# Axion Dark Matter: What is it and Why Now?

Francesca Chadha-Day1,∗, John Ellis2,3,∗, David J. E. Marsh2,∗

###### Abstract

The axion has emerged in recent years as a leading particle candidate to
provide the mysterious dark matter in the cosmos, as we review here for a
general scientific audience. We describe first the historical roots of the
axion in the Standard Model of particle physics and the problem of charge-
parity invariance of the strong nuclear force. We then discuss how the axion
emerges as a dark matter candidate, and how it is produced in the early
Universe. The symmetry properties of the axion dictate the form of its
interactions with ordinary matter. Astrophysical considerations restrict the
particle mass and interaction strengths to a limited range, which facilitates
the planning of experiments to detect the axion. A companion review discusses
the exciting prospect that the axion could indeed be detected in the near term
in the laboratory.

A review prepared for Science Advances

1 Institute for Particle Physics Phenomenology, Department of Physics, Durham
University,
Durham DH1 3LE, United Kingdom
2 Theoretical Physics and Cosmology Group, Department of Physics, King’s
College London, Strand,
London WC2R 2LS, United Kingdom
3 Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland;
National Institute of Chemical Physics & Biophysics, Rävala 10, 10143 Tallinn,
Estonia
∗ To whom correspondance should be addressed

KCL-PH-TH/2021-20, CERN-TH-2021-045, IPPP/20/91

##  1 Introduction

The Standard Model of particle physics provides a successful description of
the visible matter in the Universe, from stars to the inner workings of atoms
and nuclei. It categorises the fundamental constituents of matter, the quarks
and leptons, and the electromagnetic, weak and strong nuclear forces between
them. The Standard Model is a mathematically consistent quantum field theory
that allows theorists to calculate accurate predictions, which have in many
cases been verified experimentally with a precision below the per-mille level
at the Large Hadron Collider (LHC) and other particle accelerators. The
crowning success of the Standard Model was the discovery of the Higgs boson in
2012 [1, 2], a particle of a novel type whose existence was predicted in 1964
[3] in order to solve theoretical problems associated with the masses of
vector bosons, see also [4]. Precise experimental measurements in the decades
preceding 2010 verified many other predictions of the Standard Model,
including the existence and mass of the top quark. They also confirmed the
necessity of the Higgs boson and enabled its mass to be estimated numerically.
Following its discovery, experiments have confirmed that it has zero spin,
unlike any fundamental particle known previously, and interactions with other
particles that are proportional to their masses, as predicted by the Standard
Model.

Despite these manifold successes, particle physicists are restless and
dissatisfied with the Standard Model, because it has many theoretical
shortcomings and leaves many pressing experimental questions unanswered. Here
we review the axion [5, 6, 7], a hypothetical particle beyond the Standard
Model that addresses some of these unresolved issues and is the focus of
growing experimental interest.

One of these issues is how matter and antimatter are (and are not)
distinguished at the level of fundamental particles. For each matter particle
in the Standard Model, special relativity and quantum mechanics require [8]
the existence of a corresponding anti-matter particle with identical mass and
spin but opposite charge. Uncharged particles such as photons may be their own
anti-particles. The existence of every anti-particle in the Standard Model has
been confirmed experimentally by observing their production in the collisions
of ordinary particles. However, astrophysical and cosmological observations
tell us that most of the visible material in the Universe is composed of the
same matter particles as us on Earth, and that there are no large
concentrations of antimatter. This cosmological matter-antimatter asymmetry is
thought to be due to differences in the interactions of elementary particles
and antiparticles [9], which violate certain symmetries that distinguish
particle from antiparticle. These are charge conjugation (denoted by $C$) and
its combination with parity reversal (denoted by $P$, also known as spatial
inversion symmetry). A form of $CP$ violation was discovered in the laboratory
over 50 years ago [10], and by now it has been observed in many decays via the
weak interactions. These observations can be accommodated within the Standard
Model with six quarks [11], albeit without a profound explanation. However,
this Standard Model mechanism is insufficient to explain the cosmological
matter-antimatter asymmetry and, moreover, it is a puzzle that $CP$ violation
does not appear also in the strong nuclear interaction: this is known as the
“strong-$CP$ problem”. As we discuss below, this puzzle could be resolved via
the hypothetical axion particle. In this review, we reserve the term ‘axion’
for the particle arising from the solution to the strong-$CP$ problem
described below, also known as the ‘QCD axion’. This will be the main topic of
this review. We will also touch briefly on ‘axion-like particles’, which do
not solve the strong-$CP$ problem. As a result, the properties of axion-like
particles are less theoretically constrained.

The failure to explain the cosmological matter-antimatter asymmetry and the
$CP$ symmetry of the strong force are just two of the Standard Model’s
failures to describe the Universe. Some might regard them as “cosmetic”
problems, since they could be solved by fiat within the context of the
Standard Model. There is, however, a much more pressing existential problem.

Multiple astronomical observations tell us that the visible matter described
by the Standard Model provides just a small part of the total density of the
Universe. There is much more invisible “dark matter” out there [12] that
remains to be explained. This is gross inadequacy of the Standard Model, since
the dark matter cannot be explained without appealing to substantial amounts
of new physics. The existence of dark matter is inferred from its
gravitational effects, and astrophysical observations suggest that it is
“cold”, i.e., it has been moving very slowly for much of the history of the
Universe, and there are experimental upper limits on how strongly it interacts
with the visible matter. Moreover, the agreement between independent
cosmological observations (including galaxy clustering and the temperature
anisotropies in the cosmic microwave background, CMB, radiation [13]) and the
corresponding theory of structure formation also tells us that dark matter
must have been present since early in the history of the Universe, a year or
so after the Big Bang or even earlier. Jim Peebles’ share of the 2019 Nobel
Prize in Physics was for the theory of cosmological structure formation behind
this understanding, and the evidence it provides for the existence of cold
dark matter [14]. Beyond these basic facts relating to the temperature and
longevity of the dark matter, we have very little information about its nature
and properties. It may well consist of one or more types of fundamental
particle, though part or all of it might consist of macroscopic lumps of some
invisible form of matter such as black holes. The axion has become one of the
prime particle candidates for providing dark matter.

Historically, the first type of particle dark matter that was suggested was
some sort of weakly-interacting massive particle, abbreviated to WIMP, such as
a massive neutrino. Minimal versions of this hypothesis within the Standard
Model were excluded experimentally, leading particle theorists to consider
possible candidates in extensions of the Standard Model, e.g., based on
supersymmetry [15]. However, this is not the only possibility. If the dark
matter particle is a low-mass boson, it may populate the Universe in a
coherent wave-like state that is also slow-moving and hence “cold”. The axion
is the prototype for such wave-like dark matter.

For many years following the proposals of the axion and WIMP in the late
1970s, axion dark matter was a minority interest among particle physicists,
who were mainly focused on high-energy collider physics. Their primary
motivation was the search for the Higgs boson, but WIMP searches were also an
important consideration that could often be pursued in parallel with the Higgs
search. The search for the Higgs boson culminated in its discovery at the LHC
in 2012, but WIMP searches at the LHC and elsewhere [16] have proved fruitless
so far. These two developments have contributed to growing theoretical and
experimental interest in the axion. On the one hand, the Higgs is an existence
proof for an apparently elementary spin-zero boson undergoing spontaneous
symmetry breaking, something that was somewhat controversial and unknown
previously. This proof of principle invalidated one possible theoretical
prejudice against the existence of the axion, which shares these properties
with the Higgs. Perhaps more importantly, the non-appearance of any WIMP at
the LHC and in other direct searches has diminished enthusiasm for that
candidate for particle dark matter, though some hope still springs eternal. In
parallel, there has been growing realization that axion-like particles (ALPs)
appear quite generically in extensions of the Standard Model, e.g., those with
their roots in string theory, see Fig. 1. For these reasons, the axion has now
become a favoured theoretical candidate for dark matter, motivating this
review.

A companion article reviews the prospects for axion and ALP searches driven by
experimental and technological advances at the so-called “precision frontier”
of particle physics. These advances make axion dark matter a viable candidate
for discovery, which until recently appeared impossible, and are important
factors driving the growing interest in the field.

What is known about dark matter (DM)? • Cosmic density (strong evidence:
cosmic microwave background anisotropies [13]). Expressed as a fraction of the
total density of the Universe, DM makes up 26% of the Universe, compared to 6%
in ordinary matter, and 68% in vacuum energy. • Local density (strong
evidence: Milky Way stellar motions). The local density of dark matter is
around 0.3 to 0.4 GeV cm-3, equivalent to one proton every few cubic
centimetres, or one solar mass per cubic lightyear. The density is measured on
average over a relatively large fraction of the galaxy. The actual density at
the precise location of the Earth could be significantly different. This is
particularly relevant to axions, as discussed below. The local density is
around $10^{5}$ times the average cosmic density. • Local velocity dispersion
(strong evidence: Milky Way stellar motions). The velocity dispersion of DM is
around $\sigma_{v}=200\text{ km s}^{-1}$, and our local motion with respect to
the galactic rest frame is in the direction of the constellation Cygnus. • No
preferred galactic length scale (strong evidence: galaxy clustering and
evolution). DM must be non-relativistic ($v\sim c$ would allow DM to move
significant distances during galaxy formation), and have negligible pressure
(which would imprint sound waves during galaxy formation). This discounts
standard model neutrinos and other “hot” or “warm” DM. For bosons, the de
Broglie wavelength (which can be modelled as an effective pressure) must be
small compared to the galaxy clustering scale. • Early appearance of DM
(strong evidence: galaxy clustering). DM had to be present, and gravitating,
in the Universe long before the cosmic microwave background formed, and its
gravitational influence began before the Universe was one year old. For light
bosonic DM (like the axion) this corresponds to the latest epoch of particle
creation ($t_{\textup{cold}}$ in Fig. 4). • Lack of significant interactions
(strong evidence: the “Bullet Cluster” [17]). DM cannot interact with itself
or ordinary matter too strongly.

Figure 1: Many extensions of the Standard Model predict additional massive
bosons, beyond the $W,Z$ and Higgs bosons of the Standard Model. They might be
scalar (even under both P and CP transformations), pseudoscalar (odd under
both P and CP) or vector particles. The prototype for a pseudoscalar boson is
the axion, which is “ultralight” with mass $m\ll 1\text{ eV}$. Other proposals
include pseudoscalar axion-like particles (ALPs), hidden photons, and scalar
particles such as moduli and dilatons. Any of these might provide the
astrophysical dark matter.

##  2 Enter the Axion

###  2.1 The Strong $CP$ Problem

The strong $CP$ problem may be posed simply as follows: why have we observed
no electric dipole moment (EDM) for the neutron? The neutron EDM is a quantity
that would violate Charge Parity ($CP$) symmetry. This is in contrast to most
of the laws of fundamental physics, which look the same after a $CP$
transformation.

To understand the effect of a $CP$ transformation on the neutron, we can
instead think about how a neutron would behave under a time reversal
transformation. This is because if quantum field theory is Lorentz invariant
(i.e., if it is consistent with Einstein’s theory of Special Relativity), then
all particles and processes must be invariant under the $CPT$ combination of
discrete transformations, where $T$ reverses the arrow of time (this is known
as the $CPT$ theorem of quantum field theory). So, if we find something that
violates $T$ symmetry, it must also violate $CP$ symmetry in such a way that
the combination $CPT$ is not violated.

As an illustration, let us consider the effect of time reversal on the energy
of a neutron with an electric dipole moment ${\bf d}$ in an electric field E.
This energy is given by the Hamiltonian

$\mathcal{H}=-{\bf d}\cdot{\bf E}.$ (1)

If the neutron has a permanent EDM, this EDM must be aligned with the
neutron’s spin. (This follows from the Wigner-Eckart theorem in quantum
mechanics.) When we reverse the direction of time, the direction of the
neutron’s spin is reversed, and therefore the direction of the EDM must also
be reversed under a time reversal transformation $T$. However, the electric
field direction remains the same under a time reversal transformation. So we
can see from equation (1) that if the neutron has a non-zero permanent EDM,
then the energy of a neutron in an electric field will change under a $T$
transformation. Hence a neutron electric dipole moment would violate $T$
symmetry and therefore also $CP$ symmetry.

Figure 2 shows the $P$ and $T$ transformations of a permanent electric or
magnetic dipole moment (which must be aligned with a particle’s spin), of an
electric field and of a magnetic field. We see that an electric dipole moment
violates both $T$ and $P$ symmetry, as the spin transforms with the opposite
sign to the electric field under both these transformations. Conversely, a
magnetic dipole moment does not violate $P$ or $T$ symmetry as the spin
transforms in the same way as the magnetic field.

Figure 2: Effect of time and parity transformations an electric or magnetic
dipole moment (proportional to the spin shown by the red arrow), on an
electric field and on a magnetic field.

Some experimentally observed processes within the Standard Model do violate
$CP$ symmetry, i.e., the process and the corresponding CP transformed process
occur at different rates. These $CP$-violating processes are generated by the
weak interaction. However, $CP$ violation has not been observed in any
experiment on the strong interaction, which describes the forces that bind
quarks together to form protons, neutrons and other hadrons in the theory of
Quantum Chromodynamics (QCD). This is very surprising because the Standard
Model predicts that the structure of the QCD vacuum itself should violate
$CP$. The $CP$ violation of the QCD vacuum can be parameterized by an angle
denoted $\theta$.

Furthermore, a neutron EDM should also receive a contribution from the effect
of the weak interaction (the interaction responsible for radioactive decays)
on the quarks. Thus the total neutron EDM can be expressed using the total
$CP$-violating angle $\bar{\theta}$:

$\bar{\theta}={\rm strong}\,{\rm interaction}\,{\rm contribution}+{\rm
weak}\,{\rm interaction}\,{\rm contribution}$ (2)

Within the Standard Model, the electric dipole moment of the neutron is
proportional to $\bar{\theta}$, with $|{\bf d}|\sim 3.6\times
10^{-16}\bar{\theta}\,e\,{\rm cm}$. The two contributions making up
$\bar{\theta}$ are entirely unrelated within the Standard Model and are simply
set by hand. Note that, although $\bar{\theta}$ is the sum of these two
parameters, only the total $\bar{\theta}$ is experimentally observable. There
is no way to measure the strong interaction and weak interaction contributions
to $\bar{\theta}$ individually. Experiment tells us that $|{\bf d}|<1.8\times
10^{-26}e\,{\rm cm}$ [18], implying that $\bar{\theta}<5\times 10^{-11}$.
Thus, a very precise cancellation between unrelated parameters is required to
explain the observations. This could merely be a coincidence, or it could be a
first hint of new fundamental physics.

###  2.2 The Axion Solution

Strong $CP$ Problem and the Axion • The Standard Model contains a constant,
$\bar{\theta}$, which is known to be an angle and thus take values between 0
and $2\pi$. Symmetry specifies no preferred angle. • The value of this angle
determines the neutron electric dipole moment, $|{\bf d}|=3.6\times
10^{-16}\bar{\theta}\,e\,{\rm cm}$. Measurements are consistent with
$\bar{\theta}\lesssim 10^{-10}$, which suggests some missing symmetry
principle. • The axion theory introduces a new field, $a(t,x)$, such that
$\bar{\theta}\propto a(t,x)$, and for which the value $a=0$ is energetically
favourable.

The axion is the most popular solution to the strong $CP$ problem. It appeared
first in a model proposed by Roberto Peccei and Helen Quinn in 1977 [5]. Their
idea was to promote $\bar{\theta}$ to a dynamical quantity, rather than simply
a constant parameter as in the Standard Model. In practice, this means adding
a new field - the axion field - to the Standard Model that couples to the
strong nuclear force in the same way as $\bar{\theta}$. As we shall see, the
axion field relaxes to a value such that $\bar{\theta}$ is zero and hence the
total neutron EDM vanishes.

The behaviour of particles is determined by their symmetries. Emmy Noether
showed that these symmetries correspond to conservation laws, such as
conservation of energy or charge. In the Standard Model, symmetries lead to
forces between particles: the force “communicates” the symmetry from place to
place between particles. Forces are in turn mediated by bosonic particles: the
force is a field, and quantum mechanics associates a particle with every field
(e.g. the photon for the electromagnetic field and electromagnetic gauge
symmetry). Thus, to add the axion to the Standard Model, we introduce a new
symmetry, called the Peccei-Quinn (PQ) symmetry [5], which is a global axial
$U(1)$ symmetry. ‘Global’ means that the symmetry transformation is the same
everywhere and for all time - in contrast to the gauge symmetries associated
with interactions in the Standard Model, whose transformations can be made
independently at different points and at different times. ‘Axial’ means that
the symmetry transformation acts differently on left-handed and right-handed
particles. Roughly speaking, a particle’s handedness tells us about the
relative orientation of its spin and its velocity. Finally, a $U(1)$ symmetry
transformation is one that is mathematically equivalent to a rotation about a
single axis.

Introducing the PQ symmetry corresponds to introducing a new bosonic field
beyond the single Higgs boson of the Standard Model. At high temperatures,
such as in the early universe, the PQ symmetry is evident in all particle
interactions. However, there is a phase transition that “hides” the PQ
symmetry when the temperature falls low enough (the bosonic force carriers of
the symmetry become heavy and cannot be thermally excited). Similar phenomena
of “spontaneous symmetry breaking” are central to our understanding of many
areas of macroscopic physics, such as superconductivity, and there are also
examples in particle physics, such as the Higgs mechanism for generating
particle masses.

To visualize spontaneous symmetry breaking, consider a ball rolling in the
“sombrero” potential shown on the left of Fig. 3. If the ball has enough
energy (as at high temperature), it is able to roll over the hill in the
centre of the potential and occupies equally all areas of the circular valley
in the “brim” of the potential. Now imagine that the ball loses energy (as at
low temperature), slows down and comes to rest. It will choose at random to
sit in one particular position in the potential well - even though every
position in the circular minimum of the potential well is exactly equivalent.
This arbitrary choice (a “choice” made the random thermal state of the early
Universe) is spontaneous symmetry breaking. Notice now that, while it would
take a lot of energy to get the ball over the potential hill again, we can
push the ball around the circle of the potential well with the smallest of
nudges. This is a generic feature of spontaneous symmetry breaking. In
particle physics, it corresponds to the appearance following spontaneous
symmetry breaking of a massless particle, which is called a Nambu-Goldstone
boson [19, 20]. The Nambu-Goldstone boson of the spontaneously broken Peccei-
Quinn symmetry is the axion [6, 7]. It is represented by a field, $a$, which
is proportional to the problematic $\bar{\theta}$ angle of the strong-$CP$
problem, making the angle dynamical rather than a fixed and mysterious
constant. (Achieving this remarkable theoretical sleight of hand to make
$a\propto\bar{\theta}$ is described briefly in what follows, and in more
detail in the Supplemental Material.)

Figure 3: “Sombrero” potential of the Peccei-Quinn field $\Phi$ is shown
schematically before (left) and after (right) the QCD phase transition. The
axion corresponds to the angular direction of this potential, shown by the
orange line. The state of the field is given by a point in the potential. Low
energy configurations are favoured. For illustration, the potential on the
right is shown for a scenario with a large amount of PQ symmetry breaking.
More details are given in appendix A.2.

We have not yet solved the Strong $CP$ problem, as the massless axion field
could a priori take any value. The next part of the story is the QCD phase
transition (strictly speaking, a cross over) that occurs as the temperature
falls. When the temperature is sufficiently low, QCD becomes strongly-coupled
and confines quarks and gluons into the bound-state protons, neutrons and
other hadrons that we see today. This phase transition breaks the PQ symmetry
by distorting the sombrero potential as seen on the right in Fig. 3. The
potential now has discrete minima and the energy is minimised by the axion
field taking the value of one of these minima. Thus, after the QCD cross over,
the axion field rolls to the newly created minimum point, which is where the
contribution of $\bar{\theta}$ to the neutron EDM vanishes, setting the net
neutron EDM to zero. (The reason this minimum has the right $CP$ properties is
discussed in the Supplemental Material.) Notice that now, to make the ball
move around the sombrero potential we would need to push it away and up from
its minimum point. The energy required to move the classical axion field a
small distance away from the minimum can be modelled as an effective potential
$V(a)=m_{a}^{2}a^{2}/2$. Upon quantisation, we interpret the parameter $m_{a}$
in the classical potential as the mass of the axion particle.

The axion mass can be computed in terms of well-understood physics of the
strong nuclear force by considering the axion mixing with the neutral pion - a
bound state of quarks with the same quantum numbers as the axion. The axion’s
interactions with the neutral pion mean that the pion’s mass generates a small
mass for the axion - this effect is only possible because the axion and the
pion have the same quantum numbers. This leads to the following relation for
the axion mass $m_{a}$:

$m_{a}f_{a}\sim m_{\pi}f_{\pi},$ (3)

where $f_{a}$ is proportional to the energy at which the PQ symmetry is
spontaneously broken, $m_{\pi}$ is the pion mass, and $f_{\pi}$ is a constant
that is known from the rate of decay of the pion via the weak interaction. (A
deeper understanding of the origin of the axion mass relies on the theory of
“instantons”, strong-interaction effects that are discussed briefly in the
Supplemental Material.) Using the experimental measurements of these pion
properties we can calculate the axion mass. The axion mass, as most scales in
particle physics, is typically given in electronvolts (eV), where $1\text{
eV}\approx 1.8\times 10^{-36}\text{ kg}$ which assumes $\hslash=c=1$, but we
restore units of $\hslash$ and $c$ in formulae for clarity. Another useful
conversion is between mass in electronvolts and Compton frequency, such that
$0.25\text{ Hz}\approx 10^{-15}\text{ eV}$. The axion mass is:

$m_{a}=(5.70\pm 0.007)\,\mu\text{eV}\left(\frac{10^{12}\text{
GeV}}{f_{a}}\right)\,,$ (4)

where the error includes experimental and theoretical contributions appearing
in a detailed calculation of axion-pion mixing [21]. The constant $f_{a}$ is
related to the scale of spontaneous breaking of the PQ symmetry (see
Supplemental Material), and is measured in GeV. What values could this take?
The electroweak scale, $\approx 250\text{ GeV}$, was one natural choice, but
is excluded experimentally. Other scales in particle physics include the grand
unified scale, ${\cal O}(10^{16})\text{ GeV}$, and the Planck scale, $\sim
10^{19}\text{ GeV}$, which is the highest scale where ordinary quantum field
theory could possibly remain valid before quantum gravity becomes important.
These considerations give only very rough guidance as to the value of $f_{a}$
and the mass of the axion: below we narrow their possible ranges.

The axion mechanism may be implemented in a wide variety of extensions to the
Standard Model, which are the subject of much current research [22]. In any
axion model, we must introduce a new complex field $\Phi=\chi{\rm
e}^{i\theta}$, which gains a non-zero vacuum expectation value that breaks the
PQ symmetry spontaneously. After this spontaneous symmetry breaking, the axion
is related to the phase of $\Phi$ by $a=Nf_{a}\theta$, where $N$ is an
integer, the “colour anomaly”, which varies depending on the detailed
realisation of the Peccei-Quinn mechanism. Figure 3 is shown for $N=4$.

For the axion to solve the strong $CP$ problem, the model must also include
quarks that are charged under the PQ symmetry, which in turn mediate an
interaction between the axion and the gluon force carriers of the strong
nuclear force. There are two ways of achieving this. In one class of axion
models, the Standard Model quarks are charged under the PQ symmetry [23, 24]:
these models have $N=6$. This means we have to add an extra Higgs doublet to
the model in order to allow all of the Standard Model interactions to obey the
PQ symmetry. In another class of models, we instead add extra heavy,
electrically neutral quarks to the theory [25, 26]. Only these extra quarks
are charged under the PQ symmetry: the canonical version of this model has
$N=1$. The precise values of the axion’s mass and interaction strengths depend
on these details of the model (see Supplemental Material).

###  2.3 Interactions

The axion does not interact only with quarks and gluons, but also with the
other particles in the Standard Model. We know that these interactions must be
very weak for the axion to have evaded detection so far.

Particle interactions are governed by their symmetries, and the axion’s
interactions are set by its pseudoscalar nature. A pseudoscalar field changes
sign under a parity transformation, i.e., when looking at the Universe in a
mirror. If we assume that the overall interaction is unchanged by a parity
transformation, we find that only certain interactions are allowed for a
pseudoscalar particle. These can be expressed (schematically) by the non-
relativistic Hamiltonian:

$\mathcal{H}=\sqrt{\frac{\epsilon_{0}}{\mu_{0}}}g_{a\gamma\gamma}\int a{\bf
E}\cdot{\bf B}dV+g_{aff}\hslash c\triangledown a\cdot\hat{{\bf
S}}+\sqrt{\epsilon_{0}(\hslash c)^{3}}g_{\rm EDM}a\hat{{\bf S}}\cdot{\bf E},$
(5)

where $a$ is the axion field measured in units energy; $g_{a\gamma\gamma}$ is
the axion’s coupling to photons, $\gamma$ (i.e. to electromagnetism), measured
in units of inverse energy, $g_{aff}$ is the axion’s coupling to matter
particles, which are fermions, $f$ (the couplng depends on the particle in
question, we write just one case for simplicity); $g_{\rm EDM}$ is the
strength of a nuclear EDM induced by the axion; ${\bf E}$ and ${\bf B}$ are
the electric and magnetic fields, and $\hat{{\bf S}}$ is the direction of the
spin of the matter particle in question. $\epsilon_{0}$ and $\mu_{0}$ are the
permittivity and permeability of free space respectively - constants
associated with electromagnetism, $\hslash$ is the reduced Planck’s constant,
which parameterizes the size of quantum effects, and $c$ is the speed of
light.

These interactions are very different from the interactions of a scalar
particle, which can couple directly ${\bf E}^{2}-{\bf B}^{2}$ (the scalar
Maxwell term) and to the masses of matter particles. This means that light
scalar fields can mediate extra long-range forces, and their interactions with
Standard Model particles are therefore very tightly constrained by the non-
observation of such extra forces. In contrast, the couplings of pseudoscalar
particles to ${\bf E}\cdot{\bf B}$ and to the spins of matter particles [27,
28] make them much harder to detect.

What would the axion’s interactions with photons and with matter particles
look like experimentally? Via its interaction with ${\bf E}\cdot{\bf B}$,
axion DM would look like an additional electrical current or anomalous
magnetic field [29]. More generally, Maxwell’s equations of electromagnetism
are modified by the addition of the axion field. Further details on how axions
modify Maxwell’s equations of electromagnetism are discussed in the companion
experimental review. Via its coupling to nuclear and electron spins, axion DM
would cause these spins to precess, as they would in a magnetic field, but now
with an anomalous magnetization caused by the invisible presence of the axion
field [30, 31].

What are the strengths of the axion’s interaction with Standard Model
particles? How large are the coupling constants $g_{i}$ in the Hamiltonian Eq.
(5)? We know that the interactions must be very weak, or we would have found
axions already. In fact, we expect the axion’s couplings to be inversely
proportional to the scale of symmetry breaking:

$g_{i}\sim\frac{1}{f_{a}}\,,$ (6)

This is a consequence of one of the most fundamental ideas in particle
physics, namely Effective Field Theory [32], which tells us to expect the
axion’s interactions to scale inversely with the energy scale at which the
symmetry giving rise to it originates. As this energy $f_{a}$ could be very
high, the axion’s interactions could be very weak, as required by the
experimental constraints. Comparing equations (3) and (6), we see that the
axion’s couplings obey:

$g_{i}\propto m_{a}.$ (7)

This is true for most models of the QCD axion, whose couplings are generally
proportional to its mass, though this relationship can be broken in some
specific models of the QCD axion. Also, as we will see later, the axion is
just one particle in the broader class of ALPs discussed in more detail below.
Such ALPs need not in general solve the strong $CP$ problem or couple to
gluons. This means their mass could take any value and need not be
proportional to their couplings, and the constants of proportionality differ
wildly for different ALPs. ALPs could therefore be very weakly coupled and
hence extremely difficult to detect experimentally. Alternatively, Nature may
provide us with more strongly coupled ALPs that can be detected more readily,
as discussed below. By contrast, for a canonical QCD axion of a given mass, we
can predict the approximate size of the couplings to Standard Model particles,
providing a definite target for experimental searches. Further details are
given in the supplemental materials.

The interaction between DM axions or ALPs with particles and forces in the
Standard Model leads to a wide variety of ways to search for them. Initially,
in the 1980’s up to 2010 or so, experiments were few and far between. There
was just one viable method, the microwave cavity _haloscope_ , and the axion
interactions are too feeble to produce a signal measurable with technology of
the time. Now, the landscape has changed. Microwave photon detection and
cavity design have allowed the haloscope concept to break ground to exclude
regions of the QCD axion parameter space, and expand the search over a wider
frequency range [33, 34, 35]. In tandem, new methods to detect axions have
been conceived of and developed, including magnetic resonance [36, 31, 37, 38,
39], broadband antennas [40], dielectrics and metamaterials [41, 42], and
lumped circuit technology [43, 44]. These new technologies are at various
stages of maturity, with some only existing on paper, others being prototyped,
and others already making competitive measurements of axion parameter space
and excluding theoretical models. Technologies now in development should be
able to cover almost all of the viable axion parameter space in the coming
decades: the fuel for the growth of interest in this field. The companion
experimental review covers this topic in detail.

As well as searching directly for axion dark matter in the experiments
described briefly above and in the companion experimental review, the effects
of axions and ALPs may be seen indirectly in telescope observations of stars,
galaxies, and galaxy clusters. These astrophysical systems offer extreme
environments that would be impossible to replicate on Earth, and are therefore
ideal places to search for new physics.

The interaction $a{\bf E}\cdot{\bf B}$ means that axions or ALPs can
interconvert with photons in the presence of a background magnetic field. This
process is mathematically similar to neutrino oscillations, with the crucial
difference that the strength of the mixing depends on the size of the external
magnetic field. The possibility of interconversion between ALPs and photons in
space places the strongest current bounds on $g_{a\gamma\gamma}$ for very low
mass ALPs ($m_{a}\lesssim 10^{-8}$ eV).

One such bound arises from observations of distant point sources such as
Active Galactic Nuclei shining through foreground galaxy clusters. Galaxy
clusters host strong magnetic fields over very large distances. If ALPs exist,
some of the photons from these point sources would convert into ALPs as they
move through the galaxy cluster. From our point of view, some of the light
from point sources shining through galaxy clusters would go missing. The non-
observation of this effect can be used to place bounds on the axion photon
coupling (e.g. [45]), requiring $g_{a\gamma\gamma}\lesssim 10^{-12}{\,}{\rm
GeV}^{-1}$ for ALP masses $m_{a}\lesssim 10^{-11}$ eV. However, it should be
noted that this method relies on the accuracy of our knowledge of the galaxy
cluster magnetic field.

We can also bound $g_{a\gamma\gamma}$ using observations of SN 1987A, a
supernova observed in 1987 in the nearby (astronomically speaking) Large
Magellanic Cloud. A supernova is an explosion at the end of a star’s life,
which produces vast quantities of neutrinos and photons. If ALPs exist, a
supernova would also produce them copiously via nuclear interactions between
ALPs and the supernova constituents. Some of the ALPs produced by SN 1987A
would have been converted into $\gamma$ \- ray photons in the Milky Way’s
magnetic field. No such extra $\gamma$ \- ray photons have been observed,
which allows us to set a bound $g_{a\gamma\gamma}<5.3\times 10^{-12}{\,}{\rm
GeV}^{-1}$ for ALP masses $m_{a}\lesssim 4.4\times 10^{-10}$ eV (e.g.[46]). As
we will see below, SN 1987A can also be used to constrain the interactions of
higher-mass axions.

The constraints outlined above apply only to rather low mass ALPs. Photons
travelling through the plasmas of galaxies and galaxy clusters acquire a low
effective mass, but axions or ALPs with masses higher than this effective mass
cannot mix efficiently with the photon. For this reason, astrophysical axion-
photon mixing constrains the ALP parameter space, but not the standard QCD
axion for which $g_{a\gamma\gamma}\propto m_{a}$.

The coupling between axions and photons would also allow axions to be
copiously produced in stars. This would create an additional cooling mechanism
for the star and would therefore alter the course of stellar evolution. This
effect can also be used to constrain $g_{a\gamma\gamma}$, requiring
$g_{a\gamma\gamma}<6.6\times 10^{-11}{\,}{\rm GeV}^{-1}$ over a very wide
range of masses [47], thus constraining both the QCD axion and a more general
ALP. Our nearest star, the Sun, would also copiously produce axions, which
could be observed experimentally via controlled axion-photon conversion in the
laboratory. This is the aim of the CERN Axion Solar Telescope (CAST)
experiment [48], and other so-called “helioscopes”, discussed in the companion
experimental review.

##  3 Axion Dark Matter Waves

The advent of quantum mechanics in the early 20th century taught us that for
every particle there is a wave, and for every wave there is a particle. The
wavelength, $\lambda$, and frequency, $f$, of the wave are related to the
particle mass $m$ and velocity $v$ (in the non-relativistic limit) by:

$\lambda=\frac{h}{mv}\,,\quad hf=E=mc^{2}+\frac{1}{2}mv^{2}\,,$ (8)

where $h$ is Planck’s constant. When the particle is relatively heavy compared
to the axion, as in the case of the electron, the wavelength is small, and so
in our day-to-day lives we do not notice the particle behaving as a wave.
Electrons are also fermions (particles of half integer spin). This means that
they must obey the ‘exclusion principle’ - more than one electron cannot
occupy the same state, and so _collective_ wavelike behaviour does not occur
(except when they bind to form pairs with integer spin, as in
superconductors).

On the other hand, for light and massless particles, the wavelength of the
matter waves can be large, and their frequency low. If these particles are
also bosons (particles of zero or integer spin, see Fig, 1), then many
particles can occupy the same state. When the occupation number is
macroscopic, the bosons can be described using classical, rather than quantum,
field theory. This gives rise to macroscopic wavelike behaviour. Historically,
we were first aware of photons in their guise as the classical electromagnetic
radio waves of Hertz. Our technology to observe electromagnetic waves begins
in this realm of wavelengths larger than 1 mm or so, and frequencies lower
than around 1 THz.

The axion field, $a$, is just one massive bosonic field among many
possibilities, as also is its parent Peccei-Quinn field, $\Phi$. The energy
density of dark matter in the Universe is one of the most macroscopic
quantities imaginable, which in Einstein’s theory of General Relativity
affects the very geometry of spacetime and the expansion of the Universe. The
theory of axion DM is the theory of classical waves in the fields $\Phi,a$,
and their dynamics under the influence of gravity. We begin this section by
considering the origin of the energy density in these fields in the very early
Universe.

What is axion DM and how did it get here? • Axion DM consists of the energy
stored in spatial and temporal gradients, and potential energy, of the axion
field, $a(t,x)$. • Initial fluctuations of the axion field arose from a phase
transition involving spontaneous symmetry breaking that occurred in the early
Universe.

###  3.1 Two Roads to Axion Dark Matter

Observations of the CMB tell us that to a very good first approximation the
early Universe was _homogeneous, isotropic, flat_ , and _hot_. Under these
approximations, Einstein’s equations of General Relativity reduce to a single
ordinary differential equation, the Friedmann equation (see Supplementary
Material). This equation relates the temperature of the plasma in the early
Universe to its expansion rate, expressed in terms of the Hubble parameter
$H(t)$.

Figure 4: Production of axion DM. The initial conditions epoch prior to the
hot phase serves to smooth the Universe on the largest scales. The axion
dynamics and resulting DM density depends on when the Peccei-Quinn (PQ) phase
transition at the temperature $T_{\textup{PQ}}$ takes place relative to the
onset of the hot phase of the evolution of the Universe at $T_{\textup{hot}}$,
leading to two scenarios, A and B. Axion production occurs during the period
between the two times $t_{\textup{dyn}}$ and $t_{\textup{cold}}$, and is
complete in around 1 year at most. After $t_{\textup{cold}}$, axions are
described by the Cold Dark Matter model on length scales larger than their de
Broglie wavelength. The axions then “free stream” up to the time of formation
of the Cosmic Microwave Background (CMB), and beyond into the epoch of galaxy
formation and the present day. In Scenario A, the axion mass is not fixed by
the observed dark matter density, since there is an additional free parameter,
$\theta(t_{\rm hot})$. In Scenario B there is no additional free parameter and
the mass can in principle be predicted. (Some images adapted from Refs. [49,
50])

The isotropy of the CMB on the largest scales also provides evidence that this
hot, dense state of the early Universe must have been preceded by a far more
mysterious epoch that sets the large-scale initial conditions of the Universe.
That such an epoch must have existed is the only relevant point at present:
which theory describes it (for example inflation or a cyclic Universe) has no
bearing on our story (although in the case of inflation, the axion could be
used to test key predictions of the theory). After the initial conditions of
the Universe were set, the well understood “hot Big Bang” phase began, at some
finite time $t_{\textup{ hot}}$ when the temperature of the Universe was at
its maximum, $T_{\textup{ hot}}$.When cosmologists refer to the “age of the
Universe” or “time after the Big Bang”, this is in reference to the time since
$t_{\textup{ hot}}$, and not dependent on how much time there was, or was not,
before this (for example, the Universe could be infinitely old in its past).
Thus, in the box “What is known about dark matter?”, the “Creation of DM”
refers to the amount of time $\Delta t=t-t_{\textup{ hot}}$. The key
distinction for the origin of axion dark matter is whether the Peccei-Quinn
phase transition occurs for $T_{\textup{ PQ}}>T_{\textup{ hot}}$ or
$T_{\textup{ PQ}}<T_{\textup{ hot}}$. This is summarised in Fig. 4. The fact
that the oscillations of the axion field caused by spontaneous symmetry
breaking leads to a viable DM candidate was first realised in 1983 by three
separate groups working independently [51, 52, 53]. This happy accident, a DM
candidate “for free” from a particle postulated only to solve the strong-$CP$
problem, is now a main virtue of the theory. There are two scenarios for
spontaneous symmetry breaking that we must consider, shown schematically in
Fig. 4.

In Scenario A, ($T_{\textup{ PQ}}>T_{\textup{ hot}}$), the Peccei-Quinn phase
transition occurs in the epoch during which the initial conditions were set,
before the hot Big Bang. This is the simplest case to describe mathematically,
as this epoch serves to smooth the Universe on the largest scales, in
accordance with observation. This removes all “memory” of the details of the
phase transition (e.g., all the spatial gradients in the axion field), and
replaces all that complexity with just two quantities: a single random number,
the initial value of the axion field, $\theta(t_{\textup{ hot}})$, and the
axion mass, $m_{a}$. With these two inputs, the evolution of the classical
axion field, and the subsequent DM density, is determined by a simple wave
equation in which the expansion of the Universe acts as a friction term
damping the waves (see Supplemental Material).

In Scenario B ($T_{\textup{ PQ}}<T_{\textup{ hot}}$), the Peccei-Quinn phase
transition occurred during the hot Big Bang phase, _after_ the initial
conditions epoch ended. Thus the axion field is not smoothed, leaving a messy
field configuration. The physics involves the Kibble mechanism [54] for the
formation and subsequent disposal of “topological defects” in the Peccei-Quinn
field (the very same mechanism occurs in the superfluid phase transition of
Helium [55]). Topological defects are axion field configurations that occur
between regions where the axion field takes different values, which may
subsequently decay into axion DM. There is also more model dependence in this
case, as different models of the axion lead to different kinds of topological
defect. The dynamics in Scenario B is highly non-linear. However, in Scenario
B the random number, $\theta(t_{\textup{ hot}})$, of Scenario A is averaged
over in the initial distribution and is no longer a free parameter. In
Scenario B large density variations in the initial state of the axion field
lead to the formation of compact DM objects known as “miniclusters” [56].
These objects offer new opportunities to discover the axion by astrophysical
means, but also lead to an increased theoretical uncertainty in the local
axion density for DM detection in the laboratory.

In both Scenarios A and B an important change happens to the axion field when
the expansion rate of the Universe (which is falling during cosmic time) drops
below the natural frequency of axion oscillations, which is determined by its
mass. At this time, $t_{\textup{dyn}}$, the axion field becomes highly
dynamical (the wave equation becomes underdamped) and begins to oscillate.
Prior to $t_{\textup{dyn}}$, there were no non-relativistic axions: on the
length and time scales of the expansion of the Universe, the axion field was
effectively static. This is because the expansion of the universe is
responsible for damping the axion field oscillations.

Thus it is at $t_{\textup{dyn}}$ that we think of axion DM as being produced.
In Scenario A, the axion field undergoes damped simple harmonic motion, while
in Scenario B, we think of axion field oscillations being emitted from the
decaying topological defects. This epoch of particle production lasts only for
a relatively short period, as the large scale energy density in the axion
field, stored up from the phase transition, is converted into field
oscillations, which in quantum theory are equivalent to axion particles. From
$t_{\textup{dyn}}$ up to some later time $t_{\textup{cold}}$ the number of
axion particles grows. Then, from $t_{\textup{cold}}$ up to the present day
the number of axion particles in any sufficiently large “comoving” volume (a
volume that increases in proportion with the expansion of the Universe) in the
Universe is conserved , i.e., the axion number density is simply diluted by
the expansion of the Universe just like any other particle would be. Axion
cold dark matter is born, and starts to form cosmic structures such as
galaxies.

###  3.2 Cosmic Structure Formation

Because of the very large number of axions needed to supply the observed DM
density, there are a large number of axions occupying each quantum state. This
means that axion DM is described to a very good approximation by the
_classical_ scalar field $a(t,x)$. Averaged over cosmic scales (distances of
more than 10 Megaparsecs), the axion field is very uniform, and its energy
density is stored in harmonic oscillations:
$\rho=\frac{1}{2}[\dot{a}^{2}/(\hslash
c^{3})+m_{a}^{2}a^{2}/(\hslash^{3}c^{3})]$, with $a=a_{0}\cos
m_{a}c^{2}t/\hslash$, thus $\rho=m_{a}^{2}a_{0}^{2}/(2\hslash^{3}c^{3})$. The
value of $a_{0}$ is determined by the early evolution of the axion field as
described above. Crucially, for the axion to behave as cold DM with weak self-
interactions, we require $a_{0}\ll f_{a}$, which permits the harmonic motion
approximation for small displacements. The frequency of oscillation of the
field is determined by the particle mass and is given by the Compton
frequency.

On smaller scales, gravity causes the axion field to develop inhomogeneities,
and it clusters into so-called “DM halos”. On larger scales, the halos are
linked by filaments and sheets in the so-called “cosmic web”. The process of
structure formation begins very early in the history of the Universe: the
gravitational potential wells of the DM halos began to form at very high
temperatures, when ordinary matter was still completely ionized (i.e., before
the CMB formed). The development of inhomogeneities in the axion field is
governed by gravity, and the (small) axion self-interactions. The equations
describing this evolution are known as the Gross-Pitaevski-Poisson equations,
a form of non-linear Schrödinger equation for the mean field occupation, where
the potential is determined by the density of the field itself via the Poisson
equation (see Supplemental Material).

It is here that axion DM begins to take on some unique characteristics
compared to heavier particle DM such as WIMPs. These features are:

  * •

The De Broglie wavelength. Gradient energy dominates over gravitational energy
on small scales. The axion field is uniform on small scales and there is a
minimum mass for a DM halo [57].

  * •

Axion Stars. At very high density, the axion forms a kind of soliton (a stable
wavepacket-like field configuration) supported by an equilibrium between
gravity and gradients [58, 59].

  * •

Wave turbulence and interference. In the structures of the cosmic web, axion
waves have dynamic velocities. Where there are coherent flows, this leads to
interference patterns in filaments. In thermalised/virialised environments
(halos) the velocities are Maxwell-Boltzmann distributed, and are described by
wave turbulence [60, 61].

A numerical simulation of axion structure formation is shown in Fig. 5, where
the above mentioned effects can be seen. Similar phenomena showing the
formation of solitons and “incoherent” turbulent solitons occur in certain
non-linear optics systems [62].

Inside the Milky Way, at the location of the Earth, the axion field is in the
turbulent regime. It is given as:

$a=\frac{1}{2}\left(\Psi
e^{-im_{a}c^{2}t/\hslash}+\Psi^{*}e^{+im_{a}c^{2}t/\hslash}\right)\,$ (9)

The amplitude field $\Psi$ is Rayleigh distributed (up to small corrections)
with a coherence length and time governed by the de Broglie wavelength at the
local galactic orbital velocity, and an amplitude fixed by the local DM
density. The overall frequency $f$ is determined by Eq. (8), with the velocity
drawn from the Maxwell-Boltzmann distribution, leading to a frequency
dispersion $\Delta f/f\approx 10^{-6}$. The frequency dispersion leads a
natural linewidth for axion DM.

Figure 5: Zoom in to part of the Cosmic Web formed by gravitational
interactions and the interference of axion waves. The dense, almost spherical
nodes of the web are dark matter halos, which host galaxies like our own. The
inset shows a volume rendering of such a halo. Notice the interference of
coherent waves outside of dark matter halos forming interference fringes.
Inside halos, there are self bound solitonic objects at high density (“axion
stars”), and turbulent waves in the outer regions (“incoherent solitons”). In
order to make the unique wave effects visible on cosmic scales (kiloparsecs,
kpc), a very low particle mass, $m_{a}\approx 10^{-22}\text{ eV}$, is used in
this simulation. The same physics operates and the same effects occur,
however, for all types of bosonic dark matter, with the length scale set by
the particle de Broglie wavelength. Figure adapted from work presented in
Refs. [63, 64].

##  4 What is the Axion Mass?

The QCD axion has a single free parameter in the low-energy theory, the decay
constant $f_{a}$, which in turn sets the axion particle mass (at zero
temperature), $m_{a}$, according to Eq. (3). In order to try to detect the
axion in the laboratory, one would like to know the approximate mass scale, so
that experiments can be appropriately targeted to the mean oscillation
frequency, $f=m_{a}c^{2}/h$, which in turn determines the frequency of any
radiation or other associated phenomenon (as discussed elsewhere in this
review and in the experimental companion review). Currently, astrophysical
considerations provide the best clues to the range of allowed values for the
axion mass, summarised in Fig. 6, which we now elaborate on.

Beginning at the lightest end of the scale are bounds on so-called “fuzzy” DM.
When the particle mass is very low, the de Broglie wavelength is
cosmologically large, and affects the formation and internal dynamics of
galaxies. Galaxies cannot form below the de Broglie wavelength, which
suppresses galaxy formation in the early Universe (where low-mass galaxies
form first). Conservative constraints on the axion mass in this scenario are
$m\gtrsim 10^{-22}\text{ eV}$, although the bounds can be tightened by up to
three orders of magnitude including more data, and better knowledge of the
structure formation process and galactic dynamics [65, 66]. This limit also
applies to other forms of bosonic DM (not just axions, see Fig, 1), and is the
fundamental lower limit on DM particle mass. If DM is to be lighter than
$10^{-22}\text{ eV}$, observationally it can be only a sub-dominant fraction
of the total DM density, with current bounds around the 2% level [67].

Figure 6: Constraints on the axion mass, measured in electronvolts (eV). Red
arrows and the shaded region indicate exclusions. The region outside that
marked “The QCD Axion” corresponds to more exotic axion-like particles (ALPs).
“Fuzzy dark matter” shows a simulation of cosmic structure formation with very
low particle mass DM (adapted with permission from Ref. [68]), at the lower
limit of what is acceptable observationally for the dominant DM component.
“Black hole spins” shows the ergoregion and horizons of spacetime around a
spinning black hole, the geometry of which allows very light bosonic particles
to extract angular momentum from the black hole. These last two constraints
apply under the assumption that axion self-interactions are negligible. The
SN1987A constraint applies only to the QCD axion, where the axion-nuclear
coupling is proportional to the particle mass, and not to ALPs (image is a
composite [69]).

The lack of a significant observed length scale in cosmic structure formation
is one of the key facts known about DM, as described in the Box “What is known
about DM?”. Similarly, there is no evidence of a preferred time scale related
to the formation of DM, and only a bound on when this must have been complete.
Demanding that axions or other ultralight bosons (see next section) were
formed early enough sets $t_{\rm cold}\gtrsim 1$ year, and similarly bounds
$m\gtrsim 10^{-22}\text{ eV}$.

The next important constraint on the particle mass results from a property
known as “black hole superradiance”. When the Compton wavelength of a boson is
resonant with the radius of the “ergoregion” of a spinning (Kerr) black hole,
the so-called “Penrose process” [70] operates, and vacuum fluctuations of the
boson extract angular momentum from the black hole, reducing its rate of spin.
(The ergoregion is a region close to a spinning black hole in which objects
are necessarily forced to co-rotate with the black hole.) The radius of the
ergoregion is fixed by the black hole mass and spin, while the spin of the
black hole can be inferred, for example, from the Doppler shift in the
emission of X-rays from its accretion disk. Rapidly spinning black holes would
quickly spin down if a resonant boson existed, and so the observation of
spinning and old black holes places exclusions on boson masses. The most
robust, and also most important, range of these exclusions comes from stellar
mass black holes with spins measured from X-ray images, and for spin-0 bosons
excludes the range $1\times 10^{-13}\mathrm{\,eV}\leq m\leq 2\times
10^{-11}\mathrm{\,eV}$ [71, 72], with a similar range excluded for massive
spin-1 (vector) bosons. The importance of this bound is that it overlaps the
range of the masses allowed for the QCD axion with spontaneous symmetry
breaking scale below the Planck scale, $10^{19}\text{ GeV}$. Since it is
widely believed that symmetry breaking above the Planck scale is forbidden in
quantum gravity and string theory [73], superradiance leads to a solid lower
bound to the mass range of the QCD axion.

An upper bound on the QCD axion mass can be derived using the relationship
between the mass and the coupling strength to nuclear matter, $g_{aNN}$, and
is derived from observations of the supernova SN1987A [74]. The axion-nucleon
coupling leads to emission of axions from neutrons and protons inside a
collapsing supernova core in the Bremstrahhlung process, $N+N\rightarrow
N+N+a$. The produced axions interact only very weakly with the surrounding
nuclear matter, and so can escape from the supernova core, leading to excess
cooling compared to the case without axions. Neutrinos are also produced
inside supernovae, and provide a known cooling channel. If axions are also
produced, then the supernova cools more quickly and thus has less time to emit
neutrinos, leading to a lower neutrino flux. The neutrinos produced in SN1987A
were observed in neutrino detectors on Earth, and the duration and timing of
the neutrino burst is known, and is consistent with predictions of standard
supernova models. Thus, the axion production rate must be low, leading to an
upper bound on $g_{aNN}$, and thus an upper bound on $m_{a}$ (see Eq. 7),
which is $m_{a}\lesssim 2\times 10^{-2}\text{ eV}$.

The superradiance and SN1987A bounds thus limit the QCD axion natural
frequency, of critical importance for direct searches, to the range:

$\displaystyle 10\text{ kHz}\lesssim f\lesssim 10\text{ THz}\,.$ (10)

The QCD axion mass can be further narrowed down by appeal to the dark matter
density, $\rho_{a}$, in the two cosmological scenarios discussed in the
previous section. Observationally, we have the restriction that the axion
density cannot be larger than the observed dark matter density (although it
could be smaller, if the axions are not all of the dark matter).

The important difference between Scenarios A and B is one of predictability.
In both cases, the dark matter density depends inversely on the axion mass. In
Scenario A, there are two continuous free parameters $\theta_{i}$ and $m_{a}$.
Thus it is possible to reproduce the correct cosmic density of axions for
(almost) any value of the mass $m_{a}$, with smaller values of the mass having
$\theta(t_{\textup{ hot}})\ll 1$. Without a prediction for the axion mass, the
experimental task of searching for the axion could be long and arduous. This
scenario is probed by experiments operating in the 10 kHz to 10 GHz regime.

In Scenario B, on the other hand, there is only one continuous parameter,
$m_{a}$, and the discrete parameter $N$. In principle, the observed cosmic DM
density then predicts the value of $m_{a}$. The problem with trying to make
such a prediction is the computational complexity: the non-linear dynamics of
the axion field in Scenario B cannot be solved exactly. Unfortunately,
computational limitations even with modern supercomputers prevent a complete,
end-to-end, _ab initio_ simulation, and the resulting predictions for the mass
vary considerably depending on the theoretical methods employed.

Demanding that the axion is all the DM, the predicted value of the mass is in
the range $0.025\text{ meV}\lesssim m_{a}\lesssim 0.5\text{ meV}$ when $N=1$
(where the range covers approximately the predicted regions from the
simulations of Refs. [75, 76], which span those also of other groups). The
case $N>1$ predicts a slightly higher range $0.1\text{ meV}\lesssim
m_{a}\lesssim 20\text{ meV}$ (the upper bound comes from SN1987A) [77]. Lower
values of the axion mass are excluded in Scenario B. The theoretical
uncertainty is large, but in general such predictions provide motivation to
search for axion dark matter using experiments in the 10 GHz to 10 THz
frequency range. This promise to _predict_ the axion mass uniquely in this
scenario is driving many theoretical investigations, and is happening now due
to advances in computer simulation techniques.

##  5 ALPs Everywhere

It has been realized since the proposal of the axion that light spin-zero
particles with properties resembling those of the axion may appear in a number
of different theoretical contexts. Such axion-like particles (ALPs) may either
be pseudoscalar bosons, whose couplings are odd under parity transformations
(reversing spatial directions), or scalar bosons, whose couplings are even
under parity. The latter are sometimes referred to loosely as dilatons or
moduli, and may be thought of as lower-mass siblings of the Higgs boson of the
Standard Model. Generically, both classes of ALPs possess, like axions,
couplings with fermion-antifermion pairs, and also with pairs of photons,
gluons, and the other vector bosons via their field strengths, of the forms
${\bf E}\cdot{\bf B}$ for pseudoscalars and $|{\bf E}|^{2}+|{\bf B}|^{2}$ for
scalars (where ${\bf E}$ and ${\bf B}$ are generalised for the other forces of
the Standard Model). ALPs are distinguished from the QCD axion of our previous
discussion, since ALPs do not solve the strong-$CP$ problem. ALPs do, however,
still make excellent dark matter candidates.

There is a single scalar Higgs boson in the Standard Model associated with the
generation of particle masses via breaking of the underlying electroweak
symmetry. However, including another symmetry-breaking Higgs field requires
the appearance of an additional scalar particle as well as a pseudoscalar
partner particle. The original axion model is an example of this feature, but
it can be generalized by including more Higgs fields of different types. This
scenario occurs in many theories, including supersymmetry, and grand unified
theories.

Another theoretical framework where ALPs appear is provided by string theory
[78]. Generic four-dimensional particle physics models obtained from string
theory, by the compactification of extra dimensions of space on small
manifolds, contain many additional pseudoscalar and scalar fields, which may
have (very) small masses. A useful and famous analogy for compact dimensions
of space is to imagine our world as a tightrope. From the perspective of large
humans, the rope is one-dimensional, and one can only move along its length.
However, a small creature such as an ant can circle the “extra dimension” of
the rope’s circumference. In quantum mechanics, length is inversely
proportional to energy, and thus the small length scales of extra dimensions
of space tend to be associated to large energy scales, possibly even the
Planck scale of quantum gravity. The sizes and shapes of the compact spaces
are unspecified a priori, since General Relativity and Quantum Mechanics tell
us that the compact dimensions must vary in size and shape at different
places. These changes in size and shape can be described, from a four-
dimensional perspective, by a collection of new fields, analogous to the
Newtonian gravitational field, which in General Relativity describes the
curvature of spacetime. Futhermore, there must be some dynamical energy
principle that fixes the average size and shape of the compact dimensions; as
the potential in Fig. 3 fixes the axion angle and also gives it a mass, so
these dynamics give masses to the ALPs. This new dynamics is by nature
extremely speculative, and (due to the vast array of possibilities for the
topology of the compact space) even the most predictive approaches to this
problem currently make only statistical statements about the masses of the
resulting ALPs. However, the existence of extra dimensions of space is one of
the central predictions of string theory and, since it leads to the existence
of ALPs, they too can be considered a prediction of the theory.

Although the forms of ALP couplings are similar to those of the original
axion, their strengths are quite model-dependent, and even less is known about
them than about the ALP masses. For this reason, there is great interest in
designing experimental searches for axions that are sufficiently general to
also be sensitive to wide ranges of the parameter spaces of ALP couplings to
both fermions and gauge bosons.

We also mention the growing interest in the possible existence of light vector
(spin 1) bosons. In the Standard Model there is, on the one hand, the photon
whose mass is thought to be exactly zero and is very tightly constrained by
experimental observations, and on the other hand the massive vector bosons
responsible for the weak interactions, with no vector bosons of intermediate
mass. However, it is possible that there might be additional light vector
bosons that are very weakly coupled to the particles of the Standard Model,
so-called ‘dark’ or ‘hidden’ vector bosons, as also appear in some string
theory models, for example. Their couplings to Standard Model particles would
take different forms from those of ALPs. For example, couplings to pairs of
photons or gluons would be forbidden, and their couplings to fermion-
antifermion pairs would be vector- or axial-vector-like. For this reason,
different types of experimental strategies are required to search for them,
and we do not discuss them further in this review.

Finally, we discuss the possibility of axion-like quasiparticles appearing in
certain exotic materials in solid state physics. Quasiparticles such as
phonons (quanta of vibrational modes of solids) and magnons (quanta of
magnetization) have been staples of condensed matter theory since the earliest
days of quantum mechanics. For a quasiparticle to behave as an axion, it must
be a pseudoscalar, which requires the parent solid to have certain specific
symmetry properties under parity and time-reversal. The would-be axion
quasiparicle should also have the trademark axion-like couplings, in
particular to ${\bf E}\cdot{\bf B}$, giving rise to axion electrodyanmics. The
possibility of axion electrodynamics in solids was identified already by
Wilczek, one of the fathers of the axion theory, in the 1980s [79], but only
recently has the quest to realise this idea in the laboratory picked up pace.

Mixing of parallel electric and magnetic fields occurs in general in
magnetoelectric materials, and in particular in topological insulators. The
existence of the axion quasiparticle requires the breaking of time-reversal
symmetry, and is provided by magnetic ions and antiferromagnetic order.
Candidate materials to host the axion quasiparticle include Bi(Fe)2Se3 and
Mn2Bi2Te5 [80, 81]. The axion quasiparticle has yet to be observed in the lab,
although various experiments have been designed to hunt for it, and there is
growing interest in these candidate materials with possible applications to
high-speed computer memory [82]. Furthermore, as discussed in the experimental
companion to this review, axion quasiparticles could also prove useful in the
detection of axion dark matter.

##  6 Summary, Conclusions, and Musings

The dark matter zeitgeist is in flux. For many years experimental and
theoretical efforts were directed at WIMPs and similar particle candidates.
The technology to detect WIMPs was readily available in the form of the LHC,
and the experimental programme of direct WIMP searches also developed rapidly
[15, 16]. Theories attempting to explain the origin of the Higgs mass seemed
to predict naturally the existence of WIMP dark matter, with mass comparable
to that of the Higgs itself, providing a very tantalising prospect. The
experimental searches for WIMPs performed excellently, yet to date have found
nothing. While there are still important parts of the parameter space left to
explore, initial optimism has been dampened by experimental cold water.

Almost in parallel to the WIMP story, there was, somewhat behind the scenes,
the ever-present and developing story of the axion. Also proposed in the 1970s
as a sticking plaster for problems in the Standard Model [5, 6, 7], and with a
dark matter candidate for free [51, 52, 53], that is where the shared story of
axions and WIMPs ends. The axion is far lighter and more weakly coupled than
the WIMP, and initially experimental searches for the axion made slow and
painful progress towards the theoretically defined goals. All of this is
changing at present and the axion’s star is in ascendance. Theorists are
working hard trying to hem in the predictions of the axion model, while new
technological advances offer prospects for detection in the near future (see
the companion experimental review).

Axion dark matter is an ultralight bosonic field, which manifests itself on
Earth primarily as exotic oscillating phenomena violating $CP$ symmetry. The
primary and defining oscillation is in electric dipole moments, the neutron
EDM being of particular interest. The axion also acts as an oscillating source
in Maxwell’s equations, and can be detected as an anomalous electric or
magnetic field. The oscillations in the axion field were set up in the early
Universe by a process of symmetry breaking and phase transition, and the
discovery of the axion could shed light on those extremely high energy
processes in the most distant cosmic past.

The parameter space of axion models is relatively well defined. Astrophysical
considerations hem in the oscillation frequency (and thus the particle mass)
to a range between kHz and THz, tightening to GHz to THz given some
assumptions about the phase transition (see Fig. 4). Intense theoretical
efforts are being made at present to try and tighten this range still further,
limited by computational resources. Experimental searches for the axion often
rely on resonant effects [29], and it is the task of tuning these experimental
“radios” over the wide range of available frequencies that makes axion
searches so long and laborious. If the theorists could predict the axion mass
accurately, which might be within reach, the experimental search could be
tuned, and accomplished within a very short period. However, we should temper
our optimism. The theoretical predictions rely on a large number of
assumptions, which could be violated in Nature.

Axion theorists are also intensely engaged in thinking about the axion
couplings, $g$, to ordinary matter [22]. Astrophysical constraints, for
example from stellar evolution, often provide a good guide for how large these
couplings can be in general, without theoretical bias. The axion theory
predicts $g\propto m_{a}$, but the constant of proportionality varies for
different models. A given experiment or observation constrains the value of
the constant of proportionality, and it is for theorists to interpret which,
if any, of the available axion models has been excluded. Again, efforts to pin
down the range of allowed couplings are starting to reach some degree of
consensus, just in time for the data onslaught that is coming our way from new
experiments.

As described in the companion experimental review, this onslaught of data is
already beginning, and should reach a conclusion in the next decade or so.
Experiments with resonant circuits and nuclear magnetic resonance operate in
the kHz range and below, microwave cavities in the GHz range, and the use of
dielectrics and novel materials can push into near THz and above. If these
experiments succeed, the next task will be to identify the axion precisely.
One would like to measure the axion via all its interactions, Eq. (5).
Fortunately, once one experiment succeeds and finds the right axion “radio
station” (i.e. the value of the axion mass), all the others will quickly be
able to tune in and measure the different couplings of the axion. For example,
with the axion photon and electron couplings in hand, it should be possible to
determine whether the axion speaks to the Standard Model via new heavy quarks
[25, 26], or via the Higgs boson [23, 24] (see Supplemental Material).

An axion dark matter experiment in reality measures a combination of the axion
coupling and the dark matter density, so another key task will be to use
overlapping experiments that do not rely on the dark matter density to break
this degeneracy. Fortunately, ideas are available using nuclear magnetic
resonance, electric dipole measurements, searches for new forces, and
combinations of searches for axions from the Sun and other astrophysical
sources (see the experimental companion to this review). Axion searches also
offer a wealth of information about the frequency structure of the dark
matter, which would provide a detailed view of dark matter in our galactic
neighbourhood, and open a new field of dark matter astronomy [83]. This
combination of astrophysics and particle physics could eventually determine
not just if the axion is indeed the dark matter, but exactly what fraction of
the dark matter it is and how it is distributed. If the axion is just one part
of the dark matter, a new series of more challenging problems arise, since all
dark matter searches are harder when aimed at sub-components.

Fortunately, theorists are prepared for such an eventuality. Many theories
possessing the axion also possess a zoo of related ALPs, all or none of which
could also contribute to the dark matter, and some of which could even be
related to hidden dimensions of spacetime [78]. If the axion is discovered, it
will open up the doors to experimentally testing these far reaching theories.
The interest in these related particles, and experimental searches for them,
is also reaching new heights, but that is a story too varied for this review.

The story of the axion, like much of modern particle physics, began in the
1970s, rooted in ideas of symmetry. The Standard Model is a symmetry success
story, but other theories based on symmetry have failed. Does Nature really
play by these rules? The popularity of the axion has risen as a prime
candidate to explain dark matter, the cosmic glue that in many ways is
responsible for our own existence. What composes this mysterious substance,
and how did it get here? The discovery of the axion seems tantalisingly close,
promising answers to these deepest of questions.

## Acknowledgements

We were saddened to hear during the completion of this paper of the deaths of
Toshihide Maskawa and Steven Weinberg, whose deep theoretical insights
underlie the topics reviewed here.

We are indebted to Marco Gorghetto for providing an image used in Fig. 4, to
Matthew Stott for providing an image used in Fig. 6, and to Philip Mocz who
provided the images used in Fig. 5, and to all authors who have allowed us to
reproduce images from their published work. The Chandra composite image in
Fig. 6 has credit to the following sources, X-ray: NASA/CXC/SAO/PSU [84];
Optical: NASA/STScI [85]; Millimeter: ESO/NAOJ/NRAO/ALMA [86]. We thank Yannis
Smertzidis (CAPP), Ian Williams (University of Surrey), and SungWoo Youn
(CAPP) for comments on the draft. FC-D is supported by the UK STFC grant
ST/P001246/1 and Stephen Hawking Fellowship EP/T01668X/1. The work of JE was
supported in part by the UK STFC via grant ST/T000759/1, and in part by the
Estonian Research Council via grant MOBTT5. DJEM was previously supported by
the Alexander von Humboldt Foundation and the German Federal Ministry of
Education and Research, and is now supported by a UK STFC Ernest Rutherford
Fellowship.

### Data and Materials Availability

All data needed to evaluate the conclusions in the paper are present in the
paper and/or the Supplementary Materials.

### Author Contributions

All authors contributed equally to the writing and editing of the manuscript.

### Competing Interests

The authors declare no competing interests.

## References

  * [1] Aad, G. _et al._ Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC.  _Phys. Lett. B_ 716, 1–29 (2012).  1207.7214.
  * [2] Chatrchyan, S. _et al._ Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC.  _Phys. Lett. B_ 716, 30–61 (2012).  1207.7235.
  * [3] Higgs, P. W.  Broken Symmetries and the Masses of Gauge Bosons.  _Phys. Rev. Lett._ 13, 508–509 (1964).
  * [4] Englert, F. & Brout, R.  Broken Symmetry and the Mass of Gauge Vector Mesons.  _Phys. Rev. Lett._ 13, 321–323 (1964).
  * [5] Peccei, R. D. & Quinn, H. R.  CP conservation in the presence of pseudoparticles.  _Phys. Rev. Lett._ 38, 1440–1443 (1977).
  * [6] Weinberg, S.  A new light boson?  _Phys. Rev. Lett._ 40, 223–226 (1978).
  * [7] Wilczek, F.  Problem of strong P and T invariance in the presence of instantons.  _Phys. Rev. Lett._ 40, 279–282 (1978).
  * [8] Dirac, P. A. M.  The quantum theory of the electron.  _Proc. Roy. Soc. Lond. A_ 117, 610–624 (1928).
  * [9] Sakharov, A. D.  Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe.  _Pisma Zh. Eksp. Teor. Fiz._ 5, 32–35 (1967).
  * [10] Christenson, J. H., Cronin, J. W., Fitch, V. L. & Turlay, R.  Evidence for the $2\pi$ Decay of the $K_{2}^{0}$ Meson.  _Phys. Rev. Lett._ 13, 138–140 (1964).
  * [11] Kobayashi, M. & Maskawa, T.  CP Violation in the Renormalizable Theory of Weak Interaction.  _Prog. Theor. Phys._ 49, 652–657 (1973).
  * [12] Rubin, V. C. & Ford, W. K., Jr.  Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions.  _Astrophys. J._ 159, 379–403 (1970).
  * [13] Aghanim, N. _et al._ Planck 2018 results. VI. Cosmological parameters.  _Astron. Astrophys._ 641, A6 (2020).  1807.06209.
  * [14] Peebles, P. J. E.  Large scale background temperature and mass fluctuations due to scale invariant primeval perturbations.  _Astrophys. J. Lett._ 263, L1–L5 (1982).
  * [15] Jungman, G., Kamionkowski, M. & Griest, K.  Supersymmetric dark matter.  _Phys. Rept._ 267, 195–373 (1996).  hep-ph/9506380.
  * [16] Aprile, E. _et al._ Dark Matter Search Results from a One Ton-Year Exposure of XENON1T.  _Phys. Rev. Lett._ 121, 111302 (2018).  1805.12562.
  * [17] Clowe, D., Gonzalez, A. & Markevitch, M.  Weak lensing mass reconstruction of the interacting cluster 1E0657-558: Direct evidence for the existence of dark matter.  _Astrophys. J._ 604, 596–603 (2004).  astro-ph/0312273.
  * [18] Abel, C. _et al._ Measurement of the permanent electric dipole moment of the neutron.  _Phys. Rev. Lett._ 124, 081803 (2020).  2001.11966.
  * [19] Nambu, Y.  Axial vector current conservation in weak interactions.  _Phys. Rev. Lett._ 4, 380–382 (1960).
  * [20] Goldstone, J.  Field Theories with Superconductor Solutions.  _Nuovo Cim._ 19, 154–164 (1961).
  * [21] Grilli di Cortona, G., Hardy, E., Pardo Vega, J. & Villadoro, G.  The QCD axion, precisely.  _JHEP_ 01, 034 (2016).  1511.02867.
  * [22] Di Luzio, L., Giannotti, M., Nardi, E. & Visinelli, L.  The landscape of QCD axion models.  _Phys. Rept._ 870, 1–117 (2020).  2003.01100.
  * [23] Dine, M., Fischler, W. & Srednicki, M.  A Simple Solution to the Strong CP Problem with a Harmless Axion.  _Phys. Lett. B_ 104, 199–202 (1981).
  * [24] Zhitnitsky, A. R.  On Possible Suppression of the Axion Hadron Interactions. (In Russian).  _Sov. J. Nucl. Phys._ 31, 260 (1980).
  * [25] Kim, J. E.  Weak-interaction singlet and strong $\mathrm{CP}$ invariance.  _Phys. Rev. Lett._ 43, 103–107 (1979).  URL https://link.aps.org/doi/10.1103/PhysRevLett.43.103.
  * [26] Shifman, M., Vainshtein, A. & Zakharov, V.  Can confinement ensure natural cp invariance of strong interactions?  _Nuclear Physics B_ 166, 493–506 (1980).  URL https://www.sciencedirect.com/science/article/pii/0550321380902096.
  * [27] Moody, J. & Wilczek, F.  NEW MACROSCOPIC FORCES?  _Phys. Rev. D_ 30, 130 (1984).
  * [28] Arvanitaki, A. & Geraci, A. A.  Resonantly Detecting Axion-Mediated Forces with Nuclear Magnetic Resonance.  _Phys. Rev. Lett._ 113, 161801 (2014).  1403.1290.
  * [29] Sikivie, P.  Experimental tests of the ’invisible’ axion.  _Phys. Rev. Lett._ 51, 1415–1417 (1983).
  * [30] Barbieri, R., Cerdonio, M., Fiorentini, G. & Vitale, S.  AXION TO MAGNON CONVERSION: A SCHEME FOR THE DETECTION OF GALACTIC AXIONS.  _Phys. Lett. B_ 226, 357–360 (1989).
  * [31] Budker, D., Graham, P. W., Ledbetter, M., Rajendran, S. & Sushkov, A. O.  Proposal for a Cosmic Axion Spin Precession Experiment (CASPEr).  _Phys. Rev. X_ 4, 021030 (2014).  1306.6089.
  * [32] Weinberg, S.  Nonlinear realizations of chiral symmetry.  _Phys. Rev._ 166, 1568–1577 (1968).
  * [33] Asztalos, S. J. _et al._ SQUID-Based Microwave Cavity Search for Dark-Matter Axions.  _Phys. Rev. Lett._ 104, 041301 (2010).  0910.5914.
  * [34] Braine, T. _et al._ Extended Search for the Invisible Axion with the Axion Dark Matter Experiment.  _Phys. Rev. Lett._ 124, 101303 (2020).  1910.08638.
  * [35] Semertzidis, Y. K. _et al._ Axion Dark Matter Research with IBS/CAPP (2019).  1910.11591.
  * [36] Graham, P. W. & Rajendran, S.  New observables for direct detection of axion dark matter.  _Phys. Rev. D_ 88, 035023 (2013).  1306.6088.
  * [37] Crescini, N. _et al._ Operation of a ferromagnetic axion haloscope at m_a=58 $\mu$ eV.  _European Physical Journal C_ 78, 703 (2018).  1806.00310.
  * [38] Marsh, D. J. E., Fong, K. C., Lentz, E. W., Šmejkal, L. & Ali, M. N.  Proposal to Detect Dark Matter using Axionic Topological Antiferromagnets.  _Phys. Rev. Lett._ 123, 121601 (2019).  1807.08810.
  * [39] Aybas, D. _et al._ Search for Axionlike Dark Matter Using Solid-State Nuclear Magnetic Resonance.  _Phys. Rev. Lett._ 126, 141802 (2021).  2101.01241.
  * [40] Horns, D. _et al._ Searching for WISPy cold dark matter with a dish antenna.  _Journal of Cosmology and Astroparticle Physics_ 2013, 016 (2013).  1212.2970.
  * [41] Caldwell, A. _et al._ Dielectric Haloscopes: A New Way to Detect Axion Dark Matter.  _Phys. Rev. Lett._ 118, 091801 (2017).  1611.05865.
  * [42] Egge, J., Knirck, S., Majorovits, B., Moore, C. & Reimann, O.  A first proof of principle booster setup for the MADMAX dielectric haloscope.  _Eur. Phys. J. C_ 80, 392 (2020).  2001.04363.
  * [43] Kahn, Y., Safdi, B. R. & Thaler, J.  Broadband and Resonant Approaches to Axion Dark Matter Detection.  _Phys. Rev. Lett._ 117, 141801 (2016).  1602.01086.
  * [44] Salemi, C. P. _et al._ Search for Low-Mass Axion Dark Matter with ABRACADABRA-10 cm.  _Phys. Rev. Lett._ 127, 081801 (2021).  2102.06722.
  * [45] Reynolds, C. S. _et al._ Astrophysical limits on very light axion-like particles from Chandra grating spectroscopy of NGC 1275 (2019).  1907.05475.
  * [46] Brockway, J. W., Carlson, E. D. & Raffelt, G. G.  SN1987A gamma-ray limits on the conversion of pseudoscalars.  _Phys. Lett. B_ 383, 439–443 (1996).  astro-ph/9605197.
  * [47] Ayala, A., Domínguez, I., Giannotti, M., Mirizzi, A. & Straniero, O.  Revisiting the bound on axion-photon coupling from Globular Clusters.  _Phys. Rev. Lett._ 113, 191302 (2014).  1406.6053.
  * [48] Anastassopoulos, V. _et al._ New CAST Limit on the Axion-Photon Interaction.  _Nature Phys._ 13, 584–590 (2017).  1705.02290.
  * [49] Vaquero, A., Redondo, J. & Stadler, J.  Early seeds of axion miniclusters.  _JCAP_ 04, 012 (2019).  1809.09241.
  * [50] Armengaud, E. _et al._ Physics potential of the International Axion Observatory (IAXO).  _JCAP_ 2019, 047 (2019).  1904.09155.
  * [51] Preskill, J., Wise, M. B. & Wilczek, F.  Cosmology of the invisible axion.  _Phys. Lett. B_ 120, 127–132 (1983).
  * [52] Abbott, L. F. & Sikivie, P.  A cosmological bound on the invisible axion.  _Phys. Lett. B_ 120, 133–136 (1983).
  * [53] Dine, M. & Fischler, W.  The not-so-harmless axion.  _Phys. Lett. B_ 120, 137–141 (1983).
  * [54] Kibble, T. W. B.  Topology of Cosmic Domains and Strings.  _J. Phys. A_ 9, 1387–1398 (1976).
  * [55] Zurek, W. H.  Cosmological Experiments in Superfluid Helium?  _Nature_ 317, 505–508 (1985).
  * [56] Hogan, C. J. & Rees, M. J.  Axion miniclusters.  _Phys. Lett. B_ 205, 228–230 (1988).
  * [57] Khlopov, M., Malomed, B. & Zeldovich, I.  Gravitational instability of scalar fields and formation of primordial black holes.  _Mon. Not. Roy. Astron. Soc._ 215, 575–589 (1985).
  * [58] Seidel, E. & Suen, W. M.  Oscillating soliton stars.  _Phys. Rev. Lett._ 66, 1659–1662 (1991).
  * [59] Guzman, F. S. & Urena-Lopez, L. A.  Gravitational cooling of self-gravitating Bose-Condensates.  _Astrophys. J._ 645, 814–819 (2006).  astro-ph/0603613.
  * [60] Schive, H.-Y., Chiueh, T. & Broadhurst, T.  Cosmic structure as the quantum interference of a coherent dark wave.  _Nature Physics_ 10, 496–499 (2014).  1406.6586.
  * [61] Hui, L., Joyce, A., Landry, M. J. & Li, X.  Vortices and waves in light dark matter.  _JCAP_ 01, 011 (2021).  2004.01188.
  * [62] Picozzi, A. & Garnier, J.  Incoherent soliton turbulence in nonlocal nonlinear media.  _Phys. Rev. Lett._ 107, 233901 (2011).  URL https://link.aps.org/doi/10.1103/PhysRevLett.107.233901.
  * [63] Mocz, P. _et al._ Galaxy formation with BECDM – I. Turbulence and relaxation of idealized haloes.  _Mon. Not. Roy. Astron. Soc._ 471, 4559–4570 (2017).  1705.05845.
  * [64] Mocz, P. _et al._ First star-forming structures in fuzzy cosmic filaments.  _Phys. Rev. Lett._ 123, 141301 (2019).  1910.01653.
  * [65] Marsh, D. J. E. & Niemeyer, J. C.  Strong Constraints on Fuzzy Dark Matter from Ultrafaint Dwarf Galaxy Eridanus II.  _Phys. Rev. Lett._ 123, 051103 (2019).  1810.08543.
  * [66] Rogers, K. K. & Peiris, H. V.  Strong bound on canonical ultra-light axion dark matter from the Lyman-alpha forest.  _Phys. Rev. Lett._ 126, 071302 (2021).  2007.12705.
  * [67] Hlozek, R., Marsh, D. J. E. & Grin, D.  Using the Full Power of the Cosmic Microwave Background to Probe Axion Dark Matter.  _Mon. Not. Roy. Astron. Soc._ 476, 3063–3085 (2018).  1708.05681.
  * [68] Veltmaat, J., Niemeyer, J. C. & Schwabe, B.  Formation and structure of ultralight bosonic dark matter halos.  _Phys. Rev. D_ 98, 043509 (2018).  1804.09647.
  * [69] URL https://chandra.harvard.edu/photo/2017/sn1987a/.
  * [70] Penrose, R. & Floyd, R.  Extraction of rotational energy from a black hole.  _Nature_ 229, 177–179 (1971).
  * [71] Arvanitaki, A., Dimopoulos, S., Dubovsky, S., Kaloper, N. & March-Russell, J.  String axiverse.  _Phys. Rev. D_ 81, 123530 (2010).  0905.4720.
  * [72] Stott, M. J. & Marsh, D. J. E.  Black hole spin constraints on the mass spectrum and number of axionlike fields.  _Phys. Rev. D_ 98, 083006 (2018).  1805.02016.
  * [73] Arkani-Hamed, N., Motl, L., Nicolis, A. & Vafa, C.  The String landscape, black holes and gravity as the weakest force.  _JHEP_ 06, 060 (2007).  hep-th/0601001.
  * [74] Raffelt, G. G.  Astrophysical axion bounds.  _Lect. Notes Phys._ 741, 51–71 (2008).  hep-ph/0611350.
  * [75] Klaer, V. B. & Moore, G. D.  The dark-matter axion mass.  _JCAP_ 11, 049 (2017).  1708.07521.
  * [76] Gorghetto, M., Hardy, E. & Villadoro, G.  More Axions from Strings.  _SciPost Phys._ 10, 050 (2021).  2007.04990.
  * [77] Hiramatsu, T., Kawasaki, M., Saikawa, K. & Sekiguchi, T.  Axion cosmology with long-lived domain walls.  _JCAP_ 01, 001 (2013).  1207.3166.
  * [78] Svrcek, P. & Witten, E.  Axions In String Theory.  _JHEP_ 06, 051 (2006).  hep-th/0605206.
  * [79] Wilczek, F.  Two applications of axion electrodynamics.  _Phys. Rev. Lett._ 58, 1799–1802 (1987).  URL https://link.aps.org/doi/10.1103/PhysRevLett.58.1799.
  * [80] Li, R., Wang, J., Qi, X.-L. & Zhang, S.-C.  Dynamical axion field in topological magnetic insulators.  _Nature Physics_ 6, 284–288 (2010).  0908.1537.
  * [81] Zhang, J. _et al._ Large Dynamical Axion Field in Topological Antiferromagnetic Insulator Mn2Bi2Te5.  _Chinese Physics Letters_ 37, 077304 (2020).  URL https://doi.org/10.1088%2F0256-307x%2F37%2F7%2F077304.
  * [82] Cheng, Y., Peng, B., Hu, Z., Zhou, Z. & Liu, M.  Recent development and status of magnetoelectric materials and devices.  _Physics Letters A_ 382, 3018–3025 (2018).
  * [83] O’Hare, C. A. J. & Green, A. M.  Axion astronomy with microwave cavity experiments.  _Phys. Rev._ D95, 063017 (2017).  1701.03118.
  * [84] Frank, K. A. _et al._ Chandra Observes the end of an era in SN 1987a.  _Astrophys. J._ 829, 40 (2016).  1608.02160.
  * [85] Orlando, S., Miceli, M., Pumo, M. L. & Bocchino, F.  Supernova 1987A: a Template to Link Supernovae to their Remnants.  _Astrophys. J._ 810, 168 (2015).  1508.02275.
  * [86] Indebetouw, R. _et al._ Dust Production and Particle Acceleration in Supernova 1987A Revealed with ALMA.  _Astrophys. J. Lett._ 782, L2 (2014).  1312.4086.
  * [87] Dematteis, G., Grafke, T., Onorato, M. & Vanden-Eijnden, E.  Experimental evidence of hydrodynamic instantons: The universal route to rogue waves.  _Phys. Rev. X_ 9, 041057 (2019).  URL https://link.aps.org/doi/10.1103/PhysRevX.9.041057.
  * [88] Coleman, S.  _Aspects of Symmetry_ (Cambridge University Press, 1988).
  * [89] Vafa, C. & Witten, E.  Parity Conservation in QCD.  _Phys. Rev. Lett._ 53, 535 (1984).
  * [90] Georgi, H. & Randall, L.  Flavor conserving cp violation in invisible axion models.  _Nuclear Physics B_ 276, 241–252 (1986).  URL https://www.sciencedirect.com/science/article/pii/0550321386900222.

##  Appendix A Supplemental: Theoretical Methods

###  A.1 The Axion and Quantum Chromodynamics

We look here in more detail at the strong $CP$ problem and the axion. When we
write down the Lagrangian density for Quantum Chromodynamics (QCD) we must
include all terms allowed by the symmetries of the theory. This means that the
Lagrangian density should include the $CP$-violating term In the supplemental
material, we use natural units with $c=\hslash=1$.

$\mathcal{L}\supset\bar{\theta}\frac{g_{3}^{2}}{32\pi^{2}}G_{\mu\nu}\cdot\tilde{G}^{\mu\nu},$
(11)

where $g_{3}$ is the coupling constant of the strong force, $G_{\mu\nu}$ is
the gluon field strength tensor,
$\tilde{G}_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}G^{\rho\sigma}$ is its dual,
and $\bar{\theta}$ includes contributions from the quark masses and from the
structure of the vacuum itself:

$\bar{\theta}=\theta+{\rm arg}\,{\rm det}(M),$ (12)

where $\theta$ is the QCD vacuum angle and $M$ is the quark mass matrix. We
seek an explanation for why $\bar{\theta}$ is so small.

To this end, we introduce the global Peccei-Quinn symmetry $U(1)_{PQ}$. The
spontaneous breaking of this symmetry yields a massless Nambu-Goldstone boson,
the axion. This means that under a $U(1)_{PQ}$ transformation with parameter
$\epsilon$ the axion field transforms like:

$a(x)\xrightarrow{}a(x)+\epsilon f_{a}.$ (13)

When the $U(1)_{PQ}$ symmetry is exact, the axion must be exactly massless, so
that adding $\epsilon f_{a}$ to the axion field results in the same energy
density.

The axion interacts with gluons via the Lagrangian density

$\mathcal{L}\supset\frac{a}{f_{a}}\frac{g_{3}^{2}}{32\pi^{2}}G_{\mu\nu}\cdot\tilde{G}^{\mu\nu}.$
(14)

Notice that this has exactly the same form as the problematic strong
$CP$-violating term (11), but the coefficient of $G\cdot\tilde{G}$ is now a
dynamical field. This is what allows the total coefficient of
$G\cdot\tilde{G}$ to relax to zero in the ground state of the theory. How this
term is generated is discussed in more detail in the next subsection. We note
in passing that the axion’s coupling to photons takes the same form
$\mathcal{L}\supset\frac{g_{a\gamma\gamma}}{4}F\cdot\tilde{F}=g_{a\gamma\gamma}{\bf
E}\cdot{\bf B}$, where $F$ is the electromagnetic field strength tensor.

The axion’s coupling to gluons also ensures that the $U(1)_{PQ}$ symmetry is
anomalous. An anomalous symmetry is a symmetry that would be conserved in a
classical theory but is broken by quantum effects. Put another way, the
theory’s Lagrangian is invariant under the symmetry, but the path inegral
measure is not. This means that the axion is no longer massless, but is a
pseudo-Nambu-Goldstone boson. The potential energy, and therefore the mass, of
the axion field is generated by QCD instantons. Instantons are non-trivial
solutions to the classical equations of motion, whose effects are not captured
by perturbative quantum field theory methods. Instantons have been observed
experimentally in the context of fluid dynamics [87].

The effects of instantons in QCD can be accounted for by a $\theta$-dependent
change to the vacuum energy, leading to an effective potential for the axion.
In the dilute instanton gas approximation (see e.g. Ref. [88]) applicable to
high-temperature QCD, the potential is:

$V(a(x))\propto-{\rm cos}\left(\bar{\theta}+\frac{a(x)}{f_{a}}\right).$ (15)

Therefore, minimising the potential energy of the axion field sets the total
coefficient of $G\cdot\tilde{G}$ to zero. The mass of the axion is found from
a Taylor expansion of $V(a(x))$. The constant of proportionality is the QCD
topological susceptibility, which is temperature-dependent and can be computed
by instanton methods, in effective field theory, or using non-perturbative
lattice QCD.

How do we know that the minimum of the axion potential is at precisely the
value that leads to no $CP$ violation in strong interactions? One way to see
this is via the Vafa-Witten theorem, which tells us that in vector-like
theories, such as QCD, dynamical parity-violating terms are zero in the ground
state of the theory [89]. The neutron EDM violates $P$ as well as $CP$, and
the axion field make the total neutron EDM a dynamical variable. Therefore, in
the absence of weak interactions, the Vafa-Witten theorem guarantees that the
axion field potential minimum will be at zero neutron EDM. However, the theory
of quarks is not quite vector-like because the quarks are also charged under
the chiral weak interaction. This means that the minimum of the axion
potential is slightly shifted from the $CP$ conserving value. The magnitude of
this effect can be calculated using effective field theory, and is found to
lead to an effective $CP$ violating angle
$\bar{\theta}\simeq\mathcal{O}(10^{-17})$ [90], far below current experimental
bounds.

###  A.2 Invisible axion models

So far, we have described in only general terms how the axion solves the
strong $CP$ problem. Now, we would like to be more specific. The key property
of the axion is its coupling to gluons described above. In order to solve the
strong $CP$ problem, this coupling should be generated by the chiral anomaly
mediated by the coupling between the axion and fermions charged under the
strong interactions, leading to the triangle Feynman diagram in Fig. S1. In
the absence of the chiral anomaly, this diagram would have zero amplitude. In
the Standard Model, the fermions charged undeer the strong interactions are
the quarks, $u,d,c,s,t,b$.

In the original axion model [6, 7], the Peccei-Quinn field $\Phi$ is a second
Higgs field, with one field giving mass to up-type quarks, and another to
down-type quarks. The axion field is then the complex phase of the combined
Higgs field. However, given the known vacuum expectation value of the Higgs
field, this model is ruled out by experimental data from colliders. This led
to the development of “invisible” axion models. In this section we will
describe the first two invisible axion models A review of these and more
general models can be found in Ref. [22].. In these models, the Peccei-Quinn
field is not related to the Higgs field, and so its vacuum expectation value
$v_{\rm PQ}$ is a free parameter. If $v_{\rm PQ}$ is high enough, the axion’s
interactions are weak enough to have evaded experiments to date The axion
decay constant is given by $f_{a}=v_{\rm PQ}/N$, where $N$ is the (model
dependent) color anomaly of the Peccei-Quinn symmetry.

Figure S1: The colour anomaly of the Peccei-Quinn symmetry. The current of the
PQ symmetry is $J_{\mu}^{\rm PQ}=\partial_{\mu}a$. Due to the chiral
interaction between the axion and a quark, $\mathcal{Q}$ (under which the
left- and right-handed Weyl spinors carry opposite charge), this current is
not conserved in the quantum theory. The loop diagram mediates a non-vanishing
interaction between axions and gluons, $A^{\mu}$.

We describe first the Kim [25] Shifman-Vainshtein-Zakharov [26] (KSVZ) model.
The KSVZ model extends the Standard Model by the introduction of two new
fields. The first is the Peccei-Quinn field $\Phi$, which is uncharged under
all the Standard Model interactions. The second is a new heavy vector fermion
$\mathcal{Q}$, which is charged under the strong interactions, i.e., a new
exotic quark.The historical KSVZ model also has an electrical charge for the
new quark, with a value inspired by grand unified theories. For simplicity, we
do not discuss further this option. The Lagrangian of this sector is

$\mathcal{L}=(\partial_{\mu}\Phi)(\partial^{\mu}\Phi)-\frac{\lambda}{4}(|\Phi|^{2}-v_{\rm
PQ}^{2})^{2}+i\bar{\mathcal{Q}}(\partial_{\mu}-ig_{3}A_{\mu})\gamma^{\mu}\mathcal{Q}+(y_{Q}\bar{\mathcal{Q}}_{L}\mathcal{Q}_{R}\Phi+y_{Q}^{\dagger}\bar{\mathcal{Q}}_{R}\mathcal{Q}_{L}\Phi^{\dagger})\,.$
(16)

The first term is the kinetic term of the PQ field. The second term is the
potential of the PQ field, with the characteristic symmetry breaking form,
with $\lambda$ the PQ self-interaction strength. In the vacuum, the field
takes the form $\Phi=(v_{\rm PQ}+\chi)e^{ia/v_{\rm PQ}}$, with $\chi$ a
massive field and $a$ the massless Nambu-Goldstone boson arising from breaking
of the rotational symmetry in the complex plane of $\Phi$, i.e., $U(1)$
symmetry. The third term is the quark kinetic term, with the $SU(3)$ covariant
derivative: $A_{\mu}$ is the matrix gluon field, an element of $SU(3)$ (which
appears in the right-hand side of the Feynman diagram). Note that the new
quark does not couple to the Higgs boson, and so has no mass term in the
Lagrangian.

The fourth term is where the real action of the PQ mechanism resides: the
coupling between the PQ field and the new quark, with a Yukawa coupling
constant $y_{Q}$. This term gives a mass to the new quark when $\Phi$ gains a
vacuum expectation value: $\Phi$ acts like the Higgs for the new quark. If the
product $y_{Q}v_{\rm PQ}$ is large, then the new quark is too heavy to have
been observed in any collider experiment so far. This interaction term also
leads to a coupling between the axion and the new heavy quark, leading to the
required chiral anomaly (this gives the left-hand side of the Feynman
diagram).

The Dine-Fischler-Srednicki [23] Zhitnitsky [24] (DFSZ) model is slightly more
subtle to describe, since it involves changes to the Higgs sector of the
Standard Model, splitting the Higgs into up- and down-type doublets. This
additional subtlety avoids the introduction of the new quark, $\mathcal{Q}$,
of the KSVZ model, and instead mediates the coupling between axions and gluons
using the quarks of the Standard Model. We do not discuss this model in
detail, since it requires detailed background on the Higgs and the Standard
Model. From the perspective of low-energy phenomenology, the DFSZ model
differs from the KSVZ model in that it possesses a direct coupling between the
axion and the electron, $g_{aee}$, whereas the KSVZ model acquires this
coupling only at loop order, and so it is suppressed in magnitude. Since the
electron and the quarks of the Standard Model also carry electric charge, the
DFSZ axion acquires its coupling to electromagnetism via loop diagrams
mediated by the Standard Model fermions, in addition to its mixing with the
neutral pion, thus leading to a different strength of coupling,
$g_{a\gamma\gamma}$ than that in the KSVZ model (see below). These two
differences, in the values of $g_{aee}$ and $g_{a\gamma\gamma}$ , will allow
(in the event that the axion is discovered in the future) for the KSVZ and
DFSZ models (and other axion variants) to be distinguished by measurements as
described in the experimental companion to this review.

Many axion search strategies depend on the axion’s mass and coupling to
electromagnetism. These are related by [21]

$g_{a\gamma\gamma}=\frac{\alpha_{em}}{2\pi
f_{a}}\left(\frac{\mathcal{E}}{N}-1.92(4)\right)=\left(0.203(3)\frac{\mathcal{E}}{N}-0.39(1)\right)\frac{m_{a}}{{\rm
GeV}^{2}},$ (17)

where the numbers in brackets denote theoretical uncertainties, $\alpha_{em}$
is the electromagnetic fine structure constant, $\mathcal{E}$ is the
electromagnetic anomaly of the PQ symmetry and $N$ the colour anomaly. The
model dependent coupling coefficient
$C_{\gamma}=\frac{\mathcal{E}}{N}-1.92(4)$ is often used. The first term in
Eq. (17) is generated by the mixing between the axion and the neutral pion,
and is model-independent (so long as the axion has a coupling to gluons and
solves the strong $CP$ problem: the coupling can be absent for an ALP), while
$\mathcal{E}$ and $N$ in the second term are model-dependent. For example, in
the DFSZ model $\frac{\mathcal{E}}{N}=\frac{8}{3}$ (generated by the fermions
charged under both electromagnetism and PQ), whereas for the KSVZ model
described here $\frac{\mathcal{E}}{N}=0$. In principle,
$\frac{\mathcal{E}}{N}$ could be large, leading to a large value for
$|g_{a\gamma\gamma}|$ (there is some maximum set by unitarity). Alternatively,
$\frac{\mathcal{E}}{N}$ could by chance be such that the two terms in Eq. (17)
approximately cancel, leading to a very small value for $|g_{a\gamma\gamma}|$.
However, such a model would need to be deliberately constructed so as to hide
the axion, and in general we take the DFSZ model to give the smallest
canonical value of $|g_{a\gamma\gamma}|$. Precisely outlining the allowed
range of $g_{a\gamma\gamma}$ in viable models is an active area of research.

###  A.3 Equations of Axion Dark Matter

The Friedmann equation:

$\frac{1}{R(t)^{2}}\left(\frac{dR}{dt}\right)^{2}\equiv
H(t)^{2}=\frac{8\pi^{3}G_{N}}{90}g_{\star}(T)T^{4}\,,$ (18)

where $t$ is time “since the Big Bang”, $T$ is temperature, $G_{N}$ is the
Newton constant, $g_{\star}(T)$ counts the number of relativistic particles at
temperature $T$ (with different factors for bosons and fermions), and we have
defined the Hubble rate $H(t)$ from the cosmic scale factor $R(t)$. The
evolution of temperature with time, and also $g_{\star}(T)$, can be found from
elementary thermodynamic considerations. At the beginning of the hot Big Bang,
the temperture was at its maximum value, $T_{\textup{hot}}$, the time was
$t_{\textup{hot}}$, and the value of $R(t_{\textup{hot}})$ is chosen by
normalisation.

In the simple case described by Scenario A, the axion field evovles according
to the Klein-Gordon equation:

$\frac{d^{2}a}{dt^{2}}+3H(t)\frac{da}{dt}+m_{a}(T)^{2}a=0\,,$ (19)

with initial condition $\theta(t_{\textup{ hot}})=\theta_{i}$,
$\dot{\theta}(t_{\textup{ hot}})=0$, and the axion number density is

$n_{a}=\frac{1}{m_{a}(T)}\left[\frac{1}{2}\left(\frac{da}{dt}\right)^{2}+\frac{1}{2}m_{a}(T)^{2}a^{2}\right]\,.$
(20)

The temperature dependence of the axion mass, $m_{a}(T)$, can be computed from
the topological susceptibility as described briefly above. At temperatures
above the QCD phase transition the mass grows with temperature, roughly
described by a power law, $m_{a}\propto T^{4}$, while at low temperature it
has the asymptotic value given in Eq. (3). The Hubble parameter acts as a
friction term in the Klein-Gordon equation, which falls over time as the
Universe expands and cools. The axion field begins damped harmonic motion at
the epoch when $H(T)\approx m_{a}(T)$ (labelled $t_{\textup{dyn}}$ in Fig. 4).
At some slightly later time, $t_{\rm cold}$, the oscillation period of the
axion field becomes large compared to the Hubble rate, and the Klein-Gordon
equation enters an attractor solution in which the number density is conserved
modulo the expansion of the Universe, i.e., $n_{a}(t)/R(t)^{3}=\text{
const.}$. The axion dark matter density is then given by
$\rho_{a}=m_{a}n_{a}$, and is a function of the zero-temperature axion mass
and the initial field value.

At late times the axion is described by the Gross-Piatevski-Poisson equations.
These equations arise as the non-relativistic limit of the Klein-Gordon
equation for the axion field in a weakly perturbed Friedmann-Robertson-Walker
spacetime, under a phase approximation, $a=\sqrt{2/m_{a}}(\psi
e^{im_{a}t/\hslash}+\psi^{*}e^{-im_{a}t/\hslash})$. The “wave function” $\psi$
obeys:

$\displaystyle i\hslash\frac{\partial\psi}{\partial
t}+\frac{\hslash^{2}}{2m_{a}}\nabla^{2}\psi-\frac{\lambda_{a}}{2m_{a}}|\psi|^{2}\psi-
mV_{N}\psi=0\,,$ (21) $\displaystyle\nabla^{2}V_{N}=4\pi
G_{N}(m_{a}|\psi|^{2}+\rho)\,,$ (22)

where $\lambda_{a}$ is the dimensionless axion self-interaction coupling
constant. In the dilute instanton gas approximation,
$\lambda_{a}=-m_{a}^{2}/f_{a}^{2}$. The mass density of axions to leading
order is $m_{a}|\psi|^{2}$. The Newtonian gravitational potential is $V_{N}$,
and $\rho$ represents the density of all other gravitating matter.