frame coefficients change smoothly, and from the explicit expression (11) we
see that this transition happens on a timescale set by

$\Delta t\sim\frac{2\eta}{\gamma}\,.$ (14)

Intuitively, there is a finite “resonance band” of width $\Delta\Omega\sim
2\eta$ in which this transition is active, and (14) is the time spent moving
through it.

A non-adiabatic transition, with $z\lesssim 1$, is qualitatively different
(see the right panel of Fig. 6). Before passing through the resonance (i.e.
before the “scattering event”), the system remains in an instantaneous
eigenstate and the Schrödinger frame coefficients change smoothly. During the
transition, however, the system will partly evolve into the other
instantaneous eigenstate, not dissimilar from particle production via high-
energy scattering. After this event, the system exists in a linear combination
of eigenstates and they oscillate among one another at a frequency set by the
energy difference $E_{+}-E_{-}\sim\gamma t$. These oscillations eventually
decay away on a timescale again set by (14), so that the Schrödinger (dressed)
frame coefficients have a well-defined limit as $t\to\infty$. Finally, if the
transition is extremely non-adiabatic, $z\ll 1$, the system has no time to
respond to the changing Hamiltonian, and so its state is unaffected by the
transition.

####  2 Multi-State Transitions

A qualitatively new feature appears when multiple, nearly degenerate states
are involved in a transition. To illustrate this, we study a three-state
extension of (8), described by the following dressed frame Hamiltonian

$\mathcal{H}_{\mathrm{D}}(t)=\frac{\gamma t}{2}\begin{pmatrix}1&0&0\\\
0&{\scalebox{0.75}[1.0]{$-$}}1&0\\\
0&0&{\scalebox{0.75}[1.0]{$-$}}1\end{pmatrix}+\begin{pmatrix}0&\eta_{12}&\eta_{13}\\\
\eta_{12}&0&\eta_{23}\\\ \eta_{13}&\eta_{23}&0\end{pmatrix}.$ (15)

As I described in §3, this type of multi-state transition will be relevant for
vector clouds, as they tend to have multiple states that are nearly degenerate
in energy and connected via the gravitational perturbation. Indeed, (15)
describes any transition which involves a single state interacting with two
other states that have the same azimuthal angular momentum and unperturbed
energy. However, our focus on the three-state system is primarily pedagogical,
as vector cloud transitions tend to involve four or more states. Fortunately,
we will find that the main features of these more complicated multi-state
transitions can be explained using the simpler three-state model.

Figure 7:  Evolution of the instantaneous eigenstates of a coupled three-state
system.

It is again helpful to plot the instantaneous energy eigenvalues as a function
of time, shown in Fig. 7. At late times, we see that there are two
instantaneous energy eigenstates, $|E_{1}(t)\rangle$ and $|E_{2}(t)\rangle$,
whose energy difference approaches a _constant_ as $t\to+\infty$. This is
significant because, if the motion is non-adiabatic enough to excite the
system into a combination of these two instantaneous eigenstates, there can be
large, coherent oscillations with a _fixed_ frequency set by the interaction
strength $\eta_{23}$. Such oscillations are indeed seen in the right panel of
Fig. 8. This is impossible in a two-state transition (8), as the difference in
energies necessarily diverges as $t\to+\infty$, and any oscillatory behavior
induced by non-adiabaticity rapidly increases in frequency and decays away, as
seen in the right panel of Fig. 6.

Let us understand this transition more quantitatively. At very early times,
the first term in (15) again dominates. However, there is now a degenerate
subspace that is only lifted by $\eta_{23}$ and we cannot ignore it. It will
thus be convenient to explicitly diagonalize this subspace. I therefore
introduce a new basis of states $|\tilde{a}\rangle$, which is related to the
dressed frame basis $|a\rangle$ by

$|\tilde{1}\rangle=|1\rangle\,,\quad|\tilde{2}\rangle=\frac{1}{\sqrt{2}}\left(-|2\rangle+|3\rangle\right),\quad\text{and}\quad|\tilde{3}\rangle=\frac{1}{\sqrt{2}}\left(|2\rangle+|3\rangle\right).$
(16)

In this basis, the dressed frame Hamiltonian becomes

$\tilde{\mathcal{H}}_{\mathrm{D}}=\begin{pmatrix}\gamma
t&\tilde{\eta}_{12}&\tilde{\eta}_{13}\\\ \tilde{\eta}_{12}&-\gamma
t-\eta_{23}&0\\\ \tilde{\eta}_{13}&0&-\gamma t+\eta_{23}\end{pmatrix},$ (17)

where we have defined the effective couplings

$\tilde{\eta}_{12}=\frac{1}{\sqrt{2}}(\eta_{12}-\eta_{13})\qquad\text{and}\qquad\tilde{\eta}_{13}=\frac{1}{\sqrt{2}}(\eta_{12}+\eta_{13})\,.$
(18)

We see that the role of the coupling $\eta_{23}$ is to break the degeneracy
between $|\tilde{2}\rangle$ and $|\tilde{3}\rangle$, and force
$|\tilde{1}\rangle$ to become nearly degenerate with $|\tilde{2}\rangle$ at a
different time than it becomes degenerate with $|\tilde{3}\rangle$. As we will
argue in §3, we can thus treat this three-state transition as a combination of
two-state transitions.

In the asymptotic past, the instantaneous energy eigenstates are then

$\hskip-5.0pt|E_{1}({\scalebox{0.75}[1.0]{$-$}}\infty)\rangle=\left(\begin{array}[]{@{\mkern
3mu} r @{\mkern 3mu}}.0mu\lx@intercol\hfil 1\mkern 3.0\\\
.0mu\lx@intercol\hfil 0\mkern 3.0\\\ .0mu\lx@intercol\hfil 0\mkern
3.0\end{array}\right),\hskip
4.0pt|E_{2}({\scalebox{0.75}[1.0]{$-$}}\infty)\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{@{\mkern-2mu}
r @{\mkern 3mu}}.0mu\lx@intercol\hfil 0\mkern 3.0\\\
.0mu\lx@intercol\hfil-1\mkern 3.0\\\ .0mu\lx@intercol\hfil 1\mkern
3.0\end{array}\right),\hskip
4.0pt|E_{3}({\scalebox{0.75}[1.0]{$-$}}\infty)\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{@{\mkern
3mu} r @{\mkern 3mu}}.0mu\lx@intercol\hfil 0\mkern 3.0\\\
.0mu\lx@intercol\hfil 1\mkern 3.0\\\ .0mu\lx@intercol\hfil 1\mkern
3.0\end{array}\right),$ (19)

and from Fig. 7 we see that they permute in the asymptotic future,

$\hskip-5.0pt|E_{1}(+\infty)\rangle\propto|E_{2}({\scalebox{0.75}[1.0]{$-$}}\infty)\rangle,\hskip
4.0pt|E_{2}(+\infty)\rangle\propto|E_{3}(-\infty)\rangle,\hskip
4.0pt|E_{3}(+\infty)\rangle\propto|E_{1}({\scalebox{0.75}[1.0]{$-$}}\infty)\rangle\,.$
(20)

Such a cyclic permutation is present in any transition which involves one
state going into many degenerate states, as is visually apparent from Fig. 7.

Figure 8:  Adiabatic (_left_) and non-adiabatic (_right_) evolution of the
three-state system. Specifically, the adiabatic and non-adiabatic evolution
are simulated with $(\eta_{12},\eta_{13},\eta_{23})=(2.5,1.5,0.25)$ and
$(\eta_{12},\eta_{13},\eta_{23})=(4.75,0.75,3.5)$, respectively, in units of
$\sqrt{\gamma}$. The final state in the right panel oscillates with a
frequency of about $2\eta_{23}$.

Where do the large, coherent oscillations in Fig. 8 come from? Let us focus on
the transition mediated by $\tilde{\eta}_{12}$, with associated LZ parameter
$\tilde{z}_{12}\equiv\tilde{\eta}_{12}^{2}/\gamma$. Note that, even if all of
the original LZ parameters are very large, $z_{ab}=\eta_{ab}^{2}/\gamma\gg 1$,
the transition may still be non-adiabatic if there is a cancellation between
$\eta_{12}$ and $\eta_{13}$ in (18), such that $\tilde{z}_{12}\lesssim 1$. I
assume that the transition is fast enough ($\tilde{z}_{12}\sim 1$) to excite
$|E_{2}(t)\rangle$, but slow enough ($\tilde{z}_{13}\gg 1$) that we can ignore
$|E_{3}(t)\rangle$. If the initial state is
$|\psi({\scalebox{0.75}[1.0]{$-$}}\infty)\rangle\propto|E_{1}({\scalebox{0.75}[1.0]{$-$}}\infty)\rangle$,
then at late times it becomes

$\displaystyle|\psi(t)\rangle\xrightarrow{\ t\to\infty\ }$ $\displaystyle
e^{-i\gamma
t^{2}/2-i\eta_{23}t}\sqrt{1-e^{-\pi\tilde{z}_{12}}}\,|E_{1}(\infty)\rangle$
(21) $\displaystyle+e^{-i\gamma
t^{2}/2+i\eta_{23}t+i\delta}e^{-\pi\tilde{z}_{12}/2}\,|E_{2}(\infty)\rangle\,,$

where the phase difference $\delta$ depends on $\gamma$ and $\eta_{ab}$. We
thus see that, in the asymptotic future, there are oscillations in the
Schrödinger frame populations with frequency $2\eta_{23}$. For example, we
have

$|c_{2}(t)|^{2}=\frac{1}{2}-A_{12}\cos(2\eta_{23}t+\delta)\,,$ (22)

where $A_{12}=e^{-\pi\tilde{z}_{12}/2}\sqrt{1-e^{-\pi\tilde{z}_{12}}}$. This
amplitude achieves its maximum $A_{12}=\tfrac{1}{2}$ when
$\tilde{z}_{12}=\tfrac{1}{\pi}\log 2$, and we see that these oscillations
disappear both when the transition is either very adiabatic, so that the state
$|E_{2}(t)\rangle$ is never excited, or very non-adiabatic, so that the state
$|E_{1}(t)\rangle$ does not survive.

These slow oscillations will persist even if the state $|E_{3}(t)\rangle$ is
excited by the transition, i.e. when $\tilde{z}_{12}\sim\tilde{z}_{13}\sim 1$.
In this case, there will then be transient oscillations, like those seen in
the two-state system and in the right panel of Fig. 6, which will eventually
decay and leave only these fixed frequency oscillations. However, if the
transition is extremely non-adiabatic, such that both $\tilde{z}_{12}$ and
$\tilde{z}_{13}\ll 1$, the system will always almost entirely occupy the state
$|E_{3}(+\infty)\rangle=|E_{1}({\scalebox{0.75}[1.0]{$-$}}\infty)\rangle$ and
the system will evolve very little in time.

The adiabatic and (mildly) non-adiabatic evolution of this three-state system
are shown in the left and right panels of Fig. 8, respectively. The slow
oscillations created by a somewhat non-adiabatic transition are seen on the
right. On the left, the completely adiabatic transition creates a final state
that is a linear superposition of Schrödinger frame states, i.e. energy
eigenstates. It is a special feature of this three-state system that the
relative final state populations are independent of the coupling $\eta_{23}$.
Indeed, if four or more states are involved in the transition, the final state
populations do depend on the couplings $\eta_{ab}$, since the asymptotic
energy eigenstates clearly depend on how the dressed frame Hamiltonian’s
degenerate subspace is lifted.

We find that multi-state transitions generically yield final states that are
superpositions of the dressed basis states (and thus the energy eigenstates).
Almost all of the transitions that we will analyze in Section 4 will be
adiabatic, so this superposition can be easily approximated by an eigenvector
of the coupling matrix $\eta_{ab}$, restricted to the degenerate subspace. As
we mentioned at the outset of this section, these multi-state transitions
qualitatively distinguish scalar and vector clouds, and we will exploit this
to infer the spin of the boson from gravitational-wave observations in §3 and
§3.

####  3 An S-Matrix Approach

In §1, we found that a two-state system initially occupying the lowest
instantaneous energy eigenstate
$|\psi({\scalebox{0.75}[1.0]{$-$}}\infty)\rangle\propto|E_{-}({\scalebox{0.75}[1.0]{$-$}}\infty)\rangle$
could transition into the other instantaneous eigenstate with an occupation
“probability” (12) with a well-defined asymptotic limit. Given the system’s
causal structure—the nature of the transition provides a well-defined notion
of both asymptotic past and asymptotic future—it is tempting to interpret (12)
as defining the element of a scattering matrix,

$|\langle
E_{+}(+\infty)|E_{-}({\scalebox{0.75}[1.0]{$-$}}\infty)\rangle|^{2}=|S_{+-}|^{2}=e^{-2\pi
z}\,.$ (23)

However, a rigorous definition of the scattering matrix must be more subtle,
as the three-state system does not necessarily have a well-defined, static
final state. For instance, what is the scattering matrix describing the
coherent oscillations in the right panel of Fig. 8?

The goal of this section is three-fold. We first generalize our discussion of
the LZ transition beyond the simplified toy models like (8) and toward a
description of the boson cloud throughout the entire binary inspiral. We will
then introduce an “interaction picture,” leading to an interaction Hamiltonian
that is well-localized in time. This is crucial for constructing a well-
defined S-matrix. We will leverage this localization to argue that different
resonances during the inspiral decouple from one another, allowing us to
ignore all but the finite number of states involved in a particular resonance.
The behavior of the boson cloud during the full inspiral can thus be described
as a series of localized “scattering events,” each of which can be analyzed
individually.

##### Adiabatic decoupling

Let us first consider the simplest inspiral configurations: large, equatorial,
quasi-circular orbits for which the Hamiltonian (1) takes the form

$\mathcal{H}_{ab}=E_{a}\delta_{ab}+\eta_{ab}(t)e^{-i\Delta
m_{ab}\varphi_{*}(t)}\,,$ (24)

where $\Delta m_{ab}=m_{a}-m_{b}$ is the difference in azimuthal angular
momentum between the states $|a\rangle$ and $|b\rangle$ and $\eta_{ab}(t)$ is
the matrix of cross-couplings. The latter evolves slowly in time. Unlike in
the previous sections, we will not truncate (24) to a finite set of states
$\\{|a\rangle\\}$, but instead argue that such a truncation is possible. The
indices in this equation should thus be understood to run over all states in
the atomic spectrum.

As before, we may define a time-dependent unitary transformation

$\mathcal{U}_{ab}(t)=e^{-im_{a}\varphi_{*}(t)}\delta_{ab}\,,$ (25)

that moves our system into a frame that rotates along with the binary
companion. This isolates the “slow” motion responsible for LZ transitions and
discards the perturbation’s distracting fast motion. In this dressed frame,
the Hamiltonian is given by

$\left(\mathcal{H}_{\mathrm{D}}\right)_{ab}=\delta_{ab}\left(E_{a}-m_{a}\dot{\varphi}_{*}\right)+\eta_{ab}\,.$
(26)

This is analogous to (6), though here we have not shifted by an overall
reference energy.

Because the gravitational perturbations $\eta_{ab}$ are always small compared
to the energy eigenvalues $E_{a}$, the instantaneous eigenstates
$|E_{i}(t)\rangle$ are almost always111111This argument fails for states that
form a nearly-degenerate subspace that is predominantly lifted by the
gravitational perturbation $\eta_{ab}$, as in the three-state toy model in §2.
As we saw there, the perturbation cannot be ignored and the instantaneous
eigenstates far away from the transition do not asymptote to a single dressed
basis state. However, the following discussion also applies to these
degenerate subspaces—as long as the system is not excited into a subspace, it
decouples from the dynamics. well-approximated by the gravitational atom’s
energy eigenstates—that is, $|E_{i}(t)\rangle\approx|a\rangle$ for some
dressed state $a$. We will denote the state of the cloud by $|\psi(t)\rangle$.
If the system evolves adiabatically, its population in each instantaneous
eigenstate, $|\langle E_{i}(t)|\psi(t)\rangle|^{2}$, remains unchanged. Since
these instantaneous eigenstates are well-approximated by the dressed states
$|a\rangle$, the cloud’s population in each dressed state, $|\langle
a|\psi(t)\rangle|^{2}$, similarly remains unchanged. This means that, unless
the system already has an appreciable population in a particular dressed
state, we can simply ignore it.121212Strictly speaking, this is only true at
leading order, and this state does have a small effect on the states we do not
ignore. However, this small effect can be incorporated using standard
perturbative techniques.

This will always be the case unless there is a point in time when a subspace
becomes nearly degenerate with the subspace the system occupies, in which case
the approximation $|E_{i}(t)\rangle\approx|a\rangle$ fails. Because of the
assumed hierarchy between the energies $E_{a}$ and the couplings $\eta_{ab}$,
this can only happen when the instantaneous frequency $\Omega(t)$ is such that
the diagonal elements for two or more states are approximately equal,

$\dot{\varphi}_{*}\equiv\pm\Omega(t)\approx\frac{E_{a}-E_{b}}{m_{a}-m_{b}}\,.$
(27)

To determine which states we need to keep track of in a given frequency range,
we therefore simply need to find for which states this _resonance condition_
is met and include them with the states that the system already occupies.

Critically, the width of the resonance defined by (27) is roughly set by
$|\Delta\Omega|\sim\eta_{ab}$. Once these states go through a resonance, they
very quickly decouple and the magnitudes of their coefficients become non-
dynamical. Hence, if the resonances are widely separated, we may consider them
as a sequence of events and analyze them individually. Associated to each
event is thus an S-matrix, which describes how the system evolves through the
resonance.

This logic can be extended to more general orbital configurations. As we
discuss in Appendix 10.B, each pair of states is generically connected by a
perturbation that oscillates at multiple frequencies. The dressed frame
transformation (25) must then be generalized beyond a simple unitary
transformation in order to find a slowly varying dressed frame Hamiltonian
like (26). This can be done, but since it requires a technology that would
distract from our main story, we do not describe it here. Instead, we relegate
this generalization to Appendix 10.B. Fortunately, the above discussion also
applies there, _mutatis mutandis_. To avoid getting bogged down by unnecessary
technical details, we thus continue with the simplified setup described by
(24).

##### Interaction picture

We must define a scattering matrix with respect to a set of states which do
not evolve in the asymptotic past or future. Since the dressed frame’s
instantaneous eigenstates are stationary (up to a phase) under adiabatic
motion, it is natural to define an _interaction frame_ by expanding the
general state $|\psi(t)\rangle$ as

$|\psi(t)\rangle=\sum_{i}e_{i}(t)\,e^{-i\int^{t}\\!\mathrm{d}t^{\prime}\,E_{i}(t^{\prime})}|E_{i}(t)\rangle\,,$
(28)

where the sum runs over all instantaneous eigenstates. From our previous
discussion, we know that this expansion is approximately the same as the
expansion in terms of dressed states

$|\psi(t)\rangle=\sum_{a}c_{a}(t)\,e^{-i\int^{t}\\!\mathrm{d}t^{\prime}\,(E_{a}-m_{a}\dot{\varphi}_{*}(t^{\prime}))}|a\rangle\,,$
(29)

at least far from any resonance. We can think of (29) as the analog of the
interaction picture in textbook quantum field theory in which one works with
free particle states, while (28) is the analog of the renormalized state
basis. While either can be used, our focus will be on (28) since the dynamics
are clearer in that basis.

Figure 9: Interaction Hamiltonian for the two-state system as a function of
time. We see that the interaction is localized over a width $4\eta/\gamma$
near the resonance at $\Omega(t)=\Delta E$.

The Schrödinger equation simplifies in the instantaneous eigenstate basis (28)
to

$i\frac{\mathrm{d}e_{i}}{\mathrm{d}t}=\mathcal{H}_{ij}^{{\mathrm{int}}}(t)\,e_{j}(t)\,,$
(30)

where we have defined the _interaction Hamiltonian_ 131313In general, we
should also include the adiabatic phase factor $e^{i\delta}$ in (31), with
$\delta=i\int^{t}\\!\mathrm{d}t^{\prime}\,\langle
E_{i}(t^{\prime})|\partial_{t^{\prime}}E_{i}(t^{\prime})\rangle$. However, we
can normalize the instantaneous eigenstates $|E_{i}(t^{\prime})\rangle$ to be
real and set $\delta=0$.

$\mathcal{H}_{ij}^{\mathrm{int}}=\begin{dcases}\frac{i\langle
E_{i}(t)|\dot{\mathcal{H}}_{\mathrm{D}}(t)|E_{j}(t)\rangle}{E_{i}(t)-E_{j}(t)}\exp\left(i\int^{t}\\!\mathrm{d}t^{\prime}\,(E_{i}(t^{\prime})-E_{j}(t^{\prime}))\right)&i\neq
j\,,\\\ \qquad\qquad\,\,0&i=j\,.\end{dcases}$ (31)

Clearly, this term is relevant only when two or more energy levels become
nearly degenerate. Furthermore, since eigenvalues repel, this interaction is
only relevant for a short amount of time. This can be seen quantitatively for
the two-state system from §1, where the interaction Hamiltonian is explicitly

$\mathcal{H}_{+-}^{\mathrm{int}}(t)=-\frac{i\gamma}{4\hskip
0.5pt\eta}\frac{(\tau+\sqrt{1+\tau^{2}})^{2i\eta^{2}/\gamma}}{1+\tau^{2}}\,e^{(2i\eta^{2}/\gamma)\tau\sqrt{1+\tau^{2}}}\,,\quad\text{with}\quad\tau=\frac{\gamma
t}{2\eta}\,.$ (32)

As pictured in Fig. 9, this interaction is well localized in time about the
resonance and becomes irrelevant on a timescale $\Delta t\sim 2\eta/\gamma$,
cf. (14).

The fact that this interaction quickly turns off allows us to define a notion
of stationary asymptotic states—the instantaneous eigenstates with their
dynamical phases extracted—and thus a scattering matrix

$S=U^{\mathrm{int}}(\infty,-\infty)\,,$ (33)

where $U^{\mathrm{int}}(t,t_{0})$ is the time-evolution operator associated to
the interaction Hamiltonian $\mathcal{H}^{{\mathrm{int}}}$. In fact, since the
interaction is negligible between resonances, we can define a scattering
matrix associated to each resonance,

$S_{i}=U^{\mathrm{int}}(t_{i}+\Delta t_{\mathrm{w}},t_{i}-\Delta
t_{\mathrm{w}})\,,$ (34)

where $t_{i}$ is the time at which the $i$-th resonance occurs and $\Delta
t_{\mathrm{w}}$ is some multiple of the characteristic interaction timescale
(14). For virtually all resonant transitions, this interaction timescale is
much shorter than the time it takes for a binary to move to the next
resonance. To good approximation, we may then think of the scattering matrix
$S_{i}$ as evolving the cloud from the asymptotic past of the $i$-th resonance
to its asymptotic future. As a result, each scattering event is effectively
isolated from its neighbouring resonances and the cloud’s evolution throughout
the inspiral is well described by combining the $S-$matrices for all previous
resonances (see Fig. 1),141414We must note that this separation into isolated
events discards potentially crucial phase information, especially if we only
consider the S-matrix magnitudes in anything other than the interaction
picture. This phase information would be needed to characterize possible
coherence effects. Fortunately, if the resonances are widely separated, it is
very unlikely that any type of coherence can be achieved and it is thus a
useful approximation to simply drop this information. However, since such
coherent behavior is more likely for nearby resonances, it may be necessary to
consider these resonances as a single event.

$U^{\mathrm{int}}(t,-\infty)\approx\prod_{i=1}^{k}S_{i}\,,\mathrlap{\qquad
t_{k+1}>t>t_{k}\,,}$ (35)

for any time $t$ between the $k$-th and $(k+1)$-th resonance.

There is one more simplifying approximation we can make. Note that the
instantaneous eigenstates asymptote to dressed frame states far from the
resonance, cf. (28) and (29). This implies that, unless the system occupies
multiple instantaneous eigenstates which oscillate among one another at some
fixed frequency as in §2, the _magnitudes_ of the S-matrix elements in the
dressed frame basis will have well-defined limits far from the resonance. In
fact, their magnitudes will equal those of the instantaneous eigenstate basis.
This is useful, since it then is not necessary to compute the interaction
Hamiltonian (31) at all if we are only interested in the relative populations
of the states that the cloud occupies after a resonance. We can instead work
entirely in the dressed frame basis, in which the Hamiltonian takes a simple
form, and analyze the resonance there.

##### General S-matrix

Since we will only need to consider the orbit shortly before and after a
resonance, we may take the orbital frequency $\Omega(t)$ to be roughly linear
in time (3). This relies on the assumption that the resonance bandwidth,
$\Delta\Omega\sim\eta$, is much smaller than the orbital frequency itself, cf.
(15), which we argued in §3 is true for all the transitions we consider.
Similarly, we may also assume that the level mixings $\eta_{ab}(t)$ are time-
independent.

Using (3), the dressed frame Hamiltonian (26) can be written as

$\left(\mathcal{H}_{\mathrm{D}}\right)_{ab}=\mathcal{A}_{ab}+\mathcal{B}_{a}\delta_{ab}t\,,$
(36)

where we have defined

$\displaystyle\mathcal{A}_{ab}$ $\displaystyle\equiv\delta_{ab}\left(E_{a}\mp
m_{a}\Omega_{0}\right)+\eta_{ab}\,,$ (37) $\displaystyle\mathcal{B}_{a}$
$\displaystyle\equiv\mp m_{a}\gamma\,.$

This is known as the _generalized Landau-Zener problem_ [Brundobler:1993smat].
We will be specifically interested in the magnitudes of the S-matrix in this
basis, defined by

$|S_{ab}|=\lim_{t\to\infty}|U_{ab}(t,-t)|\,,$ (38)

where $U_{ab}$ is the time-evolution operator associated with (36).
Unfortunately, analytic solutions for this problem are only known in special
cases. However, a small amount of progress can be made by noting that the
Schrödinger equation for this system is invariant under a collection of
symmetry transformations [Brundobler:1993smat], and thus so are the S-matrix
elements. The invariant combinations of parameters made from
$\mathcal{A}_{ab}$ and $\mathcal{B}_{a}$ are the generalized Landau-Zener
parameters

$z_{ab}\equiv\frac{|\mathcal{A}_{ab}|^{2}}{|\mathcal{B}_{a}-\mathcal{B}_{b}|}=\frac{\eta_{ab}^{2}}{\gamma|\Delta
m_{ab}|}\,,$ (39)

which extends the original definition (13) to the general, multi-state system.
The S-matrix elements must then be functions of these parameters and their
combinations.

###  4 Summary

In this chapter, we studied the evolutions of the boson clouds when they are
parts of binary systems. We showed that the gravitational perturbation sourced
by a binary companion greatly enriches the dynamics of the system, by inducing
mixing between different energy eigenstates of the cloud. Interestingly, just
like the hydrogen atom under the influence of an external perturbation, these
level mixings must obey certain selection rules, which are different for the
scalar and vector clouds. At certain critical resonance frequencies, the level
mixings become non-perturbative and are significantly enhanced. In order to
investigate the dynamics of the clouds at these resonance frequencies, it was
crucial that we carefully incorporated the gradual increase of the orbital
frequency due to the shrinking motion of the orbit. In that case, we showed
that the associated dynamics are analogous to the Landau-Zener transition in
quantum mechanics.

We found that the transitions in the scalar case typically involve only two
well-separated energy eigenstates, while those for the vector cloud can
simultaneously involve multiple nearly-degenerate states. As a result, in the
scalar case, the Landau-Zener transition typically forces the initial cloud to
completely populate the state that it couples to. On the other hand, for the
vector cloud, multiple levels can be occupied in the final state. The
simultaneous occupation of multiple end states would result in neutrino-like
oscillations between different vector eigenstates — an effect that persists
even after the cloud exits the resonance bands. Furthermore, a key parameter
that dictates the nature of the transition is the Landau-Zener parameter (13),
which quantifies the adiabaticity of the transition. As demonstrated in Figs.
6 and 8, an adiabatic transition would lead to a smooth transition to the
final states, while a non-adiabatic transition would result in further
oscillatory features in the final states.

The phenomenological implications of these rich dynamics will be explored in
detail in the next chapter. Crucially, we shall find that the backreaction of
these effects on the orbit are sensitive to the mass and intrinsic spins of
the boson fields. Furthermore, we saw in Figs. 5 and 7 that the dynamics at
each of these resonances can be described effectively by an S-matrix. As the
binary experiences multiple resonances throughout its inspiral, the total
evolution of the cloud can therefore be succinctly described by multiplying
successive scattering matrices. Motivated by its various analogy with particle
collider physics, we therefore refered to these types of binary systems as
“gravitational colliders.”

## Observational Signatures

I have described the foundations underlying the dynamics of the gravitational
atoms when they are parts of binary systems. This chapter is dedicated to
exploring the rich phenomenologies that arise from these “gravitational
collider”. In particular, I will describe the effects of these dynamics on the
gravitational waves emitted by the cloud and the binary system. Crucially, I
will also explain how these signals can carry vital information about the
masses and intrinsic spins of the ultralight bosons. This chapter is also
based on the works in [Baumann:2018vus, Baumann:2019ztm].

###  1 Overview and Outline

One of the central goals in this chapter is to include the backreaction of the
cloud’s dynamics on the orbit, which was not studied in the previous chapter.
As we will see, this backreaction can significantly affect the binary’s
orbital motion and associated gravitational-wave signals. For instance, any
change in the cloud’s angular momentum (which is most prominent during a
resonance) must be balanced by a change in the orbital angular momentum of the
binary, so that the total angular momentum is conserved. Depending on the
orientation of the orbit and the nature of the transition, the binary can
either absorb angular momentum from the cloud or release it. The resulting
backreaction can either lead to a floating, sinking, or kicked orbit. In all
cases, the induced changes in the orbital motion produce significant
dephasings in the gravitational waves emitted by the binary, relative to the
evolution without a cloud.111For intermediate and extreme mass ratio
inspirals, with the cloud carried by the larger black hole, the backreaction
on the orbit can be a dramatic effect. If angular momentum is transfered from
the cloud to the binary, long-lived floating orbits are naturally induced,
creating a new source of continuous monochromatic gravitational waves. In
contrast, when the binary transfers angular momentum to the cloud, it can
experience a sudden kick instead. This greatly enriches the dynamics of the
binary system, for instance by producing eccentric orbits. While these kicks
are difficult to model quantitatively, they are distinctive qualitative
signatures of the existence of a gravitational atom in a binary system. See
Fig. 1 for an illustration of evolution of the orbital frequency as the binary
transitions through multiple successive Landau-Zener transitions. Importantly,
these dephasings are correlated with the positions of the resonances and
therefore probe the spectral properties of the cloud.

Figure 1: Illustration of the gravitational atom in a binary system. During
the inspiral, the companion induces level mixings between different states of
the cloud. This mixing is strongest when the orbital frequency $\Omega(t)$,
which slowly increases due to gravitational-wave emission, matches the energy
gap between the coupled states. During these resonant transitions, which occur
at the frequencies $\Omega_{i}$, the change in the state of the cloud produces
a backreaction on the binary’s orbital dynamics. These effects either induce
floating or sinking orbits across these resonance bands, thus leaving distinct
imprints in the gravitational waves emitted by the binary. Moreover, minute
time-dependent finite-size effects can further distinguish the nature of the
boson cloud, including its intrinsic spin.

Even though the backreaction on the orbit is a smoking gun for the existence
of boson clouds, many degeneracies between the scalar and vector case still
remain, since the transfer of angular momentum is not very sensitive to the
internal structure of the clouds. To break these degeneracies, we need to
probe the clouds through their time-dependent finite-size effects on the
gravitational waveforms. This is possible because, as we found in Chapter 5,
the many nearly degenerate states of the vector field can result in neutrino-
like oscillations between different eigenstates, with characteristic
frequencies set by the energy splitting between the states. Crucially, this
behaviour is absent in the scalar case. The oscillations persist after the
resonance and are therefore a key characteristic of clouds carrying intrinsic
spin. Although these effects are difficult to model in detail, they are
correlated with the position of the resonances, and therefore represent robust
signatures of the boson clouds.

The continuous and monochromatic gravitational waves emitted by the cloud
would also be affected when the clouds are in binary systems. For instance,
the Landau-Zener transition can result in a complete population of a decaying
mode in the spectrum, resulting in a depletion (or significant attenuation) of
the energy density stored in the cloud. Furthermore, this monochromatic signal
can experience Doppler modulation due to the orbital motion of the binary. In
short, both the signals from the cloud and the binary are unique imprints of
the cloud and provide powerful new probes of physics beyond the Standard
Model. Our findings motivate revisiting current waveform models and
continuous-wave searches in order to target the unique signatures of
ultralight particles in binary gravitational-wave searches.

The outline of this chapter is as follow: in Section 2, I describe the
backreaction of time-dependent boson clouds on the orbital motion. I also
provide an estimate of the finite-size effects of the cloud, as well as their
time dependence induced by the Landau-Zener transitions. In Section 3, I
discuss how the binary system affects the continuous gravitational-wave
emission from the cloud. In Section 4, I explore the observational
consequences of boson clouds in a binary system, and study the effects that
can help us to unravel their atomic structure. Finally, in Appendix 11, I
provide further details on the angular momentum transfer between the cloud and
the orbit.

###  2 Signals from the Binary

In the previous chapter, we saw that Landau-Zener transitions can
significantly redistribute energy and angular momentum between different
states of the cloud. The mass and spin of the cloud in a general superposition
of states, at small $\alpha$, are given by

$\displaystyle M_{c}(t)$
$\displaystyle=M_{c,0}\left(|c_{1}|^{2}+|c_{2}|^{2}+\cdots+|c_{N}|^{2}\right),$
(1) $\displaystyle\textbf{S}_{c}(t)$
$\displaystyle=S_{c,0}\left(m_{1}|c_{1}|^{2}+m_{2}|c_{2}|^{2}+\cdots+m_{N}|c_{N}|^{2}\right)\hat{\textbf{z}}\,,$

where $S_{c,0}\equiv M_{c,0}/\mu$ and $\hat{\textbf{z}}$ is a unit vector
along the spin-axis of the black hole, cf. Fig. 2, and the $c_{i}$ account for
the population of the different states. We have ignored the
$\hat{{\mathbf{x}}}$ and $\hat{{\mathbf{y}}}$ components of the spin, as they
decouple for equatorial orbits. Any change in the energy and angular momentum
of the cloud must be balanced by an associated change of the binding energy
and orbital angular momentum of the binary’s orbit. As we will see, this
greatly enriches the dynamics of the system and the gravitational-wave signals
emitted from the binary.

####  1 Floating, Sinking, and Kicked Orbits

To analyze the effect an LZ transition has on the binary orbit, it is most
convenient to consider the transfer of angular momentum between the cloud and
the binary. Ignoring the intrinsic spins of the black holes, conservation of
angular momentum implies

$\frac{{\rm d}}{{\rm
d}t}\left(\textbf{L}+\textbf{S}_{c}\right)=\textbf{T}_{\rm gw}\,,$ (2)

where L is the orbital angular momentum and $\textbf{T}_{\rm gw}$ is the
torque due to gravitational-wave emission from the binary. Since (2) is a
vectorial equation, the dynamics associated to angular momentum transfer can
in general be very complicated, and depend on the relative orientations of the
different angular momenta. For simplicity, I will only consider equatorial
orbits and ignore precession [Porto:2016pyg]. Hence, the magnitudes of the
vectors in (2) obey the relation

$\frac{{\rm d}}{{\rm d}t}\left(L\pm S_{c}\right)=T_{\rm gw}\,,$ (3)

where, as usual, the upper (lower) sign denotes co-rotating (counter-rotating)
orbits. Specializing further to quasi-circular orbits, we find that the
orbital frequency evolves as (see Appendix 11 for a derivation)

$\displaystyle\frac{{\rm d}\Omega}{{\rm d}t}$
$\displaystyle=\gamma\left(\frac{\Omega}{\Omega_{r}}\right)^{11/3}\\!$ (4)
$\displaystyle\pm
3R_{J}\,\Omega_{r}\\!\left(\frac{\Omega}{\Omega_{r}}\right)^{4/3}\\!\frac{{\rm
d}}{{\rm
d}t}\left[m_{1}|c_{1}|^{2}+m_{2}|c_{2}|^{2}+\cdots+m_{N}|c_{N}|^{2}\right],$

where $\gamma$ was defined in (16), and $R_{J}$ is the ratio of the spin of
the cloud to the orbital angular momentum of the binary at the resonance
frequency,

$R_{J}\equiv\left(\frac{S_{c,0}}{M^{2}}\right)\frac{(1+q)^{1/3}}{q}(M\Omega_{r})^{1/3}\,.$
(5)

Far from the transition, the occupation densities in (4) are constant and the
instantaneous frequency during the inspiral is well-approximated by the
standard quadrupole formula, cf. (15). During the transition, on the other
hand, the angular momentum transfer between states with different $m$ exerts
an additional torque on the binary. Crucially, the direction of the torque
depends on various factors, including the orientation of the orbit, the signs
of the azimuthal quantum numbers, and whether the occupation densities are
growing or decaying in time. For simplicity, we set
$m_{2}=m_{3}=\cdots=m_{N}$,222This turns out to be the case for any transition
induced by the gravitational quadrupole $\ell_{*}=2$. such that near the
resonance (4) simplifies to

$\displaystyle\frac{{\rm d}\Omega}{{\rm d}t}\simeq\gamma\mp 3\hskip
1.0pt\Delta m\hskip 2.0ptR_{J}\,\Omega_{r}\,\frac{{\rm d}|c_{1}(t)|^{2}}{{\rm
d}t}\,,$ (6)

where we have defined $\Delta m\equiv m_{2}-m_{1}$ and used conservation of
the occupation densities during the transitions.

By choosing the initial condition $|c_{1}(-\infty)|^{2}=1$, the time
derivatives of $|c_{1}(t)|^{2}$ must decrease during the transition. When
$\Delta m<0$, the cloud loses angular momentum to the orbit, forcing a co-
rotating inspiral to stall and counter-rotating inspiral to shrink faster.
These phenonema are reversed between co-rotating and counter-rotating orbits
when $\Delta m>0$. To investigate the detailed dynamics of these backreaction
effects, we must solve (6) numerically. The solution in Figs. 2 and 3 display
the floating and sinking orbits, whose properties we describe below.

Figure 2: Evolution of the orbital frequencies (left) and the occupation
densities (right) of an unbackreacted orbit (dashed lines) and a floating
orbit (solid lines). Shown is a transition near the resonance frequency
$\Omega_{r}=5\mu\alpha^{2}/144$ with $\alpha=0.07$ and $q=0.1$. We also
assumed that the parent black hole spun maximally, $S_{c,0}/M^{2}=1$, before
it grew the cloud. The transition begins when the binary enters the resonance
band, denoted by the thin gray band about $\Omega=\Omega_{r}$.

##### Floating orbits

When the torque acts against the shrinking of the orbit, we find that LZ
transitions naturally induce floating orbits [Press:1972zz].333This floating
mechanism is distinct from the one studied in [Zhang:2018kib], which assumes
$\gamma=0$. It is also different from the analysis in [Cardoso:2011xi], which
relies on modifications of General Relativity. This is shown in Fig. 2 for a
two-state transition, where the orbital frequency increases more slowly in the
resonance band. More precisely, the rate of change in the frequency of
gravitational waves emitted by the binary, $f_{\rm gw}$, during the transition
satisfies (cf. Appendix 11)

$\left(\frac{{\rm d}f_{\rm gw}}{{\rm d}t}\right)_{\rm
float}=\frac{1}{1+3R_{J}\,|\Delta m\Delta
E|/(4\eta)}\left(\frac{\gamma}{\pi}\right),$ (7)

which is clearly smaller than the unperturbed rate, $\gamma/\pi$.
Interestingly, in the large backreaction limit, (7) asymptotes to zero and
never turns negative.444While the chirp rate (7) was derived assuming perfect
adiabaticity, this observation remains true even in numerical simulations.
However, if the system is tuned to be nearly non-adiabatic, there may be small
oscillations in the instantaneous frequency. The parameters for which these
oscillations occur are not physically realizable and so we do not pursue them
further. This means that the orbit can at most float, but never grows during a
transition. Furthermore, the frequency of the gravitational waves emitted
during the floating phase, $f_{\rm float}$, is directly related to the
resonance frequency

$f_{{\mathrm{float}}}=\frac{\Omega_{r}}{\pi}=\frac{1}{\pi}\left|\frac{\Delta
E}{\Delta m}\right|\,.$ (8)

A measurement of this approximately monochromatic gravitational-wave signal
thus provides a direct probe of the spectral properties of the boson cloud. In
Section 4, we will describe the phenomenological consequences of these
floating orbits in more detail. In particular, we will discuss how the
duration of the floating depends on the parameters of the system.

The description above remains qualitatively unchanged for multi-level
transitions, as they typically satisfy $m_{2}=\dots=m_{N}$. Moreover, since
multi-level transitions necessarily involve nearly degenerate excited states,
$E_{2}\approx\cdots\approx E_{N}$, the expression (8) remains an excellent
approximation to the gravitational-wave frequency emitted by the corresponding
floating orbits. The only quantitative difference is that $\eta$ must be
replaced by an effective coupling $\eta_{\mathrm{eff}}$ that characterizes the
multi-state transition, which is well-approximated by the Pythagorean sum of
the diagonalized couplings $\tilde{\eta}_{1a}$, cf. (17). We discuss its
precise definition in Appendix 11.

Another key feature of these floating orbits is that the adiabaticity of the
transition is enhanced. In analogy to the LZ parameter (13), the degree of
adiabaticity of the evolution can now be quantified by the parameter
$z^{\prime}\equiv\eta^{2}/\dot{\Omega}$. Floating orbits, with
$\dot{\Omega}<\gamma$, thus enhance the adiabaticity of the transition,
$z^{\prime}>z$. The predictions made by adiabatically following the
instantaneous eigenstates of the system, such as the final-state occupation
densities, are therefore robust.

##### Sinking and kicked orbits

Figure 3:  Sinking (_left_) and kicked (_right_) orbits, with their
corresponding occupation densities below. As in Figure 2, this transition is
near the resonance $\Omega_{r}=5\mu\alpha^{2}/144$ with $\alpha=0.07$.
However, we flip the sign of (6) so that backreaction causes the orbit to
either sink ($q=1$) or kick ($q=0.1$).

The transition can also exert a torque that increases the orbital frequency of
the binary, and therefore temporarily accelerates the shrinking of the orbit.
If the effect is weak, we call it a sinking orbit (see the left panel in Fig.
3). As long as the adiabaticity of the transition is preserved, the chirp rate
of the emitted gravitational waves is 555While (9) naively admits solutions
that are divergent and negative, this simply indicates the breakdown of the
adiabatic approximation in those corresponding regions of parameter space, cf.
(10).

$\left(\frac{{\rm d}f_{\rm gw}}{{\rm d}t}\right)_{\rm
sink}=\frac{1}{1-3R_{J}|\Delta m\Delta
E|/(4\eta)}\left(\frac{\gamma}{\pi}\right),$ (9)

which differs from (7) by an important sign. Compared to the unperturbed
orbits, these sinking orbits spend a much shorter amount of time in the region
where the LZ transition is efficient.

A sufficiently strong “kick,” on the other hand, can destroy any initial
adiabaticity. Using $\dot{\Omega}\lesssim\eta^{2}$ as a diagnostic, we
estimate that adiabaticity is preserved whenever

$R_{J}\lesssim\frac{4\eta\left(1-\gamma/\eta^{2}\right)}{3\left|\Delta m\Delta
E\right|}\simeq\frac{4\hskip 1.0pt\eta}{3\left|\Delta m\Delta E\right|}\,,$
(10)

where the second relation assumes an initial adiabatic transition,
$\gamma\ll\eta^{2}$. As we discussed in Section 3, the resulting non-adiabatic
transition yields drastically different final states, most prominently through
the presence of oscillations in the occupation densities of final state. These
oscillations transfer angular momentum between different states of the cloud,
even after the transition. As shown in bottom panel of Fig. 3, this effect
also modulates the quasi-circular motion of the binary, leaving dramatic
oscillatory features in the binary’s gravitational-wave signal. Having said
that, we caution that (6) assumes that the binary motion is always well-
described by a quasi-circular orbit. This need not be the case—strong non-
adiabatic transitions in which $\dot{\Omega}\gtrsim\Omega^{2}$ can induce
significant eccentricity (or even unbind the orbit). The full phenomenology of
the kicked orbits can therefore be much richer than what was described above,
and deserves a more detailed investigation. We will focus thus primarily on
sinking orbits, where this backreaction is weak enough to preserve both
adiabaticity and the quasi-circularity of the initial orbit.

####  2 Finite-Size Effects

In addition to the backreactions on the orbit during resonances, the cloud can
also reveal itself in the gravitational-wave signal through subtle
modifications in the waveform due to various finite-size effects. These
effects were elaborated in §1 and §2, where I described their impacts on the
dynamics of the binary system and the associated gravitational waveform. In
the following, we describe how these effects are manifested specifically for
the cloud.

##### Spin-induced quadrupole

The spinning motion of the cloud generates a hierarchy of mass multipole
moments. For simplicity, we consider a cloud that occupies a _single_
eigenstate. Its mass quadrupole moments then experience $\alpha$-suppressed
oscillations with frequency $\mu$, which are responsible for the cloud’s well-
studied decay via gravitational-wave emission [Arvanitaki:2010sy,
Yoshino:2014]. For example, denoting the spherical harmonic decomposition of
the mass multipole moments by $Q_{\ell m}$, the axisymmetric and non-
axisymmetric quadrupole moments of the scalar $|2\hskip 0.5pt1\hskip
0.5pt1\rangle$ state are

$Q_{c}\equiv Q_{20}=-6M_{c}r_{c}^{2}\quad\text{and}\quad Q_{2,\pm
2}\sim\,\alpha^{2}Q_{c}\hskip 1.0pte^{\pm 2i\mu t}\,.$ (11)

Since $\mu\gg\Omega(t)$, these oscillations average out over an orbital period
and therefore do not affect the dynamics of the binary significantly. This is
the generic behavior of all clouds occupying a single eigenstate—finite-size
effects in these cases are dominated by their time-independent component.

In §1, we saw that these spin-induced moments would perturb the dynamics of
the binary system and leave imprints in their gravitational waveforms. For the
dominant axisymmetric quadrupole moment, this backreaction is often
parameterized through the relation (16), where $\kappa$ is the dimensionless
quadrupole parameter. A rigorous computation of $\kappa_{c}$, which is the
quadrupole parameter for the cloud, would require incorporating the
backreaction on the spacetime geometry, which is beyond the scope of this
work.666Exact solutions with (complex) scalar hair around spinning black holes
have been studied numerically in the literature, see e.g. [Herdeiro:2014goa].
In principle, these quasi-stationary spacetimes can also have $\kappa\gg 1$.
Instead, we will assume that the cloud dominates the contribution to the
multipole moments of the cloud, and estimate $\kappa_{c}$ by comparing the
cloud’s mass quadrupole, $Q_{c}$, to $J^{2}/M$, where $M$ and $J$ are the
total mass and angular momentum of cloud. We choose to normalize $\kappa_{c}$
with $(M,J)$ of the cloud, instead of $(M_{c},J_{c})$ of the cloud itself,
because i ) the quantities $(M,J)$ can be directly measured through
gravitational-wave observations, and ii ) they are conserved throughout the
evolution (up to small losses due to gravitational-wave emission). The
parameters $(M,J)$ therefore also coincide with the mass and angular momentum
of the initial black hole prior to the formation of the cloud.

From the stress-energy tensor of the scalar field, we find for the $\ket{211}$
state

$\kappa_{c}(\alpha)\,\geq\,-\frac{Q_{c}(\alpha)}{M^{3}}\sim
10^{3}\left(\frac{M_{c}(\alpha)/M}{0.1}\right)\left(\frac{0.1}{\alpha}\right)^{4}\,,$
(12)

where we have imposed the weak cosmic censorship condition, $J\leq M^{2}$
[Penrose:1969pc], to obtain the lower bound. The effect on the gravitational-
wave phase then scales as

$\kappa_{c}(\alpha)\hskip 1.0ptv^{4}\hskip
1.0pt\chi^{2}\,\gtrsim\,10^{-2}\left(\frac{M_{c}(\alpha)/M}{0.1}\right)\left(\frac{v}{\alpha/2}\right)^{4}\,,$
(13)

where we used $\chi\simeq 1$, and assumed that the initial black hole is
rapidly rotating (which is required for the cloud to form in the first place).
Notice that, in the regime of validity of the perturbative expansion, the
relative velocity satisfies $v\lesssim\alpha/2$.777For $q\lesssim 1$, the
virial theorem implies $v\simeq(\alpha/2)\sqrt{r_{c}/R_{*}}$ and hence
$v\lesssim\alpha/2$ in the regime of validity of the multipole expansion,
$r_{c}<R_{*}$. For $v>\alpha/2$, the companion experiences a smaller amount of
the cloud according to Gauss’ law. Even though larger relative velocities may
seem favorable, the reduction in the effective mass of the cloud will
dominate, leading to negligible finite-size effects once the companion enters
the cloud.

##### Tidal deformations

In addition to the spin-induced multipole moments, the cloud may also acquire
induced multipoles in the presence of a gravitational perturbation. This tidal
deformation effect is often parameterized through the relation (18), where
$\lambda$ is the dimensionless Love numebr. Since the cloud is much less
compact than an isolated black hole, but carries a significant fraction of the
mass and angular momentum of the system, we can, in principle, have large Love
numbers for the cloud system. Although a detailed computation is beyond the
scope of this work, on dimensional grounds we expect $\lambda_{c}$, which is
the Love number of the cloud, to scale as

$\lambda_{c}(\alpha)\,\sim\,\left(\frac{M_{c}(\alpha)}{M}\right)\left(\frac{r_{c}}{2r_{g}}\right)^{4}\,\sim\,10^{7}\left(\frac{M_{c}(\alpha)/M}{0.1}\right)\left(\frac{0.1}{\alpha}\right)^{8}\,.$
(14)

The parameter $\lambda_{c}$ enters in the phase of the waveform at 5PN order,
and its imprint on the signal from the binary therefore scales as

$\lambda_{c}(\alpha)\hskip 2.0ptv^{10}\sim
10^{-6}\left(\frac{M_{c}(\alpha)/M}{0.1}\right)\left(\frac{\alpha}{0.1}\right)^{2}\left(\frac{v}{\alpha/2}\right)^{10}\,,$
(15)

where $v\lesssim\alpha/2$ in the regime of validity of the multipole
expansion. While this PN order seems high, the lack of standard model
background in Einstein’s theory offers a powerful opportunity to probe the
dynamics of vacuum spacetimes in General Relativity, through precision
gravitational-wave data.

####  3 Time-Dependent Dephasings

In the previous subsection, we focused on clouds that occupy a single
eigenstate. In particular, we found that these finite-size effects in these
cases are time independent. However, as we saw in Section 3, Landau-Zener
transitions can lead to the simultaneous occupation of different energy levels
for the vector field. We will therefore relate the occupation densities
$|c_{a}(t)|^{2}$ to the time-dependences of the multipole moments of the cloud
in this subsection.

##### Superpositions and oscillations

For scalar clouds, a typical adiabatic Landau-Zener transition forces an
initial state to fully transfer its population to another state with a
different shape. As a result, the effective parameters $\kappa$ and $\lambda$
evolve in time. Interesting multipolar time dependences are possible if the
cloud occupies _multiple_ eigenstates, which is typical for vector clouds. In
a general time-dependent superposition, the axisymmetric quadrupole moment is
approximately

$Q_{c}(t)=\sum_{a}|c_{a}(t)|^{2}\,Q_{c,a}+\sum_{\begin{subarray}{c}a\neq
b\end{subarray}}\delta_{m_{a},m_{b}}\,|c_{a}(t)c_{b}(t)|\,Q_{ab}\cos(\Delta
E_{ab}\,t)\,,$ (16)

where we have discarded terms that are either subleading in $\alpha$ or
rapidly oscillating. The first term represents the weighted sum of the
individual axisymmetric moments $Q_{c,a}$ of each state $|a\rangle$, while the
second represents additional contributions that arise from interference
effects, which occur between different states with the same azimuthal angular
momenta. Crucially, the $Q_{ab}$’s are not $\alpha$-suppressed and may lead to
large oscillations in $Q_{c}(t)$ at frequencies set by the energy differences
$\Delta E_{ab}$. Furthermore, there can be similar interference effects in the
non-axisymmetric moments if the superposition involves states with different
azimuthal quantum numbers, as may happen after a kick-induced non-adiabatic
transition.

There are, in fact, two sources of oscillatory behavior in (16). As we
discussed in §2, a mildly non-adiabatic multi-state transition can excite
long-lived oscillations in the occupation densities $|c_{a}(t)|$, with
frequencies set by the gravitational perturbation $\eta_{ab}$. It is clear
from (16) that the multipole moment will inherit these oscillations. On the
other hand, even if the $|c_{a}(t)|$ are constant, there are still
oscillations in the multipole moment since the system occupies multiple, non-
degenerate energy eigenstates, each with their own spatial profile. These are
similar to neutrino oscillations,888The oscillations are neutrino-like in the
following sense: after the resonance, the system occupies an instantaneous
eigenstate (like a “flavor eigenstate” for neutrinos) which, away from the
interaction, becomes a superposition of energy eigenstates (like “mass
eigenstates” for neutrinos). and are present in the final states of both
adiabatic and non-adiabatic multi-state transitions. Notice that their
frequencies depend only on the energy differences $\Delta E_{ab}$, which have
been calculated [Baumann:2019eav].

While we have only explicitly demonstrated this for the spin-induced
quadrupole moment, these qualitative features also apply for the higher-order
spin-induced moments and the tidally-induced moments. In summary, whenever the
cloud occupies a superposition of energy eigenstates, the mass moments and
tidally-induced moments oscillate with unsuppressed amplitude at frequencies
that are slow compared to the orbital frequency.999In principle, these
oscillating quadrupoles are also new sources of gravitational waves emitted by
the cloud. While we leave the study of their emission rates to future work, we
note that their frequencies are typically very small and so we do not expect
that this is an efficient decay channel for the cloud. A detailed study of the
impact that these dynamical finite-size effects have on the orbit, and
ultimately on the waveform, is beyond the scope of this work. Yet, it is clear
that they serve as important relics that can help us decode the nature of an
LZ transition that occurred in the past. As we will discuss in §3, correlating
these oscillatory finite-size effects with the dephasing induced by floating
and sinking orbits will serve as a powerful probe of the mass and spin of the
ultralight particles.

##### Cloud depletion

Since a transition can populate decaying states, the energy and angular
momentum of the cloud can also be reabsorbed by the central black hole over
the typical decay timescales of these modes. The mass of the cloud
approximately evolves as

$\displaystyle\frac{M_{c}(t)}{M_{c,0}}$
$\displaystyle=\exp\left(\sum_{a}2\,\Gamma_{a}\\!\int^{t}_{0}\\!{\rm
d}t^{\prime}\,|c_{a}(t^{\prime})|^{2}\right)\equiv\exp\big{[}-2\mathcal{A}(t,0)\big{]}\,,$
(17)

where the quantity $\mathcal{A}$ is called the depletion estimator. Since the
spin-induced and tidally-induced moments scale linearly with the mass of the
cloud, cf. (13) and (14), this decay can introduce additional time dependence
in the various effects discussed above, which in turn also affect the
waveforms. As we have seen in Chapter 3, the depletion is generally slower
than the oscillation timescale discussed in (16). During the transition, they
may therefore appear as modulations of the dominant finite-size effects.
Importantly, this depletion persists even after the cloud has passed the
resonance and can then be observed even away from resonances, thereby
providing another unique signature of the boson clouds.101010The total amount
of depletion during a resonance was estimated in [Baumann:2018vus] and
[Berti:2019wnn], assuming a constant resonance frequency. As we show in this
thesis, LZ transitions lead to a much longer depletion time which overwhelms
the results in [Baumann:2018vus, Berti:2019wnn] in almost all scenarios.

###  3 Signals from the Cloud

We described in §2 that the real boson cloud is a source of continuous, nearly
monochromatic gravitational-wave signal [Arvanitaki:2010sy, Yoshino:2014]. We
elaborate on the detectability of this signal here, and discuss how it can
develop new time dependences when it is in a binary system.

The frequency of these gravitational waves is determined by the mass of the
boson field,

$\displaystyle
f_{c}\simeq\frac{\mu}{\pi}=484\,\text{Hz}\,\left(\frac{\mu}{10^{-12}\text{
eV}}\right).$ (1)

This is shown in Fig. 4 for physically motivated values of $\alpha$ and $M$,
and compared to the frequency bands of ground-based
[TheLIGOScientific:2014jea, TheVirgo:2014hva, Somiya:2011np,
Sathyaprakash:2012jk, Evans:2016mbw] and space-based [AmaroSeoane:2012km,
Kawamura:2011zz, Graham:2017pmn] detectors, as well as pulsar timing arrays
(PTAs) [Kramer:2013kea, McLaughlin:2013ira, Hobbs:2013aka, Smits:2008cf]. We
see that ground-based experiments probe $M\lesssim 3\times 10^{3}M_{\odot}$,
while LISA would give us access to $M\gtrsim 3\times 10^{3}M_{\odot}$.

On the other hand, the root-mean-square strain amplitude depends on the
intrinsic spin of the boson, cf. more discussion in §2. For concreteness, we
consider the strain amplitude of the dominant scalar $\ket{211}$ mode. From
(15), we find that [Brito:2014wla, Yoshino:2014]

$\displaystyle h_{c}\simeq 2\times
10^{-26}\left(\frac{M}{3M_{\odot}}\right)\left(\frac{M_{c}(\alpha)/M}{0.1}\right)\left(\frac{\alpha}{0.07}\right)^{6}\left(\frac{10\,\text{kpc}}{d}\right),$
(2)

where $d$ is the (Euclidean) distance of the source. Since the signal is
emitted continuously, its detectability depends on the total observation time
$T_{\rm obs}$. The signal-to-noise ratio (SNR) of the cloud signal is

$\displaystyle\text{SNR}=h_{c}\,\langle F^{2}_{+}\rangle^{1/2}\frac{T_{\rm
obs}^{1/2}}{\sqrt{S_{n}(f_{c})}}\,,$ (3)

where $S_{n}(f_{c})$ is the (one-sided) noise spectral density at the
frequency $f_{c}$, and $\langle F^{2}_{+}\rangle=\langle
F^{2}_{\times}\rangle$ is the angular average of the square of the detector
pattern functions $F_{+,\times}$ for each gravitational-wave polarization.
Using (2), we get

$\text{SNR}\simeq 13\,\langle F^{2}_{+}\rangle^{1/2}\left(\frac{T_{\rm
obs}}{1\text{ yr}}\right)^{1/2}\left(\frac{10^{-23}\text{
Hz}^{-\frac{1}{2}}}{\sqrt{S_{n}(f_{c})}}\right)\left(\frac{M}{3M_{\odot}}\right)\left(\frac{M_{c}/M}{0.1}\right)\left(\frac{\alpha}{0.07}\right)^{6}\left(\frac{10\,\text{kpc}}{d}\right),$
(4)

which is strongly dependent on $\alpha$. We see that cloud systems with
$\alpha<0.07$ and $M<10^{2}\,M_{\odot}$ may be detectable only within our own
galaxy, while those with larger values of $\alpha$, and bigger masses, may be
observed at extra-galactic distances. Further discussion and detailed
forecasts can be found in [Arvanitaki:2014wva, Arvanitaki:2016qwi,
Brito:2017zvb, Riles:2017evm].

Figure 4: Frequency of the monochromatic signal for physically motivated
ranges of $\alpha$ and $M$. The vertical dotted lines illustrate the typical
observational bands of various gravitational-wave observatories.

####  1 Resonant Attenuation

When the cloud is part of a binary system, the cloud may deplete over time,
according to:

$\frac{h_{c}(t)}{h_{c}(t_{0})}=\frac{M_{c}(t)}{M_{c}(t_{0})}\sim\exp\big{[}-2\mathcal{A}(t,t_{0})\big{]}\,,$
(5)

where $\mathcal{A}$ is the depletion estimator introduced in (17). This decay
is most prominent if the Landau-Zener transition forces the initial cloud to
fully populate a final decaying state. Depending on the state that is being
occupied, the depletion can occur over sufficiently short timescales for
moderate values of $\alpha$. Observing this time-dependent change in the
monochromatic signal would therefore provide a tentative hint of the presence
of the cloud and that it had experienced a Landau-Zener transition in its
earlier binary evolution.

####  2 Doppler Modulation

In the presence of a companion, the gravitational-wave frequency $f_{c}$ will
also be modulated by the orbital motion.111111This is the same effect that
famously led to the discovery of the Hulse-Taylor binary pulsar, where the
frequency of the emitted electromagnetic radiation was found to be modulated
by the orbital motion [Hulse:1974eb]. In particular, if the orbital plane of
the binary is oriented such that the cloud has a non-vanishing radial velocity
along the line-of-sight, the orbital motion will induce a periodic Doppler
shift of the gravitational-wave signal,

$\frac{\Delta f_{c}}{f_{c}}=\frac{v_{r}}{c}=\left(\frac{M_{*}}{M_{\rm
tot}}\right)\frac{R_{*}\Omega}{c}\sin\iota\,,$ (6)

where $v_{r}$ is the radial component of the velocity of the cloud along the
line-of-sight, and $\iota$ is its angle relative to the normal of the orbit.
While detecting the periodic modulation of the frequency is experimentally
(and computationally) challenging, this Doppler effect would be very
convincing evidence that the cloud is part of a binary system. At the same
time, it will open a new avenue to infering parameters of a binary system at
low orbital frequencies, by monitoring the signal from the cloud with
continuous searches. Similar searches are being performed for neutron stars in
binaries [Aasi:2014erp].

###  4 Unraveling the Atomic Structure

So far, we have studied simplified models for the evolution of boson clouds in
black hole binaries. This allowed us to identify a number of interesting
dynamical effects that, in principle, can have important observational
consequences. In this section, we investigate under which conditions these
effects are accessible to current and future gravitational-wave observations.
Although a comprehensive study of the rich phenomenology is beyond the scope
of this thesis, we will identify a few robust observational signatures that
are worth exploring further.

We will begin, in §1, with a discussion of the most likely state of the cloud
before it experiences any resonances. This initial state is prepared via
superradiance and is sufficiently long-lived to be subsequently observed with
ground-based and space-based observatories. We will entertain the possibility
that all of the interesting physics of the resonances and the associated
backreaction on the orbit occurs within the sensitivity of gravitational-wave
experiments. In §2, we start by determining the regions in parameter space for
which resonant transitions can occur within the frequency bands of present and
future detectors. For this range of parameters, we will then determine which
effects can be observed due to the existence of one or several resonant
transitions. In §3, we discuss the states of the boson cloud after each
resonance transition and explain how they can be probed during the inspiral.

####  1 Initial States of the Cloud

As described in §2, superradiant growth occurs when the angular velocity of
the black hole is larger than the angular phase velocity of a quasi-bound
state. For future convenience, we repeat the superradiance condition here,

$\Omega_{\mathrm{H}}>\frac{\omega}{m}\,.$ (1)

The resulting boson cloud will first populate the dominant growing mode, with
$m=1$, until the superradiance condition saturates at
$\Omega_{\mathrm{H}}=\omega/m\approx\mu$. This mode remains stable for a long
time, as it only decays via gravitational-wave emission on a much longer
timescale. After that, the fastest growing $m=2$ mode will continue to
superradiantly drain mass and angular momentum from the black hole, until it
too becomes stable when $\Omega_{\mathrm{H}}=\omega/2$. While this process
can, in principle, continue towards larger values of $m$, each subsequent
growth occurs over timescales that are much longer than the previous one, cf.
(29) and (32), so that only the first few $m$ modes can be produced on
astrophysical timescales.

Both the mass $M_{c,0}$ and the angular momentum $S_{c}\equiv m\hskip
1.0ptS_{c,0}$ contained in each growing mode increase during superradiance,
until $\Omega_{\mathrm{H}}=\omega/m$. The spin of the black hole at saturation
is

$\frac{a}{M}=\frac{4m(M\omega)}{m^{2}+4(M\omega)^{2}}=\frac{4\alpha}{m}+\mathcal{O}(\alpha^{3})\,.$
(2)

Using angular momentum conservation and the relation $M_{c,0}=\omega\hskip
1.0ptS_{c,0}$ [Bekenstein:1973mi], we can estimate the final mass and angular
momentum stored in each mode. The results are summarized in Table 1. These
estimates are rather conservative, as they ignore the re-absorption of the
lower $m$ modes, which could spin up the black hole and further enhance the
amplitude of the cloud. Within this approximation, we find that $M_{c,0}$ and
$S_{c,0}$ are suppressed by a power of $\alpha$ for modes with $m>1$. In the
following, we will refer to the $m=1$ modes as “ground states,” and call the
$m>1$ modes “excited states.” However, we stress that this terminology is only
meant to distinguish between the growing modes, and that the ground state is
not the lowest frequency mode of the cloud (see Figs. 1 and 4).

|  $m=1$ |  $m=2$ |  $m=3$ |  $\cdots$
---|---|---|---|---
Scalar: $|n\hskip 0.5pt\ell\hskip 0.5ptm\rangle$ |  $|2\hskip 0.5pt1\hskip 0.5pt1\rangle$ |  $|3\hskip 0.5pt2\hskip 0.5pt2\rangle$ |  $|4\hskip 0.5pt3\hskip 0.5pt3\rangle$ |  $\cdots$
Vector: $|n\hskip 0.5pt\ell\hskip 0.5ptj\hskip 0.5ptm\rangle$ |  $|1\hskip 0.5pt0\hskip 0.5pt1\hskip 0.5pt1\rangle$ |  $|2\hskip 0.5pt1\hskip 0.5pt2\hskip 0.5pt2\rangle$ |  $|3\hskip 0.5pt2\hskip 0.5pt3\hskip 0.5pt3\rangle$ |  $\cdots$
$M_{c,0}/M$ |  $\alpha-4\alpha^{2}$ |  $\alpha^{2}$ |  $2\alpha^{2}/9$ |  $\cdots$
$S_{c,0}/M^{2}$ |  $1-4\alpha$ |  $\alpha$ |  $2\alpha/9$ |  $\cdots$
Table 1:  List of the dominant growing modes for scalar and vector fields.
Shown also are the mass $M_{c,0}$ and angular momentum $S_{c,0}$ extracted by
each mode in the limit $\alpha\ll 1$. For the $m=1$ modes, we assumed that the
black hole was initially maximally spinning.

##### Ground states

It is natural to first consider the fastest growing modes, with $m=1$. These
are the states $|2\hskip 0.5pt1\hskip 0.5pt1\rangle$ and $|1\hskip
0.5pt0\hskip 0.5pt1\hskip 0.5pt1\rangle$ for the scalar and vector clouds,
respectively. In the limit $\alpha\ll 1$, the typical growth timescales of
these states are

$\displaystyle\Gamma^{-1}_{211}$ $\displaystyle\simeq\frac{10^{6}\,\hskip
0.5pt{\mathrm{yrs}}}{\tilde{a}}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.019}{\alpha}\right)^{9}\mathrlap{\qquad\,\,\,\text{(scalar)}\,,}$
(3) $\displaystyle\Gamma^{-1}_{1011}$ $\displaystyle\simeq\frac{10^{6}\,\hskip
0.5pt{\mathrm{yrs}}}{\tilde{a}}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.0033}{\alpha}\right)^{7}\mathrlap{\qquad\text{(vector)}\,,}$

where $\tilde{a}\equiv a/M\leq 1$ is the dimensionless spin of the black hole.
We see that for $\alpha\gtrsim 0.019$ (scalar) and $\alpha\gtrsim 0.0033$
(vector) the clouds grow quickly on the timescale that is shorter than the
typical merger time ($\sim 10^{6}\,\hskip 0.5pt{\mathrm{yrs}}$). Since these
clouds emit continuous, monochromatic gravitational waves, they gradually
deplete over timescales of order [Arvanitaki:2010sy, Baumann:2018vus,
Yoshino:2014, Baryakhtar:2017ngi, East:2017mrj, East:2018glu]

$\displaystyle T_{211}$ $\displaystyle\simeq
10^{8}\,\text{yrs}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.07}{\alpha}\right)^{15}\mathrlap{\qquad\
\text{(scalar)}\,,}$ (4) $\displaystyle T_{1011}$ $\displaystyle\simeq
10^{8}\,\text{yrs}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.01}{\alpha}\right)^{11}\mathrlap{\,\,\qquad\text{(vector)}\,,}$

where we have assumed that $M_{c}\simeq\alpha M$. Requiring these modes to be
stable on astrophysical timescales, $T\gtrsim 10^{8}$ years, puts upper bounds
on the values of $\alpha$. For stellar-mass black holes, $M\sim 60M_{\odot}$,
we impose $\alpha\lesssim 0.07$ (scalar) and $\alpha\lesssim 0.01$ (vector),
while for supermassive black holes, $M\sim 10^{6}M_{\odot}$, the range can
increase to $\alpha\lesssim 0.18$ (scalar) and $\alpha\lesssim 0.04$ (vector).

Figure 5: Relevant timescales for the ground states and first excited states
of the scalar (_left_) and vector (_right_), as functions of $\alpha$ and for
fixed $M=60M_{\odot}$. Note that both the growth time and the lifetime scale
linearly with $M$. We have indicated the timescales that are shorter than
typical merger times ($\sim 10^{6}$ years) and general astrophysical processes
($\sim 10^{8}$ years), as well as the age of the Universe ($\sim 10^{10}$
years).

##### Excited states

To probe larger values of $\alpha$, we must instead consider excited states
with $m\geq 2$, which are much longer lived. These excited states also take
much longer to form than the ground states. For example, the typical growth
timescales for the scalar $|3\hskip 0.5pt2\hskip 0.5pt2\rangle$ and vector
$|2\hskip 0.5pt1\hskip 0.5pt2\hskip 0.5pt2\rangle$ modes are

$\displaystyle\Gamma^{-1}_{322}$
$\displaystyle\simeq\frac{10^{6}\,\text{yrs}}{\tilde{a}}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.11}{\alpha}\right)^{13}\mathrlap{\qquad\
\ \,\text{(scalar)}\,,}$ (5) $\displaystyle\Gamma^{-1}_{2122}$
$\displaystyle\simeq\frac{10^{6}\,\text{yrs}}{\tilde{a}}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.046}{\alpha}\right)^{11}\mathrlap{\,\,\qquad\text{(vector)}\,.}\hskip
20.0pt$

At the same time, these excited states are also much more stable than the
ground states, depleting via gravitational-wave emission over the timescales

$\displaystyle T_{322}$ $\displaystyle\simeq
10^{8}\,{\mathrm{yrs}}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.22}{\alpha}\right)^{20}\mathrlap{\,\,\,\,\,\qquad\text{(scalar)}\,,}$
(6) $\displaystyle T_{2122}$ $\displaystyle\simeq
10^{8}\,{\mathrm{yrs}}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.08}{\alpha}\right)^{16}\mathrlap{\,\,\qquad\,\,\,\text{(vector)}\,.}\hskip
20.0pt$

Demanding the cloud to be stable on astrophysical timescales, $T\gtrsim
10^{8}\,{\mathrm{years}}$, leads to $\alpha\lesssim 0.22$ (scalar) and
$\alpha\lesssim 0.08$ (vector) for $M\sim 60M_{\odot}$, and $\alpha\lesssim
0.44$ (scalar) and $\alpha\lesssim 0.19$ (vector) for $M\sim
10^{6}M_{\odot}$.121212While the gravitational-wave emission rate of the
scalar $|3\hskip 0.5pt2\hskip 0.5pt2\rangle$ mode is known [Yoshino:2014], the
analogous rate for the vector $|2\hskip 0.5pt1\hskip 0.5pt2\hskip
0.5pt2\rangle$ mode is not. Nevertheless, since the emission rate for the
$\ell=0$ modes of both the scalar and vector states share the same
$\alpha$-scaling, we assume that a similar relation also applies for the
scalar and vector excited states. In other words, we assume that $|2\hskip
0.5pt1\hskip 0.5pt2\hskip 0.5pt2\rangle$ decays at the same rate as $|2\hskip
0.5pt1\hskip 0.5pt1\rangle$. The only difference between the lifetimes
$T_{211}$ and $T_{2122}$ in (4) and (6) is then that we take the initial mass
of the cloud to be $M_{c,0}=\alpha M$ in the former and $M_{c,0}=\alpha^{2}M$
in the latter. Beyond these limits, we must consider excited states with
larger $m$. However, we do not expect these higher excited states to be
qualitatively different, and so we will only focus on the phenomenology of the
$m=1$ and $m=2$ states.

The cloud’s history is summarized in Figure 5. The black hole will first grow
the ground state (blue, solid). We require that this process takes place on a
short enough timescale ($\lesssim 10^{6}\,{\mathrm{yrs}}$) to be observable,
which sets a lower bound on $\alpha$. This ground state will then decay via
gravitational-wave emission (blue, dashed). We find an upper bound on $\alpha$
by demanding that this state is suitably long-lived ($\gtrsim
10^{8}\,{\mathrm{yrs}}$). For larger $\alpha$, we are more likely to observe
the cloud in an excited state, which grows (yellow, solid) and decays (yellow,
dashed) via the same mechanisms, but on longer timescales.

####  2 Resonant Transitions

The states presented above can live long enough to be accessible to
gravitational-wave observatories. Resonant transitions can then occur in band
during the binary’s inspiral phase. As a consequence, there will be a series
of distinct signatures which can reveal the nature of the gravitational atom.
We will first describe the observational characteristics of a single
transition, and then discuss how observing multiple, sequential transitions
can be used to further elucidate the properties of the boson clouds.

##### Single transition

Consider a transition between states with principal quantum numbers $n_{a}$
and $n_{b}$. The orbital frequency at the resonance, $\Omega_{r}$, implies the
following frequency of the emitted gravitational waves

$f_{\rm res}=\frac{\Omega_{r}}{\pi}=1.3\times
10^{-2}\,\text{Hz}\,\left(\frac{60M_{\odot}}{M}\right)\left(\frac{\alpha}{0.07}\right)^{3}\varepsilon_{{\mathrm{B}}}\,,$
(7)

where we have introduced the transition-dependent quantity

$\varepsilon_{{\mathrm{B}}}\equiv\frac{36}{5}\frac{2}{|\Delta
m|}\\!\left|\frac{1}{n_{a}^{2}}-\smash{\frac{1}{n_{b}^{2}}}\right|,$ (8)

and chosen the normalization such that $\varepsilon_{{\mathrm{B}}}\equiv 1$
for the first Bohr transition of the scalar ground state, $|2\hskip
0.5pt1\hskip 0.5pt1\rangle\to|3\hskip
0.5pt1\,{\scalebox{0.75}[1.0]{$-$}}{1}\rangle$. The resonance frequency for
this specific transition, as a function of $M$ and $\alpha$, is shown in Fig.
6. Indicated are the ranges for which the signal falls into the frequency
bands of current and future gravitational-wave observations.

Figure 6: Resonance frequency of the Bohr transition $|2\hskip 0.5pt1\hskip
0.5pt1\rangle\to|3\hskip 0.5pt1\hskip 0.5pt\hskip
0.5pt{{\scalebox{0.75}[1.0]{$-$}}1}\rangle$, as a function of $M$ and
$\alpha$. This is representative for all Bohr transitions, since the frequency
(7) is relatively insensitive to the choice of transition, under which
$f_{\mathrm{res}}\to\varepsilon_{\mathrm{B}}f_{\mathrm{res}}$. Depending on
the values of $M$ and $\alpha$, the resonance may fall in the band of pulsar
timing arrays (PTA), the LISA observatory, proposed deciHertz experiments, or
currently operating ground-based LIGO/Virgo detectors.  Figure 7: Total time
$\Delta t_{\mathrm{tot}}$ spent in the sinking Bohr transition $|2\hskip
0.5pt1\hskip 0.5pt1\rangle\to|3\hskip 0.5pt1\hskip 0.5pt\hskip
0.5pt{{\scalebox{0.75}[1.0]{$-$}}}1\rangle$. On the left, we plot this as
function of $\alpha$ and $q$, for fixed mass $M=60M_{\odot}$. On the right, we
instead fix $\alpha=0.07$ and plot the total time as a function of $M$ and
$q$. For both, we provide the corresponding resonance frequency
$f_{\mathrm{res}}$ on the top axis. We plot constant contours of the ratio
$|\Delta t_{\mathrm{c}}/\Delta t|$ as dashed light blue lines. The cloud’s
backreaction on the orbit becomes stronger as $q$ decreases, shortening the
time it takes for the binary to move through the transition. In blue, we
indicate the region in parameter space where the adiabatic condition (10) is
violated and the backreaction is strong enough that we lose predictive control
and the estimation (9) is inapplicable.

An important additional characteristic is the time spent within the resonance
band. Given the resonance bandwidth of $\Delta\Omega\sim 2\eta$, and using (7)
and (9), we can estimate this as

$\Delta t_{\rm tot}\,\simeq\,\Delta t\pm\Delta t_{c}\,,$ (9)

where $+/-$ represents floating/sinking orbits, respectively. Here, $\Delta
t\simeq 2\eta/\gamma$ is the time it takes the inspiral to move through the
resonance in the absence of backreaction, whereas $\Delta t_{c}\simeq
3R_{J}|\Delta m\Delta E|/2\gamma$ is the additional contribution from the
presence of the cloud,

$\displaystyle\Delta t$
$\displaystyle\simeq\,4\,\text{yrs}\,\left(\frac{M}{60M_{\odot}}\right)\frac{1}{(1+q)^{2/3}}\left(\frac{0.07}{\alpha}\right)^{8}\left(\frac{R_{ab}}{0.3}\right)\varepsilon_{\mathrm{B}}^{-8/3}\,,$
(10) $\displaystyle\Delta t_{c}$
$\displaystyle\simeq\,1\,\text{yr}\,\left(\frac{M}{60M_{\odot}}\right)\frac{(1+q)^{2/3}}{q^{2}}\left(\frac{S_{c,0}}{M^{2}}\right)\left(\frac{0.07}{\alpha}\right)^{7}\left(\frac{|\Delta
m|}{2}\right)^{2}\varepsilon_{\mathrm{B}}^{-7/3}\,.$ (11)

In (10), we used (22) to extract the value of $\eta_{ab}$ as a function of
$R_{ab}$. To assess the strength of the backreaction, it is useful to take the
ratio between these two timescales,

$\frac{\Delta t_{c}}{\Delta
t}\simeq\frac{1}{4}\frac{(1+q)^{4/3}}{q^{2}}\left(\frac{\alpha}{0.07}\right)\left(\frac{S_{c,0}}{M^{2}}\right).$
(12)

In all cases, we find that the backreaction crucially depends on the angular
momentum stored in the cloud, $S_{c,0}$, as well as the mass ratio, $q$.
Notice that, for excited states, $S_{c,0}$ is suppressed by a power of
$\alpha$, and the backreaction is typically weaker. Alternatively, we can also
estimate the size of the backreaction through the difference in the number of
orbital cycles spent in the resonance band with and without the cloud

$\displaystyle\Delta N_{c}$ $\displaystyle=f_{\rm res}\Delta
t_{c}\,\simeq\,10^{5}\,\frac{(1+q)^{2/3}}{q^{2}}\left(\frac{S_{c,0}}{M^{2}}\right)\left(\frac{0.07}{\alpha}\right)^{4}\left(\frac{|\Delta
m|}{2}\right)\varepsilon^{-4/3}_{\mathrm{B}}\,.$ (13)

Since binary searches are sensitive to $\gtrsim\mathcal{O}(1)$ difference in
the number of orbital cycles between signal and template waveforms, the
estimate (13) suggests that backreaction on the orbit can dramatically alter
the gravitational-wave signal, introducing a substantial dephasing with
respect to the waveform model without the cloud. The effect is especially
prominent for $q\ll 1$, yielding a significant dephasing for intermediate
(IMRIs) and extreme mass ratio inspirals (EMRIs), where a small black hole
perturbs a larger cloud.131313Requiring these resonant transitions to be
adiabatic, cf. (1), weakly constrains the mass ratio $q\gtrsim\alpha^{5}$.

Figure 8: Total time $\Delta t_{\mathrm{tot}}$ spent in the floating Bohr
transition $|3\hskip 0.5pt2\hskip 0.5pt2\rangle\to|2\hskip 0.5pt0\hskip
0.5pt0\rangle$, for a parent black hole of mass $M=60M_{\odot}$, as a function
of $\alpha$, $f_{\mathrm{res}}$, and $q$. Increasing the fiducial mass $M$
reduces the resonance frequency and enhances the floating time, according to
$f_{\mathrm{res}}\to(60M_{\odot}/M)f_{\mathrm{res}}$ and ${\Delta
t_{\mathrm{tot}}\to(M/60M_{\odot})\Delta t_{\mathrm{tot}}}$, for fixed
$\alpha$. We plot constant contours of the ratio $\Delta t_{\mathrm{c}}/\Delta
t$ as dashed light blue lines, indicating where the inspiral spends an
additional $1\%$, $10\%$, and $100\%$ amount of time in the transition.

The dephasing due to the backreaction on the orbit is a robust signature of
boson clouds in binary systems. However, whether the orbit floats or sinks
depends on its orientation (co-rotating or counter-rotating with respect to
the black hole’s spin), and the nature of the transition; see the discussion
below (6). For example, for the transition $|2\hskip 0.5pt1\hskip
0.5pt1\rangle\to|3\hskip 0.5pt1\hskip 0.5pt\hskip
0.5pt{{\scalebox{0.75}[1.0]{$-$}}1}\rangle$, which occurs for counter-rotating
orbits, the cloud absorbs angular momentum from the orbit, which therefore
shrinks faster, thus reducing the time it takes for the binary to move through
the transition. We plot the total time it takes for the inspiral to cross this
resonance in Fig. 7. Notice that backreaction dominates for $q\ll 1$, when the
cloud contains a large fraction of the total angular momentum. Especially in
this limit, there is a danger that the system moves through the transition too
quickly, violating the condition (10) and hence destroying the validity of
(9). In that case, the binary’s orbit can receive a significant kick from the
cloud (cf. Fig. 3), potentially making the orbit highly eccentric. As
discussed in Section 2, the cloud’s dynamics then becomes highly non-adiabatic
and depends sensitively on the backreacted dynamics of the orbit. A precise
characterization of this interesting region in parameter space thus requires
either different analytic techniques or direct numerical simulations.

For excited states, on the other hand, Bohr transitions can also occur for co-
rotating orbits. For example, during the transition $|3\hskip 0.5pt2\hskip
0.5pt2\rangle\to|2\hskip 0.5pt0\hskip 0.5pt0\rangle$,141414Since we only focus
on Bohr transitions, we will assume that the initial frequency of this binary
is higher than the resonance frequencies of the hyperfine $|3\hskip
0.5pt2\hskip 0.5pt2\rangle\to|3\hskip 0.5pt2\hskip 0.5pt0\rangle$ and fine
$|3\hskip 0.5pt2\hskip 0.5pt2\rangle\to|3\hskip 0.5pt0\hskip 0.5pt0\rangle$
transitions, such that they were missed by the binary inspiral. the cloud
loses angular momentum, thereby producing a floating orbit. Fig. 8 illustrates
the total time it takes for the binary inspiral to move through this
transition. As a reference we choose the fiducial value $M=60M_{\odot}$, but
the plot can be easily scaled to any value of $M$. The thick dashed line
(labeled 100%) denotes parameters for which the presence of the cloud doubles
the time spent in the resonance region. During this transition, the inspiral
floats and the orbital frequency $\Omega(t)$ remains roughly constant at the
resonance value (8). The backreaction time again dominates for $q\ll 1$ but,
unlike for the sinking orbits, this regime is still under analytic control. As
before, this effect is particularly relevant for IMRIs and EMRIs, in which
case the float can be extremely long-lived—much longer than the typical
timescale of gravitational-wave observatories ($\sim 10$ years). They may thus
serve as another source of continuous monochromatic gravitational waves.

As we have demonstrated, for favorable values of $M$ and $M_{*}$, the cloud
provides large corrections to the inspiral at a frequency that falls within
the bands of future gravitational-wave observatories. A measurement of this
resonance frequency, together with a measurement of $q$ and $M$, would then
provide a strong constraint on the allowed values of $\alpha$, and thus the
boson’s mass $\mu$. However, it is important to note that the determination of
a single resonance frequency does not uniquely fix the boson mass, since
additional information is needed to identify which specific transition
occurred. This degeneracy is broken if we observe multiple transitions.

##### Multiple transitions

As seen in Figs. 7 and 8, the binary passes through the transition relatively
quickly for moderate values of $\alpha$ and $q$. Since subsequent resonance
frequencies may be nearby, this region of parameter space suggests that we may
observe _multiple_ resonant transitions in a given observational window. In
Fig. 9, we show two representative examples for the evolution of the scalar
cloud, starting from the ground state $|2\hskip 0.5pt1\hskip 0.5pt1\rangle$
and first excited state $|3\hskip 0.5pt2\hskip 0.5pt2\rangle$, respectively.
Notice that, because the gravitational atom has a finite _ionization energy_ ,
cf. (25) and (27), there is a maximum frequency at which resonant transitions
can occur,

$f_{\mathrm{max}}\simeq
0.2\,\,{\mathrm{Hz}}\left(\frac{60M_{\odot}}{M}\right)\left(\frac{\alpha}{0.07}\right)^{3}.$
(14)

Together with the selection rules discussed in §2, this explains why the
transition trees151515The dominant $\ell_{*}=2$ perturbation mediates all of
the resonances displayed in these transition trees, except for the $|3\hskip
0.5pt1\,\hskip 0.5pt{\scalebox{0.75}[1.0]{$-$}}1\rangle\to|6\hskip
0.5pt2\,\hskip 0.5pt{\scalebox{0.75}[1.0]{$-$}}2\rangle$ and
$|3100\rangle\to|62j\,\hskip 0.5pt{\scalebox{0.75}[1.0]{$-$}}1\rangle$
resonances, which are instead induced by the weaker $\ell_{*}=3$ perturbation.
This is because, after experiencing an earlier resonance in these cases, the
binary has an orbital frequency that already exceeds all resonance frequencies
that can be excited from the newly prepared $|n=3,\ell=1\rangle$ states by
$\ell_{*}=2$. A further transition is nevertheless still possible through
$\ell_{*}=3$, which supports perturbations with $|\Delta m|=1$ (see Appendix
9.C) and hence induces transitions at higher frequencies, cf. (8). pictured in
Figs. 9 and 10 terminate at a particular _end state_.161616In principle, there
is also a continuum of states above this ionization energy where the cloud
becomes unbound from the black hole. However, since the gravitational
perturbation is very weak, it is unlikely these states can be appreciably
occupied by the sort of resonant transitions we described in §2.  A more
detailed numerical analysis may then be needed to fully incorporate all of the
relevant physics.

Figure 9: Evolution of the $|2\hskip 0.5pt1\hskip 0.5pt1\rangle$ (top) and
$|3\hskip 0.5pt2\hskip 0.5pt2\rangle$ (bottom) states, during a counter-
rotating and co-rotating inspiral, respectively (for $\alpha=0.07$, $q=1$ and
$M=60M_{\odot}$). Each history contains a series of floating ( ) and sinking (
) orbits, separated by periods of “normal” inspiral evolution ( ), with only
weak perturbative mixing. As we discuss in the main text, the $|2\hskip
0.5pt0\hskip 0.5pt0\rangle$ state has a large decay width. Unless the binary
moves quickly to the next transition, which occurs only for $q\gg 1$, this
forces the cloud to deplete before experiencing the next resonance. We
indicate this by the reduced opacity of the states after $|2\hskip
0.5pt0\hskip 0.5pt0\rangle$.

As a rough guide as to whether or not multiple transitions can be observed, we
estimate the time it takes the inspiral to move from one resonance frequency
$f_{{\rm res}}^{(i)}$ to another $f_{{\rm res}}^{(j)}$. Assuming quasi-
circular adiabatic evolution, we find

$\Delta T_{i\to
j}=2.5\,\text{yrs}\,\left(\frac{M}{60M_{\odot}}\right)\frac{(1+q)^{1/3}}{q}\left(\frac{0.07}{\alpha}\right)^{8}\left(\varepsilon_{{\mathrm{B}},(i)}^{-8/3}-\varepsilon_{{\mathrm{B}},(j)}^{-8/3}\right),$
(15)

where $\varepsilon_{{\mathrm{B}},(i)}$ represents the parameter (8) associated
to the $i$-th transition. For example, we have
$\varepsilon_{{\mathrm{B}},(1)}=1$ and $\varepsilon_{{\mathrm{B}},(2)}=6/5$
for the transitions $|2\hskip 0.5pt1\hskip 0.5pt1\rangle\to|3\hskip
0.5pt1\,\hskip 0.5pt{\scalebox{0.75}[1.0]{$-$}}1\rangle$ and $|3\hskip
0.5pt1\,\hskip 0.5pt{\scalebox{0.75}[1.0]{$-$}}1\rangle\to|6\hskip
0.5pt2\,\hskip 0.5pt{\scalebox{0.75}[1.0]{$-$}}2\rangle$, respectively.
Comparing (15) with (10), we find that, as long as the binary moves through a
single Bohr transition in a reasonable amount of time (say $1$ year), we
should expect to observe a second Bohr transition on a similar timescale.
Furthermore, since the rate at which the binary sweeps through the frequency
accelerates, we should generically expect to observe several transitions.

Observing the dephasing of sequential Bohr transitions allows us to break
degeneracies among the different parameters. For instance, let us consider
successive Bohr transitions, labeled $i$ and $i+1$, between states with
principal quantum numbers $n_{a}$, $n_{b}$, and $n_{c}$. The azimuthal angular
momentum differences between the states are $\Delta m_{ab}$ and $\Delta
m_{bc}$. The ratio of the resonance frequencies then is

$\frac{f_{{\mathrm{res}}}^{(i+1)}}{f_{{\mathrm{res}}}^{(i)}}=\frac{\varepsilon_{{\mathrm{B}},(i+1)}}{\varepsilon_{{\mathrm{B}},(i)}}=\left|\frac{\Delta
m_{ab}}{\Delta
m_{bc}}\right|\left(\frac{n_{a}}{n_{c}}\right)^{2}\left|\frac{n^{2}_{c}-n_{b}^{2}}{n_{b}^{2}-n_{a}^{2}}\right|,$
(16)

which crucially depends only on integer quantities.171717Note that the
presence of fine or hyperfine corrections does not affect our conclusions. An
observed sequence of these ratios then provides a fingerprint with which a
particular transition history can be identified. For instance, the sequence of
successive frequency ratios {1.2, 1.33, 1.03, 4.27, 1.005} of the counter-
rotating history in Fig. 9 is clearly different from that of the co-rotating
history, {1.35, 5, 1.02}. Even though some of the ratios in these two
histories are nearly equal, they never appear in the same order, and so we may
use this sequence as a unique identifier for each history. This, combined with
a measurement of the black hole mass $M$, can then be used to infer the boson
mass $\mu$. However, there is a caveat. As illustrated in Figs. 9 and 10, the
cloud may deplete considerably before reaching the next resonance.
Fortunately, as we discuss next, this can also be used as a unique signature
of the boson cloud.

##### Cloud depletion

Throughout the above discussion, we have implicitly assumed that the cloud
does not appreciably evolve away from the resonant transitions. However, in
the later stages of the evolution, the cloud may occupy a rapidly decaying
mode and is quickly reabsorbed back into the black hole, before arriving at
the next resonance. After that, the effects of the cloud on the gravitational-
wave signal become negligible.

Using (17), we can estimate the number of $e$-folds of decay between two
successive transitions. At leading-order in $\alpha$, we find 181818In order
to obtain the $\alpha$-scaling in (17) , we also used the fact that the
central black hole is slowly spinning after experiencing superradiance, cf.
(2).

$|\Gamma|\Delta T_{i\to
j}=\mathcal{C}\,\frac{(1+q)^{1/3}}{q}\alpha^{2j+2\ell-2}\left(\varepsilon_{{\mathrm{B}},(i)}^{-8/3}-\varepsilon_{{\mathrm{B}},(j)}^{-8/3}\right),$
(17)

where $\Gamma$ is the decay rate of the state that is occupied after the
$i$-th transition. Typically, the dimensionless coefficient
$\mathcal{C}\lesssim\mathcal{O}(0.1)$ for the states of interest, and can
depend sensitively on the state’s quantum numbers. Depletion is significant
when the ratio in (17) is greater than $1$.

For scalar fields, the fastest decaying modes have $\ell=0$, in which case the
estimator in (17) scales inversely with $\alpha$. A cloud that populates these
states will therefore almost always deplete before it reaches the next
resonance. The cloud only survives when $q\gtrsim\alpha^{-3}$, namely when the
cloud is on the smaller black hole, in which case the system moves quickly
enough to the next transition. Similarly, for vector clouds, a fast decaying
state with $\ell=0$ and $j=1$ becomes occupied along the transition tree in
Fig. 10.191919Notice that, because we are assuming that the central black hole
has a spin that saturates at the superradiance condition, the state
$|2011\rangle$ turns into a decaying mode, despite having $m>0$. Unless $q\gg
1$, it therefore suffers the same fate as the scalar cloud. However, a vector
cloud still experiences more resonances over a wider range of mass ratios than
a scalar cloud.

Although a rapid depletion of the cloud prevents us from exploring later
resonant transitions, it is a unique feature of gravitational atoms in binary
systems [Baumann:2018vus] that helps to distinguish them from other exotic
compact objects like boson stars, e.g. [Sennett:2017etc]. Using (17), we see
that, for $q\ll 1$, a cloud that populates a fast-decaying state will
typically deplete before it reaches the next resonance. Nevertheless, these
IMRIs and EMRIs are precisely the binaries for which we only expect to see a
single transition in a reasonable observational period (15). Furthermore, they
are also the binaries for which backreaction is most significant. This
suggests that, to probe these types of binary systems, we will typically need
to hunt for long floats or strongly kicked orbits, rather than multiple
correlated transitions.

Figure 10:  Evolution of the $|1\hskip 0.5pt0\hskip 0.5pt1\hskip
0.5pt1\rangle$ (top) and $|2\hskip 0.5pt1\hskip 0.5pt2\hskip 0.5pt2\rangle$
(bottom) states during a counter-rotating inspiral (for $\alpha=0.01$ and
$\alpha=0.07$, respectively). In both cases, $q=1$ and $M=60M_{\odot}$. Notice
that the ground state of the vector cloud $|1\hskip 0.5pt0\hskip 0.5pt1\hskip
0.5pt1\rangle$ only experiences a single transition. The excited state
$|2\hskip 0.5pt1\hskip 0.5pt2\hskip 0.5pt2\rangle$, on the other hand, mimics
the history of the scalar ground state $|2\hskip 0.5pt1\hskip 0.5pt1\rangle$.
Yet, we see that vector transitions involve superpositions of many states. The
large decay width of the state $|2\hskip 0.5pt0\hskip 0.5pt1\hskip
0.5pt1\rangle$ makes it unlikely for the cloud to survive after that point,
unless $q\gg 1$. As in Fig. 9, we indicate this by reducing the opacity of the
states after the $|2\hskip 0.5pt0\hskip 0.5pt1\hskip 0.5pt1\rangle$ state.

####  3 Post-Resonance Evolution

As argued above, observing multiple successive resonances provides a detailed
mapping of the spectral properties of the cloud, allowing us to extract the
mass $\mu$ of the ultralight boson. However, since this fingerprint depends
only on the energy differences between states, it can fail to distinguish
between scalar and vector clouds, which mainly differ by the number of states
involved in a transition. Fig. 10 shows transitions of the vector cloud,
starting from the ground state $|1\hskip 0.5pt0\hskip 0.5pt1\hskip
0.5pt1\rangle$ and the first excited state $|2\hskip 0.5pt1\hskip 0.5pt2\hskip
0.5pt2\rangle$, respectively. While the dynamics of the vector ground state is
rather unique (involving only a single transition), distinguishing between
scalar and vector clouds for other histories is more difficult. In particular,
the vector state $|2\hskip 0.5pt1\hskip 0.5pt2\hskip 0.5pt2\rangle$ in Fig. 10
has a similar transition history as the scalar state $|2\hskip 0.5pt1\hskip
0.5pt1\rangle$ in Fig. 9. It is thus non-trivial to disentangle these two
histories using only information about their resonance frequencies. As we
discuss here, studying finite-size effects can help us break this
degeneracy.202020In principle, this degeneracy can also be lifted through
measurements of $\partial_{0}\Omega$ and $\Delta t_{\rm tot}$ across a
resonance, as the mode $|2\hskip 0.5pt1\hskip 0.5pt2\hskip 0.5pt2\rangle$
carries less angular momentum than the mode $|2\hskip 0.5pt1\hskip
0.5pt1\rangle$ (for the same value of $\alpha$), and this should have a weaker
effect on the orbit.

##### Finite-size effects

There is a major qualitative difference between the evolution of scalar and
vector clouds that we can exploit to distinguish them: vector transitions
typically involve more than two states, while scalar transitions do not. As
discussed in §3, the result of these multi-state vector transitions is a
superposition of states, each with their own shape and characteristic
frequency. There are then large oscillations in the shape of the vector cloud,
at frequencies which depend on the energy differences $\Delta E_{ab}$ between
the occupied states. Detecting these oscillatory finite-size effects is thus a
smoking gun of particles with spin.212121We expect the same phenomena to apply
also to higher-spin particles. Namely, for a given $n$, $\ell$ and $m$, there
will be a larger number of states, and the finite-size effects can oscillate
at a larger number of frequencies.

We can illustrate this effect quantitatively using the first Bohr transition
of the vector excited state, $|2\hskip 0.5pt1\hskip 0.5pt2\hskip
0.5pt2\rangle\to|3\hskip 0.5pt1\hskip 0.5ptj\hskip 0.5pt0\rangle$. After the
transition, the state evolves into a superposition with has negligible
population in the $|3\hskip 0.5pt1\hskip 0.5pt1\hskip 0.5pt0\rangle$ mode:
222222The gravitational perturbation only mediates degenerate subspace
transitions between $|3\hskip 0.5pt1\hskip 0.5pt0\hskip 0.5pt0\rangle$ and
$|3\hskip 0.5pt1\hskip 0.5pt2\hskip 0.5pt0\rangle$. As a consequence, the
final state is an equally weighted sum of these two states, independent of
both $q$ and $\alpha$.

$|\psi(t)\rangle\approx\frac{1}{\sqrt{2}}\hskip
2.0pte^{-iE_{3100}t}\left(|3\hskip 0.5pt1\hskip 0.5pt0\hskip
0.5pt0\rangle+e^{-i\Delta E_{20}t}|3\hskip 0.5pt1\hskip 0.5pt2\hskip
0.5pt0\rangle\right),$ (18)

where we have defined $\Delta E_{20}\equiv E_{3120}-E_{3100}$. The dominant
energy difference is set by the fine-structure splitting, cf. (27) and (28),
and so the frequency of these oscillations is roughly

$\frac{\Delta E_{20}}{2\pi}\approx 6\times
10^{-5}\,{\mathrm{Hz}}\,\left(\frac{60M_{\odot}}{M}\right)\left(\frac{\alpha}{0.07}\right)^{5}\,.$
(19)

At the same time, the cloud’s axisymmetric quadrupole moment inherits these
oscillations, which can be parameterized as

$\kappa_{c}(t)=\kappa_{0}\left(1-2\sqrt{2}\cos(\Delta E_{20}\,t)\right)\,,$
(20)

with $\kappa_{c}$ defined in §2 and we have ignored corrections that are
suppressed by $\alpha$. The $\kappa_{0}$ value represents the quadrupole
moment of the state $|3\hskip 0.5pt1\hskip 0.5pt0\hskip
0.5pt0\rangle+i|3\hskip 0.5pt1\hskip 0.5pt2\hskip 0.5pt0\rangle$. For
$\alpha=0.07$ and $M=60M_{\odot}$, the cloud’s mass distribution oscillates
between bulging at the black hole’s spin axis and bulging along its equatorial
plane at a period of roughly $4.6$ hours. Since it then takes $1.2$ years to
evolve towards the next transition, cf. (15), we could potentially observe
$2.2\times 10^{3}$ of these cycles. If, in addition, we include the relatively
fast depletion of the $|3\hskip 0.5pt1\hskip 0.5pt0\hskip 0.5pt0\rangle$ mode,
these oscillations decay on a timescale of about a month.

Because the frequency of these oscillations is so heavily $\alpha$-suppressed,
observing them is a challenge. For example, a single oscillation of the
$|3\hskip 0.5pt2\hskip 0.5ptj\,\hskip
0.5pt{\scalebox{0.75}[1.0]{$-$}}1\rangle$ superposition produced by the vector
ground state can take decades for $\alpha=0.01$ and $M=60M_{\odot}$. This is
far outside present observational timescales. Similarly, the binary might
merge or encounter another resonance before the cloud has a chance to complete
a single cycle. This is typically the case for the end state of a transition
tree, like the $|8\hskip 0.5pt2\hskip 0.5ptj\,\hskip
0.5pt{\scalebox{0.75}[1.0]{$-$}}1\rangle$ superposition for the vector excited
state in Fig. 10. Though we can avoid these issues by considering supermassive
black holes and/or higher excited states where larger values of $\alpha$ are
allowed, we are more likely to detect these oscillations for the earlier Bohr
transitions. For instance, while the superpositions $|6\hskip 0.5pt2\hskip
0.5ptj\,\hskip 0.5pt{\scalebox{0.75}[1.0]{$-$}}1\rangle$ and $|7\hskip
0.5pt2\hskip 0.5ptj\,\hskip 0.5pt{\scalebox{0.75}[1.0]{$-$}}1\rangle$
oscillate $\sim\\!40$ times slower than the $|3\hskip 0.5pt1\hskip
0.5ptj\hskip 0.5pt0\rangle$ with frequency (19), they still execute hundreds
of cycles, which may be detected with high-precision templates.

Since finite-size effects enter at higher post-Newtonian orders, they are
difficult to measure during the early stages of the inspiral when the binary’s
relative velocity is small [Porto:2016pyg]. However, boson clouds can have
much larger multipole moments compared to black holes in isolation
[Baumann:2018vus], which can greatly enhance our chances of detecting them
through gravitational-wave precision measurements, even before the system
achieves high velocities in the merger phase. As we argued in §3, the
axisymmetric moments of the cloud roughly scale as $Q_{c}\sim M_{c}r_{c}^{2}$,
where $M_{c}$ and $r_{c}$ are the mass and Bohr radius of the cloud,
respectively. For an excited initial state, the dimensionless quadrupole
moment is then of the order
$\kappa_{c}\sim(M_{c}/M)\tilde{a}^{-2}\alpha^{-4}$, which is much larger than
the corresponding moment for the pure Kerr black hole, $\kappa=1$. The effects
of these large, fluffy clouds are amplified near the merger, and we might
expect that we can infer detailed information about the shape of the end
states in a transition tree by accurately modeling the finite-size effects in
this phase.

##### Decaying shapes

In §3, we described how finite-size effects can receive further time-dependent
changes when the cloud occupies decaying states. While this depletion
typically occurs over much longer timescales than the orbital period, it can
be significant when the decaying states have small angular momentum. For
examples, the decay times of the $|2\hskip 0.5pt0\hskip 0.5pt0\rangle$ and
$|2\hskip 0.5pt0\hskip 0.5pt1\hskip 0.5pt1\rangle$ modes—the main depletion
channels for the histories depicted in Figs. 9 and 10—are

$\displaystyle\left|\Gamma_{200}^{-1}\right|$ $\displaystyle\simeq
1\,\text{yr}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.014}{\alpha}\right)^{6}\mathrlap{\qquad\,\,\,\text{(scalar)}\,,}$
(21) $\displaystyle\left|\Gamma_{2011}^{-1}\right|$ $\displaystyle\simeq
1\,\text{yr}\left(\frac{M}{60M_{\odot}}\right)\left(\frac{0.045}{\alpha}\right)^{8}\mathrlap{\qquad\,\,\,\text{(vector)}\,.}$

We see that, even for moderate values of $\alpha$, these depletion effects can
be significant and are observable within the typical lifetimes of
gravitational-wave observatories. As pointed out in [Baumann:2018vus], this
change in the contribution from finite-size effects can also strongly indicate
the presence of a boson cloud.

###  5 Summary

In this chapter, we combined our findings in the previous two chapters and
explored the phenomenological implications of the gravitational collider.
Firstly, we incorporated the backreaction of the Landau-Zener transition on
the binary orbit, which was omitted in our analysis in Chapter 4. We found
that the transfer of angular momentum between the cloud and the orbit during
each resonance can cause large corrections to the gravitational-wave signal
from the binary. Notably, there is a dephasing with respect to the frequency
evolution without a cloud, arising from transient floating and sinking orbits,
cf. Figs. 2 and 3. For a floating orbit, the orbital frequency of the binary
is approximately constant and has a mean value that is given by the resonance
frequency. Since the adiabaticity of the orbital evolution is enhanced in this
case, the corresponding Landau-Zener transition involves a smooth and
prolonged transition between the initial and the final states of the cloud. On
the other hand, if the backreaction generates a sinking orbit, the orbital
frequency of the binary temporarily increases across the resonance frequency.
If this backreaction is sufficiently strong, the Landau-Zener transition can
turn an initially adiabatic orbit to evolve non-adiabatically. As a result,
the oscillatory features that we discovered in the previous chapter could be
imprinted on the binary motion, significantly enriching the orbit evolution of
the binary. The orbit could also receive a strong kick from the cloud, which
could potentially generate large orbit eccentricities and even unbind the
orbit.

Furthermore, we saw that time-dependent finite-size effects provide additional
information about the available states in the gravitational atom which, due to
the extended nature of the cloud, may be observed during the early inspiral
phase. Similarly, strong mixing with decaying modes during the resonant
transition can also deplete the cloud as it approaches merger. The induced
depletion of the cloud can have important phenomenological consequences, both
for the monochromatic signal emitted by the cloud and the finite-size effects
imprinted in the waveforms of the binary signal. The characteristic time
dependence of the signals thus becomes a very distinctive feature of the
dynamics of boson clouds in binary systems, which would not be present in
other scenarios, e.g. a boson star. Figures 9 and 10 illustrate a few such
examples for the evolutions of the scalar and vector clouds in binary systems.
Crucially, observing these signatures would allow us to infer the microscopic
properties of the ultralight boson fields. In particular, the resonance
frequencies determine the boson’s mass and we must observe the final state of
a transition to distinguish particles of different spin. A precise
reconstruction of a gravitational-wave signal can therefore help us not only
to detect new ultralight bosons, but also distinguish between the scalar and
vector cases.

## Searching for Dark Compact Objects

So far, we focused on the dynamics and the signatures of gravitational atoms
in binary systems. However, other types of dark compact objects can also arise
in many scenarios for physics beyond the Standard Model. In order to search
for any of these compact objects, we would need to perform matched filtering
on the observational data with their associated template waveforms.111Coherent
burst search methods [Klimenko:2015ypf, Lynch:2015yin, Cornish:2014kda] have
been developed to detect transient gravitational waves in a model-independent
way. Nevertheless, they only capture loud and short-duration events, such as
the near-merger regime of binary coalescences, and are insensitive to weak and
long-duration binary inspirals [Abbott:2016ezn, Abbott:2019heg,
LIGOScientific:2018mvr]. Nevertheless, modeling accurate waveforms for each of
these general222In this chapter, I use the word ‘general’ to refer to any
binary system involving at least one non-standard astrophysical compact
object. In contrast, we use the word ‘standard’ to refer to binary systems
involving only black holes and neutron stars. binary signals is a laborious
task. Fortunately, the complicated microphysics of all of these objects are
encoded in various universal finite-size effects, which we discussed in some
detail in Section 2. In this chapter, I describe work based on the
collaborative work [Chia:2020psj], where we assess the extent to which
existing template banks can be used to search for gravitational-wave signals
emitted by general binary coalescences.

###  1 Overview and Outline

To address the problem described above, we compute the so-called effectualness
[Damour:1997ub, Buonanno:2009zt] of existing template banks to general
waveforms. The effectualness describes how much signal-to-noise ratio is
retained when we compute the overlap between a signal and the best-fitting
template waveform in a bank. In addition to the usual mass and spin
parameters, these general waveforms incorporate the effects of the spin-
induced quadrupole moment and the compactness of the binary components. A
detection of these general signals would therefore represent a discovery of
new physics in binary systems. Our template bank is designed to resemble those
used by the LIGO/Virgo collaboration [Brown:2012qf, DalCanton:2017ala,
Roy:2017oul, 2018arXiv181205121M], demonstrating that, if these new signals
exist, they could remain undetected. When constructing our template bank, we
follow closely the geometric-placement method presented in [Roulet:2019hzy].
Our work is complementary to [Venumadhav:2019tad, Venumadhav:2019lyq] which
further optimized the LIGO/Virgo search pipelines. Instead, we hope to broaden
the searches beyond these standard binary black hole and binary neutron star
signatures.

We emphasize that our work is in contrast to several proposed tests of new
physics in binary systems [Krishnendu:2017shb, Krishnendu:2018nqa,
Kastha:2018bcr, Kastha:2019brk], which seek to measure or constrain plausible
parametric deviations in observed waveforms. This a priori assumes a
successful detection of the new binary system. Detection is typically achieved
through matched filtering with current template banks, which necessarily means
that the waveform deviations are small. Our focus is instead on the
detectability of these plausible new binary signals, including those that
incur large deviations from the binary black hole template waveforms.

Figure 1: Overview of the different methods used to search for gravitational
waves emitted by a general binary system with the LIGO and Virgo detectors.
The signals of low-mass binary systems are typically weaker and last longer
than those of high-mass systems. The horizontal dashed line schematically
illustrates the detector noise level, below which coherent burst searches
become quickly insensitive. In this chapter we focus on low-mass binary
inspirals which are only detectable with matched filtering.

The ability to generate accurate template waveforms is a crucial prerequisite
to achieving our goal. We therefore restrict ourselves to general low-mass
binary systems, where the total mass of the binary is $M\lesssim
10\,M_{\odot}$, for the following reasons:

  * •

In the LIGO and Virgo detectors, the inspiral regime only dominates for low-
mass binary systems. The binary inspiral is an interesting regime because
analytic results of the PN dynamics are readily available. This provides us
with a well-defined framework to construct precise waveforms that incorporate
additional physics, such as finite-size effects.

  * •

The inspiraling signals of these systems last up to several minutes
(corresponding to hundreds or thousands of cycles) and are typically very
weak. They are therefore hard to detect with coherent burst searches. Matched
filtering is the optimal and only realistic avenue to search for them (cf.
Fig. 1 for a comparison of these different search techniques).

By assuming that the PN dynamics are valid up until the merger regime, we have
implicitly ignored other plausible effects that may occur even in the early
inspiraling regime such as: Roche-lobe mass transfer [Paczynski1971]; third-
body perturbation [Kozai, LIDOV1962719, Lim:2020cvm]; floating, sinking, or
kicked orbits [Baumann:2019ztm]; dark matter environmental effects
[Blas:2016ddr, Edwards:2019tzf, Kavanagh:2020cfn]; new fifth forces
[Hook:2017psm, Kopp:2018jom, Wong:2019yoc]; and strong gravitational dynamics
[Bezares:2017mzk, Palenzuela:2017kcg, Bezares:2018qwa]. Despite these
limitations, our general waveforms still capture a wide class of new types of
binary systems which have been overlooked in the literature.

The structure of this chapter is as follows: In Section 2, I discuss the
construction the waveform for a general binary inspiral. Specifically, the
impact of various finite-size effects on the waveform will be incorporated. In
Section 3, I describe the development of a template bank that is
representative of those used in standard search pipelines. I then show the
effectualness of our general waveform to this template bank.

###  2 General Inspiral Waveforms

In this section, we construct the template waveforms for general binary
inspirals. We first motivate several classes of dark compact objects that
could exist in our Universe (§1). We then review how an astrophysical object’s
multipole structure, imprints itself on the phase of the gravitational waves
emitted by the binary system (§2). Finally, we describe how an object’s size,
parameterized through its compactness, affects the cutoff frequency of our
template waveforms (§3). These waveforms are generalizations of those
constructed for binary systems with black holes and neutron stars. Their
effectualness to existing template banks will be studied in Section 3.

####  1 Dark Compact Objects

In Section 2, I discussed two broad classes of dark matter based on their mass
ranges, which in turn determine whether they behave as “waves” or “particles”
over astrophysical length scales. In the former case, the Compton wavelengths
of the fields can be larger than the sizes of astrophysical black holes and
can therefore form gravitational atoms. In the latter case, other types of
dark compact objects can arise. Here I briefly discuss a few such examples.

Particle-like dark matter can accrete around black holes and form high-density
minispikes, with overdensities that peak near the black hole horizon
[Gondolo:1999ef, Sadeghian:2013laa]. In the simplest scenario where the
accretion process is uninterruped by other astrophysical processes, these
spikes would evolve from an initial dark matter halo distribution, such as the
NFW profile, to a profile that is several orders of magnitude denser. Since
the accretion occurs over long astrophysical timescales, in reality,
astrophysical baryonic effects and self-annihilations of the dark matter
particles could flatten these spikes [Merritt:2002vj, Sadeghian:2013laa,
Ferrer:2017xwm]. Despite these uncertainties, the density profiles of these
spikes could still be much larger than the average dark matter density in our
Universe. Like the gravitational atom, these minispikes make black holes
interesting probes of dark matter. Indeed, one of the best ways of probing the
true radial profiles of these spikes is through their signatures in binary
systems, which could potentially allow us to infer the microscopic properties
of the dark matter particles [Hannuksela:2019vip, Kavanagh:2020cfn].

A myriad of bosonic configurations can arise in physics beyond the Standard
Model scenarios that propose the existence of additional massive boson fields
in our Universe. These objects differ from the gravitational atoms in that
they have regular boundary conditions at their centers, while the
gravitational atoms satisfy the “only-ingoing” condition at the black hole
horizon. These objects are often called boson stars if they are made up of
complex fields [Kaup1968, Ruffini1969, Breit:1983nr, Colpi1986,
Liebling:2012fv] (see e.g. [Schunck:2003kk] for a review), and are stabilized
by the fact that they possess conserved charges. On the other hand, if the
objects are made of real fields, they can form solitons [Lee1987-2, Lee1987-3,
Lee1987-4, Lynn:1988rb] or oscillons [Seidel1991, Copeland:1995fq,
Braaten:2015eeu], with the oscillons being denser, more compact, and therefore
subject to more strong relativistic effects. Unlike boson stars, these real
bosonic configurations are time-dependent and therefore dissipates through
gravitational-wave radiation. All of these boson states can be formed either
through the Jean’s instability of a relic initial density in the early
Universe or through gravitational cooling processes, see e.g.
[Bianchi:1990mha, Seidel:1993zk]. Depending on the precise microphysical
properties of the underlying bosons, these configurations would have
distinctive mass-radius relations. By measuring the mass, radius, and other
finite-size effects in theirn binary waveforms, we could potentially detect
the presence of these compact objects and even infer the microscopic
properties of the boson fields.

Finally, we briefly mention that primordial black holes could exist in our
Universe [HawkingPBH]. Unlike astrophysical black holes that are formed
through collapses of stars, primordial black holes are proposed to had been
formed in the early Universe, such as through critical collapse of curvature
perturbations due to sharp features in the primordial curvature power spectrum
(see e.g. [Sasaki:2018dmp] for a review). Importantly, the masses of
primordial black holes do not have to obey the Tolman-Volkoff-Oppenheimer
lower bound $\gtrsim 3M_{\odot}$, which only applies to black holes that are
formed through stellar collapses [Tolman1939]. As such, while it can be
difficult to distinguish between primordial black holes and those of
astrophysical origins if they are heavy, a detection of low-mass black holes
could indicate the presence of these putative new objects. Since the finite-
size effects of black holes are known in detail, e.g. the spin-induced
quadrupole moment $\kappa=1$ and tidal Love number $\lambda=0$, cf. Section 2,
one can further distinguish primordial black holes from other putative low-
mass compact objects through precise measurements of their finite-size effects
in binary waveforms.

####  2 Dephasing from Quadrupole Moment

In Section 2, we described how a general astrophysical object can source a
series of multipole moments [Geroch:1970cd, Hansen:1974zz, Thorne:1980ru]. For
objects that are spherically symmetric when they are not spinning, the
dominant finite-size effect is given by the spin-induced quadrupole moment,
which is often parameterized through the relation (16). There, the $\kappa$
parameter is the dimensionless quadrupole parameter and $\chi$ is the
dimensionless spin of the object. Importantly, the value of $\kappa$ depends
sensitively on the detailed properties of the object. It is instructive to
recall what the typical values of $\kappa$ are for known examples. In
particular, we know 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]. For more speculative objects, such as superradiant boson
clouds and boson stars, $\kappa$ can be as large as $\sim 10^{2}-10^{3}$. It
can even develop oscillatory time-dependences or vary significantly with
$\chi$ [Baumann:2018vus, Baumann:2019ztm, Ryan:1996nk, Herdeiro:2014goa].
Absent a specific compact-object model in mind, we will henceforth treat
$\kappa$ as a free constant parameter, with the requirement that $\kappa\geq
1$.

When the object is part of a binary system, the precise effect of $Q$ on the
gravitational-wave signal is known in the early-inspiral, post-Newtonian
regime of the coalescence [Barker75, Poisson:1997ha, nrgrs, Porto:2008jj,
Hergt:2010pa, Buonanno:2012rv, Bohe:2015ana, Marsat:2014xea,
Krishnendu:2017shb]. This effect was discussed in Section 2; for future
convenience, we repeat some of the key points here. To simplify our analysis,
we restrict ourselves to binary orbits which are quasi-circular, and assume
that the binary components’ spins are parallel to the (Newtonian) orbital
angular momentum vector of the binary. In the Fourier domain, the
gravitational-wave strain is [Cutler:1994ys]

$\tilde{h}(f;\bm{p})=\mathcal{A}(\bm{p})\,f^{-7/6}\,e^{i\psi(f;\bm{p})}\,\theta\big{(}f_{\rm
cut}(\bm{p})-f\big{)}\,,$ (1)

where $f$ is the gravitational-wave frequency, $\bm{p}$ is the set of
intrinsic parameters of the binary, $\mathcal{A}$ is the strain
amplitude,333The amplitude is independent of $f$ at leading Newtonian order.
We will ignore higher-order PN corrections to $\mathcal{A}$, as they do not
substantially affect the overlap between different waveforms. As a result, the
constant $\mathcal{A}$ disappears in the normalized inner product (see Section
3 later). $\psi$ is the phase, $\theta$ is the Heaviside theta function, and
$f_{\rm cut}$ is the cutoff frequency of our general waveform. Schematically,
the phase evolution reads

$\displaystyle\psi(f;\bm{p})=$ $\displaystyle\,\,2\pi
ft_{c}-\phi_{c}-\frac{\pi}{4}+\frac{3}{128\hskip 1.0pt\nu\hskip
1.0ptv^{5}}\Big{(}\psi_{\rm NS}+\psi_{\rm S}\Big{)}\,,$ (2)

where $t_{c}$ is the time of coalescence, $\phi_{c}$ is the phase of
coalescence, $\nu=m_{1}m_{2}/M^{2}$ is the symmetric mass ratio,
$M=m_{1}+m_{2}$ is the total mass of the binary, and $v=(\pi Mf)^{1/3}$ is the
circular orbital velocity. The quantities $\psi_{\rm NS}$ and $\psi_{\rm S}$
represent the non-spinning and spinning phase contribution, respectively.
Because the overlap between waveforms are especially sensitive to phase
coherence [Cutler:1992tc], we will retain terms in the phase up to 3.5PN order
— these terms are fully known in the literature; see for example [Arun:2004hn,
Wade:2013hoa, Mishra:2016whh, Krishnendu:2017shb]. The quadrupole parameter
$\kappa$ in (16) contributes to $\psi_{\rm S}$ through the interaction between
$Q$ and the tidal field sourced by the binary companion. It first appears at
2PN order [Poisson:1997ha, Wade:2013hoa]

$\psi_{\rm
S}\supset-50\sum_{i=1}^{2}\,\left(\frac{m_{i}}{M}\right)^{2}\kappa_{i}\hskip
1.0pt\chi_{i}^{2}\,v^{4}\,,$ (3)

where the subscript $i=1,2$ represents each of the binary components. It also
appears in the phase at 3PN order, though in a slightly complicated manner
[Krishnendu:2017shb]

$\displaystyle\psi_{\rm S}\supset\frac{5}{84}\sum_{i=1}^{2}\sum_{j\neq
i}\,\bigg{[}$ $\displaystyle 15609\left(\frac{m_{i}}{M}\right)^{2}$ (4)
$\displaystyle+27032\hskip 1.0pt\frac{m_{i}m_{j}}{M^{2}}+9407\hskip
1.0pt\left(\frac{m_{j}}{M}\right)^{2}\bigg{]}\left(\frac{m_{i}}{M}\right)^{2}\kappa_{i}\hskip
1.0pt\chi_{i}^{2}\,v^{6}\,.$

We have simplified (3) and (4) by enforcing $\chi_{i}$ to be aligned with the
orbital angular momentum ($0<\chi_{i}\leq 1$) or anti-aligned with it
($-1\leq\chi_{i}<0$). By incorporating these dephasing effects into template
waveforms, we can potentially detect the presence of new astrophysical objects
through observations of a binary’s inspiral. As we will show, if the dephasing
is large enough, these signals could even be missed by current LIGO/Virgo
template-bank searches (see Fig. 2 for an illustration of a dephased
waveform).

In principle, the gravitational waves emitted during the binary’s merger
regime also provide information about the quadrupolar structure of the
objects. However, the detailed dynamics of this regime are often sensitive to
the microphysics of the objects and can only be resolved accurately through
numerical relativity simulations. To preserve analytic control over our
waveform (1), we will ignore the merger regime throughout this chapter. This
is achieved in practice by restricting ourselves to low-mass binary systems
with $M\lesssim 10M_{\odot}$, whose merger frequencies typically lie above the
upper bound of the observational windows of ground-based detectors (see §3
later for more detailed discussions). Crucially, these binary systems would
inspiral within the detector bands over a large number of orbiting cycles,
making matched filtering with our general inspiral waveform the optimal way of
searching for them, cf. Fig. 1.

While we have only focused on the object’s quadrupole moment so far, other
types of finite-size effects, such as the object’s higher-order spin-induced
moments and tidal deformabilities [Chia:2020yla, Binnington:2009bb,
Damour:2009vw, Landry:2015zfa, Pani:2015hfa], can also contribute to the phase
(2). Nevertheless, these additional terms only start to appear at $3.5$PN
[Levi:2014gsa, Marsat:2014xea, Krishnendu:2017shb] and $5$PN [nrgr,
Flanagan:2007ix] orders, respectively, and are therefore subdominant compared
to (3) and (4). In particular, these higher-PN terms are suppressed in the
early inspiral of the coalescence, and only become non-negligible in the
strong gravity, near merger regime.444Since the leading-order term in (2)
scales as $\sim v^{-5}$, roughly speaking, orbital parameters that appear at
$\lesssim 2.5$PN order affect the phase predominantly in the inspiralling
stage, when the number of inspiraling cycles is large, while those with
$\gtrsim 2.5$PN become more prominent near merger; see for example
[Harry:2018hke]. By focusing on the binary’s early inspiral regime we can
therefore ignore these higher-order effects. For concreteness, we will set
these quantities to their corresponding values for black holes [Hansen:1974zz,
Thorne:1980ru, Landry:2015zfa, Pani:2015hfa].555Note however that we retain
the $\kappa$-dependence in the 3.5PN phasing term [Krishnendu:2017shb] in
Section 3.

Figure 2: Comparison between the waveforms of a binary black hole inspiral and
a general binary inspiral. In both cases, the binary components’ masses and
spins are $m_{1}=m_{2}=2M_{\odot}$ and $\chi_{1}=\chi_{2}=0.8$. No amplitude
modulation is observed as we assume that the components’ spins are aligned
with the orbital angular momentum of the binary. One of the binary components
is assumed to be a black hole $(\kappa_{1}=1)$, while the other can be a more
general object $(\kappa_{2}\equiv\kappa\geq 1)$. The vertical dashed line
represents the cutoff of our general waveform, which happens earlier in the
inspiral (lower frequency) for binary components which are less compact (i.e.
as $\mathcal{C}$ decreases).

####  3 Cutoff Frequency from Small Compactness

While the parameter $\kappa$, described in §2, characterizes the deformation
of an astrophysical object’s shape, it does not carry information about its
size. This is instead described by the compactness parameter

$\mathcal{C}\equiv\frac{m}{r}\,,$ (5)

where $r$ is the equatorial radius of the object. Black holes, which are the
most compact known astrophysical objects, have $0.5\leq\mathcal{C}\leq 1$,
where the lower and upper bound corresponds to the compactness of a
Schwarzschild and an extremal Kerr black hole respectively. For neutron stars,
$\mathcal{C}$ can range between $\sim 0.1-0.3$ [Ozel:2016oaf, Abbott:2018exr].
On the other hand, objects that arise in many BSM scenarios can be much more
diffuse, with $\mathcal{C}\ll 0.1$ (for example see [AmaroSeoane:2010qx,
Croon:2018ybs]).

The compactness of each binary component can significantly affect the cutoff
frequency of the waveform (1) because together they dictate the binary
separation at which the merger occurs. For binary systems with highly-compact
objects, such as black holes and neutron stars, the merger frequencies must be
deduced through detailed numerical relativity studies, as the full non-linear
dynamics of the binary merger must be taken into account (see for example
[Bernuzzi:2015rla, Bohe:2016gbl] for the fitting formulae for the merger
frequencies of binary black hole and binary neutron star waveforms). Roughly
speaking, these strong-gravity effects become important when the orbital
separation, $R$, is smaller than the binary’s innermost stable circular orbit
(ISCO), $r_{\rm ISCO}\approx 6M$ [Blanchet:2006zz, Schafer:2009dq]. However,
if the binary components have sufficiently small $\mathcal{C}$’s, the binary
would have already merged at $R<r_{\rm ISCO}$. In this case, the strong-
gravity regime is not reached at merger, making the analytic PN approximation
still a valid description of the dynamics.

Pledging ignorance to the merger dynamics of binary systems with small-
compactness objects, we will terminate our waveform (1) when the binary
touches, i.e. when $R=r_{1}+r_{2}$.666Some papers use $R=r_{\rm ISCO}$ as a
merger condition for low-compactness binary systems. However, the notion of an
ISCO ceases to exist for objects with $\mathcal{C}\lesssim 1/6\approx 0.17$,
as this fictitious ISCO would be located in the interior of the object. This
leads to a factor of $\approx 6$ underestimation in $f_{\rm cut}$, which can
substantially reduce the frequency range over which the SNR could be
accummulated.  Using Kepler’s third law, our cutoff frequency is therefore

$f_{\rm
cut}=\frac{1}{\pi}\sqrt{\frac{m_{1}+m_{2}}{\left(m_{1}/\mathcal{C}_{1}+m_{2}/\mathcal{C}_{2}\right)^{3}}}\hskip
8.0pt,$ (6)

where we have ignored PN corrections to (6). Furthermore, we have neglected
the influence of the binary components’ quadrupoles and higher-order multipole
moments on (6), which can be important when the binary separation is small.
These additional corrections would deepen the gravitational potential of the
binary, thereby increasing the cutoff frequency towards larger values
[Blanchet:2006zz, Schafer:2009dq]. In other words, (6) underestimates the
actual touching frequency of the binary, and may thus be viewed as a
conservative cutoff of our waveform frequency. The impact of this cutoff on
our waveform is schematically illustrated in Fig. 2. In the special case where
both of the binary components have the same compactness,
$\mathcal{C}_{1}=\mathcal{C}_{2}=\mathcal{C}$, (6) becomes

$f_{\rm cut}\simeq
1440\,\text{Hz}\left(\frac{\mathcal{C}}{0.2}\right)^{3/2}\left(\frac{4M_{\odot}}{M}\right)\,.$
(7)

For low-mass binary systems ($M\lesssim 10M_{\odot}$), the observational lower
bound of ground-based detectors, $f\gtrsim 10\,\text{Hz}$, implies that we can