, our results can be interpreted as
constraints on the properties of ALP domain walls. The excluded (model-
dependent) ALP parameter space covers masses in the range $\sim
10^{-15}-10^{-7}$ eV [30]. A new analysis procedure is currently being
developed to improve on the sensitivity to narrower domain wall widths with
respect to our previous work [30]. The new analysis is based on a preselection
of candidate signals groups occurring within a given time window. Then each
group is tested for consistency with the domain wall crossing model. This
allows an efficient scrutiny of the data compared to the previous method. In
Ref. [30] all possible domain wall configurations were scanned and their
agreement with the data were evaluated. The new procedure results in a
comparatively less computer-intensive routine.

###  IV.2 Axion stars

Instead of the ALP dark matter being primarily in the form of topological
defects such as domain walls as discussed above in Sec. IV.1, it may be the
case that inhomogeneities in the dark matter distribution provide seeds that
enable ALP self-interactions or gravity to attract together a large local
density of ALPs, thereby forming a spherical bound-state (see Refs. [92, 93,
94, 95] and references therein). Such spherical bound states are referred to
as axion or ALP stars, and they may constitute a significant fraction of the
dark matter density [92]. Under certain conditions, ALP stars are stable under
perturbations and radiative decay [96, 97, 98, 99], and can be efficiently
formed in the early Universe [100, 101, 102, 103]. The extensive theoretical
studies of ALP stars to determine their evolution, stability, and
characteristic radii and masses for different models of the ALP self-
interaction establish that ALP stars are, indeed, a plausible dark matter
scenario [92, 104, 105]. As described in Ref. [106], ALP stars are a
compelling search target for the GNOME, since there is a range of masses and
radii for ALP stars (not ruled out by existing empirical observations) for
which the terrestrial encounter rate can be sufficiently high (at least once
per year) that a detection would be feasible. ALP stars can interact with
atomic spins via the linear or quadratic gradient interactions [Eqs. (6) and
(9)]. Thus if an ALP star passed through Earth, it would cause oscillatory
pseudo-magnetic field pulses in GNOME sensors that, in principle, could be
within the sensitivity range of GNOME [106].

####  IV.2.1 Theoretical description

Figure 5: Upper diagram: schematic depiction of a GNOME sensor passing through
an ALP star, indicating with the long black arrow the trajectory of the sensor
along $\boldsymbol{v}$ (parallel to $\boldsymbol{k}$, along
$\hat{\boldsymbol{x}}$), the initial direction of the sensitive axis of the
GNOME sensor (blue arrow, along $\hat{\boldsymbol{x}}$), and the axis of
Earth’s rotation (dashed red arrow, $45^{\circ}$ to the $x$-axis). Lower plot:
example signal due to the linear spin coupling [Eq. (19)] from an ALP star
passage corresponding to the above schematic. For this example, the ALP star
radius $R\approx 200R_{E}$, $v=10^{-3}c$, and at $t=0$ the sensor is located
at ${\left(x_{0},y_{0},z_{0}\right)}={\left(2R,0,R/2\right)}$, where the
origin is defined to be the center of the ALP star. The ALP field oscillation
frequency is chosen to be $\omega=2\pi\times 10\,{\rm Hz}$, and we assume
$\omega\approx\omega_{c}$, neglecting the binding energy. The ALP-star signal
in the plot is based on the exponential model, Eq. (16). The inset plot shows
an $\approx 1~{}{\rm s}$ long segment of the simulated signal.

Axion or ALP stars held together via self-interactions are called _oscillons_
[107] or _axitons_ [108]. There have been a number of theoretical studies of
such composite systems, and a few general features are observed. Models of
oscillons and axitons demonstrate that they can persist in a stable or quasi-
stable regime [109, 110, 111], however if the ALP density is too large, they
tend to radiate ALP waves and lose mass [92]. Sufficiently dilute ALP stars
can be stable over long time scales [109, 110, 112]. The ALP fields bound in
the form of oscillons have a characteristic field oscillation frequency
$\omega$ which is somewhat smaller than the ALP Compton frequency
$\omega_{c}$. This stems from the fact that, while free ALPs oscillate at the
Compton frequency $\omega_{c}$, because the energy of the ALPs in the star is
reduced by a binding energy $\epsilon_{b}$, the oscillation frequency of ALPs
in the oscillon is given by

$\displaystyle\omega=\omega_{c}-\epsilon_{b}/\hbar~{}.$ (14)

This is a key difference between the ALP domain walls considered in Sec. IV.1
and the ALP stars considered here: the ALP field of the domain wall does not
oscillate, being a topological defect related to the ALP field potential
energy, whereas the ALP field of the star does oscillate, since most of the
field energy is in the form of kinetic energy. The mass $M$ of an oscillon is
$M\approx N{\left(m_{a}-\epsilon_{b}/c^{2}\right)}$, where $N$ is the number
of ALPs in the star.

Alternatively, axion stars could be held together by gravity [113, 114, 115,
116, 96, 97, 98, 99]. However, in this case it may be expected that
gravitational tidal forces due to the Sun, Earth, and other bodies in the
solar system could cause distortions and disruptions of the axion star [117],
complicating the details of the shape profile in ways that are difficult to
predict. Thus, for simplicity, here we focus on the oscillon or axiton models
where self-interactions stronger than gravity dominate.

####  IV.2.2 Signal model

A number of different approximate analytic wavefunctions for ALP stars have
been studied in the literature (see Ref. [95] for a review). Two of the more
widely used radial wavefunctions $\Psi(r)$, where $r$ is the radial distance
from the center of the ALP star, are the Gaussian approximation,

$\displaystyle\Psi(r)=\sqrt{\frac{N\kappa^{3}}{\pi^{3/2}R^{3}}}e^{-\kappa^{2}r^{2}/(2R^{2})}\,,$
(15)

and the exponential approximation,

$\displaystyle\Psi(r)=\sqrt{\frac{N\kappa^{3}}{\pi R^{3}}}e^{-\kappa r/R}\,,$
(16)

where $R$ is the characteristic radius of the ALP star enclosing 99% of its
mass and $\kappa$ is a numerical parameter ($\kappa=2.8$ for the Gaussian
wavefunction and $\kappa=4.2$ for the exponential wavefunction [95]). In Ref.
[95] it is argued that, in fact, a more accurate description of an ALP star in
many relevant scenarios is given by a “linear-plus-exponential” wave function:

$\displaystyle\Psi(r)=\sqrt{\frac{N\kappa^{3}}{7\pi
R^{3}}}{\left(1+\frac{\kappa r}{R}\right)}e^{-\kappa r/R}~{}.$ (17)

In any case, the ALP field as observed from the reference frame of a GNOME
sensor is given by

$\displaystyle\varphi(\boldsymbol{r},t)=\Psi(r)\cos{\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega
t+\theta\right)}\,,$ (18)

where $\boldsymbol{k}$ is the ALP wave vector, $\boldsymbol{r}$ is the
distance from the center of the ALP star to the sensor, and $\theta$ is a
random phase. Since the ALP field in the star is essentially a Bose condensate
with long coherence time [92], we can treat the phase $\theta$ as constant
throughout the duration of the passage of the GNOME sensor through the ALP
star, and for simplicity in the following discussion we set $\theta=0$.

The expected signal in a GNOME sensor can be calculated based on the exotic
pseudo-magnetic fields $\boldsymbol{\Upsilon}_{l}=\boldsymbol{\nabla}\varphi$
and $\boldsymbol{\Upsilon}_{q}=\boldsymbol{\nabla}\varphi^{2}$ obtained from
the expression (18), where (as above) the subscripts denote the linear or
quadratic fields:

$\displaystyle\boldsymbol{\Upsilon}_{l}$
$\displaystyle={\left(\boldsymbol{\nabla}\Psi\right)}\cos{\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega
t\right)}-\boldsymbol{k}\Psi(r)\sin{\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega
t\right)}\,,$ $\displaystyle={\frac{\partial\Psi}{\partial
r}}\cos{\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega
t\right)}\hat{\boldsymbol{r}}-\frac{m_{a}v}{\hbar}\Psi(r)\sin{\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega
t\right)}\hat{\boldsymbol{k}}\,,$ (19)

and

$\displaystyle\boldsymbol{\Upsilon}_{q}$
$\displaystyle={\left(\boldsymbol{\nabla}\Psi^{2}\right)}\cos^{2}{\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega
t\right)}-2\boldsymbol{k}\Psi(r)\cos{\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega
t\right)}\sin{\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega t\right)}\,,$
$\displaystyle=2\Psi(r){\frac{\partial\Psi}{\partial
r}}\cos^{2}{\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega
t\right)}\hat{\boldsymbol{r}}-\frac{2m_{a}v}{\hbar}\Psi(r)\cos{\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega
t\right)}\sin{\left(\boldsymbol{k}\cdot\boldsymbol{r}-\omega
t\right)}\hat{\boldsymbol{k}}\,.$ (20)

There are two generally non-orthogonal components of the signal along
$\hat{\boldsymbol{k}}$ and $\hat{\boldsymbol{r}}$, which change in magnitude
as the GNOME sensor passes through the ALP star. Note that the sensitive axis
$\hat{\boldsymbol{m}}$ changes in time as Earth rotates, so this can introduce
additional time dependence of the signal which is
$\propto\hat{\boldsymbol{m}}\cdot\boldsymbol{\Upsilon}$. Figure 5 shows an
example of a simulated signal in a GNOME sensor passing through an ALP star
for a particular set of parameters. Both the direction and magnitude of
$\boldsymbol{\Upsilon}$ change as the GNOME sensor passes through the ALP
star, as well as the direction of $\hat{\boldsymbol{m}}$, leading to the
somewhat complicated time-dependence.

The GNOME collaboration is currently developing an analysis method to search
for ALP stars exhibiting this type of signal using the excess power statistic
[118].

###  IV.3 Q-balls

Another class of compact composite dark matter objects are _Q-balls_ or _Q-
stars_ [119, 120, 121, 122]. Q-balls are nontopological solitons: bound states
of a complex scalar field $\varphi$. If complex scalar field $\varphi$ obeys
global $U(1)$ symmetry, and has a non-zero net charge density Q, the self-
interaction of the scalar field makes it energetically favorable for the field
to fragment into clumps, called Q-balls [123, 124]. In some models, Q-balls
could comprise the entirety of dark matter [125, 126]. There are also various
theoretical extensions of the Q-ball concept, involving, for example, multiple
scalar fields [127] or local $U(1)$ symmetries instead of global [128]. The
distinction between Q-balls and ALP stars is somewhat subtle. Q-balls, because
of the global $U(1)$ symmetry, are true solitons since they are stable due to
the conserved charge associated with $\varphi$. On the other hand, oscillons,
our model for ALP stars, represent “quasi-solitons” because, while under
particular conditions they can be very long-lived, they are technically
unstable. The Q-ball is composed of a complex field, whereas the oscillon is
composed of a real field. For both Q-balls and oscillons, an attractive
potential is required for formation. As we see in the following, the
differences between the underlying theoretical models describing oscillons and
Q-balls lead to distinct experimental signatures in GNOME.

####  IV.3.1 Theoretical description

The complex scalar field of a Q-ball takes the form [119]

$\displaystyle\varphi(\boldsymbol{r},t)=\Psi(r)e^{i\omega t}\,,$ (21)

where $\omega\lesssim\omega_{c}$ is the angular oscillation frequency. The
stability of the Q-ball is a consequence of its conserved charge $Q$

$\displaystyle
Q=\frac{\omega}{\hbar^{2}c^{3}}\int\left|\varphi(\boldsymbol{r},t)\right|^{2}d^{3}\boldsymbol{r}~{}.$
(22)

The necessary conditions for Q-ball formation are that $Q\neq 0$ averaged over
the whole space and the existence of a self-interaction potential $U(\Psi)$
possessing at least two distinct minima at $\Psi=0$ and at $\Psi=\Psi_{0}$
[119, 120, 121, 122]. In this case, there are regions of space with different
vacuum energy values, and the regions where $\Psi=\Psi_{0}$ can deform but not
disappear because of the conserved charge $Q$. A characteristic feature of the
Q-ball is that the potential energy of the field, $U{\left(\Psi\right)}$, is
nonzero in the Q-ball’s transitional surface region where the field goes from
$\Psi=0$ to $\Psi=\Psi_{0}$. Therefore the Q-ball’s energy is minimized when
its surface area is minimized, leading to a spherical bound-state.
Furthermore, it is energetically favorable for the constituent particles
corresponding to the field $\varphi$ (ALPs in our considered case) to remain
within the Q-ball in the case where $\omega\lesssim\omega_{c}$, since ALPs
inside the Q-ball have energy $\hbar\omega$ while those outside the Q-star
have energy $\hbar\omega_{c}$. The values of $\omega^{2}$ and $\omega_{c}^{2}$
are proportional to $\partial^{2}U/\partial\Psi^{2}$ at the respective
potential minima inside ($\Psi=\Psi_{0}$) and outside ($\Psi=0$) the Q-ball,
and can thus be different [119, 122]. The condition $\omega\lesssim\omega_{c}$
ensures stability of the Q-star with respect to radiative decay via ALP
emission.

####  IV.3.2 Signal model

Figure 6: Upper diagram: schematic depiction of a GNOME sensor passing through
an astronomically large Q-ball (radius $\approx 100R_{E}$), indicating with
the long black arrow the trajectory of the sensor along $\boldsymbol{v}$
(parallel to $\boldsymbol{k}$, along $\hat{\boldsymbol{x}}$, $v\approx
10^{-3}c$), the initial direction of the GNOME-sensor sensitive axis (blue
arrow, along $\hat{\boldsymbol{x}}$), and the axis of Earth’s rotation (dashed
red arrow, $45^{\circ}$ to the $x$-axis). The purple curve below shows the
profile function of the Q-ball. Lower plot: example signal due to the
quadratic spin coupling from a Q-ball passage corresponding to the above
schematic.

The precise form of $\Psi(r)$ depends on details of the potential function,
and, although there are special forms of $U(\Psi)$ admitting analytic
solutions [129, 130, 131], in general numerical computation is required.
However, there have been a number of analyses, for example Refs. [132, 133,
134], that have derived useful approximate radial profile functions that
closely correspond to numerical solutions. For simplicity of mathematical
description, here we adopt the form given by Ref. [132],

$\displaystyle\Psi(r)=\frac{C_{0}}{\sqrt{1+C_{1}\cosh{\left(\alpha
r\right)}}}\,,$ (23)

where $C_{0}$, $C_{1}$, and $\alpha$ are constants determined by fitting to
numerical solutions for various potentials and values of $Q$ and $\omega$. In
contrast to the cases of ALP domain walls and ALP stars as described in Secs.
IV.1 and IV.2, for the complex-valued field $\varphi(r)$ forming Q-balls [Eq.
(21)], any couplings of spins to $\varphi(r)$ _must_ be quadratic in
$\varphi(r)$ [135]. This is required in order to respect the global $U(1)$
symmetry that endows the Q-ball with its charge: from a mathematical point-of-
view, interactions must involve products of $\varphi(r)$ and $\varphi(r)^{*}$
to give a real-valued energy. Therefore we consider the pseudo-magnetic field
$\boldsymbol{\Upsilon}_{q}$ arising from the quadratic interaction,

$\displaystyle\boldsymbol{\Upsilon}_{q}$
$\displaystyle=\boldsymbol{\nabla}{\left|\varphi(r,t)\right|}^{2}=\boldsymbol{\nabla}\Psi^{2}(r)\,,$
(24) $\displaystyle=\alpha C_{0}^{2}C_{1}\frac{\sinh{\left(\alpha
r\right)}}{\left[1+C_{1}\cosh{\left(\alpha
r\right)}\right]^{2}}\hat{\boldsymbol{r}}\,.$ (25)

Figure 6 shows a schematic example of the signal that would be measured by a
GNOME sensor passing through a Q-ball. In contrast to the ALP star considered
in Sec. IV.2, the signal of the Q-ball does not exhibit oscillatory behavior,
and is in fact more similar to the signature of an ALP domain wall considered
in Sec. IV.1. However, there is the notable difference that for a Q-ball there
should be two correlated time-separated signals corresponding to entry and
exit from the $\Psi=\Psi_{0}$ region, offering a unique signature pattern that
can be exploited for noise rejection.

Note that the signal pattern in GNOME corresponding to a Q-ball encounter
(Fig. 6) closely resembles that from two consecutive domain-wall encounters
(Fig. 4), or a domain-wall encounter with a quadratic coupling in the case of
a domain wall that is “thick” compared to the size of Earth. This suggests
that our data analysis strategies applied to domain-wall searches [71, 91, 30]
can be adapted to the case of Q-balls. Development of an analysis algorithm
based on this concept is under way.

###  IV.4 Dark-matter field fluctuations

Figure 7: Visualisation of the dark matter field oscillating at its Compton
frequency with fluctuations of the envelope function.

Unlike the compact dark matter objects described so far, in this section we
consider the more commonly assumed model for UBDM, namely that the UBDM field
is spread more evenly throughout the galactic halo volume and not concentrated
in large, compact objects. Due to their ultralight nature and (if they
constitute a sizable fraction of the dark matter) their enormous number
density, ALPs can be treated as classical plane waves. In the standard halo
model (SHM), ALPs are virialized in the gravitational potential of the Milky
Way, and their energy is given by $\hbar\omega\approx m_{a}c^{2}+m_{a}v^{2}/2$
where $\omega$ is the observed ALP field oscillation frequency and $v\sim
10^{-3}c$ is the ALP velocity. Due to the ALPs’ randomized velocities,
different ALP field modes interfere with one another resulting in a net field
that stochastically fluctuates on a characteristic time scale given by the ALP
field coherence time $\tau{{}_{\mbox{\scriptsize coh}}}\sim\hbar/(m_{a}v^{2})$
[24]. Given that the virial velocity of dark matter in the Milky Way galaxy at
the location of our solar system is $\approx 10^{-3}c$ [136, 137],
$\tau{{}_{\mbox{\scriptsize coh}}}\sim 10^{6}\times(2\pi/\omega_{c})$. This
means that the ALP field fluctuations occur on a time scale that is a factor
of a million slower than the ALP field oscillations. The stochastically
varying properties of the virialized ALP field are well-described by the
Rayleigh distribution [25, 138], which, notably, also describes thermal
(chaotic) light. If the coherence length of the virialized ALP field,
$\lambda{{}_{\mbox{\scriptsize coh}}}$, is large compared to the diameter of
Earth, GNOME magnetometers could detect common mode fluctuations of the ALP
dark matter field.

There are several experiments aiming to detect the ultralight dark-matter
field through an observation of a resonant spin coupling to ALPs at their
Compton frequency [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Such
measurements can also be performed with GNOME and Advanced GNOME sensors.
Given the close analogy between the behavior of virialized ALP fields and
thermal light, we can take full advantage of the fact that GNOME is a network
of spatially distributed sensors by implementing a detection scheme similar to
Hanburry-Brown-and-Twiss intensity interferometry [139] as recently proposed
in Ref. [63]. The quadratic ALP interaction with spins [Eq. (9)] leads to a
signal in GNOME magnetometers related to the intensity of the ALP field,
$\varphi^{2}(\boldsymbol{r},t)$. A measurement of the stochastic intensity-
like fluctuations of the dark matter field amplitude can be performed using
GNOME by correlating time series from distant sensors in order to detect the
common-mode signal. Instead of detecting fast oscillations at
$\approx\omega_{c}$, the slowly varying (at characteristic frequency $\sim
10^{-6}\omega_{c}$) ALP field envelope function would be measured. In this
way, the probed ALP mass range could be extended to masses several orders of
magnitude larger in comparison to resonant measurements performed with the
same bandwidth-limited sensors: GNOME’s $\sim 100$-Hz-bandwidth sensors could
search for ALPs with Compton frequencies up to $\sim 100$ MHz [63].

####  IV.4.1 Theoretical description

At each point in space, the dark matter field can be described as a
superposition of plane waves representing individual modes of the ALP field
(i.e., summing over all the individual ALPs composing the field),

$\displaystyle\varphi(\boldsymbol{r},t)=\sum_{n=1}^{N}\frac{A}{\sqrt{N}}\cos(\omega_{n}t-\boldsymbol{k}_{n}\cdot\boldsymbol{r}+\theta_{n})\,.$
(26)

The amplitude is defined to be $A=\hbar\sqrt{2\rho_{\textrm{dm}}}/(m_{a}c)$ so
that the average ALP field energy density corresponds to the average dark
matter density $\rho_{\textrm{dm}}$. The $n$-th ALP has a random velocity
$\boldsymbol{v}_{n}$ corresponding to the wave vector
$\boldsymbol{k}_{n}=m_{\varphi}\boldsymbol{v}_{n}/\hbar$. Based on the SHM,
the probability distribution function of the velocities, $\boldsymbol{v}_{n}$,
follows a displaced Maxwell-Boltzmann distribution:

$\displaystyle
f_{\textrm{lab}}(\boldsymbol{v})\approx\frac{1}{\pi^{3/2}v_{0}^{3}}\textrm{exp}\left[-\frac{(\boldsymbol{v}-\boldsymbol{v}_{\textrm{lab}})^{2}}{v_{0}^{2}}\right]\,,$
(27)

where $\left|\boldsymbol{v}_{\textrm{lab}}\right|\approx 233$ km/s is the
Sun’s velocity with respect to the Galactic rest frame and $v_{0}\approx 220$
km/s is the velocity dispersion of the ALPs.

The $n$-th oscillation frequency is mostly determined by the Compton frequency
$\omega_{c}=m_{\varphi}c^{2}/\hbar$, but shifted due to the relativistic
Doppler effect for particle waves. For $v_{n}/c\ll 1$ the effect can be
approximated by [27]

$\displaystyle\omega_{n}=\omega_{c}\left(1+\frac{\boldsymbol{v}_{n}^{2}}{2c^{2}}\right).$
(28)

The fluctuations of the dark matter field in time arise from the random
directions, phases, and spread in frequencies of constituent waves. See Fig. 7
for a visualisation of the ALP dark matter field amplitude as a function of
time.

Figure 8: Auto-correlation of a simulated ALP dark matter field $\varphi$, the
low-frequency component of its square $\varphi_{s}^{2}$ associated with
stochastic fluctuations, and two gradient components perpendicular and
parallel to $\boldsymbol{v}_{\mathrm{lab}}$: namely
$\boldsymbol{\nabla}_{\perp}\varphi_{s}^{2}$ and
$\boldsymbol{\nabla}_{\parallel}\varphi_{s}^{2}$, where the subscript $s$
denotes the low-frequency component associated with stochastic fluctuations.

####  IV.4.2 Signal model

Assuming the quadratic coupling Hamiltonian [Eq. (9)], the pseudo-magnetic
field is given by $\boldsymbol{\Upsilon}_{q}=\boldsymbol{\nabla}\varphi^{2}$
and the signal observed in a GNOME sensor is
$\propto\hat{\boldsymbol{m}}\cdot\boldsymbol{\Upsilon}_{q}$. As the detectors
have limited bandwidth $\Delta\omega$, for the case where
$\Delta\omega\ll\omega_{c}$ the fast oscillations at $2\omega_{c}$ are not
observable in the measurement. The measured pseudo-magnetic field can be
approximated by the near-dc component

$\displaystyle\boldsymbol{\Upsilon}_{q}=\boldsymbol{\nabla}\varphi^{2}\approx\frac{A^{2}}{2N}\sum_{n,m=1}^{N}\boldsymbol{k}_{nm}\sin(\omega_{nm}t-\boldsymbol{k}_{nm}\cdot\boldsymbol{r}+\theta_{nm}),$
(29)

where $\omega_{nm}=\omega_{n}-\omega_{m}$,
$\boldsymbol{k}_{nm}=\boldsymbol{k}_{n}-\boldsymbol{k}_{m}$, and
$\theta_{nm}=\theta_{n}-\theta_{m}$. Such a signal can be well characterized
by two parameters: coherence time and amplitude.

For a solitary magnetometer or comagnetometer, a UBDM signature would be
difficult to distinguish from comparably large background noise. Our strategy
for analysis of GNOME data, presently underway, is to measure the cross-
correlation between data measured at different stations. For stations with
aligned sensitive axes and separations $\lesssim\lambda{{}_{\mbox{\scriptsize
coh}}}$ and time delays $\lesssim\tau{{}_{\mbox{\scriptsize coh}}}$, UBDM
quadratically coupled to atomic spins could generate a common-mode signal that
might be distinguishable from uncorrelated background noise for sufficiently
strong interactions. Figure 8 shows the auto-correlation function of simulated
ALP signals, which is equivalent to the cross-correlation between stations
with aligned sensitive axes within a coherence patch (ignoring any
contribution from uncorrelated noise). The width of the correlation feature is
$\approx\tau{{}_{\mbox{\scriptsize coh}}}$ in relative time shift units. This
technique of intensity interferometry will enable a search for the quadratic
coupling to ALPs over a mass range $\sim 10^{-14}$ eV – $10^{-9}$ eV using
GNOME [63].

###  IV.5 Solar axion halo

Another theoretical possibility, closely related to the axion star model
described in Sec. IV.2, is that tidal shearing forces may enable astrophysical
bodies such as Earth and the Sun to capture UBDM in their gravitational fields
and form a local halo [140, 141, 142, 143, 144], see Fig. 9. In these models,
the Earth or Sun effectively acts as a seed in the formation process of the
axion star. If such a process occurs, there would be a substantial overdensity
of the ALP field near these bodies as compared to the average dark matter
density as discussed in Refs. [140, 142]. Possible mechanisms for capture of
dark matter by dense astrophysical bodies have been studied, for example, in
Refs. [145, 146, 147, 148, 149, 150], and continue to be actively
investigated.

Figure 9: Upper diagram: Schematic diagram of Earth moving through a solar ALP
halo. The pseudo-magnetic field associated with a solar ALP halo coupling to
atomic spins has both a radial component in the $-\hat{\boldsymbol{r}}$
direction due to the spatial gradient and a transverse component due to the
“ALP wind” directed along the ALP halo’s relative velocity with respect to the
lab frame, $\hat{\boldsymbol{k}}$. Lower plot: example signal due to the
linear spin coupling. Signal is shown over four days, exhibiting an envelope
function due to the daily modulation of the signal due to the rotation of the
GNOME sensor’s sensitive axis with respect to the pseudo-magnetic field
$\boldsymbol{\Upsilon}$. Inset shows the fast oscillation at the Compton
frequency.

####  IV.5.1 Theoretical description

Here we assume the model of a gravitationally bound solar ALP halo described
in Refs. [140, 142], and also assume that the halo is non-rotating (i.e., at
rest with respect to the Sun). For a solar halo, the ALP field amplitude at
the position of Earth exponentially decays over a characteristic length scale
given by

$\displaystyle R_{\star}\approx\frac{\hbar^{2}}{G_{N}M_{\odot}m_{a}^{2}}\,,$
(30)

where $G_{N}$ is Newton’s gravitational constant and $M_{\odot}$ is the Sun’s
mass. For the solar ALP halo to extend to the position of Earth, we require
$R_{\star}\gtrsim 1\,\textrm{AU}$. This imposes the requirement that
$m_{a}c^{2}\lesssim 10^{-14}~{}{\rm eV}$. The ALP field oscillates at
$\approx\omega_{c}$ with a coherence time $\tau{{}_{\mbox{\scriptsize
coh}}}\gtrsim m_{a}R_{\star}^{2}/\hbar$. For $m_{a}c^{2}\lesssim
10^{-14}~{}{\rm eV}$, the coherence time is longer than $\sim 10^{7}$ s;
therefore, we can treat oscillations of the ALP field as effectively single
frequency (i.e., monochromatic). Thus the ALP field can be described as [140,
142]:

$\displaystyle\varphi(\boldsymbol{r},t)\approx\varphi_{0}\cos{\left(\omega_{c}t-\boldsymbol{k}\cdot\boldsymbol{r}+\theta\right)}e^{-r/R_{\star}}\,,$
(31)

where $\varphi_{0}$ is a constant determined by the overall energy density in
the solar ALP halo and $\theta$ is a random phase, constant over the coherence
time and coherence length. For simplicity, in this example we set $\theta=0$.

####  IV.5.2 Signal model

The exotic pseudo-magnetic field due to a solar ALP halo for the linear spin
interaction is given by

$\displaystyle\boldsymbol{\Upsilon}_{l}=\boldsymbol{\nabla}\varphi(\boldsymbol{r},t)=\varphi_{0}e^{-r/R_{\star}}{\left[\boldsymbol{k}\sin{\left(\omega_{c}t-\boldsymbol{k}\cdot\boldsymbol{r}\right)}-\frac{\hat{\boldsymbol{r}}}{R_{\star}}\cos{\left(\omega_{c}t-\boldsymbol{k}\cdot\boldsymbol{r}\right)}\right]}\,,$
(32)

and for the quadratic spin interaction is given by

$\displaystyle\boldsymbol{\Upsilon}_{q}=\boldsymbol{\nabla}\varphi^{2}(\boldsymbol{r},t)=\varphi_{0}^{2}e^{-2r/R_{\star}}{\left[\boldsymbol{k}\sin{\left(2\omega_{c}t-2\boldsymbol{k}\cdot\boldsymbol{r}\right)}-\frac{2\hat{\boldsymbol{r}}}{R_{\star}}\cos^{2}{\left(\omega_{c}t-\boldsymbol{k}\cdot\boldsymbol{r}\right)}\right]}~{}.$
(33)

We note that in both cases there are two components of $\boldsymbol{\Upsilon}$
in the lab frame [142]: (1) a radial component from the spatial ALP gradient
directed toward the Sun’s position and (2) a transverse component along
$\boldsymbol{k}$ due to the ALP wind interaction [6, 5]. Because the relative
velocity of Earth with respect to the Sun is dominated by its orbital motion
($\sim 100$ times faster than the velocity component due to Earth’s rotation
about its axis), it turns out that all GNOME sensors have approximately the
same $\boldsymbol{k}$ and thus see the same field. The amplitudes of the
radial and transverse components of $\boldsymbol{\Upsilon}$ are relatively
constant in time, so the signal observed in a particular GNOME sensor has a
predictable daily modulation due to the time dependence of
$\hat{\boldsymbol{r}}\cdot\hat{\boldsymbol{m}}$ and
$\hat{\boldsymbol{k}}\cdot\hat{\boldsymbol{m}}$ caused by the rotation of
Earth. An example of the time dependence of such a signal is shown in the
lower plot of Fig. 9.

Analysis of GNOME data to search for a solar ALP halo is in progress. The
strategy is to search for a persistent single-frequency signal in cross-
correlation data between stations, and then use the predicted daily modulation
to distinguish between a signal from a solar ALP halo and systematic
backgrounds.

###  IV.6 Exotic Low-mass Fields (ELFs) emitted from black hole mergers

In Ref. [56], a subset of the present authors considered a possibility that
bursts of ultrarelativistic scalar fields can be produced in the course of
some high energy astrophysical event such as a supernova explosion, a binary
black hole merger, or in conjunction with a fast radio burst via some as yet
unknown coherent process666Note that due to the energy-time uncertainty
relation, if the high-energy astrophysical event generating the ELF has
duration $\tau_{0}$, the corresponding spread in energies of the emitted ELF
wave packet is $\gtrsim\hbar/\tau_{0}$. Thus the spread in frequencies at the
source is $\Delta\omega_{0}\gtrsim 2\pi/\tau_{0}$. Our discussion here assumes
coherent production, which requires $\Delta\omega_{0}\ll\omega$. To interpret
results in terms of the model presented here and in Ref. [56], we must
restrict considerations to the regime where the relativistic energy
$\varepsilon=\hbar\omega$ of the ELF satisfies $\varepsilon\gg\hbar/\tau_{0}$,
as noted in Ref. [151]. (see, for example, Ref. [152]). These “exotic low-mass
fields” (ELFs) would thus have large mode occupation numbers, enabling their
treatment as classical waves with oscillation frequency given by
$\hbar\omega=\sqrt{m^{2}c^{4}+p^{2}c^{2}}\approx pc$. Such ELFs can be
searched for with GNOME.

####  IV.6.1 Theoretical description

Consider a scenario where some energy $\Delta E$ is radiated isotropically in
the form of a Gaussian ELF burst with duration $\tau_{0}$ at the source. The
Klein-Gordon equation for $\varphi(r)$ can be solved to obtain

$\displaystyle\varphi(t)\approx$
$\displaystyle\frac{A_{0}}{R}\sqrt{\frac{\tau_{0}}{\tau}}\exp{\left(-\frac{(t-t_{s})^{2}}{2\tau^{2}}\right)}$
$\displaystyle\times\cos{\left(\omega_{0}(t-t_{s})-\frac{\omega_{0}}{4\delta
t}(t-t_{s})^{2}\right)}\,,$ (34)

where $\omega_{0}$ is the central frequency of the wave packet, $\tau$ the
duration of the pulse as measured by sensors on Earth, $t_{s}=R/v_{g}$
represents the transit time of the ELF pulse from a source at distance $R$
from Earth, and $v_{g}$ the group velocity. The delay time between an
electromagnetic or gravitational wave trigger and an ELF pulse is $\delta
t=R/v_{g}-R/c$. In terms of the total energy released, the amplitude $A_{0}$
is given by

$A_{0}\approx\frac{1}{\pi^{1/4}}{\left(\frac{1}{\omega_{0}}\sqrt{\frac{c\Delta
E}{2\pi\tau_{0}}}\right)}\,.$ (35)

Equations (34) and (35) describe a Gaussian wavepacket which disperses with a
chirp rate

$\frac{d\omega(t)}{dt}=-\frac{1}{\tau_{0}\tau}=-\frac{\omega_{0}}{2\delta
t}\,.$ (36)

Notably, the chirp rate is constrained by the central frequency and the delay
from the electromagnetic or gravitational wave trigger.

Figure 10: Example signal of an ELF measured by a GNOME sensor based on Eq.
(34), with $\omega_{0}=2\pi\times(0.5~{}{\rm Hz})$, $\tau=10$ s, and $\delta
t=10$ s.

####  IV.6.2 Signal model

As a consequence of the couplings described in Eqs. (3) to (9), the existence
of scalar fields with a form such as that described by Eq. (34) would lead to
pseudo-magnetic fields in GNOME due to interaction terms proportional to
$\boldsymbol{S}\cdot\boldsymbol{\nabla}\varphi(\boldsymbol{r},t)$ and
$\boldsymbol{S}\cdot\boldsymbol{\nabla}[\varphi(\boldsymbol{r},t)]^{2}$. A
signal due to an ELF event (Fig. 10) would therefore: (1) originate from the
same source as the electromagnetic or gravitational wave trigger; (2) have a
spatial pattern in the GNOME network given by the projection of the ELF
velocity vector on the axes of the sensors in the network; and (3) have a
characteristic chirp pattern given by Eq. (36). Using these characteristics,
we have designed an analysis algorithm to search for ELF signals in GNOME
data. Measurements from the GNOME sensors are projected into the time-
frequency plane with a resolution chosen to match the expected signal
dispersion. We then construct a network-wide test statistic and search for
signals consistent with ELFs in the GNOME data in intervals of time shortly
after reported binary black hole merger and fast radio burst events.

Figure 11: Accessible masses for ELFs from sources located at distances R =
1500 Mpc and R = 0.01 Mpc. The left and right edges of the bands are set by
the delay times $10\,\rm{s}<\delta t<10\,\rm{hr}$.

We show in Fig. 11 the range of masses that are accessible by GNOME given the
bandwidth of the magnetometers, assuming an observation time of 10 hours after
the trigger. Thus GNOME can act as a “telescope” to detect ELF signals from
sources that generate ALP bursts of sufficient intensity.

###  IV.7 Other potential candidates

There are many other exotic physics scenarios that can potentially be searched
for using data from the GNOME experiment, demonstrating the versatility of the
general approach. For example, as already noted, it may be that there are
relatively strong interactions between particles in the dark sector. Such
interactions could cause dark matter particles to coalesce into large
composite “blobs” [33]. In contrast to the axion star, Q-ball, and solar
axion-halo scenarios already considered above, which are composite states of
ultralight particles, there could be dark matter blobs formed from heavy
fermions or bosons that exert long-range forces mediated by other fundamental
constituents of the dark sector [33, 153]. Other possible search targets for
the GNOME experiment are ALP waves emanating from black hole superradiance
[154] or from the collapse of axion stars [152]. In related work, unshielded
magnetometer networks have been used to search for signals from dark (or
hidden) photons [155, 156] and axions [157].

Yet another possible search target for GNOME are UBDM streams, in particular
those gravitationally focused by other bodies in the solar system.
Cosmological simulations indicate that up to $10^{12}$ small-velocity-
dispersion (compared to that in the SHM) streams of DM may be present in the
vicinity of the solar system [158]. The deflection experienced by these
particles depends on the mass distribution of the source of gravitational
focusing, is independent of the DM particle mass, and is inversely
proportional to the square of its velocity. For UBDM, its wave nature has to
be considered for a full description of the spectral signatures due to
gravitational focusing [159] Generally, for deflection of particles by the
gravitational potential of the Earth, efficient focusing on Earth’s surface
(where GNOME operates) occurs for particles with velocities of approximately
10 to 20 km/s [160, 161]. In this case, flux enhancements of up to $10^{9}$
are predicted. Gravitational deflection by other objects in the solar system,
such as Jupiter, can also generate flux enhancements on Earth. This requires
larger stream velocities of approximately $10^{-3}$ c and results in smaller
enhancements of only $10^{6}$ [162]. Again, the networked approach of GNOME
offers several advantages when searching for such gravitationally focused
streams. Namely, the small size of such stream foci, on the order of a few km
for deflection in Earth’s gravitational potential and much less for that of
Jupiter, leads to transient signals lasting only about 10 s. This can be
difficult to distinguish from noise when only a single detector is employed.
For long streams, there is a chance of the detecting the enhanced DM flux
daily with one or more detectors over the course of several weeks. Obviously,
the likelihood of encountering the high flux regions increases with the number
of sensors in a network. A network also allows for precisely determining the
spatial direction of the stream, which can help to better understand the
origins of such streams when detected, although this can be a complex endeavor
as upstream sources of gravitational deflection also have to be considered.
Furthermore, sensors at approximately the same latitudes could produce
correlated signals from the same focused-DM source (Fig. 12). The geographical
locations of GNOME sensors make it viable for such correlations.

Figure 12: Example of a low velocity UBDM stream (red) gravitationally focused
into a small region on Earth’s surface, the trajectories are approximately
following predictions made in Ref. [160]. As Earth rotates around its axis,
the focus follows a path (yellow) along a fixed latitude.

While GNOME is primarily a tool for detection of transient “new-physics”
signals, its sister experiment, the Cosmic Axion Spin-Precession Experiment
(CASPEr) [9, 11, 16], is searching for oscillating (but not necessarily
transient) effects of galactic dark matter and dark matter concentrated in the
Sun- and Earth-bound halos. CASPEr operates simultaneously in two locations,
Boston University (USA) and the Helmholtz Institute, Johannes Gutenberg
University, Mainz (Germany). The central idea of CASPEr is that it is a
nuclear magnetic resonance (NMR) experiment, where the role of the transverse
oscillating magnetic field “$\boldsymbol{B}_{1}$” is relegated to the ambient
Axion/ALP field either through its coupling to gluons (CASPEr-electric) or via
its gradient interactions [see Eqs. (6) and (9); CASPEr-gradient]. While
CASPEr will reach up to Compton frequencies $\approx$600 MHz, it has already
ventured into the much lower-frequency range [12, 13], from near DC to several
hertz. The low-frequency range has also been explored by various other
experiments [10, 17, 18, 26, 20].

The Advanced-GNOME comagnetometers [32, 84] are ideally suited to contribute
to “CASPEr-like” searches in the Compton frequency range from near-DC to
several hundred hertz, with the sensitivity that is estimated to be
competitive to the best previous searches. The results of the first
experiments in this direction may be expected already at the early stages of
Advanced GNOME, as a search in this mode does not require the full network
capabilities. However, as with GNOME, we expect that the network will be
immediately useful in vetoing spurious signals from various laboratory
inferences, which will likely be different in different laboratories.
Moreover, the network offers interesting possibilities of correlating the
stochastic behavior of the signal in different stations that may offer an
additional tool for discriminating the new-physics signals.

##  V Conclusion and outlook

The idea of GNOME is to carry out synchronous measurements using optical
atomic magnetometers and comagnetometers operating within magnetically
shielded environments in distant locations. The GNOME experiment searches for
a class of signals different from that probed by most other experiments,
namely transient and stochastic effects that could arise from ALP fields of
astrophysical origin passing through Earth during a finite time. By analyzing
the correlation between the signals from multiple, geographically separated
sensors, it is possible to probe a wide variety of exotic physics scenarios.

When the idea of GNOME was conceived, ALP domain walls were the sole targets
of the search [31, 28]. As the collaboration expanded and matured, we
gradually realized that there are many more interesting physics scenarios that
can be investigated with this tool, as, we hope, we have been able to convince
the reader of this “GNOME science-case” article. Extrapolating from the first
10 years of the GNOME collaboration’s work, we may only expect the science
case to continue its expansion, hopefully leading to a discovery of “new”
physics.

The advent of GNOME was soon followed by the establishment of networks of
atomic clocks [89] introduced to search for scalar UBDM whose effect is to
produce apparent oscillation of fundamental constants, rather than causing
spin precession. At the same time, it was realized that certain UBDM
candidates like the relaxion [163, 164] may have mixed intrinsic parity and
displaying both the scalar and pseudoscalar properties. This provides a
motivation for building hybrid-sensor networks (for example, (co)magnetometer-
clock networks) to cross-correlate the signals and enhance the background-
suppression capabilities. We expect this cross-network approach to become
commonplace in future work. Moreover, GNOME is planning future science runs to
be synchronized with gravitational-wave detector networks such as LIGO/VIRGO
in order to carry out “exotic” multimessenger astronomy [56].

It appears that the science case for search networks such as GNOME is mostly
limited by our current scientific horizons and there are, in fact, many other
interesting and unexpected signals we could be looking for, had we better
imagination. But could we perhaps look for such signals before we realize what
they are? It appears that modern machine-learning approaches indeed offer us
such an exciting opportunity. It might work along the following lines: a
machine-learning system is trained on the data from GNOME (and/or an expanded
hybrid-sensor network) and establishes the “normal” dynamic state of the
network. It is then further trained on a variety of new-physics scenarios. As
a result of the training that can continue for the lifetime of the experiment,
the artificial intelligence (AI) system would be able to recognize unusual
events or patterns on the network. While this may look like a remote prospect,
one should recognize that the task at hand is not too different from that of
detection of unidentified threats, which is routinely done at present by
automated security systems.

In conclusion of our overview of the “GNOME science”, we mention that, apart
from fundamental physics, magnetometer networks may also be useful in more
“down-to-Earth” applications such as, for example, the study of magnetic
signal patterns in urban environments and what they reveal about cities [165,
166].

## Acknowledgements

The authors are grateful to Konstantin Zioutas for bringing the gravitational
focusing effects to the attention of the GNOME collaboration. The GNOME
collaboration thanks the Mainz Institute for Theoretical Physics (MITP) of the
Cluster of Excellence “Precision Physics, Fundamental Interactions, and
Structure of Matter” (PRISMA+ EXC 2118/1) for its hospitality and its partial
support during the completion of this work. This work was additionally
supported by the U.S. National Science Foundation under grants PHY-1307507,
PHY-1707875, PHY-1707803, PHY-2110370, PHY-2110388, and PHY-2207546; by the
German Research Foundation (DFG) under grant no. 439720477 and within the
German Excellence Strategy (Project ID 39083149); by COST (European
Cooperation in Science and Technology); ZDG acknowledges institutional funding
provided by the Institute of Physics Belgrade through a grant by the Ministry
of Science, Technological Development and Innovations of the Republic of
Serbia. The work of SP and GL is supported by the National Science Centre of
Poland within the Opus program (grant 2019/34/E/ST2/00440) and in part by the
Excellence Initiative – Research University of the Jagiellonian University in
Kraków. The work of YVS was supported by the Australian Research Council under
the Discovery Early Career Researcher Award DE210101593. The work of AD was
supported in part by the Heising-Simons Foundation. The work of JE was
supported by the World Premier International Research Center Initiative (WPI),
MEXT, Japan and by the JSPS KAKENHI Grant Numbers 21H05451 and 21K20366. RF
and DFJK acknowledge support by the NSF-BSF grant No. 2016635.

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