cmr  See Cover.pdf

# Probing Particle Physics with Gravitational Waves

Horng Sheng Chia

(16 oktober 2020)

Publications

This thesis is based on the following publications:

  * [Baumann:2019eav]

D. Baumann, H. S. Chia, J. Stout, and L. ter Haar, “The Spectra of
Gravitational Atoms”, JCAP 12 (2019) 006, arXiv:1908.10370 [gr-qc].

Presented in Chapter 4.

  * [Baumann:2018vus]

D. Baumann, H. S. Chia, and R. A. Porto, ‘‘Probing Ultralight Bosons with
Binary Black Holes", Phys. Rev. D 99, 044001 (2019), arXiv:1804.03208 [gr-qc].
Selected for Editors’ Suggestion and Featured in Physics.

Presented in Chapters 5 and 6.

  * [Baumann:2019ztm]

D. Baumann, H. S. Chia, R. A. Porto, and J. Stout, ‘‘Gravitational Collider
Physics", Phys. Rev. D 101, 083019 (2020), arXiv:1912.04932 [gr-qc].

Presented in Chapters 5 and 6.

  * [Chia:2020psj]

H. S. Chia and T. D. P. Edwards, ‘‘Searching for General Binary Inspirals with
Gravitational Waves", JCAP 11 (2020) 033, arXiv:2004.06729 [astro-ph.HE].

Presented in Chapter 7.

###### Contents

  1. 0 Introduction
  2. 1 Gravitational-Wave Astrophysics
    1. 1 Theoretical Foundations
      1. 1 General Relativity
      2. 2 Black Holes
      3. 3 Gravitational Waves
    2. 2 Compact Binary Coalescences
      1. 1 Dynamics of Binary Systems
      2. 2 Gravitational Waveforms
    3. 3 Gravitational-Wave Observatories
      1. 1 Current Detectors
      2. 2 Future Detectors
  3. 2 Particle Physics at the Weak-Coupling Frontier
    1. 1 Physics Beyond the Standard Model
      1. 1 The Standard Model
      2. 2 Motivations for New Physics
    2. 2 The Weak-Coupling Frontier
      1. 1 Particle Dark Matter
      2. 2 Ultralight Bosons
    3. 3 Gravitational Atoms
      1. 1 Black Hole Superradiance
      2. 2 Hydrogenic Bound States
  4. 3 The Spectra of Gravitational Atoms
    1. 1 Overview and Outline
    2. 2 Scalars and Vectors around Kerr
      1. 1 Tensor Representations
      2. 2 Separable Equations of Motion
      3. 3 Summary of Results
    3. 3 Analytical Computation of the Spectra
      1. 1 Matched Asymptotic Expansion
      2. 2 Scalar Field Solutions
      3. 3 Vector Electric Modes
      4. 4 Vector Magnetic Modes
    4. 4 Numerical Computation of the Spectra
      1. 1 Continued Fraction Method
      2. 2 Nonlinear Eigenvalue Problem
      3. 3 Chebyshev Interpolation
      4. 4 Solving Without Separability
    5. 5 Summary
  5. 4 Gravitational Collider Physics
    1. 1 Overview and Outline
    2. 2 Gravitational Atoms in Binaries
      1. 1 Metric Perturbation
      2. 2 Gravitational Level Mixings
      3. 3 Dynamical Perturbation
    3. 3 The Gravitational Collider
      1. 1 Two-State Transitions
      2. 2 Multi-State Transitions
      3. 3 An S-Matrix Approach
    4. 4 Summary
  6. 5 Observational Signatures
    1. 1 Overview and Outline
    2. 2 Signals from the Binary
      1. 1 Floating, Sinking, and Kicked Orbits
      2. 2 Finite-Size Effects
      3. 3 Time-Dependent Dephasings
    3. 3 Signals from the Cloud
      1. 1 Resonant Attenuation
      2. 2 Doppler Modulation
    4. 4 Unraveling the Atomic Structure
      1. 1 Initial States of the Cloud
      2. 2 Resonant Transitions
      3. 3 Post-Resonance Evolution
    5. 5 Summary
  7. 6 Searching for Dark Compact Objects
    1. 1 Overview and Outline
    2. 2 General Inspiral Waveforms
      1. 1 Dark Compact Objects
      2. 2 Dephasing from Quadrupole Moment
      3. 3 Cutoff Frequency from Small Compactness
    3. 3 Detectability of General Inspirals
      1. 1 Binary Black Hole Template Bank
      2. 2 Effectualness to Inspiral Waveforms
    4. 4 Summary
  8. 7 Conclusions and Outlook
  9. 8 Details of the Spectral Analysis
    1. 8.A Tensor Spherical Harmonics
    2. 8.B Details of the Analytical Treatment
    3. 8.C Details of the Numerical Treatment
  10. 9 Gravitational Perturbations
    1. 9.A Three-Body Analogy
    2. 9.B Fictitious Dipole
    3. 9.C Tidal Moments of Inclined Orbits
    4. 9.D Higher-Order Couplings
  11. 10 Landau-Zener Transition
    1. 10.A The Two-State Model
    2. 10.B Adiabatic Floquet Theory
  12. 11 Backreaction on the Orbit
    1. 11.A Orbital Dynamics
    2. 11.B Backreaction Time

## Introduction

The direct detection of gravitational waves has offered us a new window onto
our Universe [Abbott:2016blz, TheLIGOScientific:2017qsa]. These ripples of
spacetime, generated by some of the most powerful astrophysical events, carry
vital information about physics and phenomena that are typically inaccessible
by other observational means. This breakthrough has come at a time when many
challenges in foundational aspects of physics, cosmology, and astrophysics
remain unresolved. For instance, despite decades of active research, we still
do not know what dark matter is [Aghanim:2018eyx]. In this thesis, I explore
novel ways of probing new physics beyond the Standard Model of particle
physics using current and future gravitational-wave observations.

Over the past several years, the LIGO and Virgo detectors have observed the
gravitational waves emitted by multiple binary black hole and binary neutron
star systems [LIGOScientific:2018mvr, Venumadhav:2019lyq, Nitz:2019hdf]. These
detections have advanced our understanding of astrophysics and, at the same
time, offer a wealth of new scientific opportunities. For example, by
observing the gravitational waves of binary black holes, previously-unknown
populations of astrophysical black holes were unveiled, with masses that are
heavier than those expected from X-ray binaries [Remillard:2006fc]. The
observed binary signals have also motivated rapid advances in the field of
“precision gravity” [Porto:2016zng, Blanchet:2006zz, Porto:2016pyg,
Levi:2018nxp, Schafer:2018kuf, Bern:2019crd], which aims to construct highly-
accurate template waveforms in order to reliably extract binary signals from
noisy data. Furthermore, the simultaneous observations of gravitational waves
and electromagnetic radiation from merging neutron stars inaugurated the field
of multi-messenger astronomy [TheLIGOScientific:2017qsa,
Cowperthwaite:2017dyu, Troja:2017nqp]. These neutron star signals have
provided strong evidences for their role in manufacturing Nature’s heavy
elements, such as gold and platinum [Kasen:2017aa, Pian:2017aa]; in the
future, they will offer independent measurements of the expansion rate of our
Universe [Schutz:1986aa]. The examples given here represent only a few of the
many achievements that have been made by the current network of gravitational-
wave detectors. Moreover, next generation detectors will observe over a
greater range of frequencies and with much better sensitivities, therefore
having the potential to reveal much more about our Universe.

While it is clear that gravitational-wave observations will continue to
transform astrophysics in many ways, it is less obvious how they can advance
our understanding of particle physics. In the past, searches for new physics
have been dominated by the high-energy frontier, whereby heavy particles are
excited through scattering processes in particle colliders. While this way of
probing new physics has been very successful, culminating in the development
of the Standard Model [Aad:2012tfa, Chatrchyan:2012xdj, Tanabashi:2018oca], it
relies on the new particles having appreciable couplings with the particles
involved in the collision processes. Traditional collider experiments are
therefore blind to “dark sectors” that interact very weakly with ordinary
matter, even if the putative new degrees of freedom are light. However, by
virtue of the equivalence principle, all forms of matter and energy must
interact through gravity. Gravitational waves are therefore excellent probes
of physics in the dark sector, as long as the associated time-dependent
quadrupole moments are sufficiently large for the signals to be realistically
detectable. This is the case if new types of compact objects exist in the dark
sector and are present in inspiraling binary systems. Indeed, as I will
elaborate shortly, various types of dark compact objects naturally arise in
proposed extensions to the Standard Model. As long as these objects can be
formed and are stable over astrophysical timescales, their impacts on the
dynamics of binary systems and the associated gravitational waves could
provide vital clues about new physics that is otherwise hard to detect. This
way of probing new physics is complementary to other probes of particle
physics in the weak-coupling frontier [Essig:2013lka, Berlin:2018bsc], which
broadly seeks to develop creative methods to detect particles that couple very
weakly to ordinary matter.

One of the most well-motivated classes of particles at the weak-coupling
frontier are ultralight bosons. These bosons are attractive examples of new
physics because they are excellent candidates for dark matter and may resolve
some outstanding problems of the Standard Model [Peccei:1977hh,
Weinberg:1977ma, Wilczek:1977pj, Svrcek:2006yi, Arvanitaki:2009fg,
Acharya:2010zx, Cicoli:2012sz, delAguila:1988jz, Goodsell:2009xc,
Camara:2011jg]. These particles are characterized by their extremely small
masses, corresponding to Compton wavelengths that can be of the order of
astrophysical length scales. Remarkably, if the Compton wavelengths of these
fields are comparable to or larger than the sizes of rotating black holes,
they can be spontaneously amplified by extracting the energies and angular
momenta of these black holes. This amplification mechanism, commonly known as
black hole superradiance [Zeldovich:1971a, Zeldovich:1972spj,
Starobinsky:1973aij, Starobinsky:1974spj, Bekenstein:1973mi], generates clouds
of boson condensates that are gravitationally bounded to the black holes at
their centers [Arvanitaki:2009fg, Arvanitaki:2010sy]. Since these bound states
resemble the proton-electron structure of the hydrogen atom, they are often
called “gravitational atoms.” This analogy is in fact not purely qualitative;
rather, the equations that govern both types of atoms are actually
mathematically identical.111For example, in the hydrogen atom, the electron
wavefunction is bounded to the proton at its center through the $1/r$
Coulombic electrostatic potential. Similarly, in the gravitational atom, the
boson cloud is attracted to the central black hole through the $1/r$ Newtonian
gravitational potential. As a result, the eigenstates of the gravitational
atom have a Bohr-like energy spectrum as the hydrogen atom. Black hole
superradiance is an excellent way of probing ultralight particles at the weak-
coupling frontier because it only relies on the gravitational coupling between
the black hole and the boson, with no assumptions made about other types of
interactions of the fields. Furthermore, all properties of the gravitational
atoms, such as their energy spectra and superradiant growth timescales, can be
computed accurately from first principles, see e.g. [Detweiler:1980uk,
Dolan:2007mj, Pani:2012vp, Pani:2012bp, Baryakhtar:2017ngi, Endlich:2016jgc].
Indeed, a central goal of this thesis is to fully explore the qualitative and
quantitative aspects of these boson clouds [Baumann:2019eav]. Importantly, we
will find that these computations reveal subtle features of the clouds that
can have significant impacts on the clouds’ phenomenologies.

Figure 1: Illustration of the gravitational atom in a binary system. During
the inspiral, the binary companion induces level mixings between different
eigenstates of the cloud. This mixing becomes non-perturbative when the
orbital frequency, which slowly increases due to gravitational-wave emission,
matches the energy difference between the coupled states. During these
resonant transitions (denoted by the colored ellipses), the cloud can
significantly backreact on the binary’s orbit and affect the binary’s
gravitational-wave emissions. Furthermore, various types of finite-size
effects associated to the cloud can also leave distinctive imprints on the
phases of the binary waveforms.

The gravitational-wave signatures of these boson clouds are most significant
when they are parts of binary systems [Baumann:2018vus, Baumann:2019ztm]. In
particular, as I will describe in this thesis, the gravitational perturbation
sourced by a binary companion can force the cloud to evolve in a highly
dynamical manner. Due to the structural similarity between the hydrogen and
the gravitational atoms, the dynamics of the cloud in a binary system is
analogous to that of an atom shone by a laser beam. Many of the classic
results in atomic physics, such as the presences of level mixings and
selection rules, therefore also apply to the boson cloud. Moreover, these
level mixings are dramatically enhanced when the orbital frequency of the
binary matches the energy difference between different eigenstates in the
spectrum. These resonance transitions can significantly backreact on the
orbit, thereby affecting the gravitational waves emitted by the binary system
(see Fig. 1 for an illustration). Since the clouds have large spatial extents,
they can also affect in the gravitational waves of the binary through various
so-called “finite-size effects.” A notable example is the spin-induced
quadrupole moment of the cloud [Geroch:1970cd, Hansen:1974zz, Thorne:1980ru],
which characterizes its deformation from spherical symmetry due to its
rotational motion. In addition, when the tidal force exerted by the binary
companion deforms the shape of the cloud, the orbit would deviate from its
normal evolution due to the associated change in the orbital binding energy
and affect the flux emitted to asymptotic infinity [Flanagan:2007ix,
Vines:2011ud]. Since the imprints of these finite-size effects on the binary
waveforms are known in detail, a precise measurement of these effects in the
waveforms provide clean probes of the nature of the gravitational atoms.

The gravitational waves emitted by a binary system presents us with a unique
window of opportunity to detect the existence of putative ultralight bosons in
our Universe. Furthermore, these signatures offer us rare probes of the
microscopic properties of the boson fields that form the clouds
[Baumann:2018vus, Baumann:2019ztm]. In particular, the mass of the boson field
is directly related to the resonance frequency of the gravitational waves
emitted by the binary during the resonant transitions described above. On the
other hand, the intrinsic spin of the boson field can be infered through
interesting time-dependences in the finite-size imprints on the waveforms.
This way of probing new particles is in fact directly analogous to the
discipline of particle collider physics. This is because in an ordinary
collider, a particle’s mass is determined by the energy at which the particle
appears as a resonant excitation, while its spin is measured via the angular
dependence of the final state. The analogy is made further apparent by the
fact that the dynamics of the gravitational atoms near these resonance
frequencies can be quantified by an S-matrix, making these systems effectively
“gravitational colliders” [Baumann:2018vus, Baumann:2019ztm]. Given its
similarity with particle collider physics, we refer to this nascent discipline
of probing ultralight bosons with binary systems as gravitational collider
physics.

In addition to the gravitational atoms, other types of compact objects can
also arise in many scenarios of new physics at the weak-coupling frontier. For
example, if the new particles have Compton wavelengths that are much shorter
than the sizes of astrophysical black holes, they behave essentially as point-
like particles and can accrete around black holes to form high-density spikes
[Gondolo:1999ef, Ferrer:2017xwm]. More exotic types of compact objects, such
as boson stars [Kaup1968, Ruffini1969, Breit:1983nr, Colpi1986,
Liebling:2012fv], could also exist in our Universe. In all of these cases,
when the dark compact objects are parts of binary systems, they can affect the
binaries’ gravitational-wave emissions through various dynamical effects, such
as the finite-size effects described above for the gravitational atoms. To
realistically search for these unusual types of binary systems, one would
typically need to match the noisy observational data to template waveforms,
which include the effects of the putative new physics [Thorne1980Lectures,
Dhurandhar:1992mw, Cutler:1992tc]. This matched-filtering technique is,
however, extremely sensitive to the phase coherence between the signal and
template waveforms. A mismodelling of the signal template waveform can easily
degrade its detectability [Cutler:1992tc]. In this thesis, I present an
analysis whereby we assess the extent to which ordinary binary black hole
template waveforms, which are used in the LIGO and Virgo search pipelines,
would allow us to search for these binary systems. This represents an
important first step towards understanding the limitations of existing search
strategies to detect new physics with the gravitational waves of binary
systems.

The above discussion clearly demonstrates the potential of probing new physics
using gravitational waves. While this thesis focuses primarily on the
signatures from binary systems, other types of gravitational-wave signals,
such as continuous monochromatic waves and stochastic gravitational-wave
background, could also offer important clues about physics beyond the Standard
Model. Since gravitational-wave science is a precision science, it is
imperative that we continue the strong interplay between the developments of
new theoretical ideas and novel search strategies. Current detectors are
rapidly improving their observational capabilities, and we are seeing a flurry
of proposals for developing future gravitational-wave detectors. Now is an
opportune time to explore ways of maximizing the discovery potentials of these
gravitational-wave measurements. While a detection of new physics would
certainly transform our understanding of particle physics, even non-
observations can be informative, as they would place meaningful bounds on new
physics. All in all, I believe that this thesis represents an important step
towards our goal of using gravitational-wave observations to probe particle
physics at the weak-coupling frontier.

##### Outline of this thesis

The rest of this thesis is organized as follows. In Chapter 1, I describe
several aspects of gravitational-wave astrophysics. These include a discussion
on the theoretical foundations of General Relativity and a summary of current
and future gravitational-wave observatories. One of the main goals in this
chapter is to elaborate on the dynamics of binary systems and the associated
gravitational-wave emissions. In Chapter 2, I provide a broad overview of the
weak-coupling frontier in particle physics. After a brief description of the
Standard Model and motivations for new physics, I discuss several classes of
weakly-coupled particles. I also introduce the concept of superradiance and
the gravitational atom in this chapter, which will serve as important
background material for subsequent chapters. While writing these two chapters,
I had in mind a broad audience who may wish to learn the basics of
gravitational-wave astophysics and particle physics. The particle physicists
may therefore find Chapter 1 helpful for delving further into the
gravitational-wave literature, while the general relativists and
astrophysicists may find the review on physics beyond the Standard model in
Chapter 2 useful.

The remaining chapters present the main results of this thesis. In Chapter 3,
I describe the analytic and numeric computations of the spectra of the
gravitational atoms. In particular, I focus on the spectra of the scalar and
vector gravitational atoms, demonstrating how our computations reveal
important qualitative differences between the two types of atoms. In Chapter
4, I describe the dynamics of the gravitational atoms in binary systems. I pay
special attention to the cloud’s evolution when the orbital frequency of the
binary matches the resonance frequencies associated to the cloud. I further
elaborate on how these resonant dynamics are equivalent to the Landau-Zener
transitions in quantum mechanics, with the physics effectively encoded in an
“S-matrix.” In Chapter 5, I discuss the rich phenomenologies associated to the
gravitational atoms in binary systems. These include the backreaction of the
resonances on the orbit and various types of time-dependent finite-size
imprints on the binary’s waveform. I also describe how the scalar and vector
atoms can be distinguished through precise reconstructions of these
gravitational waveforms. In Chapter 6, I present an analysis that quantify the
extent to which binary black hole template waveforms can be used to detect new
dark compact objects in binary systems. I first construct a phenomenological
waveform that represents a wide range of new binary signals, and then compute
the overlap between the binary black hole template waveforms and the
phenomenological waveforms. Finally, I conclude and provide an outlook on
future directions in Chapter 7.

A number of appendices contains technical details that had been omitted in the
main text. In Appendix 8, I provide further details on the analytic and
numeric computations of the spectra of the gravitational atoms. In Appendix 9,
I elaborate on the gravitational perturbation sourced by a binary companion on
the boson clouds. In Appendix 10, I present a formalism that encodes the
dynamics of the Landau-Zener transition for an arbitary binary orbit
configuration. Finally, Appendix 11 includes the derivation of the
backreaction of the Landau-Zener transition on the orbit.

##### Notation and conventions

In this thesis, I adopt the ‘mostly plus’ signature $(-,+,+,+)$ for the
metric. The covariant derivative with respect to the metric is represented by
the symbol $\nabla$. Greek letters are used for spacetime indices
($\mu,\nu,\ldots$), while Latin letters will either stand for spatial indices
($i,j,\ldots$), label indices ($i,k,\ldots$), or vierbein indices
($a,b,\ldots$). To avoid confusion, I will sometimes wrap label indices in
parentheses. Unless stated otherwise, I work in natural units with
$G=\hbar=c=1$.

I will adopt the Boyer-Lindquist coordinates $\\{t,r,\theta,\phi\\}$ for the
Kerr black hole. Denoting mass and specific angular momentum by $M$ and $a$,
respectively, the line element is

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

where $\Delta\equiv r^{2}-2Mr+a^{2}$ and $\Sigma\equiv
r^{2}+a^{2}\cos^{2}\theta$. The roots of $\Delta$ determine the inner and
outer horizons, located at $r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$, and the angular
velocity of the black hole at the outer horizon is $\Omega_{H}\equiv
a/2Mr_{+}$. Dimensionless quantities, defined with respect to the black hole
mass, are labeled by tildes; e.g. $\tilde{a}\equiv a/M$ and
$\tilde{r}_{\pm}\equiv r_{\pm}/M$.

The gravitational radius of a black hole is given by $r_{g}\equiv GM/c^{2}$.
On the other hand, the gravitational fine-structure constant is $\alpha\equiv
r_{g}/\lambda_{c}$, where $\lambda_{c}\equiv\hbar/(\mu c)$ is the (reduced)
Compton wavelength of a boson field with mass $\mu$. Quantities associated to
the boson clouds will be denoted by the subscript $c$. For example, the mass
and angular momentum of the clouds are $M_{c}$ and $S_{c}$, respectively. I
use the subscript $*$ for quantities related to the binary companion, e.g.
$M_{*}$ is its mass and $\\{R_{*},\iota_{*},\varphi_{*}\\}$ represent its
spatial coordinates in the Fermi frame of the cloud.

The eigenstates of the scalar and vector atoms are denoted by $|\hskip
0.5ptn\hskip 0.5pt\ell\hskip 0.5ptm\rangle$ and $|\hskip 0.5ptn\hskip
0.5pt\ell\hskip 0.5ptj\hskip 0.5ptm\rangle$, with the integers
$\\{n,\ell,j,m\\}$ labeling the principal, orbital angular momentum, total
angular momentum, and azimuthal angular momentum numbers, respectively.
Following the convention in atomic physics, we have $n\geq\ell+1$. I refer to
vector modes that acquire a factor of $(-1)^{j}$ under a parity transformation
as ‘electric modes,’ and those that acquire a factor of $(-1)^{j+1}$ as
‘magnetic modes.’ These modes are to be distinguished from odd and even modes,
which, by my convention, receive a factor of $-1$ and $+1$ under parity,
respectively.

## Gravitational-Wave Astrophysics

Gravitational-wave astrophysics is a remarkably rich subject that involves
theoretical studies of General Relativity and pioneering developments in many
experimental techniques. Indeed, the direct detections of gravitational waves
by the LIGO and Virgo observatories would not have been possible without the
significant theoretical and experimental advancements in the field over the
past several decades. At the same time, the field remains an active area of
research with further developments required in order to maximally exploit the
scientific opportunities offered by the current and future network of
gravitational-wave detectors.

In this chapter, I will describe both the theory and the observations
underlying gravitational-wave astrophysics. In Section 1, I review the
foundational aspects of General Relativity and two of its most famous
predictions: black holes and gravitational waves. In Section 2, I then
describe the relativistic dynamics of compact binary systems and the
associated gravitational-wave emissions. The literature on this subject is
vast and can appear highly technical; my goal is to describe only the key
qualitative features of these computations and include a summary of the state
of the art. In Section 3, I summarize the status of current gravitational-wave
detectors and their scientific achievements over the past few years. I also
provide a broad overview of many existing proposals for future gravitational-
wave observatories.

###  1 Theoretical Foundations

General Relativity has revolutionized our understanding of gravity
[Einstein1915]. It supersedes Newtonian gravity because many of the somewhat
ad hoc features in the latter are explained by the fundamental principles of
General Relativity. Furthermore, the predictions of General Relativity are in
excellent agreement with virtually all observations made in cosmology and
astrophysics, many of which cannot be accounted for in Newtonian physics
[Will:2014kxa]. I will first review the basic elements of General Relativity,
and then discuss black holes and gravitational waves as two of its most
important solutions.

####  1 General Relativity

General Relativity was developed out of the need for incorporating relativity
into gravity. In Special Relativity, space and time are treated on equal
footing. In addition, physical quantities are invariant under Lorentz
transformations between different inertial frames. Nevertheless, these
properties are absent in Newtonian gravity. To understand this concretely, let
us consider the Poisson equation

$\partial_{i}^{2}U=4\pi\rho\,,$ (1)

where $\partial_{i}\equiv\partial/\partial x^{i}$ is a spatial derivative and
$U$ is the gravitational potential sourced by an energy density $\rho$. Space
and time are clearly treated differently in (1) because time derivatives are
absent. A naive generalization of the Laplacian $\partial_{i}^{2}$ to the
d’Alembertian $\equiv-\partial_{0}^{2}+\partial_{i}^{2}$ wouldn’t be
sufficient either because $\rho$ is not a Lorentz scalar. How, then, can one
construct a relativistic theory of gravity?

A seemingly unrelated puzzle in Newtonian gravity offered Einstein the vital
clue to tackle this problem. In Newtonian mechanics, the force experienced by
a test particle in a gravitational potential is

$m_{I}\frac{{\rm d}^{2}x^{i}}{{\rm d}t^{2}}=-m_{G}\partial^{i}U\,,$ (2)

where $m_{I}$ is the particle’s inertial mass and $m_{G}$ is its gravitational
mass. These two masses, a priori, need not be the same: the former describes
the particle’s inertia to changing its motion, while the latter is a charge
that quantifies its coupling to a gravitational field. Nevertheless, $m_{I}$
is found to be equal to $m_{G}$ in experiments and astrophysical observations.
Is there an explanation for this “fine-tuning” between these two types of
masses?

Einstein intuited that the equality between the inertial and gravitational
masses is not a coincidence, but rather reflects a fundamental property of
gravity. The exact equality $m_{I}=m_{G}$ has profound consequences, as it
implies that all test particles, regardless of their masses, experience the
same gravitational acceleration. This is often formulated as the weak
equivalence principle, which states that the motion of a test particle in a
gravitational field is independent of its mass and composition. All test
particles with the same initial conditions must therefore travel on the same
preferred class of trajectories called geodesics or free-falling trajectories.
This universal behaviour of gravity makes it qualitatively distinct from other
conventional forces, such as electromagnetism, where particles with different
amounts of charge accelerate at different rates and hence propagate on
different trajectories. In addition, the weak equivalence principle implies
that gravity as experienced by a free-falling test particle necessarily
vanishes. To guide our intuition, I use Einstein’s famous thought experiment
of an observer enclosed in a free-falling elevator. In this setup, the
observer cannot distinguish whether she is free-falling on Earth or is
floating in outer space because the walls of the elevator do not exert any
reaction force. Since gravity is clearly absent in the latter case, one is led
to conclude that this is similarly the case in the former.

Because the motion of test particles is independent of their masses, Einstein
proposed what we experience as gravity is instead a fundamental property of
the background spacetime on which test particles propagate. But what property
of spacetime exactly? A crucial feature of the weak equivalence principle is
that it is only applicable to test particles, but not spatially-extended
objects.111More precisely, by a spatially-extended object I mean an object
whose size is comparable or larger than the typical length scale on which the
gravitational field in its vicinity varies. By contrast, a test particle has a
size which is much smaller than this variation length scale and can therefore
treated as point-like. While a free-falling test particle cannot experience
gravitation locally, an extended object does through tidal forces. The fact
that gravity manifests itself differently in a “local” region and a “global”
region of spacetime is reminiscent of curvature in geometry, whereby a
globally curved surface appears flat in any local region. These insights led
Einstein to conclude that gravity is a manifestation of curved spacetime.

Our discussion so far has been guided by the weak equivalence principle and by
considering the motion of particles in a gravitational field. However,
multiple generalizations of the weak equivalence principle can be formulated.
For instance, one version stipulates that the universal coupling to gravity
applies to all forms of matter and energy-momentum, not just to test
particles. Another version, the so-called strong equivalence principle, states
that the laws of Special Relativity are restored in all local free-falling
frames. The strong equivalence principle therefore makes precise what I meant
above by a “locally flat region in curved spacetime” — it refers to a local
inertial frame whereby the background spacetime is Minkowskian. In what
follows, these different versions of the equivalence principle will be
referred universally as the “equivalence principle.”

The equivalence principle motivates generalizing the Minkowski spacetime in
Special Relativity to a curved spacetime in General Relativity. One of the
central objects in curved geometries is the proper length between two distinct
points on the spacetime

${\rm d}s^{2}=g_{\mu\nu}(x)\,{\rm d}x^{\mu}{\rm d}x^{\nu}\,,$ (3)

where $x^{\mu}$ is a choice of coordinate and $g_{\mu\nu}$ is the spacetime
metric. In a local inertial frame, $g_{\mu\nu}$ reduces to the Minkowski
metric, $\eta_{\mu\nu}=\text{diag}(-1,+1,+1,+1)$. In that case, Special
Relativity requires the proper length to remain invariant under Poincaré
transformations. However, this principle no longer holds in General Relativity
because the isometries of Minkowski spacetime are generally absent. Instead,
the line element (3) is viewed as a geometric object in General Relativity,
whereby its proper length is invariant under coordinate transformations. As a
result, under the coordinate transformation $x^{\mu}\to x^{\prime\mu}$, the
metric $g_{\mu\nu}$ must transform as

$g_{\mu\nu}(x^{\prime})=\frac{\partial x^{\rho}}{\partial x^{\prime\hskip
1.0pt\mu}}\frac{\partial x^{\sigma}}{\partial x^{\prime\hskip
1.0pt\nu}}g_{\rho\sigma}(x)\,.$ (4)

Crucially, unlike a Poincaré transformation, (4) is a not a symmetry of
General Relativity. Rather it represents the infinite amount of freedom we
have in choosing the coordinates to describe the metric, with no changes
whatsover in the underlying geometry of spacetime. Given its moral similarity
with gauge transformations in field theory, I occasionally refer (4) as the
gauge transformation of $g_{\mu\nu}$.

The idea of invariance under a coordinate transformation is not restricted to
geometric objects such as a line element. In fact, all laws of physics and
physical observables are necessarily independent of the way in which we choose
our coordinates to represent them. This is the foundational principle of
General Relativity and is known as general covariance or diffeomorphism
invariance. Mathematically, the laws of physics and physical observables must
be tensorial, which means the forms of their expressions must remain unchanged
under coordinate transformations. To illustrate the power of general
covariance in constraining the possible forms of physical laws, it is
instructive to consider the equation of motion of a test particle free-falling
in a curved spacetime. This is given by the geodesic equation:

$\frac{{\rm d}v^{\mu}}{{\rm
d}\tau}+\Gamma^{\mu}_{\alpha\beta}v^{\mu}v^{\nu}=0\,,$ (5)

where $\tau$ is the particle’s proper time, $v^{\mu}\equiv{\rm d}x^{\mu}/{\rm
d}\tau$ is its four-velocity, and
$\Gamma^{\mu}_{\alpha\beta}=g^{\mu\rho}(\partial_{\alpha}g_{\rho\beta}+\partial_{\beta}g_{\rho\alpha}-\partial_{\rho}g_{\alpha\beta})/2$
is the Levi-Civita symbol. The first term ${\rm d}v^{\mu}/{\rm d}\tau$
acquires an additional term under a coordinate transformation. The Levi-Civita
symbol is therefore introduced in (5) to cancel this term and ensure the
geodesic equation is tensorial.

The equation (5) states that geodesics on curved spacetimes are equivalent to
free-falling trajectories in gravity, which I argued above are special
trajectories due to the weak equivalence principle. In Minkowski spacetime,
where $\Gamma_{\alpha\beta}^{\mu}=0$, (5) restores the classic result that
test particles move along straight lines in the absence of any forces. The
Levi-Civita symbol, which depends on $g_{\mu\nu}$, therefore determines how a
free-falling trajectory bends according to the geometry of its background
spacetime. In §1, we will see how the geodesic equation reduces to Newton’s
equation (2) in the non-relativistic limit. Crucially, $m_{I}$ and $m_{A}$
make no appearance whatsoever in the geodesic equation. As such, $m_{I}=m_{A}$
is naturally enforced in (2). Newtonian gravity and the weak equivalence
principle are hence naturally incorporated into General Relativity.

It is also important to examine the equation that governs the motion of a
spatially-extended object in a curved spacetime. For simplicity, I model this
object as consisting of two neighbouring test particles separated by the
vector $\zeta^{\mu}$. As both particles travel along their respective
geodesics, the separation $\zeta^{\mu}$ changes according to the curvature of
spacetime. More precisely, the evolution of $\zeta^{\mu}$ is given by the
geodesic deviation equation:

$\frac{\mathrm{D}^{2}\zeta^{\mu}}{{\rm
d}\tau^{2}}+{R^{\mu}}_{\nu\rho\sigma}v^{\nu}\zeta^{\rho}v^{\sigma}=0\,,$ (6)

where $\mathrm{D}u^{\mu}/{\rm d}\tau={\rm d}u^{\mu}/{\rm
d}\tau+{\Gamma^{\mu}}_{\alpha\beta}v^{\alpha}u^{\beta}$ is the covariant
derivative of an arbitrary vector $u^{\mu}$ along an integral curve of the
four-velocity $v^{\mu}$ (i.e. an observer’s worldline) and
${R^{\mu}}_{\nu\rho\sigma}=\partial_{\rho}\Gamma^{\mu}_{\nu\sigma}-\partial_{\sigma}\Gamma^{\mu}_{\nu\rho}+\Gamma^{\tau}_{\nu\sigma}\Gamma^{\mu}_{\tau\rho}-\Gamma^{\tau}_{\nu\rho}\Gamma^{\mu}_{\tau\sigma}$
is the Riemann tensor. In Minkowski spacetime, we have
${R^{\mu}}_{\nu\rho\sigma}=0$ and $\zeta^{\mu}$ is either fixed or changes at
a constant rate. This is simply a manifestation of the fact that geodesics in
the Minkowski spacetime are straight lines. However, in a curved spacetime,
${R^{\mu}}_{\nu\rho\sigma}\neq 0$, the separation $\zeta^{\mu}$ can
accelerate. In other words, geodesics which are initially parallel to one
another can either converge or diverge as they evolve. This relative
acceleration is a manifestation of tidal forces in gravity. The Riemann tensor
is therefore essential in describing the curvature of spacetime and its
gravitational field.

Through the geodesic equation (5) and the geodesic deviation equation (6), I
have made precise the qualitative descriptions about the motions of particles
and extended objects in curved spacetime. However, I have not discussed what
sources curvature of spacetime to begin. This is described by the Einstein
field equation

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi T_{\mu\nu}\,,$ (7)

where $R_{\mu\nu}\equiv{R^{\alpha}}_{\mu\alpha\nu}$ is the Ricci tensor,
$R\equiv{R^{\alpha}}_{\alpha}$ is the Ricci scalar, and $T_{\mu\nu}$ is the
energy-momentum tensor. The left-hand side of (7) is also known as the
Einstein tensor, $G_{\mu\nu}$. Since $G_{\mu\nu}$ represents the geometry of
spacetime, while $T_{\mu\nu}$ describes the matter sector, the Einstein field
equation dictates the way in which spacetime and energy-momentum interact with
each other. In fact, this interaction is unique because the contracted Bianchi
identity, $\nabla_{\mu}G^{\mu\nu}=0$, simultaneously enforces conservation of
the energy-momentum tensor, $\nabla_{\mu}T^{\mu\nu}=0$. While matter sources
the curvature of spacetime, the spacetime in turn constrains the way in which
matter can evolve. To quote Wheeler’s famous dictum: “Matter tells spacetime
how to curve, spacetime tells matter how to move.”

Finally, I explicitly discuss how various principles of General Relativity are
embedded in the Einstein field equation. Relativity is clearly obeyed by (7)
because it consists of tensors defined on spacetime. In addition, since
$T_{\mu\nu}$ encodes all forms of energy-momentum, (7) demonstrates their
universal coupling in gravity, thereby manifesting the equivalence principle.
Furthermore, the contracted Bianchi identity enforces four constraints on
$g_{\mu\nu}$, which reflects the inherit redundancies in $g_{\mu\nu}$ due to
our arbitrary choice of coordinates. As such, only six of the ten components
of $g_{\mu\nu}$ are independent when $T_{\mu\nu}\neq 0$. As we shall see in
§3, this is no longer the case in vacuum, $T_{\mu\nu}=0$, where the metric
only contains two independent degrees of freedom.

####  2 Black Holes

Black holes are a remarkable prediction of General Relativity. Once considered
a pure mathematical curiosity, they are now known to be ubiquitous in our
Universe, even playing significant roles in astrophysical environments. In
this subsection, I will review theoretical aspects of black holes. I will pay
special attention to aspects which are relevant for this thesis.

Black holes are vacuum solutions of the Einstein field equation (7),

$R_{\mu\nu}=0\,.$ (8)

The static and spherically symmetric black hole solution was discovered by
Schwarzschild in 1916, shortly after the Einstein’s field equation was
published [Schwarzschild1916, Droste1917]. The line element is

${\rm d}s^{2}=-\left(1-\frac{2M}{r}\right){\rm
d}t^{2}+\left(1-\frac{2M}{r}\right)^{-1}{\rm d}r^{2}+r^{2}\left({\rm
d}\theta^{2}+\sin^{2}\theta{\rm d}\phi^{2}\right)\,,$ (9)

where $M$ is the mass of the black hole. The Schwarzschild solution (9) has
two special locations: a singularity at $r=0$ that represents a divergence of
the spacetime curvature, and the event horizon at $r=2M$. The event horizon is
defined as the null surface on which any trajectory in the black hole
interior, $r<2M$, are unable to escape to the exterior, $r>2M$. The
gravitational pull of black holes is therefore so strong that its escape
velocity exceeds the speed of light. The apparent singularity at $r=2M$ in (9)
is merely a coordinate singularity. It can be removed by analytically
extending the metric through coordinate systems which are adapted to null
radial geodesics, such as the Eddington-Finkelstein coordinates.

Remarkably, Birkhoff’s theorem states that (9) is the unique solution of all
spherically-symmetric vacuum spacetimes [Birkhoff, Jebsen]. The Schwarzschild
metric is therefore also applicable to the exterior regions of many
astrophysical objects, as spherical symmetry is often an excellent
approximation. After all, it is for this reason that (9) could be used to
accurately postdict the perihelion precession of Mercury around the Sun
[Einstein1916mercury] and predict the bending of light by the moon in a solar
eclipse [Eddington1920]. Furthermore, since staticity is not assumed in
Birkhoff’s theorem, (9) remains applicable to a radially pulsating star. The
metric in the interior of a star, on the other hand, depends on its energy-
momentum tensor and can be solved using the Tolman-Volkoff-Oppenheimer
equations [Tolman1939, OV1939].

While the Schwarzschild solution was found shortly after the formulation of
General Relativity, the solution for a rotating black hole was only derived by
Roy Kerr half a century later [Kerr1963] (see [Kerr:2007dk] for a personal
account of his discovery). In what follows, I will express the Kerr solution
in Boyer-Lindquist coordinates [BL1967], as it conveniently reduces to the
Schwarzschild metric (9) when the black hole spin vanishes. With a slight
abuse of notation, I also denote these coordinates by $\\{t,r,\theta,\phi\\}$.
The line element of a Kerr black hole with mass $M$ and angular momentum $J$
then reads

${\rm d}s^{2}=-\frac{\Delta}{\Sigma}\left({\rm d}t-a\sin^{2}\theta\,{\rm
d}\phi\right)^{2}+\frac{\Sigma}{\Delta}{\rm d}r^{2}+\Sigma\hskip 1.0pt{\rm
d}\theta^{2}+\frac{\sin^{2}\theta}{\Sigma}\left(a\hskip 1.0pt{\rm
d}t-(r^{2}+a^{2})\,{\rm d}\phi\right)^{2}\,,$ (10)

where $\Delta\equiv r^{2}-2Mr+a^{2}$ and $\Sigma\equiv
r^{2}+a^{2}\cos^{2}\theta$. The spin parameter $a\equiv J/M$ is bounded by
$0\leq a\leq M$. The roots of $\Delta$ determine the inner and outer horizons,
located at $r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$. The inner horizon is inaccessible
by an external observer, and can therefore be neglected for all practical
purposes. The off-diagonal component $g_{0\phi}$ in (10) captures the
rotational frame-dragging effect, also called the Lense-Thirring effect [Lens-
Thirring1918]. From the point of view of an observer at infinity, a free-
falling test particle with no angular momentum would co-rotate with the black
hole. When this test particle approaches $r_{+}$, it co-rotates at the angular
velocity of the black hole, $\Omega_{\rm H}=a/2Mr_{+}$.

In general, an observer can counteract the frame-dragging effect by acquiring
angular momentum in the direction opposite to the black hole spin. However,
near the black hole, there exists an ergoregion in which all observers and
even light must co-rotate with the black hole. More formally, this is a region
where a timelike Killing vector field at infinity turns spacelike. The
ergoregion therefore lies within $r_{+}<r<r_{\rm erg}$, where $r_{\rm
erg}=M+\sqrt{M^{2}-a^{2}\cos^{2}\theta}$ is a root of $g_{00}$ in (10). That a
timelike vector must turn spacelike in the ergoregion is interesting because a
test particle in this region can have negative energy, at least from the point
of view of an observer at infinity. The Penrose process [Penrose:1969pc] is a
thought experiment whereby a particle in the ergoregion disintegrates into a
positive and a negative energy component, with the latter falling into the
black hole. By local conservation of energy and momentum, the former must
emerge from the ergoregion with larger energy and angular momemtum. As such,
energy and angular momentum are extracted from the rotating black hole.
Remarkably, there exists a limit to the amount of energy which can be
extracted through this process. This is because the mass of the black hole can
be decomposed into [Christodoulou:1970]

$M^{2}=M^{2}_{\rm irr}+\frac{J^{2}}{4M_{\rm irr}^{2}}\,,$ (11)

where $M_{\rm irr}$ is the black hole’s irreducible mass. Heuristically, the
irreducible mass can be interpreted as the “rest mass” of the black hole,
while the second term in (11) is its “rotational energy,” with only the latter
extractable through the Penrose process. Though rather contrived, the Penrose
process was the first concrete realization of an energy-extraction mechanism
from a rotating black hole. As we shall see in Section 3, a wave analog called
black hole superradiance can trigger an instability on a rotating black hole,
efficiently extracting energy and angular momentum from a rotating black
hole.222I emphasize, however, that the analogy between the Penrose process and
superradiance is qualitative at best. In particular, the ergoregion plays no
role in superradiance. One can view this distinction as a difference in the
scales involved: a particle in the Penrose process is point-like, whereas to
trigger superradiance, the Compton wavelength of the wave must be larger than
the size of the black hole. Crucially, superradiance can occur spontaneously
and is therefore realizable in astrophysical black holes.

The Schwarzschild and Kerr black holes are also distinct in their multipolar
structures. As a consequence of spherical symmetry, a non-rotating black hole
only sources a monopole. A Kerr black hole, on the other hand, generates an
infinite tower of multipole moments. Generically, the multipole moments of a
stationary object in General Relativity is organized through its mass moments,
$M_{\ell}$, and current moments, $S_{\ell}$, where $\ell$ denotes all non-
negative integers [Geroch:1970cd, Hansen:1974zz]. For an object that is
reflection symmetric about its equatorial plane, like the Kerr black hole, the
odd-$\ell$ mass moments and even-$\ell$ current moments vanish identically.
Black holes are special in that all of its multipole moments can be succinctly
described through the simple relation [Hansen:1974zz, Thorne:1980ru]

$M_{\ell}+iS_{\ell}=M\left(ia\right)^{\ell}\,.$ (12)

The interpretation of this hierarchy of moments is clear: $M_{0}=M$ is the
black hole’s mass, $S_{1}=Ma$ is its spin, $M_{2}=-Ma^{2}$ is its mass
quadrupole, etc. Importantly, the multipolar structure of the black hole is
uniquely fixed by $M$ and $a$. This is qualitatively distinct from a general
astrophysical object, where the quadrupole and higher-order multipole moments
typically depend on additional parameters that describe the internal structure
of the object.

The existence of the solutions (9) and (10) does not preclude other black hole
solutions. However, the uniqueness theorems [Israel1967, Carter:1971zc,
Robinson:1975bv] state that the Schwarzschild and Kerr metrics are indeed the
only solutions of black holes.333For electrovacuum spacetimes, black holes can
also carry electric and magnetic charges (see [Chrusciel:2012jk] for a review
on more general uniqueness theorems when other types of matter fields are
present). Nevertheless, I will ignore charged black holes throughout this
thesis, as they do not seem to be astrophysically relevant. I also do not
consider black hole solutions in other spacetime dimensions. As an
astrophysical object collapses and forms a black hole, all of its parameters
but $M$ and $a$ would disappear, thereby making (12) the unique multipolar
structure of all black holes in our Universe. This remarkable simplicity of
black holes is often phrased more colloquially as black holes have “no hair.”

Astrophysical black holes are often immersed in complicated environments and
perturbed by surrounding matter. It is therefore important to examine the
stability of black holes under these perturbations. To a good approximation, I
describe the mode stability of black holes by considering their response when
perturbed by an incident null wave.444There exists many notions of stability
in General Relativity. In addition to mode stability, there are also linear
and non-linear stabilities. See e.g. [Teukolsky:2014vca] for their differences
and a summary of the stabilities of Kerr black holes. In early pioneering
works, it was shown that all modes decay for a Schwarzschild black hole
[ReggeWheeler, Vishveshwara, Zerilli1970a]. These stable modes, also known as
quasi-normal modes, correspond to the ringing spectra of the black hole. On
the other hand, the situation is richer for Kerr black holes. Through the
Teukolsky equation [Teukolsky:1973ha, Press:1973zz, Teukolsky:1974yv], it was
found that Kerr black holes are only stable when its angular velocity
satisfies $\Omega_{\rm H}<\omega/m$, where $\omega$ and $m$ are the frequency
and azimuthal number of the mode — conserved quantities due to the Kerr
isometries. On the other hand, for a highly rotating black hole with

$\Omega_{\rm H}>\frac{\omega}{m}\,,$ (13)

the Kerr black hole is unstable to a mode perturbation sourced by a massive
field [Damour:1976kh, Detweiler:1980uk, Brito:2015oca]. This is precisely the
superradiance phenomenon which I alluded to earlier. This instability, of
course, does not prevent the formation of astrophysical black holes. Rather,
as I shall elaborate in Section 3, it provides a robust mechanism of
coherently enhancing fields around black holes, making black holes interesting
laboratories of physics beyond the Standard Model.

####  3 Gravitational Waves

The Einstein field equation (7) involves a set of coupled partial differential
equations which are hard to solve for generic dynamical spacetimes.
Fortunately, analytic time-dependent solutions are attainable when a weak
metric perturbation is considered on a fixed background spacetime. In this
subsection, I first review the linear perturbation theory of General
Relativity. I then describe gravitational waves as plane wave solutions of the
linearized Einstein equations. I end by discussing how these waves are
generated through accelerating quadrupole moments.

To simplify the following discussion, the background spacetime is fixed to be
Minkowski space. This is an excellent approximation in many realistic
scenarios, including the background spacetimes of gravitational-wave detectors
and inspiraling binary systems. The metric can therefore be written as

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}\,,$ (14)

where $h_{\mu\nu}$ is a perturbation about Minkowski spacetime. In linear
perturbation theory, $h_{\mu\nu}$ is assumed to be small. In order for this
perturbative scheme to be valid, we must choose a local inertial frame wherein
the components of $h_{\mu\nu}$ are much smaller than unity.

The metric perturbation transforms as a spin-2 representation of the Poincaré
group. It also transforms non-trivially under an infinitesimal coordinate
transformation $x^{\mu}\to x^{\prime\mu}=x^{\mu}+\xi^{\mu}$, where $\xi^{\mu}$
is an arbitrary function of $x^{\mu}$. In particular, the gauge transformation
(4) implies that

$h_{\mu\nu}\to
h^{\prime}_{\mu\nu}=h_{\mu\nu}-\left(\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}\right)\,,$
(15)

at leading order in $\xi$, with $h^{\prime}_{\mu\nu}\equiv
h_{\mu\nu}(x^{\prime})$. Physical observables must therefore remain invariant
under the gauge transformation (15) in linearized General Relativity. For
future convenience, I define the trace-reversed field, $\bar{h}_{\mu\nu}\equiv
h_{\mu\nu}-\eta_{\mu\nu}h/2$. The gauge transformation (15) then becomes
$\bar{h}_{\mu\nu}\to\bar{h}^{\prime}_{\mu\nu}=\bar{h}_{\mu\nu}-\xi_{\mu\nu}$,
where
$\xi_{\mu\nu}\equiv\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}-\eta_{\mu\nu}\partial_{\alpha}\xi^{\alpha}$.
We saw in (7) that the contracted Bianchi identity imposes four constraints on
$g_{\mu\nu}$. A similar gauge condition must therefore be imposed on
$\bar{h}_{\mu\nu}$ to remove its gauge redundancies. A convenient choice is
the so-called harmonic gauge, $\partial^{\mu}\bar{h}_{\mu\nu}=0$, which
imposes four gauge conditions on $\bar{h}_{\mu\nu}$, as desired, and
dramatically simplifies the linear expansion of (7) to

$\square\,\bar{h}_{\mu\nu}=-16\pi T_{\mu\nu}\,,$ (16)

where $\square\equiv\eta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}$ is
the flat-space d’Alembertian. The linearized Einstein equation (16) therefore
describes the sourcing of $\bar{h}_{\mu\nu}$ through a weak source
$T_{\mu\nu}$. The harmonic gauge is also convenient because it automatically
enforces $\partial_{\mu}T^{\mu\nu}=0$, thereby preserving some structural
similarity between the linear and full non-linear Einstein equations.

Before describing solutions of (16), it is critical to investigate the
solutions of $\xi_{\nu}$ that enforce the harmonic gauge. Acting with a
partial derivative on the the gauge transformation of $\bar{h}_{\mu\nu}$, we
obtain
$\partial^{\mu}\bar{h}_{\mu\nu}\to\partial^{\mu}\bar{h}^{\prime}_{\mu\nu}=\partial^{\mu}\bar{h}_{\mu\nu}-\square\xi_{\nu}$.
The harmonic gauge is therefore imposed through the inhomogeneous solution
$\xi_{\nu}=-\square^{-1}\left(\partial^{\mu}\bar{h}^{\prime}_{\mu\nu}\right)$,
where $\square^{-1}$ formally denotes the Green’s function operator. In
addition, we see that the harmonic gauge is preserved by a residual condition

$\square\xi_{\nu}=0\quad\implies\quad\square\xi_{\mu\nu}=0\,,$ (17)

where $\xi_{\mu\nu}$ is defined in the paragraph below (15). We see that these
residual conditions are simple homogeneous wave equations. As such, the most
general solution of $\xi_{\nu}$ consists of a superposition of the
inhomogeneous solution described above and a plane wave solution
$\xi_{\nu}=C_{\nu}\,e^{ik_{\alpha}x^{\alpha}}$, where $C_{\nu}$ is an
arbitrary constant vector and $k^{\nu}$ is null.555The most general solution
to the wave equation is
$\xi_{\nu}=C^{+}_{\nu}\,e^{+ik_{\alpha}x^{\alpha}}+C^{-}_{\nu}\,e^{-ik_{\alpha}x^{\alpha}}$.
Here, I impose the condition that the wave propagates at a definite direction,
such that either one of the terms vanishes. Without loss of generality, I
choose $C_{\mu}^{-}=0$.  Crucially, $\xi_{\nu}$ is an arbitrary function and
hence no physical content can be attributed to $C_{\nu}$. As we shall see
shortly, this gauge freedom in $C_{\nu}$ can be exploited to remove additional
redundancies in vacuum solutions of $\bar{h}_{\mu\nu}$.

I now describe propagating plane wave solutions to (16) in the exterior region
of a source. Setting $T_{\mu\nu}=0$, the linearized Einstein equation
simplifies to the homogeneous wave equation $\square\bar{h}_{\mu\nu}=0$. Its
solution reads

$\bar{h}_{\mu\nu}=\text{Re}\big{(}H_{\mu\nu}e^{ik_{\alpha}x^{\alpha}}\big{)}\,,$
(18)

where $H_{\mu\nu}$ is a symmetric constant tensor. Because $k^{\nu}$ is null,
gravitational waves propagate at the speed of light in vacuum. The harmonic
gauge gives $k^{\mu}H_{\mu\nu}=0$, thereby forcing the components of
$H_{\mu\nu}$ to lie in the plane transverse to the direction of propagation.

At this stage, one may erroneously conclude that $H_{\mu\nu}$ contains six
independent degrees of freedom. Nevertheless, because
$\square\left(\bar{h}_{\mu\nu}-\xi_{\mu\nu}\right)=0$ is also a solution to
the equation of motion in vacuum, cf. (17), one can freely choose the
components of $C_{\nu}$ above to impose four additional constraints on
$\bar{h}_{\mu\nu}$. An obvious choice is one that sets four of the components
of $\bar{h}_{\mu\nu}$ to zero. This can be achieved through the transverse and
traceless (TT) gauge

$\bar{h}_{0\mu}=0\,,\qquad h=0\,.$ (19)

The transverse gauge, along with the harmonic gauge, only imposes three
independent constraints. The traceless condition implies that
$\bar{h}_{\mu\nu}=h_{\mu\nu}$. To impose the TT gauge onto (18), we can define
the projection operator

$\Lambda_{ij,kl}\equiv P_{ik}P_{jl}-\frac{1}{2}P_{ij}P_{kl}\,,\qquad
P_{ij}(\hat{\textbf{n}})\equiv\delta_{ij}-n_{i}n_{j}\,,$ (20)

where $\hat{\textbf{n}}$ is a unit vector in the propagation direction of the
gravitational wave. The solution in the transverse and TT gauge then is
$h^{\rm TT}_{ij}=\Lambda_{ij,kl}\bar{h}_{kl}$. Without loss of generality, I
take the wave to propagate in the $z$-direction, such that

$h^{\rm TT}_{ij}=\begin{pmatrix}\phantom{+}h_{+}&h_{\times}&0\phantom{+}\\\
\phantom{+}h_{\times}&-h_{+}&0\phantom{+}\\\
\phantom{+}0&0&0\phantom{+}\end{pmatrix}_{ij}\cos\left[\omega(t-z)\right]\,,$
(21)

where $\omega$ is the gravitational-wave frequency. The components $h_{+}$ and
$h_{\times}$ in (21) therefore capture the two independent physical degrees of
freedom of a gravitational wave. That a massless spin-2 field contains only
two degrees of freedom is of course a classic result of Wigner’s study of
massless representations of the Poincaré group [Wigner1939]. Nevertheless, it
is important to appreciate that Einstein did not notice these gauge artifacts
at the time the solution was derived [Einstein1916, Weyl1922]. It was
Eddington who later explicitly showed the unphysical nature of these gauge
artefacts, as he found them to propagate at velocities that depend on the
choice of coordinates; or as Eddington himself put it, travel at “the speed of
thought” [Eddington1922].

Despite the discovery of the solution (21) in the 1920s, confusions over
whether gravitational waves are physical persisted in the subsequent decades.
For instance, Einstein believed (21) to be an artifact of the linear
approximation, and that no gravitational-wave solution would be found in the
full theory. Upon finding a singularity in a non-linear cylindrical wave
solution in 1936, he and Rosen initially concluded that gravitational waves
were therefore unphysical, only to later retract from this conclusion once
they understood that it was a coordinate singularity [EinsteinRosen1937].
Indeed, a recurring source of confusion over the decades had been the issue of
coordinate or gauge dependences. For example, the harmonic and TT gauges,
though mathematically convenient, lack clear physical interpretations. In
addition, it was found that the energy-momentum tensor of a gravitational
field is not invariant under the gauge transformation (15), thereby casting
further doubt on the physical nature of gravitational waves. These issues were
discussed extensively in the historic Chapel Hill Conference, which had led to
the resolutions of many of these problems [Pirani:1956wr, Bondi:1957dt,
Bondi:1958aj]. Here, I will focus only on the issue of whether energy-momentum
tensor of gravitational waves is physical.

The fact that the energy-momentum tensor of gravitational waves, $t_{\mu\nu}$,
is not gauge invariant is a direct manifestation of the equivalence principle:
one can choose a local inertial frame at any point in spacetime, where the
gravitational field vanishes locally. While $t_{\mu\nu}$ is not well-defined
locally, it certainly carries energy and momentum in an extended spacetime
volume. In other words, one can show that the averaged energy-momentum tensor
over a spacetime region, $\langle t^{\mu\nu}\rangle$, is indeed gauge-
invariant. Mathematically, this is because the averaging procedure allows us
to perform integration by parts, which cancels the $\xi_{\nu}$-dependent terms
that arise from the gauge transformation (15). Physically, the averaging
provides a meaningful separation between the short-wavelength metric
perturbation and the long-wavelength background metric in (14). Since $\langle
t^{\mu\nu}\rangle$ is gauge invariant, it is convenient to impose the harmonic
and TT gauges, in which case one finds

$\langle t^{\mu\nu}\rangle=\frac{1}{32\pi}\langle\partial^{\mu}h^{{\rm
TT}ij}\partial^{\nu}h^{\rm TT}_{ij}\rangle\,.$ (22)

The averaged energy and momentum density are $\langle t^{00}\rangle=\langle
t^{0z}\rangle=\langle\dot{h}_{+}^{2}+\dot{h}_{\times}^{2}\rangle/16\pi$.
Gravitational waves are therefore physical and can do work. In the limit where
the averaged spacetime volume is taken to be infinite, a rigorous definition
of total energy and momentum of the spacetime can be obtained. These general-
relativistic quantities are called the ADM energy and momentum [ADM1962].

The plane-wave solution (21) provides the foundation for more complicated
gravitational-wave solutions. I now describe radiative solutions of (16) which
are sourced by $T_{\mu\nu}\neq 0$. This inhomogeneous equation can be solved
with a retarded Green’s function, which imposes a “no incoming radiation”
boundary condition. In addition, since we are interested in the wave solution
in the exterior of the source, I apply the TT gauge through the projection
operator (20) to remove the gauge redundancies in $h_{\mu\nu}$. The integral-
solution therefore reads

$h_{ij}^{\rm TT}(t,\textbf{x})=4\,\Lambda_{ij,kl}(\hat{\textbf{n}})\int{\rm
d}^{3}\textbf{y}\,\frac{T_{kl}\left(t-|\textbf{x}-\textbf{y}|,\textbf{y}\right)}{|\textbf{x}-\textbf{y}|}\,.$
(23)

For simplicity, I will henceforth restrict myself to cases where an observed
gravitational wave is far from the source, i.e.
$r\equiv|\textbf{x}|\gg|\textbf{y}|$, as is typically the case in realistic
astrophysical scenarios. This allows for a further simplication
$|\textbf{x}-\textbf{y}|\approx r(1-\textbf{y}\cdot\hat{\textbf{n}}/r)$, where
terms that decay as $\mathcal{O}(|\textbf{y}|^{2}/r^{2})$ or faster can be
ignored.

Generally, the source term in (23) has a complicated spatial structure and
moves in a dynamical manner. As such, it is more convenient to systematically
study its properties through a multipole expansion. By performing a Taylor
expansion about small values of $|\textbf{y}|$ in (23), we find

$\displaystyle\hskip-3.0pt\int{\rm
d}^{3}\textbf{y}\,T_{kl}\left(t_{r}+\textbf{y}\cdot\hat{\textbf{n}},\textbf{y}\right)=\sum_{p=0}^{\infty}\frac{n^{i_{1}}\cdots
n^{i_{p}}}{p!}\frac{{\rm d}^{p}}{{\rm d}t^{p}}\left[\int{\rm
d}^{3}\textbf{y}\,y^{i_{1}}\cdots y^{i_{p}}T_{kl}(t,\textbf{y})\right]_{\rm
ret},$ (24)

where $t_{r}\equiv t-r$ is the retarded time and the subscript means the
quantities are evaluated at $t=t_{r}$. The term within the bracket in (24) are
the tensorial moments of the stress tensor $T_{ij}$. The physical
interpretation of these terms, however, are clearer when they are rewritten in
terms of the mass moments and current moments:666These definitions are the
non-relativistic, though gauge-dependent, versions of the general-relativistic
moments described in §2.

$\displaystyle M^{i_{1}\cdots i_{p}}(t_{r})$ $\displaystyle\equiv\int{\rm
d}^{3}\textbf{y}\,y^{i_{1}}\cdots y^{i_{p}}\,T_{00}(t_{r},\textbf{y})\,,$ (25)
$\displaystyle S_{k}^{i_{1}\cdots i_{p}}(t_{r})$ $\displaystyle\equiv\int{\rm
d}^{3}\textbf{y}\,y^{i_{1}}\cdots y^{i_{p}}\,T_{0k}(t_{r},\textbf{y})\,.$

For example, the dominant source term in (24) can be re-expressed as

$\int{\rm
d}^{3}\textbf{y}\,T_{ij}(t_{r},\textbf{y})=\frac{1}{2}\left[\ddot{M}_{ij}(t)\right]_{\rm
ret}\,,$ (26)

where an overdot represents a derivative with respect to $t$. The relation
(26) is obtained through $\partial_{\mu}T^{\mu\nu}=0$, which is a consequence
of the harmonic gauge, and integration by parts where the boundary terms
vanish due to the finite spatial support of $T_{ij}$. Using the same
techniques, one find that the higher-order terms in (24) can be written as a
sum of time-derivatives of the higher-order mass and current moments (25). The
mass and current moments (25) therefore completely characterize the multipolar
structure of the source.

Because the projection operator (20) satisfies
$\Lambda_{ij,kl}\,\delta_{kl}=0$, it is conventional to remove the trace
component of the moments in (23). Defining the trace-free quadrupole moment
$Q_{ij}\equiv M_{ij}-\delta_{ij}M_{kk}/3$, the leading-order gravitational-
wave solution is therefore

$h_{ij}^{\rm
TT}(t,\textbf{x})=\frac{2}{r}\Lambda_{ij,kl}(\hat{\textbf{n}})\,\left[\ddot{Q}_{kl}(t)\right]_{\rm
ret}\,.$ (27)

This clearly shows that an accelerating mass quadrupole is the dominant source
of gravitational waves. Importantly, monopole and dipole radiations are absent
in General Relativity. While I have only shown this to be the case in linear
theory, the conclusion remains unchanged in the full non-linear theory. In
particular, the absence of monopole radiation is a consequence of Birkhoff’s
theorem [Birkhoff, Jebsen], where all spherically spacetimes must admit the
Schwarzschild solution which is time independent, cf. §2. On the other hand,
dipole radiation is absent because, unlike electromagnetism, there are no
opposite charges in gravity.

Using (27), we can evaluate the power emitted by the accelerating source,
$P\equiv\int{\rm d}^{3}\textbf{x}\,\langle\hskip 1.0pt\dot{t}^{00}\hskip
1.0pt\rangle$, where the averaging is performed over timescales that are much
longer than the temporal variation of $Q_{ij}$. In addition, averaging the
power emission over all gravitational-wave propagation directions, we find the
averaged power loss

$P=\frac{1}{5}\Big{\langle}\dddot{Q}_{ij}\dddot{Q}_{ij}\Big{\rangle}_{\rm
ret}\,.$ (28)

This is the famous Einstein quadrupole formula [Einstein1918]. Similar
expressions for the linear and angular momenta losses can also be obtained.
Interestingly, while (28) was derived about a hundred years ago, its validity
remained controversial until the 1980s, even after various conceptual issues
of gravitational wave had been resolved at that time [Ehlers1976].
Observationally, the discovery of the Hulse-Taylor binary pulsar in 1974 had
allowed for the first test of this formula through a measurement of the
binary’s orbital decay [Hulse:1974eb]. By the early 1980s, the orbital decay
of the Hulse-Taylor binary was found to be in excellent agreement with (28),
therefore providing strong observational confirmation of its validity
[Taylor1982]. See [Kennefick:1997kb] for the history of this controversy.

###  2 Compact Binary Coalescences

Compact binary systems are one of the loudest and most important sources of
gravitational waves in our Universe. Their inspiral motions can be highly
relativistic, thereby sourcing large accelerating quadrupoles. Furthermore,
binary systems are unique in that accurate computations of their gravitational
waveforms, especially in the early inspiraling regime, are attainable. This
makes an observed waveform a rich source of information about the binary’s
dynamics and the physics of its components.

I now describe the dynamics of a binary system in General Relativity. This
includes both the conservative and dissipative effects on the orbital
evolution. My main focus is on the dynamics in the early inspiraling stage,
though I also briefly comment on the subsequent merger and ringdown regimes
(see Fig. 1 for the different regimes of a binary coalescence). I also discuss
the gravitational waves emitted by binary systems. My primary goal there is to
understand how various physical effects impact the waveforms of the
gravitational waves emitted by the binary systems.

Figure 1: Illustration of the inspiral (blue), merger (green), and ringdown
(red) stages of a binary coalescence and its corresponding gravitational
waveform. Figure adapted from [Antelis:2016icm].

####  1 Dynamics of Binary Systems

The dynamics of a two-body system in Newtonian mechanics is well understood
since the publication of the Principia. Nevertheless, relativistic corrections
to a binary’s Newtonian evolution are notoriously hard to compute, making them
an active research problem to this date. After a brief description of how
General Relativity reduces to Newtonian gravity, I review key facts of
Keplerian orbits. I then discuss different perturbative schemes that have been
developed to describe the relativistic dynamics of the two-body motion. A
technical description of these perturbative methods is beyond the scope of my
discussion. Rather, I aim to provide a qualitative sketch of the different
physical effects at play, including those incurred by the binary components on
the orbit. Crucially, my discussion includes binary systems whose components
consist of general compact objects and not just black holes.

In what follows, I consider perturbative expansions of physical quantities,
including the relative acceleration of the binary, $a^{i}$, its total energy,
$E$, and its orbital angular momentum, $L^{i}$. It is hence convenient to
schematically denote these quantites by $X\in\\{a^{i},E,L^{i}\\}$, such that

$X=X_{\rm N}+X_{\rm 1PN}+X_{\rm SO}+X_{\rm S_{1}S_{2}}+X_{\rm Q}+X_{\rm
T}+\cdots\,,$ (1)

where the subscripts denote the Newtonian, first post-Newtonian, spin-orbit,
spin-spin, quadrupole, and tidal deformability, respectively. I focus on these
terms because they represent qualitatively new effects that impact the orbital
dynamics.

##### Newtonian dynamics

General Relativity reduces to Newtonian gravity when i) the gravitational
field is weak, and ii) the sources move non-relativistically. More precisely,
the former means that the background metric is well approximated by Minkowski
spacetime; the latter implies that the spatial component of the four-velocity
$v^{\mu}={\rm d}x^{\mu}/{\rm d}\tau=(t,v^{i})$ is suppressed, $v^{i}\ll 1$.
With these approximations, the geodesic equation (5) reduces to the Newtonian
equation of motion

$\frac{{\rm d}^{2}x^{i}}{{\rm d}t^{2}}=-\partial^{i}U\,,$ (2)

where the proper time $\tau$ is well-approximated by the coordinate time $t$
and $U=-h_{00}/2$ is the gravitational potential. In its classic “$F=ma$”
representation, (2) would instead be written as (2), which includes the
inertial and gravitational masses. However, (2) clearly demonstrates these
masses are exactly the same, as required by the equivalence principle.
Similarly, the Einstein field equation (7) reduces to the Poisson equation (1)
in the weak-field and non-relativistic limit, in which case the time-time
component of (7) dominates and the energy density is $\rho\equiv-T_{00}$.

Newton’s equation (2) applies to binary systems whose components are well-
approximated by point particles. We denote the masses of these components by
$m_{1}$ and $m_{2}$. To solve the two-body problem, it is convenient to work
in the center-of-mass frame, where the coordinates are given by the binary’s
barycenter position and the relative binary separation,
$R^{i}=R\hat{R}^{i}$.777 The binary separation, $R$, though well-defined in
Newtonian gravity, is ambiguous in General Relativity. In the rest of our
discussion, physical quantities will still be written as functions of $R$,
though we emphasize that these results are only valid in the center-of-mass
frame. This gauge dependence can be eliminated by replacing $R$ with gauge-
invariant quantities, such as the orbital frequency of the binary through the
relativistic generalization of Kepler’s law. In this picture, the two-body
system reduces effectively to a one-body description, whereby a fictitious
particle with reduced mass, $m_{1}m_{2}/(m_{1}+m_{2})$, moves in the
gravitational potential sourced by the total mass $M_{\rm tot}=m_{1}+m_{2}$.
The potential experienced by this fictitious particle is therefore $U=-M_{\rm
tot}/R$, while the relative acceleration, $a^{i}=\ddot{R}^{i}$, is given by

$a^{i}_{\rm N}=-\frac{M_{\rm tot}}{R^{2}}\hat{R}^{i}\,,$ (3)

where the subscript denotes Newtonian. We have simplified the problem, which
initially has six components of spatial coordinates, to a system with only
three independent degrees of freedom. This is possible because of conservation
of linear momentum, which implies that the motion of the barycenter is uniform
and therefore trivial. Other conserved quantities include the energy and
orbital angular momentum of the binary system, which are

$\displaystyle E_{\rm N}$ $\displaystyle=\nu M_{\rm
tot}\left(\frac{1}{2}v^{2}-\frac{M_{\rm tot}}{R}\right),$ (4) $\displaystyle
L^{i}_{\rm N}$ $\displaystyle=\nu M_{\rm tot}\,\epsilon^{ijk}R_{j}v_{k}\,,$

where $\nu\equiv m_{1}m_{2}/M^{2}_{\rm tot}$ is the symmetric mass ratio,
$v\equiv|v^{i}|$ is the relative velocity of the binary, and $\epsilon_{ijk}$
is the Levi-Civita symbol. Crucially, conservation of $L^{i}_{\rm N}$ dictates
that both its magnitude and direction do not evolve. The orbital plane, which
is perpendicular to $L^{i}_{\rm N}$, is therefore fixed at all times.
Depending on the ratio of $E_{\rm N}$ and $|L^{i}_{\rm N}|$, the orbit can
have non-trivial eccentricity, $e$, with $e=0$ for a circular orbit and
$)<e<1$ for an elliptical orbit.

Finally, we note that the orbital velocity of the binary and its gravitational
potential are intrinsically related to each other. In particular, the virial
theorem states that the time-averaged (or equivalently the orbit-averaged)
behaviour of the binary satisfies

$v^{2}=\frac{M_{\rm tot}}{R}\,.$ (5)

As we shall now see, relativistic corrections to the Newtonian dynamics can be
computed through a perturbative expansion in $v$ or in $M_{\rm tot}/R$. The
relation (5) will therefore be important for power counting the higher-order
terms that appear in the expansions.

##### Relativistic dynamics

During the early inspiral stage of a binary, the gravitational field is weak
and its motion remains slow. A perturbative expansion that includes
relativistic dynamics can therefore be performed in this regime. When the
binary components are treated as point particles, two choices of expansion
parameters are possible: either in $v$, known as the post-Newtonian (PN)
expansion, or in Newton’s constant $G$ (equivalently $M_{\rm tot}/R$ in units
of $G=1$), known as the post-Minkowskian (PM) expansion.888Post-Newtonian
terms at the $n$-th PN order are $v^{2n}$ suppressed compared to the leading
Newtonian result. On the other hand, $n$-th order PM terms represent
corrections to Minkowski spacetime at powers of $G^{n}$. In fact, the virial
theorem (5) implies that both expansions are a double series — the
perturbative series in either parameter would recover partial results of the
other, cf. Fig. 2.

In the PN expansion, a systematic expansion in $v\ll 1$ is performed on the
metric perturbation $h_{\mu\nu}$, cf. (14). More precisely, this involves
taking the non-relativistic limit in $T_{\mu\nu}$ and solving the linearized
Einstein equation (16) iteratively in powers of $v$. In the point-particle
approximation, the sizes of the binary components are taken to be comparable
to their gravitational radii, $r_{c}\sim m$, in which case the virial theorem
(5) implies that $r_{c}\sim R\hskip 1.0ptv^{2}$. On the other hand, the
gravitational waves emitted by a binary has a typical wavelength $\lambda_{\rm
gw}\sim R\hskip 1.0ptv^{-1}$. These scaling mean that the PN expansion can be
viewed as an expansion that relies on the hierarchy $r_{c}\ll R\ll\lambda_{\rm
gw}$. A crucial feature of the PN expansion is that time derivatives are
suppressed compared to spatial derivatives,
$\partial_{0}h_{\mu\nu}=v^{i}\partial_{i}h_{\mu\nu}\ll\partial_{i}h_{\mu\nu}$.
As a result, retardation effects are treated as small corrections to
instantaneous effects. This has important ramifications for the region of
validity of the PN series: for small retarded times, we have

$h_{\mu\nu}(t-r)=h_{\mu\nu}(t)-r\hskip 1.0pt\partial_{0}h_{\mu\nu}+\cdots\,.$
(6)

Since $\partial_{0}h_{\mu\nu}\sim h_{\mu\nu}/\lambda_{\rm gw}$, the series (6)
diverges as the wave propagates over distances $r\gtrsim\lambda_{\rm gw}$.
This motivates a separation of regions in the binary problem: the near zone,
$r\ll\lambda_{\rm gw}$, is a region where the PN expansion holds, while the
wave zone, $r\gg\lambda_{\rm gw}$, is where the approximation becomes invalid.
By disregarding putative retardation effects, the PN expansion is only well
suited to describe the conservative dynamics of the binary. As we shall see
shortly, this shortcoming is compensated by the PM expansion, which is valid
in both regions.

The conservative dynamics of a point-particle binary system is symmetric under
time reversal. As such, only terms that are suppressed by even powers of $v$
contribute to the relativistic corrections to the Newtonian dynamics. The
first post-Newtonian correction was performed by Lorentz and Droste
[Lorentz1937], though the credit often goes to Einstein, Infeld and Hoffman
[EIH1938] who generalized the result to a $N$-body system. It was found that
the 1PN correction to the relative acceleration is

$a^{i}_{\rm 1PN}=-\frac{M_{\rm
tot}}{R^{2}}\left\\{\left[(1+3\nu)v^{2}-(4+2\nu)\frac{M_{\rm
tot}}{R}-\frac{3}{2}\nu\dot{R}^{2}\right]\hat{R}^{i}-(4-2\nu)\dot{R}\hskip
1.0ptv^{i}\right\\}\,.$ (7)

From (5), we see that terms in $a^{i}_{\rm 1PN}$ are $v^{2}$-suppressed
relative to the Newtonian acceleration (3). A qualitatively new feature of (7)
is the presence of a force component along $v^{i}$, which perturbs the orbit
from its Keplerian nature. In fact, this is precisely the origin of orbital
precessions in General Relativity, such as that of Mercury around the Sun. It
is also instructive to show the 1PN contributions to the energy and orbital
angular momentum:

$\displaystyle E_{\rm 1PN}$ $\displaystyle=\nu M_{\rm
tot}\left\\{\frac{3}{8}(1-3\nu)v^{4}+\frac{M_{\rm
tot}}{2R}\left[(3+\nu)v^{2}+\nu\dot{R}^{2}+\frac{M_{\rm
tot}}{R}\right]\right\\},$ (8) $\displaystyle L^{i}_{\rm 1PN}$
$\displaystyle=L^{i}_{\rm N}\left[\frac{1}{2}(1-3\nu)v^{2}+(3+\nu)\frac{M_{\rm
tot}}{R}\right].$

Comparing these expressions with their Newtonian counterparts (4), we see that
the mass ratio $\nu$ first explicitly appears in these conserved quantities,
up to an overall factor of $\nu M_{\rm tot}$. As we shall discuss in §2, a
measurement of the 1PN contribution to the phases of gravitational waveforms
would therefore allow us to infer the masses of both binary components. In
addition, we find that $L^{i}_{\rm 1PN}$ is parallel to $L^{i}_{\rm N}$ — a
1PN correction therefore does not change the orientation of the orbital plane.

Figure 2: Summary of known results for the conservative dynamics of point-
particle binary systems in the PN and PM expansions. In particular, results up
to 4PN and 3PM are fully known. Figure adapted from [Bern:2019crd].

Through heroic efforts over the past few decades, higher-order PN corrections
to the conservative dynamics of point-particle binaries were computed via a
myriad of techniques [ADM1962, Damour:1987b, Blanchet:2006zz, Buonanno:1998gg,
nrgr]. Interestingly, it was found that up to 3PN order, the qualitative
conclusions that we drew from the 1PN correction above remain unchanged
[Ohta1973, Jaranowski:1997ky, Blanchet:2000nv, Damour:2001bu, Blanchet:2004ek,
Foffa:2011ub, Schafer:2018kuf, nrgr]. This is despite the presence of various
technical challenges posed at high PN orders, such as the emergence of
ultraviolet (UV) divergences due to the point-particle
approximation.999Logarithms that scale as $\propto\ln R$ generically appear in
these high PN terms, as a result of regularizing the UV divergences.
Nevertheless, as mentioned in footnote 7, $R$ is a gauge-dependent quantity
with ambiguous physical interpretation. Using (the PN-corrected) Kepler’s law,
$R$ can be replaced by the orbital frequency of the binary, which is gauge
invariant. In this case, one can show that these logarithms disappear, see
e.g. [Blanchet:2006zz]. To date, the point-particle dynamics has been
completed up to 4PN order [Damour:2014jta, Damour:2016abl, Marchand:2017pir,
Bernard:2017ktp, Foffa:2019rdf, Foffa:2019yfl]. In this case, a qualitatively
new phenomenon emerges: due to the non-linearities of General Relativity,
gravitational waves that were emitted by the binary in the past can
backscatter onto the orbit in the future. This phenomenon, known as the tail
effect, first appears at 4PN order and contributes to the dynamics in a manner
that is non-local in time [Blanchet:1987wq, Galley:2015kus, Marchand:2017pir,
Foffa:2019yfl]. As a result, logarithmic terms of the form $v^{8}\ln(v)$
appear in the conserved quantities $E_{\rm 4PN}$ and $L^{i}_{\rm 4PN}$. See
[Bernard:2017ktp] for a summary of the explicit expressions of $a^{i},E$ and
$L^{i}$ up to 4PN order.

While the PN expansion is well-suited to describe the conservative dynamics in
the near region, it is no longer valid in the wave region. This is especially
important because wave retardation effects, which contribute to the
dissipative dynamics of the orbit, are missed in the PN series. This
shortcoming is compensated by the PM expansion, which is valid in both the
near zone and the wave zone. A classic treatment of the PM expansion involves
the Landau-Lifshitz form of the Einstein field equation [Landau1980Classical].
In particular, by introducing the inverse gothic (pseudo) metric,
$\textsf{h}^{\mu\nu}\equiv\eta^{\mu\nu}-\sqrt{-g}\hskip 1.0ptg^{\mu\nu}$, (7)
can be written as

$\square\,\textsf{h}^{\mu\nu}=-16\pi\tau^{\mu\nu}\,,\qquad\tau^{\mu\nu}=(-g)\left(T^{\mu\nu}+\Lambda^{\mu\nu}\right)\,,$
(9)

where $\square$ is the d’Alembertian in Minkowski spacetime, $\tau_{\mu\nu}$
is the effective (pseudo) energy-momentum tensor, and $\Lambda_{\mu\nu}$ is a
tensor which consists of $\textsf{h}_{\mu\nu}$ and its derivative starting at
the quadratic order $\sim(\partial\textsf{h})^{2}$ (see [Landau1980Classical]
for its explicit expression). The Landau-Lifshitz form (9) must be
supplemented by the constraint $\partial_{\mu}\tau^{\mu\nu}=0$ to enforce the
Bianchi identity. Because $\Lambda_{\mu\nu}$ only depends on
$\textsf{h}_{\mu\nu}$, but not on $T_{\mu\nu}$, it is sometimes known
heuristically as “the energy-momentum tensor” of gravitational waves, though
we described in §3 that it is ill-defined without proper averaging. Expanding
(9) to 1PM order about a Minkowski background, the contribution from
$\Lambda_{\mu\nu}$ vanishes and $\textsf{h}_{\mu\nu}$ reduces to the trace-
reversed metric $\bar{h}_{\mu\nu}$ introduced in §3 — we have recovered the
linearized Einstein equation (16). Higher-order PM expansions to the
conservative dynamics, to which $\Lambda_{\mu\nu}$ does contribute, have been
pursued over the past few decades [Bertotti:1956pxu, Kerr:1959zlt,
Westpfahl:1979gu, Bel:1981be, Damour:2016gwp]. The state-of-the-art for the
conservative dynamics is recently achieved at 3PM order [Bern:2019nnu,
Bern:2019crd]. As demonstrated in Fig. 2, results in the PM expansion are
complementary to those obtained in the PN series.

The PM expansion is complementary to the PN expansion in the near zone.
However, it is a necessity to describe the wave zone. By matching the
conservative dynamics in the near zone with the gravitational waves
propagating in the wave zone, we can infer the impact of a binary’s evolution
on the observed gravitational waves (see §3 for more details at 1PM order).
Conversely, this matching also allows us to study the backreaction of
radiative effects on the orbit. The leading-order radiative-reaction
acceleration is [Iyer:1995rn, Iyer:1993xi]101010This expression is unique up
to an additional gauge transformation in the radiative potential in the
metric, see e.g. [poisson_will_2014] for a pedagogical discussion. Physical
quantities, such as the orbit-averaged binding energy, are obviously not
affected by such a gauge transformation.

$a^{i}_{\rm RR}=\frac{8\nu}{5}\frac{M_{\rm
tot}^{2}}{R^{3}}\left[\left(3v^{2}+\frac{17}{3}\frac{M_{\rm
tot}}{R}\right)\dot{R}\hat{R}^{i}-\left(v^{2}+\frac{3M_{\rm
tot}}{R}\right)v^{i}\right].$ (10)

The physics of this radiative-reaction force is equivalent to the quadrupolar
emission of gravitational waves (28). Comparing (10) with the Newtonian
acceleration (3), we see that the dominant radiative-reaction force appears at
2.5PN order. Energy and angular momentum of the binary are therefore no longer
conserved beyond this PN order, as they can dissipate through gravitational
waves. In fact, simple dimensional analysis would have allowed us to predict
the order at which dissipation first occurs: applying the quadrupole formula
(28) to a binary system, we find that the power emitted by the binary is

$P=\frac{32}{5}\nu^{2}\left(M_{\rm tot}\Omega\right)^{10/3}\,,$ (11)

where $\Omega$ is the orbital frequency of a circular binary and $v=(M_{\rm
tot}\Omega)^{1/3}$. Since acceleration is related to power through $P=\nu
M_{\rm tot}a^{i}_{\rm RR}v^{i}$, we find $a_{\rm RR}\sim M_{\rm
tot}R^{-2}v^{5}$, which is 2.5PN suppressed compared to the Newtonian
acceleration. Since gravitational-wave dissipation is not symmetric under time
reversal, radiative-reaction forces necessarily appear as odd powers of $v$.
As it stands, relativistic corrections to (10) have been computed up to 3.5PN
order [Pati:2002ux, Nissanke:2004er, Konigsdorffer:2003ue]. The dissipative
contribution of the tail effect, like its conservative counterpart, appears at
4PN order.

##### Spins and finite-size effects

So far, we have approximated the binary components as point-like particles.
However, an astrophysical object generally has non-vanishing spin angular
momentum. Depending on the internal structure of the object, it can also
possess non-trivial quadrupole and higher-order multipole moments.
Furthermore, it can acquire induced multipole moments in the presence of a
gravitational perturbation. I will now describe how these different physical
effects impact the dynamics of the orbit.

The dynamical evolution of a binary system is greatly enriched when its
components have spins. We denote the spin vectors by $S^{i}_{1}\equiv
m_{1}^{2}\chi_{1}^{i}$ and $S^{i}_{2}\equiv m_{2}^{2}\chi_{2}^{i}$, where
$\chi^{i}_{1,2}$ are the dimensionless spin parameters, which are bounded by
$0\leq|\chi^{i}_{1,2}|\leq 1$. Unless both spins are parallel to the orbital
angular momentum of the binary, the mutual gravitational force of the binary
exerts a torque on the spins. This torque is phenomenologically interesting
because it leads to precessions of the spins and the orbital plane. To
understand this concretely, we consider the evolution of the total spin
vector, $S^{i}=S^{i}_{1}+S^{i}_{2}$, which satisfies [Barker75,
HartleThorne1985, Kidder:1992fr, Kidder:1995zr]

$\frac{{\rm d}S^{i}}{{\rm d}t}=\frac{\epsilon^{ijk}}{2R^{3}}L^{j}_{\rm
N}\Big{[}4S^{k}+3\sigma^{k}\Big{]}+\frac{3\hskip
1.0pt\epsilon^{ijk}}{R^{3}}\Big{[}(\hat{R}^{l}S_{2}^{l})(\hat{R}^{j}S_{1}^{k})+(\hat{R}^{l}S_{1}^{l})(\hat{R}^{j}S_{2}^{k})\Big{]}\,,$
(12)

where $\sigma^{i}\equiv(m_{2}/m_{1})S_{1}^{i}+(m_{1}/m_{2})S_{2}^{i}$ is a
mass-weighted spin parameter. The first term in (12) is the spin-orbit
interaction and the second term is the spin-spin interaction. We see that the
total spin generally evolves in a very complicated manner. Nevertheless, the
total angular momentum, $J^{i}=L^{i}+S^{i}$, is conserved—at least up to 2.5PN
order, cf. (10). This implies that the orbital angular momentum satisfies
${\rm d}L^{i}/{\rm d}t=-{\rm d}S^{i}/{\rm d}t$, where both $L^{i}$ and $S^{i}$
precess around the fixed $J^{i}$. To gain better intuition for the precessing
motion, we simplify (12) by taking $S^{i}_{2}=0$, such that the spin-spin
interaction vanishes. In this case, the precession equations are

$\frac{{\rm d}L^{i}}{{\rm d}t}=\Omega_{\rm
p}\,\epsilon^{ijk}\hat{J}^{j}L^{k}\,,\qquad\frac{{\rm d}S_{1}^{i}}{{\rm
d}t}=\Omega_{\rm p}\,\epsilon^{ijk}\hat{J}^{j}S_{1}^{k}\,,$ (13)

where $J^{i}\equiv J\hat{J}^{i}$ and $\Omega_{\rm p}=J(1+3M_{\rm
tot}/m_{1})/2R^{3}$ is the precession frequency. When the total angular
momentum is dominated by the orbital angular momentum, we find that
$\Omega_{\rm p}\sim\Omega\hskip 1.0ptv^{2}<\Omega$ and the precession
timescale is therefore longer than the orbital period. Crucially, the
precession of $L^{i}$ leads to orbital precession of the binary. Since
gravitational waves are not emitted isotropically from a binary, a precessing
orbit generates interesting modulations in the amplitude and phase of the
observed waveform. Spin-spin precession can also occur when both $S^{i}_{1}$
and $S^{i}_{2}$ are non-vanishing. In that case, we infer from dimensional
analysis, $|S^{i}|\sim\chi_{i}v|L^{i}|<|L^{i}|$, that the spin-spin precession
timescale is even longer than that induced by the spin-orbit interaction.

It is instructive to consider how these spin effects contribute to the binding
energy and orbital angular momentum of the binary. For the spin-orbit
interaction, we have111111The explicit expressions for $E_{\rm SO}$ and
$L_{\rm SO}^{i}$ actually depend on the representative world line chosen for
the binary system. This is similar to the ambiguity of the dipole moment of an
object, which vanishes if the object’s representative worldline is chosen to
be that of its center of mass. Here, we present results evaluated along the
world line that tracks the binary’s centre-of-mass, as defined in the local
comoving frame; see e.g. [poisson_will_2014] for a detailed discussion.
[Barker75, HartleThorne1985, Kidder:1992fr, Kidder:1995zr]

$\displaystyle E_{\rm SO}$ $\displaystyle=\frac{1}{R^{3}}L^{i}_{\rm
N}\sigma^{i}\,,$ (14) $\displaystyle L^{i}_{\rm SO}$ $\displaystyle=\nu\hskip
1.0pt\epsilon^{ijk}\epsilon^{klm}\left[\frac{M_{\rm
tot}}{R}\hat{R}^{j}\hat{R}^{l}\Big{(}2m_{1}^{2}\hskip
1.0pt\chi^{m}_{1}+2m_{2}^{2}\hskip
1.0pt\chi^{m}_{2}+\sigma^{m}\Big{)}-\frac{1}{2}v^{j}v^{l}\sigma^{m}\right]\,.$

A direct comparison between (14) and (4) indicates that the spin-orbit
interaction occurs at 1.5PN order. This can be understood intuitively from the
fact that $L^{i}_{\rm N}$ couples to the gravitomagnetic potential of the
Riemann tensor, which scales as $M_{\rm tot}v/R$ relative to the Newtonian
potential. In addition, we find that $L^{i}_{\rm SO}$ is not necessarily
parallel to $L^{i}_{\rm N}$. As the dominant component $L^{i}_{\rm N}$
precesses around $J^{i}$, $L^{i}_{\rm SO}$ can introduce an additional
nutation motion. On the other hand, the spin-spin contribution to the energy
and orbital angular momentum are [Barker75, HartleThorne1985, Kidder:1992fr,
Kidder:1995zr]

$\displaystyle E_{\rm S_{1}S_{2}}$ $\displaystyle=\frac{\nu^{2}M_{\rm
tot}^{4}}{R^{3}}\left[3(\hat{R}^{i}\chi^{i}_{1})(\hat{R}^{i}\chi^{i}_{2})-\left(\chi^{i}_{1}\chi^{i}_{2}\right)\right],$
(15) $\displaystyle L^{i}_{\rm S_{1}S_{2}}$ $\displaystyle=0\,.$

That this is a 2PN effect can be inferred straightforwardly from dimensional
analysis, $|S^{i}|\sim\chi_{i}v|L^{i}|<|L^{i}|$. As it stands, higher-order
corrections to the the spin-orbit interaction have been computed up to 3.5PN
[Faye:2006gx, nrgrss, Bohe:2015ana] and 2PM orders [Bini:2017xzy,
Bini:2018ywr]. On the other hand, the spin-spin interactions are known up to
4PN order [Levi:2016ofk] and 2PM order [Vines:2017hyw, Bern:2020buy] (see
[Porto:2016pyg, Levi:2018nxp] for reviews and further references therein).

In addition to the mass and spin parameters, a general astrophysical object is
also characterized by its multipolar structure [Geroch:1970cd, Hansen:1974zz,
Thorne:1980ru]. For simplicity, we only consider objects that are spherically
symmetric when they are not spinning, such that they only acquire non-trivial
multipole moments when their spins do not vanish.121212This need not be the
case, as a general astrophysical object can inherit higher-order permanent
multipole moments, which are present even when the object is not spinning. In
this case, the dominant moment is given by the axisymmetric spin-induced
quadrupole, $Q$, which is often parameterized through [Poisson:1997ha]

$Q_{a}\equiv-\kappa_{a}\,m_{a}^{3}\chi_{a}^{2}\,,$ (16)

where $a=1,2$, the dimensionless quadrupole parameter, $\kappa$, quantifies
the amount of shape deformation due to the spinning motion. The larger the
(positive) value of $\kappa$, the more oblate the object is. Crucially, the
value of $\kappa$ depends sensitively on the internal structure of the object.
For instance, we saw in (12) that Kerr black holes have $\kappa=1$
[Hansen:1974zz, Thorne:1980ru], while $2\lesssim\kappa\lesssim 10$ for neutron
stars, with the precise value depending on the nuclear equation of state
[Laarakkers:1997hb, Pappas:2012ns]. As we shall see in later sections, more
speculative objects such as superradiant boson clouds and boson stars can have
$\kappa$ as large as $10^{3}$.

In a binary system, the quadrupole of a binary component couples to the tidal
field sourced by its binary companion. This interaction provides an additional
contribution to the binding energy of the orbit [Barker75, Poisson:1997ha]:

$E_{Q}=-\frac{1}{2}\nu M_{\rm tot}\left(\frac{M_{\rm
tot}}{R}\right)^{3}\sum_{a=1}^{2}\left(\frac{m_{a}}{M_{\rm
tot}}\right)^{2}\kappa_{a}\left[3\left(\hat{R}^{i}\chi_{a}^{i}\right)^{2}-(\chi^{i}_{a})^{2}\right]\,.$
(17)

Comparing this with the binding energy from the monopole (4), it is clear that
this quadrupolar interaction is suppressed by $(M_{\rm tot}/R)^{2}$ and
therefore contributes at 2PN order. That a quadrupole moment can also induce
precession effects is well known in Newtonian mechanics, see e.g.
[goldstein:mechanics]. Nevertheless, similar to the spin-spin interaction
above, a quadrupole-induced precession occurs over timescales that are longer
than the spin-orbit precession, and are therefore often neglected. While we
only considered the quadrupole, higher-order multipoles can also contribute to
the binding energy, though their effects are suppressed.

In addition to the spin-induced multipole moments, an astrophysical object can
also develop additional multipole moments when it is perturbed by a
gravitational field. In the case of a binary system, this occurs when the
tidal field sourced by a binary companion, $\mathcal{E}_{ij}$, deforms the
shape of the object. In reponse, the object acquires a tidally-induced
quadrupole moment, $\delta Q_{ij}$. When the tidal perturbation is adiabatic,
the linear response of the object’s moment to the tidal field is parameterized
through the relation [Flanagan:2007ix]

$\delta Q_{ij,a}\equiv-\lambda_{a}\hskip 1.0ptm_{a}^{5}\mathcal{E}_{ij}\,,$
(18)

where $\lambda$ is the object’s (dimensionless) tidal deformability parameter,
also known as the Love number. Remarkably, the Love numbers of a black hole
vanish identically [Chia:2020yla, Binnington:2009bb, Damour:2009vw,
Gurlebeck:2015xpa, Landry:2015zfa, Pani:2015hfa]. This conclusion holds for
arbitrary values of black hole spin, for both the electric-type and magnetic-
type perturbations, and to all orders in the multipole expansion of the tidal
field [Chia:2020yla]. For neutron stars, $\lambda$ depends on the precise
nuclear equation of state — a measurement of this parameter through a phase
binary neutron star waveform is indeed an active area of research, see §2 for
more discussion.

As binding energy is transfered to the binary components in order to deform
their shapes, the orbit shrinks at an accelerating rate. This tidal effect
therefore provides an additional channel to deepen the gravitational potential
between the components. It contributes to $E$ in (1) as follows
[Flanagan:2007ix, Vines:2011ud]:

$E_{\rm T}=-\frac{3}{2}\nu^{2}M_{\rm tot}\left(\frac{M_{\rm
tot}}{R}\right)^{6}\,\sum_{a=1}^{2}\left(\frac{m_{a}}{M_{\rm
tot}}\right)^{3}\lambda_{a}\,.$ (19)

This shows that the tidal deformability affects the orbital evolution at 5PN
order, which is a much higher-order effect than those described above.
However, this does not necessarily mean that this effect is negligible, as
$\lambda$ can be very large. As we shall see in §2, a high-PN order only
implies that it is suppressed in the early inspiral stage of the binary but
can be significant near the binary merger. Higher-order corrections of the
tidal effects on the orbital dynamics have also been computed [Vines:2011ud,
Banihashemi:2018xfb].

##### Extreme mass ratio inspirals

The PN and PM expansions assume that the self-gravity of a binary system is
weak, so that the background spacetime is well-approximated by the Minkowski
metric. However, this is no longer true when one of the binary components is
much heavier than the other. In this case, strong gravitational effects of the
heavier object are important even at leading order. The heavier object is
usually taken to be a Kerr black hole because, in contrast to other
astrophysical objects, it has no maximum mass. The mass ratio of these types
of binary systems can easily be as small as $\nu\sim 10^{-5}$, and they are
therefore commonly called extreme mass ratio inspiral (EMRI) systems. The
smallness of the mass ratio can be leveraged to systematically expand the
evolution equations in $\nu\ll 1$. Crucially, this perturbative expansion is
complementary to the PN and PM expansions, as the results obtained at each
order in $\nu$ necessarily restores partial results in $v$ and $G$ (see Fig. 2
for a similar relationship between the PN and PM series).

In the test particle limit, $\nu\to 0$, the motion of the smaller object
traces a geodesic around the Kerr black hole [Bardeen1972]. However, the
trajectory is no longer a geodesic when the self gravity of the object is non-
negligible. This is because the object’s motion now becomes a source of
gravitational-wave emission, which backreacts on its orbit through a
radiation-reaction force. These finite-$\nu$ corrections can be computed
through the gravitational self-force formalism (see [Barack:2018yvs] for a
non-specialist review on this subject). To highlight some of the qualitative
distinctions between the self-force theory and the PN and PM formalisms, we
generalize the linearized Einstein equation (16) from Minkowski spacetime to
an arbitrary background spacetime $g_{\mu\nu}^{(0)}$:

$\square\,\bar{h}_{\mu\nu}+2R^{(0)}_{\mu\alpha\nu\beta}\bar{h}^{\alpha\beta}=-16\pi
T_{\mu\nu}\,,$ (20)

where $R^{(0)}_{\mu\nu\rho\sigma}$ is the Riemann tensor of the background
metric and $\bar{h}_{\mu\nu}$ is the trace-reversed metric perturbation, but
now defined relative to $g_{\mu\nu}^{(0)}$. Modeling the smaller object as a
point particle that sources $T_{\mu\nu}$ and denoting its coordinates by
$z^{\mu}$, we can derive its equation of motion at first order in perturbation
theory. This is known as the MiSaTaQuWa equation [Mino:1996nk, Quinn:1996am]

$\frac{\text{D}^{2}z^{\mu}}{{\rm
d}\tau^{2}}=-\frac{1}{2}g_{(0)}^{\mu\nu}\left(2h^{\rm
tail}_{\nu\rho\sigma}-h^{\rm
tail}_{\rho\sigma\nu}\right)u^{\rho}u^{\sigma}\,,$ (21)

where $h^{\rm tail}_{\mu\nu\rho}\equiv m\int{\rm
d}\tau\,\nabla_{\rho}G^{+}_{\mu\nu\alpha\beta}u^{\alpha}u^{\beta}$ is the
gravitational tail field generated by object’s radiation reaction,
$u^{\mu}\equiv{\rm d}z^{\mu}/{\rm d}\tau$ is the body’s four-velocity, and
$G^{+}_{\mu\nu\alpha\beta}$ is the retarded Green’s function. The tail
perturbation $h^{\rm tail}_{\mu\nu\rho}$, which depends on the object’s entire
worldline history, is exactly the same as that described in the PN expansion
above. In that case, this effect only contributes at 4PN order. However, we
see in (21) that the tail effect already appears at leading order and is, in
fact, the only source of perturbation. This means the curvature of the
background spacetime $g^{(0)}_{\mu\nu}$ is so strong that it can efficiently
backscatter the gravitational waves emitted by the object’s past to its
future. The self force of an EMRI is therefore a radiation-reaction force that
is dominated by the non-local tail effects.

The derivation of (21) marked only the beginning of the self-force program.
For instance, it was shown that second order in perturbation theory must be
achieved in order to construct template waveforms that are accurate enough to
detect EMRIs through matched-filtering searches [Hinderer:2008dm]. At this
perturbative order, the point-particle approximation leads to unphysical UV
divergences and therefore introduces new technical challenges. This is
certainly reminiscent of the similar problem of the point-particle
approximation in the post-Newtonian expansion at 3PN order. In addition, the
equations involved are coupled partial differential equations that are
challenging to solve numerically. These problems motivated further
developments in the self-force theory [Barack:2018yvs]. At present, the
equation of motion has been pursued up to the second order [Gralla:2012db,
Pound:2012nt, Pound:2017psq, Pound:2019lzj]. An overaching goal of the current
self-force program is to develop numerically efficient methods to evolve the
equation over long timescales, so that the gravitational waveforms emitted by
EMRI systems can be readily obtained.

In addition to the formalism described above, the dynamics of EMRIs can also
be computed using the Teukolsky equation, which computes the linear
perturbations on a fixed background black hole geometry (see e.g.
[Sasaki:2003xr] for a review). Schematically, this approach involves solving
the Teukolsky equation perturbatively in $\nu$, with the smaller object acting
as a source term in the inhomogeneous equation [Teukolsky:1973ha,
Mano:1996vt]. This approach is convenient because the Teukolsky equations are
a set of ordinary differential equations, instead of partial differential
equations, making them more computationally tractable. In addition, since the
Teukolsky equation provides solutions for the gravitational waves radiated
towards infinity, it is most useful for computing quantities such as the
energy flux of the binary system. Indeed, this method has provided powerful
consistency checks and even new results for the PN series in the radiative
sector. For instance, at linear order in $\nu$ , recent endeavours had been
able to compute the energy flux up to a remarkable 22PN order for a
Schwarzschild black hole [Fujita:2012cm] and 11PN order for a Kerr black hole
[Fujita:2014eta]. To derive the self-force correction to the orbit’s
conservative dynamics in this approach, one would have to employ metric
reconstruct methods in the binary’s near zone [Shah:2012gu, Keidl:2010pm]. The
black hole perturbation theory formalism is therefore, in many ways, an
important and complementary method of computing the self-force dynamics of
EMRI systems.

Finally, we remark that the gravitational waves emitted by EMRIs provide a
unique probe of the spacetime geometry of Kerr black holes. Specifically, by
measuring the mass, spin, and higher-order moments of the black hole, we can
directly test the “no hair theorem” for black holes. A violation of this
consistency condition, such as the measurement of a much larger quadrupole
moment, would imply the presence of a more exotic compact object
[Ryan:1995wh].

##### Binary mergers and ringdown

So far, we have focused on the inspiral regime of the binary’s evolution,
where analytic predictions are attainable. Those perturbative schemes,
however, break down in the last few orbits when strong non-linear
gravitational effects dominate. In this case, numerical relativity is
necessary to resolve the detailed dynamics, which include both the merger and
ringdown stages, cf. Fig. 1 for the different parts of a binary coalescence.

The goal in numerical relativity is to solve the ten coupled nonlinear partial
differential equations of the Einstein field equation (7). Among the many
technical challenges involved, the constraints imposed by the Bianchi
identity, $\nabla_{\mu}G^{\mu\nu}=0$, also known as the Hamiltonian and
momentum constraints in the $3+1$ ADM formalism, must be enforced throughout
the binary’s evolution. In addition, choosing the appropriate gauge is
essential in order to guarantee well-posedness of the binary setup, lest
numerical errors due to finite resolution grow quickly in time. After decades
of attempts and technical developments, stable numeric evolutions of binary
black hole mergers were finally achieved in 2005 [Pretorius:2005gq,
Baker:2005vv, Campanelli:2005dd]. Ever since this breakthrough, many similar
simulations were performed over a wider range of parameters and through
different methods, see e.g. [Kidder:2000yq, Loffler:2011ay, Moesta:2013dna,
Brugmann:2008zz, Clough:2015sqa]. Because these simulations are
computationally expensive, they resolve at most the last $100$ orbits—see e.g.
[Jani:2016wkt, Healy:2019jyf, Boyle:2019kee] for a catalogue of binary black
hole simulations. Importantly, these results include both the merger and
ringdown portions of the binary’s evolution. In §2, we describe how these
numerical results are combined with the analytic, inspiral part of the
gravitational waveform, thereby constructing waveforms which incorporate the
full evolution of the binary.

Last but not least, while the quasi-normal ringdown spectra emitted after a
binary merger are natural outputs of numerical relativity simulations, they
can also be computed analytically using black hole perturbation theory
[Teukolsky:1973ha]. Schematically, the ringdown strain can be expressed
parameterically as a linear superposition of damped sinusoids
[Teukolsky:1973ha, Press:1971wr, Chandrasekhar:1975zza]

$h(t)=\sum_{p=0}^{\infty}A_{p}\,e^{-i(\omega_{p}t+\phi_{p})-t/\tau_{p}}\,,$
(22)

where $p$ denotes the overtone number, the quantities
$\\{A_{p},\omega_{p},\phi_{p},\tau_{p}\\}$ are the associated mode amplitude,
frequency, phase, and decay timescale, respectively, and $t$ represents the
time after a reference start time at merger. The lowest overtone, $p=0$, is
the longest-lived mode and has the largest amplitude. Crucially, the ringdown
frequencies and decay timescales only depend on the mass and spin of the final
black hole, and therefore offer a clean probe of the “no hair theorem” of Kerr
black holes described in §2 [Dreyer:2003bv]. On the other hand, the amplitudes
and phases depend on the geometry of the binary before merger, which dictates
the degree to which these modes are excited [Lim:2019xrb].

####  2 Gravitational Waveforms

In §1, I described how various analytic approximations have been adopted to
compute the dynamics of binary systems to high perturbative orders. These are
not pure academic exercises — rather, they are foundational in our efforts to
search for the gravitational waves emitted by binary systems. This is because
gravitational-wave signals are typically weak and buried in noisy data. The
optimal way of recovering these signals is to matched filter the data with
very precise template waveforms [Thorne1980Lectures, Dhurandhar:1992mw,
Cutler:1992tc], which can only be constructed through a detailed understanding
of the dynamics at play. As we shall see, a minimum accuracy of 3.5PN order in
the phase of template waveforms is necessary to successfully perform matched
filtering. My primary goal here is to sketch the methods used in constructing
template waveforms. I will also demonstrate how gravitational waveforms are
rich sources of information about the physics of the binary systems.

I first focus on the inspiral stage of the binary’s evolution. For simplicity,
I restrict myself to binaries in quasi-circular orbits. This is often a
reasonable simplification because gravitational-wave emissions can efficiently
damp away orbital eccentricities [PeterMatthews1963, Peter1964]. I also assume
that the orbit evolves adiabatically, i.e. its tangential velocity dominates
over the speed at which the orbital radius shrinks. This is an excellent
approximation for the binary’s inspiral stage and stops being valid only near
its merger. In this case, Kepler’s law implies that the velocity of the orbit
is $v=(M_{\rm tot}\Omega)^{1/3}$ and the quadrupole formula for a binary
system (11) is

$P=\frac{32}{5}\nu^{2}v^{10}\,.$ (23)

In order to construct waveforms up to a given PN order, $P$ must also be
computed up to the same order (see discussion later). Higher-order radiative
multipoles of the binary, such as its mass octupole and current quadrupole,
must then be taken into account, cf. §3 for a related discussion. In addition,
radiative tail effects, which begin to contribute to $P$ at 1.5PN, must be
included.131313The PN counting can sometimes be confusing and warrants further
clarification. For instance, we mentioned in §1 that the tail effect first
appears at 4PN order in the binary’s acceleration. In the dissipative sector,
however, the PN counting has to be compared with the leading-order radiation-
reaction acceleration, which appears at 2.5PN order. Since the latter
contributes to the dominant energy flux (23), the dissipative tail effect
therefore appears as a 1.5PN correction in $P$. At present, the point-particle
energy flux is nearly completed up to 4PN order [Blanchet:1995fg,
Blanchet:2001aw, Marchand:2020fpt], while corrections from spin effects have
been achieved up to various degrees of precision, see [Porto:2016pyg,
Levi:2018nxp] for reviews.

Since $E$ is derived in the near zone, while $P$ is obtained in the wave zone,
cf. §1, we match the physics of these regions through the energy balance
argument

$P=-\left[\hskip 1.0pt\frac{{\rm d}E}{{\rm d}t}\hskip 1.0pt\right]_{\rm
ret}\,.$ (24)

Solving (24) allows us to determine the evolution of an orbit and its
corresponding gravitational waveform. More precisely, the binary separation
varies as $[\hskip 1.0pt\dot{R}\hskip 1.0pt]_{\rm ret}=-P/({\rm d}E/{\rm
d}R)$, while the phase of the gravitational waveform evolves as $[\hskip
1.0pt\dot{\phi}_{\rm gw}\hskip 1.0pt]_{\rm ret}=2\pi f=2v^{3}/M_{\rm
tot}$.141414Here, we used the fact that for the dominant quadrupolar emission
(23) generates waves with a frequency that is twice the orbital frequency of
the binary, $f=2f_{\rm orb}$, with $\Omega=2\pi f_{\rm orb}$ and
$v=(M\Omega)^{1/3}$. However, higher-order radiative multipoles in $P$ can
generate waves with harmonics of $f_{\rm orb}$, i.e. $f=kf_{\rm orb}$ with
$k\geq 1$, though their amplitudes are suppressed [Blanchet:1996pi]. By
ignoring these higher harmonics, we are implicitly working with a restricted
waveform.  Using Kepler’s law, these evolution equations can be written as
differential equations in $[\hskip 1.0pt\dot{v}\hskip 1.0pt]_{\rm ret}$ and
$[\hskip 1.0pt\dot{\phi}_{\rm gw}\hskip 1.0pt]_{\rm ret}$, whose solutions are

$\displaystyle t_{r}(v)$ $\displaystyle=t_{c}+\int^{v_{c}}_{v}{\rm
d}v^{\prime}\,\frac{{\rm d}E(v^{\prime})/{\rm d}v^{\prime}}{P(v^{\prime})}\,,$
(25) $\displaystyle\phi_{\rm gw}(v)$ $\displaystyle=\phi_{c}+\frac{2}{M_{\rm
tot}}\int^{v_{c}}_{v}{\rm d}v^{\prime}v^{\prime 3}\,\frac{{\rm
d}E(v^{\prime})/{\rm d}v^{\prime}}{P(v^{\prime})}\,,$

where $\phi_{c}$ and $t_{c}$ represent the phase and the retarded time at a
reference velocity $v_{c}$. The time-domain gravitational waveforms as
observed by detectors at time $t$ are
$h(t)=h_{+}(t)F_{+}+h_{\times}(t)F_{\times}$, where $F_{+,\times}$ are the
detector’s antenna-pattern functions. The polarizations of the gravitational
waves schematically read

$\displaystyle h_{+}(t)=\mathcal{A}_{+}(t_{r})\cos\left[\phi_{\rm
gw}(t_{r})\right]\,,\qquad
h_{\times}=\mathcal{A}_{\times}(t_{r})\sin\left[\phi_{\rm
gw}(t_{r})\right]\,,$ (26)

where recall that $t_{r}\equiv t-r$, and $\mathcal{A}_{+,\times}$ are the
strain amplitudes — cf. (27) for the relationship between $\mathcal{A}$ at
leading order and the acceleration of the quadrupole moment. Higher PN
corrections to $\mathcal{A}$, which also generate higher harmonics of $f$ (see
footnote 14), can be found in e.g. [Blanchet:1996pi].

In principle, the solutions in (25) can be obtained straightforwardly by
substituting $E$ and $P$, each expanded to the required PN order, into the
integrands. However, since the PN expansion is only an asymptotic series,
there is an ambiguity in the way in which the ratio $({\rm d}E/{\rm d}v)/P$
can be treated. For instance, do we keep this ratio unexpanded, such that the
integrand is given by a ratio of polynomials in $v$? Or, do we perform a
Taylor expansion to the required PN order, with the denominator eliminated?
Unfortunately, there is no clear resolution to this problem. This has led to
multiple treatments of this ratio, thereby resulting in different PN waveform
approximants, such as the Taylor T1, Taylor T2, Taylor T3, Taylor T4, and
Taylor Et models; see e.g. [Damour:2000zb, Buonanno:2009zt] for a summary of
their differences. Resummation methods have also been proposed to achieve
better convergence [Damour:1997ub, Damour:2007xr]. In any case, it was found
that these waveform approximants are in good agreement for low-mass binary
system, though appreciable deviations can occur for large-mass binaries
[Buonanno:2009zt].

Since gravitational-wave data analysis is most conveniently performed in the
frequency domain, it is desirable to Fourier transform (26). This is not as
straightforward as it may seem, because the signal only lasts for a finite
period of time $-\infty<t<t_{c}+r$, while a Fourier transformation requires
the integral domain to be $-\infty<t<\infty$. A common approach is to use the
stationary phase approximation [Thorne1980Lectures, Cutler:1994ys], whereby
the integral is evaluated at the value of $t$ that dominates the Fourier
integral. More precisely, this is the time that minimizes the phase of the
integrand, which oscillates quickly otherwise and therefore provides
suppressed contributions. In this approximation, the strain amplitude in the
frequency domain is schematically

$\tilde{h}(f)=\mathcal{A}(f)f^{-7/6}e^{i\psi(f)}\,,$ (27)

where we extracted an overall factor of $f^{-7/6}$ for convenience. At leading
Newtonian order, the amplitude $\mathcal{A}$ is then independent of $f$ and
reads

$\mathcal{A}_{+,\times}=\frac{1}{\pi^{2/3}}\left(\frac{5}{24}\right)^{1/2}\frac{\mathcal{M}^{5/6}}{r}g_{+,\times}(\iota)\,,$
(28)

where $\mathcal{M}\equiv\nu^{3/5}M_{\rm tot}$ is the chirp mass of the binary,
$r$ is the (luminosity) distance between the source and the detector, and the
functions $g_{+}=(1+\cos^{2}\iota)/2$ and $g_{\times}=\cos\iota$ describe the
angular distributions of the gravitational-wave polarizations, with $\iota$
being the inclination angle between the orbit’s (Newtonian) orbital angular
momentum vector and the detector’s line of sight. Since higher-order
corrections to $\mathcal{A}$ do not substantially affect matched filtering,
they will be neglected for simplicity. On the other hand, the phase $\psi$ in
the stationary-phase approximation is

$\psi_{+}(f)=2\pi ft_{c}-\phi_{\rm gw}(f)-\frac{\pi}{4}\,,$ (29)

where $\phi_{\rm gw}$ is given by (25) and $\psi_{\times}=\psi_{+}+\pi/2$.
Unlike in $\mathcal{A}$, high PN terms must be retained in $\psi$ because
match-filtering is extremely sensitive to the phase coherence between
waveforms. A commonly-adopted frequency-domain waveform is the TaylorF2
waveform, where higher-order PN terms in the denominator of $\phi_{\rm gw}$
are Taylor expanded, so that the $v$-integral can performed analytically, see
e.g. [Arun:2008kb]. The phase then is

$\displaystyle\psi^{\rm(F2)}_{+}(f)=$ $\displaystyle\,\,2\pi
ft_{c}-\phi_{c}+\frac{3}{128\hskip 1.0pt\nu\hskip 1.0ptv^{5}}\Big{[}\psi_{\rm
PP}+\psi_{\rm S}+\psi_{\rm T}\Big{]}-\frac{\pi}{4}\,,$ (30)

where $\psi_{\rm PP},\psi_{\rm S}$ and $\psi_{\rm T}$ are the point-particle,
spin-dependent, and the tidal contributions to the phase, respectively. Up to
3.5PN order, $\psi_{\rm PP}$ is [Arun:2004hn]

$\displaystyle\hskip-5.0pt\psi_{\rm PP}=$ $\displaystyle
1+\left(\frac{3715}{756}+\frac{55}{9}\nu\right)v^{2}-16\pi
v^{3}+\bigg{(}\frac{15293365}{508032}+\frac{27145}{504}\nu+\frac{3085}{72}\nu^{2}\bigg{)}v^{4}$
(31)
$\displaystyle+\pi\left(\frac{38645}{756}-\frac{65}{9}\nu\right)\Big{[}1+3\log(v)\Big{]}v^{5}+\bigg{[}\bigg{(}\frac{11583231236531}{4694215680}-\frac{640}{3}\pi^{2}$
$\displaystyle-\frac{6848}{21}\gamma_{E}\bigg{)}+\left(\frac{2255\pi^{2}}{12}-\frac{15737765635}{3048192}\right)\nu+\frac{76055}{1728}\nu^{2}-\frac{127825}{1296}\nu^{3}$
$\displaystyle-\frac{6848}{21}\log(4v)\bigg{]}v^{6}+\pi\left(\frac{77096675}{254016}+\frac{378515}{1512}\nu-\frac{74045}{756}\nu^{2}\right)v^{7}\,,\hskip-10.0pt$

where $\gamma_{E}$ is the Euler-Mascheroni constant. We see that the leading-
order $v$-dependence in $\psi$ is $v^{-5}$, which formally diverges as $v\to
0$. Since matched filtering is only successful if the error in the waveform’s
phase is $\lesssim\mathcal{O}(1)$ radians, simple power counting suggests that
terms up to 2.5PN order in $\psi$ must be retained. Nevertheless, since the
coefficients of powers of $v$ can exceed order unity, as clearly shown in
(31), a 2.5PN-accurate phase remains insufficient. Early estimates have
instead shown that a minimum of 3.5PN accuracy is required for successful
matched-filtering searches [Cutler:1992tc].

To extract the parameters of the binary, notice that the leading
$v$-dependence in (30) is proportional to the chirp mass of the binary, $(\nu
v^{5})^{-1}\propto\left(\mathcal{M}f\right)^{-5/3}$. A simultaneous
measurement of $\psi$ and $\mathcal{A}$ would therefore allow us to infer the
luminosity distance $r$, up to uncertainties in the orbit’s inclination angle
$\iota$. Moreover, a 1PN measurement of $\psi$ would allow us to determine
$\nu$, thereby breaking the degeneracy between the component masses in
$\mathcal{M}$. Interesting, we find that odd powers and logarithms of $v$ are
present in (31), despite their absence in the point-particle conservative
dynamics up to 3.5PN order, cf. §1. These contributions arise from higher-
order corrections to $P$, such as the dissipative tail effect.

For binary systems with spinning components, precession effects lead to
modulations in the amplitude and the phase of the waveforms
[Apostolatos:1994mx, Apostolatos:1995pj, Buonanno:2002fy]. For simplicity, we
assume that the objects’ spins are parallel to the (Newtonian) orbital angular
momentum of the binary, i.e. it is either aligned or anti-aligned, such that
precession effects may be ignored. In this case, the spin-dependent phase can
be decomposed as $\psi_{\rm S}=\psi_{\rm SO}+\psi_{\rm S_{1}S_{2}}+\psi_{\rm
S^{2}}+\psi_{\rm Q}$, which are the spin-orbit, spin-spin, self-spin and spin-
induced quadrupolar contributions. At their respective leading orders, these
terms are [Poisson:1995ef, Poisson:1997ha, Mikoczi:2005dn, Arun:2008kb]

$\displaystyle\psi_{\rm SO}$
$\displaystyle=\sum_{a=1}^{2}\left[\frac{113}{12}\left(\frac{m_{a}}{M_{\rm
tot}}\right)^{2}+\frac{25}{4}\nu\right]\chi^{i}_{a}\hat{L}^{i}_{\rm
N}\,v^{3}\,,$ (32) $\displaystyle\psi_{\rm S_{1}S_{2}}$
$\displaystyle=\nu\left[-\frac{3605}{24}(\chi_{1}^{i}\hat{L}_{\rm
N}^{i})(\chi_{2}^{i}\hat{L}_{\rm
N}^{i})+\frac{1235}{24}\chi_{1}^{i}\chi_{2}^{i}\right]v^{4}\,,$
$\displaystyle\psi_{\rm S^{2}}$
$\displaystyle=-\frac{5}{48}\sum_{a=1}^{2}\left(\frac{m_{a}}{M_{\rm
tot}}\right)^{2}\left[7\chi^{2}_{a}-\left(\chi_{a}^{i}\hat{L}_{\rm
N}^{i}\right)^{2}\right]v^{4}\,,$ $\displaystyle\psi_{\rm Q}$
$\displaystyle=-50\sum_{a=1}^{2}\,\left(\frac{m_{a}}{M_{\rm
tot}}\right)^{2}\kappa_{a}\hskip 1.0pt\chi_{a}^{2}\,v^{4}\,.$

where $\hat{L}_{\rm N}^{i}$ is the unit vector of the orbital angular
momentum, $L^{i}_{\rm N}=L_{\rm N}\hat{L}^{i}_{\rm N}$. We see that the spin-
orbit interaction first appears at 1.5PN order, while the spin-spin and spin-
induced quadrupole terms first appear at 2PN order, which are expected from
their effects on the conservative dynamics, cf. §1. Crucially, there exists a
new self-spin term $\psi_{\rm S^{2}}$ at 2PN order, which arises from the
current-quadrupolar contribution to $P$ in the dissipative sector
[Mikoczi:2005dn].151515The spin-spin interaction that contributes to the phase
is often denoted by spin(1)-spin(2), in order to distinguish it from these
self-spinning terms, which are instead labeled by spin(1)-spin(1) and
spin(2)-spin(2). Higher-order corrections to these different effects can be
found in [Mishra:2016whh, Krishnendu:2017shb]. Through precise measurements of
the phases of waveforms, we can therefore determine the spins and quadrupole
moments of the binary components.

The microphysical properties of non-spinning binary components can still be
probed through their tidal deformabilities. In this case, the leading-order
phase imprint is [Flanagan:2007ix]

$\psi_{\rm T}=-24\sum_{a=1}^{2}\left(\frac{m_{a}}{M_{\rm
tot}}\right)^{4}\left(1+11\nu\frac{M_{\rm
tot}}{m_{a}}\right)\lambda_{a}\,v^{10}\,.$ (33)

While this 5PN effect may naively seem negligible, it is only suppressed in
the early inspiral regime of the binary. Specifically, we saw in (30) that the
leading $v$-dependence scales as $v^{-5}$. Terms that appear at $\gtrsim
2.5$PN can therefore become appreciable when $v$ approaches unity, before the
adiabatic approximation breaks down. The dephasing (33) can therefore be
appreciable near the merger regime of the binary.

The analytic waveform described above only applies to the binary’s inspiral
stage, where $v$ is small and the adiabatic approximation holds. However, as
the binary approaches merger, the adiabatic approximation breaks down and
numerical relativity becomes necessary. Since it is important to construct
template waveforms that are valid throughout the inspiral, merger, and
ringdown stages, certain methods must be introduced in order to match the
analytic and numeric waveforms. One of the most commonly adopted methods
involves constructing phenomenological waveforms [Husa:2015iqa, Khan:2015jqa],
whereby the Fourier-domain analytic waveform is matched with numerical
relativity waveforms in an overlapping region. The matching in this
overlapping region is performed by introducing phenomenological terms and
finding the best-fitting parameters. An alternative approach is the effective-
one-body (EOB) method, which maps the conservative PN dynamics of a binary
system to an effective system that involves a test particle orbiting a
deformed Kerr metric [Buonanno:1998gg, Buonanno:2000ef]. In this case,
numerical relativity results are used to calibrate certain free coefficients
that exist in the associated effective Hamiltonian, see e.g.
[Taracchini:2013rva]. Both the phenomenological and EOB waveforms have been
employed extensively in existing gravitational-wave searches. A major goal in
waveform modeling is to refine these models in order to increase their
accuracy and include additional physics such as spins and finite-size effects,
e.g. [Pratten:2020ceb, Thompson:2020nei, Matas:2020wab, Ossokine:2020kjp].

###  3 Gravitational-Wave Observatories

In the previous sections, I reviewed theoretical aspects of black holes,
gravitational waves, and the relativistic dynamics of binary systems.
Remarkably, these predictions of General Relativity were all confirmed by the
LIGO observatories in a binary black hole merger event in September 2015
[Abbott:2016blz] (see Fig. 3 for the signal waveforms). This discovery was
followed by multiple detections of binary coalescences, including the historic
discovery of a binary neutron star merger [TheLIGOScientific:2017qsa]. Here I
review the status of current detectors and highlight some of their key
scientific achievements to this date. In addition, I provide a summary of
prospective future gravitational-wave detectors.

####  1 Current Detectors

The currently operating gravitational-wave detectors include the two Advanced
LIGO detectors in Hanford and Livingston, United States
[TheLIGOScientific:2014jea], as well as the Advanced Virgo detector in Pisa,
Italy [TheVirgo:2014hva]. These observatories are L-shaped Michelson
interferometers whose laser arms are $4\hskip 1.0pt$km and $3\hskip 1.0pt$km
long, respectively. With advanced engineering designs, such as Fabry-Perot
cavities in the detector arms, LIGO and Virgo are able to measure differential
changes in arm lengths to a precision of
$\mathcal{O}(10^{-23})\,\text{Hz}^{-1/2}$ in their most sensitive frequency
range, approximate at $30\hskip 1.0pt\text{Hz}-2\hskip 1.0pt\text{kHz}$. A
representative noise curve of Advanced LIGO at design sensitivity is shown in
Fig. 4. After the first detection of a binary black hole merger by LIGO in the
first observing run (O1), the Virgo detector joined the subsequent runs O2 and
O3. The addition of Virgo in this global network of observatories has allowed
for significant improvements in the sky localization and signal detectability
of gravitational-wave sources.

Figure 3: The gravitational waveforms of GW150914, the first binary black hole
merger event observed by LIGO. The top row demonstrates the strain data in the
LIGO detectors, while the second row illustrates the best-fitting template
waveform for the signal. Figure adapted from [Abbott:2016blz].

The scientific achievements of LIGO and Virgo thus far are hard to overstate.
To begin, the excellent agreement between the signal and template waveforms of
binary black holes is yet another confirmation of Einstein’s theory of General
Relativity as the foundational description of gravity, even in the strongly
dynamical regime. These observations have also provided a unique probe of
various properties of black holes. For instance, by observing the quasi-normal
modes in the ringdown stage of merged black holes, we were able to infer the
nature of the black hole horizons [Dreyer:2003bv, Meidam:2014jpa, Isi:2019aib,
Cardoso:2019rvt]. The observed binary black hole signals have also revealed
previously unknown populations of astrophysical black holes, as the masses of
these black holes lie beyond the expected mass range of those found in X-ray
binaries [Remillard:2006fc]. A catalogue of binary black hole signals that
were observed in O1 and O2, including both highly significant and marginal
events, was published by the LIGO/Virgo collaboration
[LIGOScientific:2018mvr]. See also [Venumadhav:2019lyq, Nitz:2019hdf] for
additional binary events reported by independent research groups.

The observation of the first binary neutron star coalescence
[TheLIGOScientific:2017qsa] has also brought about new scientific
opportunities. Through a precise measurement of the tidal deformability
parameter in the phase of the waveform, cf. §2, we were able probe the nature
of nuclear matter in the high-density regime [Abbott:2018exr]. Furthermore,
the electromagnetic emissions from the neutron star merger, which accompany
the gravitational-wave radiations in the earlier inspiral regime, offered an
unprecedented opportunity for multi-messenger astronomy. For example, by
comparing the relative propagation speeds between the gravitational waves and
light in these events, significant constraints were put on various modified
theories of gravity [Creminelli:2017sry, Ezquiaga:2017ekz, Baker:2017hug]. In
addition, the electromagnetic spectrum of merged neutron stars provided strong
evidence of their roles in manufacturing Nature’s heavy elements, such as gold
and platinum [Kasen:2017aa, Pian:2017aa]. Moreover, the electromagnetic
counterpart provided a ‘standard siren’ that allowed for a distance
calibration to the source, thereby offering an independent measurement of the
local Hubble parameter [Schutz:1986aa].

At the time of writing, the LIGO/Virgo O3 run was just completed. In addition
to finding a new binary neutron star [Abbott:2020uma], the collaboration