discovered a binary black hole whose component masses are appreciably
asymmetric, with the more massive black hole about four times heavier than its
lighter counterpart [LIGOScientific:2020stg]. This is a qualitatively new
signal because the asymmetric mass ratio allowed for a significant measurement
of a higher-order radiative multipole moment in the gravitational-wave
emission. As evident, every observing run thus far has brought about new
scientific opportunities and this momentum has no sign of stopping. As it
stands, O4 and O5 are planned to start operating in 2022 and 2025,
respectively, each scheduled to observe for at least one calendar year
[Aasi:2013wya]. The detectors should by then have been upgraded to Advanced
LIGO Plus and Advanced Virgo Plus, whose sensitivities are few times better
than those of current detectors. Plans for a more ambitious upgrade of the
LIGO detectors, called the LIGO Voyager, are also well underway
[Adhikari:2019zpy]. GEO 600, which is an operating 600m laser interferometer
in Hanover, Germany is unlike to contribute significantly to the network of
gravitational-wave detectors. Nevertheless, it will be tuned to focus on high
frequency narrow-band sensitivity at a few kilohertz, possibly contributing to
the understanding of neutron star merger physics where the signal may be loud
[Dooley:2015fpa].

In addition to ground-based laser interferometers, there exists a global
network of pulsar timing arrays (PTAs) which aims to detect a stochastic
background of gravitational waves emitted by supermassive black hole
inspirals. This collaboration, called the International Pulsar Timing Array
[IPTA:2013lea], consists of groups in Europe, North America and Australia
[Kramer:2013kea, McLaughlin:2013ira, Hobbs:2013aka]. PTAs are sensitive to
gravitational waves in the nanohertz frequency regime, approximately within
$10^{-10}\hskip 1.0pt\text{Hz}-10^{-8}\hskip 1.0pt\text{Hz}$, see e.g. Fig. 1
of [Hazboun:2019vhv]. The separation between PTAs and the observed pulsars
play the same role as the detector arms in laser interferometers, but can now
be as long as $10^{19}\,\text{m}$. Gravitational waves that propagate in the
intermediate region would modulate the arrival times of the signals emitted by
these pulsars, which are known to be very stable, periodic sources of radio
waves. These arrays therefore seek to measure relative phase shifts between
different pulsar arrival signals due to the modulations from gravitational
waves [DetweilerPTA]. While no such gravitational-wave signal has been
measured so far, a breakthrough is on the horizon [Lommen_2015]. The Square
Kilometer Array (SKA), which is a trans-continental PTA with better
sensitivities, is also expected to begin its observations a few years later
[Smits:2008cf].

####  2 Future Detectors

The successes of LIGO and Virgo and the promise of PTAs mark only the
beginning of gravitational-wave astronomy. New ground-based detectors have
already been constructed and will join LIGO and Virgo in future observing
runs. In addition, there is now a flurry of proposals for developing next-
generation gravitational-wave detectors. In what follows, I will summarize
these prospective future gravitational-wave detectors, separating them into a
few broad categories for convenience. A few representative noise curves in
each category are illustrated in Fig. 4.

Figure 4: The sensitivity curves of current and future gravitational-wave
detectors. Figure adapted from [Kuns:2019upi]. For the PTA sensitivity curves,
which have relatively low strain sensitivities and lie in the nanohertz
regime, see e.g. [Hazboun:2019vhv].

  * •

Current-generation ground-based detectors: in addition to the upgrades of LIGO
and Virgo described above, two new Michelson interferometers are already
constructed and will soon be operating: the Kamioka Gravitational Wave
Detector (KAGRA) in Japan [Somiya:2011np] and LIGO-India
[Unnikrishnan:2013qwa]. These detectors, whose arm lengths are $3$ km and $4$
km, respectively, will observe in the same frequency range as LIGO and Virgo,
therefore contributing to a wider network of ground-based observatories.
Crucially, these additional detectors will significantly improve the network’s
overall angular resolution and sensitivity to gravitational-wave sources. Like
LIGO and Virgo, these detectors are expected to operate throughout the 2020s.

  * •

Next-generation ground-based detectors: in the 2030s, the Einstein Telescope
is planned to be constructed in continental Europe. This is conceived as an
underground laser interferometer with the shape of an equilateral triangle,
each arm being $10$ km long [Sathyaprakash:2012jk]. In the United States, the
Cosmic Explorer is envisioned to be a L-shaped Michelson interferometer with
$40$ km arm lengths [Reitze:2019iox]. These detectors aim to have
sensitivities that are an order-of-magnitude better than current advanced
detectors.

  * •

Milli-hertz space-borne detectors: after decades of funding delay, the Laser
Interferometer Space Antenna (LISA) has recently been selected by the European
Space Agency as a mission to be launched in the early 2030s
[2017arXiv170200786A]. With laser path lengths that are longer than $10^{6}$
km, this engineering marvel will probe gravitational waves in the milli-hertz
regime, $10^{-4}\hskip 1.0pt{\rm Hz}-10^{-2}\hskip 1.0pt{\rm Hz}$. Prospects
for its realistic launch have also been bolstered by the success of the LISA
Pathfinder, a precursor to the actual mission [2018PhRvD..98j2005A]. Similar
space-based missions have also been proposed in China, including TianQin
[Luo:2015ght] and Taiji [Guo:2018npi]. More futuristic milli-hertz detectors,
such as the Big Bang Observer [Crowder:2005nr], have also been envisioned.

  * •

Deci-hertz detectors: another unexplored gravitational-wave window is the
deci-hertz frequency range, $10^{-2}\hskip 1.0pt{\rm Hz}-1\hskip 1.0pt{\rm
Hz}$, which lies between the LISA and ground-based detector sensitivity bands.
A primary detection method uses laser interferometry with ensembles of cold
atoms playing the role of free-falling test masses [Dimopoulos:2008sv,
Harms:2013raa, Canuel:2016pus, Badurina:2019hst]. Space-based laser
interferometers, such as the Deci-hertz Interferometer Gravitational Wave
Observatory (DECIGO) in Japan [Kawamura:2011zz] and TianGO in China
[Kuns:2019upi], have also been proposed.

This summary clearly demonstrates the exciting prospect of expanding the
network of gravitational-wave detectors in the near future and even several
decades later. Together, these observatories would measure gravitational waves
over ten orders of magnitude in frequency. The future of gravitational-wave
astronomy is bright.

## Particle Physics at the Weak-Coupling Frontier

In the previous chapter, I focused on General Relativity and its consequences
for the gravitational sector of our Universe. This discussion excluded matter
and the other known forces in Nature, which are instead described by the
Standard Model of particle physics. Despite the excellent agreement between
the Standard Model and the outcomes of many particle physics experiments, it
is indisputable that physics beyond the Standard Model exists in our Universe.
This new physics often involves the presence of new particles that couple very
weakly to ordinary matter, making them hard to detect in particle colliders.
Fortunately, by virtue of the equivalence principle, all particles must
interact through gravity. The central goal of this thesis is to explore how
gravitational waves could offer us universal probes of new physics.

In this chapter, I provide an overview of the weak-coupling frontier in
particle physics. In Section 1, I review the Standard Model and summarize its
various shortcomings. In Section 2, I discuss several classes of new particles
which are proposed to address these problems and, at the same time, are found
to necessarily couple weakly to ordinary matter. In Section 3, I discuss how
rotating black holes are novel probes of putative ultralight bosons. This way
of detecting new particles is attractive because it only relies on gravity —
the weakest of all forces. The discussion in this section will serve an
important basis for Chapters 3, 4, and 5.

###  1 Physics Beyond the Standard Model

The Standard Model is the quantum field theory that governs the fundamental
interactions (except gravity) between all known particles in our Universe (see
Fig. 1). Despite its excellent agreement with the outcomes of particle physics
experiments, there is now overwhelming evidence that it is far from a complete
description of Nature.

####  1 The Standard Model

The Standard Model is built upon the isometries of Minkowski spacetime and a
set of non-Abelian gauge symmetries. The spacetime symmetry dictates that the
particles of the theory transform as unitary irreducible representations of
the Poincaré group [Wigner1939]. On the other hand, the gauge symmetry governs
the way in which these particles interact with one another. Specifically, the
gauge group of the Standard Model is $SU(3)_{C}\otimes SU(2)_{L}\otimes
U(1)_{Y}$, where $SU(3)_{C}$ is associated to quantum chromodynamics (QCD),
the theory of the strong nuclear force [Gross:1973id, Gross:1973ju,
Politzer:1973fx], and $SU(2)_{L}\otimes U(1)_{Y}$ describes the electroweak
sector [Glashow:1961tr, Weinberg:1967tq, Salam:1968rm]. More precisely, the
subscript $C$ denotes the color charge of QCD, while $L$ and $Y$ stands for
left-handedness and hypercharge respectively. The left-handedness of
$SU(2)_{L}$ is especially interesting because it implies that the right-handed
particles do not couple to this subgroup. The Standard Model is therefore
chiral, i.e. it is not invariant under a parity transformation, P. As we shall
see, the theory is also not invariant under time reversal, T, and charge
conjugation, C. Nevertheless, it must respect the combined CPT symmetry of any
local quantum field theory in Minkowski spacetime.

The particle content of the Standard Model can be organized into four broad
categories: i) the gauge bosons, which mediate the forces; ii) the quarks,
which interact via both the strong and electroweak forces; iii) the leptons,
which only couple to the electroweak sector; and iv) the Higgs boson, which
mediates the spontaneous symmetry breaking of the electroweak sector. The
gauge bosons are massless vector (spin-1) fields while the quarks and leptons
are spin-1/2 fermions. The Higgs boson is the only scalar (spin-0) field in
theory. See Fig. 1 for a summary of the Standard Model particles.

The gauge group of the Standard Model itself does not inform us how these
particles interact with one another. Instead, we must also know the
representations under which they transform in this group. For instance, the
gauge bosons transform as adjoint representations of each of the $SU(3)_{C}$,
$SU(2)_{L}$, and $U(1)_{Y}$ subgroups. The self-interactions of the gauge
bosons are therefore given by the structure constant of the subgroup’s Lie
algebra. On the other hand, the matter fields transform as fundamental (and
anti-fundamental) representations of the respective subgroups. For instance,
in QCD, the quarks are organized into two types of triplet representations:
the up-type quarks, $(u,c,t)$, and the down-type quarks, $(d,s,b)$. On the
other hand, in the electroweak sector, the left-handed components of the
quarks are organized into three generations of doublet representations:
$(u,d)_{L}$, $(c,s)_{L}$, and $(t,b)_{L}$. By contrast, the right-handed
quarks transform as trivial representations under $SU(2)_{L}$, though they
have non-vanishing hypercharges. Similarly, the left-handed leptons are
organized into three flavor generations: the electron and its neutrino
$(\nu_{e},e^{-})_{L}$, its muonic counterpart $(\nu_{\mu},\mu^{-})_{L}$, and
its tau-flavour counterpart $(\nu_{\tau},\tau^{-})_{L}$. Most notably, there
are only right-handed electron, muon, and tau leptons in the Standard Model,
but no right-handed neutrinos. Because all quarks and leptons are Dirac
spinors (except for the neutrinos, which could be either Dirac or Majorana
fermions), they have antiparticle counterparts with opposite charges. Finally,
the Higgs field also transforms as a doublet representation under
$SU(2)_{L}\otimes U(1)_{Y}$.

Figure 1: The Standard Model particles in a nutshell. The red, blue, purple,
and gray categories represents the quarks, leptons, gauge bosons, and the
Higgs boson, respectively.

The symmetries and particle content uniquely fix the Lagrangian of the
Standard Model.111Here, we focus only on the renormalizable part of the
Standard Model Lagrangian. In other words, only operators with mass dimensions
four or below are considered in (1). Higher-dimensional operators can be
included in an effective field theory treatment, and are suppressed by the
scale of new physics [Weinberg:1978kz, Weinberg:1979sa, Buchmuller:1985jz,
Lehman:2014jma, Henning:2015alf].  Schematically, it reads

$\mathcal{L}_{\rm
SM}=-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}+i\bar{\Psi}\gamma^{\mu}D_{\mu}\Psi+\left(\bar{\Psi}_{i}Y_{ij}\Psi_{j}H+h.c.\right)+\left|D_{\mu}H\right|^{2}-V(H)\,,$
(1)

where $X_{\mu\nu}$ represents the gauge field strengths, $D_{\mu}$ is the
covariant derivative with respect to the gauge fields, $\Psi$ denotes the
quarks and leptons in their appropriate representations, $Y_{ij}$ is the
Yukawa matrix, $h.c.$ denotes the Hermitian conjugate, $H$ is the complex
Higgs field, and $V(H)=m_{H}^{2}|H|^{2}+\lambda_{H}|H|^{4}$ is the Higgs
potential. Crucially, the charges of the matter fields are appropriately
assigned such that the Lagrangian (1) is invariant under the gauge
transformations. The interactions between the matter fields and the gauge
fields are implicitly encoded in $D_{\mu}$. Since the weak interaction only
couples to left-handed particles and right-handed antiparticles, it violates C
and P maximally. Furthermore, it also violates CP because the Yukawa matrix in
the quark sector contains complex elements.

At high energies, the gauge symmetry of the Standard Model is fully restored.
In this limit, all particles, except the Higgs boson, are massless. However,
below the electroweak scale at $246$ GeV, the Higgs boson acquires a non-
trivial vacuum expectation value, thereby triggering spontaneous symmetry
breaking [Higgs:1964pj, Guralnik:1964eu, Higgs:1964ia, Englert:1964et] in the
electroweak sector $SU(2)_{L}\otimes U(1)_{Y}\to U(1)_{EM}$, where $U(1)_{EM}$
is the gauge group of quantum electrodynamics (QED). Two important
implications emerge from this phenomenon. Firstly, all fermions, except the
neutrinos, become massive, with their masses proportional to the sizes of the
corresponding elements in $Y_{ij}$. Secondly, the gauge bosons split into the
massive $W^{\pm}$ and $Z$ bosons, which mediate the weak nuclear force, and
the photon $A$. Crucially, the large masses of the $W^{\pm}$ and $Z$ bosons at
$\approx 100$ GeV mean that the weak interaction is a short-range force. This
short-range behavior is effectively modeled as point-like in Fermi theory,
which is the effective field theory of the weak nuclear force at low energies.
On the other hand, $A$ is massless and mediates the usual long-range
electromagnetic force in QED.

The strong nuclear force displays characteristic features that are absent in
the electroweak sector. One of the key properties of the theory is asymptotic
freedom, which means the coupling strength of the strong force, $g_{s}$,
increases at high energies and decreases at low energies. This is
qualitatively distinct from QED, where the electromagnetic coupling strength
runs in the opposite manner. As a result, there exists a typical energy scale,
also known as the QCD scale $\approx 100$ MeV, below which $g_{s}$ exceeds
order unity and QCD becomes strongly interacting. At low energies, quarks and
gluons therefore hadronize to form baryons and mesons, which are composite
particles with no color charges. In fact, quarks and gluons are believed to be
so strongly bound that they cannot be observed as free particles. This
confinement behaviour, though not proven mathematically, has been demonstrated
by numerical simulations in lattice QCD, see e.g. [Kondo:2014sta,
Ratti:2018ksb] for recent reviews. Another important feature of QCD is its
topologically non-trivial vacuum manifold. In particular, vacuum solutions in
the theory are separated by energy barriers constructed from gluon gauge field
configurations. Tunnelling events through these energy barriers are known as
instantons [tHooft:1976snw, tHooft:1976rip], and can contribute to important
non-perturbative dynamics in QCD.

The success of the Standard Model is ultimately grounded in its excellent
agreement with the outcomes of many experiments. Over the past few decades,
particle physics had been predominantly probed through the high-energy
frontier, in which heavy particles are produced through relativistic
collisions of light elementary particles. This way of probing new physics has
allowed us to observe virtually all of the particles and forces predicted by
the Standard Model. Impressively, the Large Hadron Collider (LHC) has
confirmed the validity of the theory up to approximately $10$ TeV
[Aad:2012tfa, Chatrchyan:2012xdj, Tanabashi:2018oca], which is nearly two
orders of magnitude larger than the electroweak scale. Furthermore, the
Standard Model has been shown to be correct by many high-precision
measurements of the theory. This is demonstrated most impressively through the
anomalous magnetic moment of the electron, $a_{e}$, whose theoretical and
experimental results have been compared to an exquisite precision
[Aoyama:2014sxa]:

$\displaystyle a_{e}$
$\displaystyle=0.001159652181643(25)\,,\qquad\text{(theory)}$ (2)
$\displaystyle a_{e}$
$\displaystyle=0.00115965218073(028)\,,\qquad\text{(experiment)}\,.$

Other Standard Model parameters, though measured to a lesser degree of
precision, have also demonstrated excellent agreements between theory and
experiment [Tanabashi:2018oca].

While the Standard Model successfully describes the strong, weak, and
electromagnetic forces, gravity is notably absent in the theory. As we have
seen in §1, the latter is instead described by the Einstein field equation
(7), whose dynamical degree of freedom is the metric $g_{\mu\nu}$. These four
known fundamental forces of Nature are represented succinctly by the action

$S=\int{\rm d}^{4}x\sqrt{-g}\,\left[\frac{1}{2}M_{\rm
pl}^{2}R+\mathcal{L}_{\rm SM}\right]\,,$ (3)

where $M_{\rm pl}=(\hbar c/8\pi G)^{1/2}=2.4\times 10^{18}\,\text{GeV}$ is the
(reduced) Planck scale. The first term is called the Einstein-Hilbert action,
which yields the Einstein field equation when it is varied with respect to
$g_{\mu\nu}$. Unlike $\mathcal{L}_{\rm SM}$, the Einstein-Hilbert term is not
renormalizable. A ultraviolet completion is therefore necessary in order to
describe gravity at high energies. In fact, both terms in the action (3) are
only empirically verified at low energies $\lesssim 10$ TeV and should
therefore be viewed as an effective field theory of a high-energy theory,
plausibly one that unifies all of the four forces. New physics that appear at
high energies could be incorporated in (3) through higher-dimensional
operators that respect diffeomorphism invariance and the symmetries of the
Standard Model. From dimensional analysis, these operators must be suppressed
by the energy scales at which new physics appear. As a result, their
phenomenological implications at low energies are also typically suppressed.

####  2 Motivations for New Physics

Despite the successes of the Standard Model, multiple observational evidences
point towards the need for new physics. In addition, theoretical
considerations suggest the presence of several flaws in the theory, which
could be resolved through the introduction of new physical principles. In this
subsection, I will present a summary of key motivations for physics beyond the
Standard Model.

A stunning realization of modern physics is that most of the matter in our
Universe is dark. Through various astrophysical and cosmological observations,
we have discovered that the Standard Model particles only constitute $15\%$ of
the total matter in our Universe, while the remaining $85\%$ is a mysterious
form of dark matter (see e.g. [Bertone:2004pz] for a review). Dark matter
cannot be the Standard Model particles because it is non-baryonic, i.e. it
doesn’t seem to interact with the strong, weak, and electromagnetic forces at
all. Nevertheless, thanks to the equivalence principle, we have been able to
detect it through its gravitational interactions. For instance, dark matter
halos are known to be responsible for the flattening of galaxy rotation curves
[Rubin:1970zza, Roberts1975, Rubin:1980zd, Persic:1995ru], such as that of the
Milky Way [Iocco:2015xga]. Dark matter is also essential in explaining the
intricate features of the acoustic oscillations of the cosmic microwave
background radiation [Aghanim:2018eyx]. Although we understand what dark
matter does in our cosmos, we still do not know what it is. Given the
intricate structure of the Standard Model, it is not unreasonable to speculate
that dark matter also consists of a rich spectrum of particles, possibly even
mediated by many hidden forces. In Section 2, I will review several important
classes of dark matter. Indeed, a central goal of this thesis is explore how
different types of dark matter can be probed through the gravitational waves
emitted by astrophysical systems, such as binary black holes.

Matter, which includes both the Standard Model particles and dark matter, only
makes up $30\%$ of the total energy budget of our Universe. Remarkably, the
majority of the Universe’s energy density is contained in a form of negative-
pressure fluid that is commonly known as dark energy. Dark energy is a
critical component of our cosmic menu because it drives the accelerated
expansion of our Universe [Riess:1998cb, Perlmutter:1998np]. In the
$\Lambda$CDM model [Aghanim:2018eyx], dark energy is well-described by a
cosmological constant, i.e. a uniform energy density that permeates all of
spacetime. This effect can be incorporated into the action (3) through the
inclusion of a dimension-zero operator,222That is, this term does not depend
on any fields or derivatives. $\Lambda_{cc}^{4}\sim(1\,\text{meV})^{4}$.
Because this constant term is used to renormalize the zero-point energy in a
quantum field theory, the cosmological constant can be interpreted as the
energy density of vacuum. Classically, the relatively small value of
$\Lambda_{cc}$ compared to other known energy scales in our Universe poses no
conceptual problem. Nevertheless, as we shall shortly describe, in quantum
field theory this presents us with a severe theoretical challenge.

Although the Standard Model is an extremely successful theory, one of its
predictions is actually wrong. Specifically, it predicts that all neutrinos
are massless, which contradicts many experimental outcomes that showed the
presence of neutrinos masses (at least for two of the neutrino species). For
instance, neutrino-oscillation experiments [Fukuda:1998mi, Ahmad:2001an,
Ahmad:2002jz] have confirmed that the flavour eigenstates of neutrinos do not
coincide with their mass eigenstates, which is only possible if they have non-
vanishing masses. One of the ways in which neutrinos can acquire masses is
through the introduction of right-handed sterile neutrinos, which only couple
to the Standard Model neutrinos [Abazajian:2012ys]. In this case, the
smallness of the total neutrino masses, i.e. at $\sim 50$ meV, can be
explained through their couplings to very heavy sterile neutrinos, in what is
usually called the see-saw mechanism [Minkowski:1977sc, Yanagida:1979as,
Mohapatra:1979ia, Schechter:1980gr]. Alternatively, without enlarging the
particle content of the theory, the neutrino masses can be explained by
including the Weinberg operator [Weinberg:1979sa] into the Standard Model
Lagrangian (1). Schematically, this operator reads

$\mathcal{L}_{\rm
SM}\supset\frac{1}{\Lambda_{\nu}}\bar{\Psi}H^{\dagger}H\Psi+h.c.\,,$ (4)

where $\Psi$ denotes the left-handed lepton fields and $\Lambda_{\nu}$ is a
new high-energy scale. Interestingly, by choosing $\Lambda_{\nu}\sim 10^{14}$
GeV, which is close to the energy scales of Grand Unified Theories
[Georgi:1974sy, Pati:1974yy, Fritzsch:1974nn], electroweak symmetry breaking
would lead to the correct total neutrinos masses. Furthermore, the Weinberg
operator is in fact the only dimension-five operator in the effective field
theory of the Standard Model [Weinberg:1978kz, Weinberg:1979sa,
Buchmuller:1985jz, Lehman:2014jma, Henning:2015alf]. It is remarkable that the
only flawed prediction of the Standard Model is naturally corrected by
incorporating the only leading non-renormalizable term into (1).

Another shortcoming of the Standard Model is its inability to explain the
dominance of matter over antimatter in our Universe. From the Sakharov
conditions [Sakharov_1991], we know that baryogenesis must be driven by non-
equilibrium dynamics and violate C, CP, and baryon number (see e.g.
[Cline:2006ts] for a pedagogical review on the subject). Intriguingly, all of
these conditions are actually satisfied by the Standard Model: the Universe is
out of equilibrium as it cools, C- and CP-violating terms are present in the
electroweak sector, and baryon number is violated by an anomaly [Adler:1969gk,
tHooft:1976rip]. However, the amount of C, CP, and baryon number violations in
the Standard Model are insufficient to explain the observed matter-antimatter
asymmetry observed in our Universe. Furthermore, baryogenesis is necessarily a
strong first-order phase transition, while the electroweak phase transition is
of second-order nature [Kajantie:1996qd]. The need to explain the matter-
antimatter asymmetry in our Universe is therefore another strong motivation
for introducing physics beyond the Standard Model.

One of the most profound observations in cosmology is the apparent acausal
correlations between different regions of our Universe. For example,
measurements of the cosmic microwave background (CMB) have shown that the
average temperatures of widely-separated parts on our celestial sphere are
identical [Penzias:1965wn]. This is despite the fact that, in the hot Big Bang
paradigm, different regions of the early Universe did not have enough time to
thermalize with one another before recombination, which is the epoch when the
CMB was emitted. Furthermore, the statistics of the CMB density fluctuations
were found to display subtle correlations over length scales that were larger
than the horizon at recombination [Smoot:1992td, Spergel:2003cb]. The most
popular resolution to this horizon problem involves the introduction of an
inflationary epoch before the hot Big Bang, when the Universe expanded at an
accelerated rate [Guth:1980zm, Linde:1981mu, Albrecht:1982wi] (see
[Baumann:2009ds] for a pedagogical introduction). In this picture, different
patches of the CMB were actually in causal contact with one another in the
pre-inflationary epoch. Crucially, the statistics of quantum mechanical
fluctuations that were generated during inflation agree excellently with those
of the primordial density fluctuations inferred from the CMB [Guth:1982ec,
Bardeen:1983qw]. That inflation offers a causal mechanism to generate the
correct initial conditions of our Universe makes it a leading paradigm in
modern cosmology. Importantly, none of the Standard Model particles including
the Higgs, at least in its minimal form in (1), can be responsible for driving
the inflationary epoch.

So far, we have described observational evidences for physics beyond the
Standard Model. However, a number of theoretical considerations also hint at
the presence of new physics. One of the key conceptual problems of the
Standard Model concerns the unstable hierarchy between the electroweak scale
and other higher energy scales. Specifically, the hierarchy problem is the
theoretical observation that loop corrections to the Higgs boson mass,
$m_{H}$, are quadratically divergent with respect to the masses of putative
heavy particles that appear above the electroweak scale (see e.g.
[Schwartz:2013pla] for details of this computation). The parameter $m_{H}$ is
therefore extremely sensitive to physics at the ultraviolet (UV). In fact, it
would be natural for $m_{H}$ to be lifted by UV physics to a value that is of
the order of the new high scale. This lifting can, of course, be avoided if
there exists counterterms in the UV theory that renormalize these divergences.
Nevertheless, the precision at which these counterterms must be fine-tuned in
order to maintain the small Higgs mass at $m_{H}=125$ GeV can be unusually
high. For instance, assuming $M_{\rm pl}$ is the only energy scale above the
electroweak scale, as is the case in the action (3), the counterterms must be
fine tuned at the level of $\sim(m_{H}/M_{\rm pl})^{2}\sim 10^{-34}$. This
fine-tuning problem suggests the presence of new physics near the electroweak
scale, which motivated the construction of many beyond the Standard Model
proposals such as low-scale supersymmetry [Dimopoulos:1981zb, Sakai:1981gr,
Dimopoulos:1981yj, Ibanez:1981yh] and large extra dimensions
[ArkaniHamed:1998rs, Randall:1999ee, Randall:1999vf]. Importantly, the
hierarchy problem does not apply to the Standard Model fermions and gauge
bosons, as their relatively small masses are protected by approximate chiral
and gauge symmetries respectively, which become exact when their masses vanish
[tHooft:1979rat]. On the other hand, no such approximate symmetry manifestly
exists for the Higgs boson.

An even more severe fine-tuning problem exists for the cosmological constant.
To begin, quantum corrections to $\Lambda_{cc}$ would generate UV divergences
that scale quartically with respect to a higher energy scale. Comparing this
with the quadratic scaling for the Higgs mass, $\Lambda_{cc}$ is much more
sensitive to corrections by UV physics.333Heuristically, these UV scalings can
be understood from the fact that the degree of divergence of any operator in a
Lagrangian is equal to the spacetime dimension minus the operator’s mass
dimension [Weinberg:1995mt]. The UV divergence of the $\Lambda^{4}$ term,
which is a dimension-zero operator, therefore scales with the forth power of a
high energy scale. On the other hand, $m_{H}^{2}H^{\dagger}H$, which is a
dimension-two operator, scales quadratically.  Furthermore, the fine-tuning
problem is exacerbated by the fact that we believe we know what the UV physics
above $\Lambda_{cc}$ is — it’s the Standard Model! Unfortunately, loop
corrections to $\Lambda_{cc}$ from the vacuum fluctuations of the Standard
Model particles do not cancel each other. As a result, the relevant
counterterms must already be fine tuned at least at the level of
$\sim(\Lambda_{cc}/m_{H})^{4}\sim 10^{-56}$. In the limit where the
cosmological constant is sensitive to physics at the Planck scale, the fine
tuning would be at a staggering level of $\sim(\Lambda_{cc}/M_{\rm
pl})^{4}\sim 10^{-120}$. The apparent instability of $\Lambda_{cc}$ to a
miniscule change to its value is known as the cosmological constant problem
[Weinberg:1988cp, Padmanabhan:2002ji]. The lack of generally-accepted
theoretical mechanisms that resolve this issue has led to anthropic
reasonings, which posit that the observed value of $\Lambda_{cc}$ in our
Universe is the result of a selection effect in a much larger Multiverse
[Weinberg:1987dv, Bousso:2000xa].

The smallness of another parameter in the Standard Model also offers a
tentative hint for new physics. In QCD, the following P- and CP-violating term
should, in principle, be present in the Lagrangian (5),

$\mathcal{L}_{\rm SM}\supset\theta\hskip
1.0pt\frac{g_{s}^{2}}{32\pi^{2}}G_{\mu\nu}\tilde{G}^{\mu\nu}\,,$ (5)

where $G_{\mu\nu}$ is the gluon field strength,
$\tilde{G}_{\mu\nu}\equiv\epsilon_{\mu\nu\rho\sigma}G^{\rho\sigma}/2$ is its
dual field strength, with $\epsilon_{\mu\nu\rho\sigma}$ the fully
antisymmetric tensor, and $\theta$ is called the strong CP phase. The term (5)
is usually ignored in the Lagrangian because it is a total divergence and
therefore does not affect the classical equation of motion. However, it plays
an important role in the quantum mechanical structure of the QCD vacuum and
hence cannot be ignored. A priori, $\theta$ can take any value between $0$ and
$2\pi$. Since all other dimensionless parameters in the Standard Model have
values that are $\mathcal{O}(1)$, one naturally expects this to also be the
case for $\theta$ as well. However, experimental measurements of the neutron
electric dipole moment constrain this parameter to be $|\theta|<10^{-10}$
[Baker:2006ts, Afach:2015sja], which is many orders of magnitude smaller than
the generic expectation. This lack of observed CP violation in the strong
interaction is also known as the strong CP problem. Despite the apparent
similarity with the hierarchy and cosmological constant problems, we emphasize
that the smallness of $\theta$ is of a distinct nature, as it does not
represent any sensitivity to loop corrections introduced by physics in the
ultraviolet. In §2, I will describe one attractive resolution to this problem,
which involves promoting $\theta$ to a dynamical field, such that the
smallness of $\theta$ is naturally achieved by the field relaxing towards the
minimum of its potential.

Finally, the Standard Model does not include gravity. A complete description
of quantum gravity is perhaps the most ambitious goal in theoretical physics.
In §1, we described how the action (3) should only be viewed as an effective
field theory that is valid at low energies $\lesssim 10$ TeV. This is
especially true for the Einstein-Hilbert action, which is non-renormalizable
and must therefore be ultraviolet-completed at high energies. Perhaps the most
promising candidate of quantum gravity is string theory, which posits that the
particles that we observe are excitations of elementary strings. String theory
is a remarkably rich subject and has dominated much of theoretical physics
over the past few decades (see [Green:1987sp, Green:2012pqa,
Polchinski:1998rq, Polchinski:1998rr, Becker:2007zj] for classic textbooks).
Indeed, the phenomenology that arises in string theory could potentially
resolve many of the deficiencies of the Standard Model described above
[stringpheno2012, Baumann:2014nda]. For the purpose of this thesis, it
suffices to remark only on a few key features of the theory. Firstly, all
versions of string theory require the presence of extra compact spatial
dimensions, whose topological properties are generally incredibly rich
[Candelas:1985en, Douglas:2006es]. Secondly, the quantized string spectrum
necessarily includes many new particles, such as antisymmetric tensor fields.
The Kaluza-Klein reduction of these higher-dimensional fields over a compact
manifold would generate new particles in our observable four-dimensional
Universe, possibly forming a component of dark matter. Finally, string theory
is also a theory that includes solitonic objects, such as D-branes, world-
sheet instantons, and gauge-theory instantons [tHooft:1976rip, Dine:1986zy,
Becker:1995kb, Kallosh:1995hi, Polchinski:1995mt]. These objects would
introduce non-perturbative effects, which affect the low-energy physics by
giving these new particles small but non-vanishing masses.

###  2 The Weak-Coupling Frontier

In §2, I described multiple evidences for physics beyond the Standard Model in
our Universe. Proposed solutions to these problems often involve the
introduction of new degrees of freedom that have escaped our detections thus
far. Over the past few decades, searches for new particles have been dominated
by the high-energy frontier, where heavy particles appear as resonances in
scattering processes of colliders, such as the LHC. While this way of probing
new physics have helped to uncover the fundamental building blocks of Nature
[Aad:2012tfa, Chatrchyan:2012xdj, Tanabashi:2018oca], it relies on having
appreciable interactions with the Standard Model particles involved in the
collision processes. In other words, traditional collider experiments are
blind to “dark sectors” which couple very weakly to ordinary matter, even if
the associated new particles are very light [Essig:2013lka, Berlin:2018bsc].
Roughly speaking, if these new particles interact with the Standard Model in a
way that is weaker than the weak nuclear force, they can easily escape
collider searches because of the small interaction cross sections (see Fig.
2). In that case, we must find creative ways of detecting new light particles
at the weak-coupling frontier. In what follows, I will motivate several
generic classes of new particles, which I universally refer to as “dark
matter”, although these particles could also simultaneously solve the other
problems described in §2. I will also briefly mention how they are being
searched for by many different types of experiments, and point the reader to
references for existing constraints.

Figure 2: Particle colliders are blind to new particles that are either too
heavy or too weakly-coupled to be produced in sufficient numbers. Very massive
particles (up to $10^{14}$ GeV) can be created in the “cosmological collider”
that was active during inflation [Chen:2009zp, Baumann:2011nk, Noumi:2012vr,
Flauger:2013hra, Arkani-Hamed:2015bza, Arkani-Hamed:2018kmz, Baumann:2019oyu].
In this thesis, I explore the weak-coupling frontier through the gravitational
waves emitted by binary systems. In particular, in Chapters 3, 4, and 5, I
focus on the signatures of ultralight bosons within the mass range
$[10^{-20},10^{-10}]$ eV, which can form boson clouds around rotating
astrophysical black holes. Remarkably, the dynamics of these clouds in binary
systems are described by an “S-matrix” and are sensitive to both the masses
and intrinsic spins of the bosons. In direct analogy with ordinary colliders,
I will therefore refer to these binary systems as “gravitational colliders”.
In Chapter 6, I discuss how other types of weakly-coupled particles could form
dark compact objects in our Universe. These objects can similarly be probed
through the gravitational waves emitted when they are parts of binary systems.

####  1 Particle Dark Matter

In this subsection, I focus on dark matter that behaves like “particles”, in
that they do not display wave-like properties over astrophysical length
scales. As a rough fiducial guide, I consider dark matter that is heavier than
the total mass of neutrinos $\sim 1$ meV, which are the lightest particles in
the Standard Model, to fall within this category. These particles would have
Compton wavelengths that are shorter than $0.1$ mm, or equivalently have de
Broglie wavelengths that are shorter $0.1$ m if they follow the typical
velocity distribution of dark matter in astrophysical environments. The
discussion below will be guided by the types of forces—weak, electromagnetic
and gravitational—through which dark matter could couple to the Standard
Model.

A popular class of dark matter are the weakly-interacting massive particles
(WIMPs) — hypothetical particles that interact with ordinary matter through
the weak nuclear force (see [Arcadi:2017kky] for a recent review). One of the
most attractive features of the WIMP paradigm is that the predicted thermal
relic abundance agrees with the observed dark matter density in our Universe
[Srednicki:1988ce, Gondolo:1990dk]. Furthermore, since the interactions of
WIMPs are naturally tied to the electroweak scale, they can be easily embedded
into many theories that are proposed to address other problems of the
electroweak sector, such as the hierarchy problem. While the details of WIMPs
can vary with different theories, a generic feature of all models is the
presence of an interaction that mediates two-two scattering processes with the
Standard Model particles. Schematically, this is captured by the effective
operator

$\mathcal{L}_{\chi}\supset c_{\chi}\chi\chi\hskip 1.0pt\psi\psi\,,$ (1)

where $\chi$ is the dark matter particle, $\psi$ denotes a Standard Model
particle, and $c_{\chi}$ is the interaction coefficient, which is typically
suppressed by the electroweak scale. Indeed, (1) provides the basis for most
of the current experimental searches for WIMPs. For instance, direct-detection
methods are designed to detect the elastic scattering process
$\chi+\psi\to\chi+\psi$ through the recoils of stable nuclei [Akerib:2016vxi,
Cui:2017nnn, Aprile:2018dbl]. Interestingly, these experiments will soon
approach the “neutrino floor”, below which they would instead observe the
background atmospheric and solar neutrinos and hence become insensitive to
WIMPs with weaker couplings [Schumann:2019eaa]. On the other hand, indirect-
detection searches aim to detect the final products of inelastic scattering
(annihilation) processes $\chi+\chi\to\psi+\psi$, where $\psi$ can be gamma
rays, cosmic rays, or neutrinos [FermiLAT2009, Adriani:2008zr, Abbasi:2012ws].
Through these observations, we are beginning to constrain the annihilation
cross section of WIMP dark matter [Gaskins:2016cha].

Dark matter could also potentially carry small amounts of electric charge.
These charged dark matter or milli-charged particles [Goldberg:1986nk,
Cheung:2007ut, Feldman:2007wj], would couple feebly with the Standard Model
particles through the electromagnetic force. Denoting the ratio of the dark
matter electric charge to the elementary charge by $Q$, the relevant terms in
the dark matter Lagrangian are

$\mathcal{L}_{Q}\supset-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+QeJ^{\mu}_{\rm
D}A_{\mu}\,,$ (2)

where $A_{\mu}$ is the electromagnetic field, $F_{\mu\nu}$ is its field
strength, $e$ is the elementary charge, and $J_{\rm D}^{\mu}$ is the charge
current of dark matter. Because charged dark matter must be invariant under a
$U(1)_{EM}$ gauge transformation, it must either be a complex scalar or a
Dirac fermion, in which case the interaction in (2) is directly analogous to
scalar and spinor QED, respectively. Unlike the electric charges of the
Standard Model particles, $Q$ need not be quantized. In addition, unless the
charged dark matter is chiral, the value of $Q$ does not interfere with
anomaly cancellation of the Standard Model. Constraints for this type of dark
matter have been obtained through astrophysical and collider searches, see
e.g. [Dimopoulos:1989hk, Davidson:2000hf, Essig:2013lka, Berlin:2018bsc].
Interestingly, dark matter with $Q\sim 1$ could still contribute a sizable
fraction to the total dark matter density, if the dark matter mass falls
within the approximate range $[10^{5},10^{11}]$ GeV [DeRujula:1989fe,
Chuzhoy:2008zy].

In the previous example, dark matter was electrically charged and therefore
coupled to electromagnetism. However, the dark sector could also interact with
ordinary matter through the electromagnetic force if photons kinetically mix
with a hypothetical dark photon [Holdom:1986eq, Okun:1982xi, Galison:1983pa].
Denoting the dark photon by $A^{\prime}_{\mu}$, the most general
renormalizable Lagrangian is

$\mathcal{L}_{A^{\prime}}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+eJ_{\rm
SM}^{\mu}A_{\mu}-\frac{1}{4}F^{\prime}_{\mu\nu}F^{\prime\mu\nu}-\frac{\epsilon}{2}F^{\prime}_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\mu^{2}A^{\prime}_{\mu}A^{\prime\mu}\,,$
(3)

where $J_{\rm SM}$ is the charge current sourced by the Standard Model
particles, $F^{\prime}_{\mu\nu}$ is the dark photon field strength, and
$\epsilon\ll 1$ is the kinetic mixing parameter. The physical interpretation
of the kinetic mixing is clearer if we perform the gauge transformation
$A_{\mu}\to A_{\mu}-\epsilon A^{\prime}_{\mu}$, in which case the mixing term
is removed while the second term in (3) generates an additional source term
$\epsilon eJ^{\mu}_{\rm SM}A^{\prime}_{\mu}$. This interaction is therefore
directly analogous to that in (2), except that it is the Standard Model
fermions (and not the photon) that couple to the dark sector through
suppressed electric charges. A recent review on this subject, which includes
discussions of massless dark photons and mixings with the $U(1)_{Y}$
hypercharge gauge group, can be found in [Fabbrichesi:2020wbt]. In §2, I will
describe current constraints on dark photons in the $\mu$-$\epsilon$ parameter
space. Importantly, we shall find that existing constraints apply
predominantly to heavy particle-like dark photons but are largely absent in
the ultralight regime.

Finally, dark matter particles can also gravitate and form astrophysical bound
states. For instance, they could be accreted around black holes over
astrophysical timescales and develop high-density minispikes [Gondolo:1999ef,
Ferrer:2017xwm]. In addition, they could also form dark compact objects such
as boson stars and solitons [Kaup1968, Ruffini1969, Breit:1983nr, Colpi1986,
Lee1987-2, Lee1987-3] through a wide variety of astrophysical processes
[Seidel:1993zk, Grasso:1990zg, Schunck:2003kk]. If these objects are part of
binary systems, they would greatly enrich the binaries’ dynamical evolution,
such as through the finite-size effects that we described in Section 2. This
would in turn leave significant imprints in the gravitational waves emitted by
the binary system, thereby allowing us to probe the internal structure of
these objects through the observed waveforms. In Chapter 6, I will elaborate
on probing particle dark matter through these dark compact objects and their
associated gravitational-wave signatures.

####  2 Ultralight Bosons

Dark matter could also consist of new degrees of freedom that are light, so
light that their Compton and de Broglie wavelengths are of the order of
astrophysical length scales. This type of dark matter is qualitatively
distinct from the particle dark matter described above, as they can display
wave-like phenomenology over large scales, which are hard to observe if their
masses are large. The Compton wavelength and the de Broglie wavelength of a
massive field, denoted by $\lambda_{c}$ and $\lambda_{db}$ respectively, are

$\lambda_{c}\simeq 2\hskip 1.0pt\text{km}\left(\frac{10^{-10}\hskip
1.0pt\text{eV}}{\mu}\right)\,,\quad\lambda_{db}\simeq 3000\hskip
1.0pt\text{km}\left(\frac{10^{-10}\hskip
1.0pt\text{eV}}{\mu}\right)\left(\frac{200\hskip 1.0pt\text{km
s}^{-1}}{u}\right)\,,$ (4)

where $\mu$ and $u$ are the field’s mass and average velocity. We therefore
see that dark matter with masses smaller than order $10^{-10}$ eV would
display wave-like properties over scales that are larger than the sizes of
solar-mass black holes. Boson fields with such small masses are especially
interesting because can occupy a narrow region in phase space in large
numbers. The resulting boson state is therefore a classical coherent
oscillation that acts like a pressureless fluid, which is precisely the
property of cold dark matter, albeit only over length scales that are longer
than the corresponding de Broglie wavelength.444Ultralight bosons with masses
of the order of $10^{-22}$ eV or smaller, also known as fuzzy dark matter,
have been studied extensively in the literature [Hu:2000ke, Hui:2016ltb]. This
mass range is interesting because it corresponds to bosons having de Broglie
wavelengths that are longer than order 10 kpc, which are the length scales
probed by most astrophysical and cosmological observations. It is therefore
possible to distinguish them from cold dark matter through their distinct
phenomenologies at these large scales. In order to probe the wave-like
properties of ultralight bosons that are heavier than $10^{-22}$eV, one would
need to observe astrophysical events at shorter scales. In Section 3, we shall
see how these heavier bosons can trigger novel effects around astrophysical
black holes. In this subsection, I discuss several well-motivated classes of
ultralight boson fields. I organize our discussion according to the intrinsic
spins of the boson fields.

Theoretical solutions to the strong CP problem are surprisingly scarce.
Perhaps the most appealing proposal involves the introduction of a new global
$U(1)$ symmetry, also called the Peccei-Quinn symmetry, to the Standard Model
[Peccei:1977hh, Weinberg:1977ma, Wilczek:1977pj]. In this scenario, the
Goldstone boson that arises from spontaneous breaking of the Peccei-Quinn
symmetry removes $\theta$ in (5) through the boson’s dynamical evolution to
the minimum of its potential. This Goldstone mode is commonly known as the
axion, $a$, and the energy scale at which Peccei-Quinn symmetry breaking
occurs, $f$, is called the axion decay constant. Below the Peccei-Quinn and
electroweak scales, the effective action for the axion at leading-order in
$a/f$ is

$\mathcal{L}_{a}=-\frac{1}{2}\partial_{\mu}a\,\partial^{\mu}a+\frac{a}{f}\frac{g_{s}^{2}}{32\pi^{2}}G_{\mu\nu}\tilde{G}^{\mu\nu}+\frac{\partial_{\mu}a}{f}j^{\mu}_{a}+\frac{1}{4}\,a\,g_{a\gamma\gamma}F_{\mu\nu}\tilde{F}^{\mu\nu}\,,$
(5)

where $j^{\mu}_{a}$ is an axial current sourced by the quarks
[Srednicki:1985xd], $F_{\mu\nu}$ is the electromagnetic field strength,
$\tilde{F}_{\mu\nu}$ is its dual, and $g_{a\gamma\gamma}\propto f^{-1}$ is a
coupling constant. The Lagrangian (5) enjoys the continuous shift symmetry
$a\to a+\text{const}$, as is the case for all Goldstone modes. However, non-
perturbative effects introduced by the instantons in QCD break this continuous
shift symmetry to a discrete shift symmetry [tHooft:1976snw, tHooft:1976rip,
Sikivie:1982qv]. The effective potential of the axion below the QCD scale,
which is obtained by integrating out the quarks and gluons in the second and
third terms of (5), therefore has the following structure:

$\mathcal{L}_{a}\supset-\Lambda_{a}^{4}\Big{[}1-\cos\left(a/2\pi
f\right)\Big{]}+\cdots\,,$ (6)

where $\Lambda_{a}$ is proportional to the QCD scale and we have ignored
multi-instanton contributions. The potential (6) clearly has the discrete
symmetry $a\to a+2\pi nf$, for arbitrary integers $n$. Crucially, the minimum
of the potential is located at $a=0$. The axion can therefore resolve the
strong CP problem by dynamically relaxing to its potential minimum.

In the late Universe, the QCD axion oscillates around the minimum of its
potential with a small amplitude. In that case, we can approximate the
potential (6) by performing a Taylor expansion about $a=0$ to quadratic order.
The quadratic potential provides a notion of the QCD axion mass, which is
[diCortona:2015ldu, DiVecchia:1980yfw]

$\mu=5.7\times 10^{-6}\,\text{eV}\left(\frac{10^{12}\hskip
1.0pt\text{GeV}}{f}\right)\,.$ (7)

Crucially, this shows that $\mu$ and $f$ are inversely proportional to each
other. Stellar-cooling constraints have already placed a lower bound on
$f\gtrsim 10^{10}$ GeV [Friedland:2012hj, Giannotti:2017hny], which is much
higher than the electroweak scale. As a result, the QCD axion must be
ultralight. In the limit where $f$ exceeds the energy scales of Grand Unified
Theories $\sim 10^{16}$ GeV, the axion mass can even be as small as $\lesssim
10^{-10}$ eV. At present, most of our constraints on the QCD axion rely on the
photon-axion coupling term in (5). These constraints are summarized in Fig. 3,
where the yellow band illustrates the predicted parameter ranges for the QCD
axion. Note that, despite the precise relationship between $\mu$ and $f$ in
(7), the finite thickness of the QCD axion window in Fig. 3 reflects an order-
one model dependence in the coefficient $g_{a\gamma\gamma}$ [Kim:1979if,
Shifman:1979if, Dine:1981rt, Zhitnitsky:1980tq]. See e.g. [Wu:2019exd,
Barth:2013sma] for similar constraints that rely on the axion-electron and
axion-neutron couplings.

Figure 3: Illustration of the constraints on the axion in the
$g_{a\gamma\gamma}$-$\mu$ parameter space. Details of these various
experimental constraints can be found in e.g. [Carosi:2013rla]. The yellow
diagonal band denotes the predicted window for the QCD axion. Importantly,
this figure shows it is challenging to constrain these new particles in the
small-mass and weak-coupling regime. Figure adapted from [Ringwald:2012cu].

Ultralight axions are also generic predictions of string theory and many
beyond-the-Standard-Model scenarios with extra dimensions.555These axions are
commonly called axion-like particles, or ALPs in short, in order to
distinguish them from the QCD axion. Nevertheless, such a distinction is not
important in this thesis. Unless stated otherwise, I will therefore broadly
refer all of them simply as “axions.” In particular, they arise from the
Kaluza-Klein reduction of $p$-form gauge potentials (i.e. massless
antisymmetric tensor fields with $p$ number of indices) over $p$-cycles of
compact spatial dimensions [Witten:1984dg, Svrcek:2006yi]. In string theory,
these $p$-form fields arise naturally as part of the particle spectrum of the
quantized string. These axions can be expressed more precisely as

$a_{i}(x^{\mu})=\int_{\Sigma_{p}}{\rm
d}^{m}y\,A_{p}\left(x^{\mu},y^{m}\right)\,,$ (8)

where $x^{\mu}$ and $y^{m}$ are the coordinates of the four-dimensional
spacetime and the compact space respectively, $A_{p}$ is the $p$-form gauge
field, and $\Sigma_{p}$ denotes the compact manifold with non-trivial
$p$-cycles. The label $i$ indexes the list $i=1,2,\cdots N$, where $N$ is the
number of non-trivial $p$-cycles on $\Sigma_{p}$. The relation (8) therefore
illustrates that the number of axions is equal to the number of non-trivial
$p$-cycles on $\Sigma_{p}$.666To unpack this rather complicated language, it
is instructive to consider the example of a torus, which is a two-dimensional
surface with two 1-cycles (the two circles that cannot be continuously
deformed to one other) and one 2-cycle (the torus’s interior volume). The
Kaluza-Klein reduction of a 1-form, which is a higher-dimensional analog of
the electromagnetic gauge field, would therefore produce two axions, while a
2-form would instead yield a single axion. In string theory, the compact
manifolds are six-dimensional surfaces with rich topologies and have large
values of $N$ [Candelas:1985en, Douglas:2006es]. As a result, a natural
prediction of string theory is the existence of many axions, possibly dozens
or even hundreds of them, in our Universe [Arvanitaki:2009fg, Acharya:2010zx,
Cicoli:2012sz].

The axions which we just described enjoy a continuous shift symmetry to all
orders in perturbation theory, as a direct consequence of the higher-
dimensional gauge invariance of the $p$-forms. However, like the QCD axion,
non-perturbative effects break this continuous shift symmetry to a discrete
shift symmetry. In string theory, these non-perturbative effects can arise
from a myriad of sources, including D-branes, world-sheet instantons, gauge-
theory instantons, and so on [tHooft:1976rip, Dine:1986zy, Becker:1995kb,
Kallosh:1995hi, Polchinski:1995mt]. The low-energy effective action of these
axions therefore takes the following form:

$\mathcal{L}_{a}=-\frac{1}{2}K_{ij}\,\partial_{\mu}a^{i}\partial^{\mu}a^{j}-\sum_{i=1}^{N}\Lambda_{i}^{4}\Big{[}1-\cos\left({Q_{i}}^{j}a_{j}/2\pi
f\right)\Big{]}\,,$ (9)

where $K_{ij}$ is the internal metric that describes the couplings between the
axions, $\Lambda_{i}$ are dynamically generated scales, and ${Q_{i}}^{j}$
parameterizes the effective decay constant $\sim f/|Q_{i}^{j}|$ of each of the
axion. Crucially, $\Lambda_{i}\propto M_{s}\hskip 1.0pte^{-S_{i}}$ is
exponentially suppressed by the non-perturbative effects, where $S_{i}\gg 1$
is the sum of the action of these effects and $M_{s}$ is a high-energy scale,
typially taken to be the string scale. As a result, the axion masses are
generically expected to be extremely small (see e.g. [Demirtas:2018akl] for
the distribution of axion mass for a given choice of compact manifold).
Crucially, unlike (7), the masses need not be related to decay constants at
all. The plausible parameter ranges for these axions are therefore
significantly wider than for the QCD axion. For instance, the constraints
outside the QCD axion window in Fig. 3 should be viewed as applicable to these
axions as well. Similar constraints obtained via the axion-electron and axion-
neutron couplings [Wu:2019exd, Barth:2013sma] are also applicable. Crucially,
in all of these cases, there are virtually no constraints in the ultralight
and weakly-interacting regime.

Figure 4: Illustration of the constraints on dark photons. Details of these
observational constraints can be found in e.g. [Essig:2013lka]. Similar to the
axions, the the small-mass and weak-coupling parameter regime is virtually
unconstrained by these observations. Figure adapted from [Essig:2013lka].

While we have focused on ultralight scalar fields so far, ultralight vector
fields can also arise in many scenarios of physics beyond the Standard Model.
Dark photons, which we have described in §1, constitute a well-motivated class
of such new particles. These vector fields can acquire their masses through
the Higgs mechanism, in which case $\mu\sim\epsilon\,m_{H}$ and $m_{H}$ is the
Higgs scale associated to either the electroweak sector or the dark sector.
The masses are naturally suppressed by the fact that both $m_{H}$ for the dark
sector and $\epsilon$ can be extremely small [delAguila:1988jz,
Goodsell:2009xc, Camara:2011jg]. In addition, if dark photon couples to a
strongly-interacting dark sector, the associated dynamical symmetry breaking
would naturally generate a vector mass that is exponentially suppressed
compared to a high-energy scale [Reece:2018zvv]. In any case, a small vector
mass is technically natural because gauge symmetry is restored in the massless
limit, making ultralight vector fields ubiquitous in many particle physics
models. Current constraints on the dark photon in the $\epsilon$-$\mu$
parameter space are shown in Fig. 4 (see [Fabbrichesi:2020wbt] for a recent
review on these experimental constraints). Crucially, this figure shows that
dark photons with low masses, i.e. $\mu\lesssim 10^{-6}$ eV, are hard to
detect by existing experiments.

Finally, we remark that many other types of ultralight bosons with spins have
been proposed in the literature. For instance, dark radiation, which are
ultralight vector fields that completely decouple from the Standard Model
gauge group, also arise in string compactifications and many physics beyond
the Standard Model scenarios [Goodsell:2009xc, Camara:2011jg]. In these cases,
none of the constraints shown in Fig. 4 would apply. Ultralight massive tensor
fields could also exist in theories that modify gravity, in which case they
must couple to Einstein gravity in a non-linear manner [Hinterbichler:2011tt,
deRham:2014zqa]. The only realistic way of detecting these new ultralight
bosons would therefore be through their gravitational couplings, such as in
cosmology or through black holes.

###  3 Gravitational Atoms

In §2, we saw that ultralight bosons are well-motivated particles that exist
in many proposed extensions to the Standard Model. Interestingly, if these
bosons have masses from $10^{-10}$ eV to $10^{-20}$ eV, they can be
spontaneously amplified around rotating astrophysical black holes (see Fig. 2
for a comparison with other energy scales in our Universe). This is the case
because the wave amplification mechanism, also known as black hols
superradiance, is most efficient when the Compton wavelength of the field is
of the order of the gravitational radius of the black hole. Bosons in the
stated mass range have Compton wavelengths that range from about $1$ km to
$10^{10}$ km, which coincide with the sizes of known black holes in our
Universe. Superradiance would generate in a coherent bosonic configuration
that is gravitationally bounded to the rotating black hole. Remarkably, these
bound states resemble the proton-electron structure of the hydrogen atom, and
are therefore often called “gravitational atoms”. I will first describe the
qualitative features of the superradiance phenomenon and then sketch some of
the key properties of the gravitational atoms.

####  1 Black Hole Superradiance

Superradiance is the phenomenon where the amplitude of a bosonic wave is
magnified due to the presence of dissipation.777By contrast, superradiant
amplification cannot occur for fermionic fields [Unruh:1973bda, Unruh:1974bw,
Manogue1988]. This is because the amplification process increases the
occupation number of a single state, which is nevertheless forbidden by the
Pauli exclusion principle. In fact, it is a purely kinematic effect that
pervades many areas of physics, though in subtly different guises (see
[Bekenstein:1998nt, Brito:2015oca] for reviews). Superradiance was first
understood in the context of Cherenkov radiation [Cerenkov:1934zz], which is
emitted by a medium due to the superluminal inertial motion of a charged
particle that propagates within it. More precisely, this radiation is emitted
when the linear velocity of the particle, $v$, exceeds the phase velocity of
the radiation in the medium [Frank:1937fk]

$v>\frac{c}{N}\,,$ (1)

where $c$ is the speed of light in vacuum (restored for clarity) and $N>1$ is
the medium’s index of refraction. In this case, the electromagnetic
interaction between the charged particle and the atoms in its surrounding
region provides the source of transfering energy between the particle and the
medium. As the particle’s superluminal motion triggers the formation of a
radiation shock front, its kinetic energy is transfered to excite the nearby
atoms in the medium. Cherenkov radiation is therefore the radiation that is
spontaneously emitted by the relaxation of these excited atoms. Crucially, the
amplitude of this emission can be large due to the bosonic nature of photons,
which allows for a large occupation number of the excited state.

Superradiance can also be triggered by the rotational motion of an
axisymmetric body. This was first conceived by Zeldovich through his thought
experiment of the scattering an electromagnetic wave onto a rotating metal
cylinder [Zeldovich:1971a, Zeldovich:1972spj]. In this example, the conducting
nature of the cylinder’s surface provides a mechanism of transfering of energy
and angular momentum between the cylinder and the wave. In general, the
incident wave would be partially reflected and partially absorbed by the
cylinder. However, if the cylinder is highly rotating, the incident wave can
instead extract energy and angular momentum from the cylinder, resulting in a
reflected wave that has a larger amplitude. This only occurs when the angular
velocity of the cylinder, $\Omega_{\rm c}$, exceeds the angular phase velocity
of the incident wave

$\Omega_{\rm c}>\frac{\omega}{m}\,,$ (2)

where $\omega$ and $m$ are the frequency and azimuthal angular momentum number
of the wave. Indeed, this kinematic condition is merely the rotational analog
of (1). While we have described rotational superradiance in a scattering
process, electromagnetic waves can also be spontaneously emitted due to the
quantum-mechanical fluctuations of these modes in the vacuum [Unruh:1974bw].
Although rotational superradiance is theoretically well-understood, this way
of amplifying light has not been experimentally realized because it is
difficult to spin cylinders up to the order of the frequency of
electromagnetic waves. Instead, rotational superradiance has been detected
through the amplification of phonons by draining vortex flows in water tanks
[Torres:2016iee], which have been proposed to be analog systems for black
holes [Unruh:1980cg, Weinfurtner:2010nu].

Remarkably, rotational superradiance can also be excited by highly-rotating
black holes [Starobinsky:1973aij, Starobinsky:1974spj, Bekenstein:1973mi]. In
fact, black holes are in many ways ideal objects to trigger this amplification
process. To begin, the event horizon of a black hole is a perfect absorber and
therefore provides the most efficient mechanism of transfering energy and
angular momentum with its environment. Furthermore, this process only relies
on the gravitational interaction between the black hole and the fields in its
vicinity. By virtue of the equivalence principle, all boson fields in our
Universe that satisfy the superradiance condition can therefore experience
this amplification.888In practice, however, not all bosons would amplify
appreciable over reasonable astrophysical timescales. See §2 for more details
about the rates of amplification. Crucially, these fields can even be
spontaneously amplified through their quantum-mechanical fluctuations in the
vacuum [Unruh:1974bw]. The kinematic condition for black hole superradiance is
similar to (2), except the angular velocity is replaced by that of the black
hole:

$\Omega_{\rm H}>\frac{\omega}{m}\,,$ (3)

where $\omega$ and $m$ are now the frequency and azimuthal angular momentum of
any boson field and not necessarily those of light. The condition (3)
demonstrates that only modes that co-rotate with the black hole ($m>0$) can
grow, while counter-rotating modes ($m<0$) would be decay into the black hole.
Crucially, because the Kerr metric (10) is perfectly axisymmetric, these
growing and decaying modes do not mix with one another around isolated black
holes. Recall that (3) was also mentioned in §2 in the context of the
Teukolsky equation, which describes the perturbation of a black hole by a
massless wave. Here I emphasize that (3) also applies to massive
perturbations, with $\omega$ dependent on the boson mass (see §2 for more
details).

It is perhaps counter-intuitive that waves can extract energy and angular
momentum from a black hole — isn’t the event horizon supposed to be a one-way
membrane in which everything that enters it cannot escape? Rotating black
holes are in this sense special because, from the point of view of an
asymptotic observer at infinity, ingoing waves that satisfy (3) have negative
energies, which are equivalent to fluxes of outgoing positive energy.
Superradiance is therefore qualitatively similar to the Penrose process
described in §2, in which case the energy of a black hole is extracted because
particles can have negative energies inside the black hole’s ergoregion.
Indeed, the very idea that black holes can actually lose energy has proven to
be very influential in the history of physics. For instance, superradiance was
in fact the key inspiration behind the formulation of Hawking radiation
[Hawking:1974sw] (see [sunyaev2004zeldovich] for Thorne’s personal account on
visiting Zeldovich along with Hawking in the early 1970s). The latter is
nevertheless qualitatively distinct from superradiance in two important ways:
Hawking emission is a quantum-mechanical phenomenon and it can be radiated
even for Schwarzschild black holes.

####  2 Hydrogenic Bound States

We described that black hole superradiance can in principle occur for both
massive and massless boson fields. However, this process is only efficient for
massive boson fields because the boson’s mass, $\mu$, sources an additional
potential barrier at distances far away from the black hole (see e.g. Fig. 7
of [Arvanitaki:2010sy]). In the superradiant-scattering picture, this barrier
would reflect waves that have already been amplified back to the black hole
and hence trigger more superradiance. As a result, the wave’s amplitude
further increases while the black hole spins further down. This process would
repeat continuously until the angular velocity of the black hole saturates the
inequality (3). The final state of this black-hole instability is a cloud of
boson condensate that is gravitationally bounded to a slowly-rotating black
hole. Depending on the efficiency of the superradiant extraction, the mass of
the cloud can range up to about $30\%$ of the mass of the black hole
[Christodoulou:1970]. This way of extracting energy and angular momentum
efficiently from a black hole is a natural realization of the “black-hole
bomb” thought experiment [Press:1972zz]. Importantly, this instability does
not occur for a massless field, which does not source a reflection barrier and
therefore only experiences superradiance as a single event.

The key parameter underlying the superradiance phenomenon of a massive field
is the ratio of the gravitational radius of the rotating black hole, $r_{g}$,
to the (reduced) Compton wavelength of the field, $\lambda_{c}$. This is often
called the gravitational fine-structure constant, whose typical values are of
order

$\alpha\equiv\frac{GM\mu}{\hbar c}\simeq
0.04\left(\frac{M}{60M_{\odot}}\right)\left(\frac{\mu}{10^{-13}\,{\mathrm{eV}}}\right)\,.$
(4)

This parameter plays a similar role as the fine-structure constant of QED and
dictates virtually all properties of the boson clouds, including their sizes
and energy spectra. As we shall see, superradiance occurs when $\alpha$ is
smaller than order unity. From (4), we find that ultralight bosons with masses
from $10^{-20}$ eV to $10^{-10}$ eV are therefore efficiently amplified by
astrophysical black holes in our Universe, which are believed to have masses
approximately from $1M_{\odot}$ to $10^{10}M_{\odot}$.

Although superradiance can occur for bosons for any spin, in what follows I
only focus on boson clouds made of scalar and vector fields. I will also
assume that the self interactions of the fields are negligible, such that the
only interaction in these systems is the gravitational coupling between the
boson’s mass and the Kerr metric. There is an extensive literature on
superradiance of minimally-coupled scalar [Detweiler:1980uk, Dolan:2007mj,
Arvanitaki:2010sy, Yoshino:2012kn, Dolan:2012yt, Okawa:2014nda, Yoshino:2014,
Baumann:2018vus, Brito:2014wla, Arvanitaki:2014wva, Arvanitaki:2016qwi,
Brito:2017wnc, Brito:2017zvb] and vector fields [Witek:2012tr, Pani:2012vp,
Pani:2012bp, Endlich:2016jgc, East:2017ovw, East:2017mrj, East:2018glu,
Baryakhtar:2017ngi, Cardoso:2018tly, Dolan:2018dqv, Baumann:2019eav,
Siemonsen:2019cr]. Here I only review some of the key properties of these
clouds that are known in the literature before the works presented in
subsequent chapters were pursued.

##### Scalar clouds

Scalar boson clouds are the most well-studied examples of the gravitational
atom. They are bound-state solutions of the Klein-Gordon equation of the
scalar field $\Phi$ around the spacetime of a rotating black hole

$\left(g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}-\mu^{2}\right)\Phi(t,\textbf{r})=0\,,$
(5)

where $g_{\alpha\beta}$ is the Kerr metric (10). As described above, these
solutions are remarkably similar to the states of the hydrogen atom in quantum
mechanics. To make this manifest, it is useful to consider the ansatz
[Arvanitaki:2010sy]

$\Phi(t,\bm{r})=\begin{dcases}\frac{1}{\sqrt{2\mu}}\left[\psi(t,\bm{r})\,e^{-i\mu
t}+\psi^{*}(t,\bm{r})\,e^{+i\mu t}\right]&\text{$\Phi$ real}\,,\\\\[4.0pt]
\frac{1}{\sqrt{2\mu}}\,\psi(t,\bm{r})\,e^{-i\mu t}&\text{$\Phi$
complex}\,,\end{dcases}$ (6)

where $\psi$ is a complex scalar field which varies on a timescale that is
longer than $\mu^{-1}$, and is therefore often called the non-relativistic
field. Substituting this ansatz into (5) and expanding in powers of $\alpha$,
the Klein Gordon equation reduces to a Schrödinger-like equation

$i\frac{\partial}{\partial
t}\psi(t,\textbf{r})=\left(-\frac{1}{2\mu}\nabla^{2}-\frac{\alpha}{r}+\Delta
V\right)\psi(t,\textbf{r})\,,$ (7)

where $\Delta V$ represents higher-order corrections in $\alpha$. At leading
order, this is identical to the Schrödinger equation for the hydrogen atom,
whose eigenfunctions are labeled by the principal “quantum” number $n$, the
orbital angular momentum number $\ell$, and the azimuthal angular momentum
number $m$.999The rigorous definitions of these quantum numbers, in the
context of the (approximate) isometries of the Kerr metric, will be discussed
in §1 in detail. These quantum numbers satisfy the inequalities $n\geq\ell+1$,
$\ell\geq 0$, and $\ell\geq|m|$. The bound-state solutions of the scalar cloud
at leading order are thus given by

$\psi_{n\ell m}(t,\textbf{r})=R_{n\ell}(r)Y_{\ell
m}(\theta,\phi)\,e^{-i(\omega_{n\ell m}-\mu)t}\,,$ (8)

where $Y_{\ell m}$ are the scalar spherical harmonics and $R_{n\ell}$ are the
hydrogenic radial functions

$R_{n\ell}(r)=A\,e^{-\sqrt{\mu^{2}-\omega^{2}}\hskip
1.0ptr}r^{\ell}L^{(2\ell+1)}_{n-\ell-1}\left(2\sqrt{\mu^{2}-\omega^{2}}\hskip
1.0ptr\right)\,,$ (9)

with $A$ the amplitude of the field and $L$ the associated Laguerre
polynomial. For notational simplicity, we will denote the normalized
eigenstates (8) by $|n\hskip 0.5pt\ell\hskip 0.5ptm\rangle$, with $\langle
n\hskip 0.5pt\ell\hskip 0.5ptm|n^{\prime}\hskip 0.5pt\ell^{\prime}\hskip
0.5ptm^{\prime}\rangle=\delta_{nn^{\prime}}\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}$.
The overall amplitude $A$, determined by the total mass of the cloud, will be
restored when necessary. Notice that for small values of $\alpha$, following
the analogy with the hydrogen atom, the radial profile peaks at the “Bohr
radius”

$r_{\mathrm{c}}\equiv(\mu\alpha)^{-1}\,.$ (10)

The cloud is thus concentrated at a distance which is larger than both the
gravitational radius of the black hole and the Compton wavelength of the
field.

Despite the similarity between the hydrogen and the gravitational atoms, there
is an important difference between them. While the former has wavefunctions
that are regular at $r=0$, the latter must satisfy purely ingoing boundary
conditions at the black hole’s event horizon. As a consequence, the
eigenstates of a boson cloud have eigenfrequencies which are generally
complex,

$\omega=E+i\hskip 0.5pt\Gamma\,,$ (11)

where $E$ and $\Gamma$ denote the energies and instability rates,
respectively. Before the work presented in Chapter 3 was pursued, the explicit
expressions for $E$ and $\Gamma$ are known only at leading order. In
particular, they are [Detweiler:1980uk, Arvanitaki:2010sy]

$\displaystyle E_{n\ell m}$
$\displaystyle=\mu\left(1-\frac{\alpha^{2}}{2n^{2}}+\mathcal{O}\left(\alpha^{3}\right)\right),$
(12) $\displaystyle\Gamma_{n\ell m}$
$\displaystyle=2\tilde{r}_{+}C_{n\ell}\,g_{\ell
m}(\tilde{a},\alpha,\omega)\,(m\Omega_{\mathrm{H}}-\omega_{n\ell m})\hskip
1.0pt\alpha^{4\ell+5}\,,$ (13)

where the numerical coefficients $C_{n\ell}$ and $g_{\ell m}$ can be found in
§3, while $\tilde{r}_{+}\equiv r_{+}/M$ and $\tilde{a}\equiv a/M$ are the
dimensionless black hole outer horizon and spin. Following the analogy with
the hydrogen atom, we will refer (12) as the Bohr $(\Delta n\neq 0)$ energy
levels of the gravitational atom. From (13), we clearly see that a scalar mode
only grows when the superradiance inequality (3) is satisfied. The dominant
growing mode is the $|n\hskip 0.5pt\ell\hskip 0.5ptm\rangle=|2\hskip
0.5pt1\hskip 0.5pt1\rangle$ state, with $\Gamma_{211}\propto\alpha^{9}/M$ (see
Fig. 5 for an illustration of this dominant $2p$ scalar configuration).
Because $\Gamma_{211}$ depends so sensitively on $\alpha$, the growth
timescale of the scalar cloud can easily vary from few minutes to over
astrophysical timescales. In particular, the typical growth timescale is

$\Gamma^{-1}_{211}\simeq\frac{10^{6}\,\text{yrs}}{\tilde{a}}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.019}{\alpha}\right)^{9}\,.$
(14)

By demanding the scalar cloud to grow appreciably within the age of the
Universe, we can place an observationally-relevant lower bound of
$\alpha\gtrsim 0.005$. In addition, the superradiance condition $m\hskip
1.0pt\Omega_{\rm H}>\omega$ also places an upper bound of $\alpha<0.5$, which
is obtained by assuming a maximally-spinning black hole. A black hole can
therefore in principle probe bosons with masses over two orders of magnitude,
although the amplification process is more efficient for larger values of
$\alpha$.

Figure 5: Illustration of the dominant growing modes of the scalar ($2p$) and
vector ($1s$) gravitational atoms. The small arrows denote the intrinsic spin
of the vector field, which allows for superradiant growth even for states
without orbital angular momentum.

While the spectra (12) and (13) are identical for both real or complex scalar
fields, their stability properties may be different. Because the energy-
momentum tensor of a real scalar field is always time dependent and non-
axisymmetric, clouds made out of real fields are a continuous source of
nearly-monochromatic gravitational waves [Arvanitaki:2010sy]. The power of
this emission is given by [Arvanitaki:2010sy, Yoshino:2014]

$\displaystyle
P_{n\ell}=D_{n\ell}\left(\frac{M_{c}}{M}\right)^{2}\alpha^{4\ell+10}\,,$ (15)

where $M_{c}\simeq\mu A^{2}$ is the mass of the cloud and the coefficient
$D_{n\ell}$ can be found in [Yoshino:2014]. Complex fields, on the other hand,
can have certain configurations that suppress this gravitational-wave
emission. This occurs when the real and imaginary components of the field are
in the same eigenstate, but their relative phase is such that they produce
time-independent and axisymmetric configurations. Whether this is realized in
a specific case depends on the initial conditions of the superradiant growth
of the cloud. In either case, since the timescale for this gravitational-wave
emission is much longer than the timescale of the superradiance instability,
the former doesn’t inhibit the formation of the cloud [Yoshino:2014]. For
instance, the typical lifetime of the $\ket{211}$ mode for a real scalar field
is [Arvanitaki:2010sy, Yoshino:2014]

$T_{211}\simeq
10^{8}\,\text{yrs}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.07}{\alpha}\right)^{15}\,,$
(16)

which is much longer than its typical growth timescale (14). However,
depending on the values of $M$ and $\alpha$, the clouds may still deplete on
cosmological and astrophysical timescales.

Our discussion above summarizes the leading-order behaviour of the scalar
cloud. However, an important gap in the literature was that modes with
different values of $\ell$ and $m$ remain degenerate in (12). To fully
understand the qualitative features of the scalar atom, it is important that
we pursue higher-order corrections to $E$, up to the order in $\alpha$ at
which the degeneracies between all modes in the spectrum are broken. This is
not just a pure academic exercise; rather, we shall find in this thesis that
these small energy splittings would play crucial roles in determining the
evolution of the clouds when they are part of binary systems. The rigorous
analytic and numeric computations of these corrections will be shown in
Chapter 3. Continuing the analogy with the hydrogen atom, we will refer to the
energy splittings that differ by $\Delta\ell\neq 0$ and $\Delta m\neq 0$ as
fine and hyperfine splittings, respectively. These results are summarized in
§3, and the full scalar spectrum is illustrated in Fig. 1.

##### Vector clouds

The vector clouds are bound-stated solutions of the Proca equation of a
massive vector field $A_{\mu}$ around the Kerr metric,

$\left(g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}-\mu^{2}\right)A^{\mu}=0\,,$
(17)

which must be supplemented by the Lorenz condition $\nabla_{\mu}A^{\mu}=0$.
Because the Lorenz condition completely determines $A^{0}$, we may focus
entirely on the spatial components $A^{i}$. Just like in the scalar case, it
is convenient to introduce the analog of the ansatz (6) for $A^{i}$, where the
non-relativistic field is now denoted by $\psi^{i}$. Subtituting this ansatz
into (17) and expanding in powers of $\alpha$, the Proca equation becomes
[Baryakhtar:2017ngi]

$i\frac{\partial}{\partial
t}\psi^{i}(t,\textbf{r})=\left[-\frac{\delta^{il}}{2\mu}\nabla^{2}-\delta^{il}\,\frac{\alpha}{r}+\Delta
V^{il}\right]\psi^{l}(t,\textbf{r})\,,$ (18)

where the terms in (18) share the same interpretation as their scalar analogs
in (7). Since spherical symmetry is restored when $\alpha\ll 1$, the vector
eigenstates can still be characterized by definite angular momenta numbers in
this limit. It is therefore straightforward to express the leading-order
eigenfunctions as follows

$\displaystyle\psi_{n\ell jm}^{i}(t,\textbf{r})$
$\displaystyle=R_{n\ell}(r)Y^{i}_{\ell,jm}(\theta,\phi)\,e^{-i(\omega_{n\ell
jm}-\mu)t}\,,$ (19)

where $R_{n\ell}$ is the hydrogenic radial function (9) and the angular
function $Y_{\ell,jm}^{i}$ is the pure-orbit vector spherical harmonic
[Baryakhtar:2017ngi, Thorne:1980ru]. Compared to the scalar solution (8), we
see the vector cloud has a more elaborate angular structure. In particular,
the eigenstates of massive vector fields are now labeled by four “quantum
numbers” $\\{n,\ell,j,m\\}$, where $n$ and $\ell$ have the same interpretation
as in the scalar case, and $j=\ell\pm 1$, $\ell$ is the total angular momentum
with $m$ its azimuthal component. That $j$ can take three different values for
a given $\ell$ is a direct manifestation of the fact that a massive vector
field has three independent degrees of freedom. Since (19) acquires a factor
of $(-1)^{\ell+1}$ under a parity transformation, the $j=\ell\pm 1$ states are
called electric-type modes, while the $j=\ell$ states are magnetic-type. For
notational simplicity, we will denote the normalized states (19) by $|n\hskip
0.5pt\ell\hskip 0.5ptj\hskip 0.5ptm\rangle$.

Like the scalar atom, the boundary condition at the black hole horizon implies
the vector clouds have complex eigenfrequencies. Prior to the computations
performed in Chapter 3, the energies and instability rates of the vector
quasi-bound states are found to be [Baryakhtar:2017ngi]

$\displaystyle E_{n\ell jm}$
$\displaystyle=\mu\left(1-\frac{\alpha^{2}}{2n^{2}}+\mathcal{O}\left(\alpha^{3}\right)\right),$
(20) $\displaystyle\Gamma_{n\ell jm}$
$\displaystyle\propto\left(m\Omega_{H}-\omega_{n\ell
jm}\right)\alpha^{2\ell+2j+5}\,.$ (21)

At leading order, the vector energy is the same as its scalar counterpart
(12). This is to be expected, since both the scalar and vector fields have the
same Coulomb-like potential in their equations of motion, cf. (7) and (18). On
the other hand, the vector instability rate is only known up to the
superradiance condition and the $\alpha-$scaling, which depends on both $\ell$
and $j$. Through this $\alpha$-scaling, we find that the dominant growing mode
for the vector field is $|1\hskip 0.5pt0\hskip 0.5pt1\hskip 0.5pt1\rangle$,
which has $\Gamma_{1011}\propto\alpha^{7}/M$ (see Fig. 5 for an illustration
of this dominant $1s$ vector configuration). This is enhanced, by two powers
of $\alpha$, with respect to the dominant growing mode for the scalar field.
In particular, the typical growth timescale of the dominant vector mode is

$\Gamma^{-1}_{1011}\simeq\frac{10^{6}\,\text{yrs}}{\tilde{a}}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.0033}{\alpha}\right)^{7}\,.$
(22)

This shows that vector fields with $\alpha\gtrsim 0.0005$ can grow
significantly within the age of the Universe. Combined with the upper bound
$\alpha<0.5$ derived from the superradiance condition, we find a black hole
can probe ultralight vector fields with masses spanning over three orders of
magnitude, which is significantly wider than the scalar case. As in the scalar
caase, the vector clouds could also dissipate through the monochromatic
gravitational-wave emission if their configurations have time-dependent and
non-axisymmetric components. Unfortunately, an analytic expression like (15)
for the vector emission power is not known. Nevertheless, there had been
attempts to estimate the lifetime of the dominant $\ket{1011}$ mode through
numerical relativity results and analytic determination of its
$\alpha$-scaling [Baryakhtar:2017ngi, East:2017mrj, East:2018glu]. In this
case, it was found that the typical lifetime of the vector $1s$ mode is

$T_{1011}\simeq
10^{8}\,\text{yrs}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.01}{\alpha}\right)^{11}\,.$
(23)

Compared to (16), the dominant vector cloud dissipates much faster than its
scalar counterpart for the same values of $M$ and $\alpha$.

The coefficient of (21) was not known in the literature because the Proca
equation is hard to solve near the black hole horizon, which is necessarily
for a rigorous computation for $\Gamma_{n\ell jm}$. Previous attempts had at
least been made to calculate the coefficient for the dominant $1s$ mode with
different approximation schemes [Pani:2012bp, Endlich:2016jgc,
Baryakhtar:2017ngi], though these results did not agree with one other (see
Table 1 of [Cardoso:2018tly] for a summary). In Chapter 3, I resolve this
issue by rigorously deriving the analytic expression for $\Gamma_{n\ell jm}$
that is valid for all vector modes. In addition, like the scalar case, I also
derive the analytic results for the higher-order corrections to the vector
energies (20), up to the order in $\alpha$ at which the degeneracies between
all modes in the vector spectrum are lifted (see Fig. 4 for an illustration of
the vector spectrum). Numeric computations of both the vector energies and
instability rates are also demonstrated, in which case we find excellent
agreement between the analytic and numeric results. These results, along with
those obtained for the scalar atom, are summarized in §3. Importantly, they
represent the state of the art and complete our qualitative and quantitative
understandings of both the scalar and vector clouds.

Before proceeding to subsequent chapters, it is important to emphasize the key
qualitative distinctions betweeen the scalar and the vector atoms. Firstly,
because the scalar field has no intrinsic spin, it can only excite
superradiance by having $\ell>0$. The centrifugal barrier generated by the
orbital angular momentum therefore causes the scalar modes to peak far away
form the black hole, as for the dominant $2p$ state. On the other hand, a
vector field has intrinsic spin and can trigger superradiance without relying
on orbital angular momentum. Vector states with $\ell=0$, such as the dominant
$1s$ mode, therefore have non-negligible support near the black hole and are
sensitive to strong gravity effects. A second important difference between the
two types of atoms is the plurality of the number of states in their spectra.
As we shall find in Chapter 3, the vector field contains many more nearly-
degenerate energy levels in its spectrum. This simple fact would turn out to
play an important role in distinguishing the phenomenologies of the scalar and
vector clouds when they are part of binary systems. In particular, level
transitions which are induced by the binary companion typically would only
involve two levels for the scalar atom, while multiple levels can
simultaneously participate in the vector case. As I will elaborate in Chapters
4 and 5, a two-level and a multi-level transition would leave qualitatively
different imprints in the gravitational waves emitted by the binary system.

## The Spectra of Gravitational Atoms

We saw in §2 that the leading-order behaviours of the scalar and vector
gravitational atoms resemble the Bohr-like structure of the hydrogen atom.
However, higher-order corrections to these solutions are also of interest, as
they play important roles in the dynamics of the clouds when they are part of
binary systems (Chapter 4). The evolution of the cloud in turn perturbs the
orbital motion of the binary system, thereby affecting the gravitational waves
emitted by the binary (Chapter 5). In this chapter, I present the work done in
[Baumann:2019eav], where we investigated the spectra of the scalar and vector
gravitational atoms in detail. Our principal goal was to accurately compute
the energy splittings between the eigenstates of the clouds — the analog of
the fine and hyperfine structure of the hydrogen atom — and the instability
rates for both types of fields.

###  1 Overview and Outline

When the gravitational fine structure constant (4) is small, $\alpha\ll 1$,
the spectra can be treated perturbatively as an expansion in powers of
$\alpha$. Using such an expansion and the separability of the Klein-Gordon
equation (5), we solve for the energy spectrum [Baumann:2018vus,
Baumann:2019eav] and instability rates [Detweiler:1980uk, Baumann:2019eav] of
a massive scalar field, up to the order in $\alpha$ at which the degeneracies
between all modes in the spectrum are broken (see Fig. 1 for an illustration
of the scalar field spectrum). A similar computation for the massive vector
field had been previously attempted [Endlich:2016jgc, Baryakhtar:2017ngi,
Pani:2012vp, Pani:2012bp], though not with full rigor. Until recently, the
main technical obstruction in the vector analysis had been the fact that the
Proca equation (17) could not be decomposed into separate angular and radial
equations. However, in [Krtous:2018bvk, Frolov:2018ezx], a separable ansatz
was discovered for the electric modes of a massive vector field on the Kerr
background.111In [Dolan:2018dqv], it was shown that a special class of
magnetic modes are contained within this ansatz (see Appendix 8.B). However,
it is not yet clear whether it contains every magnetic mode, nor is it clear
how to recover them. We use this ansatz to compute the spectrum and
instability rates of a Proca field perturbatively. As described in §2, a
special feature of superradiantly-generated vector clouds is that the dominant
growing mode is a $1s$ state, which has non-negligible support near the black
hole, see Fig. 5. These modes are then especially sensitive to the near-
horizon geometry of the black hole, and this sensitivity necessitate going
beyond ordinary perturbation theory to determine the spectrum.

While the magnetic modes of the Proca field are separable on the Schwarzschild
background [Rosa:2011my], a separable ansatz in the Kerr spacetime remains
elusive, and thus rigorous analytic results are difficult to achieve for this
part of the spectrum. Fortunately, all degeneracies between the states in the
spectrum are broken at linear order in the spin of the black hole, and the
Schwarzschild ansatz still separates the Proca equation at this order
[Pani:2012bp, Pani:2012vp]. Assuming that the fine and hyperfine structure of
the magnetic modes are similar to their electric counterparts, we use this
small-spin expansion to derive the magnetic spectrum perturbatively in
$\alpha$. Furthermore, we ‘derive’ their leading-order instability rates using
educated guesswork, and are thus able to attain perturbative results for the
most phenomenologically relevant aspects of the spectrum, for all modes of the
Proca field.

Figure 1: Schematic illustration of the energy spectrum for a massive scalar
field. Each state is labeled by the quantum numbers $|\hskip 0.5ptn\hskip
0.5pt\ell\hskip 0.5ptm\rangle$ as described in §2. The solid blue lines are
growing modes, while the dashed red lines are decaying modes. The dominant
scalar mode is $\ket{211}$.

To test the analytic results, and to move beyond the limit of small $\alpha$,
we compute the spectra of scalar and vector clouds numerically (see also e.g.
[Dolan:2007mj, Pani:2012bp, Cardoso:2018tly, Dolan:2018dqv]). Formulating this
as a nonlinear eigenvalue problem allow us to attain highly-accurate results
for the spectrum with little computational cost. Our method also has the
advantage that it does not rely upon a separable ansatz, which allows us to
rigorously determine the magnetic spectrum and check our analytic guesswork
for its instability rates. Our method is reliable even for very small values
of $\alpha$, where precise numeric results are typically difficult to achieve
without separability [Pani:2012bp, Cardoso:2018tly], and we apply it to both
the dominant growing modes and other phenomenologically relevant states. We
pay special attention to the numerical obstacles that can potentially destroy
the accuracy of a solution, and describe how to avoid them. This new
formulation thus provides a robust and flexible method for finding the quasi-
bound state spectra for arbitrary ultralight tensor fields about any
stationary black hole, without relying on separability of the equations of
motion.

The outline of this chapter is as follows: in Section 2, I describe the Klein-
Gordon and Proca equations around rotating black holes. I also present a
summary of our analytic and numeric results in §3. In Section 3, I discuss how
the method of matched asymptotic expansion is used to compute the spectrum of
energy eigenvalues and the associated instability rates, for both the scalar
field and the electric modes of the vector field. In addition, I describe the
computation for the energy spectra of the magnetic modes at linear order in
the black hole spin $\tilde{a}$ and motivate their conjectured instability
rates. In Section 4, I present our numerical methods. Our techniques do not
assume separability of the equations of motion and, in principle, work for
ultralight fields of arbitrary spin. Numerous technical details are relegated
to Appendix 8.

To avoid notational clutter, I will set $\mu\equiv 1$ in this chapter, so that
times and distances are measured relative to the typical oscillation timescale
and Compton wavelength of the fields. The appropriate factors of $\mu$ will be
restored when presenting explicit results, as in §3.

###  2 Scalars and Vectors around Kerr

This section outlines the fundamentals of computing the quasi-bound state
solutions of the Klein-Gordon equation (5) and Proca equation (17) around the
Kerr background. In §1, we provide a rigorous discussion about the tensor
representations of Kerr spacetime, thereby making precise the definitions of
the “quantum numbers” mentioned in §2. In §2, we review the separability of
the equations of motion in the Kerr geometry. A summary of both the analytic
and numeric results is presented in §3, with the details of their derivations
relegated to the subsequent sections.

####  1 Tensor Representations

We begin with a discussion of general tensor fields $T_{\mu\nu\dots\rho}$ on
the Kerr background. We are interested in quasi-bound states solutions, which
are ‘purely ingoing’ at the outer horizon and vanish at infinity. As in atomic
physics, these solutions will be characterized by a set of discrete ‘quantum
numbers’ describing the energy, the orbital angular momentum and the intrinsic
spin of the states. A precise definition of these quantum numbers is
complicated by the fact that the spin of black hole breaks spherical symmetry.
We will label our states by the quantum numbers they attain in the flat-space
limit, where the spin of the black holes can be ignored.

In order to define the flat-space limit, it is convenient to rescale the
temporal and radial coordinates, $t\mapsto t\mu^{-1}$ and $r\mapsto
r\mu^{-1}$, so that the metric (10) becomes

$\displaystyle\mu^{2}{\rm d}s^{2}=$
$\displaystyle-\frac{\Delta}{\Sigma}\left({\rm d}t-\alpha\hskip
1.0pt\tilde{a}\sin^{2}\theta\,{\rm d}\phi\right)^{2}+\frac{\Sigma}{\Delta}{\rm
d}r^{2}$ (1) $\displaystyle+\Sigma\hskip 1.0pt{\rm
d}\theta^{2}+\frac{\sin^{2}\theta}{\Sigma}\left(\alpha\hskip
1.0pt\tilde{a}\hskip 1.0pt{\rm d}t-(r^{2}+\alpha^{2}\tilde{a}^{2})\,{\rm
d}\phi\right)^{2}\,,$

where $\tilde{a}\equiv a/M$ and

$\displaystyle\Delta$ $\displaystyle\equiv r^{2}-2\alpha
r+\alpha^{2}\tilde{a}^{2}\,,$ (2) $\displaystyle\Sigma$ $\displaystyle\equiv
r^{2}+\alpha^{2}\tilde{a}^{2}\cos^{2}\theta\,.$

All physical quantities are now measured in units of the Compton wavelength of
the field, $\mu^{-1}$, which we henceforth set to $\mu^{-1}\equiv 1$. Notice
that the spin parameter $\tilde{a}$ in (1) always appears in the combination
$\alpha\hskip 1.0pt\tilde{a}$, so spherical symmetry is approximately restored
when $\alpha\ll 1$. In fact, in the limit $\alpha\to 0$, the line element (1)
reduces to that of Minkowski spacetime, while at linear order in $\alpha$ it
becomes the Schwarzschild solution. Spin-dependent terms, such as those
corresponding to the Lense-Thirring effect, only appear at quadratic order in
$\alpha$. As far as the dynamics of the field is concerned, we therefore
expect the effects of spin to be subleading compared to the gravitational
potential sourced by $M$.

The eigenstates of the field are labeled by a discrete set of quantum numbers
that reflect the (approximate) isometries of the background metric. To
identify these quantum numbers, we first note that the Kerr geometry has two
Killing vectors

$k_{t}\equiv-i\frac{\partial}{\partial t}\,,\qquad
k_{z}\equiv-i\frac{\partial}{\partial\phi}\,,$ (3)

representing the fact that the metric (1) is both stationary and axisymmetric.
The equations of motion for an arbitrary tensor field $T_{\mu\nu\cdots\rho}$
on the Kerr background will inherit the isometries (3). If these equations are
linear, we may decompose their solutions in terms of states with definite
frequency and azimuthal angular momentum,222Our analysis will focus entirely
on complex scalar and vector fields, as the complex representations of these
isometries are much simpler than for real fields. Because the equations of
motion (5) and (17) are linear, a real solution can be generated by simply
taking the real part of a complex solution, and so the spectra are identical.

$\displaystyle\pounds_{t}\hskip 2.0ptT_{\mu\nu\cdots\rho}$
$\displaystyle=-\omega\hskip 2.0ptT_{\mu\nu\cdots\rho}\,,$ (4)
$\displaystyle\pounds_{z}\hskip 2.0ptT_{\mu\nu\cdots\rho}$
$\displaystyle=+m\hskip 2.0ptT_{\mu\nu\cdots\rho}\,,$

where $\pounds_{t}\equiv\pounds_{k_{t}}$ and
$\pounds_{z}\equiv\pounds_{k_{z}}$ are the Lie derivatives with respect to the
isometries (3). In the Schwarzschild limit $\tilde{a}\to 0$, the geometry
gains two more Killing vector fields $k_{x}$ and $k_{y}$ which, together with
$k_{z}$, form an ${\mathrm{SO}}(3)$ algebra. We can then expand the field into
its temporal and spatial components, which we further expand in
representations of this algebra, i.e. eigentensors of the total angular
momentum operator

$\pounds^{2}\hskip 2.0ptT_{ik\cdots
l}\equiv\left(\pounds_{x}^{2}+\pounds_{y}^{2}+\pounds_{z}^{2}\right)T_{ik\cdots
l}=j(j+1)\hskip 2.0ptT_{ik\cdots l}\,,$ (5)

where $j$ denotes the total angular momentum. Furthermore, in the flat-space
limit, $\alpha\to 0$, solutions also have definite orbital angular momentum
$\ell$ (cf. Appendix 8.A), so that they can be characterized by the quantum
numbers $\omega$, $\ell$, $j$ and $m$. We will label such states by $|\hskip
0.5ptn\hskip 0.5pt\ell\hskip 0.5ptj\hskip 0.5ptm\rangle$, where we have
introduced an integer quantum number $n$ that indexes the discrete quasi-bound
state frequencies $\omega_{n}$. Since total and orbital angular momenta are
the same for the scalar field, we will label its states by $|\hskip
0.5ptn\hskip 0.5pt\ell\hskip 0.5ptm\rangle$. States still have definite total
angular momentum in the Schwarzschild limit, but indefinite orbital angular
momentum. At finite $\tilde{a}$, these states no longer have definite total
angular momentum. Nevertheless, we will still label our states by $|\hskip
0.5ptn\hskip 0.5pt\ell\hskip 0.5ptj\hskip 0.5ptm\rangle$ with the
understanding that these quantum numbers regain their physical meaning as
$\alpha\to 0$.

Finally, the Kerr metric is invariant under the parity transformation

${\cal P}:\,\,\,(\theta,\phi)\mapsto(\pi-\theta,\phi+\pi)\,.$ (6)

This helps us to further organize the spectrum into states with definite
parity. Solutions to the Klein-Gordon and Proca equations with definite parity
are invariant under (6) up to a sign, and, with our conventions, we have
$\mathcal{P}|\hskip 0.5ptn\hskip 0.5pt\ell\hskip
0.5ptm\rangle=(-1)^{\ell}|\hskip 0.5ptn\hskip 0.5pt\ell\hskip 0.5ptm\rangle$
and $\mathcal{P}|\hskip 0.5ptn\hskip 0.5pt\ell\hskip 0.5ptj\hskip
0.5ptm\rangle=(-1)^{\ell+1}|\hskip 0.5ptn\hskip 0.5pt\ell\hskip 0.5ptj\hskip
0.5ptm\rangle$ for scalars and vectors, respectively. See Appendix 8.A for a
more detailed discussion.

####  2 Separable Equations of Motion

We will now cast the Klein Gordon equation (5) and the Proca equation (17)
around Kerr black holes into sets of coupled ordinary differential equations.
These separable forms will be essential in our analytical treatment in Section
3, though they are not necessary in our numerical analyses in Section 4.

##### Massive scalar fields

We first specialize to the case of a massive scalar field around a Kerr black
hole. Since the Klein-Gordon equation is relatively simple, this will serve as
a useful illustration of our approach without the technical distractions that
arise in the vector analysis.

The Klein-Gordon equation (5) is separable in Boyer-Lindquist coordinates
through the ansatz [Brill:1972xj, Carter:1968]

$\displaystyle\Phi\left(t,\textbf{r}\right)=e^{-i\omega
t+im\phi}R(r)S(\theta)\,.$ (7)

The spheroidal harmonics $S(\theta)$ obey the differential equation333To avoid
clutter, we suppress the dependence on the angular quantum numbers, i.e.
$S\equiv S_{\ell m}$ and $\Lambda\equiv\Lambda_{\ell m}$.

$\left(-\frac{1}{\sin\theta}\frac{\mathrm{d}}{\mathrm{d}\theta}\left(\sin\theta\frac{\mathrm{d}}{\mathrm{d}\theta}\right)-c^{2}\cos^{2}\theta+\frac{m^{2}}{\sin^{2}\theta}\right)S=\Lambda
S\,,$ (8)

where $\Lambda$ is the angular eigenvalue and the spheroidicity parameter is

$c^{2}\equiv-\alpha^{2}\tilde{a}^{2}(1-\omega^{2})\,.$ (9)

The radial equation is a confluent Heun equation,444The Heun equation is a
generalization of the hypergeometric equation with four, instead of three,
regular singular points. By merging two of these singularities, we arrive at
the confluent Heun equation, which is similarly a generalization of the
confluent hypergeometric equation (15) that determines the wavefunctions of
the hydrogen atom. The angular equation (8) is also a confluent Heun equation.
For a detailed treatment, see [Andre:1995heun].

$\displaystyle
0\,=\,\frac{1}{R\,\Delta}\frac{\mathrm{d}}{\mathrm{d}r}\\!\left(\\!\Delta\frac{\mathrm{d}R}{\mathrm{d}r}\right)$
$\displaystyle\ -\frac{\Lambda}{\Delta}-\left(1-\omega^{2}\right)+\
\frac{P_{+}^{2}}{(r-r_{+})^{2}}+\frac{P_{-}^{2}}{(r-r_{-})^{2}}$ (10)
$\displaystyle\
-\frac{A_{+}}{(r_{+}-r_{-})(r-r_{+})}+\frac{A_{-}}{(r_{+}-r_{-})(r-r_{-})}\,,$

where we have defined the following coefficients

$\displaystyle A_{\pm}$ $\displaystyle\equiv
P_{+}^{2}+P_{-}^{2}+\gamma^{2}+\gamma_{\pm}^{2}\quad{\rm and}\quad
P_{\pm}\equiv\frac{\alpha(\tilde{a}m-2\hskip
1.0ptr_{\pm}\omega)}{r_{+}-r_{-}}\,,$ (11)

with

$\displaystyle\gamma^{2}$
$\displaystyle\equiv\frac{1}{4}(r_{+}-r_{-})^{2}(1-\omega^{2})\,,$ (12)
$\displaystyle\gamma_{\pm}^{2}$
$\displaystyle\equiv\left[\alpha^{2}(1-7\omega^{2})\pm\alpha(r_{+}-r_{-})(1-2\omega^{2})\right].$

We are interested in quasi-bound states, i.e. solutions to (10) that are
purely ingoing at the horizon and vanish at infinity:

$R(r)\propto\begin{cases}(r-r_{+})^{iP_{+}}&\text{as }r\to r_{+}\\\
\exp(-\sqrt{1-\omega^{2}}\hskip 1.0ptr)&\text{as }r\to\infty\end{cases}\,.$
(13)

As is typical for eigenvalue problems, these two boundary conditions can only
be simultaneously satisfied for specific values of $\omega$. Unfortunately,
while solutions to both (8) and (10) are well known, closed-form expressions
for their eigenvalues are not. However, in the limit $\alpha\ll 1$, we can
construct a perturbative expansion of the spectrum. This perturbative approach
is complicated by the presence of terms in (10) that diverge as $r\to r_{+}$.
These terms represent _singular perturbations_ in $\alpha$, i.e. if we naively
expand them in powers of $\alpha$, an infinite number of terms become relevant
as $r\to r_{+}$. Luckily, this is not a disaster. The role of these singular
terms is simply to modify the characteristic scale on which $R(r)$ varies. As
we will see explicitly in Section 3, the radial function satisfies

$\frac{1}{R}\frac{\mathrm{d}R}{\mathrm{d}r}\sim\begin{cases}\alpha^{-1}&\text{as
}r\to r_{+}\\\ \alpha&\text{as }r\to\infty\end{cases}\,.$ (14)

This motivates splitting the interval $[r_{+},\infty)$ into a ‘near region’
and a ‘far region’ (see Fig. 2). In the near region, the $(r-r_{+})^{-2}$ pole
in (10) dominates and an approximate solution to the differential equation for
$R(r)$ can be obtained by dropping the subleading terms. In the far region, on
the other hand, the non-derivative terms are dominated by the constant
$(1-\omega^{2})$ term. By dropping the sub-leading contributions to (10) in
each region, we can then construct approximate analytic solutions in the near
and far regions, order-by-order in $\alpha$. These solutions are then matched
in the ‘overlap region,’ allowing a determination of the frequency eigenvalue
$\omega$, as an expansion in powers of $\alpha$. The details of this _matched
asymptotic expansion_ will be presented in Section 3. For the angular problem,
the perturbative solutions are valid over the entire angular domain and no
matching is necessary.

Figure 2: Illustration of the near and far regions used in our perturbative
treatment for the massive scalar field, where $r_{\pm}$ are the inner and
outer horizons of the black hole, $\lambda_{c}=\mu^{-1}$ is the Compton
wavelength of the field, and $r_{c}=\left(\mu\alpha\right)^{-1}$ is the
typical Bohr radius of the quasi-bound state. The two asymptotic solutions are
matched in the overlap region.

To gain an intuitive understanding of the behavior of the solutions at leading
order, it is instructive to first consider the equation of motion obeyed by
the far-zone radial function, $R^{\rm far}_{0}$. At leading order, this reads

$\left[-\frac{1}{2r^{2}}\frac{{\rm d}}{{\rm d}r}\left(r^{2}\frac{{\rm d}}{{\rm
d}r}\right)-\frac{\alpha}{r}+\frac{\ell(\ell+1)}{2r^{2}}+\frac{1-\omega^{2}}{2}\right]R^{\text{far}}_{0}=0\,.$
(15)

This is analogous to the time-independent Schrödinger equation for the
hydrogen atom, where the radial-gradient term arises from the kinetic energy,
the $1/r$ term corresponds to a Coulomb-like central potential with
gravitational coupling constant $\alpha$, the $1/r^{2}$ term is its
centrifugal barrier with orbital angular momentum number $\ell$, and the
constant term captures the exponential behavior of the quasi-bound states at
$r\to\infty$. We therefore expect the energy spectrum of all modes to be Bohr-
like at leading order, with the radial function

$R^{\text{far}}_{0}(r)\propto
e^{-\sqrt{1-\omega^{2}}\,r}r^{\ell}L^{(2\ell+1)}_{n-\ell-1}\big{(}2\sqrt{1-\omega^{2}}\,r\big{)}\,,$
(16)

as in (9). The exponential behavior in (16) implies that the solution has a
typical Bohr radius given by $r_{c}\equiv(\mu\alpha)^{-1}$, as we found in
(10). Since (15) only captures the physics in the far region, it does not
describe physical effects that depend on the boundary condition at the event
horizon, such as the instability rates of the energy eigenstates. To study
these effects, we match the near- and far-zone solutions. By pushing this
procedure to higher orders in $\alpha$, we determine the complete spectrum of
the scalar field. A schematic illustration of this spectrum appears in Fig. 1.

##### Massive vector fields

The task of solving for the quasi-bound state solutions of vector fields is
more involved because the matching involves three different regions, and the
equation of motion is not obviously separable. The latter problem was
addressed in [Frolov:2018ezx] and [Pani:2012bp] where separable ansätze were
found for the electric modes and magnetic modes (in the limit of small black
hole spin), respectively. In Section 3, we use these results to derive the
spectrum of vector quasi-bound states perturbatively.

Consider the following ansatz for a vector field on the Kerr background
[Krtous:2018bvk, Frolov:2018ezx]

$\displaystyle A^{\mu}=B^{\mu\nu}\nabla_{\nu}Z\,,\quad{\rm with}\quad
Z(t,{\bf{r}})=e^{-i\omega t+im\phi}R(r)S(\theta)\,,$ (17)

where $R$ and $S$ are the radial and angular functions.555We emphasize that
$R$ and $S$ are not the actual radial and angular profiles of the vector
field, since the tensor $B^{\mu\nu}$ depends on both $r$ and $\theta$.
However, we still retain the terminology ‘radial’ and ‘angular function,’ and
denote them by $R$ and $S$, to draw a direct parallel with the scalar. The
precise relationships between $R$, $S$ and the profile of the vector field
$A_{\mu}$ in the far zone will be given in §3.  The polarization tensor
$B^{\mu\nu}$ is defined by

$\displaystyle
B^{\mu\nu}\left(g_{\nu\sigma}+i\lambda^{-1}h_{\nu\sigma}\right)=\delta^{\mu}_{\sigma}\,,$
(18)

where $\lambda$ is generally a complex parameter, which we will refer to as
the angular eigenvalue, and $h_{\mu\nu}$ is the principal tensor of the Kerr
spacetime [Frolov:2017kze]. 666To simplify many of the following equations, we
take the angular eigenvalue $\lambda$ to be the inverse of that in
[Krtous:2018bvk, Frolov:2018ezx]. An explicit expression for $B^{\mu\nu}$ is
given in Appendix 8.B. The three independent degrees of freedom of the vector
field can be organized in terms of $j=\ell\pm 1,\ell$. Since the $j=\ell\pm 1$
modes acquire a factor of $(-1)^{j}$ under a parity transformation, they are
called _electric modes_ , while the $j=\ell$ modes are the magnetic modes
[Thorne:1980ru]. In Section 3, we will show that the ansatz (17) captures all
of the electric modes of the vector field.777In [Dolan:2018dqv], it was found
that the ansatz (17) restores at least a subset of the magnetic modes in a
special limit (see Appendix 8.B for further details). It remains unclear,
however, how to write all magnetic modes in separable form. In this thesis, I
will instead utilize a different ansatz, which, in the limit of small black
hole spin, provides a separable equation for all of the magnetic modes.

Substituting (17) into the Proca equation, we obtain differential equations
for $S(\theta)$ and $R(r)$. The angular equation reads [Dolan:2018dqv]

$\displaystyle\frac{1}{\sin\theta}\frac{{\rm d}}{{\rm
d}\theta}\\!\left(\sin\theta\frac{{\rm d}S}{{\rm d}\theta}\right)+$
$\displaystyle\left(c^{2}\cos^{2}\theta-\frac{m^{2}}{\sin^{2}\theta}+\Lambda\right)S$
(19)
$\displaystyle=\frac{2\alpha^{2}\tilde{a}^{2}\cos\theta}{\lambda^{2}q_{\theta}}\left(\sin\theta\frac{{\rm
d}}{{\rm d}\theta}+\lambda\,\sigma\cos\theta\right)S\,,$

where the spheroidicity parameter $c^{2}$ was defined in (9), and

$\displaystyle q_{\theta}$ $\displaystyle\equiv
1-\alpha^{2}\tilde{a}^{2}\lambda^{-2}\cos^{2}\theta\,,$ (20)
$\displaystyle\sigma$ $\displaystyle\equiv\omega+\alpha\hskip
1.0pt\tilde{a}\hskip 1.0pt\lambda^{-2}(m-\alpha\tilde{a}\hskip
1.0pt\omega)\,,$ $\displaystyle\Lambda$
$\displaystyle\equiv\lambda\left(\lambda-\sigma\right)+2\alpha\tilde{a}\hskip
1.0ptm\omega-\alpha^{2}\tilde{a}^{2}\omega^{2}\,.$

The radial equation becomes

$\displaystyle 0=$ $\displaystyle\frac{{\rm d}^{2}R}{{\rm
d}r^{2}}+\left(\frac{1}{r-r_{+}}+\frac{1}{r-r_{-}}-\frac{1}{r-\hat{r}_{+}}-\frac{1}{r-\hat{r}_{-}}\right)\frac{{\rm
d}R}{{\rm d}r}$ (21)
$\displaystyle+\bigg{(}-\frac{\Lambda}{\Delta}-(1-\omega^{2})+\frac{P_{+}^{2}}{(r-r_{+})^{2}}+\frac{P_{-}^{2}}{(r-r_{-})^{2}}-\frac{A_{+}}{(r_{+}-r_{-})(r-r_{+})}$
$\displaystyle\ \ \ \ \ \
+\frac{A_{-}}{(r_{+}-r_{-})(r-r_{-})}-\frac{\phantom{\mathcal{B}}\lambda\hskip
0.5pt\sigma
r}{\Delta\left(r-\hat{r}_{+}\right)}-\frac{\phantom{\mathcal{B}}\lambda\hskip
0.5pt\sigma r}{\Delta\left(r-\hat{r}_{-}\right)}\bigg{)}\,R\,,$

where $\hat{r}_{\pm}\equiv\pm i\lambda$ depend on the angular eigenvalue and
the parameters $P_{\pm}$ and $A_{\pm}$ were defined in (11). The presence of
the additional poles at $r=\hat{r}_{\pm}$, makes this equation considerably
more complicated than the corresponding equation (10) in the scalar case.

In principle, the task of determining the spectrum of electric modes of the
vector quasi-bound states is the same as for the scalar. However, due to the
additional poles at $r=\hat{r}_{\pm}$, the widths of the near and far regions
are now much smaller and these regions no longer overlap. To match the
asymptotic expansions in the near and far regions, we must introduce an
intermediate region888Physically, the necessity of the intermediate region
arises because the dynamics in the near and far regions depend on the angular
momentum of the vector field in distinct ways. As we will see in Section 3,
while the dynamics in the near region is sensitive to the total angular
momenta of the field (including its intrinsic spin), the dynamics in the far
region depends only on its orbital angular momentum. The intermediate region
thus smoothly interpolates between these two behaviors.  that overlaps with
both (see Fig. 3) and construct a solution that serves as a bridge. The
matching is then performed in a two-step procedure, first between the far and
intermediate regions, and then between the intermediate and near regions.

Figure 3:  Illustration of the different regions used in our perturbative
treatment for the massive vector field. The additional poles at
$\hat{r}_{\pm}$ along the imaginary axis reduce the radii of convergence of
the near- and far-zone solutions, so that there is no region where they
overlap. Matching therefore requires the intermediate region.

In the Schwarzschild limit, solutions have definite total angular momentum and
parity. Because the vector spherical harmonic $Y^{i}_{j,\,j\hskip 0.5ptm}$ has
opposite parity to $Y^{i}_{j\pm 1,\,j\hskip 0.5ptm}$ and the scalar harmonic
$Y_{j\hskip 0.5ptm}$, it completely decouples from all other angular modes.
This magnetic mode,

$A^{i}(t,\textbf{r})=r^{{\scalebox{0.75}[1.0]{$-$}}1}R(r)\,Y^{i}_{j,\,j\hskip
0.5ptm}(\theta,\phi)\,e^{-i\omega t}\,,$ (22)

is thus completely separable in the Schwarzschild limit [Rosa:2011my], and
[Pani:2012vp, Pani:2012bp] showed that this persists to linear order in
$\tilde{a}$. We can thus use this ansatz to determine the magnetic spectrum in
the limit of small spin.

Figure 4: Schematic illustration of the energy spectrum for a massive vector
field. Each state is labeled by the quantum numbers $|\hskip 0.5ptn\hskip
0.5pt\ell\hskip 0.5ptj\hskip 0.5ptm\rangle$. The dominant vector mode is
$\ket{1011}$.

Substituting (22) into the Proca equation and expanding to first order in
$\tilde{a}$, the radial function $R(r)$ satisfies [Pani:2012vp, Pani:2012bp]

$\displaystyle 0=\frac{{\rm d}^{2}R}{{\rm
d}r^{2}}+\left(\frac{1}{r-\check{r}_{+}}-\frac{1}{r-\check{r}_{-}}\right)\frac{{\rm
d}R}{{\rm d}r}\
+\bigg{(}-\frac{\check{\Lambda}}{\Delta}-\left(1-\omega^{2}\right)$ (23)
$\displaystyle\ +\
\frac{\check{P}_{+}^{2}}{(r-\check{r}_{+})^{2}}+\frac{\check{P}_{-}^{2}}{(r-\check{r}_{-})^{2}}-\frac{\check{A}_{+}}{(\check{r}_{+}-\check{r}_{-})(r-\check{r}_{+})}+\frac{\check{A}_{-}}{(\check{r}_{+}-\check{r}_{-})(r-\check{r}_{-})}\bigg{)}R\,,$

where $\check{\Lambda}=j(j+1)$ and $\check{P}_{\pm},\check{A}_{\pm}$,
$\check{r}_{\pm}$ represent $P_{\pm}$, $A_{\pm}$, $r_{\pm}$ expanded to linear
order in $\tilde{a}$, respectively. Specifically, the position of the inner
and outer horizons, $\check{r}_{-}=0$ and $\check{r}_{+}=2\alpha$, have
shifted in this approximation, and we expect that (23) does not accurately
describe the near-horizon behavior of these magnetic modes. When expanded to
linear order in $\tilde{a}$, the scalar radial equation (10) differs from (23)
only in its $\mathrm{d}R/\mathrm{d}r$ coefficient. In Section 3, we will
discuss the error this small-spin approximation introduces. To compute the
magnetic spectra for arbitrary spin, we must still solve the Proca equation
(17) numerically. For a schematic illustration of the vector field spectrum,
see Fig. 4.

####  3 Summary of Results

In Sections 3 and 4, we derive the spectra of scalar and vector quasi-bound
states around Kerr black holes in detail. Here, we summarize our main results.

We write the frequency eigenvalues as

$\omega\hskip 1.0pt\equiv\hskip 1.0ptE+i\hskip 0.5pt\Gamma\hskip
2.0pt\equiv\hskip 1.0pt\mu\,\sqrt{1-\frac{\alpha^{2}}{\nu^{2}}}\,,$ (24)

where the real part represents the energy $E$ and the imaginary part
determines the instability rates $\Gamma$. At leading order, we expect the
energy spectrum to be Bohr-like, so that the real part of $\nu$ is an integer
$n$. As we will see below, it is also convenient to work with $\nu$ because
the fine and hyperfine structure can be read off directly from its
$\alpha$-expansion.

Figure 5: Numeric results of the energy eigenvalues for scalar field
eigenstates $\ket{n\ell m}$, for $\tilde{a}=0.5$. The lower panels show the
ratio of the numeric results to the analytical predictions in (25). The fine
and hyperfine structures are clearly seen in ${\mathrm{Re}}\,\nu$, and we find
that (25) is an excellent approximation for $\alpha\lesssim 0.2$.

For the scalar field, the energy eigenvalues are

$\displaystyle E_{n\ell m}$
$\displaystyle=\mu\left(1-\frac{\alpha^{2}}{2n^{2}}-\frac{\alpha^{4}}{8n^{4}}+\frac{f_{n\ell}}{n^{3}}\hskip
2.0pt\alpha^{4}+\frac{h_{\ell}}{n^{3}}\hskip 2.0pt\tilde{a}m\hskip
1.0pt\alpha^{5}+\cdots\right),$ (25) $\displaystyle{\mathrm{Re}}\,(\nu_{n\ell
m})$ $\displaystyle=n+f_{n\ell}\hskip 1.0pt\alpha^{2}+h_{\ell}\hskip
1.0pt\tilde{a}m\hskip 1.0pt\alpha^{3}+\cdots\,,$

where the principal quantum number $n$ are integers that satisfy
$n\geq\ell+1$, and

$\displaystyle f_{n\ell}$
$\displaystyle\equiv-\frac{6}{2\ell+1}+\frac{2}{n}\,,$ (26) $\displaystyle
h_{\ell}$
$\displaystyle\equiv\frac{16}{2\ell\left(2\ell+1\right)\left(2\ell+2\right)}\,.$

The first three terms of $E_{n\ell m}$ describe the constant mass term, the
hydrogen-like Bohr energy levels and the relativistic corrections to the
kinetic energy. The terms proportional to $f_{n\ell}$ and $h_{\ell}$ are the
fine-structure $(\Delta\ell\neq 0)$ and hyperfine-structure $(\Delta m\neq 0)$
splittings, respectively. As the second line in (25) suggests, these
splittings can be more easily extracted from the real part of $\nu_{n\ell m}$.
Numeric results for ${\mathrm{Re}}\,\nu$, and their comparison with the
perturbative approximations (25), are shown in Fig. 5 for representative
scalar modes; cf. Fig. 1 for a schematic illustration of the scalar spectrum.

Figure 6: Numeric results of the energy eigenvalues for vector field
eigenstates $\ket{n\ell jm}$, for $\tilde{a}=0.5$. The lower panels show the
ratio of the numeric results to our perturbative results in (27).

For the vector field, we study the electric and magnetic modes separately
using the ansätze (17) and (22). However, we find that the energy eigenvalues
for _all_ vector modes can be written as

$\displaystyle E_{n\ell jm}$
$\displaystyle=\mu\left(1-\frac{\alpha^{2}}{2n^{2}}-\frac{\alpha^{4}}{8n^{4}}+\frac{f_{n\ell
j}}{n^{3}}\hskip 2.0pt\alpha^{4}+\frac{h_{\ell j}}{n^{3}}\hskip
2.0pt\tilde{a}m\hskip 1.0pt\alpha^{5}+\cdots\right),$ (27)
$\displaystyle{\mathrm{Re}}\,(\nu_{n\ell jm})$ $\displaystyle=n+f_{n\ell
j}\hskip 1.0pt\alpha^{2}+h_{\ell j}\hskip 1.0pt\tilde{a}\hskip
1.0ptm\alpha^{3}+\cdots\,,$

with the coefficients

$\displaystyle f_{n\ell j}$ $\displaystyle=-\frac{4\left(6\hskip
1.0pt\ell\hskip
1.0ptj+3\ell+3j+2\right)}{\left(\ell+j\right)\left(\ell+j+1\right)\left(\ell+j+2\right)}+\frac{2}{n}\,,$
(28) $\displaystyle h_{\ell j}$
$\displaystyle=\frac{16}{\left(\ell+j\right)\left(\ell+j+1\right)\left(\ell+j+2\right)}\,,$

for $j=\ell\pm 1,\ell$. Notably, we find that the spectrum of the vector field
is qualitatively similar to the scalar case (25); cf. Fig. 4 for a schematic
illustration of the vector spectrum. Numeric results for ${\mathrm{Re}}\,\nu$
for representative electric and magnetic modes, and their comparison with
(27), are shown in Fig. 6.

Figure 7: Superradiant growth rates for the dominant vector mode $\ket{1011}$,
for different values of the black hole spin $\tilde{a}$. The growth timescale
is for a Proca field of mass $\mu=\hbar/10^{-9}\,{\mathrm{yr}}\approx
2.1\times 10^{-14}\,{\mathrm{eV}}$. The points are numeric data, while the
solid lines are the approximation (32). The lower panel shows the ratio of our
numerics and analytics, and includes the decaying regime, $\Gamma<0$, at
larger values of $\alpha$.

We also compute the instability rates for these scalar and vector quasi-bound
states. For the scalar field, we find (see also [Detweiler:1980uk,
Baryakhtar:2017ngi, Pani:2012bp])

$\Gamma_{n\ell m}=2\tilde{r}_{+}C_{n\ell}\,g_{\ell
m}(\tilde{a},\alpha,\omega)\,(m\Omega_{H}-\omega_{n\ell m})\hskip
1.0pt\alpha^{4\ell+5}\,,$ (29)

where we have defined

$\displaystyle C_{n\ell}$
$\displaystyle\equiv\frac{2^{4\ell+1}(n+\ell)!}{n^{2\ell+4}(n-\ell-1)!}\left[\frac{\ell!}{(2\ell)!(2\ell+1)!}\right]^{2}\,,$
(30) $\displaystyle g_{\ell m}(\tilde{a},\alpha,\omega)$
$\displaystyle\equiv\prod^{\ell}_{k=1}\left(k^{2}\left(1-\tilde{a}^{2}\right)+\left(\tilde{a}m-2r_{+}\hskip
1.0pt\omega\right)^{2}\right).$ (31)

The dominant growing mode is $|\hskip 0.5ptn\hskip 0.5pt\ell\hskip
0.5ptm\rangle=|\hskip 0.5pt2\hskip 0.5pt1\hskip 0.5pt1\rangle$, and hence
$\Gamma_{211}\propto\mu\,\alpha^{8}$ [Dolan:2007mj, Detweiler:1980uk],999For
the normalization of the dominant growing mode, we find $C_{21}=1/48$, which
differs from the result in [Detweiler:1980uk] by a factor of $2$. This
discrepancy was also observed in the appendices of [Baryakhtar:2017ngi,
Pani:2012bp]. where we have used $\Omega_{H}\sim\mu\alpha^{-1}$. Numeric
results for the growth rates $\Gamma$ of the fastest growing scalar states are
shown in Fig. 8, along with their comparison to the analytic approximations
(29).

Figure 8: Superradiant growth rates for selected scalar and vector modes, for
$\tilde{a}=0.5$. The growth timescale is for
$\mu=\hbar/10^{-9}\,{\mathrm{yr}}\approx 2.1\times 10^{-14}\,{\mathrm{eV}}$.
The points represent numeric data, while the solid lines are our perturbative
predictions (29) and (32).

The instability rates for the vector modes take a very similar form. In
particular, we find that they are

$\Gamma_{n\ell jm}=2\tilde{r}_{+}C_{n\ell
j}\,g_{jm}(\tilde{a},\alpha,\omega)\left(m\Omega_{H}-\omega_{n\ell
jm}\right)\alpha^{2\ell+2j+5}\,,\quad\text{for $j=\ell\pm 1,\ell$}\,,$ (32)

where the coefficient,

$\displaystyle C_{n\ell j}\equiv$
$\displaystyle\frac{2^{2\ell+2j+1}(n+\ell)!}{n^{2\ell+4}(n-\ell-1)!}$ (33)
$\displaystyle\times\left[\frac{(\ell)!}{(\ell+j)!(\ell+j+1)!}\right]^{2}\left[1+\frac{2\left(1+\ell-j\right)\left(1-\ell+j\right)}{\ell+j}\right]^{2}\,,$

is valid for all $j=\ell\pm 1,\ell$, and the function $g_{jm}$ is obtained by
replacing $\ell$ by $j$ in (31). The $\alpha$-scaling in (32) was also found
in [Pani:2012vp, Pani:2012bp, Endlich:2016jgc, Baryakhtar:2017ngi]. Notice
that (32) is also valid for the scalar rate (29), if we set $j=\ell$ and
multiply $C_{n\ell j}$ by a factor of $j^{2}/(j+1)^{2}$. The dominant vector
growing mode is $|\hskip 0.5ptn\hskip 0.5pt\ell\hskip 0.5ptj\hskip
0.5ptm\rangle=|\hskip 0.5pt1\hskip 0.5pt0\hskip 0.5pt1\hskip 0.5pt1\rangle$
and has $\Gamma_{1011}\propto\mu\hskip 1.0pt\alpha^{6}$ [Endlich:2016jgc,
Baryakhtar:2017ngi, Cardoso:2018tly, East:2017ovw, East:2017mrj, East:2018glu,
Pani:2012vp, Pani:2012bp, Dolan:2018dqv] (see Table 1 of [Cardoso:2018tly] for
a summary of $\Gamma_{1011}$ obtained through other methods). This is faster
than the dominant growing mode for the scalar field. In Fig. 7, we show our
numeric results for the growth rate $\Gamma_{1011}$ at different values of the
black hole spin $\tilde{a}$, and compare it with (32). In Fig. 8, we compare
the numeric results for the growth rates of other electric and magnetic vector
modes to their perturbative approximations (32). For comparison, we also
include the dominant scalar modes.

Unfortunately, the separable ansatz (22) for the magnetic mode does not
describe the Proca field in the near-horizon geometry of the Kerr black hole,
and thus cannot be used to rigorously derive a perturbative expression for the
magnetic instability rates. In our framework, the result (32) for $j=\ell$ was
obtained through an educated guess and, in the absence of numerical evidence
or an alternative approximation scheme, should be taken as purely conjectural.
Having said that, it both matches the result of [Baryakhtar:2017ngi] and is in
excellent agreement with our numeric results (c.f. Figures 8 and 9), and so we
include it as a guide for phenomenology.

###  3 Analytical Computation of the Spectra

In this section, we compute in detail the energy spectra and instability rates
of quasi-bound states of massive scalar and vector fields around rotating
black holes, using the method of matched asymptotic expansions
[bender1999advanced, holmes1998introduction]. The conceptual challenge of the
computation can be illustrated using the scalar field, while avoiding the
technical heft of the vector. We thus begin with the analysis of a scalar
field [Starobinsky:1973aij, Detweiler:1980uk] in §1 and §2, before moving to
the vector case in §3.

In what follows, it will be convenient to write the frequency eigenvalue as

$\omega\equiv\sqrt{1-\frac{\alpha^{2}}{\nu^{2}}}\,,$ (1)

where the real part of $\nu$ is an integer for the leading-order Bohr
spectrum. We will compute ${\mathrm{Im}}\,\nu$ to leading order in $\alpha$
and ${\mathrm{Re}}\,\nu$ to order $\alpha^{3}$, as all degeneracies are broken
at this order.

####  1 Matched Asymptotic Expansion

We begin by illustrating the essential features of a matched asymptotic
expansion through the simpler scalar field radial equation (10). Since
perturbative solutions of (10) cannot capture the boundary conditions at
$r=r_{+}$ and $r\to\infty$ simultaneously, a matched asymptotic expansion is
necessary to compute the eigenvalues $\nu$. Schematically, we approximate the
radial function separately in both the near and far regions (cf. Fig. 2), and
match these solutions in the region of overlap.

To define the _near region_ , it is convenient to introduce the rescaled
radial coordinate

$z\equiv\frac{r-r_{+}}{r_{+}-r_{-}}=\frac{r-\alpha\hskip
1.0pt\tilde{r}_{+}}{\alpha\left(\tilde{r}_{+}-\tilde{r}_{-}\right)}\,,$ (2)

where $\tilde{r}_{\pm}\equiv r_{\pm}/\alpha$, so that
$\tilde{r}_{\pm}\sim\mathcal{O}(1)$. In this coordinate, the inner and outer
horizons are mapped to $z=-1$ and $z=0$, respectively. As discussed in §2, the
radial function varies much more rapidly in the near region than in the far
region, $\mathrm{d}R/\mathrm{d}r\sim\alpha^{-1}$, and the reason we change
coordinates is to accommodate this rapid change in the $\alpha$-expansion. The
radial equation (10) then becomes

$\displaystyle\frac{1}{R\,z(z+1)}\frac{{\rm d}}{{\rm d}z}$
$\displaystyle\left(\\!z(z+1)\frac{{\rm d}R}{{\rm
d}z}\right)-\frac{\Lambda}{z(z+1)}$ (3)
$\displaystyle-4\gamma^{2}+\frac{P_{+}^{2}}{z^{2}}+\frac{P^{2}_{-}}{(z+1)^{2}}-\frac{A_{+}}{z}+\frac{A_{-}}{z+1}=0\,,$

where the coefficients were defined in (11) and (12). We then expand (3) in
powers of $\alpha$, with $z$ kept fixed. By comparing the dominant and
subdominant terms at each order in $\alpha$, we find that this region covers
the range $0\leq z\lesssim\alpha^{-2}$, cf. Fig. 2. Near the outer horizon at
$z=0$, the $1/z^{2}$ pole dominates and the radial solution scales as

$R(z)\,\sim\,B_{1}\hskip 1.0ptz^{iP_{+}}+B_{2}\hskip
1.0ptz^{-iP_{+}}\,,\mathrlap{\qquad z\to 0\,,}$ (4)

where $B_{1}$ and $B_{2}$ are integration constants. The purely ingoing
boundary condition at the event horizon requires that $B_{2}=0$. Similarly,
approaching the inner horizon at $z=-1$, we find that
$\lim_{z\to{\scalebox{0.5}[0.85]{$-$}}1}\,R(z)\,\sim\,(z+1)^{\pm iP_{-}}$.
However, unlike the outer horizon, the inner horizon is not physically
accessible, and hence no boundary condition has to be imposed on it.

To identify the far region, it is useful to use the alternative coordinate

$x\equiv 2\sqrt{1-\omega^{2}}\,(r-r_{+})=4\gamma z\,,$ (5)

where $\gamma$ was defined in (12). In this coordinate, the near horizon
region, $z\sim 1$, is mapped to $x\sim\alpha^{2}$, and the Bohr radius
$r_{c}\sim\alpha^{-1}$ is at $x\sim 1$. In the far region, we expand the