# Modeling horizon absorption in spinning binary black holes
using effective worldline theory

M. V. S. Saketh<EMAIL_ADDRESS>Department of Physics, University of
Maryland, College Park, MD 20742, USA  Max Planck Institute for Gravitational
Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam 14476, Germany
Jan Steinhoff<EMAIL_ADDRESS>Max Planck Institute for Gravitational
Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam 14476, Germany
Justin Vines<EMAIL_ADDRESS>Max Planck Institute for Gravitational
Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam 14476, Germany
Alessandra Buonanno<EMAIL_ADDRESS>Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam
14476, Germany  Department of Physics, University of Maryland, College Park,
MD 20742, USA

###### Abstract

The mass and spin of black holes (BHs) in binary systems may change due to the
infall of gravitational-wave (GW) energy down the horizons. For spinning BHs,
this effect enters at 2.5 post-Newtonian (PN) order relative to the leading-
order energy flux at infinity. There is currently a discrepancy in the
literature in the expressions of these horizon fluxes in the test-body limit
at 4PN order (relative 1.5PN order). Here, we model the horizon absorption as
tidal heating in an effective worldline theory of a spinning particle equipped
with tidally-induced quadrupole and octupole moments. We match the tidal
response to analytic solutions of the Teukolsky equation in a scattering
scenario, and obtain general formulae for the evolution of mass and spin. We
then specialize to the case of aligned-spin–quasi-circular binaries, obtaining
the corresponding contributions to the GW phasing through 4PN order.
Importantly, we find that the number of GW cycles due to horizon fluxes with
masses observed by LIGO-Virgo-KAGRA detectors is about 2-3 orders of magnitude
smaller than the other contributions to the phasing at the same PN order.
Furthermore, in the test-body limit, we find full agreement with results
obtained earlier from BH perturbation theory, with a small mass in an
equatorial circular orbit treated as a source perturbing the Kerr metric.
Thus, we weigh in on one side of the previous discrepancy.

##  I Introduction and summary

With ninety gravitational-wave (GW) events [1] from compact-binary
coalescences observed by the LIGO-Virgo detectors [2, 3], GW astronomy has
become an important instrument to explore our universe. While the worldwide
network of detectors has recently included the KAGRA detector [4], and will
continue to improve in sensitivity in the future [5, 6, 7, 8, 9], the accuracy
of waveform models needs to keep in step with the increasing sensitivity in
order to avoid systematic errors in parameter estimation [10]. Predictions for
GWs from the late stage of a binary inspiral and merger require full numerical
solutions of the strong-field dynamics. Here we focus instead on the early
inspiral, applying the post-Newtonian (PN, weak-field and slow motion)
approximation, and calculate horizon-absorption effects on the GW phase from
black hole (BH) binaries up to 4PN order.

The presence of the horizon leads to some interesting effects in BH binaries.
In particular, GW energy can fall into (or out of, in the case of
superradiance) the horizon, leading to a change in mass and magnitude of spin
angular momentum111We will refer to the magnitude of spin angular momentum
simply as “spin” in the rest of this work. of the BHs in a binary. The change
in these parameters is consistent with the second law of BH dynamics and
always leads to an increase (or no change) in the area of the horizon [11]. A
change in mass and spin can also happen for other compact bodies like neutron
stars through tidal heating (see, e.g., Refs. [12, 13]). Future GW detectors
may be sensitive to these dissipative effects, which can then provide a probe
for the nature of BHs (e.g., the presence of a horizon) [14, 15, 16, 17].
Horizon-absorption effects have been included in some effective-one-body (EOB)
waveform models [18, 19, 20], but not yet in the state-of-the-art models used
for LIGO-Virgo-KAGRA (LVK) data analysis.

For spinning BHs, the leading-order flux into the horizon starts at 2.5PN with
respect to the leading quadrupolar flux (of the binary) to infinity [21, 22,
23, 24, 25, 26, 27]. Through the flux-phase relation, this means that the GW
phase is also affected at 2.5PN order (see, e.g., Ref. [28, 29]). For
nonspinning BHs, the same effect starts at 4PN [21, 24]. In the test-body
limit, when there is a tiny BH orbiting a much larger spinning BH, one can
solve for the horizon energy flux of the large BH via BH perturbation-theory
(BHPT), which consists in this case of solving the Teukolsky equation with
incoming boundary conditions at the horizon, and outgoing boundary conditions
at infinity, as was done, e.g., in Ref. [21]. However, it is nontrivial to
extend this calculation to generic mass ratios. This was accomplished at
leading 2.5PN order in Ref. [30], for the case of aligned-spin circular orbits
via arguments relating the spin-aligned–quasi-circular inspiral to the case
when two spinning BHs are held at rest with respect to each other. The result
was extended further to 1.5PN orders (to absolute 4PN order) in Refs. [31, 22]
where BHPT was used along with a matching between the near zone and the
orbital zone for BHs in a binary to derive the energy and angular-momentum
fluxes across their horizons for generic mass ratios. However, for spinning
BHs, the result was inconsistent with that obtained in the test-body limit in
Ref. [21]. This discrepancy is yet to be settled in the literature and thus
the correct expression for mass and angular-momentum evolution for generic
mass ratios at 4PN is yet to be clarified. Settling this discrepancy is
crucial to derive the correct horizon-flux contribution to the waveform phase
at 4PN order, and it is one of the goals of this paper.

The problem of computing the horizon fluxes for spinning BHs was tackled
recently in an effective field theory (EFT) framework in Ref. [23] (see also,
e.g., Refs. [32, 33, 34, 35, 36, 37] for previous work in that direction and
Refs. [38, 39, 40, 41, 42, 43] for the EFT formalism including spin and tidal
effects), where the BH was treated as a point particle with tidally induced
quadrupole moments. Then in an “in-in” formalism, a parametrized expression
for the absorption cross section for a graviton being absorbed by the
effective particle was computed, whose parameters were fixed by a matching
calculation against the classical absorption cross section for gravitational
plane waves by a spinning BH(from Ref. [44]). This in turn fixes the
correlation function for the quadrupole moment, which was subsequently used to
derive the dissipative part of the Green’s function relating the tidal fields
to the quadrupole moments. Once the tidal response was fixed in this way, the
EFT was applied to compute the evolution equation for mass and spin by
suitably defining them in the worldline theory from the effective action. This
calculation was carried out at leading (2.5PN) order and the result was
consistent with earlier works in Refs. [24, 25, 26, 27].

In this work, we similarly develop an effective worldline framework to model
the dissipative dynamics of spinning BHs, but extend it by 1.5PN orders. We
follow Ref. [23] in treating the spinning BH as a point particle with tidally
induced moments, but work in a purely classical framework. We identify and fix
the dissipative part of the tidal response by comparing results from the
scattering of GWs off the particle/BH between the effective and real theories
(see also, e.g., Refs. [45, 46, 47, 48] for other works involving comparison
of such a scattering process between real and effective theories). We then
derive expressions for the evolution of mass and spin up to 1.5PN order
(relative to the leading order). We find that the resulting expressions are
consistent with earlier results obtained in the test-body limit in Ref. [21],
thus weighing in on one side of the discrepancy discussed above. We then
derive the 4PN contribution to the waveform phase due to the horizon fluxes,
while consistently including the effects due to the changing parameters (mass
and spin) of the members of the binary (extending an earlier 3.5PN result
[29]).

To accomplish this, we write down in Sec. II an effective worldline action for
a spinning particle with (gravito-electric and -magnetic) quadrupole and
octupole moments, coupled accordingly to quadrupolar and octupolar (gravito-
electric and -magnetic) tidal fields in the action. We then motivate ansätze
relating the tidal fields linearly to the multipole moments. In the absence of
spin, spherical symmetry and parity symmetry imply that a given tidal field
only induces the corresponding multipole moment (e.g., the electric-type
quadrupole $Q_{\mathcal{E}}^{\mu\nu}$ is induced only by the electric-type
tidal field $\mathcal{E}^{\mu\nu}$). However, in the presence of spin, it is
possible for octupolar tidal fields to induce quadrupolar tidal fields and
vice-versa, while still preserving parity. This is an important property of
the ansätze and turns out to be crucial for correctly modelling the
dissipative dynamics of spinning BHs. In particular, the Teukolsky equation
which governs the curvature perturbations in a Kerr background is separable in
spheroidal harmonics with spin weight -2. This feature can be modelled in the
effective theory only by including the interaction between quadrupole
(octupole) fields and octupole (quadrupole) moments.

Once we motivate general ansätze for the multipole moments, we further
specialize them by using the fact that the response tensors (relating the
tidal fields to the multipole moments) can only have a nontrivial tensor
structure due to the spin of the particle, which allows us to decompose them
into a set of basis tensors with undetermined coefficients to be fixed. For
this purpose, in Sec. III, we place the effective particle at the origin and
scatter GWs off of it, and then use the Einstein equation and the ansätze to
solve for the scattered wave and then subsequently to compute the degree of
absorption for the spheroidal $l=2,~{}3$ modes of the wave to
$\mathcal{O}(\epsilon^{7})$, where $\epsilon=GM\omega$ with $M$ being the BH
mass and $\omega$ the GW angular frequency. Comparing the degree of absorption
obtained by solving the scattering problem in the effective theory with that
obtained by solving the same problem in the actual setup of GWs scattering off
a spinning BH, using BHPT as governed by the Teukolsky equation [49], finally
fixes the response coefficients that contribute to dissipation. The response
coefficients are notably nonpolynomial in the BH spin.

Once the (dissipative part of the) tidal response is fixed, we proceed to
compute expressions for the evolution of mass and spin in the effective theory
in Sec. IV. We first derive general evolution equations for mass $m$, and spin
$J$, in terms of the tidal fields and multipole moments from the equations of
motion obtained from the action, and then derive the explicit expressions for
the special case parallel-spin–quasi-circular222In this work, by parallel-
spin–quasi-circular binaries, we always mean BHs in a binary with their spin
vectors parallel to each other and to the orbital angular momentum. binaries
to relative 1.5PN order. We show its consistency (or lack thereof) with
earlier results and then proceed to compute the effect on the waveform phase
up to 4PN with respect to the leading-order quadrupolar flux of the system to
infinity in Sec. V. With that, we conclude our work in this paper in Sec. VI.

We work with the (-,+,+,+) metric signature convention. We use greek symbols
$\mu$, $\nu$, …, for space-time indices ranging over $\\{0,1,2,3\\}$ with 0
being used for the time-component, and latin symbols $i$, $j$, …, for spatial
indices ranging over $\\{1,2,3\\}$. We use $\epsilon_{0123}=\epsilon_{123}=1$
as the convention for Levi-Civita tensor(s). We also use the multi-index
notation where $\mu_{L}=\mu_{1}\mu_{2}\dots\mu_{l}$ for conveniently
representing a string of indices where useful and use the notation
$\langle\mu_{1}\mu_{2}...\rangle$ to represent symmetrization and trace
removal of a tensor over the contained indices. We set the speed of light
$c=1$ in the work. However, we keep the dependence on the Newton constant $G$
explicit for most of the work, and mention explicitly when setting $G=1$ as
well to facilitate comparison with earlier works.

##  II Setup in effective worldline theory

In this section, we discuss the effective worldline theory for the point
particle used to model the spinning compact object. We first briefly outline
the particle’s multipole structure and how it can be used to model absorption.
We then proceed to set up an effective action for the particle including the
multipole moments, and then write down general ansätze with undetermined
coefficients for the multipole moments as a linear function of the tidal
fields consistent with axisymmetry and parity symmetry.

We model the horizon flux of the spinning BH in effective worldline theory as
tidal heating of a composite paricle with several tidally induced multipole
moments, whose degrees of freedom are contained in symmetric trace-free (STF)
tensors $Q^{\mu_{L}}_{\mathcal{E},n}$, $Q^{\mu_{L}}_{\mathcal{B},n}$, $l\geq
2$, $n\geq 0$, satisfying $Q^{\mu_{L}}u_{\mu_{i}}=0$, $1\leq i\leq l$. In the
effective action, we choose the multipole moments to couple with the tidal
fields

$\displaystyle\mathcal{E}_{\mu_{L}}$
$\displaystyle=\nabla_{\langle\mu_{L-2}}R_{\mu_{l-1}|\alpha|\mu_{l}\rangle\beta}u^{\alpha}u^{\beta},$
(1) $\displaystyle\mathcal{B}_{\mu_{L}}$
$\displaystyle=\frac{1}{2}\nabla_{\langle\mu_{L-2}}\epsilon_{|\gamma|\mu_{l-1}}{}^{\alpha\beta}R_{|\alpha\beta|\mu_{l}\rangle\delta}u^{\gamma}u^{\delta},~{}l\geq
2,$

where $u^{\mu}$ is the four-velocity and we are using the notation
$\langle\mu_{L}\rangle$ to denote symmetrization and trace-removal. In the
effective action, we choose the electric ‘$\mathcal{E}$’ (magnetic
‘$\mathcal{B}$’) multipole moments to couple with electric (magnetic) tidal
fields in accordance with the number of indices and parity as
$S_{\mathrm{tidal}}=(1/2)\sum_{l=2}^{\infty}\sum_{n}[Q_{\mathcal{E},n}^{\mu_{L}}\mathcal{E}_{\mu_{L}}+(\mathcal{E}\leftrightarrow\mathcal{B})]]$.
We will only need to consider their coupling and induction by quadrupolar
($l=2$) and octupolar ($l=3$) tidal fields to the PN order relevant in this
work. In addition, the various multipole moments are dynamical and are coupled
to each other via an internal action $S_{\mathrm{int}}$ such that the energy
tidally pumped into these modes may progressively escape into higher
order/smaller length-scale multipoles effectively leading to dissipation. If
there is a sufficiently large number of degrees of freedom, and if they are
appropriately coupled, the recurrence time becomes essentially infinite and
the system becomes effectively irreversible. We will not however explicitly
model the process of dissipation and only use that as a justification to write
down ansätze for the multipole moments in terms of the tidal fields that
allows for dissipation. Note that at the end of the day, we only intend to
mimic the BH’s horizon absorption (as tidal heating) and acquire an effective
model that may be used to study the associated dynamics. Whether there is any
physical relation to the real microscopic degrees of freedom of a BH and this
model is unknown and not directly relevant to this work. Our approach of
incorporating tidal moments in the action is slightly different at a
superficial level from the prescription used in Ref. [23] where instead a
single quadrupole moment was used but allowed to be a function of several
unknown microscopic degrees of freedom denoted by $X$. Practically however,
there is not much of a difference.

In the absence of spin, an unperturbed BH is spherically symmetric, and the
linear tides can only be induced by the fields to which they directly couple
to in the action, e.g., as333We will only ever need an ansatz for the sum of
all multipole moments for a given $l$ and parity ($\mathcal{E}/\mathcal{B}$)
for computing the evolution equations for physical quantities such as total
spin angular momentum or total linear momentum. We however allow for the
presence of multiple multipole moments with the same $l$ and parity labels
$\mathcal{E}/\mathcal{B}$ for generality.,

$\displaystyle\sum_{n=0}^{\infty}Q^{\mu_{L}}_{\mathcal{E},n}=M\sum_{m=0}^{\infty}\lambda_{\mathcal{E},m}^{l}(GM)^{2+l+m}\frac{D^{m}}{D\tau^{m}}\mathcal{E}^{\mu_{L}},$
$\displaystyle\text{and similarly for
}(\mathcal{E}\leftrightarrow\mathcal{B})$ (2)

where parity symmetry444In general, $\mathcal{E}_{\mu_{L}}$ (or
$\mathcal{B}_{\mu_{L}}$) transforms under parity as $(-1)^{l}$ or
$[(-1)^{(l+1)}]$. The multipole moments with which they explicitly couple in
the action cannot be induced by tidal fields with a different transformation
under parity if the particle is to be parity-preserving. prevents the magnetic
(electric) tidal field for the same $l$ from inducing the electric (magnetic)
multipole moment, and spherical symmetry means there is no special tensor with
which to contract the higher multipolar order tidal fields (or multiply the
lower multipolar order ones) to contribute to the ansatz for
$\sum_{n}Q^{\mu_{L}}_{\mathcal{E}(\mathcal{B}),n}$. Spherical symmetry is also
the reason there is no mixing of indices in the response tensor. We can also
identify from Eq. (II) which response coefficients are conservative and which
are dissipative simply by looking at the transformation under time-reversal.
The coefficients next to odd powers of time derivatives are dissipative and
the ones next to even powers are conservative. This is less trivial for a
spinning particle which allows for mixing of multipolar orders with the help
of the spin tensor and Pauli-Lubanski spin vector.

However, a spinning BH can have induction between different multipolar orders
(different $l$), since the spin-tensor $S^{\mu\nu}$ defined in Eq. (6), or
Pauli-Lubanski spin vector
$s^{\mu}=-[1/(2m)]\epsilon^{\mu}{}_{\nu\rho\sigma}p^{\nu}S^{\rho\sigma}$, can
appear in above relations (II). We still need to keep parity considerations in
mind as spinning BHs obey parity symmetry. Crucially, for our purposes, we
need to include the induction of octupole moments by quadrupolar fields and
vice-versa to capture the tidal heating of the spinning BHs at relative 1.5PN
order. We will write down a general ansatz for multipole moments in the
spinning case in Subsec. II.2. But first, we will write down in Subsec. II.1
an effective action for a particle with spin and aforementioned multipole
moments and derive the equations of motion for spin and four-momentum, and the
effective stress-energy tensor, following closely the prescription in Ref.
[50] but with suitable modifications to allow for the presence of tidally
induced moments. We will then write down general parametrized ansätze for the
multipole moments in terms of the tidal fields constrained by the symmetries
of the particle (axisymmetry and parity invariance), which is somewhat similar
to the approach used in Ref. [23] to fix the form of the correlation function
of the quadrupole moments.

###  II.1 Action and equations of motion for momentum and spin angular
momentum

To get the equations of motion, we will follow a direct extension of the
simple procedure given in Ref. [50] for deriving the equations of motion from
an implicit action, while including the aforementioned tidal moments. We
include the spinning degrees of freedom by attaching to the particle a body-
fixed tetrad $\epsilon_{A}{}^{\mu}$ satisfying orthonormality and
completeness. The angular velocity of the particle is then measured with the
quantity $\Omega^{\mu\nu}=\epsilon_{A}{}^{\mu}\frac{D\epsilon^{A\nu}}{D\tau}$.
It is sufficient in this work just to include the spin at leading order and
ignore spin-induced multipole moments. The worldline of the particle is
denoted by $z^{\mu}(\tau)$, where $\tau$ is the proper time. We can then write
down the action implicitly as

$\displaystyle S=\int d\tau L($ $\displaystyle
u^{\mu},\Omega^{\mu\nu},g_{\mu\nu},Q^{\mu_{L}}_{\mathcal{E},n},Q^{\mu_{L}}_{\mathcal{B},n},\dot{Q}^{\mu_{L}}_{\mathcal{E},n},\dot{Q}^{\mu_{L}}_{\mathcal{B},n},$
$\displaystyle R_{\mu\nu\rho\sigma},\nabla_{\lambda}R_{\mu\nu\rho\sigma}),$
(3)

where $u^{\mu}=dz^{\mu}/d\tau$ and we are using the notation
$\dot{a}=Da/D\tau$. Additionally, we have assumed that only the first time
derivative of each individual multipole moment needs to be included in the
action. We have also restricted ourselves to including just the quadrupolar
($l=2$) and octupolar ($l=3$) tidal fields in the action explicitly as
mentioned before. The coupling of tidal fields with higher order multipole
moments is irrelevant to the PN order of interest in this work as we will see
later. We do however keep all the higher multipolar order moments since tidal
heating requires the presence of several additional degrees of freedom into
which the system may pump energy. Additionally, as mentioned before, we choose
the multipole moments to couple directly with the corresponding tidal fields
(in accordance with number of indices and parity label) by imposing

$\displaystyle\frac{\partial
L}{\partial\mathcal{E}_{\mu_{L}}}=\frac{1}{2}\sum_{n}Q_{\mathcal{E},n}^{\mu_{L}},~{}\frac{\partial
L}{\partial\mathcal{B}_{\mu_{L}}}=\frac{1}{2}\sum_{n}Q_{\mathcal{B},n}^{\mu_{L}},\quad
l=2,~{}3$

which is equivalent to having in the action a linear combination of the form
$S_{\mathrm{tidal}}=\sum_{l=2}^{3}\sum_{n}\int
d\tau\frac{1}{2}\mathcal{E}_{\mu_{L}}Q_{\mathcal{E},n}^{\mu_{L}}+(\mathcal{E}\leftrightarrow\mathcal{B})$
in the total action. We can derive the equations of motion for momentum and
spin angular momentum directly from this implicit action. First, we consider a
general variation

$\displaystyle\delta L=$ $\displaystyle p_{\mu}\delta
u^{\mu}+\frac{1}{2}S_{\mu\nu}^{\mathrm{rot}}\delta\Omega^{\mu\nu}+\frac{\partial
L}{\partial g_{\mu\nu}}\delta g_{\mu\nu}-\frac{1}{6}J^{\mu\nu\rho\sigma}\delta
R_{\mu\nu\rho\sigma}-\frac{1}{12}J^{\lambda\mu\nu\rho\sigma}\delta\nabla_{\lambda}R_{\mu\nu\rho\sigma}$
$\displaystyle+\sum_{l=2}^{\infty}\sum_{n=0}^{\infty}\bigg{(}\frac{\partial
L}{\partial Q_{\mathcal{E},n}^{\mu_{L}}}\delta
Q_{\mathcal{E},n}^{\mu_{L}}+\frac{\partial
L}{\partial\dot{Q}_{\mathcal{E},n}^{\mu_{L}}}\delta\dot{Q}_{\mathcal{E},n}^{\mu_{L}}+\frac{\partial
L}{\partial Q_{\mathcal{B},n}^{\mu_{L}}}\delta
Q_{\mathcal{B},n}^{\mu_{L}}+\frac{\partial
L}{\partial\dot{Q}_{\mathcal{B},n}^{\mu_{L}}}\delta\dot{Q}_{\mathcal{B},n}^{\mu_{L}}\bigg{)},$
(5)

where we have defined

$\displaystyle p_{\mu}$ $\displaystyle=(\partial L/\partial
u^{\mu})|_{\Omega^{\mu\nu}},~{}S^{\mathrm{rot}}_{\mu\nu}=\frac{1}{2}(\partial
L/\partial\Omega^{\mu\nu}),$ (6) $\displaystyle
J^{\mu\nu\rho\sigma}=-6\frac{\partial L}{\partial
R_{\mu\nu\rho\sigma}}=-3\sum_{n=0}^{\infty}u^{[\mu}Q_{\mathcal{E},n}^{\nu][\rho}u^{\sigma]}+\frac{3}{2}\sum_{n=0}^{\infty}Q_{\mathcal{B},n}^{\alpha\langle\rho}\epsilon_{\beta\alpha}{}^{\mu\nu}u^{|\beta|}u^{\sigma\rangle_{R}},$
(7) $\displaystyle J^{\lambda\mu\nu\rho\sigma}=-12\frac{\partial
L}{\partial\nabla_{\lambda}R_{\mu\nu\rho\sigma}}=-6\sum_{n=0}^{\infty}u^{\langle\mu}Q_{\mathcal{E},n}^{\nu\rho\lambda}u^{\sigma\rangle_{\nabla
R}}+3\sum_{n=0}^{\infty}Q_{\mathcal{B},n}^{\alpha\langle\rho\lambda}\epsilon_{\beta\alpha}{}^{\mu\nu}u^{|\beta|}u^{\sigma\rangle_{\nabla
R}},$ (8)

where the expressions for the $J$’s follow trivially from Eq. (II.1) and we
are using $\langle abcd\rangle_{R}$ ($\langle abcd\rangle_{\nabla R}$) to
represent the symmetrization of indices according to the symmetries of the
Riemann tensor (covariant derivative of the Riemann tensor) respectively. Now,
if we consider a variation of the form $x^{\mu}\rightarrow x^{\mu}+\xi^{\mu}$
corresponding to an infinitesimal change of coordinates, we obtain the
constraint

$\displaystyle 2\frac{\partial L}{\partial g_{\mu\nu}}$
$\displaystyle=p^{\mu}u^{\nu}+S_{\mathrm{rot}}^{\mu\rho}\Omega^{\nu}{}_{\rho}+\frac{2}{3}R^{\mu}{}_{\lambda\rho\sigma}J^{\nu\lambda\rho\sigma}+\frac{1}{3}J^{\lambda\nu\tau\rho\sigma}\nabla_{\lambda}R^{\mu}{}_{\tau\rho\sigma}+\frac{1}{12}J^{\nu\lambda\tau\rho\sigma}\nabla^{\mu}R_{\lambda\tau\rho\sigma}$
$\displaystyle+\sum_{l=2}^{\infty}\frac{l}{2}\sum_{n=0}^{\infty}[\mathcal{M}_{\mathcal{E},n}^{\mu~{}\mu_{L-1}}Q^{\mathcal{E},n}_{\mu_{L-1}}{}^{\nu}+P_{\mathcal{E},n}^{\mu~{}\mu_{L-1}}\dot{Q}^{\mathcal{E},n}_{\mu_{L-1}}{}^{\nu}+(\mathcal{E}\leftrightarrow\mathcal{B})],$
(9)

where we have defined
$\mathcal{M}_{\mathcal{E}(\mathcal{B}),n}^{\mu_{L}}=(\partial
L/\partial\dot{Q}_{\mu_{L}}^{\mathcal{E}(\mathcal{B}),n})$, and
$P_{\mathcal{E}(\mathcal{B}),n}^{\mu_{L}}=(\partial L/\partial
Q_{\mu_{L}}^{\mathcal{E}(\mathcal{B}),n})$. Note that the equations of motion
for the multipole moments obtained upon variation of the action with respect
to them $(\partial L/\partial
Q^{\mu_{L}}_{\mathcal{E}(\mathcal{B}),n})-(D/D\tau)(\partial
L/\partial\dot{Q}^{\mu_{L}}_{\mathcal{E}(\mathcal{B}),n})=0$, imply that
$\mathcal{M}_{\mathcal{E}(\mathcal{B}),n}^{\mu_{L}}=\dot{P}^{\mu_{L}}_{\mathcal{E}(\mathcal{B}),n}$
on the actual worldline. Eq. (9) is a useful identity to eliminate the partial
derivative with respect to the metric later in the equations of motion for
momentum and spin angular momentum. First, we obtain the equation for spin
angular momentum easily by variation of the action with respect to the tetrad
variables $\epsilon_{A}{}^{\mu}$. We get the simple equation

$\displaystyle\delta S=\int d\tau\frac{\partial
L}{\partial\Omega^{\mu\nu}}\delta\Omega^{\mu\nu}=0\implies\frac{DS^{\mu\nu}_{\mathrm{rot}}}{D\tau}=2\Omega^{[\mu}{}_{\rho}S_{\mathrm{rot}}^{\nu]\rho},$

and we can eliminate the RHS using Eq. (9) by taking its antisymmetric part
[which leads to ($\partial L/\partial g_{\mu\nu}$) vanishing] and we get

$\displaystyle\frac{DS^{\mu\nu}_{\mathrm{rot}}}{D\tau}$
$\displaystyle=2p^{[\mu}u^{\nu]}+\frac{4}{3}R^{[\mu}{}_{\lambda\rho\sigma}J_{\lambda}{}^{\nu]\tau\rho\sigma}+\frac{2}{3}\nabla^{\lambda}R^{[\mu}{}_{\tau\rho\sigma}J_{\lambda}{}^{\nu]\tau\rho\sigma}+\frac{1}{6}\nabla^{[\mu}R_{\lambda\tau\rho\sigma}J^{\nu]\lambda\tau\rho\sigma}$
$\displaystyle+\sum_{l=2}^{\infty}\sum_{n=0}^{\infty}l\frac{D}{D\tau}[P_{\mathcal{E},n}^{[\mu}{}_{\mu_{L-1}}Q^{\nu]\mu_{L-1}}_{\mathcal{E},n}+(\mathcal{E}\leftrightarrow\mathcal{B})],$
(11)

where we have used
$\mathcal{M}_{\mathcal{E}(\mathcal{B}),n}^{\mu_{L}}=\dot{P}^{\mu_{L}}_{\mathcal{E}(\mathcal{B}),n}$,
valid on the worldline. We can now redefine the spin angular momentum as
$S^{\mu\nu}=S_{\mathrm{rot}}^{\mu\nu}-\sum_{l=2}^{\infty}\sum_{n=0}^{\infty}l[P_{\mathcal{E},n}^{[\mu}{}_{\mu_{L-1}}Q^{\nu]\mu_{L-1}}_{\mathcal{E},n}+(\mathcal{E}\leftrightarrow\mathcal{B})]$,
to get

$\displaystyle\frac{DS^{\mu\nu}}{D\tau}$
$\displaystyle=2p^{[\mu}u^{\nu]}+\frac{4}{3}R^{[\mu}{}_{\lambda\rho\sigma}J_{\lambda}{}^{\nu]\tau\rho\sigma}$
(12)
$\displaystyle+\frac{2}{3}\nabla^{\lambda}R^{[\mu}{}_{\tau\rho\sigma}J_{\lambda}{}^{\nu]\tau\rho\sigma}+\frac{1}{6}\nabla^{[\mu}R_{\lambda\tau\rho\sigma}J^{\nu]\lambda\tau\rho\sigma}.$

This is the appropriate definition of total spin angular momentum of the body
as reinforced by the fact that this also shows up in the stress energy tensor
(see Eq. 14) in the expected manner. To solve the equations of motion, we also
need to impose a “spin supplementary condition” (SSC) to ensure that it is a
spatial tensor with the right number of degrees of freedom. Here, we impose
the “covariant” or Tulczyjew-Dixon SSC at the level of the equations of motion
upon the total physical spin angular momentum as $S^{\mu\nu}p_{\mu}=0$.

The equation of motion for momentum can be obtained by variation of the action
with respect to the worldline $z^{\mu}(\tau)$, which can be done following the
covariant approach as shown in Ref. [50] to get

$\displaystyle\frac{Dp_{\mu}}{D\tau}$
$\displaystyle=-\frac{1}{2}R_{\mu\nu\rho\sigma}u^{\nu}S^{\rho\sigma}-\frac{1}{6}J^{\lambda\nu\rho\sigma}\nabla_{\mu}R_{\lambda\nu\rho\sigma}$
(13)
$\displaystyle-\frac{1}{12}J^{\tau\lambda\nu\rho\sigma}\nabla_{\mu}\nabla_{\tau}R_{\lambda\nu\rho\sigma}.$

Note that only the sum of multipole moments
$\sum_{n}Q_{\mathcal{E},n}^{\mu_{l}}$, appear in the expressions for the time
derivatives of momentum and spin angular momentum.

Finally, the stress energy tensor of the particle can be derived by varying
the effective action with respect to the metric $g_{\mu\nu}$ as
$T^{\mu\nu}=\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta g_{\mu\nu}},$ where all
dependency of the action on $g_{\mu\nu}$ needs to be taken into account during
the variation. This variation was performed in Ref. [50] to obtain

$\displaystyle T^{\mu\nu}$ $\displaystyle=T^{\mu\nu}_{\mathrm{pole-
dipole}}+T^{\mu\nu}_{\mathrm{quadrupole}}+T^{\mu\nu}_{\mathrm{octupole}},$
(14) $\displaystyle T^{\mu\nu}_{pole-dipole}$ $\displaystyle=\int d\tau
p^{(\mu}u^{\nu)}\frac{\delta^{(4)}(x-z)}{\sqrt{-g}}-\nabla_{\rho}\int d\tau
S^{\rho(\mu}u^{\nu)}\frac{\delta^{(4)}(x-z)}{\sqrt{-g}},$ (15) $\displaystyle
T^{\mu\nu}_{\mathrm{quadrupole}}$ $\displaystyle=\int
d\tau\frac{1}{3}R^{(\mu}_{~{}\lambda\rho\sigma}J^{\nu)\lambda\rho\sigma}\frac{\delta^{(4)}(x-z)}{\sqrt{-g}}-\nabla_{\rho}\nabla_{\sigma}\int
d\tau\frac{2}{3}J^{\rho(\mu\nu)\sigma}\frac{\delta^{(4)}(x-z)}{\sqrt{-g}},$
(16) $\displaystyle T^{\mu\nu}_{\mathrm{octupole}}$ $\displaystyle=\int
d\tau\Bigg{[}\frac{1}{6}\nabla^{\lambda}R^{(\mu}_{~{}\xi\rho\sigma}J_{\lambda}^{~{}~{}\nu)\xi\rho\sigma}+\frac{1}{12}\nabla^{(\mu}R_{\xi\tau\rho\sigma}J^{\nu)\xi\tau\rho\sigma}\Bigg{]}\frac{\delta^{(4)}(x-z)}{\sqrt{-g}}$
(17) $\displaystyle+\nabla_{\rho}\int
d\tau\Bigg{[}-\frac{1}{6}R^{(\mu}_{~{}\xi\lambda\sigma}J^{|\rho|\nu)\xi\lambda\sigma}-\frac{1}{3}R^{(\mu}_{~{}\xi\lambda\sigma}J^{\nu)\rho\xi\lambda\sigma}+\frac{1}{3}R^{\rho}_{~{}\xi\lambda\sigma}J^{(\mu\nu)\xi\lambda\sigma}\Bigg{]}\frac{\delta^{(4)}(x-z)}{\sqrt{-g}}$
(18) $\displaystyle+\nabla_{\lambda}\nabla_{\rho}\nabla_{\sigma}\int
d\tau\frac{1}{3}J^{\sigma\rho(\mu\nu)\lambda}\frac{\delta^{(4)}(x-z)}{\sqrt{-g}},$
(19)

which holds true even in the presence of inducible multipole moments when the
contribution of the multipole moments is included in the definitions of the
momentum and spin angular momentum. Since we are only interested in linear
tides, we can drop all nonlinear (in curvature or metric perturbation)
contributions in the quadrupolar and octupolar stress energy tensor to get a
simpler truncated stress energy tensor as

$\displaystyle T^{\mu\nu}$ $\displaystyle=\int d\tau
p^{(\mu}u^{\nu)}\delta^{(4)}(x-z)-\nabla_{\rho}\int d\tau
S^{\rho(\mu}u^{\nu)}\delta^{(4)}(x-z)-\nabla_{\rho}\nabla_{\sigma}\int
d\tau\frac{2}{3}J^{\rho(\mu\nu)\sigma}\delta^{(4)}(x-z)$
$\displaystyle+\nabla_{\lambda}\nabla_{\rho}\nabla_{\sigma}\int
d\tau\frac{1}{3}J^{\sigma\rho(\mu\nu)\lambda}\delta^{(4)}(x-z).$ (20)

This truncated stress energy tensor will be important later to solve the
problem of GWs scattering off the effective particle.

To proceed further, we need to relate the tidal fields to the multipole
moments. We will now motivate and write down general ansätze for the multipole
moments.

###  II.2 Ansätze for multipole moments

Going forward, we define the sum of all multipole moments for a given $l$ and
parity (electric or magnetic) as individual moments for convenience as
$Q_{\mathcal{E}}^{\mu_{L}}=\sum_{n}Q_{\mathcal{E},n}^{\mu_{L}}$. Now, as
mentioned earlier, when the particle obeys spherical symmetry, only the tidal
fields to which the multipole moment explicitly couples to in the action can
affect it and we have Eq. (II). However, in the presence of spin, it is
possible to have more fields in the formula for multipole moments. Since we
intend to model the spinning BH, which breaks spherical symmetry but still
respects parity, we can only have tidal fields with the same transformation
under parity as the moment in the right hand side. The transformation under
reflection is $(-1)^{l}$ for $\mathcal{E}$ fields and $(-1)^{l+1}$ for
$\mathcal{B}$ fields. Furthermore, we restrict our attention to quadrupolar
and octupolar tidal fields as the higher multipolar order tidal fields are not
relevant to the order of interest in this work. Thus, we can write a general
ansatz as

$\displaystyle Q^{\mu\nu}_{\mathcal{E}}$
$\displaystyle=M\sum_{m}^{\mathrm{\infty}}(GM)^{4+m}\lambda_{\mathcal{E},m}^{\mu\nu}{}_{\rho\sigma}\frac{D^{m}}{D\tau^{m}}\mathcal{E}^{\rho\sigma}+M\sum_{m}^{\infty}(GM)^{5+m}\zeta^{\mu\nu}_{\mathcal{B},m}{}_{\rho\sigma\gamma}\frac{D^{m}}{D\tau^{m}}\mathcal{B}^{\rho\sigma\gamma},\quad(\mathcal{E}\leftrightarrow\mathcal{B}),$
$\displaystyle Q_{\mathcal{B}}^{\mu\nu\rho}$
$\displaystyle=M\sum_{m}^{\infty}(GM)^{5+m}\eta^{\mu\nu\rho}_{\mathcal{E},m}{}_{\sigma\gamma}\frac{D^{m}}{D\tau^{m}}\mathcal{E}^{\sigma\gamma}+M\sum_{m}^{\infty}(GM)^{6+m}\Lambda^{\mu\nu\rho}_{\mathcal{B},m}{}_{\alpha\sigma\delta}\frac{D^{m}}{D\tau^{m}}\mathcal{B}^{\alpha\sigma\delta},\quad(\mathcal{E}\leftrightarrow\mathcal{B}),$
(21)

where we have rendered the response tensors dimensionless by removing factors
of $GM$. The response tensors can only have a nontrivial structure due to the
spin of the particle. Additionally, they must be orthogonal to $u^{\mu}$ and
be traceless in the upper and lower set of indices separately. One can thus
generally decompose the response tensors as linear combination of building-
block tensors made up of the spin tensor $S^{\mu\nu}$, Pauli-Lubanski spin
vector
$s^{\mu}\approx-(1/2)\epsilon^{\mu}{}_{\nu\rho\sigma}u^{\nu}S^{\rho\sigma}$,555The
Pauli Lubanski spin vector is actually defined as
$s^{\mu}=-[1/(2m)]\epsilon^{\mu}{}_{\nu\rho\sigma}p^{\nu}S^{\rho\sigma}$.
However, as $p^{\mu}=mu^{\mu}+\mathcal{O}(R)$, we can neglect the curvature-
dependent corrections when substituting in the linear tidal-response. and the
orthogonal (to four-velocity $u^{\mu}$) projection operator
$\mathcal{P}^{\mu}_{\nu}=\delta^{\mu}_{\nu}+u^{\mu}u_{\nu}$. While infinitely
many such combinations may be written, only a handful of them are linearly
independent and we can generally decompose the response tensors
$\lambda,~{}\zeta,~{}\eta$, as (see, e.g., Ref. [23] where the correlation
function for the quadrupole moment was fixed)

$\displaystyle\lambda_{\mathcal{E},n}^{\mu\nu}{}_{\rho\sigma}$
$\displaystyle=f_{\mathcal{E},n}^{0}\mathcal{P}^{\langle\mu}_{\langle\rho}\mathcal{P}^{\nu\rangle}_{\sigma\rangle}+f_{\mathcal{E},n}^{1}\hat{S}^{\langle\mu}{}_{\langle\rho}\mathcal{P}^{\nu\rangle}_{\sigma\rangle}+f_{\mathcal{E},n}^{2}\hat{s}^{\langle\mu}\hat{s}_{\langle\rho}\delta^{\nu\rangle}_{\sigma\rangle}+f_{\mathcal{E},n}^{3}\hat{s}^{\langle\mu}\hat{s}_{\langle\rho}\hat{S}^{\nu\rangle}{}_{\sigma\rangle}+f_{\mathcal{E},n}^{4}\hat{s}^{\langle\mu}\hat{s}_{\langle\rho}\hat{s}^{\nu\rangle}\hat{s}_{\sigma\rangle},~{}(\mathcal{E}\leftrightarrow\mathcal{B})$
$\displaystyle\zeta_{\mathcal{B},n}^{\mu\nu}{}_{\rho\sigma\gamma}$
$\displaystyle=\lambda_{\mathcal{E},n}^{\mu\nu}{}_{\langle\rho\sigma}\hat{s}_{\gamma\rangle}(f_{\mathcal{E},n}\rightarrow
g_{\mathcal{B},n}),~{}\eta_{\mathcal{E},n}^{\mu\nu\rho}{}_{\rho\sigma}=\hat{s}^{\langle\rho}\lambda_{\mathcal{E},n}^{\mu\nu\rangle}{}_{\rho\sigma}(f_{\mathcal{E},n}\rightarrow
h_{\mathcal{E},n}),\quad(\mathcal{E}\leftrightarrow\mathcal{B})$ (22)

where the sets of coefficients $f$, $g$, $h$ can now only depend on the spin
parameter $\chi=J/(GM^{2})$, where $J=\sqrt{(1/2)S^{\mu\nu}S_{\mu\nu}}$ is the
magnitude of spin angular momentum. Also, we have defined $\hat{S}^{\mu\nu}$
and $\hat{s}^{\mu}$ as the normalized versions of the spin-tensor
($\hat{S}^{\mu\nu}\hat{S}_{\mu\nu}=2$) and spin-vector
($\hat{s}^{\mu}\hat{s}_{\mu}=1$) so they are dimensionless and independent of
any parameter. Adding any other tensor made up of the same ingredients will be
linearly dependent on the remaining pieces, which follows simply from the
relations $\hat{S}^{\mu\nu}\hat{s}_{\nu}=0$, and
$\hat{S}^{\mu}{}_{\rho}\hat{S}^{\nu}{}_{\sigma}=-\mathcal{P}^{\mu}{}_{\sigma}\mathcal{P}^{\nu}{}_{\rho}+\mathcal{P}^{\mu\nu}\mathcal{P}_{\rho\sigma}-\mathcal{P}_{\rho\sigma}\hat{s}^{\mu}\hat{s}^{\nu}+\mathcal{P}^{\mu}{}_{\sigma}\hat{s}^{\nu}\hat{s}_{\rho}+\mathcal{P}^{\nu}{}_{\rho}\hat{s}^{\mu}\hat{s}_{\sigma}-\mathcal{P}^{\mu\nu}\hat{s}_{\rho}\hat{s}_{\sigma}$,
and thus no more free coefficients can be introduced. We have transferred all
freedom in choosing the response to the associated sets of $\chi$-dependent
parameters $f$, $g$, and $h$. We can similarly decompose the tensor
$\Lambda^{\mu\nu\rho}_{\alpha\sigma\delta}$ but it turns out to be irrelevant
to the order of interest in this work, so we just drop its contribution from
now on. We can drop most of the time-derivatives of the tidal fields in the
ansätze for the same reason and work with simplified truncated ansätze by
including terms only up to $(GM)^{5}$,

$\displaystyle Q_{\mathcal{E}}^{\mu\nu}$
$\displaystyle=(GM)^{4}\lambda_{\mathcal{E},0\rho\sigma}^{\mu\nu}\mathcal{E}^{\rho\sigma}+(GM)^{5}\lambda_{\mathcal{E},1\rho\sigma}^{\mu\nu}\frac{D}{D\tau}\mathcal{E}^{\rho\sigma}$
$\displaystyle+(GM)^{5}\nu^{\mu\nu}_{\mathcal{B},0}{}_{\langle\rho\sigma}\hat{s}_{\gamma\rangle}\mathcal{B}^{\rho\sigma\gamma},~{}(\mathcal{E}\leftrightarrow\mathcal{B}),$
$\displaystyle Q_{\mathcal{B}}^{\mu\nu\rho}$
$\displaystyle=(GM)^{5}\hat{s}^{\langle\rho}\xi_{\mathcal{E},0}^{\mu\nu\rangle}{}_{\alpha\beta}\mathcal{E}^{\alpha\beta},~{}(\mathcal{E}\leftrightarrow\mathcal{B}),$
(23)

where we have defined
$\nu^{\mu\nu}_{\mathcal{B},0}{}_{\rho\sigma}=\lambda^{\mu\nu}_{\mathcal{B}(\mathcal{E}),0}{}_{\rho\sigma}(f_{\mathcal{E}(\mathcal{B}),0}\rightarrow
g_{\mathcal{B}(\mathcal{E}),0})$, and
$\xi^{\mu\nu}_{\mathcal{E},0}{}_{\rho\sigma}=\lambda^{\mu\nu}_{\mathcal{E},0}{}_{\rho\sigma}(f_{\mathcal{E}(\mathcal{B}),0}\rightarrow
h_{\mathcal{E},0})$. The 40 coefficients,
$f_{\mathcal{E}(\mathcal{B}),0}^{i}$, $f_{\mathcal{E}(\mathcal{B}),1}^{i}$,
$g_{\mathcal{B}(\mathcal{E}),0}^{i}$, $h_{\mathcal{E}(\mathcal{B}),0}$
characterize the tidal response of the particle. Some contribute to
dissipative effects whereas others contribute only to conservative effects.
Unlike for the spinless case, it is not obvious here which ones contribute to
dissipation and which ones do not, due to coupling between different
multipolar orders. At leading order, focussing only on the quadrupolar tidal
field inducing the quadrupolar multipole moment, it can be shown (see Ref.
[23]) that the part of the tensor $\lambda^{\mu\nu\rho\sigma}_{\mathcal{E},0}$
that is antisymmetric under $\mu\nu\leftrightarrow\rho\sigma$ contributes to
dissipation. This criteria is flipped for
$\lambda^{\mu\nu}_{\mathcal{E},1\rho\sigma}$ as it is next to an odd power of
a time-derivative. This includes the coefficients
$f^{1}_{\mathcal{E}(\mathcal{B}),0}$, $f^{3}_{\mathcal{E}(\mathcal{B}),0}$,
$f^{0}_{\mathcal{E}(\mathcal{B}),1}$, $f^{2}_{\mathcal{E}(\mathcal{B}),1}$,
$f^{4}_{\mathcal{E}(\mathcal{B}),1}$. It is less trivial for the coefficients
entering tensors that mix the different multipolar orders. We will see in Sec.
III.4 which coefficients are conservative and which are dissipative among
these. We accomplish this by scattering GWs off the effective particle and
computing the degree of absorption for the $l=2$ spheroidal modes and
comparing with the analogous result obtained by solving the Teukolsky
equation. This comparison, along with demanding that the different spheroidal
modes of the Newman-Penrose scalar $\psi_{4}$ scatter independently in the
effective theory will help identify which coefficients lead to dissipation and
which only to conservative effects. Additionally, we will also be able to fix
the coefficients that contribute to dissipative effects through comparison
with results from BHPT.

##  III Fixing the unknown parameters in ansätze for the multipole moments

To fix the dissipative part of the response, we now scatter GWs off the
effective particle with mass $M$ and compute a suitable quantity indicating
the degree of absorption. We can then fix the free parameters by comparison
with the analogous quantity for actual spinning BHs, obtained by solving the
Teukolsky equation. We will only consider the effect of the linearly induced
tidal moments in the scattering of the wave in what follows, since the other
contributions to the scattered wave are expected to be purely conservative or
irrelevant (up to the order of interest). However, in this way we also neglect
the leading-order effect of nonlinearities due to the wave scattering off the
stationary gravitational background, leading to certain subtleties when
matching with the result from the Teukolsky equation which we will have to
address.

###  III.1 The GW environment

Let the unperturbed particle be at the origin at rest, $u^{\mu}=(1,0,0,0)$ and
$z^{\mu}=(\tau=t,0,0,0)$, where $\tau$ is the proper time of the particle
which is identical to the background time $t$ when undisturbed. We add to the
particle’s background a general GW perturbation
$h^{\mu\nu}=\sqrt{-g}g^{\mu\nu}-\eta^{\mu\nu}$. We choose the perturbation to
be in the harmonic and transverse-traceless (TT) gauge, satisfying
$\partial_{\mu}h^{\mu\nu}=0$, $h^{\mu\nu}u_{\nu}=0$,
$h^{\mu\nu}\eta_{\mu\nu}=0$. At zeroth order (in $G$), when the interactions
of the metric perturbation with the stationary gravitational field of the
particle are neglected, it just satisfies the flat space-time wave equation
$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}h^{\alpha\beta}=0$ (except at origin
where the particle is present). Then, we can write the general tensor wave
solution in the rest frame of the particle (defined by $u^{\mu}$) at leading
order as

$\displaystyle
h_{ij}=\sum_{l=2}^{\infty}\frac{1}{\omega^{l-2}}\bigg{(}C^{K_{l}}_{\mathcal{E},\mathrm{in}}\Pi_{ij}^{k_{l-1}k_{l}}\hat{\partial}_{K_{l-2}}+\frac{1}{\omega}C^{K_{l}}_{\mathcal{B},\mathrm{in}}\Pi_{ij}^{k_{l}m}\epsilon_{k_{l-1}mn}\hat{\partial}_{K_{l-2}n}\bigg{)}\psi_{\mathrm{in}}+(\mathrm{in}\rightarrow\mathrm{out}),$
(24)

where $\omega$ is the frequency of the wave in the rest frame of the particle
and
$\Pi_{kl}^{ij}=(1/2)(P^{i}{}_{k}P^{j}{}_{l}+P^{j}{}_{k}P^{i}{}_{l}-P^{ij}{P}_{kl})$,
with $P^{ij}=\delta^{ij}+\partial^{i}\partial^{j}/\omega^{2}$, is a
differential projection operator which ensures that the harmonic and TT gauge
conditions are satisfied. $\psi_{\mathrm{in}}=\exp[-i\omega(t+r)]/(\omega r)$
and $\psi_{\mathrm{out}}=\exp[-i\omega(t-r)]/(\omega r)$ are the incoming and
outgoing wave solutions for the $l=0$ mode of a scalar wave respectively.
$C_{\mathcal{E}/\mathcal{B},\mathrm{in}/\mathrm{out}}^{K_{l}}$ are
dimensionless symmetric trace-free (STF) tensors characterizing the amplitudes
of each $l$ mode. $\mathcal{E}$, $\mathcal{B}$ label the coefficients tuning
the electric and magnetic polarizations respectively.666In general, there are
two distinct solutions for each $\omega$, $l$, and $m$ for a massless tensor
wave (except for scalars). We split them according to parity in this work,
with $\mathcal{E}$ modes transforming as $(-1)^{l}$ and $\mathcal{B}$ modes
transforming as $(-1)^{l+1}$ under reflection. The labels are also related to
the manner in which different coefficients contribute to the tidal fields (see
Eq. (29)). The different $l$ modes and polarizations ($\mathcal{E}$,
$\mathcal{B}$) do not mix under rotations. However, that does not mean we can
tune them separately in the presence of the particle, as the particle’s
inducible multipole moments combined with its spin can couple different modes
with each other. The general solution in Eq. (24) can be understood by noting
that a basis of solutions to the homogeneous wave equation in flat space-time
(albeit allowing for irregular behaviour at the origin) can be obtained by
acting arbitrary number of spatial derivatives upon the spherically symmetric
solutions, i.e., $\psi_{\mathrm{in}}$, $\psi_{\mathrm{out}}$ and any linear
combination of them. We can then generally write the solution for $h_{ij}$ as
a linear combination of the basis solutions with undetermined tensors
contracting them to get the appropriate tensor structure. One can then
decompose them into rotationally independent pieces by splitting the
undetermined tensors into STF tensors and then separating them according to
parity ($\mathcal{E}$ and $\mathcal{B}$). Finally, one uses the projection
operator $\Pi^{ij}_{kl}$ to ensure $h^{ij}$ is traceless and satisfies the
harmonic gauge condition.

Now, we want to split the total general solution in Eq. (24) into two parts,
one that can be regarded as the “input” part of the wave which corresponds to
the part that induces the tidal moments and an “output” part which is sourced
from the induced tidal moments. Naively, one might think that the input part
can be obtained by simply setting the outgoing mode coefficients
$C_{\mathcal{E}/\mathcal{B},\mathrm{out}}^{K_{l}}=0$, but that is incorrect as
there will be an outgoing wave in general even in the absence of a particle
due to the the fact that an incoming wave packet in the distant past becomes
an outgoing wave packet in the distant future (after crossing the origin).
Also, the incoming part of the wave by itself is irregular at the origin and
thus cannot be sustained without the presence of a particle. In fact, the
correct splitting is given by the regular (input) and irregular (output) parts
of the wave respectively, as follows

$\displaystyle h_{ij}$
$\displaystyle=\sum_{l=2}^{\infty}\frac{1}{\omega^{l-2}}\bigg{(}C^{K_{l}}_{\mathcal{E},\mathrm{reg}}\Pi_{ij}^{k_{l-1}k_{l}}\hat{\partial}_{K_{l-2}}+\frac{1}{\omega}C^{K_{l}}_{\mathcal{B},\mathrm{reg}}\Pi_{ij}^{k_{l}m}\epsilon_{k_{l-1}mn}\hat{\partial}_{K_{l-2}n}\bigg{)}\psi_{\mathrm{reg}}+(\mathrm{reg}\rightarrow\mathrm{irr}),$
(25) $\displaystyle\psi_{\mathrm{reg}}$
$\displaystyle=\frac{1}{2i}(\psi_{\mathrm{out}}-\psi_{\mathrm{in}})=\frac{\exp(-i\omega
t)\sin(\omega r)}{\omega
r},\quad\psi_{\mathrm{irr}}=(\psi_{\mathrm{out}}+\psi_{\mathrm{in}})=\frac{\exp(-i\omega
t)\cos(\omega r)}{\omega r},$ (26) $\displaystyle C_{\mathrm{reg}}$
$\displaystyle=(C_{\mathrm{out}}-C_{\mathrm{in}})i,\quad
C_{\mathrm{irr}}=C_{\mathrm{out}}+C_{\mathrm{in}},$ (27)

where $\psi_{\mathrm{reg}}$ is regular at the origin and obeys the wave
equation everywhere whereas $\psi_{\mathrm{irr}}$ is irregular at the origin
and requires support from a source, such as the stress energy tensor of the
effective particle. More specifically,
$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\psi_{\mathrm{reg}}=0$, and
$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\psi_{\mathrm{irr}}=(4\pi/\omega)\delta^{(3)}(\vec{r})\exp(-i\omega
t)$. Thus, we will use the regular part of $h_{ij}$, which has finite values
at the origin to compute the tidal fields that will induce the multipole
moments and the irregular part of the fields will be related to the stress
energy tensor of the particle’s induced tidal moments through the relation

$\displaystyle\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}h_{ij}^{\mathrm{irr}}$
$\displaystyle=\sum_{l=2}^{\infty}\frac{\exp(-i\omega
t)}{\omega^{l-1}}\bigg{(}C^{K_{l}}_{\mathcal{E},\mathrm{irr}}\Pi_{ij}^{k_{l-1}k_{l}}\hat{\partial}_{K_{l-2}}$
$\displaystyle+\frac{1}{\omega}C^{K_{l}}_{\mathcal{B},\mathrm{irr}}\Pi_{ij}^{k_{l}m}\epsilon_{k_{l-1}mn}\hat{\partial}_{K_{l-2}n}\bigg{)}4\pi\delta^{(3)}(\vec{r})$
$\displaystyle=16\pi G|g|\Pi^{kl}_{ij}T_{kl}(\vec{r}),$ (28)

where we have projected out the stress energy tensor appropriately since we
are focusing on the radiative part of the field and using
$h_{ij}^{\mathrm{irr}}$ to label the irregular part of the metric perturbation
(i.e., the part generated from acting spatial derivatives on
$\psi_{\mathrm{irr}}$) . We will have to solve this relation with the stress
energy tensor corresponding to that due to the induced quadrupolar and
octupolar multipole moments given to leading order in the curvature tensor in
Eq. (20).

###  III.2 Induced multipole moments

As mentioned earlier, the regular part of the wave, which can exist without
support and satisfies the homogeneous wave equation, should be seen as the
input part of the wave. We will use this part of the metric perturbation to
compute the tidal fields which will induce the multipole moments. With the
definitions given earlier in Eqs. (1), and the formula for the regular part of
the wave given in Eq. (25), we find that the value of the tidal fields at the
origin is given by

$\displaystyle\mathcal{E}^{ij}_{\mathrm{origin}}$
$\displaystyle=-\frac{1}{5}\omega^{2}C^{ij}_{\mathcal{E},\mathrm{reg}},$ (29a)
$\displaystyle\mathcal{B}^{ij}_{\mathrm{origin}}$
$\displaystyle=\frac{i}{5}\omega^{2}C^{ij}_{\mathcal{B},\mathrm{reg}},$ (29b)
$\displaystyle\mathcal{E}^{ijk}_{\mathrm{origin}}$
$\displaystyle=\frac{1}{21}\omega^{3}C^{ijk}_{\mathcal{E},\mathrm{reg}},$
(29c) $\displaystyle\mathcal{B}^{ijk}_{\mathrm{origin}}$
$\displaystyle=\frac{-i}{21}\omega^{3}C^{ijk}_{\mathcal{B},\mathrm{reg}}.$
(29d)

The multipole moments can now be computed by substituting these fields in Eqs.
(II.2). However, we see that the nontrivial (due to spin) response tensors mix
the various components together. While this is inconvenient, a simple way to
rewrite these expressions in a basis that does not mix components is to orient
the coordinate system so that the Pauli-Lubanski spin vector is along the
z-axis and expand in spin-weighted spherical harmonics. For later convenience,
we choose spherical harmonics with spin weight ‘-2’. The simplest way to
transform to this basis from the Cartesian basis is to define

$\displaystyle\vec{m}=\frac{1}{\sqrt{2}}(\hat{\theta}+i\hat{\phi}),$
$\displaystyle\hat{\theta}=\cos(\theta)\cos(\phi)\hat{x}+\cos(\theta)\sin(\phi)\hat{y}-\sin(\theta)\hat{z},$
$\displaystyle\hat{\phi}=-\sin(\phi)\hat{x}+\cos(\phi)\hat{y},$
$\displaystyle\hat{r}=\sin(\theta)\cos(\phi)\hat{x}+\sin(\theta)\sin(\phi)\hat{y}+\cos(\theta)\hat{z},$
(30)

and then expand $Q^{ij}\bar{m}_{i}\bar{m}_{j}$ and
$O^{ijk}\bar{m}_{i}\bar{m}_{j}\hat{r}_{k}$ in spin weight -2, spherical
harmonics by projection as

$\displaystyle Q^{l=2,m}_{\mathcal{E}/\mathcal{B}}=\int
d\Omega~{}Q^{ij}_{\mathcal{E}/\mathcal{B}}\bar{m}_{i}\bar{m}_{j}\bar{Y}^{l=2,m}_{-2}(\theta,\phi),$
$\displaystyle O^{l=3,m}_{\mathcal{E}/\mathcal{B}}=\int
d\Omega~{}Q^{ijk}_{\mathcal{E}/\mathcal{B}}\bar{m}_{i}\bar{m}_{j}\hat{r}_{k}\bar{Y}^{l=3,m}_{-2}(\theta,\phi),$
(31)

where we are using $Y_{s=-2}^{lm}(\theta,\phi)$ to denote spherical harmonics
with spin weight -2. They are normalized to 1 [i.e., $\int
d\Omega~{}Y_{-2}^{lm}(\theta,\phi)\bar{Y}_{-2}^{lm}(\theta,\phi)=1$]. We
similarly define the incoming and outgoing mode coefficients in this basis as

$\displaystyle C^{l=2,m}_{\mathcal{E}/\mathcal{B}}$ $\displaystyle=\int
d\Omega~{}C^{ij}_{\mathcal{E}/\mathcal{B}}\bar{m}_{i}\bar{m}_{j}\bar{Y}_{-2}^{2m}(\theta,\phi)$
$\displaystyle C^{l=3,m}_{\mathcal{E}/\mathcal{B}}$ $\displaystyle=\int
d\Omega~{}C^{ijk}_{\mathcal{E}/\mathcal{B}}\bar{m}_{i}\bar{m}_{j}\hat{r}_{k}\bar{Y}_{-2}^{3m}(\theta,\phi),$
(32)

where we have suppressed the ‘$\mathrm{in}$’ (‘$\mathrm{out}$’) subscript for
brevity. The definitions are identical for both incoming and outgoing mode
coefficients. Note that we are using the symbol $O$ for octupole tensor in
spherical harmonic (with spin weight -2) basis. Then, we can write down the
expressions for the multipole moments obtained by substituting Eq. (29) into
the ansätze in Eq. (II.2) simply as

$\displaystyle Q^{2m}_{\mathcal{E}}$ $\displaystyle=-e^{-i\omega
t}M\omega^{2}\frac{(GM)^{4}}{30}\\{6\mathcal{F}^{0,\mathrm{reg}}_{\mathcal{E}}+3i\mathcal{F}_{\mathcal{E}}^{1,\mathrm{reg}}~{}m+(m^{2}-4)[-\mathcal{F}^{2,\mathrm{reg}}_{\mathcal{E}}-i\mathcal{F}^{3,\mathrm{reg}}_{\mathcal{E}}~{}m+\mathcal{F}^{4,\mathrm{reg}}_{\mathcal{E}}~{}(m^{2}-1)]\\},$
$\displaystyle O^{3m}_{\mathcal{B}}$ $\displaystyle=-e^{-i\omega
t}M\omega^{2}\frac{\sqrt{9-m^{2}}(GM)^{5}}{90\sqrt{7}}\\{6h^{0}_{\mathcal{E}}+3ih^{1}_{\mathcal{E}}~{}m+(m^{2}-4)[-h^{2}_{\mathcal{E}}-ih^{3}_{\mathcal{E}}~{}m+h^{4}_{\mathcal{E}}~{}(m^{2}-1)]\\}C^{2m}_{\mathcal{E},\mathrm{reg}},$
$\displaystyle Q^{2m}_{\mathcal{B}}$ $\displaystyle=ie^{-i\omega
t}M\omega^{2}\frac{(GM)^{4}}{30}\\{6\mathcal{F}^{0,\mathrm{reg}}_{\mathcal{B}}+3i\mathcal{F}_{\mathcal{B}}^{1,\mathrm{reg}}~{}m+(m^{2}-4)[-\mathcal{F}^{2,\mathrm{reg}}_{\mathcal{B}}-i\mathcal{F}^{3,\mathrm{reg}}_{\mathcal{B}}~{}m+\mathcal{F}^{4,\mathrm{reg}}_{\mathcal{B}}~{}(m^{2}-1)]\\},$
$\displaystyle O^{3m}_{\mathcal{E}}$ $\displaystyle=ie^{-i\omega
t}M\omega^{2}\frac{\sqrt{9-m^{2}}(GM)^{5}}{90\sqrt{7}}\\{6h^{0}_{\mathcal{B}}+3ih^{1}_{\mathcal{B}}~{}m+(m^{2}-4)[-h^{2}_{\mathcal{B}}-ih^{3}_{\mathcal{B}}~{}m+h^{4}_{\mathcal{B}}~{}(m^{2}-1)]\\}C^{2m}_{\mathcal{B},\mathrm{reg}},$
(33)

where
$\mathcal{F}^{i,\mathrm{reg}}_{\mathcal{E}(\mathcal{B})}=(f^{i}_{\mathcal{E}(\mathcal{B}),0}-iGM\omega
f^{i}_{\mathcal{E}(\mathcal{B}),1})C^{2m}_{\mathcal{E}(\mathcal{B}),\mathrm{reg}}+[(i\sqrt{9-m^{2}})/(3\sqrt{7})]GM\omega
g^{i}_{\mathcal{B}(\mathcal{E}),0}C^{3m}_{\mathcal{B}(\mathcal{E}),\mathrm{reg}}$,
and we see that there is no longer any mixing of different $m$ modes. This is
simply because we oriented the coordinate system so that the z-axis is along
the spin, and thus it has 0 azimuthal quantum number to add or remove from
that of the STF tensors characterizing the field. However, there is still
mixing between different $l$ modes due to the tidal response mixing the
quadrupolar and octupolar sectors. This complicates the process of defining a
degree of absorption or a scattering phase, and we will tackle this problem
later in Sec. III.4, by switching to a basis where there is no mixing of
modes.

###  III.3 Solving for the outgoing wave

Now, we can use the wave equation with the appropriate source in Eq. (III.1)
to relate the multipole moments to the irregular part of the wave. The
relevant (projected) part of the stress energy tensor in the chosen coordinate
system is given by

$\displaystyle T^{kl}\Pi^{ij}_{kl}$
$\displaystyle=\Pi^{ij}_{kl}\Big{(}\frac{1}{2}\ddot{Q}_{\mathcal{E}}^{kl}+\frac{1}{2}\epsilon^{\langle
k}{}_{mn}\dot{Q}_{\mathcal{B}}^{l\rangle
n}\partial^{m}-\frac{1}{2}\ddot{Q}_{\mathcal{E}}^{kl}{}_{m}\partial^{m}$
$\displaystyle-\frac{1}{2}\epsilon^{\langle
k}{}_{mn}\dot{Q}_{\mathcal{B}}^{l\rangle
no}\partial^{m}\partial_{o}\Big{)}\delta^{(3)}(\vec{x}),$ (34)

which is obtained by substituting the expressions for $J^{\mu\nu\rho\sigma}$
and $J^{\lambda\mu\nu\rho\sigma}$ in terms of the multipole moments given in
Eqs. (7), (8) into the stress energy tensor Eq. (20) and discarding terms with
components along $u^{\mu}$ (as they will be eliminated upon projection). Now,
substituting Eq. (34) into the RHS of Eq. (III.1) and comparing, we get the
relations

$\displaystyle e^{-i\omega t}C^{ij}_{\mathcal{E},irr}$
$\displaystyle=2G\omega^{3}Q_{\mathcal{E}}^{ij},~{}e^{-i\omega
t}C^{ij}_{\mathcal{B},irr}=2iG\omega^{3}Q^{ij}_{\mathcal{B}},$ $\displaystyle
e^{-i\omega t}C^{ijk}_{\mathcal{E},irr}$
$\displaystyle=-2G\omega^{4}Q_{\mathcal{E}}^{ijk},~{}e^{-i\omega
t}C^{ijk}_{\mathcal{B},irr}=-2iG\omega^{4}Q_{\mathcal{B}}^{ijk},$ (35)

which are simple proportionality relations and thus can be trivially
transformed to the $l,m$ basis.

Substiting Eq. (35) in Eq. (III.2), we get the relations

$\displaystyle C^{2m}_{\mathcal{E}/\mathcal{B},\mathrm{irr}}$
$\displaystyle=-\frac{\epsilon^{5}}{15}\\{6\mathcal{F}^{0,\mathrm{reg}}_{\mathcal{E}/\mathcal{B}}+3i\mathcal{F}_{\mathcal{E}/\mathcal{B}}^{1,\mathrm{reg}}~{}m+(m^{2}-4)[-\mathcal{F}^{2,\mathrm{reg}}_{\mathcal{E}/\mathcal{B}}-i\mathcal{F}^{3,\mathrm{reg}}_{\mathcal{E}/\mathcal{B}}~{}m+\mathcal{F}^{4,\mathrm{reg}}_{\mathcal{E}/\mathcal{B}}~{}(m^{2}-1)]\\},$
$\displaystyle C^{3m}_{\mathcal{B},\mathrm{irr}}$
$\displaystyle=i\frac{\sqrt{9-m^{2}}\epsilon^{6}}{45\sqrt{7}}\\{6h^{0}_{\mathcal{E}}+3ih^{1}_{\mathcal{E}}~{}m+(m^{2}-4)[-h^{2}_{\mathcal{E}}-ih^{3}_{\mathcal{E}}~{}m+h^{4}_{\mathcal{E}}~{}(m^{2}-1)]\\}C^{2m}_{\mathcal{E},\mathrm{reg}},$
$\displaystyle C^{3m}_{\mathcal{E},\mathrm{irr}}$
$\displaystyle=-i\frac{\sqrt{9-m^{2}}\epsilon^{6}}{45\sqrt{7}}\\{6h^{0}_{\mathcal{B}}+3ih^{1}_{\mathcal{B}}~{}m+(m^{2}-4)[-h^{2}_{\mathcal{B}}-ih^{3}_{\mathcal{B}}~{}m+h^{4}_{\mathcal{B}}~{}(m^{2}-1)]\\}C^{2m}_{\mathcal{B},\mathrm{reg}},$
(36)

where we have defined $\epsilon=GM\omega$. Now, we can use the relations in
Eqs. (27) to solve the the outgoing coefficients. Since we are only interested
in linear tides, and the leading-order tidal effects already start at a very
high order in $\epsilon$, we can approximate in the RHS as
$C^{lm}_{\mathcal{E}/\mathcal{B},\mathrm{reg}}\approx-2iC^{lm}_{\mathcal{E}/\mathcal{B},\mathrm{in}}+\mathcal{O}(\epsilon^{5})$.
Then we get the expressions for
$C^{lm}_{\mathcal{E}/\mathcal{B},\mathrm{out}}$ as

$\displaystyle C^{2m}_{\mathcal{E}(\mathcal{B}),\mathrm{out}}$
$\displaystyle=-C^{2m}_{\mathcal{E}(\mathcal{B}),\mathrm{in}}+\frac{2i\epsilon^{5}}{15}\\{6\mathcal{F}^{0,\mathrm{in}}_{\mathcal{E}(\mathcal{B})}+3i\mathcal{F}_{\mathcal{E}(\mathcal{B})}^{1,\mathrm{in}}~{}m+(m^{2}-4)[-\mathcal{F}^{2,\mathrm{in}}_{\mathcal{E}(\mathcal{B})}-i\mathcal{F}^{3,\mathrm{in}}_{\mathcal{E}(\mathcal{B})}~{}m+\mathcal{F}^{4,\mathrm{in}}_{\mathcal{E}(\mathcal{B})}~{}(m^{2}-1)]\\},$
$\displaystyle C^{3m}_{\mathcal{B},\mathrm{out}}$
$\displaystyle=-C^{3m}_{\mathcal{B},\mathrm{in}}+\frac{2\sqrt{9-m^{2}}\epsilon^{6}}{45\sqrt{7}}\\{6h^{0}_{\mathcal{E}}+3ih^{1}_{\mathcal{E}}~{}m+(m^{2}-4)[-h^{2}_{\mathcal{E}}-ih^{3}_{\mathcal{E}}~{}m+h^{4}_{\mathcal{E}}~{}(m^{2}-1)]\\}C^{2m}_{\mathcal{B},\mathrm{in}},$
$\displaystyle C^{3m}_{\mathcal{E},\mathrm{out}}$
$\displaystyle=-C^{3m}_{\mathcal{E},\mathrm{in}}-\frac{2\sqrt{9-m^{2}}\epsilon^{6}}{45\sqrt{7}}\\{6h^{0}_{\mathcal{B}}+3ih^{1}_{\mathcal{B}}~{}m+(m^{2}-4)[-h^{2}_{\mathcal{B}}-ih^{3}_{\mathcal{B}}~{}m+h^{4}_{\mathcal{B}}~{}(m^{2}-1)]\\}C^{2m}_{\mathcal{B},\mathrm{in}}.$
(37)

###  III.4 Degree of absorption

Now we need to compute a suitable quantity that measures the degree of
absorption. If there were no spin, there would be no mixing of different $l$
modes, and we could simply define the degree of absorption (or emission) for
each $l,m$ mode as
$1-|C^{lm}_{\mathcal{E}/\mathcal{B},\mathrm{out}}/C^{lm}_{\mathcal{E}/\mathcal{B},\mathrm{in}}|$.
We can still do the same if we instead use a different basis, formed by a
linear combination of the mode coefficients in the spherical basis. To guess
the appropriate combination (basis) in which the modes should scatter
independently, we now turn to hints from BHPT. In BHPT, the Teukolsky equation
[49] governs the behaviour of curvature perturbations in an exact Kerr
background. Specifically, it governs the behaviour of the spin weight (-2)
Teukolsky scalar ${}_{-2}\psi(t,r,\theta\phi)$ related to the standard Newman-
Penrose curvature scalar $\psi_{4}$ by
${}_{-2}\psi=(r-ia\cos(\theta))^{4}\psi_{4}$, where $a=GM\chi$. A crucial
property of the Teukolsky equation for ${}_{-2}\psi$ is that it is separable
in spheroidal harmonic basis with spin weight -2 with fixed frequency eigen
solutions with the form ${}_{-2}\psi\propto\exp(-i\omega
t)S^{lm}_{-2}(\theta,\phi,a\omega){}_{-2}R_{lm\omega}(r)$ where we are using
$S_{s=-2}^{lm}$ to denote normalized (to 1) spheroidal harmonics with spin
weight -2 and ${}_{-2}R_{lm\omega}(r)$ is an eigen solution to the radial
Teukolsky equation (see Sec. III.5). Asymptotically far away from the BH,
i.e., as $r\rightarrow\infty$, the standard Newman-Penrose curvature scalar
$\psi_{4}$ takes the form (see Eq. (2.6) in Ref. [51])

$\displaystyle\psi_{4}\rightarrow\;$ $\displaystyle\omega^{2}\sum_{l,m,P=\pm
1}\Big{\\{}K_{lmP}^{\mathrm{out}}\frac{\exp[-i\omega(t-r^{*})]}{\omega
r}+\frac{1}{16}[\mathrm{Re}(C)+12i\epsilon
P]\frac{1}{\omega^{4}r^{4}}K_{lmP}^{\mathrm{in}}\frac{\exp[-i\omega(t+r^{*})]}{\omega
r}\Big{\\}}S^{l}_{-2,m}(\theta,\phi)$
$\displaystyle+\mathcal{O}\bigg{(}\frac{1}{r^{6}}\bigg{)},$ (38)

where $r^{*}$ is the tortoise coordinate and $P=\pm 1$ is the index denoting
the transformation under parity for each mode with $P=1$ ($P=-1$) for modes
that are symmetric (anti-symmetric) under reflection. The expression for
$\mathrm{Re}(C)$ can be found in Eq. (31) of Ref. [52]. The outgoing and
incoming mode coefficients for each spheroidal mode $l,m$ and parity $P=\pm 1$
are related simply as

$\displaystyle
K^{\mathrm{out}}_{lmP}=-(-1)^{l+m}\eta_{lm}^{P}\exp(2i\delta_{P}^{lm})K^{\mathrm{in}}_{lmP},$
(39)

where $\delta_{P}^{lm}\in\mathcal{R}$ is the conservative scattering phase for
each mode and $\eta_{lm}$ is the degree of absorption/emission (i.e., the mode
coefficients do not mix under scattering). Although these scattering phases
and degree of absorption are defined for these abstract mode coefficients
$K^{\mathrm{out}/\mathrm{in}}_{lmP}$, they are directly related to the
scattering of a GW off a Kerr BH and often employed for that purpose (see for
e.g., Refs. [53, 52]). In particular, $\eta_{lm}^{P}$ characterizes the
dissipation due to energy flux into the horizon with the transmission factor
for each mode defined as

$\displaystyle T_{lm}=1-|\eta_{lm}^{\pm}|^{2}.$ (40)

Absorption or emission of energy at the horizon is only nonzero when
$|\eta_{lm}^{P}|$ differs from 1, and thus this is a suitable quantity to be
labelled as ”degree of absorption” in the real theory.

In the effective worldline theory, we can continue to use the same quantity as
the degree of absorption provided we relate the outgoing (incoming) mode
coefficients $C^{lm}_{\mathcal{E}/\mathcal{B},\mathrm{out}(\mathrm{in})}$ of
the wave-like metric perturbation to the outgoing (incoming) mode coefficients
of $\psi_{4}$ i.e., $K^{\mathrm{out}(\mathrm{in})}_{lmP}$. In other words,
relating $C^{lm}_{\mathcal{E}/\mathcal{B},\mathrm{out}(\mathrm{in})}$ to
$K^{\mathrm{out}(\mathrm{in})}_{lmP}$ will reveal the basis in which the modes
of the wave will scatter without mixing. We accomplish this by writing down
the asymptotic behaviour of $\psi_{4}$ in the effective theory using its
definition
$\psi_{4}=-R_{\mu\nu\gamma\delta}\bar{m}^{\mu}\bar{m}^{\gamma}n^{\nu}n^{\delta}$,
where $m^{\mu}$, $\bar{m}$, $n^{\mu}$, $l^{\mu}$ form a null tetrad field
satisfying $l^{2}=m^{2}=n^{2}=0=l\cdot m=n\cdot m$, $l\cdot
n=-1,m\cdot\bar{m}=1$, where $l^{\mu}$ and $n^{\mu}$ are real vectors whereas
$m^{\mu}$ is a complex null vector with $\bar{m}^{\mu}$ as its complex
conjugate. In flat space, or at leading order when the background curvature
may be treated as a small perturbation such as far away from the particle, we
can write them in term of spherical polar coordinates as
$\hat{m}=(\hat{\theta}+i\hat{\phi})/\sqrt{2}$,
$n^{\mu}=(t^{\mu}-r^{\mu})/\sqrt{2}$,
$l^{\mu}=(t^{\mu}+\hat{r}^{\mu})/\sqrt{2}$. In the effective worldline
picture, we can thus evaluate $\psi_{4}$ at large distances by using the flat
space-time null tetrad, using the linearized curvature due to the wave-like
metric perturbation in Eq. (24), yielding

$\displaystyle\psi_{4}$
$\displaystyle=\omega^{2}\frac{\exp[-i\omega(t-r)]}{\omega r}$ (41)
$\displaystyle\times\sum_{a=\mathcal{E},\mathcal{B}}\sum_{l=2}^{\infty}i^{l-2}C_{a,out}^{K_{l-2}ij}\hat{r}_{K_{l-2}}\hat{m}_{i}\hat{m}_{j}+\mathcal{O}\Big{(}\frac{1}{r^{2}}\Big{)},$

which we can rewrite as an expansion in spherical harmonics of spin weight -2
using the conventions in Eq. (III.2) as

$\displaystyle\psi_{4}$
$\displaystyle=\omega^{2}\frac{\exp[-i\omega(t-r)]}{\omega
r}\sum_{a=\mathcal{E},\mathcal{B}}[\sum_{m=-2}^{2}Y^{2m}_{-2}(\theta,\phi)C_{a,\mathrm{out}}^{lm}$
$\displaystyle+i\sum_{m=-3}^{3}Y^{3m}_{-2}(\theta,\phi)C_{a,\mathrm{out}}^{lm}]+\mathcal{O}\Big{(}\frac{1}{r^{2}}\Big{)},$
(42)

where we have dropped modes above $l=3$ (hexadecapolar and above) as they are
not relevant to the order of interest here. Also note that that our effective
theory does not couple $l\geq 4$ modes with any of the lower modes (up to the
order of interest in this work) and thus we can safely set the associated STF
tensors to zero.

We can then switch to spheroidal harmonics of spin weight -2 via the relations
$Y^{2m}_{-2}=S^{2m}_{-2}-\epsilon\chi(2\sqrt{9-m^{2}})/(9\sqrt{7})S^{3m}_{-2}$,
$Y^{3m}_{-2}=S^{3m}_{-2}+\epsilon\chi(2\sqrt{9-m^{2}})/(9\sqrt{7})S^{2}_{-2,m}$,
valid at leading order in spheroidicity $=\epsilon\chi$, which is sufficient
for our purposes. Also note that we have dropped the contribution to
$Y^{3m}_{-2}$ from $S_{-2}^{l=4,m}$. This gives us

$\displaystyle\psi_{4}$
$\displaystyle=\omega^{2}\frac{\exp[-i\omega(t-r)]}{\omega
r}\sum_{l,m,P}^{3}\mathcal{C}^{lm}_{P,\mathrm{out}}S_{-2}^{lm}(\theta,\phi)$
$\displaystyle+\mathcal{O}\bigg{(}\frac{1}{r^{2}}\bigg{)},$ (43)

where we have defined the coefficients characterizing spheroidal modes as

$\displaystyle\mathcal{C}^{2m}_{P=1,\mathrm{out}}$
$\displaystyle=\Big{(}C^{2m}_{\mathcal{E},\mathrm{out}}+i\frac{2\epsilon\chi\sqrt{9-m^{2}}}{9\sqrt{7}}C^{3m}_{\mathcal{B},\mathrm{out}}\Big{)},~{}\mathcal{C}^{2m}_{P=-1,\mathrm{out}}=\Big{(}C^{2m}_{\mathcal{B},\mathrm{out}}+i\frac{2\epsilon\chi\sqrt{9-m^{2}}}{9\sqrt{7}}C^{3m}_{\mathcal{E},\mathrm{out}}\Big{)},$
(44) $\displaystyle\mathcal{C}^{3m}_{P=1,\mathrm{out}}$
$\displaystyle=\Big{(}iC_{\mathcal{B},\mathrm{out}}^{3m}-\frac{2\epsilon\chi\sqrt{9-m^{2}}}{9\sqrt{7}}C^{2m}_{\mathcal{E},\mathrm{out}}\Big{)},~{}\mathcal{C}^{3m}_{P=-1,\mathrm{out}}=\Big{(}iC_{\mathcal{E},\mathrm{out}}^{3m}-\frac{2\epsilon\chi\sqrt{9-m^{2}}}{9\sqrt{7}}C^{2m}_{\mathcal{B},\mathrm{out}}\Big{)},$
(45)

and $P$ is used to split them according to parity, with $P=1$ for parity
symmetric modes and $P=-1$ for parity antisymmetric modes. We can now compare
with the known asymptotic form of $\psi_{4}$ from Eq. (III.4) to leading order
in $1/r$. This gives us the simple identification between the coefficients of
the outgoing modes of $\psi_{4}$ and that of the GW in effective theory as

$\displaystyle K^{\mathrm{out}}_{lmP}=\mathcal{C}^{lm}_{P,\mathrm{out}},$ (46)

where we used the fact that the tortoise coordinate $r^{*}$ asymptotes to $r$
in the limit $r\rightarrow\infty$. Now, we know the appropriate combination of
coefficients of outgoing modes in the effective theory to use. To get a
similar relation between $K^{\mathrm{in}}_{lmP}$ and the coefficients of
incoming modes in the effective theory, the simplest way is to just guess the
form by considering the limit where there is no scattering. In the real
theory, this is the limit where $\eta_{lm}^{P}=1$ and $\delta_{P}^{lm}=0$, and
we have $K^{\mathrm{up}}_{lmP}=-(-1)^{l+m}K^{\mathrm{in}}_{lmP}$. In the
effective theory, this is simply when the irregular part of the wave should
vanish or $C_{\mathcal{E}/\mathcal{B},\mathrm{irr}}^{lm}=0\implies
C_{\mathcal{E}/\mathcal{B},\mathrm{in}}^{lm}=-C_{\mathcal{E}/\mathcal{B},\mathrm{in}}^{lm}$,
which in turn implies $\mathcal{C}_{P=\pm
1,\mathrm{out}}^{2m}=-\mathcal{C}_{P=\pm 1,\mathrm{in}}^{2m}$, where
$\mathcal{C}_{P=\pm 1,\mathrm{in}}^{2m}$ is defined exactly as
$\mathcal{C}_{P=\pm 1,\mathrm{out}}^{2,m}$ in Eqs. (44),(45) except after
transforming $\mathrm{out}\rightarrow\mathrm{in}$. Thus, we can simply
identify $K^{\mathrm{in}}_{lmP}=(-1)^{l+m}\mathcal{C}_{P=\pm
1,\mathrm{in}}^{2,m}$. Then, the scattering phase relation in Eq. (39) can now
be written in the effective theory simply as

$\displaystyle\mathcal{C}^{lm}_{P,\mathrm{out}}=-\eta^{P}_{lm}\exp(2i\delta_{P}^{lm})\mathcal{C}^{lm}_{P,\mathrm{in}},$
(47)

and the degree of absorption can be defined in the effective worldline theory
as

$\displaystyle
1-\eta^{P}_{lm}=1-\bigg{|}\frac{\mathcal{C}^{lm}_{P,\mathrm{out}}}{\mathcal{C}^{lm}_{P,\mathrm{in}}}\bigg{|}.$
(48)

Note that in this way, we have defined a common quantity as the degree of
absorption valid for both real and effective setups. Thus, this quantity will
also serve for matching between the real and effective theories to fix the
unknown tidal coefficients in the next subsection.

Essentially, comparing the form of $\psi_{4}$ in the full and effective
theories at large distances has revealed to us the combination of incoming and
outgoing coefficients in the spherical basis that scatter without mixing.
However, unsurprisingly, for general choices of tidal coefficients, the
relation between the outgoing and incoming spheroidal coefficients is not
going to nicely factorize as given in Eq. (47). In fact, the tidal
coefficients that mix the spherical $l=2$, and $l=3$ modes
$g^{i}_{\mathcal{E}/\mathcal{B}}$ and $h^{i}_{\mathcal{E}/\mathcal{B}}$ have
to be chosen on the effective theory side such that the combinations given in
Eqs. (44), (45) scatter without mixing. In this work, we impose this on the
effective theory by plugging in the expressions for the spherical outgoing
modes from Eq. (37) into the LHS of Eq. (47) and demanding that it be
proportional to the RHS, i.e., $\mathcal{C}^{lm}_{P,\mathrm{in}}$. We refer to
this as imposing spheroidal separability on the effective theory and doing so
yields the constraints

$\displaystyle g_{\mathcal{B},0}^{i}$
$\displaystyle=h_{\mathcal{E},0}^{i}=\frac{2}{3}\chi f_{\mathcal{E},0},$
$\displaystyle g_{\mathcal{E},0}^{i}$
$\displaystyle=h_{\mathcal{B},0}^{i}=-\frac{2}{3}\chi f_{\mathcal{B},0}^{i},$
(49)

which greatly reduces our list of unknown variables entering the tidal
response, and fixes the ratio of the coefficients connecting the
octupole(quadrupole) fields to the quadrupole(octupole) moments, i.e., the
coefficients
$g^{i}_{\mathcal{E}/\mathcal{B},0}$($h^{i}_{\mathcal{E}/\mathcal{B},0}$), to
the coefficients connecting the quadrupole fields to quadrupole moments, i.e.,
the coefficients $f^{i}_{\mathcal{E}/\mathcal{B},0}$.

Provided these relations are true, The different spheroidal modes will be
scattered without mixing by the tidal moments in the effective worldline
theory and we can compute the degree of absorption for the various modes using
Eq. (48) with constraints in Eq. (III.4) to be777We only focus on the degree
of absorption and not on the conservative phase as the conservative phase also
gains contributions from effects other than tidally induced multipole moments
in the effective theory, for e.g., spin-induced multipole moments. A matching
of the conservative phase can only be performed when all such effects are
included in the solution of the scattering problem in the effective theory,
which we do not.

$\displaystyle
1-\bigg{|}\frac{\mathcal{C}^{2m}_{P=1,\mathrm{out}}}{\mathcal{C}^{2m}_{P=1,\mathrm{in}}}\bigg{|}$
$\displaystyle=-\epsilon^{5}m\bigg{[}\frac{2f_{\mathcal{E},0}^{1}}{5}-(m^{2}-4)\frac{2}{15}f_{\mathcal{E},0}^{3}\bigg{]}+\epsilon^{6}\bigg{\\{}\frac{4f^{0}_{\mathcal{E},1}}{5}-(m^{2}-4)\bigg{[}\frac{2f^{2}_{\mathcal{E},1}}{15}+\frac{2f^{4}_{\mathcal{E},1}}{15}(m^{2}-1)\bigg{]}\bigg{\\}}$
$\displaystyle\quad+\mathcal{O}(\epsilon^{7}),$ $\displaystyle
1-\bigg{|}\frac{\mathcal{C}^{2m}_{P=-1,\mathrm{out}}}{\mathcal{C}^{2m}_{P=-1,\mathrm{in}}}\bigg{|}$
$\displaystyle=-\epsilon^{5}m\bigg{[}\frac{2f_{\mathcal{B},0}^{1}}{5}-(m^{2}-4)\frac{2}{15}f_{\mathcal{B},0}^{3}\bigg{]}+\epsilon^{6}\bigg{\\{}\frac{4f^{0}_{\mathcal{B},1}}{5}-(m^{2}-4)\bigg{[}\frac{2f^{2}_{\mathcal{B},1}}{15}+\frac{2f^{4}_{\mathcal{B},1}}{15}(m^{2}-1)\bigg{]}\bigg{\\}}$
$\displaystyle\quad+\mathcal{O}(\epsilon^{7}),$ $\displaystyle
1-\bigg{|}\frac{\mathcal{C}^{3m}_{P=1,\mathrm{out}}}{\mathcal{C}^{3m}_{P=1,\mathrm{in}}}\bigg{|}$
$\displaystyle=1-\bigg{|}\frac{\mathcal{C}^{3m}_{P=-1,\mathrm{out}}}{\mathcal{C}^{3m}_{P=-1,\mathrm{in}}}\bigg{|}=\mathcal{O}(\epsilon^{7}),$
(50)

where we find that the degree of absorption for the spheroidal $l=3$ modes
vanishes up to $\mathcal{O}(\epsilon^{7})$ in the effective theory. Also note
that the coefficients $f^{0}_{\mathcal{E}/\mathcal{B},0}$,
$f^{2}_{\mathcal{E}/\mathcal{B},0}$, $f^{4}_{\mathcal{E}/\mathcal{B},0}$,
$f^{1}_{\mathcal{E}/\mathcal{B},1}$, $f^{3}_{\mathcal{E}/\mathcal{B},3}$ do
not appear anywhere in the degree of absorption. These coefficients thus only
add to the conservative phase, and we will see later in Eqs. (IV), (IV) that
they behave similarly with mass/spin evolution equations as well contributing
only total time derivatives to $dm/dt$ and $dJ/dt$. We can now compare Eq.
(50) with the degree of absorption derived in the real theory by solving the
Teukolsky equation for $\eta^{P}_{lm}$ and fix the coefficients that
contribute to dissipation. However, because we solved the scattering problem
in effective theory in flat space-time thus ignoring the nonlinear
interactions between the wave and the stationary gravitatonal field of the
particle, there are some subtleties in this matching process which we will
have to tackle. We will very briefly outline the computation of
$\eta^{P}_{lm}$ in the real theory by solving the Teukolsky equation and then
match with the result from the effective theory while keeping the subtleties
in mind.

###  III.5 Matching with Teukolsky solution

In the full theory, the degree of absorption defined from the
$\mathcal{O}(1/r^{6})$ expansion of $\psi_{4}$ in Eq. (III.4) as
$1-|K^{out}_{lmP}/K^{in}_{lmP}|$ can be computed from analytical solutions
[54] to the Teukolsky equation [49] as follows. Following the review [54], the
Teukolsky equation for ${}_{-2}\psi$ [$=(r-ia\cos(\theta))^{4}\psi_{4}$ in
Boyer-Lindquist coordinates] is separable with fixed-frequency solutions given
by ${}_{-2}\psi\propto e^{-i\omega
t}\,S^{lm}_{s}(\theta,\phi,a\omega)\,{}_{-2}R_{lm\omega}(r)$, where
$S^{lm}_{s}(\theta,\phi,a\omega)$ are spin-weighted spheroidal harmonics, and
${}_{s}R_{lm\omega}(r)$ is a solution to the (homogeneous) radial Teukolsky
equation, with $s=-2$,

$\displaystyle\bigg{[}\Delta^{-s}\frac{d}{dr}\left(\Delta^{s+1}\frac{d}{dr}\right)+\frac{K^{2}-2is(r-GM)K}{\Delta}$
$\displaystyle\quad+4is\omega r-{}_{s}\lambda_{lm}\bigg{]}{}_{s}R_{\ell
m\omega}(r)=0,$ (51)

where $\Delta=r^{2}+a^{2}-2GMr$, $K=(r^{2}+a^{2})\omega-am$, and
${}_{s}\lambda_{lm}$ is the spheroidal eigenvalue. The relevant physical
solutions, labelled ${}_{-2}R^{\textrm{in}}_{\ell m\omega}(r)$, satisfy the
boundary condition demanding that they consist of purely ingoing radiation at
the event horizon $r=r_{+}:=GM+\sqrt{(GM)^{2}-a^{2}}$,

${}_{-2}R^{\textrm{in}}_{\ell
m\omega}=B^{\textrm{trans}}_{{\ell}{m}{\omega}}\Delta^{2}e^{-i\tilde{\omega}r_{*}}\;\;\textrm{as}\;\;r\rightarrow
r_{+},$ (52)

where $r_{*}$ is the tortoise coordinate and $\tilde{\omega}=\omega-
ma/(2GMr_{+})$. This fixes the asymptotic behavior at radial infinity to be of
the form

$\displaystyle{}_{-2}R^{\textrm{in}}_{\ell m\omega}$
$\displaystyle=B^{\textrm{inc}}_{{\ell}{m}{\omega}}r^{-1}e^{-i\omega
r_{*}}+B^{\textrm{ref}}_{{\ell}{m}{\omega}}r^{3}e^{i\omega r_{*}}$ (53)
$\displaystyle\;\;\textrm{as}\;\;r\rightarrow\infty,$

where $B^{\textrm{inc}}_{{\ell}{m}{\omega}}$ and
$B^{\textrm{ref}}_{{\ell}{m}{\omega}}$888Note that the above form is identical
to the one used earlier in Eq. (III.4) except that the parity dependent
factors have been absorbed into $B_{lm\omega}^{\mathrm{inc}}$. As a result,
the expression for the scattering phase now contains a parity-dependent factor
[see Eq. (III.5)]. This maybe a more convenient convention for BHPT but the
former is more transparent for matching with the effective theory. are the
coefficients of the incident and reflected waves. The ratio
$B^{\textrm{ref}}_{{\ell}{m}{\omega}}/B^{\textrm{inc}}_{{\ell}{m}{\omega}}$ is
completely determined by demanding that ${}_{-2}R^{\textrm{in}}_{\ell
m\omega}(r)$ solve the radial Teukolsky equation (51) with the boundary
condition (52). We refer the reader to Ref. [54] for details of a procedure to
produce the expansion of this ratio in powers of $GM\omega$. Finally, the
relevant scattering phase shifts and transmission factor
$\eta^{P}_{lm}e^{2i\delta_{lm\omega}^{P}}$ (equivalent to
$\mathcal{C}^{lm}_{P,\mathrm{out}}/\mathcal{C}^{lm}_{P,\mathrm{in}}$ from the
effective theory above) for waves of parity $P=\pm 1$ are given, e.g. as in
Eq. (30) of Ref. [52], by

$\displaystyle\eta_{lm}^{P}\exp(2i\delta_{lm\omega}^{P})$
$\displaystyle=(-1)^{l+1}\left(\frac{\textrm{Re}(C)+12iGM\omega
P}{16\omega^{4}}\right)$
$\displaystyle\times\frac{B^{\textrm{ref}}_{{\ell}{m}{\omega}}}{B^{\textrm{inc}}_{{\ell}{m}{\omega}}},$
(54)

with $[\textrm{Re}(C)]^{2}$ as given in Eq. (31) of Ref. [52].

This yields the complete expression for the degree of absorption for $l=2$ and
$l=3$ spheroidal modes up to $\mathcal{O}(\epsilon^{7})$ to be

$\displaystyle 1-|\eta_{2m}^{P=\pm}|=$
$\displaystyle-\epsilon^{5}m\bigg{[}\frac{2\mathcal{A}_{0}^{1}}{5}-(m^{2}-4)\frac{2}{15}\mathcal{A}_{0}^{3}\bigg{]}(1+2\epsilon\pi)$
(55)
$\displaystyle+\epsilon^{6}\bigg{\\{}\frac{4\mathcal{A}^{0}_{1}}{5}-(m^{2}-4)\bigg{[}\frac{2\mathcal{A}^{2}_{1}}{15}+\frac{2\mathcal{A}^{4}_{1}}{15}(m^{2}-1)\bigg{]}\bigg{\\}}+\mathcal{O}(\epsilon^{7}),$
$\displaystyle 1-|\eta_{3m}^{P=\pm}|=$
$\displaystyle\mathcal{O}(\epsilon^{7}),$ (56)

where

$\displaystyle\mathcal{A}^{1}_{0}=\frac{16\chi}{45}(1+3\chi^{2}),~{}\mathcal{A}_{0}^{3}=-\frac{4\chi^{3}}{3},$
(57)
$\displaystyle\mathcal{A}_{1}^{0}=\frac{16}{405}(9+9\kappa+97\chi^{2}+117\kappa\chi^{2}-6\chi^{4}+54\sigma\chi^{4}+36\chi
B_{2}+108\chi^{3}B_{2}),$
$\displaystyle\mathcal{A}_{1}^{2}=-\frac{8}{135}(115\chi^{2}+135\kappa\chi^{2}+5\chi^{4}+90\kappa\chi^{4}-24\chi
B_{1}+18\chi^{3}B_{1}+48\chi B_{2}+144\chi^{3}B_{2}),$
$\displaystyle\mathcal{A}_{1}^{4}=\frac{8}{135}(20\chi^{4}+45\kappa\chi^{4}-24\chi
B_{1}+18\chi^{3}B_{1}+12\chi B_{2}+36\chi^{3}B_{2}),$

and we are using the notation
$B_{m}=\mathrm{Im}[\mathrm{PolyGamma}(0,3+im\chi/\kappa)]$, which is odd in
$\chi$, and $\kappa=\sqrt{1-\chi^{2}}$. Note that the degree of absorption
obtained from solving the Teukolsky equation in Eqs. (55), (56) has almost
exactly the same form as that obtained from the effective theory given in Eq.
(50), with the only difference being the factor of $(1+2\epsilon\pi)$ next to
the leading-order $\epsilon^{5}$ result for $l=2$ mode. This factor is missing
in the degree of absorption derived in the effective theory in Eq. (50) due to
us neglecting the leading-order nonlinear interaction between the GW and the
gravitational field of the particle. In principle, this can be also obtained
from the effective worldline theory by including the nonlinearities and
regulating any resulting divergences. However, including the leading-order
nonlinearities in this classical setup which we are using is a complicated
task, and not very illuminating. It is easier instead to just replace the
factor with $1$ on the Teukolsky side by tracing its origins to the
nonlinearities neglected in the scattering problem in the effective picture.
We establish this explicitly for the simpler case of a scalar field scattering
of a Schwarzschild BH by including the leading-order nonlinearities on the
effective theory side in Appendix. A. There are essentially two physical
processes involved in the effective theory picture, both arising from the
interaction of the external wave (gravitational or otherwise) with the
stationary gravitational field sourced by the particle. The first is that the
value of the tidal field of the wave at the origin (location of the particle)
is modified, as shown for a scalar wave in Eq. (125) by a factor of
$(1+\pi\epsilon)$. This in turn modifies the strength of the tidally induced
multipole moment subsequently affecting the irregular (output) part of the
wave and the degree of absorption. Additionally, the wave is modified in its
journey away (to) the particle to (from) infinity again due to scattering off
the particle’s gravitational field. This factor modifies the form of the wave
asymptotically far away from the particle, as shown again for a scalar wave in
Eq. (121) in such a way that the degree of absorption is further multiplied by
a factor of $(1+\pi\epsilon)$. This result has also been derived for the case
of GW amplitudes sourced by arbitrary multipole moments in Refs. [40, 55] and
is seen to modify the radiated power by the square of that factor. Together,
these two effects modify the leading-order degree of absorption by a factor
$(1+\epsilon\pi)^{2}\approx(1+2\epsilon\pi)+\mathcal{O}(\epsilon^{2})$ which
is seen in Eq. (55). We have in fact verified the presence of this factor
multiplying the leading-order degree of absorption for all different $l$ modes
(for which we had the solution) for scalar, and tensor (gravitational) fields,
specifically $l=0,~{}1,~{}2,~{}3$ modes for the scalar field and $l=2,~{}3$
for gravitational field. Thus, to match the part of the true degree of
absorption which corresponds to the flat space-time scattering in the
effective picture, it is sufficient to simply replace the $(1+2\pi\epsilon)$
factor for $1$ from Eq. (55).

Thus, dropping the $(1+2\epsilon\pi)$ factor from Eq. (55), and then comparing
it with Eq. (50), we fix the unknown coefficients to be

$\displaystyle
f^{1}_{\mathcal{E}/\mathcal{B},0}=\frac{16\chi}{45}(1+3\chi^{2}),~{}f^{3}_{\mathcal{E}/\mathcal{B},0}=-\frac{4\chi^{3}}{3},$
(58) $\displaystyle
f^{0}_{\mathcal{E}/\mathcal{B},1}=\frac{16}{405}(9+9\kappa+97\chi^{2}+117\kappa\chi^{2}-6\chi^{4}+54\kappa\chi^{4}+36\chi
B_{2}+108\chi^{3}B_{2}),$ $\displaystyle
f^{2}_{\mathcal{E}/\mathcal{B},1}=-\frac{8}{135}(115\chi^{2}+135\kappa\chi^{2}+5\chi^{4}+90\kappa\chi^{4}-24\chi
B_{1}+18\chi^{3}B_{1}+48\chi B_{2}+144\chi^{3}B_{2}),$ $\displaystyle
f^{4}_{\mathcal{E}/\mathcal{B},1}=\frac{8}{135}(20\chi^{4}+45\kappa\chi^{4}-24\chi
B_{1}+18\chi^{3}B_{1}+12\chi B_{2}+36\chi^{3}B_{2}),$ $\displaystyle
g_{\mathcal{B},0}^{i}=h^{i}_{\mathcal{E},0}=\frac{2}{3}\chi
f_{\mathcal{E},0},$ $\displaystyle
g_{\mathcal{E},0}^{i}=h^{i}_{\mathcal{B},0}=-\frac{2}{3}\chi
f_{\mathcal{B},0},$ (59)

where we have also restated the constraints obtained by imposing spheroidal
separability from Eq. (III.4). Having fixed the (dissipative) response
coefficients, we can now compute the (dissipative part of the) induced
multipole moments in any setup. In particular, we can now consider the effect
of induced tides in a binary system with two spinning BHs in the inspiral
phase. Our focus is on computing the change in mass and angular momentum due
to tidal effects in the worldline effective theory (and horizon fluxes in the
real setup). In the next section, we derive general formulae for evolution
equations of mass and spin and then specialize to the case of parallel-
spin–quasi-circular orbits, which can then be compared with earlier results
available in literature.

##  IV General expressions for evolution equations of mass and spin

In this section, we derive general formulae for computing the evolution of of
mass and spin from the equations of motion. Then, we proceed to compute them
explicitly using the now fixed response coefficients for the special case of
parallel-spin–quasi-circular inspiral to relative 1.5PN order. We conclude
this section by comparing these results with those obtained earlier in Refs.
[30, 22, 21].

We can derive the formula for mass and spin evolution from the equations of
motion for momentum and spin angular momentum respectively. For mass, we start
with the equation of motion for momentum, i.e.,

$\displaystyle\frac{Dp^{\mu}}{D\tau}$
$\displaystyle=-\frac{1}{2}R_{\mu\nu\rho\sigma}u^{\nu}S^{\rho\sigma}-\frac{1}{6}J^{\lambda\nu\rho\sigma}\nabla_{\mu}R_{\lambda\nu\rho\sigma}$
$\displaystyle-J^{\tau\lambda\nu\rho\sigma}\frac{1}{12}\nabla_{\mu}\nabla_{\tau}R_{\lambda\nu\rho\sigma},$
(60)

where we substitute the expressions for $J^{\mu\nu\rho\sigma}$ and
$J^{\lambda\mu\nu\rho\sigma}$ given in Eqs. (7), (8), yielding

$\displaystyle\frac{Dp^{\mu}}{D\tau}$
$\displaystyle=-\frac{1}{2}R_{\mu\nu\rho\sigma}u^{\nu}S^{\rho\sigma}-\frac{1}{2}Q^{\rho\sigma}\nabla^{\mu}\mathcal{E}_{\rho\sigma}-\frac{1}{2}Q_{\mathcal{B}}^{\rho\sigma}\nabla^{\mu}\mathcal{B}_{\rho\sigma}$
$\displaystyle-\frac{1}{2}Q_{\mathcal{E}}^{\rho\sigma\lambda}\nabla^{\mu}\mathcal{E}_{\rho\sigma\lambda}-\frac{1}{2}Q_{\mathcal{B}}^{\rho\sigma\lambda}\nabla^{\mu}\mathcal{B}_{\rho\sigma\lambda}.$
(61)

Now, we define in the effective worldline theory the mass $m$ simply as
$\sqrt{-p^{2}}$. Then we have

$\displaystyle p_{\mu}\frac{Dp^{\mu}}{D\tau}=-m\frac{dm}{d\tau}$
$\displaystyle\implies\frac{dm}{d\tau}=\frac{1}{2m}p^{\mu}u^{\nu}R_{\mu\nu\rho\sigma}S^{\rho\sigma}+\frac{1}{2m}Q_{\mathcal{E}}^{\rho\sigma}(p\cdot\nabla)\mathcal{E}_{\rho\sigma}$
$\displaystyle+\frac{1}{2m}Q_{\mathcal{B}}^{\rho\sigma}(p\cdot\nabla)\mathcal{B}_{\rho\sigma}+\frac{1}{2m}Q_{\mathcal{E}}^{\rho\sigma\lambda}(p\cdot\nabla)\mathcal{E}_{\rho\sigma\lambda}$
$\displaystyle+\frac{1}{2m}Q_{\mathcal{B}}^{\rho\sigma\lambda}(p\cdot\nabla)\mathcal{B}_{\rho\sigma\lambda},$
(62)

and the first term can be shown to vanish using the relation between $p^{\mu}$
and $u^{\mu}$ if we neglect terms cubic or higher powers in curvature. They
are not relevant to the relative 1.5PN order (in horizon fluxes) we are
interested in this work. Similarly, we can substitute $p^{\mu}=mu^{\mu}$ in
the terms with multipole moments to the order of interest to get

$\displaystyle\frac{dm}{d\tau}=\frac{1}{2}Q_{\mathcal{E}}^{\rho\sigma}\dot{\mathcal{E}}_{\rho\sigma}+\frac{1}{2}Q_{\mathcal{E}}^{\rho\sigma\lambda}\dot{\mathcal{E}}_{\rho\sigma\lambda}+(\mathcal{E}\leftrightarrow\mathcal{B})$
(63)

which gives us the general formula for mass evolution valid to relative 1.5PN
order. We expect this to match with the horizon energy flux up to any total
time derivatives of functions of tidal fields, which should vanish for
scattering events or for quasi periodic processes (like parallel-spin–quasi-
circular inspiral which we shall consider shortly). The quadrupolar
contribution to mass-change matches with that given in Ref. [23] if we
identify our quadrupole tensors as twice of theirs. This is because they
choose a different normalization in the tidal coupling terms. 999 In Ref.
[23], the quadrupole tidal coupling terms in the action are
$Q_{\mathcal{E}}^{\mu\nu}\mathcal{E}_{\mu\nu}+(\mathcal{E}\leftrightarrow\mathcal{B})$.
Thus, before comparing the expressions in this work with that in Ref. [23].
One must first transform as $Q_{\mathcal{E}(\mathcal{B})}^{\mu\nu}\rightarrow
2Q_{\mathcal{E}(\mathcal{B})}^{\mu\nu}$

We can similarly derive the equation for spin evolution from the equation of
motion for spin angular momentum, i.e.,

$\displaystyle\frac{DS^{\mu\nu}}{D\tau}$
$\displaystyle=2p^{[\mu}u^{\nu]}+\frac{4}{3}R^{[\mu}{}_{\tau\rho\sigma}J^{\nu\tau\rho\sigma}+\frac{2}{3}\nabla^{\lambda}R^{[\mu}{}_{\tau\rho\sigma}J_{\lambda}{}^{\nu\tau\rho\sigma}+\frac{1}{6}\nabla^{[\mu}R_{\lambda\tau\rho\sigma}J^{\nu]\lambda\tau\rho\sigma}$
$\displaystyle
J\frac{DJ}{d\tau}=\frac{1}{2}S_{\mu\nu}\frac{DS^{\mu\nu}}{D\tau}$
$\displaystyle=\frac{2}{3}R^{\mu}{}_{\tau\rho\sigma}J^{\nu\tau\rho\sigma}S_{\mu\nu}+\frac{1}{3}\nabla^{\lambda}R^{\mu}{}_{\tau\rho\sigma}J_{\lambda}{}^{\nu\tau\rho\sigma}S_{\mu\nu}+\frac{1}{12}\nabla^{\mu}R_{\lambda\tau\rho\sigma}J^{\nu\lambda\tau\rho\sigma}S_{\mu\nu}$
$\displaystyle\frac{DJ}{d\tau}$
$\displaystyle=\frac{1}{J}Q_{\mathcal{E}}^{\mu\nu}\mathcal{E}_{\mu}{}^{\rho}S_{\rho\nu}+\frac{3}{2J}O_{\mathcal{E}}^{\mu\nu\lambda}\mathcal{E}_{\mu\lambda}{}^{\rho}S_{\nu\rho}+(\mathcal{E}\leftrightarrow\mathcal{B}),$
(64)

where we have defined $J^{2}=(1/2)S^{\mu\nu}S_{\mu\nu}$ as the magnitude of
spin angular momentum of the BH. Again, the quadrupolar contribution to
evolution of spin($dJ/dt$) is identical to that in Ref. [23] once the
multipole moments are properly identified (see footnote. 9).

Now, substituting the ansätze for the multipole moments from Eqs. (III.2) into
the expressions for the evolution of mass and spin in Eqs. (63), (64). We get

$\displaystyle\frac{dm^{*}}{d\tau}$
$\displaystyle=\frac{m}{2}(Gm)^{4}\\{f^{1}_{\mathcal{E},0}(\dot{\mathcal{E}}^{\mu\nu}\mathcal{E}_{\mu}{}^{\rho}\hat{S}_{\nu\rho})+f^{3}_{\mathcal{E},0}(\dot{\mathcal{E}}_{\mu}{}^{\rho}\mathcal{E}_{\nu}{}^{\sigma}\hat{s}^{\mu}\hat{s}^{\nu}\hat{S}_{\rho\sigma})$
$\displaystyle+(Gm)[f_{\mathcal{E},1}^{0}(\dot{\mathcal{E}}^{\mu\nu}\dot{\mathcal{E}}_{\mu\nu})+f_{\mathcal{E},1}^{2}(\dot{\mathcal{E}}_{\mu}{}^{\rho}\dot{\mathcal{E}}_{\nu\rho}\hat{s}^{\mu}\hat{s}^{\nu})+f_{\mathcal{E},1}^{4}(\dot{\mathcal{E}}_{\mu\nu}\dot{\mathcal{E}}_{\rho\sigma}\hat{s}^{\mu}\hat{s}^{\nu}\hat{s}^{\rho}\hat{s}^{\sigma})$
$\displaystyle+\xi\frac{2}{3}\chi
f_{\mathcal{E},0}^{1}(\mathcal{B}_{\mu\rho\sigma}\dot{\mathcal{E}}^{\nu\rho}-\dot{\mathcal{B}}_{\mu\rho\sigma}\mathcal{E}^{\nu\rho})\hat{s}^{\mu}\hat{S}_{\nu}{}^{\sigma}+\xi\frac{2}{3}\chi
f_{\mathcal{E},0}^{3}(\mathcal{B}_{\nu\rho\lambda}\dot{\mathcal{\mathcal{E}}}_{\mu}{}^{\sigma}-\dot{\mathcal{B}}_{\nu\rho\lambda}\mathcal{E}_{\mu}{}^{\sigma})\hat{s}^{\mu}\hat{s}^{\nu}\hat{s}^{\rho}\hat{S}_{\sigma}{}^{\lambda}]\\}$
$\displaystyle+(\mathcal{\mathcal{E}}\leftrightarrow\mathcal{\mathcal{B}},~{}\xi\rightarrow-\xi),$
(65) $\displaystyle\frac{dJ^{*}}{d\tau}$
$\displaystyle=\frac{M}{2}(Gm)^{4}\\{-2f_{\mathcal{E},0}^{1}(\mathcal{E}_{ij}\mathcal{E}^{ij})+(3f_{\mathcal{E},0}^{1}-f_{\mathcal{E},0}^{3})(\mathcal{E}_{i}{}^{k}\mathcal{E}_{jk}\hat{s}^{i}\hat{s}^{j})+f_{\mathcal{E},0}^{3}(\mathcal{E}_{ij}\hat{s}^{i}\hat{s}^{j})^{2}$
$\displaystyle-(Gm)[f_{\mathcal{E},1}^{0}(\dot{\mathcal{E}}^{\mu\nu}\mathcal{E}_{\mu}{}^{\rho}\hat{S}_{\nu\rho})+f_{\mathcal{E},1}^{2}(\dot{\mathcal{E}}_{\mu}{}^{\rho}\mathcal{E}_{\nu}{}^{\sigma}\hat{s}^{\mu}\hat{s}^{\nu}\hat{S}_{\rho\sigma})$
$\displaystyle+\xi\frac{2}{3}\chi
f_{\mathcal{E},0}^{1}(\mathcal{B}_{\mu\sigma\lambda}\mathcal{E}^{\nu\rho}\hat{s}^{\mu}\hat{S}_{\nu}{}^{\sigma}\hat{S}_{\rho}{}^{\lambda})+\xi\frac{2}{3}\chi
f_{\mathcal{E},0}^{1}(\mathcal{B}_{\mu\rho\lambda}\mathcal{E}^{\nu\rho}\hat{s}^{\mu}\hat{S}_{\nu}{}^{\sigma}\hat{S}_{\sigma}{}^{\lambda})+\xi\frac{2}{3}\chi
f_{\mathcal{E},0}^{3}(\mathcal{B}_{\nu\rho\tau}\mathcal{E}_{\mu}{}^{\sigma}\hat{s}^{\mu}\hat{s}^{\nu}\hat{s}^{\rho}\hat{S}_{\sigma}{}^{\lambda}\hat{S}_{\lambda}{}^{\tau})]\\}$
$\displaystyle+(\mathcal{E}\leftrightarrow\mathcal{B},~{}\xi\rightarrow-\xi),$
(66)

where dots represent covariant derivatives w.r.t proper time, $\xi=1$, and we
have imposed the constraints obtained by demanding spheroidal separability
from Eq. (III.4). Additionally, we have

$\displaystyle m^{*}$
$\displaystyle=m-\frac{m}{4}(Gm)^{4}\\{f_{\mathcal{E},0}^{0}(\mathcal{E}^{\mu\nu}\mathcal{E}_{\mu\nu})+f_{\mathcal{E},0}^{2}(\mathcal{E}_{\mu}{}^{\rho}\mathcal{E}_{\nu\rho}^{\sigma}\hat{s}^{\mu}\hat{s}^{\nu})+f_{\mathcal{E},0}^{4}(\mathcal{E}_{\mu\nu}\mathcal{E}_{\rho\sigma}\hat{s}^{\mu}\hat{s}^{\nu}\hat{s}^{\rho}\hat{s}^{\sigma})$
$\displaystyle+(Gm)[\xi\frac{2}{3}\chi
f_{\mathcal{E},0}^{0}(\mathcal{B}_{\mu\nu\rho}\mathcal{E}^{\nu\rho}\hat{s}^{\mu})+\xi\frac{2}{3}\chi
f_{\mathcal{E},0}^{2}(\mathcal{B}_{\nu\rho\sigma}\mathcal{E}_{\mu}{}^{\sigma}\hat{s}^{\mu}\hat{s}^{\nu}\hat{s}^{\rho})+\xi\frac{2}{3}\chi
f_{\mathcal{E},0}^{4}(\mathcal{B}_{\rho\sigma\lambda}\mathcal{E}_{\mu\nu}\hat{s}^{\mu}\hat{s}^{\nu}\hat{s}^{\rho}\hat{s}^{\sigma}\hat{s}^{\lambda})]\\}$
$\displaystyle+(\mathcal{E}\leftrightarrow\mathcal{B},~{}\xi\rightarrow-\xi),$
(67) $\displaystyle J^{*}$
$\displaystyle=J-\frac{M}{4}(GM)^{5}\\{f_{\mathcal{E},1}^{1}[(\mathcal{E}^{\mu\nu}\mathcal{E}^{\rho\sigma}\hat{S}_{\mu\rho}\hat{S}_{\nu\sigma})-(\mathcal{E}^{\mu\nu}\mathcal{E}_{\mu}{}^{\rho}\hat{S}_{\nu}{}^{\sigma}\hat{S}_{\rho\sigma})]-f_{\mathcal{E},1}^{3}(\mathcal{E}_{\mu}{}^{\rho}\mathcal{E}_{\nu}{}^{\sigma}\hat{s}^{\mu}\hat{s}^{\nu}\hat{S}_{\rho}{}^{\lambda}\hat{S}_{\sigma\lambda})\\},$
(68)

and we see that all the tidal coefficients that do not enter the degree of
absorption in the RHS of Eq. (50) only shift the definition of mass and
angular-momentum by quadratic functions of fields (i.e., they only contribute
terms that are total-time derivatives to $dm/dt$ and $dJ/dt$). Whereas the
ones that do show up in degree of absorption contribute terms that cannot be
absorbed as such in total time-derivatives. Thus, for any quasi periodic setup
(like for parallel-spin–quasi-circular orbits) or in a scattering set up where
the two particles are infinitely far in the distant past or future, the
average or total change in mass/spin respectively is determined entirely by
the coefficients that contribute to dissipation, which we have already fixed
through comparison with the degree of absorption obtained in the real theory
by solving the Teukolsky equation in Sec. III.5. This is since the total-time
derivative terms either vanish (when the particles are far away) or cancel (in
a periodic setup). In deriving the above result, we have also used the fact
that the covariant time-derivative of spin tensor and vector (which enter the
ansätze) vanishes up to the relative order to which we have expanded the
expressions for $dm/dt$ and $dJ/dt$ (i.e., up to relative 1.5PN).

The expressions for mass and spin evolution in Eqs. (IV), (IV) can be compared
with those in Refs. [23, 22] by substituting the response coefficients from
Eq. (58), and we indeed find that our expressions are identical at leading
order in $GM$ (i.e., the $(GM)^{4}$ part), but differs from Ref. [22] at next
order in $GM$. A crucial difference in our expressions when compared with
those in Ref. [22] is that octupolar tidal fields do not enter their
expressions at all. Another interesting difference is that there are
$\pi^{2}$-containing coefficients in their next-to-leading order expression
for $dJ/dt$ (and subsequently in $dm/dt$), whereas $\pi^{2}$ does not enter
any of our tidal-response coefficients. However, we will see later in Sec.
IV.2 that our expression for mass and spin evolution is consistent with Ref.
[21] for the special case of a test-body in a circular orbit around a Kerr BH
to relative 1.5PN order, unlike Ref. [22].

In the next section, we specialize to the parallel-spin–quasi-circular setting
to compute the expression mass and spin evolution during inspiral up to 1.5PN
relative to the same at leading order (4PN w.r.t leading-order flux to
infinity), and compare with earlier works that produced expressions for the
same.

###  IV.1 Results for the special case of binaries with parallel spins in and
circular orbits

Figure 1: A graphic illustrating a parallel-spin–quasi-circular binary with
two BHs with masses $m_{1}$ and $m_{2}$, and spins $J_{1}$ and $J_{2}$. The
spin vectors are orthogonal to the orbital plane. The image is drawn in the
comoving frame of BH with mass $m_{1}$, with $\hat{n}$ being the unit-vector
pointing towards the other BH ($m_{2},~{}J_{2}$) and $\hat{v}$ being the unit-
vector along the other BH’s velocity. $\Omega$ is the angular velocity of the
other BH and hence of the tidal fields in this frame. $\tau$ is the proper
time of mass $m_{1}$.

Now, we consider a system of two BHs with initial masses $m_{1}$ and $m_{2}$
and spin parameters $\chi_{1}$ and $\chi_{2}$. Their spins are parallel and
orthogonal to the orbital plane and they are in a quasi-circular orbit. We can
compute the rate of change of (initial) mass $m_{1}$ and spin
$J_{1}=Gm_{1}^{2}\chi_{1}$ averaged over one orbit using the equations Eqs.
(63),(64). In this section, we will refer to the BH with initial mass $m_{1}$
and $J_{1}$ as the primary BH, and the other BH as secondary here onwards. The
tidal fields are sourced by the other BH of mass $m_{2}$ with spin parameter
$\chi_{2}$, although they are affected by nonlinear interaction with the
fields due to the primary BH ($m_{1}$, $J_{1}$). This is most conveniently
done when the tidal fields are computed in a locally flat rest frame of the
primary BH (see Fig. 1), since then the covariant derivatives of the field
w.r.t proper time can be treated simply as ordinary time derivatives . This
has already been done for the quadrupolar fields $\mathcal{E}^{\mu\nu}$ and
$\mathcal{B}^{\mu\nu}$ in Ref. [22], and we simply borrow the expressions from
there. Rewriting the expressions here here, we have

$\displaystyle\frac{1}{2}(\mathcal{E}_{11}+\mathcal{E}_{22})$
$\displaystyle=-\frac{m_{2}}{2r^{3}}\bigg{[}1+\frac{X_{1}}{2}V^{2}-6X_{2}\chi_{2}V^{3}+\mathcal{O}(V^{4})\bigg{]},$
$\displaystyle\frac{1}{2}(\mathcal{E}_{11}-\mathcal{E}_{22})$
$\displaystyle=-\frac{3m_{2}}{2r^{3}}\bigg{[}1+\frac{X_{1}-4}{2}V^{2}-2X_{2}\chi_{2}V^{3}+\mathcal{O}(V^{4})\bigg{]}\cos(2\Omega\tau),$
$\displaystyle\mathcal{E}_{12}$
$\displaystyle=-\frac{3m_{2}}{2r^{3}}\bigg{[}1+\frac{X_{1}-4}{2}V^{2}-2X_{2}\chi_{2}V^{3}+\mathcal{O}(V^{4})\bigg{]}\sin(2\Omega\tau),$
(69) $\displaystyle\mathcal{B}_{13}$
$\displaystyle=\frac{-3m_{2}}{r^{3}}V(1-X_{2}\chi_{2}V)\cos(\Omega\tau)+\mathcal{O}(V^{3}),$
$\displaystyle\mathcal{B}_{23}$
$\displaystyle=\frac{-3m_{2}}{r^{3}}V(1-X_{2}\chi_{2}V)\sin(\Omega\tau)+\mathcal{O}(V^{3}),$
(70)

where $\Omega$ is the angular velocity of the tidal field in the primary BH
frame ($m_{1}$) given by

$\displaystyle\Omega$
$\displaystyle=\sqrt{\frac{M}{r^{3}}}\bigg{[}1-\frac{1}{2}(3+\eta)V^{2}-\frac{1}{2}\bar{\chi}V^{3}+\mathcal{O}(V^{4})\bigg{]},$
(71) $\displaystyle\bar{\chi}$
$\displaystyle=X_{1}(1+X_{1})\chi+3\eta\chi_{2},$ (72)

and we are using the notation : $M=m_{1}+m_{2}$ and $X_{1}=m_{1}/M$,
$X_{2}=m_{2}/M$ are the mass-fractions. $\eta=X_{1}X_{2}$ is the symmetric
mass ratio, $r$ is the orbital separation in harmonic coordinates and
$V=\sqrt{\frac{M}{r}}$. Here, we are working in units with $G=c=1$, as was
done in Ref. [22] so that we can easily compare our results although we have
changed the notation quite a bit. Additionally, in Ref. [22], a sign factor
$\epsilon=\pm 1$ was used in front of the expression for $\Omega$, to denote
whether the secondary BH was spin aligned or antialigned w.r.t the orbital
angular momentum. However, we simply let the spin parameter(s) $\chi_{1,2}$
range over [-1,1] (instead of [0,1]) and always fix the orbital angular
momentum to be aligned along the positive z-axis without loss of generality.

The octupolar fields were not derived in Ref. [22] since they were not
relevant in their expression for the mass or spin evolution. For us, the
octupolar fields do contribute to the expressions for mass and spin evolution
as seen from Eqs. (IV), (IV). Fortunately, they (octupolar fields) are only
relevant at leading order and thus can be easily computed from the test-body
limit at leading post Newtonian order where the secondary BH (as test mass) is
orbiting the primary BH in the limit $m_{2}\ll m_{1}$. The formula for
leading-order fields does not change from this for generic mass-ratio. We get
the expressions

$\displaystyle\mathcal{E}_{ijk}$
$\displaystyle=-15\frac{m_{2}}{r^{4}}\hat{n}_{\langle
i}\hat{n}_{j}\hat{n}_{k\rangle},$ (73) $\displaystyle\mathcal{B}_{ijk}$
$\displaystyle=30\frac{m_{2}}{r^{4}}V\epsilon_{\langle
i}{}^{mn}\hat{n}_{j}\hat{n}_{k\rangle}\hat{n}_{m}\hat{v}_{n},$ (74)
$\displaystyle\hat{n}$
$\displaystyle=(\hat{n}_{1},\hat{n}_{2},\hat{n}_{3})=(\cos(\Omega\tau),\sin(\Omega\tau),0),\quad$
$\displaystyle\hat{v}$
$\displaystyle=(\hat{v}_{1},\hat{v}_{2},\hat{v}_{3})=(-\sin(\Omega\tau),\cos(\Omega\tau),0)$

where $n_{i}$ is the normal vector directed from the primary BH to the
secondary BH in harmonic coordinates, and $\hat{v}=\dot{\hat{n}}$ is the
relative velocity vector of the secondary BH w.r.t primary BH.

Now, substituting these tidal fields from Eqs. (69), (70), (73), and (74) into
the formulas for mass and spin evolution in Eqs. (IV), (IV), with the fixed
coefficients listed in Eqs. (58), (59), we get the orbit-averaged results

$\displaystyle\Big{\langle}\frac{dm_{1}}{d\tau}\Big{\rangle}$
$\displaystyle=\Omega(\Omega_{\mathrm{H}}-\Omega)C_{V},$ (75)
$\displaystyle\Big{\langle}\frac{dJ_{1}}{d\tau}\Big{\rangle}$
$\displaystyle=(\Omega_{\mathrm{H}}-\Omega)C_{V},$ (76)
$\displaystyle\Big{\langle}\frac{dA_{1}}{d\tau}\Big{\rangle}$
$\displaystyle=\frac{(dm_{1}-\Omega_{\mathrm{H}}dJ_{1})}{d\tau}\frac{8\pi}{\kappa}=\frac{-8\pi}{\kappa}(\Omega_{H}-\Omega)^{2}C_{V},$
(77)

where $\Omega_{\mathrm{H}}=\chi_{1}/[2m_{1}(1+\kappa_{1})]$, is the horizon
angular velocity of the primary BH and $A_{1}=8\pi m_{1}^{2}(1+\kappa)$, is
the horizon-surface area and

$\displaystyle C_{V}=$
$\displaystyle-\frac{16}{5}m_{1}^{2}X_{1}^{2}\eta^{2}(1+\kappa_{1})V^{12}\Big{\\{}1+3\chi_{1}^{2}+V^{2}\big{(}-3+X_{1}-\frac{51}{4}\chi_{1}^{2}+3X_{1}\chi_{1}^{2}\big{)}$
$\displaystyle+V^{3}\Big{[}\frac{-X_{1}\chi_{1}}{3}(64+60\kappa_{1}+33\chi_{1}^{2}+36\kappa_{1}\chi_{1}^{2})-\frac{3}{2}X_{2}(4+7\chi_{1}^{2})\chi_{2}-8X_{1}(1+\chi_{1}^{2})B_{2}(\chi_{1})\Big{]}\Big{\\}}.$
(78)

The contributions to the definition of mass and angular momentum due to the
conservative tidal coefficients as seen in Eqs. (IV), (68) were removed upon
averaging over an orbit. They will also not contribute in a scattering
scenario provided we can set the tidal fields to zero along the worldlines of
the particle asymptotically (in the distant past and future).

However, the results in Eqs. (75), (76), and (77) are written in terms of
gauge-dependent quantities, namely $r$ (which enters through $V$) and is the
separation between the two bodies in harmonic coordinates. Further more, it is
written in the frame of the primary BH as opposed to the more convenient PN
barycentric frame (which coincides with the primary BH frame in the test-body
limit for the secondary BH). Thus, before comparison, we convert the results
to the PN barycentric frame using the relations

$\displaystyle V$
$\displaystyle=x\Big{[}1+\frac{1}{6}(3-\eta)x^{2}+\frac{1}{6}\tilde{\chi}x^{3}+\mathcal{O}(x^{4})\Big{]},$
(79) $\displaystyle t$
$\displaystyle=\tau\Big{[}1+\frac{1}{2}(2X_{1}+3X_{2})X_{2}x^{2}+\mathcal{O}(x^{4})\Big{]},$
(80)

where $x=(M\omega_{\mathrm{orb}})^{1/3}$ with $\omega_{\mathrm{orb}}$ is the
orbital angular velocity, which is gauge invariant PN expansion parameter and
$\tilde{\chi}=(2X_{1}^{2}+3\eta)\chi_{1}+(3\eta+2X_{2}^{2})\chi_{2}$. $t$ is
the PN barycentric time and its relation to the proper time of the primary BH
$\tau$, is given in Eq. (80). These expressions have been taken from Eqs.(39)
and (40) in Ref. [22]. Then, the expressions in Eqs. (75), (76), and (77) can
be rewritten in the PN barycentric frame as an expansion in the gauge
invariant parameter $x$ as

$\displaystyle\Big{\langle}\frac{dm_{1}}{dt}\Big{\rangle}$
$\displaystyle=\Omega(\Omega_{\mathrm{H}}-\Omega)C_{x},$ (81)
$\displaystyle\Big{\langle}\frac{dJ_{1}}{dt}\Big{\rangle}$
$\displaystyle=(\Omega_{\mathrm{H}}-\Omega)C_{x},$ (82)
$\displaystyle\Big{\langle}\frac{dA}{dt}\Big{\rangle}$
$\displaystyle=\frac{(dm_{1}-\Omega_{\mathrm{H}}dJ_{1})}{dt}\frac{8\pi}{\kappa}=\frac{-8\pi}{\kappa}(\Omega_{H}-\Omega)^{2}C_{x},$
(83)

where

$\displaystyle C_{x}=$
$\displaystyle-\frac{16}{5}M^{2}X_{1}^{2}\eta^{2}(1+\kappa_{1})x^{12}\Big{(}1+3\chi_{1}^{2}+\frac{1}{4}[3(2+\chi_{1}^{2})+2X_{1}(1+3\chi_{1}^{2})(2+3X_{1})]x^{2}$
(84)
$\displaystyle+x^{3}\Big{\\{}\frac{1}{2}(-4+3\chi_{1}^{2})\chi_{2}-2X_{1}(1+3\chi_{1}^{2})(X_{1}(\chi_{1}+\chi_{2})+4B_{2}(\chi_{1}))+X_{1}\Big{[}-\frac{2}{3}(23+30\kappa_{1})\chi_{1}+(7-12\kappa_{1})\chi_{1}^{3}+4\chi_{2}$
$\displaystyle+\frac{9\chi_{1}^{2}\chi_{2}}{2}\Big{]}\Big{\\}}\Big{)},$

which can now be conveniently compared with the expressions for the same
quantities given in Eq. (45) of Ref. [22]. We find that our expression for
$C_{x}$ for generic-mass ratios is consistent with the result [22] to NLO (to
$x^{14}$) but not at NNLO (at $x^{15}$). An important visible difference is
that we have no $\pi^{2}$-containing terms at NNLO. However, as we will see in
the next subsection, it is consistent in the test body limit with earlier
results computed by solving the Teukolsky equation for the curvature
perturbation sourced by a test-body moving in a circular orbit around a
spinning BH up to relative 1.5PN order ($x^{15}$).

###  IV.2 The test body limit for circular orbits with parallel spins

In the special case where the other BH with mass $m_{2}$ becomes a test
particle, we only evaluate the quantities to leading order in $X_{2}$, mass
ratio of other particle. This is equivalent to simply setting $X_{1}=1$,
$X_{2}\rightarrow 0$ and thus $M=m_{1}\rightarrow\infty$ such that
$MX_{2}=m_{2}$ remains constant. Then the result simplifies to

$\displaystyle\frac{dm_{1}}{dt}=$
$\displaystyle\mathcal{F}_{\infty}\bigg{\\{}-\frac{x^{5}}{4}(\chi_{1}+3\chi_{1}^{3})-x^{7}(\chi_{1}+\frac{33}{16}\chi_{1}^{3})+\frac{x^{8}}{12}[6+70\chi_{1}^{2}-3\chi_{1}^{4}+6\kappa_{1}(1+13\chi_{1}^{2}+6\chi_{1}^{4})+24(\chi_{1}+3\chi_{1}^{3})B_{2}(\chi_{1})]\bigg{\\}},$
$\displaystyle\text{where
}\mathcal{F}_{\infty}=\frac{32}{5}\bigg{(}\frac{m_{2}}{M}\bigg{)}^{2}x^{10}\text{
and }\Omega\frac{dJ_{1}}{dt}=\frac{dm_{1}}{dt},$ (85)

which is consistent with the results obtained via BH perturbation theory for
the case of a tiny test particle orbiting a spinning BH in Refs. [21]. This
result has been produced by solving the Teukolsky equation for a perturbation
sourced by a tiny nonspinning BH ($M_{\mathrm{ext}}$) around a large spinning
BH $M$ and computing the energy flux down the horizon. This result has also
been reproduced in Ref. [56] and the method has been employed to push the
results for flux to infinity and horizon fluxes to a very high PN order in the
test-body limit (see Refs. [57, 56, 58]). It is thus reassuring that our
expression for mass-loss matches with this in the appropriate limits,
suggesting that our effective worldline picture is suitable for the purpose of
modelling horizon-related dissipation in spinning BHs.

##  V Effect of horizon fluxes on the waveform for circular orbits

A simple way to derive the contribution to the waveform phase from horizon
fluxes for parallel-spin–quasi-circular inspiral in the adiabatic limit is via
the stationary-phase approximation (SPA) [59, 60, 61]. This was used to
incorporate the effect of leading-order rate of change of mass (at 2.5PN) in
the waveform in the appendix of Ref. [28] and to 4 PN in Ref. [62], albeit the
horizon fluxes used in Ref. [62] (which were taken from Ref. [30]) are only
accurate to leading order and the contribution due to the changing mass in the
expression for binding energy was not taken into account [see Eq. (89)]. Here,
we use the same method (i.e., SPA) and extend the computation completely to
4PN order, while including all relevant effects due to the changing mass and
spin and compute the phase contribution to the waveform due to horizon fluxes.
We use $x$ as the gauge-invariant–PN-counting parameter to relative 1.5PN (and
absolute 4PN) order. We start from the relation

$\displaystyle
x=(M\omega_{\mathrm{orb}})^{\frac{1}{3}}=\bigg{(}M\frac{d\phi}{dt}\bigg{)}^{\frac{1}{3}},$
(86)

where $\phi$ is the orbital phase and $M=m_{1}+m_{2}$ is the total mass of the
system. The binding energy $E$ of the system is given to 1.5PN order101010It
is sufficient for us to include the expression for the binding energy to 1.5PN
order as we are only interested in computing the waveform phase contribution
due to horizon fluxes which were derived in this work to relative 1.5PN order.
(see, e.g., Ref. [28]) as

$\displaystyle E$ $\displaystyle=-\frac{M\eta
x^{2}}{2}\bigg{\\{}1+x^{2}\bigg{(}\frac{-3}{4}-\frac{\eta}{12}\bigg{)}$
$\displaystyle+x^{3}\bigg{[}\frac{8\delta\chi_{a}}{3}+\bigg{(}\frac{8}{3}-\frac{4\eta}{3}\bigg{)}\chi_{s}\bigg{]}\bigg{\\}},$
(87)

where $\delta=(m_{1}-m_{2})/M$, $\eta=m_{1}m_{2}/M^{2}$,
$\chi_{a}=(\chi_{1}-\chi_{2})/2$, $\chi_{s}=(\chi_{1}+\chi_{2})/2$. Now, we
use the energy balance law valid for circular orbits given by

$\displaystyle\dot{E}=-\mathcal{F}_{\infty}-\dot{M},$ (88)

where $\mathcal{F}_{\infty}$ is the energy flux to infinity and over-dot
represents derivative with respect to time. For noncircular orbits, one may
have to include additional Schott terms [63]. Now, we see from Eq. (V) that
$E$ is a function of $x$, the masses $m_{1}$, $m_{2}$ and the spins
$\chi_{1}$, $\chi_{2}$, and thus we can write

$\displaystyle\dot{E}$ $\displaystyle=\frac{\partial E}{\partial
x}\dot{x}+\frac{\partial E}{\partial m_{1}}\dot{m}_{1}+\frac{\partial
E}{\partial m_{2}}\dot{m}_{2}$ $\displaystyle+\frac{\partial
E}{\partial\chi_{1}}\dot{\chi}_{1}+\frac{\partial
E}{\partial\chi_{2}}\dot{\chi}_{2},$ (89)

which, along with the balance relation, yields

$\displaystyle\dot{x}$ $\displaystyle=-\bigg{(}\frac{\partial E}{\partial
x}\bigg{)}^{-1}(\mathcal{F}_{\infty}+\dot{M}+\dot{m}_{1}\partial_{m_{1}}E+\dot{m}_{2}\partial_{m_{2}}E$
$\displaystyle+\dot{\chi}_{1}\partial_{\chi_{1}}E+\dot{\chi}_{2}\partial_{\chi_{2}E}),$
(90)

where we can drop the terms arising from the spin-dependence as they do not
contribute until relative 2.5PN order in horizon fluxes. Similarly, we can
substitute the leading-order formula for $E$ in partial derivatives with
respect to $m_{1}$ and $m_{2}$ and only substitute the flux to infinity up to
1.5PN relative order to get

$\displaystyle\dot{x}=-\bigg{(}\frac{\partial E}{\partial
x}\bigg{)}^{-1}[\mathcal{F}_{\mathrm{\infty}}^{\mathrm{1.5PN}}+\dot{M}-\frac{x^{2}}{2M^{2}}(m_{2}^{2}\dot{m}_{1}+m_{1}^{2}\dot{m}_{2})].$

The flux to infinity up to 1.5PN order can be found in Ref. [28]. We can then
invert this expression and integrate to compute the time function $t(x)$ and
orbital phase $\phi(x)$ as

$\displaystyle\delta
t(x)=\int\frac{1}{\dot{x}}dx,\quad\int\frac{d\phi}{dt}dt=\int\frac{x^{3}}{M}\frac{dt}{dx}dx=\phi(x).$
(92)

Then, the orbital phase function $\phi(x)$ and the time function $t(x)$, can
be related to the waveform phase $\psi$, in Fourier domain for any spherical
mode $m$, via the relation [61]

$\displaystyle\psi_{lm}(f)=2\pi ft_{f}-m\phi(t_{f})-\frac{\pi}{4},$ (93)

where $f$ is the Fourier variable (frequency) and $t_{f}$ corresponds to the
time when the instantaneous GW frequency coincides with $f$, i.e.,

$\displaystyle\frac{dm\phi}{dt}(t_{f})=2\pi f\implies
x(t_{f})=v=\bigg{(}\frac{2\pi Mf}{m}\bigg{)}^{\frac{1}{3}}.$ (94)

$\psi_{lm}(f)$ is a useful quantity directly relevant for detectors and we
provide its correction due to the horizon fluxes explicitly as

$\displaystyle\delta\psi_{lm}(f)$ $\displaystyle=\frac{3}{128\eta
v^{5}}\frac{m}{2}\bigg{[}\sum_{n=5}^{8}\delta\psi_{n}^{\mathrm{(PN)}}v^{n}$
$\displaystyle+\sum_{n=5}^{8}\delta\psi_{n(l)}^{\mathrm{(PN)}}v^{n}log(v)\bigg{]},$
(95) $\displaystyle v$ $\displaystyle=\bigg{(}\frac{2\pi
Mf}{m}\bigg{)}^{\frac{1}{3}},$ (96)

where $\delta\psi_{lm}(f)$ is the correction to $\psi_{lm}(f)$ due to horizon
fluxes with coefficients $\delta\psi_{n}^{(PN)}$ and
$\delta\psi_{n(l)}^{(PN)}$ starting from 2.5PN ($n=5$) and up to 4PN ($n=8$)
given by

$\displaystyle\delta\psi_{5}^{(\mathrm{PN})}$
$\displaystyle=-\frac{10}{9}[(1-3\eta)\chi_{s}(1+9\chi_{a}^{2}+3\chi_{s}^{2})+\delta(1-\eta)\chi_{a}(1+3\chi_{a}^{2}+9\chi_{s}^{2})],$
(97) $\displaystyle\delta\psi_{5(l)}^{(PN)}$
$\displaystyle=3\delta\psi_{5}^{(\mathrm{PN})},$ (98)
$\displaystyle\delta\psi_{7}^{(\mathrm{PN})}$
$\displaystyle=\frac{5}{168}\\{\delta\chi_{a}[-1667-4371\chi_{a}^{2}-13113\chi_{s}^{2}+616\eta^{2}(1+3\chi_{a}^{2}+9\chi_{s}^{2})+5\eta(311+807\chi_{a}^{2}+2421\chi_{s}^{2})]$
$\displaystyle+\chi_{s}\left[840\eta^{2}\left(9\chi_{a}^{2}+3\chi_{s}^{2}+1\right)+\eta\left(38331\chi_{a}^{2}+12777\chi_{s}^{2}+4889\right)-13113\chi_{a}^{2}-4371\chi_{s}^{2}-1667\right]\\},$
(99) $\displaystyle\delta\psi_{7(l)}^{(\mathrm{PN})}$ $\displaystyle=0,$ (100)

and

$\displaystyle\delta\psi_{8}^{(\mathrm{PN})}$
$\displaystyle=\delta\psi_{8}^{(\mathrm{PN}),a}+\delta\psi_{8}^{(\mathrm{PN}),b}+\delta\psi_{8}^{(\mathrm{PN}),c}$
(101) $\displaystyle\delta\psi_{8(l)}^{(\mathrm{PN})}$
$\displaystyle=-3\delta\psi_{8}^{(\mathrm{PN})},$ (102)

with

$\displaystyle\delta\psi_{8}^{(\mathrm{PN}),a}$
$\displaystyle=-\frac{5}{27}\\{144\pi\delta(\eta-1)\chi_{a}^{3}+48\pi\delta(\eta-1)\chi_{a}+3\left(278\eta^{2}-370\eta+75\right)\chi_{a}^{4}+\left(-36\eta^{2}+213\eta-67\right)\chi_{a}^{2}$
$\displaystyle+\chi_{s}\left[-12\delta\left(\eta^{2}+190\eta-75\right)\chi_{a}^{3}+2\delta\left(10\eta^{2}+124\eta-67\right)\chi_{a}+432\pi(3\eta-1)\chi_{a}^{2}+48\pi(3\eta-1)\right]$
$\displaystyle+\chi_{s}^{2}\left[432\pi\delta(\eta-1)\chi_{a}+90\left(36\eta^{2}-62\eta+15\right)\chi_{a}^{2}-172\eta^{2}+303\eta-67\right]+3\left(82\eta^{2}-250\eta+75\right)\chi_{s}^{4}$
$\displaystyle+\chi_{s}^{3}\left[12\delta\left(21\eta^{2}-130\eta+75\right)\chi_{a}+144\pi(3\eta-1)\right]-12\left(2\eta^{2}-4\eta+1\right)\\},$
(103) $\displaystyle\delta\psi_{8}^{(\mathrm{PN}),b}$
$\displaystyle=-\frac{20}{9}\\{[\delta(2\eta-1)\kappa_{a}+(-1-2\eta^{2}+4\eta)\kappa_{s}][1+6\chi_{a}^{4}+13\chi_{s}^{2}+6\chi_{s}^{4}+\chi_{a}^{2}(13+36\chi_{s}^{2})]$
$\displaystyle-2[\kappa_{a}(1-4\eta+2\eta^{2})+\delta(1-2\eta)\kappa_{s}]\chi_{a}\chi_{s}[13+12(\chi_{a}^{2}+\chi_{s}^{2})]\\}$
(104) $\displaystyle\delta\psi_{8}^{(\mathrm{PN}),c}$
$\displaystyle=\frac{80}{9}\\{B_{2,s}[\left(2\eta^{2}-4\eta+1\right)\chi_{s}\left(9\chi_{a}^{2}+3\chi_{s}^{2}+1\right)-\delta(2\eta-1)\chi_{a}\left(3\chi_{a}^{2}+9\chi_{s}^{2}+1\right)]$
(105)
$\displaystyle+B_{2,a}[3\left(2\eta^{2}-4\eta+1\right)\chi_{a}^{3}+9\delta(1-2\eta)\chi_{a}^{2}\chi_{s}+\left(2\eta^{2}-4\eta+1\right)\chi_{a}\left(9\chi_{s}^{2}+1\right)-\delta(2\eta-1)\chi_{s}\left(3\chi_{s}^{2}+1\right)]\\}$

where we have defined $\kappa_{s}=(\kappa_{1}+\kappa_{2})/2$,
$\kappa_{a}=(\kappa_{1}-\kappa_{2})/2$, and
$B_{2,s}=[B_{2}(\chi_{1})+B_{2}(\chi_{2})]/2$,
$B_{2,a}=[B_{2}(\chi_{1})-B_{2}(\chi_{2})]/2$ for convenience. Note that the
2.5PN and 3.5PN corrections vanish for spinless case which is consistent with
the fact that horizon fluxes only start at 4PN for nonspinning BHs (and 2.5PN
for spinning case). We also find that the 4PN correction to the waveform phase
contains functions that are nonpolynomial in the spin parameters through
$\kappa=\sqrt{1-\chi^{2}}$ and
$B_{2}=\mathrm{Im}[\mathrm{PolyGamma}(0,3+i2\chi/\kappa)]$, as expected from
the expression for the horizon energy fluxes (or rate of change of masses) in
Eq. (81) at relative 1.5PN order. The Fourier phase solely due to the flux to
infinity (i.e., neglecting horizon fluxes) to 3.5PN can be found in Eq. (7)
and Appendix A in Ref. [64]. The correction to the Fourier phase
$\delta\psi_{lm}(f)$, can now be conveniently incorporated in waveform models
to include the effect of horizon fluxes to next-to-next-to-leading order (up
to 1.5PN relative, or 4PN absolute) during inspiral for quasi-circular
aligned-spin binaries.

|  $(10+10)M_{\odot}$, equal aligned spins $\chi_{1}=\chi_{2}=\chi$
---|---
0PN | 603.6
1PN | 59.4
1.5PN | $-51.4+32\chi$
2PN | $4.1-4.4\chi^{2}$
2.5PN | $-7.1+11.3\chi+\boxed{10^{-3}12.8(1+3\chi^{2})\chi}$
3PN | $2.2-6.5\chi-0.64\chi^{2}$
3.5PN | $-0.8+3.6\chi+1.3\chi^{2}-0.4\chi^{3}+\boxed{10^{-3}(8.4+22\chi^{2})\chi}$
4PN | $None$
Table 1:  Number of orbital cycles as the frequency ($f$) of the gravitational wave increases from $f=10\mathrm{Hz}$ to the frequency at the innermost stable circular orbit (ISCO) $f_{\mathrm{ISCO}}=1/(6^{3/2}M\pi)\mathrm{Hz}$ for equal masses (10 $M_{\odot}$) and equal aligned (to orbital angular momentum) spins. Recall that $\kappa=\sqrt{1-\chi^{2}}$. Here, $B_{2}=\mathrm{PolyGamma}(0,3+2i\chi/\kappa)$ lies between $0$ (at $\chi=0$) and $\pi/2$ (at $\chi=1$). The terms in boxes are contributions from horizon fluxes, and the remaining terms come from the flux to infinity, written here for comparison. | ($10$+$1.4$) $M_{\odot}$, (BH-NS), $\chi_{1}=\chi$, $\chi_{2}=0$
---|---
0PN | 3587.6
1PN | 213.5
1.5PN | $-181.5+126.2\chi$
2PN | $9.8-13.5\chi^{2}$
2.5PN | $-20.0+36.8\chi+\boxed{10^{-2}9.4(1+3\chi^{2})\chi}$
3PN | $2.3-18.6\chi-0.5\chi^{2}$
3.5PN | $-1.8+10.5\chi+3.1\chi^{2}-\chi^{3}+\boxed{10^{-2}(5.9+15.4\chi^{2})\chi}$
4PN | $None$
Table 2:  Number of orbital cycles as the frequency increases from
$f=10\mathrm{Hz}$ to the frequency at the innermost stable circular orbit
(ISCO) $f_{\mathrm{ISCO}}=1/(6^{3/2}M\pi)\mathrm{Hz}$ for a binary composed of
a non-spinning 1.4 $M_{\odot}$ neutron star (NS) and a 10 $M_{\odot}$ BH with
spin parameter $\chi$. Any contributions due to the internal structure of the
NS have not been included.

To get a qualitative idea of the relevance of horizon fluxes to the waveform,
we can look at the correction to the orbital phase $\phi(x)$ and compute how
many additional (or fewer) orbital cycles occur, as a result of the inclusion
of those effects, for some specific choices of the initial masses of the BHs.
We consider first the case of two initially equal-mass
$m_{1}=m_{2}=10M_{\odot}$ and equal-aligned-spin $\chi_{1}=\chi_{2}=\chi$ BHs,
and second the case for a binary consisting of a non spinning 1.4 $M_{\odot}$
neutron star and a 10 $M_{\odot}$ BH with spin parameter $\chi$. In the latter
case, only the BH’s horizon-flux contribution is considered and the neutron
star is treated as a structureless particle. We then compute the additional
(or fewer) number of cycles due to horizon flux(es) starting from the minimum
lower frequency of the bandwidth of LVK detectors, $\omega=\pi\times
10\mathrm{Hz}$111111Note that we are working in units where $G$=$c$=1. to that
of the innermost-stable circular orbit (ISCO)
$\omega=\omega_{\mathrm{ISCO}}=1/(6^{\frac{3}{2}}M)\mathrm{Hz}$ (in
Schwarzschild), which generally is a good approximation of the binary’s merger
frequency. We use the formula

$\displaystyle\mathcal{N}_{\mathrm{GW}}=\frac{\delta\phi[x=(M\omega_{\mathrm{ISCO}})^{\frac{1}{3}}]-\delta\phi[x=(M\pi\times
10\mathrm{Hz})^{\frac{1}{3}}]}{\pi},$

where the correction to the orbital phase function due to horizon fluxes is
obtained as shown in Eq. (92). We list the results obtained for the
aforementioned special cases in Tables 1 and 2. In Table 1, we also list the
contribution to the number of cycles due to the flux to infinity up to 3.5PN
(but with only nonspinning contributions to flux to infinity at 3PN and 3.5PN)
using the expressions for fluxes and binding energy from Ref. [28]. This is to
facilitate comparison and get a qualitative understanding of the relevance of
the horizon flux to the waveforms. Similar tables with flux-to-infinity
contributions can be found for example in Refs. [65, 66]. There are slight
numerical differences between Table 1 here and the tables in these works,
because the final result is very sensitive to the precision used for the mass
of the Sun and the gravitational constant. As clearly evident from the table,
the contribution to the number of cycles from horizon fluxes (boxed terms in
the table) is quite small when compared to the usual contributions from the
flux to infinity even at the same PN order, although it is better for larger
mass ratios. This is due to the (relatively) small numerical value of the
coefficients in the horizon fluxes when compared with analogous terms in the
fluxes to infinity when the masses are equal. This feature of small
contribution due to the horizon fluxes has already been pointed out in Ref.
[30] but it was obtained using an expression for the horizon fluxes that is
only correct at leading order. This fact (relative smallness of horizon flux
contributions) does not change much for other configurations of spins and
masses either, at least for the frequency band of LVK detectors. Although the
effect of the horizon flux on the waveform phase is small, they would need to
be included when building highly accurate waveform models for next generation
detectors on the ground and in space.

##  VI Conclusion

In this work, we set out to tackle the problem of including horizon-related
dissipation effects in spinning BHs in an effective worldline theory, which is
an important physical effect to include in precision GW predictions for future
detectors. For that purpose, we wrote down an effective action with additional
multipolar moment degrees of freedom which couple directly with tidal fields
in the action, and are tidally induced by them in accordance with the
symmetries of a Kerr BH, namely axissymmetry and parity invariance. We fixed
the remaining freedom in the ansatz relating the tidal fields to the multipole
moments by considering a scattering scenario wherein GWs were scattered off
the effective particle and the degree of absorption was compared with that
obtained from the full theory by solving the Teukolsky equation. A crucial
ingredient in being able to fix the complete dissipative part of the ansatz
through this method was to impose upon the effective theory the requirement
that the scattering be independent for different spheroidal modes of the Weyl
scalar $\psi_{4}$, which follows from the separability of the Teukolsky
equation in the full theory in spheroidal harmonics with spin weight -2.
Having fixed the relevant part (for horizon-related dissipation) of the ansatz
in this way, we used the model to compute the orbit averaged variation in mass
and spin due to horizon fluxes to relative 1.5PN order for a binary in
circular orbit with parallel spins. The mass and spin rate of change derived
using our effective model is consistent with the results obtained in the test-
body limit in Ref. [21] to relative 1.5PN order, and with Ref. [22] for
generic mass ratios up to relative 1PN order and at leading order with the
generic mass ratio results in Refs. [23, 30, 24, 26, 27, 25]. Importantly, we
have weighed in one side (specifically on the side of Ref. [21]) in the
previous discrepancy in the expression for evolution of mass in a binary in
the literature between Refs. [21] and [22]. While the source of the earlier
discrepancy is still unclear and remains to be settled, our approach suggests
that it may have something to do with including the effect of octupolar tidal
fields in the evolution of mass and spin.

Having consistently modelled the horizon-related dissipation and the
associated changes in mass, spin and area of the horizon in this manner, we
then proceeded to compute the contribution to the phasing of the waveform due
to the relative 1.5PN horizon fluxes, which is relevant to the waveform at 4PN
with respect to that of the leading-order quadrupolar flux to infinity. This
was done using the SPA valid in the adiabatic quasi-circular regime of
interest during inspiral. We found that a qualitative measure of the
contribution of the horizon fluxes, i.e., the number of cycles in the waveform
as the frequency evolves from $10\mathrm{Hz}$ to $f_{\mathrm{ISCO}}$ is very
small ( 2 to 3 orders of magnitude) compared to other contributions arising
from GW energy flux to infinity at the same PN orders for typical masses
observed by LIGO-Virgo-KAGRA detectors.

An interesting future direction will be to use the model to derive the
contribution to the waveform phasing without relying on the stationary-phase
approximation to get a result valid outside of the adiabatic regime. This can
be done for example by deriving the radiation-reaction forces due to the
tidally-induced moments obtained from first-principles instead of relying on
balance arguments. It is also of interest to consider how these results,
namely the variation in mass and spin and the contribution to waveform phasing
are affected in the presence of eccentricity or nonparallel spins. It may also
be of interest to study possible resummations for the evolution equations of
mass and spin and their contribution to the waveform phase along the lines of
Refs. [18, 19, 20], now aided by an expression valid at higher (relative 1.5)
PN orders for generic mass ratios. Finally, the approach used for modelling
the particle in this work may be extended to generic compact bodies wherein
the changes in mass and spin may occur due to tidal heating, e.g., in a
viscous fluid. Parametrizing the changes in mass, spin and the subsequent
contribution to waveform phasing for generic bodies could be very useful for
testing the predictions of general relativity, and more specifically in the
search for exotic compact bodies using next-generation GW detectors.

###### Acknowledgements.

We thank Gustav Jakobsen, Chris Kavanagh, Gustav Mogull, Raj Patil and Khun
Sang Phukon for useful discussions. We thank Chris Kavanagh in particular for
sharing with us the knowledge for generating solutions to the Teukolsky
equation.

##  Appendix A Scalar field scattering in effective worldline theory including
leading-order tail effects

In the main text, we mentioned that the leading-order tail effect, due to the
scattering of GWs off the particle’s gravitational field leads to a factor of
$(1+2\epsilon\pi)$ multiplying the leading-order degree of absorption. This is
seen clearly in the Teukolsky solution given in Eq. (55), but not in the one
derived using effective worldline theory in Eq. (50) since we only solved the
scattering problem in flat space. We motivated that this can be reproduced in
the effective theory as well by including the effect of leading-order
nonlinearities due to the gravitational field of the particle while solving
the wave equation but did not prove it. Here, we show this explicitly in the
case of a scalar field scattering off a spinless BH. In particular, we
consider the scattering of the monopole mode $l=m=0$ to
$\mathcal{O}(\epsilon^{4})$ and show that an identical factor of
$(1+2\epsilon\pi)$ multiplies the leading-order degree of absorption for this
mode when the leading-order tail effects are included.

In the effective theory, we model the spinless BH as a particle with mass $m$
with an inducible monopole moment $m_{\phi}(\tau)$ in the presence of an
external scalar field. We write an effective worldline action including a
tidal scalar monopole moment as

$\displaystyle S=-\int
d\tau(m-K_{\mathrm{Q}}m_{\phi}(\tau)\phi)-\frac{K_{\phi}}{2}\int
dtd^{3}\vec{x}\sqrt{-g}g^{\alpha\beta}\nabla_{\alpha}\phi\nabla_{\beta}\phi+\frac{1}{16\pi
G}\int d^{4}x\sqrt{-g}R,$ (107)

which is identical to the action used in Ref. [46] except we have restricted
to just including a monopole moment for simplicity.

Spherical symmetry ensures that the scalar monopole moment can only be induced
by a scalar monopole mode. Since we are only interested in dissipative
effects, we can write a general ansatz for the moment simply as

$\displaystyle
m_{\phi}(\tau)=GMK_{\phi}\sum_{n=0}^{\infty}(GM)^{n}\nu_{n}\frac{d^{2n+1}\phi}{d\tau^{2n+1}},$
(108)

The particle sources a static gravitational field given at linear order in $G$
in the rest frame as

$\displaystyle h^{00}$ $\displaystyle=-4\frac{GM}{r},~{}h^{0i}=h^{ij}=0,$
(109) $\displaystyle h^{\mu\nu}$
$\displaystyle=\sqrt{-g}g^{\mu\nu}-\eta^{\mu\nu},$ (110)

which will affect the behaviour of the scalar field through the Klein Gordon
equation. The scalar field obeys the Klein Gordon equation in curved space-
time, which to linear order in $G$ with the above metric perturbation is given
by

$\displaystyle\Box\phi$
$\displaystyle=-\ddot{\phi}+\nabla^{2}\phi=\frac{4GM}{r}\ddot{\phi}+\frac{K_{Q}}{K_{\phi}}m_{\phi}(\tau)\delta^{(3)}(\vec{r}),$
(111)
$\displaystyle\implies\omega^{2}\phi+\nabla^{2}\phi=-\frac{4\epsilon\omega}{r}\phi+\frac{K_{Q}}{K_{\phi}}m_{\phi}(\tau)\delta^{(3)}(\vec{r}),$
(112)

in the rest frame of the particle. Here we have dropped divergent terms
arising from the expansion of $(\sqrt{-g}-1)\times\delta^{(3)}(\vec{x})$ in
the Klein-Gordon equation. Such terms can also be shown to cancel amongst
themselves perturbatively but it is not relevant to our purpose. We have also
restricted our attention to a single frequency mode of the wave, i.e., we set
$\phi\sim\exp(-i\omega t)\psi(\vec{r})$. Now, we can expand the scalar field
as

$\displaystyle\phi$ $\displaystyle=\phi^{(0)}+\epsilon\phi^{(1)},$ (113)