# Axions for amateurs

David J. E. Marsha CONTACT David J. E. Marsh. Email<EMAIL_ADDRESS>a
Theoretical Particle Physics and Cosmology, King’s College London, Strand,
London, WC2R 2LS, UK

###### Abstract

Axions are an increasingly popular topic in theoretical physics, and are
sparking a global experimental effort. In the following I review the
motivations for the existence of axions, the theories underlying them, and the
methods to search for them. The target audience is an interested amateur,
physics undergraduate, or scientist in another field, and so I use no
complicated mathematics or advanced theoretical topics, and instead use lots
of analogies.

###### keywords:

axions, dark matter, haloscope, superradiance, axion electrodynamics, strong
cp problem

##  1 Invitation: a century of progress and problems

We live at an extraordinary time in scientific history: never before have we
known so much about the Universe, yet been so certain about our ignorance of
it.

###  1.1 Newton, Maxwell, and Einstein

In the world of Newton and Descartes three spatial coordinates, $x,y,z$, are
used to describe the locations of objects, including extended objects that are
imagined to be composed of infinitesimal regions of size $dx,dy,dz$. Dynamics
result from differential equations describing how the coordinates of an object
change with time. A useful example of an equation in classical mechanics is
the Navier-Stokes equation, which describes the motion of a fluid (here
assumed to be incompressible for simplicity):

$\frac{\partial\mathbf{u}}{\partial
t}+(\mathbf{u}\cdot\nabla)\mathbf{u}-\nu\nabla^{2}\mathbf{u}=-\frac{1}{\rho}\nabla
p+\mathbf{g}\,,$ (1)

where $\mathbf{u}$ is the velocity of a fluid element at location $(x,y,z)$,
$\nu$ is the viscocity, $\rho$ the fluid density, $p$ the pressure, and
$\mathbf{g}$ an external force. The fluid velocity is a vector and its
direction is specified with reference to unit vectors aligned with the
coordinate axes, $\hat{e}_{x},\hat{e}_{y},\hat{e}_{z}$, with components
$(u_{x},u_{y},u_{z})$. It appears in these equations that the coordinates
$x,y,z$ might be fixed, and the velocity of the fluid absolute.

It is possible, however, to recast the Navier-Stokes equations in ‘tensor
form’ whereby we learn that there is no special role for either the
coordinates $x,y,z$ themselves, or the absolute standard of rest. This form is
(you can find a derivation and definitions in any book on fluid mechanics):

$\frac{D\mathbf{u}}{dt}=\frac{1}{\rho}\nabla\cdot\mathbf{\sigma}+\mathbf{g}\,,$
(2)

where $D/dt$ is th convective derivative, and $\mathbf{\sigma}$ is the ‘stress
tensor’, which you can think of as a matrix $\sigma_{ij}$, which ‘points’ in
the space directions with components $\sigma_{xx},\sigma_{xy},\sigma_{xz}$,
and so on. Lagrange and Hamilton also realised that we are not bound to the
Cartesian $(x,y,z)$ coordinates, but that we can use ‘generalised coordinates’
$\mathbf{q}$ (for example, we might use cylindrical polar coordinates to
describe a fluid in a cylinder). The properties of the fluid are also
‘Galilean invariant’ (fluid experiments work the same in a stationary
laboratory, or in one moving at a constant speed e.g. on a train). So far, so
classical.

However, despite an invariance of these equations under changes of
coordinates, there still appears to be a special role for the absolute
structure of the space in which events take place. For example, we find the
distance, $D$, from one point in the fluid to another by squaring the vector
that joins them, which in Cartesian coordinates is:

$D^{2}=\Delta x^{2}+\Delta y^{2}+\Delta z^{2}\,.$ (3)

The actual distance $D$ does not change when we use different coordinates, or
if we are moving relative to the fluid.

Despite being able to cast the Navier-Stokes equations in tensor form and
change our coordinates, there are some things we can’t do. For example, the
equations are only first order in the time derivative, $\partial/\partial t$,
while they are second order in spatial derivatives via the Laplacian,
$\nabla^{2}$ (most obvious in the original form we gave in Eq. 1). This means
that we can’t make changes of coordinates that mix up space and time without
changing the form of the equations (a Galilean transformation with constant
velocity is an exception to the rule, since only derivatives of $\mathbf{u}$
appear in Navier-Stokes in tensor form). Also, our Navier-Stokes vectors and
tensors, like the fluid velocity itself $\mathbf{u}$, only ‘point’ in the
spatial directions. We don’t need a ‘timelike’ direction for $\mathbf{u}$ with
a timelike unit vector $\hat{e}_{t}$ and component $u_{t}$. Similarly, the
fluid stress tensor also has no component $\sigma_{tx}$ or others ‘pointing’
in the time direction. Space and time are fundamentally different in Newtonian
mechanics, with time flowing independently ‘from the outside’, the same for
all observers and all coordinate systems.

Einstein began to see the cracks in this picture thanks to the dawn of the
theory of electromagnetism due to Maxwell in the late 19th century. It is said
that Maxwell’s theory represented both the apex, and the end, of classical
physics. The familiar form for Maxwell’s equations for the electric field,
$\mathbf{E}$, and magnetic field $\mathbf{B}$ is:

$\displaystyle\nabla\cdot\mathbf{E}$
$\displaystyle=\frac{\rho}{\varepsilon_{0}},$ (4)
$\displaystyle\nabla\cdot\mathbf{B}$ $\displaystyle=0,$ (5)
$\displaystyle\nabla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial t}$
$\displaystyle=0,$ (6)
$\displaystyle\nabla\times\mathbf{B}-\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial
t}$ $\displaystyle=\mu_{0}\mathbf{J},$ (7)

where $\rho$ is the free electric charge density, $\mathbf{J}$ is the electric
current density, $\varepsilon_{0}$ is the vacuum permittivity (electric
constant), and $\mu_{0}$ is the vacuum permeability (magnetic constant). The
first thing that is different between these equations and the Navier-Stokes
equations is that they are the same order in space and time derivatives.
Secondly, when they are recast in tensor form the main object of concern is
the Faraday tensor, $\mathbf{F}$ which we can think of as a _four by four
matrix_ with some components _in the time direction_ , for example
$F_{tx}=E_{x}$ (for the explicit form and derivation, I enjoy the presentation
by Goldstein [1]).

Maxwell’s equations are equations for abstract objects known as _fields_ (the
electric and magnetic fields) in _spacetime_ , in contrast to the Navier-
Stokes equations that are equations for physical (tangible, touchable) fluids
composed of particles, which live in space and evolve in time. When Einstein
developed the concepts to describe Maxwell’s equations as equations in
spacetime, he was ultimately led to the conclusion that not only did he need
equations to describe fields (and fluids) evolving in spacetime, but he also
required equations to describe _spacetime itself evolving in spacetime_ : the
theory of GR. The simplest example of using GR in this way is also one of the
most profound: cosmology. 111For further reading on GR I recommend the
introductory book by Schutz [2] for practical purposes, while the ‘first
track’ in Misner, Thorne, and Wheeler [3] contains lots of thought experiments
and intuition. For those keen to do research, I enjoy Carroll [4].

When we look at very distant galaxies, we see something strange. The light
coming from those galaxies is very red. It isn’t caused by the light getting
dimmer, or any kind of filter or gas in the way. Thanks to quantum mechanics,
the light from different atoms and molecules always comes in the same colour
as seen sitting next to the atom, and each colour is in a fixed ratio to the
others. By lining up those colours of light seen from different galaxies, we
can see that the colours are systematically shifted compared to the same atoms
and molecules on Earth: all light from distant galaxies is more red than it is
here on Earth.

The only explanation for this weird fact that fits with everything else we
know about the Universe, is that the light is losing energy as it travels to
us. Red light has lower energy than blue light: that’s why you wear sunblock
against ‘ultraviolet’ (very blue) rays from the sun. Light is losing energy as
it travels to us because the space in between those distant galaxies and us is
changing: we say the Universe is expanding, but what is really happening is
that distances are getting longer relative to what they were in the past. As
light travels over these increasing distances, it loses energy, and so becomes
redder. The fabric of spacetime itself is changing over time.

In GR the fundamental object of consideration is the _spacetime metric tensor_
, which tells us how to measure the distance between events in spacetime. If
the Universe is expanding in a homogeneous and isotropic way (the same
everywhere and in all directions), then the metric tensor gives the spacetime
separation, $s^{2}$, in analogy to Eq. (3) as:

$\Delta s^{2}=-c^{2}\Delta t^{2}+a(t)^{2}(\Delta x^{2}+\Delta y^{2}+\Delta
z^{2})\,,$ (8)

where $a(t)$ is the ‘cosmic scale factor’. GR provides dynamical equations for
$a(t)$ known as the Friedmann equations. The simplest Friedmann equation is:

$\frac{da}{dt}=a\sqrt{\frac{8\pi G_{N}}{3c^{2}}\rho}\,,$ (9)

where $G_{N}$ is Newton’s gravitational constant,
$c^{2}=1/\epsilon_{0}\mu_{0}$ is the speed of light in vacuum (notice how this
is determined by physical constants in Maxwell’s equations), and $\rho$ is the
energy density. This very profound equation allows us to conceive the science
of the whole Universe called cosmology: the matter content of the Universe
affects its evolution as given by the scale factor. Vice versa, the dynamics
of $a(t)$ affects the matter density, via the continuity equation, which for a
pressureless fluid reads:

$\frac{d\rho}{dt}+\frac{3}{a}\frac{da}{dt}\rho=0\,.$ (10)

If the energy density is dominated by a fluid with constant positive pressure,
the Friedmann equation, Eq. (9), has solutions where $a(t)\propto t^{p}$, i.e.
the scale factor grows with time, which is consistent with the observed fact
going back to Hubble that distant galaxies appear to recede from us, leading
to the idea of an expanding Universe. If the Universe expands, then energy
density, $\rho$, decreases with time going into the future. If we know what
the Universe contains today, then we can also ask what it was doing in the
past. Reversing time we would see the expanding Universe contract, and so the
energy density goes up in the past. As the energy density goes up, the
Universe gets hotter and hotter, eventually reaching the state of a plasma
known as ‘the hot big bang’.

###  1.2 Modern cosmology and dark matter

GR and the theory of cosmology were developed extensively during the mid to
late twentieth century. When precise cosmological measurements began to be
made, and in particular of something called the ‘cosmic microwave background’
(CMB) in the 1990’s and early 2000’s, cosmologists began to realise that the
Universe was holding a dark secret (for a review of the development of these
measurements, see Page’s book [5]). So what is the CMB?

As we look further into space, past the galaxies that told us about the
expansion of the Universe, eventually things go black - quite literally. There
was a period around 13.5 billion years ago, called the ‘cosmic dark ages’,
when there were no stars in the entire visible Universe. Stars form when
clouds of gas are attracted by their own gravity, and the gas gets so closely
packed that a nuclear chain reaction starts. Although in the early Universe,
the _average_ the energy density was higher (since $\rho$ goes up), it was
also very smooth and flat. This is because it takes time for the inward force
of gravity to overcome the expansion of the Universe and allow galaxies to
form, in a process known as ‘hierarchical structure formation’. Before
galaxies could form, during the cosmic dark ages, the gas density is too low,
and those nuclear chain reactions that form stars couldn’t begin.

Further back still, another strange thing happens. We stop being able to see
through the Universe at all, and it becomes very bright again, but bright only
in the microwave part of the electromagnetic spectrum. This is the CMB, and it
is about 13.7 billion light years away from us.

Why can’t we see through the CMB? Well, why can’t we see through the Sun? The
Sun is made of hydrogen and helium gas. Here on Earth, you can see through
hydrogen and helium gas, but when you heat them up past about 3000 degrees,
they break apart. An atom of hydrogen is one proton (positive), and one
electron (negative). Put them together, and hydrogen is electrically neutral.
Break them apart, and you have charged particles floating around freely. Rays
of light are electromagnetic waves, and they interact with charged particles.
In a soup of free protons and electrons (a plasma, like the Sun) light gets
bounced around so much it can’t get through. If you heat hydrogen up to a 3000
degree plasma, it becomes opaque: light can’t get through.

Scientists in the mid twentieth century knew that the Universe was expanding,
and they also knew it contained a lot of hydrogen gas. They ran their
equations backwards and concluded that at some point in the past the Universe
would have been so dense that it would turn into one gigantic star: a plasma
of hot gas and electromagnetic waves, that we wouldn’t be able to see through.
These scientists also predicted, by taking the 3000 degrees it takes to break
apart hydrogen, and working out how far back in time you have to go until the
Universe is that dense and hot, and then working out how much energy the light
would lose travelling to us from that time, that this gigantic star should be
visible to us now, as microwave radiation at about -270 degrees, covering the
whole sky. And that is exactly what the CMB is. We are viewing the Universe as
the inside of a gigantic star.

This is where things get really interesting.

Because we can’t see beyond the CMB, we see it as a single surface of
microwave radiation, coming at us from all directions, all the time. This
single surface is like a photograph of the state of the Universe 13.7 billion
years ago. By looking closely at it, and trying to understand it and how it
connects to the Universe we see today, cosmologists can learn about the past,
present, and the future, of the Universe.

The cosmic microwave background is radiation left over from the big bang that
we observe filling the night sky at a cool -270 degrees Celsius: a picture of
the Universe as it was 13.7 billion years ago. Using satellites and ground-
based radio telescopes, astronomers can observe small differences in this
temperature across the sky, differences at one part in 100 thousand, and use
this to map out the distribution of matter in the very early Universe. Using
GR, specifically perturbation theory based on the Friedmann equations,
cosmologists can work out how the Universe evolved from 13.7 billion years ago
until today. This allows us do two things:

  1. 1.

Find out the matter content and initial conditions required to correctly
describe observations of the CMB.

  2. 2.

Evolve that Universe further forward and compare it to the actual Universe
observed today.

‘The actual Universe observed today’ is shorthand for observations made by
telescopes that map the locations of 100s of millions of different galaxies,
which are seen to cluster together in a grand structure called the cosmic web
(two of the many instruments to make these measurements are the Sloan Digital
Sky Survey [6] and the Dark Energy Survey [7]).

Now the critical part. The Universe we observe in the CMB and the cosmic web
of galaxies only evolve consistently one into the other if we add an extra
ingredient into the equations of GR. We must add a new form of matter in the
form of a (very) cold, (almost) pressureless fluid. This fluid is called dark
matter (DM), and its total mass is measured to a precision of 1% using the
CMB. The idea of this interplay between measurements and theory that is used
to determine the DM density consistently between the early Universe CMB, and
the late Universe cosmic web is known as hierarchical structure formation and
underpins modern cosmology [8].

The inferred average density of DM from measurements of the CMB by the
_Planck_ satellite is [9]:

$\rho_{\rm DM}=2.3\times 10^{-27}\text{ kg m}^{-3}\,,$ (11)

roughly equivalent to the mass of a single hydrogen atom in a cubic metre.
This number might seem incredibly tiny, and it is: space is very empty. This
should be compared to the average density of ordinary matter (mostly in the
form of hydrogen gas), which can be inferred in a similar way:

$\rho_{\rm matter}=4.2\times 10^{-28}\text{ kg m}^{-3}\,.$ (12)

There is about five times more DM than ordinary matter in the Universe. To
give a sense for how such a tiny number can be measured, we know that the
average density of DM in the Milky Way is about 10000 times larger than Eq.
(11). If you multiply this by the volume of the Milky Way, very roughly
$2\times 10^{17}$ cubic light years, then we get a mass more than a million
million times the mass of the Sun, which is more than the total mass of
visible stars in the Milky Way.

It is often said that we don’t know what DM is, but that is not really true.
The macroscopic theory of a cold, pressureless fluid works perfectly well to
describe cosmological measurements. Furthermore, the possibilities for
microscopic theories are severely restricted: they have to behave like the
cold fluid necessary to explain the structure of the CMB, and whatever makes
up that cold fluid had to be present in the very early Universe in the
primordial plasma. 222We focused on evidence for DM from the CMB because it is
_impossible_ to explain the CMB any other way. Modifying gravity doesn’t work
without also introducing new dark degrees of freedom, i.e. without introducing
DM. The mystery is: what is the microscopic theory of dark matter? Axions
provide one possible answer, and one that is gaining popularity among
theorists and experimentalists.

##  2 What is an axion?

###  2.1 One particle, two mysteries

####  2.1.1 Strong-$CP$ problem

Figure 1: The effect of time, $T$, and parity, $P$, transformations on an
electric or magnetic dipole moments (which must _both_ point in the same
direction as the spin given by the red arrow). We can work out the effect by
considering that $CPT$ must be conserved, and by the action of $T$ and $P$ and
on the applied electric field and on a magnetic field (recall that
$\mathbf{B}$ can be thought of as sourced by a current, so $T$ changes its
direction). (Reproduced from Ref. [10])

The story of axions begins not with DM, but with another mystery concerning
particle physics.The Standard Model of particle physics contains electrons,
and their heavier siblings, mu and tau, and the lighter cousins called
neutrinos. Then we have quarks: up, down strange, charm, top and bottom, in
three colours each, red, blue, and green. For every one of these particles,
there is an anti-particle as well. Next there are force carriers, called
bosons: the photon of electromagnetism, the gluon that holds protons and other
things together with the so-called strong nuclear force, and the $W$ and $Z$
bosons responsible for radioactivity (the weak nuclear force). Lastly there is
the Higgs boson, responsible for giving mass to the other particles.

There are three ‘discrete’ symmetries commonly considered in reference to the
Standard Model: charge conjugation (denoted by $C$), ‘parity reversal’ or
‘inversion symmetry’ (denoted by $P$) and time reversal, $T$. Each symmetry,
denote it $S$, acts on a particle state, $\psi$ as:

$S\psi=(\pm 1)\psi\,,$ (13)

if the particle state is even ($+1$) or odd ($-1$) under the symmetry. It is
believed that the combination $CPT$ should always leave any state even, i.e.
doing all of them in turn returns $(+1)$. Radioactive decays mediated by the
$W$, $Z$ and photon (unified as the ‘electroweak force’) treat particles and
anti-particles differently. Specifically, the weak interactions violate the
combination of symmetries $CP$: if states before an interaction are $CP$ even,
then afterwards the states that come out are $CP$ odd. This symmetry violation
was first observed in the 1960’s in decays of $K$-mesons [11], and is now
known to occur in many other situations involving the electroweak force.
Interactions mediated by the gluons of the strong force, however, are more
democratic, and do not distinguish between particle and anti-particle. The
mystery of why _for the same particles_ the electroweak force violates $CP$,
yet the strong force conserves it is known as the ‘strong-$CP$ problem’.

Before we get into some physics of the strong-$CP$ problem, it is useful to
begin with an analogy, which I have taken from Sikivie [12]. Let’s imagine we
want to play pool or snooker, or do anything that requires a very flat
surface. The world around us, however, is very much not flat. How can you make
a very flat surface in a room with wonky floors? The answer is to attach a
pendulum or other large weight to the surface along with a freely rotating
cantilever arm. The weight of the pendulum will pull the arm perfectly
vertical, and the attached table will be perfectly flat, like a spirit level.
To make this all stable, you will probably want some vibration isolating
springs. When you first install this pendulum and spring-loaded table, there
will be a little bit of wobble caused by the installation, but thanks to our
vibration isolating springs this wobble will go away with time, depositing
energy into the springs. If you had a very keen eye, you might be able to work
out the pendulum and springs mechanism was there by the response of the balls
moving on the table and bouncing off the cushions.

Now let’s get back to physics. If the strong force violated $CP$ like the weak
force then strongly interacting particles, like the neutron, $n$, should have
$CPn=-n$. Because the neutron should be even under $CPT$, then if $CPn=-n$
then $Tn=-n$ too, i.e. $CP$ violation implies $T$ violation. This should help
you understand how we can measure whether the strong force violates $CP$ or
not, as we now explain.

Consider an ‘electric dipole moment’ (EDM), $\mathbf{d}$, which means that
there is an energy associated to application of an electric field,
$\mathbf{E}$ to a particle. The Hamiltonian is:

$H=-\mathbf{d}\cdot\mathbf{E}\,.$ (14)

In quantum mechanics the Hamiltonian measures energy, so this equation tells
you the energy of the system i.e. there is a lower energy if the EDM,
$\mathbf{d}$, aligns with the applied $\mathbf{E}$ field. The neutron is known
to have a magnetic dipole moment (MDM), $\mu$, for which the Hamiltonian is
similarly, $H=-\mu\cdot\mathbf{B}$. Now consider what $P$ and $T$ symmetries
do to an EDM and an MDM, by their action on $\mathbf{E}$ and $\mathbf{B}$,
illustrated in Fig. 1. With some thought, this diagram should convince you
that the existence of a neutron EDM (or an EDM for any strongly interacting
particle) would violate $CP$, because it separately behaves differently from
the MDM under $P$ and $T$.

It is, however, an experimental fact that the neutron EDM is undetectably
small (i.e. zero within very small experimental errors). By exposing a large
number of neutrons to electric and magnetic fields, experimentalists can try
and measure whether there is any relative energy splitting caused by the
$\mathbf{E}$ field and measure the Hamiltonian co-efficient $\mathbf{d}$ in
Eq. (14). The value of $\mathbf{d}$ is found to be consistent with zero to
very high precision [13]. Thus, the strong force must in fact conserve $CP$,
as we stated above.

In the theory of the strong interactions, known as _quantum chromodynamics_
(QCD), the amount of $CP$ violation is expressed by a _dimensionless_ number,
$\theta$, which can be visualised classically as the base angle between the
three up and down quarks in the neutron arranged in an isosceles triangle. The
size of this angle is a _constant_ in the theory, and determines the neutron
EDM as $\mathbf{d}\propto\theta$. 333The constant of proportionality can be
estimated by dimensional analysis. An EDM has units charge times distance. The
charge we have to play with is the quark charge, $e/3$, and the distance is
the size of the neutron, $10^{-15}\text{ m}$. So we estimate the constant as
the product of these numbers, about $3\times 10^{-14}\,e\text{m}$. The value
of the neutron EDM computed using quantum field theory [14] is $d=5\times
10^{-14}\theta\,e\text{m}$: very close to our naive estimate. The observed
smallness of $\mathbf{d}$ implies $\theta\approx 10^{-10}$, i.e. in our
classical triangle picture the quarks all lie on an almost perfect straight
line. The strong $CP$ problem is that $\theta$ has two unrelated
contributions: from the electroweak force (that violates $CP$) and from the
strong force. A small value of $\theta$ implies a delicate cancellation
between two numbers.

####  2.1.2 Axions to the rescue

Figure 2: Feynman diagram that couples the axion field, $\theta$, to the
gluons, $g$ via quark, $Q$. This diagram, if it has opposite sign for left and
right handed quarks, will ensure that $\theta$ is the same $\theta$
responsible for the neutron EDM.

In the 1970’s, theoretical physicists realised this situation of having and
yet not having $CP$ symmetry in the Standard Model was a problem, and set
about trying to explain why it might be. The solution came in 1977 thanks to
Peccei and Quinn [15]. Peccei and Quinn realised that if they could ‘promote’
$\theta$ from a constant to field, which is not a constant but varies
throughout spacetime (like the fields of electromagentism that we met above),
and if they could also make $\theta=0$ somehow energetically favourable, then
the strong-$CP$ problem would solve itself dynamically. We will only have
space here to outline Peccei and Quinn’s method for achieving this trickery.

The beginning of the solution is to answer the question: ‘when is a field an
angle?’, and the answer is ‘when it is a complex number’. A complex number $z$
is first introduced to us as $z=x+iy$ with $x$ the real axis and $y$ the
imaginary axis. Since the complex numbers can thus be thought of as
coordinates on the ‘Argand plane’, with $x$ and $y$ the Cartesian coordinates,
we can also use other coordinates, in particular plane polar coordinates. In
these coordinates, $z=Re^{i\theta}$ and $x=R\cos\theta$, $y=R\sin\theta$ (do
not confuse these $x$ and $y$ with $x$ and $y$ coordinates in spacetime!).
Peccei and Quinn introduced a new field $\Phi$ to the Standard Model, which
takes on a complex value everywhere in spacetime, i.e. in polar coordinates in
the complex plane, and Cartesian coordinates in spacetime:

$\Phi=R(t,\mathbf{x})\exp[i\theta(t,\mathbf{x})]\,.$ (15)

Now we have identified the right field content, we need to try and make the
$\theta$ in Eq. (15) the same $\theta$ responsible for the neutron EDM. As
Feynman explains it [16], to every field there is a particle, and particle
physicists compute the effects of interactions between particles by drawing
squiggly little diagrams and adding up the values of stopwatches carried round
by these particles as they follow the lines on the diagrams.

The particular squiggly diagram that allowed Peccei and Quinn to make the
$\theta$ in Eq. (15) the same $\theta$ responsible for the neutron EDM is
shown in Fig. 2. The interaction we need is one between $\theta$ (which we
will now start calling the ‘axion particle’ 444The name ‘axion’ is due to
Frank Wilczek. It was Weinberg and Wilczek who, independently later in 1977
(published in 1978) [17, 18] first realised that Peccei and Quinn’s theory
predicted the existence of a particle, and computed its mass. Wilczek coined
the phrase ‘axion’ after the American detergent. The ‘axi’ comes from the
left/right handed necessity of the interaction between axions and quarks,
which physicists call ‘axial’, while the ‘on’ just sounds like a particle name
(think ‘boson’, ‘neutron’ etc.). The axion ‘cleans up the mess’ of the
strong-$CP$ problem. Weinberg’s name for the particle was the ‘Higglet’, since
it is a bit like a Higgs boson, only lighter.) and a quark, $Q$. The
interaction shown is between $\theta$ and ‘left handed’ quarks only, i.e.
those where the spin angular momentum points opposite to the linear momentum.
For Peccei and Quinn there must also be an equivalent diagram for right handed
quarks, but with an opposite sign. If we are allowed to draw these particular
diagrams then when we add up all the little stopwatches we find that the axion
field, $\theta$, couples to gluons, $g$, in just the right way that the value
of the neutron EDM is determined, at every point in spacetime, by the value of
the axion field, $\theta$. 555The actual computation requires a graduate
course in quantum field theory. You can find it in these references [19, 20].

The final ingredient in the Peccei-Quinn recipe is to make $\theta=0$
energetically favourable. Making a particular field value energetically
favourable means writing down a potential energy that depends on the value of
the field, $V=V(\Phi)$, which has a minimum at the desired value. Firstly, the
potential $V$ should very strongly favour $R=v\gg 0$, because the polar
coordinate representation Eq. (15) is only valid away from the origin of the
complex plane: at the origin $\theta$ is not defined. This part of the problem
is solved with a ‘wine bottle’ type potential, like the Higgs has in the
Standard Model, as pictured in Fig. 3. The potential should also favour
$\theta=0$, to solve the strong $CP$ problem and set the neutron EDM to zero,
the simplest option being $V(\theta)\sim\theta^{2}$, i.e. a harmonic
potential, which in quantum field theory corresponds to a mass for the axion
particle. The bit that favours $\theta=0$ should not be too big to mess with
the definition of polar coordinates: it should be a perturbation to the
leading wine bottle piece. This part of the potential comes for free due to
the strong force itself and the way that the axion interacts with mesons and
the ‘topology of the QFT vacuum’ (related to large amplitude but short-lived
fluctuations of the gluon fields called instantons).

Because the axion mass comes for free from the strong force itself, it turns
out that the only free parameter in this model is the location in field space
where $V$ is minimised in the $R$ direction, i.e. the value of what we called
$v$, which determines something called the ‘axion decay constant’, denoted
$f_{a}$. The axion mass, and the strengths of its interactions with all the
particles of the Standard Model are determined by the value of $f_{a}$. The
best current computation of the axion mass (at low temperatures) is [21]:

$m_{a}=5.7(1)\,\,\mu\text{eV}\left(\frac{10^{12}\text{ GeV}}{f_{a}}\right)\,.$
(16)

Notice that if $f_{a}$ takes the very large reference value given here, the
axion is extremely light (c.f. the electron, which has a mass $m_{e}=511\text{
keV}$). Furthermore, if $f_{a}$ increases, the axion mass goes down, and so
does the strength of all its interactions: everything goes _inversely_ with
$f_{a}$. We do not know the axion mass, and the task of discovering the axion
is to measure its mass experimentally.

Figure 3: ‘Wine bottle’ potential for the complex field $\Phi$ (the angular
direction is opened up for visualisation). The potential is minimised at a
fixed value of $R=|\Phi|=v$. We do not show the small perturbation $\Delta
V\sim\theta^{2}$ that ‘tilts the wine bottle’ and minimises it at a fixed
angle $\theta=0$. Think about how the dreggs in a real wine bottle accumulate
in the bottom of the punt when the wine is poured, and you are visualising the
axion field reaching the minimum of its potential.

####  2.1.3 Axion dark matter

Still in the 1970’s, particle physicists realised that if the unknown constant
in the axion model, $f_{a}$, was set to be close in value to a similar
parameter in the Standard Model, the Higgs vacuum expectation value
$v_{H}=246\text{ GeV}$ (the value it had to take in the original models), then
the axion would have already shown up in their experiments. Was the model of
Peccei, Quinn, Weinberg, and Wilczek dead? Shortly afterwards, between 1979
and 1981, eight different physicists working in four different groups on
either side of the iron curtain cooked up two different variations on the
model [22, 23, 24, 25]. In these variations, $f_{a}$ was allowed to be very
large, making the axion ‘invisible’ to the particle physicists’ experiments
(recall: we said above that interaction strengths are inverse with $f_{a}$).

The next step in the story came in 1983 when three different groups started to
think about what happened to this invisible axion in the early Universe [26,
27, 28]. They did this by considering the wave equation that describes the
dynamics of the axion field in an expanding Universe with scale factor $a$
that we introduced above. We will study this equation, and how it produces
axion DM in Section 3.1. In short, the early Universe naturally provides
conditions that leave the axion displaced from the minimum energy value at
$\theta=0$ (the Universe imparts some energy to the axion field). Relaxation
from the displaced value to zero leads to damped oscillations in the axion
field, and it is the energy density carried by these oscillations that can
explain DM.

The model for axion DM introduces a new parameter, the initial value of
$\theta$, and axions are capable of explaining all, or none, of the cosmic DM
for a wide range of values for the particle mass, making the experimental task
of finding axions very difficult. Nonetheless, the emergence of axions as a DM
candidate is one of the main factors that drives interest in them today.

###  2.2 Many dimensions, many axions

The origin of axions as a solution to the strong-$CP$ problem turns out to be
just one place in which they crop up in theoretical physics; another one is
_string theory_.

Trying to marry the two pioneering successes of twentieth century physics,
quantum mechanics and general relativity, turns out to be immensely difficult.
Quantum mechanics wants everything to jump around and undergo zero-point
fluctuations and for all fields to be quantised as point-like particles.
General relativity wants the fabric of spacetime to be a dynamical field
called the metric. If we quantise the metric as a particle called the
graviton, and let it fluctuate, then all hell breaks loose on small scales and
at high energies. Spacetime fluctuates and produces singularities and black
holes with ever growing number, and eventually the equations just give
infinity in answer to any question. String theory is a theory of _quantum
gravity_ that replaces particles with extended objects known as strings.
Because the strings have a finite length, then the short distance problems of
particle quantum gravity are avoided.

General relativity predicts lots of things, but there are two things it is
silent on. The first is the number of dimensions of spacetime. GR works just
fine in two, three, four, or 104 dimensions. Any more than four would seem
silly, but as Kaluza and Klein (and also Einstein) famously showed, you can
easily hide extra dimensions, and you may even get something desirable from
doing so, in terms of an ability to describe GR and electrodynamics with the
same theory. 666You can read about Kaluza-Klein theory on Wikipedia. There
could be extra dimensions of spacetime and GR gives us no guidance:
experimentally all we know is that if extra dimensions exist, then they must
be smaller than around a micron [29]. Secondly, GR does not specify the
_topology_ of spacetime. That is, if spacetime on the largest scales is closed
on itself, like a ball, or a more complex shape like a torus (doughnut) or
Riemann surface (multi-holed doughnut), GR does not give an answer for a
dynamics of why, because changes of topology are impossible in Einstein’s
theory. It is an experimental question also what the topology of the four
large dimensions we observe is. Cosmologists look for evidence of cosmic
topology, and currently we know that if the Universe is a doughnut, or a
related topology, then the size of the ‘fundamental domain’ is larger than 98%
times the size of the visible Universe [30]. If there are tiny extra
dimensions, their topology is virtually unconstrained.

Quantum field theory, the most fundamental theory of quantum mechanics we have
on which the Standard Model is based, also fails to make predictions about
certain things. QFT tells us very little about what the particle content of
the Standard Model should be. We must find out experimentally which particles
exist. The famous saying of Rabi regarding the muon ‘who ordered that?’ is now
more rightly ‘who ordered all of this?’. In particular, it is a mystery why
there should be three generations in the Standard Model corresponding to the
heavier siblings of the electron and quarks. Even invoking the anthropic
principle, that the laws of physics should be complex enough to allow life, to
explain some complexity, then the existence of the weak nuclear force seems
mysterious.

String theory provides answers on all of the above questions. String theory is
such a rigorous mathematical structure that it breaks down if its ingredients
are changed. String theory predicts that there are ten dimensions of
spacetime. 777In the weakly coupled limit. In the strong coupling limit,
degrees of freedom on the strings reorganise themselves into an ‘emergent’
11th dimension or even a 12th ‘half’ dimension in so-called M-theory and
F-theory. String theory also predicts that there are only strings of one type
(although there are different ‘dual’ descriptions), and all the particles we
observe are related to their different modes of vibration. String theory does
not predict the topology of the extra dimensions (although unlike GR it does
allow for some changes of topology), however the particle content we observe
in the four large dimensions is intimately related to topology. It is here
that axions appear in the story.

At low energies, and in ten dimensions, the dynamics of strings in string
theory reduces to what is called _supergravity_. Supergravity is a theory of
the metric (like GR), fermions (like the known particles including electrons),
and a host of other fields known as $p$-form fields. A $p$-form field is like
the four-dimensional electromagnetic vector potential,
$A=(\varphi,\mathbf{A})$ (where $\varphi$ is the electrostatic potential, so
that $\mathbf{E}=\nabla\varphi$, and $\mathbf{B}=\nabla\times\mathbf{A}$), but
a $p$-form is a rank-$p$ anti-symmetric tensor with $10^{p}$ components (not
all of them independent). So if $p=1$, think of a vector with 10 entries, if
$p=2$ think of a matrix with 100 entries, if $p=3$ think of a cube of fields
with 1000 entries, and so on.

For the metric, let’s first imagine we simply have five dimensions: our usual
four plus a circle. The equivalent to Eq. (8) is

$\Delta s^{2}=-c^{2}\Delta t^{2}+a(t)^{2}(\Delta x^{2}+\Delta y^{2}+\Delta
z^{2})+\mathcal{R}(t,\mathbf{x},\theta)^{2}\Delta\theta^{2}\,,$ (17)

where $\theta$ is the coordinate around the circle and $\mathcal{R}$ is the
radius of the circle (which can change from place to place as we move around,
if the circle is bent). Einstein’s equations give rise to equations of motion
for the circle radius, just like the Friedmann equation gave us an equation
for $a$ in Eq. (9). How do we interpret this? An extra dimension, if small
enough, would be invisible to us, just like a tightrope walker cannot move
around the thickness of the tightrope, but an ant can. So if $\mathcal{R}$ is
small, we won’t notice immediately by walking in the fifth dimension. Thanks
to GR, however, $\mathcal{R}$ is also dynamical and interacts with, for
example, the scale factor $a$ and all other fields via gravity. To make things
realistic there needs to be an extra energy penalty to fix $\mathcal{R}$ to a
small but non-zero value and hide its dynamics from us so we don’t notice. In
string theory this problem is called ‘moduli stabilisation’ and it can be done
using physics similar to what we discussed for the axion potential above.
Fluctuations of $\mathcal{R}$ with this potential energy behave like an extra
heavy particle called a modulus from the four dimensional perspective.

Now for the $p$-form fields. Well, being like the electromagnetic vector
potential, the $p$-form field also defines $p$-form electric and magnetic
fields, and these obey equations just like Maxwell’s equations, which also
have propagating wavelike solutions. In the example of the extra dimensional
circle and a 1-form just like the Maxwell field, we would need just a single
field $\phi$ to describe the dynamics of the energy stored in magnetic flux
wrapped around the circle as viewed from a four dimensional perspective (this
example is treated in some detail in the lecture notes Ref. [31]). The field
$\phi$ is known as an axion, since it has the same $CP$ properties as the
axion from the strong-$CP$ problem (inherited from the parity and time
reversal properties of the $p$-form electric and magnetic fields). This axion
does not have to couple to gluons, and might not solve the strong-$CP$
problem, although similar axions can.

String theory has six extra dimensions, not just one as in our simple circle
example, and a whole host of different $p$-form fields. The possibilities for
the topology are (possibly) endless, but for semi-realistic cases giving rise
to the Standard Model in four dimensions are thought to be given by a
topological object called a Calabi-Yau manifold, and all such topologies will
have axions [32, 33]. It is known how to construct huge numbers of different
Calabi-Yaus. The number of axions (giving the dynamics of $p$-form energy flux
in the extra dimensions) and moduli (giving the dynamics of the extra
dimensions themselves) arising in the four dimensional theory from such a
complex extra dimensional space can be large, in many cases into the hundreds,
opening up a massive playground for phenomenology known as the ‘string
axiverse’ [34, 35, 36]. Very recently, it has been possible to also compute
the masses and decay constants of the axions across large parts of the string
theory landscape, allowing even to falsify some fraction of string theory
models [37, 38], a task that has been declared in the past by detractors of
string theory to be impossible, yet axions hold the key.

###  2.3 What’s in a name?

Before we carry on, a point on nomenclature. We met the axion first as a
solution of the strong-$CP$ problem in quantum chromodynamics (QCD). We have
also met axions in string theory. There are yet more realisations of axions in
other areas of particle physics. It is common nomenclature to call all of
these particles ‘axions’, and call the axion that solves the strong-$CP$
problem (which, incidentally, is also one of the string theory axions) ‘the
QCD axion’. We adopt this nomenclature from hereon. Elsewhere you will see
authors use the convention that only the axion that solves the strong-$CP$
problem is called ‘axion’, and all the others called ‘axion-like particles’
(ALPs).

##  3 The physics of axions

The way we have so far described them, axions might seem very complicated.
They are fields in the complex plane coupled to gluons by quantum loops that
control the neutron electric dipole moment. They are energy stored in $p$-form
electric and magnetic fields threading extra dimensions of spacetime. At the
level of effective field theory, however, many aspects of axion physics are
remarkably simple. We only have space here to discuss two aspects of axion
physics. The first is the details of how axions behave in cosmology and solve
the problem of DM. The second is one way in which axions might interact with
ordinary matter and allow us to search for them.

###  3.1 Axion waves in the cosmos

Figure 4: Numerical solution of Eq. (19), on a logarithmic time axis.
Illustrated are the analytically estimated value of $t_{\rm osc}$ (vertical
dotted line), the damping envelope (dashed line), and the undamped cosine
solution (thin black line, offset for visibility). (Reproduced from Ref. [39],
published by Princeton University Press, forthcoming)

We will now work with $\phi=f_{a}\theta$, which is the ‘canonically normalised
axion field’. The wave equation for $\phi$ is:

$\frac{1}{c^{2}}\frac{\partial^{2}\phi}{\partial
t^{2}}+\frac{3}{c^{2}a}\frac{da}{dt}\frac{\partial\phi}{\partial
t}-\frac{1}{a^{2}}\nabla^{2}\phi+\frac{m_{a}^{2}c^{2}}{\hbar^{2}}\phi=0\,.$
(18)

I want you to notice a number of things about this equation. Firstly, it is a
wave equation for a _classical_ field $\phi$, just like the wave equations we
can derive for $\mathbf{E}$ or $\mathbf{B}$ from Maxwell’s equations. It
therefore has solutions that are propagating waves, if there is a source, just
like Hertz’s radio waves. The quantisation of these classical waves gives us
axion particles. Secondly, this equation is equal in derivative order in space
and in time, that means it is fully consistent with relativity and a field in
spacetime (unlike the Navier-Stokes equations which are only an effective
description of a fluid, valid above the scale of the atoms making it up).
Thirdly, like Maxwell’s equations, this equation is linear, but that is only
because I have made the simplifying assumption to drop interaction terms
between the axion and other things. It turns out that these interaction terms
are very small if $f_{a}$ is very large, and we can neglect them at a first
pass. We will return to axion interactions later. Lastly, we are still only
treating the Universe itself as homogeneous, using only the cosmic scale
factor $a$. Rest assured we can also include gravitational interactions of the
axion with ordinary matter in perturbation theory.

Now we will make another simplifying assumption. We will assume that the axion
field itself is uniform/homogeneous, i.e. it has no spatial variations, so
that $\nabla\phi=0$ and $\nabla^{2}\phi=0$. This must be true on average,
because the Universe itself is homogeneous on average, and again corrections
to this assumption can be included at a cost in mathematical complexity but
without change to the basic physical picture. The equation for the axion field
is now:

$\frac{d^{2}\phi}{dt^{2}}+3\frac{da}{dt}\frac{d\phi}{dt}+\frac{m_{a}^{2}c^{4}}{\hbar^{2}}\phi=0\,.$
(19)

This wave equation should be very familiar. It is the equation of motion for a
damped harmonic oscillator. The frequency is given by
$\omega_{a}=E_{a}/\hbar=m_{a}c^{2}/\hbar$. The damping co-efficient is given
by $\gamma=3(da/dt)/a$: the time varying expansion of the Universe acts as a
friction term on axion waves. At early times, the expansion of the Universe is
fast, and if $(da/dt)/a\gg(m_{a}c^{2})/\hbar$ the axion field is overdamped
and keeps an almost constant value over time. At later times, when the
expansion slows, and $(da/dt)/a\ll(m_{a}c^{2})/\hbar$ the axion field will
undergo damped oscillations. The numerical solution of this equation using a
realistic model for the expansion of the Universe is shown in Fig. 4, along
with parts of the solution that can be estimated analytically (for a more
detailed treatment, see Ref. [40]).

If the axion field starts off with some initial value $\phi_{i}>0$, then it
will begin to oscillate at some time $t_{\rm osc}$ when the expansion of the
Universe is slow enough that the damping becomes small. The oscillations of
the axion field carries energy density given by:

$\rho=\frac{1}{2}\left(\frac{d\phi}{dt}\right)^{2}+\frac{1}{2}m_{a}^{2}c^{2}\phi^{2}\,.$
(20)

_The fact that the oscillations of the axion field carry energy density, yet
interact very weakly with ordinary matter, makes these axion waves a candidate
to explain DM._ The axion DM is effectively created at the time when the field
begins to oscillate, which for large enough axion mass, $m>10^{-24}\text{
eV}$, happens sufficiently early to match the CMB and the cosmic web, and the
effective description is furthermore of a cold and pressureless gravitating
fluid [41].

There is one additional free parameter in the model now, besides $f_{a}$, and
that is the initial value of the axion field, $\phi_{i}$, which we can
alternatively parameterise as $\theta_{i}$, which measures how far the strong
force vacuum was from conserving $CP$ in the very early Universe (or in the
string case, the initial amount of $p$-form flux). A non-zero, albeit random,
value for $\theta_{i}$ is naturally provided by a process called spontaneous
symmetry breaking. Taking into account that, due to the dynamics of the strong
nuclear fore and the formation of protons, the ‘real’ QCD axion (as opposed to
our toy model in Eq. 19) has a time dependent mass, the relic density
parameter for QCD axion DM can be expressed in terms of these two free
parameters as:

$\rho_{a}=2.3\times 10^{-27}\text{ kg m}^{-3}\left(\frac{f_{a}}{10^{12}\text{
GeV}}\right)^{7/6}\left(\frac{\theta_{i}}{0.53}\right)^{2}\,,$ (21)

where I have chosen reference values of $f_{a}$ and $\theta_{i}$ (in radians)
that reproduce the observed value given in Eq. (11).

###  3.2 Axion electrodynamics

Axions are odd under the $CP$ symmetry: $CP\phi=-\phi$. Hamiltonians are an
energy, and so overall are even: $CPH=H$. If we want to couple the axion to
the Maxwell fields in the Hamiltonian, the axion can only couple to a $CP$ odd
combination of the fields, and the simplest such configuration is given by
$\mathbf{E}\cdot\mathbf{B}$ (which you can remind yourself is $CP$ odd with
reference to Fig. 1). Thus we can guess that the Hamiltonian for axions
coupled the Maxwell fields is:

$H_{\rm
int}=\sqrt{\frac{\epsilon_{0}}{\mu_{0}}}g\int\phi\mathbf{E}\cdot\mathbf{B}\,dV\,,$
(22)

where the factors of $\epsilon_{0}$ and $\mu_{0}$ are chosen by convention so
that $g$ is an unknown coupling constant with units inverse energy (so that
particle physicists measure $g$ in GeV-1), and the integral is taken over all
space in order to get the correct units of energy for the Hamiltonian itself.
This Hamiltonian tells us that there is an energy cost associated to regions
of space with non-zero axion field and $\mathbf{E}\cdot\mathbf{B}$. In any
region of space with non-zero $\mathbf{E}\cdot\mathbf{B}$ external source
(e.g. provided by an experimentalist), the energy can be lowered by the axion
field taking a negative value in response. Thus a static $\mathbf{B}$ field
parallel to an oscillating $\mathbf{E}$ acts as a source of axion waves as the
axion field changes sign in response to the changing $\mathbf{E}$ in order to
keep the energy at its minimum. Alternatively, in any region with non-zero
axion field and non-zero either $\mathbf{E}$ or $\mathbf{B}$, then the energy
can be lowered by creating an antiparallel $\mathbf{B}$ or $\mathbf{E}$ field.
In a region of space in which the axion oscillates, a static $\mathbf{B}$ or
$\mathbf{E}$ field will source an oscillating version of the other field: as
if from a Hertz-like oscillating charge density the axion would source radio
waves.

Now let’s see this in equations. Given a Hamiltonian, we can construct an
action (in this case simply by integrating over time), and from an action, via
a variational principle, one can derive equations of motion (for this
derivation of the ordinary Maxwell’s equations in classical mechanics, see
Ref. [1]). In this way, we find how this interaction Hamiltonian modifies
Maxwell’s equations, and the equation of motion of the axion itself, Eq. (19).
We will just look at the modified Maxwell’s equations, which read:

$\displaystyle\nabla\cdot\mathbf{E}$
$\displaystyle=\frac{\rho}{\varepsilon_{0}}-\frac{g}{\varepsilon_{0}}\mathbf{B}\cdot\nabla\phi,$
(23) $\displaystyle\nabla\cdot\mathbf{B}$ $\displaystyle=0,$ (24)
$\displaystyle\nabla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial t}$
$\displaystyle=0,$ (25)
$\displaystyle\nabla\times\mathbf{B}-\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial
t}$
$\displaystyle=\mu_{0}\mathbf{J}+g\mu_{0}\left(\mathbf{B}\cdot\frac{\partial\phi}{\partial
t}+\mathbf{E}\times\nabla\phi\right),$ (26)

The above modification of Maxwell’s equations is known as ‘axion
electrodynamics’. The axion field $\phi$ appears on the right hand side of the
sourced Maxwell equations, like a new type of ‘dark’ charge or current
density. The axion field only appears with derivatives, i.e.
$\partial\phi/\partial t$ or $\nabla\phi$. The axion field always appears
multiplied by $\mathbf{E}$ or $\mathbf{B}$. These properties bear out our
intuition from above: gradients of the axion field can act as sources for
electric and magnetic fields. If you inspect the vector properties of the
above equations, you see that the axion always sources an $\mathbf{E}$
(anti)parallel to an applied $\mathbf{B}$ and vice versa. This last fact is in
distinction to an electromagnetic wave, in which electric and magnetic fields
are perpendicular to one another. Lastly, the magnitude of the axion source
term depends on the unknown coupling constant $g$.

Axion electrodynamics has a number of other unusual properties, and is a
fascinating subject in and of itself [42] (for some solution strategies, see
Refs. [43, 44]). Recently there has been growing interest in something called
‘axion quasiparticles’. An axion quasiparticle is a system in materials
science that has the properties of axion electrodynamics, caused by something
called the magnetoelectric effect, but where the field $\phi$ is not a
fundamental particle, but instead some collective excitation of the material,
such as magnetic order [45, 46]. In such materials, one could study the
physics of axion electrodynamics, and some properties may be useful in
microelectronics. So far, materials scientists have fabricated materials with
_static_ $\phi$, but have yet to realise dynamics. My personal interest in
these materials is that, in a wonderful coincidence of physics, the axion
quasiparticles might prove useful for detecting axion dark matter [47].

##  4 The search for axions

###  4.1 Astrophysical searches

Axions can affect numerous astrophysical processes, from the lifetimes of
stars, to the large scale structure of the Universe. Reviews can be found in
Refs. [48, 40, 49]. Here we touch on just two possibilities. The two we
discuss are particularly important because they do not rely on the assumption
that axions are dark matter. The first constraint provides an upper bound on
the coupling, $g$, and the second excludes some range of values for the axion
mass, $m_{a}$.

####  4.1.1 Axions and the lifetimes of stars

As was mentioned above, the interaction Hamiltonian in Eq. (22), as well as
allowing axions to source EM waves in Maxwell’s equations, also allows EM
waves to source axions. The process is called the Primakoff process, and works
like this. Inside a star, atoms are heated up so that they are ionized, which
means there are free positively charged nuclei and negatively charged
electrons around. These free charges source electric fields by Coulomb’s law.
There are also a lot of EM waves around: it’s hot inside a star. When an EM
wave moves into the electric field of a charged particle, then we have the
$\mathbf{E}$ in the Hamiltonian from the Coulomb field, and the $\mathbf{B}$
part is provided by the oscillating part of the EM wave. This means that, to
minimise the energy, the axion field constantly moves up and down from
positive to negative values, as the $\mathbf{B}$ field in the EM wave does.
The result is that axion waves are produced.

To fix up energy conservation, the EM wave has to lose energy. We know from
the photoelectric effect that EM waves are actually quantised as photons, with
energy $E=h\nu$, and so in the quantum picture the EM wave loses energy in
these chunks, creating axions of the same energy. This extra way in which the
EM field can interact with nucleons speeds up the evolution of the star.

This phenomenon, Primakoff production of axions inside stars, has two
consequences. The first is that if axions exist, then stars should age a
little bit faster than they would otherwise. The rate of stellar ageing
depends on the size of the coupling constant, $g$, with larger $g$ leading to
more axions produced, and faster ageing. Theories of stellar structure and
lifetimes in modern astrophysics are quite advanced. In particular they can be
used to accurately predict the famous ‘Hertzsprung-Russell diagram’, that
plots stars by their temperature and brightness. In the HR diagram there is a
particular region called the horizontal branch, where stars have an almost
constant luminosity for a wide variety of temperatures. As stars grow older,
they move from the main sequence and onto the horizontal branch as they
decrease in temperature approaching the ‘asymptotic giant branch’ which leads
eventually to their death. If there is faster ageing of a star caused by axion
emission, then the time a star spends on the horizontal branch is decreased.
The fact that horizontal branch stars are known to be old allows astronomers
to place limits on the amount of extra ageing caused by axions, and thus place
an upper limit on the coupling constant $g$ [48]. 888I thank Maurizio
Giannotti for correcting my description of this process.

The second consequence of axion production in stars is that we can look for
the axions produced by the sun. The axions produced have a typical energy
determined by the internal temperature of the star, $E=k_{\rm B}T\approx
1\text{ keV}$. These axions travel to Earth. Using the inverse Primkoff
process, i.e. converting an axion wave into an oscillating electric field
inside a large magnet, turns these axions back into visible X-rays, even if
the magnet pointed at the Sun is inside a dark room. This idea is known as an
axion ‘helioscope’ and was proposed by Sikivie in 1983 [50]. A number of
helioscopes have been built, the biggest being the Cern Axion Solar Telescope
[51]. They haven’t seen anything, and this allows an upper limit on $g$ to be
set, which happens to be around the same value you get from lifetimes of
horizontal branch stars, so we wouldn’t expect to really see anything or
stellar evolution would have already shown us there was extra cooling due to
axions.

The next step in this game is to try and build a bigger helioscope to test
smaller values of $g$. The project is known as the International Axion
Observatory [52, 53], and has the opportunity to test models where $g$ is
smaller by around a factor of ten, and could discover the QCD axion if it
happens to have a mass between 3 to 100 meV, in certain models [54, 55].

####  4.1.2 Axions and black hole spins

Figure 5: Cartoon of the ‘Penrose process’. A particle falling into the
ergoregion of a rotating black hole can split in two, and leave the ergoregion
with more energy than it went in with. This process extracts angular momentum
from the black hole. If the process can repeat, the rate grows exponentially,
becoming ‘superradiant’. (Reproduced from Ref. [37])

Things famously fall into black holes and don’t escape. When things fall in,
the black hole mass will increase. If the things falling in also have angular
momentum, i.e. falling in on a curved orbit, then the black hole also gains
that angular momentum. The only thing that can decrease a black hole mass is
Hawking radiation, a very weak quantum process that would take many times the
age of the Universe to significantly reduce the mass of an astrophysical black
hole. Axions, or in fact any light boson, can change this story dramatically.

Spinning black holes have a region known as the ergoregion around them, where
spacetime itself co-rotates, and all objects must follow the spin of the black
hole. In 1971, Penrose and Floyd showed [56] that if a particle falls into
this region, then splits in two, it must gain energy as the new lighter
particle rotates faster. One of the pair falls into the black hole, and the
other gains energy is ejected from the ergoregion. Energy conservation is
balanced by the black hole losing angular momentum, and spinning a little bit
slower. This process is illustrated in Fig. 5. If this process happens over
and over by trapping the outgoing particle, then the rate grows exponentially
becoming ‘superradiant’.

Axion waves around a black hole provide for this scenario naturally if the
quantum wavelength of the axion wave, the Compton wavelength, $\lambda_{\rm
C}=h/m_{a}$ is the same size as the ergoregion itself, $r_{\rm Ergo}\approx
G_{\rm N}M_{\rm BH}$. In this case, the orbitals of the axion around the black
hole, like the orbitals of an electron in hydrogen, provide the mechnism for
the process to repeat, by trapping the axion, while the orbitals themselves
overlap the horizon of the black hole, allowing for the slow trickle of axions
into the black hole necessary for the Penrose process. The mathematics of this
process are reviewed in Ref. [57].

What this process means is that if an axion exists with $h/m_{a}=G_{\rm
N}M_{\rm BH}$ then we should expect black holes of mass $M_{\rm BH}$ to only
be spinning slowly. On the other hand, the observation of highly spinning
black holes of mass $M_{\rm BH}$ excludes the existence of axions of mass
$m_{a}=G_{\rm N}M_{\rm BH}/h$ [58]. There are many observations of spinning
black holes, and these in fact do exclude the axion from having certain values
of the mass, in particular the excluded ranges are (e.g. Ref. [37]):

$\displaystyle 10^{-13}\text{ eV}\lesssim$ $\displaystyle m_{a}\lesssim
10^{-11}\text{ eV}\quad\text{(stellar mass black holes)}\,,$ (27)
$\displaystyle 10^{-19}\text{ eV}\lesssim$ $\displaystyle m_{a}\lesssim
10^{-17}\text{ eV}\quad\text{(supermassive mass black holes)}\,.$ (28)

The exclusions ignore axion self-interactions, which if large quench the
process. Thanks to black hole superradiance, we know where not to look for
axion DM. In future, large statistical surveys of black hole spin
distributions could also reveal the existence of axions [59].

###  4.2 Microwave cavity ‘haloscope’: the archetype axion DM search

Figure 6: Sinusoidal shape of the fundamental mode of the electric field, $E$,
when confined between two conducting mirrors. In an axion haloscope, a $B$
field is applied parallel to the surface of the conductor throughout the
region of vacuum between the plates. (Reproduced from Ref. [39], published by
Princeton University Press, forthcoming)

How can we detect axion DM waves on Earth? As we have seen already, the
solution to Eq. (19) is an oscillating classical wave, with frequency
$\nu=m_{a}c^{2}/h$. From the modified Maxwell equations of axion
electrodynamics, we can see that in the presence of an external magnetic
field, this oscillating axion wave will source an oscillating electric field,
which in principle we could detect. The magnitude the induced electric
electric field is extremely tiny, even for very large magnetic fields in the
10’s of T.

However, the induced electric field can be enhanced by exploiting _resonance_
, in an idea also put forward by Sikivie in 1983 [50]. In a conductor, the
electric field must be zero, and acts like a mirror for electric fields. If we
place two conducting surfaces parallel to each other, with a vacuum in
between, the electric field must be zero at both conductors, and between them
can take any configuration with these boundary conditions. If the distance
between the plates is $L$, and the first plate is at $x=0$, then such an
electric field can be written as a sum of sine waves, known as modes:

$E=\sum_{n=1}^{\infty}E_{n}\sin\left(\frac{n\pi x}{L}\right)\,.$ (29)

The first, or fundamental mode, in such a set up is shown in Fig. 6. This is
the principle of a microwave cavity.

If axion DM waves are present, and the microwave cavity is exposed to a
magnetic field parallel to the walls, then the field induced by the axion DM
must obey the cavity boundary conditions, and can be written as in Eq. (29).
Now, because Maxwell’s equations obey relativity, then the electric field time
oscillations must be equivalent to its spatial oscillations, up to a factor of
$c$, i.e. for the fundamental mode we must have:

$E=E_{1}\sin\left(\frac{\pi x}{L}\right)\sin\left(\frac{\pi ct}{L}\right)\,,$
(30)

so that the mode has a ‘natural frequency’ $\nu=\pi c/L$. If the natural
frequency of the mode matches the axion DM frequency, i.e if $\pi
c/L=m_{a}c^{2}/h$, then the amplitude $E_{1}$ will be resonantly enhanced. We
do not know the axion mass, and thus do not know its frequency, but in this
toy model we could change the distance $L$ between the plates to change the
frequency of the fundamental mode and ‘tune’ the cavity until we reach the
right value for resonance. If the resonantly enhanced electric field, $E_{1}$,
is large enough, then it could be detected by a microwave antenna placed
between the plates, and we would see a signal appear as if out of nowhere when
the frequency is tuned just right, thus detecting axion DM, and simultaneously
measuring its mass from the resonant frequency. 999For a Contemporary Physics
article on related and parallel topic, the use of atom interferometers to
search for ultralight scalar dark matter, see Ref. [60].

To make such an idea realistic requires a few complications. Fortunately,
microwave cavities have many applications throughout physics and engineering,
and their design and optimisation is well understood. Firstly, no conductor is
perfect, and electric fields are not exactly zero on the boundaries. This
leads to losses of energy that damp the resonance, and mean that power
amplification by such means can be no larger than the ‘quality factor’ of the
cavity, $Q$, which for copper can be as large as $10^{5}$ or so, or for a
superconductor can be as large as $Q=10^{10}$. Next, we should worry about how
large the applied magnetic field can be, and how large the cavity can be.
Commercial magnets can easily get in the 10’s of Tesla with volumes in the
hundred litre range.

There should be no signal in the absence of axion DM, which means isolating
the experiment at very low temperature. A commercial dilution refrigerator can
cool such an apparatus to temperatures below 1 Kelvin, -272 Celsius. A real
microwave cavity is not a simple pair of parallel plates, but likely a
cylinder, and its modes are slightly more complex in shape, and can be tuned
for example by placing a rod in the cavity. Lastly, even with all of these
advances, the signal is still tiny, with axion DM producing a power in the
cavity in the range of $10^{-22}\text{ W}$, which is challenging to detect
even with the most sensitive antennas and amplifiers. A slightly more
realistic sketch of an axion haloscope is shown in Fig. 7.

(a) 3D representation of axion photon conversion in a cavity.

(b) Top down view of a cavity showing tuning rods.

Figure 7: Sketch of an axion haloscope. (Reproduced from Ref. [39], published
by Princeton University Press, forthcoming)

The first experiment using this design was carried out in 1987 [61]. 101010In
a remarkable coincidence, some of the first searches for another dark matter
candidate, the supersymmetric weakly interacting massive particle (WIMP), were
also carried out in this year [62]. The theory of WIMP DM production, like
that of axions, was also developed in 1983 [63, 64], after major theoretical
breakthroughs in 1981 [65]. The futures of these two models, WIMPs and axions,
were very different though, with WIMPs very much in the ascendancy throughout
the 1990’s and early 2000’s. This was due, in part, to technology: WIMP DM
direct searches developed sensitivity rapidly, and indirect searches
piggybacked off the Higgs search at Cern. Axion searches were much more
limited by technology, and ideas, until the 2010’s and later. Now, the
fortunes of axions and WIMPs have largely reversed. The modern leaders in this
field are the Axion Dark Matter eXperiment (ADMX), based in Seattle,
Washington, who have been able to push the sensitivity of the technique far
enough to test the QCD axion DM model across a range of particle mass [66]:

$2.7\,\mu\text{eV}\lesssim m_{a}\lesssim 4.2\,\mu\text{eV}\quad\text{(ADMX
exclusion of QCD axion)}\,.$ (31)

ADMX has found nothing, and has excluded the benchmark historical axion models
from composing all of the local DM density. 111111There are some subtleties if
the axion clumps into ‘miniclusters’, but this is not expected to be relevant
at the mass scale probed by ADMX, see Refs. [67, 68, 69, 70, 71] This is a
technological triumph and the leading light of axion searches at present.

###  4.3 The global axion experimental explosion

Figure 8: Top row: currently excluded range of masses for the QCD axion. Grey
shows upper and lower limits from astrophysics. Black hole spins have been
discussed here, while the other astrophysical upper limit comes from supernova
cooling. The green region is that excluded by ADMX (Eq. 31) and some other
currently operating haloscopes. Red and blue regions represent target ranges
of new technology planned for the coming decades, which may allow to search
the whole allowed range for the QCD axion. (Reproduced from Ref. [72])

The mass range probed by ADMX, Eq. (31), is narrow and does not cover anywhere
near the entire allowed range for the QCD axion mass:

$10^{-11}\text{ eV}\lesssim m_{a}\lesssim 10^{-2}\text{ eV}\quad\text{(allowed
range of QCD axion)}\,.$ (32)

The narrowness of the ADMX search range is due to the inherent resonant nature
of the haloscope method, and limitations in physical size (both large and
small) of microwave cavities with large volume fundamental mode, cooled and
subjected to large magnetic fields.

The future of axion experiments is to widen the search range using new
technology. The current status of the field is reviewed in the white paper
Ref. [72], and summarised in Fig. 8 in terms of the axion mass search range
(with a focus on US developments). The global reach of the modern program is
illustrated in Fig. 9. We end this article by briefly discussing the exciting
global future of the hunt for axions.

A microwave cavity haloscope like ADMX based on Sikivie’s original design
searches for the axion like tuning a radio. We move to one frequency, listen
for the station until we are satisfied even a weak signal is not there, then
move on to the next frequency. Steps have to be narrow, because the axion DM
station occupies a narrow region in frequency space with bandwidth
$\Delta\nu/\nu\approx 10^{-6}$. Furthermore, the frequency is intrinsically
limited to searches around 1 GHz by the microwave nature of the technology.
Nonetheless, there is still a large frequency range open to this technology,
and other haloscopes are being constructed in countries around the world, each
working at a different frequency range, and hoping that they will be the ones
lucky enough to hit the jackpot of discovering the axion. The nature of the
searches makes the experiments not in direct competition: if one experiment
hits the jackpot, it was nature that chose the frequency. This makes for a
very open and collegial field. Cavity technology has developed rapidly in
recent years, lead by innovations from around the world, for example the group
surrounding the ORGAN experiment in Australia [73], and the Centre for Axion
Precision Physics (CAPP) in South Korea [74].

Searching for axions at higher and lower frequencies than microwave haloscopes
can reach, however, has required the emergence of new ideas. At high and low
frequency, the trick has been to break the relatioship between the resonant
frequency enhancement and the size of the cavity, which we saw above requires
$\pi c/L=m_{a}c^{2}/h$.

Figure 9: Just a small selection of existing and planned axion experiments
around the world (modified from https://cajohare.github.io/AxionLimits/).

At high frequency, there are two leading ideas under development, which build
on the experience of ADMX. The first takes the cavity idea and instead
constructs a kind of open resonator, with the field mode shapes modified by
dielectric materials [75, 43]. Another similar idea uses conducting wires to
create a tuneable ‘metamaterial’ [76]. Prototypes of both of these ideas have
been created, and plans are in place to scale the prototypes to full axion DM
searches. For the dielectric case, the there is ‘MADMAX’, which is planned to
operate in Europe at DESY and Cern, and also ‘DALI’ under development in Spain
[77]. In the metamaterial case, the leading proposal is ‘ALPHA’, which is
being developed largely in the USA. The quasiparticles idea mentioned above
[47] also falls in this broad category at high frequency. An alternative
approach at high frequency is to give up on resonance and instead use a large
reflector to collect axions from a wide magnetized source region, with the
advantage that such a search is broadband [78]. Planned and operating
experiments using this technology include ‘BRASS’ in Germany [79], and ‘BREAD’
in the USA [80]. Together, these technologies may reach up to the highest
axion masses corresponding to THz frequencies.

At low frequency, a slightly different approach is taken, with the most
promising set of experiments being based on the ideas developed in Refs. [81,
82]. In this proposed ‘DM-Radio’ [83, 84], and related designs, the axion
interaction with a magnetic field produces a current in an LC circuit
resonator, which is then detected with a pickup loop. The frequency
sensitivity of this technique extends down as low as kHz.

Axions can also interact with ordinary matter via different means than axion
electrodynamics, for example coupling to electrons or nuclei directly.
Searching for axions in this way involves resonances, either nuclear magnetic
resonance (NMR), or electron spin resonance, which are present in certain
types of material. The NMR searches use precisely the technology developed
already in that advanced field, and apply it to search for axions, where they
appear as a tiny anomalous magnetic field. This scheme of experiments is
called CASPEr, and operates in Germany and the USA [85, 86]. On the side of
the electron interaction, the experiment QUAX was developed in Italy [87, 88]
(early ideas in this direction were developed in Refs. [89, 90]). These
searches for axions using material resonances and different couplings between
the axion and ordinary particles will be of key importance in a ‘post-
discovery’ world, with in particular CASPEr being able to distinguish the QCD
axion from a more general ALP, and QUAX being able to disentangle between the
two key types of QCD axion model.

##  5 Closing remarks

In this brief review we have been on a journey from the smallest scales, with
axions in extra dimensions, to the largest scales, with axions dictating the
evolution of the Universe. We have met physics from General Relativity, to the
design of microwave cavities. This is one reason to be excited by axions: they
give the physicist studying them ample opportunity to learn across a wide
range of specialisms. But the much more pressing reason to be excited by
axions is the massive growth in experiments searching for axions. Many of the
experiments mentioned above are only in the prototype stage, and will reach
design sensitivity in the next 10-20 years. This means that the coming decades
could well see the discovery of the QCD axion, if it exists. If axions are
discovered, this could shed light on the mystery of dark matter, the
strong-$CP$ problem, and even the topology of extra dimensions. There could
not be a more exciting time to come.

## Funding

I am supported by an Ernest Rutherford Fellowship from the Science and
Technologies Facilities Council (ST/T004037/1), and by a Leverhulme Research
Project (RPG-2022-145).

## Nomenclature/Notation

The gradient operator in three dimensions is $\nabla$, and in this context
$\times$ is the vector cross product. The speed of light is $c$, Planck’s
constant is $h$. Particle masses are quoted in units of electronvolts, eV,
where $1\text{ eV}=1.78\times 10^{-36}\text{ kg}$, and an atom of hydrogen is
approximately $10^{9}\text{ eV}$. Particle physicists often used units where
$\hbar=c=1$, and while I have tried my best to restore these factors, as well
as those of $\epsilon_{0}$ and $\mu_{0}$, I cannot guarantee I caught every
one.

## References

  * [1] Goldstein H. Classical mechanics. Addison-Wesley; 1980.
  * [2] Schutz B. A First Course in General Relativity. 2nd ed. Cambridge University Press; 2009.
  * [3] Misner CW, Thorne KS, Wheeler JA. Gravitation. Princeton University Press; 1973.
  * [4] Carroll SM. Spacetime and geometry. An introduction to general relativity. Addison Wesley; 2004.
  * [5] Page L. The little book of cosmology. Princeton University Press; 2020.
  * [6] Dawson KS, Schlegel DJ, Ahn CP, et al. The Baryon Oscillation Spectroscopic Survey of SDSS-III. AJ. 2013 Jan;145(1):10.
  * [7] Sevilla-Noarbe I, et al. (DES). Dark Energy Survey Year 3 Results: Photometric Data Set for Cosmology. Astrophys J Suppl. 2021;254(2):24.
  * [8] Dodelson S. Modern Cosmology. Academic Press, Elsevier Science; 2003.
  * [9] Aghanim N, et al. (Planck). Planck 2018 results. VI. Cosmological parameters. Astron Astrophys. 2020;641:A6. [Erratum: Astron.Astrophys. 652, C4 (2021)].
  * [10] Chadha-Day F, Ellis J, Marsh DJE. Axion dark matter: What is it and why now? Sci Adv. 2022;8(8):abj3618.
  * [11] Christenson JH, Cronin JW, Fitch VL, et al. Evidence for the $2\pi$ Decay of the $K_{2}^{0}$ Meson. Phys Rev Lett. 1964;13:138–140.
  * [12] Sikivie P. The Pool table analogy to axion physics. Physics Today. 1996;49:22.
  * [13] Pendlebury JM, et al. Revised experimental upper limit on the electric dipole moment of the neutron. Phys Rev D. 2015;92(9):092003.
  * [14] Crewther RJ, Di Vecchia P, Veneziano G, et al. Chiral Estimate of the Electric Dipole Moment of the Neutron in Quantum Chromodynamics. Phys Lett B. 1979;88:123. [Erratum: Phys.Lett.B 91, 487 (1980)].
  * [15] Peccei RD, Quinn HR. CP Conservation in the Presence of Instantons. Phys Rev Lett. 1977;38:1440–1443.
  * [16] Feynman RP. Quantum electrodynamics. Reading, MA: Addison-Wesley; 1998.
  * [17] Weinberg S. A New Light Boson? Phys Rev Lett. 1978;40:223–226.
  * [18] Wilczek F. Problem of Strong $P$ and $T$ Invariance in the Presence of Instantons. Phys Rev Lett. 1978;40:279–282.
  * [19] Srednicki M. Quantum Field Theory. Cambridge: Cambridge Univ. Press; 2007. Available from: https://cds.cern.ch/record/1019751.
  * [20] Zee A. Quantum Field Theory in a Nutshell; 1st ed. Princeton, NJ: Princeton Univ. Press; 2003. In a nutshell; Available from: https://cds.cern.ch/record/706825.
  * [21] Grilli di Cortona G, Hardy E, Pardo Vega J, et al. The QCD axion, precisely. JHEP. 2016;01:034.
  * [22] Kim JE. Weak Interaction Singlet and Strong CP Invariance. Phys Rev Lett. 1979;43:103.
  * [23] Shifman M, Vainshtein A, Zakharov V. Can confinement ensure natural cp invariance of strong interactions? Nuclear Physics B. 1980;166(3):493–506. Available from: https://www.sciencedirect.com/science/article/pii/0550321380902096.
  * [24] Dine M, Fischler W, Srednicki M. A Simple Solution to the Strong CP Problem with a Harmless Axion. Phys Lett B. 1981;104:199–202.
  * [25] Zhitnitsky AR. On Possible Suppression of the Axion Hadron Interactions. (In Russian). Sov J Nucl Phys. 1980;31:260.
  * [26] Dine M, Fischler W. The not-so-harmless axion. Phys Lett B. 1983 Jan;120:137–141.
  * [27] Preskill J, Wise MB, Wilczek F. Cosmology of the invisible axion. Phys Lett B. 1983 Jan;120:127–132.
  * [28] Abbott LF, Sikivie P. A cosmological bound on the invisible axion. Phys Lett B. 1983 Jan;120:133–136.
  * [29] Group PD, Workman RL, Burkert VD, et al. Review of Particle Physics. Progress of Theoretical and Experimental Physics. 2022 08;2022(8):083C01. Available from: https://doi.org/10.1093/ptep/ptac097.
  * [30] Petersen P, et al. (COMPACT). Cosmic topology. Part I. Limits on orientable Euclidean manifolds from circle searches. JCAP. 2023;01:030.
  * [31] Reece M. TASI Lectures: (No) Global Symmetries to Axion Physics. 2023 4;.
  * [32] Witten E. Some Properties of O(32) Superstrings. Phys Lett B. 1984;149:351–356.
  * [33] Candelas P, Horowitz GT, Strominger A, et al. Vacuum configurations for superstrings. Nucl Phys B. 1985;258:46–74.
  * [34] Conlon JP. The QCD axion and moduli stabilisation. JHEP. 2006;05:078.
  * [35] Svrcek P, Witten E. Axions In String Theory. JHEP. 2006;06:051.
  * [36] Arvanitaki A, Dimopoulos S, Dubovsky S, et al. String Axiverse. Phys Rev D. 2010;81:123530.
  * [37] Mehta VM, Demirtas M, Long C, et al. Superradiance in string theory. JCAP. 2021;07:033.
  * [38] Demirtas M, Gendler N, Long C, et al. PQ axiverse. JHEP. 2023;06:092.
  * [39] Marsh DJE, Ellis D, Mehta VM. Dark Matter (forthcoming). Princeton University Press; 2024.
  * [40] Marsh DJE. Axion Cosmology. Phys Rept. 2016;643:1–79.
  * [41] Hlozek R, Grin D, Marsh DJE, et al. A search for ultralight axions using precision cosmological data. Phys Rev D. 2015;91(10):103512.
  * [42] Wilczek F. Two Applications of Axion Electrodynamics. Phys Rev Lett. 1987;58:1799.
  * [43] Millar AJ, Raffelt GG, Redondo J, et al. Dielectric Haloscopes to Search for Axion Dark Matter: Theoretical Foundations. JCAP. 2017;01:061.
  * [44] Jeong J, Kim Y, Bae S, et al. Simulation of Classical Axion Electrodynamics using COMSOL Multiphysics. arXiv eprint 2303 09748. 2023;.
  * [45] Li R, Wang J, Qi XL, et al. Dynamical axion field in topological magnetic insulators. Nature Physics. 2010 mar;6(4):284–288.
  * [46] Sekine A, Nomura K. Axion Electrodynamics in Topological Materials. J Appl Phys. 2021;129(14):141101.
  * [47] Schütte-Engel J, Marsh DJE, Millar AJ, et al. Axion quasiparticles for axion dark matter detection. JCAP. 2021;08:066.
  * [48] Raffelt GG. Astrophysical axion bounds. Lect Notes Phys. 2008;741:51–71.
  * [49] Marsh DJE, Hoof S. Ultralight bosonic dark matter. Springer, eds. Jackson Kimball and van Bibber; 2023. Chapter Astrophysical Searches and Constraints, arXiv:2106.08797.
  * [50] Sikivie P. Experimental Tests of the Invisible Axion. Phys Rev Lett. 1983;51:1415–1417. [Erratum: Phys.Rev.Lett. 52, 695 (1984)].
  * [51] Andriamonje S, et al. (CAST). An Improved limit on the axion-photon coupling from the CAST experiment. JCAP. 2007;04:010.
  * [52] Armengaud E, et al. Conceptual Design of the International Axion Observatory (IAXO). JINST. 2014;9:T05002.
  * [53] Armengaud E, et al. (IAXO). Physics potential of the International Axion Observatory (IAXO). JCAP. 2019;06:047.
  * [54] Dafni T, O’Hare CAJ, Lakić B, et al. Weighing the solar axion. Phys Rev D. 2019;99(3):035037.
  * [55] Hoof S, Jaeckel J, Thormaehlen LJ. Quantifying uncertainties in the solar axion flux and their impact on determining axion model parameters. JCAP. 2021;09:006.
  * [56] Penrose R, Floyd RM. Extraction of rotational energy from a black hole. Nature. 1971;229:177–179.
  * [57] Brito R, Cardoso V, Pani P. Superradiance: New Frontiers in Black Hole Physics. Lect Notes Phys. 2015;906:pp.1–237.
  * [58] Arvanitaki A, Dubovsky S. Exploring the String Axiverse with Precision Black Hole Physics. Phys Rev D. 2011;83:044026.
  * [59] Arvanitaki A, Baryakhtar M, Dimopoulos S, et al. Black Hole Mergers and the QCD Axion at Advanced LIGO. Phys Rev D. 2017;95(4):043001.
  * [60] Buchmueller O, Ellis J, Schneider U. Large-Scale Atom Interferometry for Fundamental Physics. 2023 6;.
  * [61] DePanfilis S, Melissinos AC, Moskowitz BE, et al. Limits on the abundance and coupling of cosmic axions at 4.5$<{m}_{a}<$5.0 $\mu$ev. Phys Rev Lett. 1987 Aug;59:839–842. Available from: https://link.aps.org/doi/10.1103/PhysRevLett.59.839.
  * [62] Ahlen S, Avignone F, Brodzinski R, et al. Limits on Cold Dark Matter Candidates from an Ultralow Background Germanium Spectrometer. Phys Lett B. 1987;195:603–608.
  * [63] Weinberg S. Upper Bound on Gauge Fermion Masses. Phys Rev Lett. 1983;50:387.
  * [64] Ellis JR, Hagelin JS, Nanopoulos DV, et al. Search for Supersymmetry at the anti-p p Collider. Phys Lett B. 1983;127:233–241.
  * [65] Dimopoulos S, Georgi H. Softly Broken Supersymmetry and SU(5). Nucl Phys B. 1981;193:150–162.
  * [66] Bartram C, et al. (ADMX). Search for Invisible Axion Dark Matter in the 3.3–4.2 $\mu$eV Mass Range. Phys Rev Lett. 2021;127(26):261803.
  * [67] Hogan CJ, Rees MJ. AXION MINICLUSTERS. Phys Lett B. 1988;205:228–230.
  * [68] Ellis D, Marsh DJE, Behrens C. Axion Miniclusters Made Easy. Phys Rev D. 2021;103(8):083525.
  * [69] Kavanagh BJ, Edwards TDP, Visinelli L, et al. Stellar disruption of axion miniclusters in the Milky Way. Phys Rev D. 2021;104(6):063038.
  * [70] Hoof S, Riess J, Marsh DJE. Statistical Uncertainties of the $N_{\text{DW}}=1$ QCD Axion Mass Window from Topological Defects. 2021 8;.
  * [71] Eggemeier B, O’Hare CAJ, Pierobon G, et al. Axion minivoids and implications for direct detection. Phys Rev D. 2023;107(8):083510.
  * [72] Adams CB, et al. Axion Dark Matter. In: Snowmass 2021; 3; 2022.
  * [73] McAllister BT, Flower G, Kruger J, et al. The ORGAN Experiment: An axion haloscope above 15 GHz. Phys Dark Univ. 2017;18:67–72.
  * [74] Semertzidis YK, et al. Axion Dark Matter Research with IBS/CAPP. 2019 10;.
  * [75] Caldwell A, Dvali G, Majorovits B, et al. (MADMAX Working Group). Dielectric Haloscopes: A New Way to Detect Axion Dark Matter. Phys Rev Lett. 2017;118(9):091801.
  * [76] Lawson M, Millar AJ, Pancaldi M, et al. Tunable axion plasma haloscopes. Phys Rev Lett. 2019;123(14):141802.
  * [77] De Miguel J. A dark matter telescope probing the 6 to 60 GHz band. JCAP. 2021;04:075.
  * [78] Horns D, Jaeckel J, Lindner A, et al. Searching for WISPy Cold Dark Matter with a Dish Antenna. JCAP. 2013;04:016.
  * [79] Bajjali F, et al. First results from BRASS-p broadband searches for hidden photon dark matter. 2023 6;.
  * [80] Liu J, et al. (BREAD). Broadband Solenoidal Haloscope for Terahertz Axion Detection. Phys Rev Lett. 2022;128(13):131801.
  * [81] Chaudhuri S, Graham PW, Irwin K, et al. Radio for hidden-photon dark matter detection. Phys Rev D. 2015;92(7):075012.
  * [82] Kahn Y, Safdi BR, Thaler J. Broadband and Resonant Approaches to Axion Dark Matter Detection. Phys Rev Lett. 2016;117(14):141801.
  * [83] Brouwer L, et al. (DMRadio). Proposal for a definitive search for GUT-scale QCD axions. Phys Rev D. 2022;106(11):112003.
  * [84] Brouwer L, et al. (DMRadio). Projected sensitivity of DMRadio-m3: A search for the QCD axion below 1 $\mu$eV. Phys Rev D. 2022;106(10):103008.
  * [85] Budker D, Graham PW, Ledbetter M, et al. Proposal for a Cosmic Axion Spin Precession Experiment (CASPEr). Phys Rev X. 2014;4(2):021030.
  * [86] Jackson Kimball DF, et al. Overview of the Cosmic Axion Spin Precession Experiment (CASPEr). Springer Proc Phys. 2020;245:105–121.
  * [87] Barbieri R, Braggio C, Carugno G, et al. Searching for galactic axions through magnetized media: the QUAX proposal. Phys Dark Univ. 2017;15:135–141.
  * [88] Crescini N, et al. (QUAX). Axion search with a quantum-limited ferromagnetic haloscope. Phys Rev Lett. 2020;124(17):171801.
  * [89] Barbieri R, Cerdonio M, Fiorentini G, et al. AXION TO MAGNON CONVERSION: A SCHEME FOR THE DETECTION OF GALACTIC AXIONS. Phys Lett B. 1989;226:357–360.
  * [90] Kakhidze AI, Kolokolov IV. Antiferromagnetic axions detector. Sov Phys JETP. 1991;72:598–600.