# Stability of relativistic tidal response against small potential
modification

Takuya Katagiri  Astronomical Institute, Graduate School of Science, Tohoku
University, Aoba, Sendai 980-8578, Japan  Niels Bohr International Academy,
Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark  Hiroyuki
Nakano  Faculty of Law, Ryukoku University, Kyoto 612-8577, Japan  Kazuyuki
Omukai  Astronomical Institute, Graduate School of Science, Tohoku University,
Aoba, Sendai 980-8578, Japan

###### Abstract

The tidal response of compact objects in an inspiraling binary system is
measured by a set of tidal Love and dissipation numbers imprinted in the
gravitational waveforms. While a four-dimensional black hole in vacuum within
General Relativity has vanishing Love numbers, a black hole in alternative
theories of gravity can acquire non-vanishing Love numbers. The dissipation
numbers may quantify Planckian corrections at the horizon scale. These
properties will allow a test of classical theories of gravity in the strong-
field regime with gravitational-wave observation. Since black holes are not in
the exact vacuum environment in astrophysical situations, the following
question arises: can the environment affect the tidal response? In this paper,
we investigate the stability of the tidal response of a Schwarzschild black
hole for frequency-dependent tidal-field perturbations against a small
modification of the background. Our analysis relies on the scattering theory,
which overcomes difficulties in computing the relativistic tidal Love numbers.
The tidal Love and dissipation numbers can be understood from the property of
sufficiently low-frequency scattering waves. We show that the tidal Love
numbers are sensitive to the property of the modification. Therefore, we need
careful consideration of the environment around the black hole in assessing
the deviation of the underlying theory of gravity from General Relativity with
the Love numbers. The modification has less impact on the dissipation numbers,
indicating that quantifying the existence of the event horizon with them is
not spoiled. We also demonstrate that in a composite system, i.e., a compact
object with environmental effects, the Love and dissipation numbers are
approximately determined by the sum of the numbers of each component.

###### Contents

  1. I Introduction and summary
  2. II Tidal response of a Schwarzschild black hole
    1. II.1 Linear gravitational perturbation theory
    2. II.2 Definition: tidal Love numbers
    3. II.3 Definition: dissipation numbers
  3. III Tidal response of a Schwarzschild black hole in scattering theory
    1. III.1 Scattering waves around a Schwarzschild black hole
    2. III.2 Tidal Love and dissipation numbers from scattering waves: analytical results
      1. III.2.1 Near-region solution
      2. III.2.2 Far-region solution
      3. III.2.3 Matching of near- and far-region solutions and a tidal response from the response function
    3. III.3 Tidal Love and dissipation numbers from scattering waves: numerical results
    4. III.4 Advantage of computation of the tidal Love and dissipation numbers from the response function
  4. IV Stability of tidal Love and dissipation numbers
    1. IV.1 Modeling
    2. IV.2 Effect of a Gaussian bump on the tidal Love and dissipation numbers
    3. IV.3 Stability against combination of the potential deformation
  5. V Discussion
    1. V.1 Astrophysical implications
    2. V.2 Theoretical application: power-law correction to the effective potential
  6. A Tidal response of a Gaussian bump in the Minkowski spacetime
  7. B Potential deformation by a local matter distribution
  8. C Stability of the tidal response of a horizonless compact object
  9. D Useful formulas for special functions

##  I Introduction and summary

After the first detection of the binary black-hole merger GW150914
LIGOScientific:2016aoc ; LIGOScientific:2016vbw , the advanced Laser
Interferometer Gravitational-Wave Observatory (LIGO) LIGOScientific:2014pky
and the Virgo interferometer VIRGO:2014yos  have already detected around a
hundred coalescences of binary black holes, binary neutron stars, and binary
black hole-neutron star. This observational success has given new insights
into astrophysics Coulter:2017wya ; LIGOScientific:2017bnn ;
LIGOScientific:2018jsj , fundamental physics LIGOScientific:2016lio ;
LIGOScientific:2018dkp ; Berti:2018vdi ; Berti:2018cxi ; Barack:2018yly , and
cosmology LIGOScientific:2017adf ; Sakstein:2017xjx . The detection of
gravitational waves from the binary neutron-star merger GW170817
LIGOScientific:2017vwq , which is associated with the electromagnetic
counterpart, i.e., a short gamma-ray burst and a kilonova
LIGOScientific:2017zic ; Kasen:2017sxr , has opened up the field of multi-
messenger astronomy LIGOScientific:2017ync , and provided a stringent
constraint on the propagation speed of the graviton Baker:2017hug ;
LIGOScientific:2017zic . Future observations with improved detector
sensitivity and/or new facilities including KAGRA Somiya:2011np , may provide
the decisive answer to one of the most fundamental questions: to what extent
is General Relativity correct in the strong-gravity regime?

In binary coalescences, the tidal interaction deforms their bodies, thereby
modifying their orbital motion in the last stage of the inspiral phase. Tidal
deformability at the linear level is quantified by a set of the so-called
tidal Love numbers as the response of the object to an external static tidal
field Hinderer:2007mb ; Damour:2009vw ; Binnington:2009bb . The phase
evolution of gravitational waves from the inspiraling binary reflects the
underlying theory of gravity as well as the internal structure of the objects
through the tidal Love numbers. Measurement of them thus allows one to access
the strong-field gravity of black holes or the extremely dense inside
environment of neutron stars Flanagan:2007ix ; Vines:2011ud ;
LIGOScientific:2018cki .

Intriguingly, the Love numbers vanish exactly in theory for a black hole in
vacuum in four-dimensional General Relativity, i.e., Schwarzschild and Kerr
black holes Binnington:2009bb ; Hui:2020xxx ; LeTiec:2020bos ; LeTiec:2020spy
; Chia:2020yla ; Charalambous:2021mea . The measurement of the tidal Love
numbers of compact binaries can thus put a constraint on quantum corrections
of the event horizon Pani:2015tga ; Cardoso:2017cfl ; Maselli:2018fay ;
Addazi:2018uhd ; Cardoso:2019rvt ; Cardoso:2019nis ; Kim:2020dif ;
Narikawa:2021pak ; Nair:2022xfm  and work for testing theories of gravity in
the strong-field regime Cardoso:2017cfl ; Cardoso:2018ptl ;
Chakravarti:2018vlt ; Cardoso:2019vof ; DeLuca:2022tkm ; Brown:2022kbw . In
other words, a non-zero tidal Love number can be evidence of new physics. From
the theoretical point of view, various authors Porto:2016zng ; Penna:2018gfx ;
Charalambous:2021kcz ; Hui:2021vcv ; BenAchour:2022uqo ; Hui:2022vbh ;
Charalambous:2022rre ; Katagiri:2022vyz ; Berens:2022ebl ; Kehagias:2022ndy
have argued a connection between the vanishing of the Love numbers and a
hidden symmetric structure of the equation for linear gravitational
perturbations around a black hole.

At a practical level, the following concern still remains regarding the
usefulness of the tidal Love numbers. Although the vanishing of the Love
numbers of Schwarzschild and Kerr black holes is based on the exact vacuum
assumption, in realistic situations, tidal interactions are more or less
inevitably accompanied by the environmental effect such as the presence of
surrounding matter, giving rise to non-zero Love numbers Barausse:2014tra ;
Cardoso:2019upw ; Cardoso:2021wlq ; DeLuca:2021ite ; DeLuca:2022xlz . The
measured Love numbers would exhibit strong sensitivity to the property of
small environmental effects and its deviation from zero would be much larger
than the scale of the environment, indicating destabilization of the tidal
Love numbers. Such destabilization may interfere with the test of General
Relativity. It is thus important to theoretically assess the property of the
tidal Love number induced by the deviation from the vacuum environment around
a black hole within General Relativity in advance of future gravitational-wave
observation.111A similar problem has been discussed in the context of
quasinormal modes in Refs. Jaramillo:2020tuu ; Cheung:2021bol ; Berti:2022xfj
; Kyutoku:2022gbr  in the past few years and in Ref. Nollert:1996rf  a long
time ago.

In such a situation, a strategy within a theory/model-agnostic framework will
be useful because no one knows either the “true” theory of gravity or the
accurate model of surrounding matter fields around astrophysical black holes.
However, one immediately encounters a difficulty in evaluating the
relativistic tidal Love numbers, except for exactly solvable systems, because
it relies on the property of the hypergeometric function: the tidal Love
numbers are understood from the connection coefficient between two fundamental
solutions for a tidal field, which are constructed, respectively, around the
event horizon and around large distances, based on the analytic continuation
of the hypergeometric or associated functions Binnington:2009bb ; Hui:2020xxx
; LeTiec:2020bos ; LeTiec:2020spy ; Chia:2020yla ; Charalambous:2021mea  as
will be seen in Sec. III. Without an analytic solution capturing the physical
property of a tidal field near the event horizon, it is difficult to obtain
the corresponding coefficients in general, even in a system slightly modified
from the Schwarzschild or Kerr background. Furthermore, even if one tries to
construct local solutions analytically, for example, with the Frobenius
method, there is another problem stated in the following.

One here must address a more fundamental problem: a potential ambiguity in
computation of the relativistic tidal Love numbers Damour:2009vw ; Kol:2011vg
; Gralla:2017djj . To see this, let us consider the asymptotic behavior of the
quadrupolar mode $\Phi_{2}$ of the static gravitational perturbation to an
asymptotically flat spacetime at infinity, which schematically takes the form,

$\lim_{r\to\infty}\Phi_{2}\propto
r^{3}\left[1+\mathcal{O}\left(1/r\right)\right]+\frac{\kappa}{r^{2}}\left[1+\mathcal{O}\left(1/r\right)\right],$
(1)

where $\kappa$ is the ratio of the coefficient of the decaying term in $r$ to
that of the growing term, corresponding to the quadrupolar tidal Love number
Hinderer:2007mb ; Damour:2009vw ; Binnington:2009bb  (see the precise
definition in Sec. II). However, if one considers the subleading contribution
of the term of $r^{3}$, say,
$r^{3}[1+\cdots+\delta_{5}/r^{5}+\mathcal{O}(1/r^{6})]$, that ratio changes,
i.e., $\kappa\to\kappa+\delta_{5}$. That is, the coefficient of the decaying
term degenerates with that of the subleading term of the growing term.222The
Newtonian tidal Love numbers are unambiguously computed because solutions of
the Laplace equation have no such subleading terms poisson_will_2014 .  This
ambiguity leads to the discrepancy between the theoretical definition within
the linear gravitational perturbation theory and the observational one
appearing in gravitational-waveform models based on the post-Newtonian
approximation (see the details in Ref. Gralla:2017djj ).

An analytic continuation of the multipolar index from an integer to a generic
number allows us to avoid the aforementioned degeneracy Hui:2020xxx ;
LeTiec:2020bos ; LeTiec:2020spy ; Chia:2020yla ; Charalambous:2021mea ;
Creci:2021rkz ; however, for static perturbations, another problem, i.e., the
gauge-dependence of the tidal Love numbers Gralla:2017djj , still remains. On
the other hand, as pointed out in Ref. Creci:2021rkz , together with the
analytic continuation, the extraction of them from frequency-dependent
perturbations bypasses the gauge ambiguity because of the imposition of
boundary conditions that capture both the physical properties of the compact
object and the perturbation field (see also the details in Ref. Chia:2020yla
). Despite these efforts, there still remains, in general, yet another
subtlety of the definition: perturbation fields in generic systems do not
necessarily take the simple form of Eq. (1) at large distances, instead
include the logarithmic term Kol:2011vg ; Hui:2020xxx ; Charalambous:2021mea ;
DeLuca:2022tkm . Recently, to avoid these issues, several authors have
attempted to evaluate the linear response in terms of scattering amplitude
Creci:2021rkz ; Ivanov:2022qqt  and the worldline effective field theory
approach Ivanov:2022hlo .

In this paper, we study the property of the destabilization effect on the
tidal Love numbers, introduced by tiny deformation to the Schwarzschild
background. To overcome the difficulties in computing the relativistic tidal
Love numbers, our analysis relies on the scattering theory of linear
frequency-dependent gravitational fields around a Schwarzschild black hole. We
also discuss stability of another quantity characterizing frequency-dependent
tidal response, i.e., the dissipation number LeTiec:2020bos ; LeTiec:2020spy ;
Chia:2020yla ; Charalambous:2021mea ; Ivanov:2022hlo  (see the precise
definition in Sec. II), which quantifies the absorption of the external tidal
field into the event horizon and is imprinted in gravitational waveforms from
an inspiraling binary Tagoshi:1994sm ; Alvi:2001mx ; Poisson:2004cw ;
Porto:2007qi ; Chatziioannou:2012gq ; Maselli:2017cmm ; Cardoso:2019rvt .
Black holes have non-zero dissipation numbers, while the dissipation numbers
vanish for non-rotating horizonless objects. This property may allow one to
constrain Planckian corrections at the horizon scale Addazi:2018uhd ;
Cardoso:2019rvt .

In Sec. II, after reviewing the relativistic tidal Love numbers of a
Schwarzschild black hole for a static tidal field, we introduce the
dissipation numbers.

We first show in Sec. III that the tidal Love and dissipation numbers can be
extracted from a response function (47) that is defined from a low-frequency
scattering wave, giving another physical interpretation for them as a property
of the scattering wave Chia:2020yla ; Creci:2021rkz ; Ivanov:2022qqt . We
further derive the formulas for the Love and dissipation numbers, i.e., Eqs.
(50) and (51), in terms of the response function, which allows us to provide
further insights into them from the viewpoint of the scattering problem.
Following Refs. Chia:2020yla ; Creci:2021rkz ; Ivanov:2022qqt ; Ivanov:2022hlo
, our discussion overcomes the potential gauge-ambiguity in computation of the
relativistic tidal Love numbers by defining them in terms of scattering waves
under the boundary conditions that capture the physical property of a compact
object in the strong-field regime and of a scattering wave in the weak-field
regime. In addition, we use the analytic continuation of the multipole number
from an integer to a generic number, which resolves the aforementioned
degeneracy Creci:2021rkz .

In Sec. IV, we study the tidal response of the Schwarzschild black hole with a
slight potential modification in the shape of a Gaussian small bump (see Fig.
2). The tidal Love and dissipation numbers are calculated from the response
function numerically. We find that the bump leads to a non-zero Love number,
while it has less impact on the dissipation number. The deviation from the
vanishing Love numbers is much larger than the scale of the Gaussian bump and
is quite sensitive to the property of the bump, i.e., the location, height,
and width, while the dissipation number is not. This shows that the tidal Love
numbers are easily destabilized by a tiny deviation from the exact vacuum
environment, while the dissipation numbers are stable.

Why does a tiny modification to the Schwarzschild background destabilize the
tidal Love numbers, but not the dissipation numbers? This comes from the fact
that the tidal response consists of both those of the black hole and the
Gaussian bump. In a composite system, i.e., a Schwarzschild black hole with
environmental effects, the tidal Love and dissipation numbers are
approximately determined by the linear combination of the numbers of each
component.

As shown in Appendix A, a Gaussian bump in the Minkowski spacetime causes a
non-zero Love number and zero dissipation number. In fact, the former takes a
value close to that of the Schwarzschild black hole with the same Gaussian
bump and shares the qualitatively same feature, e.g., the dependence on the
location of the bump (see Figs. 9(a) and 9(b)). The latter has less impact on
the dissipation numbers of the black hole.

Furthermore, we analyze in Sec. IV the tidal response of a Schwarzschild black
hole with a combination of the two of Gaussian bump and/or dip (see Fig. 6).
It is demonstrated in Appendix B that such a combined modification can be
realized due to the presence of local matter. We show that the tidal Love and
dissipation numbers of the composite system are almost the same as the linear
combination of the numbers for each modification. It is also found that the
dissipation numbers are still stable.

In Sec. V, we discuss the astrophysical implication of our results. The
destabilization indicates that a test of theories of gravity with the tidal
Love numbers may require careful consideration of the environment in which
black holes are immersed. From another viewpoint, a non-zero Love number
allows us to catch a glimpse of the extreme property of matter fields around a
black hole through gravitational-wave observation. In yet another context, the
destabilization of the Love numbers may hinder constraining the matter
equation of state in neutron stars because the destabilization also occurs
even for horizonless compact objects such as a neutron star as seen in
Appendix C. For an inspiraling binary, the environmental effect varies with
time as the orbital separation decreases. Consequently, the tidal response
induced by the environment varies with time, while that arising from
modification in theories of gravity remains constant. Thus, the extraction of
the constant component from the time-varying tidal response will be an
important step in testing theories of gravity in the strong-field regime. We
also discuss theoretical application to “parametrized” formalism for the
effective potential as in Refs. Cardoso:2019mqo ; McManus:2019ulj ;
Volkel:2022aca .

In Appendix C, we discuss the case of horizonless compact objects that have a
reflective boundary at a slightly larger radius than the Schwarzschild
radius.333Horizonless compact objects have been discussed as one possibility
of signatures of a quantum correction in the strong-gravity regime, see Refs.
Cardoso:2016rao ; Cardoso:2016oxy ; Cardoso:2019apo ; Agullo:2020hxe ;
Sago:2021iku .  We find that a Gaussian bump destabilizes the tidal Love
numbers and still maintains the vanishing of the dissipation numbers. The
value of the tidal Love number of the total system is mostly determined by the
Gaussian bump. Thus, it may be challenging to distinguish horizonless compact
objects from black holes in the presence of environmental effects in terms of
the Love numbers. On the other hand, the dissipation numbers are stable, and
therefore tells their difference.

##  II Tidal response of a Schwarzschild black hole

In this section, we review linear gravitational perturbation theory of a
Schwarzschild black hole, and then introduce relativistic tidal Love and
dissipation numbers based on Refs. Hinderer:2007mb ; Damour:2009vw ;
Binnington:2009bb .

###  II.1 Linear gravitational perturbation theory

To discuss the tidal response of a Schwarzschild black hole to an external
tidal field, we review linear gravitational perturbation theory around the
black hole. In spherical polar coordinates $(t,r,\theta,\varphi)$, the metric
of the Schwarzschild black hole spacetime is given by

$g_{\mu\nu}^{(0)}dx^{\mu}dx^{\nu}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),$ (2)

where $M$ is the Arnowitt-Deser-Misner mass. Here, the coordinate range is
restricted to $-\infty<t<\infty$, $2M<r<\infty$, $0\leq\theta<\pi$, and
$0\leq\varphi<2\pi$. The linearly perturbed spacetime metric is given by

$g_{\mu\nu}=g_{\mu\nu}^{(0)}+h_{\mu\nu},$ (3)

where the perturbation field $h_{\mu\nu}$ satisfies
$|h_{\mu\nu}|\ll|g_{\mu\nu}^{(0)}|$.

Given the spherical symmetry of the background spacetime (2), each component
of $h_{\mu\nu}$ can be decomposed in terms of the tensor spherical harmonics.
The perturbation field can be separated into even- and odd-parity modes,
$h_{\mu\nu}^{\rm(even)}$ and $h_{\mu\nu}^{\rm(odd)}$, respectively, subject to
the parity transformation ${\bf
P}:(\theta,\varphi)\to(\pi-\theta,\varphi+\pi)$. There is no mixing between
the even- and odd-parity modes because the Schwarzschild metric (2) is
invariant under the parity transformation ${\bf P}$; one can therefore treat
each parity mode independently.

With appropriate gauge fixing, i.e., the so-called Regge-Wheeler gauge
Regge:1957td , the harmonic modes of $h_{\mu\nu}$ in the frequency domain can
be described by

$\left(h_{\ell
m}^{\rm(even)}\right)_{\mu\nu}=\begin{pmatrix}\left(1-\dfrac{2M}{r}\right)H_{0}^{\ell
m}Y^{\ell m}&H_{1}^{\ell m}Y^{\ell m}&0&0\\\ H_{1}^{\ell m}Y^{\ell
m}&\left(1-\dfrac{2M}{r}\right)^{-1}H_{2}^{\ell m}Y^{\ell m}&0&0\\\
0&0&r^{2}K^{\ell m}Y^{\ell m}&0\\\ 0&0&0&r^{2}\sin^{2}\theta K^{\ell m}Y^{\ell
m}\end{pmatrix},$ (4)

and

$\left(h_{\ell m}^{\rm(odd)}\right)_{\mu\nu}=\begin{pmatrix}0&0&h_{0}^{\ell
m}S_{\theta}^{\ell m}&h_{0}^{\ell m}S_{\varphi}^{\ell m}\\\ 0&0&h_{1}^{\ell
m}S_{\theta}^{\ell m}&h_{1}^{\ell m}S_{\varphi}^{\ell m}\\\ h_{0}^{\ell
m}S_{\theta}^{\ell m}&h_{1}^{\ell m}S_{\theta}^{\ell m}&0&0\\\ h_{0}^{\ell
m}S_{\varphi}^{\ell m}&h_{1}^{\ell m}S_{\varphi}^{\ell m}&0&0\end{pmatrix},$
(5)

where $Y^{\ell m}$ are the scalar spherical harmonics, and $(S_{\theta}^{\ell
m},S_{\varphi}^{\ell m}):=(-\partial_{\varphi}Y^{\ell
m}/\sin\theta,\sin\theta\partial_{\theta}Y^{\ell m})$. This form is valid only
for $\ell\geq 2$. Here, the Fourier transformation with respect to the time
variable has been performed as

${F}\left(r;\omega\right)=\int_{-\infty}^{\infty}dt\tilde{F}\left(t,r\right)e^{i\omega
t},$ (6)

where ${F}$ and $\tilde{F}$ correspond to $H_{0}^{\ell m},H_{1}^{\ell
m},H_{2}^{\ell m},K^{\ell m},h_{0}^{\ell m}$, and $h_{1}^{\ell m}$ in the
frequency domain and those in the time domain, respectively.

The vacuum linearized Einstein equation, $\delta R_{\mu\nu}=0$, leads to two
independent equations, respectively, for the even- and odd-parity modes, which
take the following unified form in the frequency domain Regge:1957td ;
Zerilli:1970wzz :

$\left(1-\frac{2M}{r}\right)\frac{d}{dr}\left[\left(1-\frac{2M}{r}\right)\frac{d}{dr}\Phi_{\ell
m}^{\pm}\left(r;\omega\right)\right]+\left(\omega^{2}-V_{\ell
m}^{\pm}\right)\Phi_{\ell m}^{\pm}\left(r;\omega\right)=0,$ (7)

with

$\begin{split}V_{\ell
m}^{+}:=&\left(1-\frac{2M}{r}\right)\left[\frac{2\lambda^{2}\left(1+\lambda\right)r^{3}+6\lambda^{2}Mr^{2}+18\lambda
M^{2}r+18M^{3}}{r^{3}\left(\lambda r+3M\right)^{2}}\right],\\\ V_{\ell
m}^{-}:=&\left(1-\frac{2M}{r}\right)\left[\frac{\ell\left(\ell+1\right)}{r^{2}}-\frac{6M}{r^{3}}\right],\end{split}$
(8)

where $\lambda:=(\ell-1)(\ell+2)/2$. Here, $\Phi_{\ell m}^{+}$ and $\Phi_{\ell
m}^{-}$ are gauge invariant variables for the even- and odd-parity modes,
respectively, and are defined by 1974AnPhy..88..323M

$\begin{split}\Phi_{\ell
m}^{+}:=&\frac{r(r-2M)}{\left(\lambda+1\right)\left(\lambda
r+3M\right)}\left(H_{2}^{\ell m}-r\frac{dK_{\ell m}}{dr}+\frac{\lambda
r+3M}{r-2M}K^{\ell m}\right),\\\ \Phi_{\ell
m}^{-}:=&\frac{r}{\lambda}\left[r^{2}\frac{d}{dr}\left(\frac{h_{0}^{\ell
m}}{r^{2}}\right)+i\omega h_{1}^{\ell m}\right].\end{split}$ (9)

Equations for $\Phi_{\ell m}^{+}$ and $\Phi_{\ell m}^{-}$ in Eq. (7) are
called the Zerilli/Regge-Wheeler equations, respectively Zerilli:1970wzz ;
Regge:1957td . One can reconstruct all the nonvanishing components of
$(h_{\ell m}^{\rm(even)})_{\mu\nu}$ and $(h_{\ell m}^{\rm(odd)})_{\mu\nu}$
from $\Phi_{\ell m}^{+}$ and $\Phi_{\ell m}^{-}$, respectively,

$\begin{split}H_{0}^{\ell m}=&\frac{1}{\lambda
r+3M}\left\\{\left[\frac{\lambda\left(\lambda
r^{2}+6M^{2}\right)}{r\left(\lambda
r+3M\right)}+\frac{3M^{2}}{r^{2}}+\lambda^{2}-\omega^{2}r^{2}\frac{\lambda
r+3M}{r-2M}\right]\Phi_{\ell m}^{+}\right.\\\
&\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left[\frac{M\left(\lambda
r+3M\right)}{r}-\lambda\left(r-2M\right)\right]\frac{d\Phi_{\ell
m}^{+}}{dr}\right\\},\\\ H_{1}^{\ell m}=&-i\omega\frac{\lambda
r\left(r-2M\right)-M\left(\lambda r+3M\right)}{\left(r-2M\right)\left(\lambda
r+3M\right)}\Phi_{\ell m}^{+}-i\omega r\frac{d\Phi_{\ell m}^{+}}{dr},\\\
H_{2}^{\ell m}=&H_{0}^{\ell m},\\\ K^{\ell
m}=&\frac{\lambda\left(\lambda+1\right)r^{2}+3\lambda
Mr+6M^{2}}{r^{2}\left(\lambda r+3M\right)}\Phi_{\ell
m}^{+}+\left(1-\frac{2M}{r}\right)\frac{d\Phi_{\ell m}^{+}}{dr},\end{split}$
(10)

and

$\begin{split}h_{0}^{\ell
m}=&\left(1-\frac{2M}{r}\right)\frac{d}{dr}\left(r\Phi_{\ell m}^{-}\right),\\\
h_{1}^{\ell m}=&-\frac{i\omega r^{2}}{r-2M}\Phi_{\ell m}^{-}.\end{split}$ (11)

###  II.2 Definition: tidal Love numbers

In this subsection, the tidal Love numbers are introduced within a
relativistic framework. Assuming that the tidal field is weak and slowly
varying in time, we can apply linear static gravitational perturbation theory.
To define the tidal Love numbers for the static perturbation, we first
introduce the notion of induced multipole moments and tidal moments of a
generic metric function following Refs. RevModPhys.52.299 ; Hinderer:2007mb ;
Cardoso:2017cfl .

In asymptotically Cartesian and mass centered coordinates
$(t,r,\theta,\varphi)$, induced multipole moments and tidal moments of any
static, spherically symmetric, and asymptotically flat spacetime can be
extracted from the asymptotic behavior of the metric components in the
asymptotically flat region:

$\begin{split}g_{tt}=&-1+\frac{2M}{r}-\sum_{\ell\geq
2}\left[\frac{2}{\ell\left(\ell-1\right)}r^{\ell}\left(\mathcal{E}_{\ell}Y^{\ell
0}+\left(\ell>\ell^{\prime}\right)\right)\right.\\\
&\qquad\qquad\qquad\qquad\left.-\frac{2}{r^{\ell+1}}\left(\sqrt{\frac{4\pi}{2\ell+1}}{\cal
M}_{\ell}Y^{\ell 0}+\left(\ell>\ell^{\prime}\right)\right)\right],\\\
g_{t\varphi}=&\sum_{\ell\geq
2}\left[\frac{2}{3\ell\left(\ell-1\right)}r^{\ell+1}\left(\mathcal{B}_{\ell}S_{\varphi}^{\ell
0}+\left(\ell>\ell^{\prime}\right)\right)+\frac{2}{r^{\ell}}\left(\sqrt{\frac{4\pi}{2\ell+1}}\frac{{\cal
S}_{\ell}}{\ell}S_{\varphi}^{\ell
0}+\left(\ell>\ell^{\prime}\right)\right)\right],\end{split}$ (12)

where we have defined $M$ as the Arnowitt-Deser-Misner mass of the central
gravitational source; ${\cal E}_{\ell}$ and ${\cal B}_{\ell}$ as even- and
odd-parity components of tidal moments, respectively, which correspond to the
amplitudes of even- and odd-parity modes of the external tidal field,
respectively; ${\cal M}_{\ell}$ and ${\cal S}_{\ell}$ as the induced mass
multipole moments and induced current multipole moments, respectively. The
notation of $\left(\ell>\ell^{\prime}\right)$ denotes the contribution of
$\ell^{\prime}$ ($<\ell$) poles.

We now define components of the tidal Love numbers for even- and odd-parity
modes, respectively, as Cardoso:2017cfl

$\begin{split}\kappa_{\ell}^{E}:=&-\frac{\ell\left(\ell-1\right)}{2r_{0}^{2\ell+1}}\sqrt{\frac{4\pi}{2\ell+1}}\frac{{\cal
M}_{\ell}}{{\cal E}_{\ell}},\\\
\kappa_{\ell}^{B}:=&-\frac{3\ell\left(\ell-1\right)}{2\left(\ell+1\right)r_{0}^{2\ell+1}}\sqrt{\frac{4\pi}{2\ell+1}}\frac{{\cal
S}_{\ell}}{{\cal B}_{\ell}},\end{split}$ (13)

where $r_{0}$ is the radius of the central gravitational source. These are
also called electric-type and magnetic-type tidal Love numbers, respectively.
The tidal Love numbers correspond to the dimensionless ratio of the
coefficient of the growing part in $r$ to that of the decaying part of the
asymptotic behavior of the metric components in Eq. (12).

For the Schwarzschild black hole, the tidal Love numbers can be read off from
$\Phi_{\ell m}^{\pm}$ Hui:2020xxx ,

$\begin{split}\left.\Phi_{\ell
m}^{\pm}\right|_{\omega=0,r\to\infty}&\sim\left(\frac{r}{2M}\right)^{\ell+1}\biggl{\\{}1+\mathcal{O}\left(2M/r\right)\\\
&\qquad\qquad\qquad\quad+2\frac{\left(\ell+2\right)\left(\ell+1\right)}{\ell\left(\ell-1\right)}\kappa_{\ell}^{E/B}\left(\frac{r}{2M}\right)^{-2\ell-1}\left[1+\mathcal{O}\left(2M/r\right)\right]\biggr{\\}}.\end{split}$
(14)

Note that $\Phi_{\ell m}^{\pm}$ are required to be regular at the black hole
horizon $r=2M$. The well-known intriguing result is that the horizon-regular
solutions of $\Phi_{\ell m}^{\pm}$ have no decaying series in $r$, i.e.,
$\Phi_{\ell m}^{\pm}|_{r\gg 2M}\propto[r/(2M)]^{\ell+1}[1+{\cal O}(2M/r)]$,
indicating the vanishing of the tidal Love numbers, i.e.,
$\kappa_{\ell}^{E}=\kappa_{\ell}^{B}=0$ Hinderer:2007mb ; Damour:2009vw ;
Binnington:2009bb .

###  II.3 Definition: dissipation numbers

We here also introduce the relativistic dissipation numbers. Now, an external
tidal field is assumed to be weak and has a low frequency $\omega\ll 1/(2M)$.
The dissipation numbers of a Schwarzschild black hole can then be read off
from $\Phi^{\pm}_{\ell m}$ at large distances as follows. Under the
requirement of no outgoing tidal field from the event horizon, the variables
$\Phi^{\pm}_{\ell m}$ with a low frequency $\omega\ll 1/(2M)$ at large
distances $2M\ll r\ll 1/\omega$ take the following forms (see Sec. III.2):

$\begin{split}\left.\Phi_{\ell m}^{\pm}\right|_{2M\ll r\ll
1/\omega}&\sim\left(\frac{r}{2M}\right)^{\ell+1}\biggl{\\{}1+\mathcal{O}\left(2M/r\right)\\\
&\qquad\qquad\qquad\quad+2\frac{\left(\ell+2\right)\left(\ell+1\right)}{\ell\left(\ell-1\right)}F_{\ell}^{E/B}\left(\omega\right)\left(\frac{r}{2M}\right)^{-2\ell-1}\left[1+\mathcal{O}\left(2M/r\right)\right]\biggr{\\}},\end{split}$
(15)

with the functions $F_{\ell}^{E/B}$ of $\omega$ defined by

$F_{\ell}^{E/B}=\kappa_{\ell}^{E/B}+\frac{i}{2}\nu_{\ell}^{E/B}(2\omega
M)+{\cal O}\left((2\omega M)^{2}\right).$ (16)

Here, $\kappa_{\ell}^{E/B}$ are the electric-type/magnetic-type tidal Love
numbers of the Schwarzschild black hole for each $\ell$-pole mode, and then
$\kappa_{\ell}^{E}=\kappa_{\ell}^{B}=0$ Hinderer:2007mb ; Damour:2009vw ;
Binnington:2009bb .

The quantities $\nu_{\ell}^{E/B}$ correspond to electric-type/magnetic-type
dissipation numbers, which quantify the dissipation of the tidal field into
the event horizon. Schwarzschild black holes have non-zero dissipation numbers
owing to the presence of the event horizon LeTiec:2020bos ; LeTiec:2020spy ;
Chia:2020yla ; Charalambous:2021mea ; Ivanov:2022hlo . This is also the case
for Kerr black holes Chia:2020yla ; Charalambous:2021mea ; Ivanov:2022hlo
.444Kerr black holes have non-zero dissipation numbers even for static
perturbations because of a relative motion with a static environment, which is
sourced by rotation Chia:2020yla ; Charalambous:2021mea ; Ivanov:2022hlo . On
the other hand, the dissipation numbers of a non-rotating horizonless object
vanish due to the absence of the event horizon. Those properties allow one to
quantify the existence of the event horizon in the context of testing
classical theories of gravity in the strong-field regime Pani:2015tga ;
Cardoso:2017cfl ; Maselli:2018fay ; Addazi:2018uhd ; Cardoso:2019rvt ;
Kim:2020dif ; Narikawa:2021pak .

##  III Tidal response of a Schwarzschild black hole in scattering theory

In astrophysical applications such as to binary systems, the gravitational
field generated by objects varies with time. A time-varying weak tidal field,
i.e., a gravitational wave, around a Schwarzschild black hole is described by
a solution of the Zerilli/Regge-Wheeler equations (7). In this section, we
discuss the frequency-dependent tidal response in terms of the scattering
theory of gravitational waves around a Schwarzschild black hole. In
particular, we show that the tidal Love and dissipation numbers of a
Schwarzschild black hole can be extracted from a response function that is
defined from the property of scattering waves. Although we here focus only on
the odd-parity mode, we expect to be able to extract tidal responses from
scattering waves even for the even-parity mode thanks to the existence of
isospectrality Chandrasekhar:1975zza . We have also checked that the results
in this section are valid for both scalar and vector-field perturbations.

###  III.1 Scattering waves around a Schwarzschild black hole

We here introduce basics of scattering gravitational waves around a
Schwarzschild black hole by focusing on the Regge-Wheeler equation in the
frequency domain,

$\left(1-\frac{r_{H}}{r}\right)\frac{d}{dr}\left[\left(1-\frac{r_{H}}{r}\right)\frac{d\Phi}{dr}\right]+\left\\{\omega^{2}-\left(1-\frac{r_{H}}{r}\right)\left[\frac{\ell(\ell+1)}{r^{2}}-\frac{3r_{H}}{r^{3}}\right]\right\\}\Phi=0,$
(17)

where $r_{H}:=2M$. Note that this is identical to one of Eq. (7). Now, let us
consider the situation where a monochromatic plane wave with a frequency
$\omega$ propagates along the $x$-axis toward a Schwarzschild black hole and
is scattered off by its effective potential.

Analogous to the problem in quantum mechanics, the scattering wave $\Phi$ at
large distances schematically takes the form of a superposition of the
incident plane wave along the $x$-axis and the outgoing spherical wave:555One
can set $m=0$ in the spherical harmonics $Y_{\ell m}$ without loss of
generality because of the spherical symmetry of the background.

$\lim_{r\to\infty}\sum_{\ell=2}^{\infty}\frac{\Phi}{r}Y_{\ell
0}\left(\theta\right)\sim e^{-i\omega x}+f(\theta)\frac{e^{i\omega
r_{*}}}{r_{*}},$ (18)

where $r_{*}$ is the tortoise coordinate defined as

$r_{*}:=r+r_{H}\ln\left(\frac{r}{r_{H}}-1\right),$ (19)

and $\theta$ is the angle measured with respect to the $x$-axis. Here, the
coefficient $f(\theta)$ is called a scattering amplitude, which can be
expanded in terms of the Legendre polynomial:

$f\left(\theta\right)=\sum_{\ell=2}^{\infty}\left(2\ell+1\right)\frac{e^{2i\delta_{\ell}\left(\omega\right)}-1}{2i\omega}P_{\ell}\left(\cos\theta\right).$
(20)

Here, $\delta_{\ell}$ is a complex-valued phase shift. By performing the
partial-wave decomposition of the incident plane wave, the asymptotic
expression for each $\ell$-pole mode of the scattering wave in Eq. (18) can be
written as a superposition of ingoing and outgoing spherical waves:

$\lim_{r\to\infty}\frac{\Phi}{r}\sim\frac{2\ell+1}{2i\omega
r_{*}}\left[e^{2i\delta_{\ell}}e^{i\omega
r_{*}}-\left(-1\right)^{\ell}e^{-i\omega r_{*}}\right].$ (21)

The quantity $e^{2i\delta_{\ell}}$ is called an S-matrix and corresponds to
the ratio of the amplitude of the outgoing spherical wave to that of the
ingoing spherical wave. Therefore, the S-matrix, and, in particular, the phase
shift $\delta_{\ell}$, have information on “strength” of the scattering.

###  III.2 Tidal Love and dissipation numbers from scattering waves:
analytical results

We here show analytically how tidal Love and dissipation numbers are imprinted
in scattering waves. For later convenience, by introducing new parameters,

$\bar{\omega}:=\omega r_{H},~{}~{}z:=\omega r,$ (22)

we rewrite the Regge-Wheeler equation (17) as

$\left(1-\frac{\bar{\omega}}{z}\right)\frac{d}{dz}\left[\left(1-\frac{\bar{\omega}}{z}\right)\frac{d\Phi}{dz}\right]+\left\\{1-\left(1-\frac{\bar{\omega}}{z}\right)\left[\frac{\ell(\ell+1)}{z^{2}}-\frac{3\bar{\omega}}{z^{3}}\right]\right\\}\Phi=0.$
(23)

In the following, we solve Eq. (23) with the matched asymptotic expansion
method Bender_Orszag_1999 . This strategy utilizes the analytical property of
the hypergeometric functions NIST:DLMF , in which an analytic continuation of
$\ell$ from an integer to generic numbers plays an important role. Henceforth,
we assume $\ell$ to be a generic number.

First, we assume that a tidal field has low frequencies, i.e.,
$\bar{\omega}\ll 1$. For the inspiral of a black-hole binary system, this
assumption can be justified except for a short pre-merger phase. For such a
low-frequency field, one can divide the exterior of the event horizon into two
regions, i.e., the near $(\bar{\omega}<z\ll 1)$ and far $(z\gg\bar{\omega})$
regions. We then solve Eq. (23) and obtain an approximate solution in each
region. Finally, by matching the solutions near and far regions in an
intermediate region $(\bar{\omega}\ll z\ll 1)$, we construct the global
analytic solution for the scattering wave approximately.

####  III.2.1 Near-region solution

We here derive an approximate solution of Eq. (23) in the near region
$(\bar{\omega}<z\ll 1)$. By introducing an auxiliary function,

$X_{\rm
N}\left(z\right):=\left(1-\frac{\bar{\omega}}{z}\right)^{-i\bar{\omega}}\left(\frac{\bar{\omega}}{z}\right)^{-\ell}\Phi,$
(24)

and a new coordinate variable,

$x:=1-\frac{\bar{\omega}}{z},$ (25)

the Regge-Wheeler equation (23) can be reduced to the following equation for
$X_{\rm N}$:

$x\left(1-x\right)\frac{d^{2}X_{\rm N}}{dx^{2}}+\left[\gamma_{\rm
N}-\left(\alpha_{\rm N}+\beta_{\rm N}+1\right)x\right]\frac{dX_{\rm
N}}{dx}-\left[\alpha_{\rm N}\beta_{\rm
N}+\frac{x^{2}-3x+3}{\left(x-1\right)^{3}}\bar{\omega}^{2}\right]X_{\rm N}=0,$
(26)

where

$\begin{split}\alpha_{\rm
N}:=&\ell+3+i\bar{\omega}+\mathcal{O}\left(\bar{\omega}^{2}\right),\\\
\beta_{\rm
N}:=&\ell-1+i\bar{\omega}+\mathcal{O}\left(\bar{\omega}^{2}\right),\\\
\gamma_{\rm N}:=&1+2i\bar{\omega}.\end{split}$ (27)

For the range of $z\ll\bar{\omega}^{1/3}$, in the last large bracket on the
left-hand side of Eq. (26), the second term is much smaller than the first
term $\alpha_{\rm N}\beta_{\rm N}$. By neglecting this term, the general
solution of Eq. (26) can be written in terms of the Gaussian hypergeometric
function around $x=0$ NIST:DLMF :

$X_{\rm N}=A_{\rm in,N}x^{-2i\bar{\omega}}~{}_{2}F_{1}\left(\alpha_{\rm
N}-\gamma_{\rm N}+1,\beta_{\rm N}-\gamma_{\rm N}+1;2-\gamma_{\rm
N};x\right)+A_{\rm out,N}~{}_{2}F_{1}\left(\alpha_{\rm N},\beta_{\rm
N};\gamma_{\rm N};x\right),$ (28)

where $A_{\rm in,N}$ and $A_{\rm out,N}$ are functions of $\bar{\omega}$.

Reconstructing the original variable $\Phi$ with Eq. (24), the general
solution of Eq. (23) in the near region $(\bar{\omega}<z\ll 1)$ is obtained as

$\begin{split}\left.\Phi_{\rm N}\right|_{\bar{\omega}<z\ll 1}=&A_{\rm
in,N}\left(1-\frac{\bar{\omega}}{z}\right)^{-i\bar{\omega}}\left(\frac{\bar{\omega}}{z}\right)^{\ell}~{}_{2}F_{1}\left(\alpha_{\rm
N}-\gamma_{\rm N}+1,\beta_{\rm N}-\gamma_{\rm N}+1;2-\gamma_{\rm
N};1-\bar{\omega}/z\right)\\\ &+A_{\rm
out,N}\left(1-\frac{\bar{\omega}}{z}\right)^{i\bar{\omega}}\left(\frac{\bar{\omega}}{z}\right)^{\ell}~{}_{2}F_{1}\left(\alpha_{\rm
N},\beta_{\rm N};\gamma_{\rm N};1-\bar{\omega}/z\right).\end{split}$ (29)

The first and second terms correspond to the ingoing and outgoing waves at the
horizon, respectively. We impose the ingoing-wave boundary condition $A_{\rm
out,N}=0$ at the horizon:

$\begin{split}\left.\Phi_{\rm N}\right|_{\bar{\omega}<z\ll 1}=&A_{\rm
in,N}\left(1-\frac{\bar{\omega}}{z}\right)^{-i\bar{\omega}}\left(\frac{\bar{\omega}}{z}\right)^{\ell}~{}_{2}F_{1}\left(\alpha_{\rm
N}-\gamma_{\rm N}+1,\beta_{\rm N}-\gamma_{\rm N}+1;2-\gamma_{\rm
N};1-\bar{\omega}/z\right).\end{split}$ (30)

Following the conventional manner Binnington:2009bb ; Cardoso:2017cfl ;
Hui:2020xxx ; Chia:2020yla ; Charalambous:2021mea , it can be seen that the
tidal Love numbers of the Schwarzschild black hole exactly vanish, while the
dissipation numbers are not zero. To see this, we investigate the asymptotic
behavior of $\Phi_{\rm N}$ in Eq. (30) at large distances $z\gg\bar{\omega}$.
For the coordinate domain of the near region $z\ll 1$, the near-region
solution (30) at large distances takes the form,

$\begin{split}\left.\Phi_{\rm N}\right|_{\bar{\omega}\ll z\ll 1}=&A_{\rm
in,N}\frac{\Gamma\left(2-\gamma_{\rm N}\right)\Gamma\left(\alpha_{\rm
N}+\beta_{\rm N}-\gamma_{\rm N}\right)}{\Gamma\left(\alpha_{\rm N}-\gamma_{\rm
N}+1\right)\Gamma\left(\beta_{\rm N}-\gamma_{\rm
N}+1\right)}\left(\frac{z}{\bar{\omega}}\right)^{\ell+1}\left[1+i\bar{\omega}\ln
z+\mathcal{O}(\bar{\omega}/z,\bar{\omega}^{2})\right]\\\
&\times\left\\{1+\mathcal{O}\left(\bar{\omega}/z\right)+\mathcal{K}_{\ell}\left(\bar{\omega}\right)\left(\frac{z}{\bar{\omega}}\right)^{-2\ell-1}\left[1+\mathcal{O}\left(\bar{\omega}/z\right)\right]\right\\},\end{split}$
(31)

where the coefficient of the decaying term is defined as

${\cal
K}_{\ell}\left(\bar{\omega}\right):=\frac{\Gamma\left(-2\ell-1\right)\Gamma\left(\ell+3-i\bar{\omega}\right)\Gamma\left(\ell-1-i\bar{\omega}\right)}{\Gamma\left(2\ell+1\right)\Gamma\left(-\ell+2-i\bar{\omega}\right)\Gamma\left(-\ell-2-i\bar{\omega}\right)}.$
(32)

Here, we have used the relations of the hypergeometric functions, Eqs. (72)
and (73).

Comparison of Eq. (31) with Eq. (15) tells that the real part of the function
${\cal K}_{\ell}$ is related to the ($\ell$th-pole) magnetic-type tidal Love
number in the static limit $\bar{\omega}\to 0$. In fact, one can analytically
show that the function ${\rm Re}[{\cal K}_{\ell}]$ indeed vanishes when
$\bar{\omega}\to 0$:

$\begin{split}\lim_{\bar{\omega}\to 0}{\rm Re}\left[{\cal
K}_{\ell}\right]&=\frac{\left(-1\right)^{\ell}\Gamma\left(\ell-1\right)^{2}\Gamma\left(\ell+3\right)}{2\Gamma\left(2\ell+1\right)\Gamma\left(2\ell+2\right)\Gamma\left(-\ell-2\right)}=0,\end{split}$
(33)

due to the presence of $1/\Gamma(-\ell-2)$ from the relation $1/\Gamma(-n)=0$
for $n=0,1,2,\cdots$. We have here used the relation of the gamma functions,
Eq. (76). The imaginary part, ${\rm Im}[{\cal K}_{\ell}]$, also vanishes in
the limit $\bar{\omega}\to 0$. Furthermore, comparing Eq. (31) with Eq. (15),
we can see that the ($\ell$th-pole) magnetic-type dissipation number is
related to the quantity ${\rm Im}[{\cal K}_{\ell}]/\bar{\omega}$ at
$\bar{\omega}\to 0$:

$\lim_{\bar{\omega}\to 0}\frac{{\rm Im}\left[{\cal
K}_{\ell}\right]}{\bar{\omega}}=\frac{\Gamma\left(\ell-1\right)^{2}\Gamma\left(\ell+3\right)^{2}}{\Gamma\left(2\ell+1\right)\Gamma\left(2\ell+2\right)},$
(34)

which is a positive value, for example, $0.20$ for the quadrupolar ($\ell=2$)
mode, giving $\nu_{2}^{B}=0.0333$.

####  III.2.2 Far-region solution

We here derive an approximate solution of Eq. (23) in the far region
$(z\gg\bar{\omega})$. Introducing an auxiliary function,

$Y_{\rm F}(z):=e^{-iz}\left(\frac{z}{\bar{\omega}}\right)^{-\ell-1}\Phi,$ (35)

and a coordinate variable,

$y:=-2iz,$ (36)

Eq. (23) leads to

$y\frac{d^{2}Y_{\rm F}}{dy^{2}}+(\beta_{\rm
F}-y)\left(1+\epsilon_{1}(z)\right)\frac{dY_{\rm F}}{dy}-\alpha_{\rm
F}\left(1+\epsilon_{2}(z)\right)Y_{\rm F}=0,$ (37)

with

$\displaystyle\alpha_{\rm F}:=$ $\displaystyle\ell+1-i\bar{\omega},$ (38)
$\displaystyle\beta_{\rm F}:=$ $\displaystyle 2\ell+2.$ (39)

Here, we have defined

$\begin{split}\epsilon_{1}(z):=&\frac{\bar{\omega}}{2\left(\ell+1+iz\right)\left(z-\bar{\omega}\right)},\\\
\epsilon_{2}(z):=&i\bar{\omega}\frac{z\left[\ell^{2}-4+\bar{\omega}\left(i+2\bar{\omega}\right)\right]-\left(\ell^{2}-4\right)\bar{\omega}-z^{2}\left(i+3\bar{\omega}\right)}{2\left(\ell+1-i\bar{\omega}\right)z\left(z-\bar{\omega}\right)^{2}}.\end{split}$
(40)

The absolute values of $\epsilon_{1}(z)$ and $\epsilon_{2}(z)$ are both much
smaller than unity in the far region ($z\gg\bar{\omega}$) for
$|\bar{\omega}(\ell^{2}-4)/(2(\ell+1))|\ll z^{2}$. By neglecting
$\epsilon_{1}(z)$ and $\epsilon_{2}(z)$, Eq. (37) takes the form of the
differential equation for the confluent hypergeometric functions. The general
solution of Eq. (37) can be well approximated by a linear combination of two
independent confluent hypergeometric functions around $z=0$ NIST:DLMF :

$Y_{\rm
F}=A_{+}M\left(\ell+1-i\bar{\omega},2\ell+2,-2iz\right)+A_{-}U\left(\ell+1-i\bar{\omega},2\ell+2,-2iz\right),$
(41)

where $M(~{},~{},-2iz)$ and $U(~{},~{},-2iz)$ are confluent hypergeometric
functions, which are Kummer’s and Tricomi’s functions, respectively; $A_{+}$
and $A_{-}$ are functions of $\bar{\omega}$.

Reconstructing the original variable $\Phi$, we obtain the general solution of
Eq. (23) in the far region $(z\gg\bar{\omega})$:

$\begin{split}\left.\Phi_{\rm
F}\right|_{z\gg\bar{\omega}}=&A_{+}\left(\frac{z}{\bar{\omega}}\right)^{\ell+1}e^{iz}M\left(\ell+1-i\bar{\omega},2\ell+2,-2iz\right)\\\
&+A_{-}\left(\frac{z}{\bar{\omega}}\right)^{\ell+1}e^{iz}U\left(\ell+1-i\bar{\omega},2\ell+2,-2iz\right).\end{split}$
(42)

Using the relations of the confluent hypergeometric functions, Eqs. (74) and
(75), the asymptotic behavior at infinity $z\to\infty$ takes the following
form:

$\begin{split}\lim_{z\to\infty}\left.\Phi_{\rm
F}\right|_{z\gg\bar{\omega}}=&A_{+}\frac{(-1)^{\ell+1}}{(-2i)^{\ell+1+i\bar{\omega}}\bar{\omega}^{\ell+1}}\frac{\Gamma\left(2\ell+2\right)}{\Gamma\left(\ell+1-i\bar{\omega}\right)}\\\
&\times\left\\{e^{2i\delta_{\ell}}e^{i\left(z+\bar{\omega}\ln\left(z/\bar{\omega}-1\right)\right)}\left[1+\mathcal{O}\left(1/z\right)\right]-\left(-1\right)^{\ell}e^{-i\left(z+\bar{\omega}\ln\left(z/\bar{\omega}-1\right)\right)}\left[1+\mathcal{O}\left(1/z\right)\right]\right\\},\end{split}$
(43)

where $\delta_{\ell}(\bar{\omega})$ is the complex-valued phase shift defined
by

$\begin{split}e^{2i\delta_{\ell}}=&\frac{\Gamma\left(\ell+1-i\bar{\omega}\right)}{\Gamma\left(\ell+1+i\bar{\omega}\right)}\left[1+2i\bar{\omega}\ln\left(2\bar{\omega}\right)+\mathcal{O}\left(\bar{\omega}^{2}\right)\right]\\\
&+(-1)^{\ell+1}\frac{\Gamma\left(\ell+1-i\bar{\omega}\right)}{\Gamma\left(2\ell+2\right)}\frac{A_{-}}{A_{+}}\left\\{1+\bar{\omega}\left[\pi+2i\ln\left(2\bar{\omega}\right)\right]+\mathcal{O}\left(\bar{\omega}^{2}\right)\right\\}.\end{split}$
(44)

Equation (43) implies that the leading behavior of the far-region solution
(42) at infinity consists of a superposition of outgoing and ingoing spherical
waves, i.e.,

$\lim_{z\to\infty}\left.\Phi_{\rm F}\right|_{z\gg\bar{\omega}}\propto
e^{2i\delta_{\ell}}e^{iz_{*}}-\left(-1\right)^{\ell}e^{-iz_{*}},$ (45)

where $z_{*}:=z+\bar{\omega}\ln(z/\bar{\omega}-1)$ is the asymptotic
expression for the tortoise coordinate (19) normalized by $\omega$.

To match the far-region and near-region solutions, we investigate the
asymptotic behavior of $\Phi_{\rm F}$ in Eq. (42) in the intermediate region
$(\bar{\omega}\ll z\ll 1)$, which is given by

$\begin{split}\left.\Phi_{\rm F}\right|_{\bar{\omega}\ll z\ll
1}=&A_{+}\left(\frac{z}{\bar{\omega}}\right)^{\ell+1}\left[1+\mathcal{O}\left(\bar{\omega}/z\right)\right]\left\\{1+\mathcal{O}\left(z\right)+{\cal
F}_{\ell}\left(\bar{\omega}\right)\left(\frac{z}{\bar{\omega}}\right)^{-2\ell-1}\left[1+\mathcal{O}(z)\right]\right\\}\end{split}.$
(46)

Here, a response function is introduced as Chakrabarti:2013xza ; Chia:2020yla
; Creci:2021rkz ,

${\cal
F}_{\ell}:=i\frac{\left(-1\right)^{\ell}}{2^{2\ell+1}\bar{\omega}^{2\ell+1}}\frac{\Gamma\left(2\ell+1\right)}{\Gamma\left(\ell+1-i\bar{\omega}\right)}\frac{A_{-}}{A_{+}}.$
(47)

In Eq. (44), writing $A_{-}/A_{+}$ in terms of ${\cal F}_{\ell}$, we obtain

$\begin{split}e^{2i\delta_{\ell}}=&\frac{\Gamma\left(\ell+1-i\bar{\omega}\right)}{\Gamma\left(\ell+1+i\bar{\omega}\right)}\left[1+2i\bar{\omega}\ln\left(2\bar{\omega}\right)+\mathcal{O}\left(\bar{\omega}^{2}\right)\right]\\\
&+i\frac{2^{2\ell+1}\bar{\omega}^{2\ell+1}\Gamma\left(\ell+1-i\bar{\omega}\right)^{2}}{\Gamma\left(2\ell+1\right)\Gamma\left(2\ell+2\right)}{\cal
F}_{\ell}\left\\{1+\bar{\omega}\left[\pi+2i\ln\left(2\bar{\omega}\right)\right]+\mathcal{O}\left(\bar{\omega}^{2}\right)\right\\}.\end{split}$
(48)

We have here used the relation of the gamma function, Eq. (77).

Equation (46) shows that the asymptotic behavior of the far-region solution
$\Phi_{\rm F}$ in the intermediate region $(\bar{\omega}\ll z\ll 1)$ consists
of a linear combination of the growing and decaying terms with increasing $z$.
The response function ${\cal F}_{\ell}$ corresponds to the ratio of their
coefficients. Comparison of Eq. (46) with Eq. (15) suggests that the response
function ${\cal F}_{\ell}$ captures the tidal response. In the following, we
see that this is indeed the case.

####  III.2.3 Matching of near- and far-region solutions and a tidal response
from the response function

The global solution is constructed by matching the near-region solution (30)
with the far-region solution (43). We then show that the function ${\cal
K}_{\ell}$ in Eq. (32) can be extracted from the response function ${\cal
F}_{\ell}$ in Eq. (47). We further derive the formulas for the Love and
dissipation numbers in terms of the response function.

From the asymptotic behaviors of the near-region (31) and far-region (46)
solutions, for successful matching of the two solutions it is required that
the response function ${\cal F}_{\ell}$ in Eq. (47) coincides with the
function ${\cal K}_{\ell}$ in Eq. (32) up to the overall factor, i.e.,

${\cal F}_{\ell}={\cal K}_{\ell}.$ (49)

This means that the response function ${\cal F}_{\ell}$ captures the magnetic-
type tidal Love and dissipation numbers through ${\cal K}_{\ell}$. We thus
obtain simple formulas for the tidal Love and dissipation numbers in terms of
${\cal F}_{\ell}$:

$\kappa_{\ell}^{B}=\frac{\ell\left(\ell-1\right)}{2\left(\ell+2\right)\left(\ell+1\right)}\lim_{\bar{\omega}\to
0}{\rm Re}\left[{\cal F}_{\ell}\right],$ (50)

and

$\nu_{\ell}^{B}=\frac{\ell\left(\ell-1\right)}{\left(\ell+2\right)\left(\ell+1\right)}\lim_{\bar{\omega}\to
0}\frac{{\rm Im}\left[{\cal F}_{\ell}\right]}{\bar{\omega}}.$ (51)

These formulas are useful for several extensions to systems that are slightly
modified from the Schwarzschild spacetime due to the presence of some matter
fields or modification of theories of gravity. If the modification is only in
the vicinity of a black hole, the far region is less affected. For this
reason, one can still use the same response function (47) and can safely
compute the Love and dissipation numbers from Eqs. (50) and (51) without the
ambiguities once we obtain the values of $A_{+}$ and $A_{-}$ in Eq. (42) by
numerical approach, e.g., by fitting the far-region solution (42) with the
numerical solution.666In Refs. Creci:2021rkz ; Ivanov:2022qqtX , the
computation of scattering-wave amplitudes of Schwarzschild spacetimes in terms
of scalar waves has been conducted, and has then been connected to gauge-
invariant quantities within an effective field theory framework, thereby
obtaining the Love and dissipation numbers in a gauge-invariant manner.

With the matching condition (49), Eq. (48) yields the analytic expression for
the phase shift in terms of ${\cal K}_{\ell}$:

$\begin{split}e^{2i\delta_{\ell}}=&\frac{\Gamma\left(\ell+1-i\bar{\omega}\right)}{\Gamma\left(\ell+1+i\bar{\omega}\right)}\left[1+2i\bar{\omega}\ln\left(2\bar{\omega}\right)+\mathcal{O}\left(\bar{\omega}^{2}\right)\right]\\\
&+i\frac{2^{2\ell+1}\bar{\omega}^{2\ell+1}\Gamma\left(\ell+1-i\bar{\omega}\right)^{2}}{\Gamma\left(2\ell+1\right)\Gamma\left(2\ell+2\right)}{\cal
K}_{\ell}\left\\{1+\bar{\omega}\left[\pi+2i\ln\left(2\bar{\omega}\right)\right]+\mathcal{O}\left(\bar{\omega}^{2}\right)\right\\},\end{split}$
(52)

This shows that tidal Love and dissipation numbers are imprinted in the phase
shift in principle. However, the Love and dissipation numbers are at least at
${\cal O}(\bar{\omega}^{2\ell+1})$ and ${\cal O}(\bar{\omega}^{2\ell+2})$,
respectively, indicating they degenerate with the subleading corrections in
the first line on the right-hand side in Eq. (52). This means that the
extraction of a tidal response from the phase shift is technically
challenging.

###  III.3 Tidal Love and dissipation numbers from scattering waves: numerical
results

We here numerically show that the matching condition (49) is indeed satisfied,
and then demonstrate that ${\rm Re}[{\cal F}_{\ell}]$ and ${\rm Im}[{\cal
F}_{\ell}]/(\omega r_{H})$ go to, respectively, zero and the value given in
Eq. (34) in the limit $\bar{\omega}\to 0$ by extrapolating them from a non-
zero frequency.

In the numerical analysis, we use Mathematica, and integrate the Regge-Wheeler
equation (17) from a radius slightly outside of the event horizon,
$r=2M(1+10^{-5})$, to large distances $r=220M$ under the ingoing-wave boundary
condition at the horizon. By fitting the far-region solution (42) with the
numerical solution in the region of $200M\leq r\leq 220M$, we read off the
coefficients $A_{+}$ and $A_{-}$, and determine the response function (47). We
have obtained the qualitatively same results for various upper limits of the
integration and the locations of the fitting region with various widths.

(a) Response function and the Love number

(b) Response function and the dissipation number

Figure 1: (Left) The functions ${\rm Re}[{\cal F}_{2}]/12$ (black solid line)
and ${\rm Re}[{\cal K}_{2}]/12$ (orange dashed line), which correspond to the
quadrupolar magnetic-type tidal Love number $\kappa_{2}^{B}$ in the static
limit $\omega\to 0$. (Right) the functions ${\rm Im}[{\cal F}_{2}]/(6\omega
r_{H})$ (black solid line) and ${\rm Im}[{\cal K}_{2}]/(6\omega r_{H})$
(orange dashed line), which correspond to the quadrupolar magnetic-type
dissipation number $\nu_{2}^{B}$ in the limit $\omega\to 0$. The value of
${\rm Im}[{\cal F}_{2}]/(6\omega r_{H})$ is almost constant $0.0333$ in a good
agreement with the analytic result from Eq. (34)

Figures 1(a) and 1(b) demonstrate the behavior of the functions ${\rm
Re}[{\cal F}_{2}]/12$ and ${\rm Im}[{\cal F}_{2}]/(6\omega r_{H})$, which
correspond, respectively, to the quadrupolar magnetic-type tidal Love number
$\kappa_{2}^{B}$ and dissipation number $\nu_{2}^{B}$ in the limit $\omega\to
0$. The overlapping of ${\rm Re}[{\cal F}_{2}]/12$ (black solid line) and
${\rm Re}[{\cal K}_{2}]/12$ (orange dashed line) means that the matching
condition (49) is indeed satisfied. In Fig. 1(a), we can see that the function
${\rm Re}[{\cal F}_{2}]/12$ approaches zero within the numerical error as
$\omega M$ becomes sufficiently small, showing the vanishing of the Love
number. This is consistent with the analytic consideration in the previous
section. In Fig. 1(b), ${\rm Im}[{\cal F}_{2}]/(6\omega r_{H})$ is almost
constant $0.0333$ in good agreement with the analytic result from Eq. (34).

###  III.4 Advantage of computation of the tidal Love and dissipation numbers
from the response function

The computation of a tidal response in terms of scattering waves allows one to
overcome the difficulties stated in Sec. I. The evaluation of the tidal Love
and dissipation numbers does not fully rely on the property of the
hypergeometric function. As already seen in Eq. (31), the conventional manner
relies on the analytic continuation between the fundamental solutions around
the event horizon and around large distances in the near region.

If one considers a tiny modification to the Schwarzschild background in the
strong-field regime as in the following section, it is in general difficult to
derive the exact solution in the near region in the form of a well-known
function. Therefore, it is hard to obtain the coefficient corresponding to
${\cal K}_{\ell}$ in Eq. (32). On the other hand, perturbation fields in the
far region are less affected by such modification near the horizon; therefore,
a far-region solution can be still described by Eq. (42). Here, the analytic
continuation of $\ell$ from an integer to a generic number makes the
distinction between the growing term in $r$ and the decaying one. To end,
computing a tidal response boils down to the problem to evaluate the static
limit of the response function ${\cal F}_{\ell}$ in Eq. (47) once we obtain
the values of $A_{+}$ and $A_{-}$ in the far-region solution (42) numerically,
and then obtain the Love and dissipation numbers through the formulas for
$\kappa_{\ell}^{B}$ and $\nu_{\ell}^{B}$ shown in Eqs. (50) and (51).

The gauge ambiguity is bypassed by computing the tidal Love and dissipation
numbers in terms of the response function. This is because the Love and
dissipation numbers as the property of sufficiently low-frequency scattering
waves are determined under the boundary conditions that capture the physical
property of a compact object in the strong-field regime and that of scattering
waves in the weak-field regime Chia:2020yla ; Creci:2021rkz ; Ivanov:2022qqt ;
Ivanov:2022hlo .

##  IV Stability of tidal Love and dissipation numbers

In this section, we discuss the effect of a small deformation of the effective
potential in the Regge-Wheeler equation on the tidal Love and dissipation
numbers by using the formulas (50) and (51). We then show that a small
deformation of the potential in the form of a Gaussian small bump gives rise
to non-zero values sensitive to the property of the bump and its deviation
from zero is much larger than the scale of the bump, exhibiting
destabilization of the Love numbers. The dissipation numbers, on the other
hand, are stable. We also analyze stability against a potential modification
that consists of combination of Gaussian bump and dip. In Appendix C, we
discuss stability of tidal Love and dissipation numbers in the case where one
imposes a reflective boundary condition at a radius slightly outside the
Schwarzschild radius as a proxy of horizonless compact objects.

###  IV.1 Modeling

Figure 2: The unperturbed potential $\delta V=0$ (black line) and deformed
potential by the Gaussian bump (54) with $h=10^{-3}/M^{2}$, $\sigma=M$, and
$a=20M$ (orange line). Here, $V_{\rm RW}$ is the Regge-Wheeler potential;
$V_{\rm RW}=(1-2M/r)[\ell(\ell+1)/r^{2}-6M/r^{3}]$. The inset shows the
enlargement around the bump.

Let us consider the Regge-Wheeler equation with its effective potential
slightly deformed:

$\left(1-\frac{2M}{r}\right)\frac{d}{dr}\left[\left(1-\frac{2M}{r}\right)\frac{d\Phi}{dr}\right]+\left\\{\omega^{2}-\left(1-\frac{2M}{r}\right)\left[\frac{\ell(\ell+1)}{r^{2}}-\frac{6M}{r^{3}}\right]-\delta
V\right\\}\Phi=0.$ (53)

As the potential deformation, we here consider a Gaussian small bump,

$\delta V=h\exp\left[-\dfrac{\left(r-a\right)^{2}}{2\sigma^{2}}\right],$ (54)

which is characterized by its height $h(\ll 1/M^{2})$, width $\sigma$, and
location of the peak $a$. The explicit form of the deformed potential, as an
example, is presented in Fig. 2 for the quadrupolar mode $\ell=2$.

A small bump can be realized by some spherically symmetric local matter
distribution Cheung:2021bol . Such a configuration may not be necessarily
realistic in astrophysical situations but can be expected to provide a certain
insight into the environmental effect on the tidal Love numbers by matter
surroundings including accretion disks Cardoso:2019upw .

###  IV.2 Effect of a Gaussian bump on the tidal Love and dissipation numbers

(a) Response function and the Love number

(b) Response function and the dissipation number

Figure 3: The functions ${\rm Re}[{\cal F}_{2}]/12$ (the left panel) and ${\rm
Im}[{\cal F}_{2}]/(6\omega r_{H})$ (the right panel) which correspond to,
respectively, the quadrupolar magnetic-type tidal Love number $\kappa_{2}^{B}$
and dissipation number $\nu_{2}^{B}$ in the limit $\omega\to 0$, showing
$\kappa_{2}^{B}\simeq-96$ and $\nu_{2}^{B}\simeq 0.0328$, in the presence of
the Gaussian bump with $h=10^{-3}/M^{2}$, $\sigma=M$, and $a=20M$ (orange
line). The black line corresponds to the unperturbed potential case ($\delta
V=0$) and is the same as those in Figs. 1(a) and 1(b).

(a) For various locations

(b) For various heights

Figure 4: The quadrupolar magnetic-type tidal Love numbers with the opposite
sign, i.e., $-\kappa_{2}^{B}$, for various locations $a$ and heights $h$ of
the Gaussian bump (54), (Left) for the locations from the radius of the
innermost stable circular orbit $a=6M$ to $a=50M$ with $h=10^{-3}/M^{2}$ and
$\sigma=M$, and (Right) for the heights from $h=10^{-6}/M^{2}$ to
$h=10^{-2}/M^{2}$ with $a=20M$ and $\sigma=M$.

We first discuss the behavior of the response function ${\cal F}_{\ell}$ in
Eq. (47). Figures 3(a) and 3(b) give the functions ${{\rm Re}[\cal F}_{2}]/12$
and ${\rm Im}[{\cal F}_{2}]/(6\omega r_{H})$. The values on the vertical axis
correspond to the quadrupolar magnetic-type tidal Love number $\kappa_{2}^{B}$
and dissipation number $\nu_{2}^{B}$, respectively. Figure 3(a) shows that the
tidal Love number takes a negative value due to the small modification of the
effective potential as $\kappa_{2}^{B}\simeq-96$. This implies that a tiny
deviation from the exact vacuum environment gives a nonzero Love number whose
deviation from zero is much larger than the scale of the small potential
modification. On the other hand, Fig. 3(b) shows $\nu_{2}^{B}\simeq 0.0328$
even in the presence of the bump, implying that the modification has less
impact on the imaginary part of the response function.

We next compute the Love and dissipation numbers from the formulas (50) and
(51) in the current deformed system. Figures 4(a) and 4(b) give the
quadrupolar magnetic-type tidal Love numbers with the opposite sign, i.e.,
$-\kappa_{2}^{B}$, for various values of the location $a$ and height $h$,
respectively, of the Gaussian bump (54). It is demonstrated that the tidal
Love number blows up nonlinearly/linearly with increasing location/height.
This implies that the Love numbers are sensitive to the properties of small
modifications to the effective potential, exhibiting destabilization of the
tidal Love numbers. This destabilization comes from the property of the Love
number of the Gaussian bump as seen in Appendix A. The nonlinear blow-up with
respect to the location is consistent with the analytic result for a thin
shell in Ref. Cardoso:2019upw . We comment that the absolute value of the Love
number, i.e., $|\kappa_{2}^{B}|$, grows as $\sigma$ increases, indicating that
a wider bump renders a larger tidal deformation.

(a) For various locations

(b) For various heights

Figure 5: The quadrupolar magnetic-type dissipation numbers $\nu_{2}^{B}$ for
various locations $a$ and heights $h$ of the Gaussian bump (54) (orange line)
and those of the Schwarzschild black hole without the bump, i.e.,
$\nu_{2}^{B}|_{\delta V=0}$ ($=0.0333$), (black line). The parameter set is
the same as that in Figs. 4(a) and 4(b).

Figures 5(a) and 5(b) show that the quadrupolar magnetic-type dissipation
number $\nu_{2}^{B}$ is smaller for a bump at a more distant location and with
a higher height. This implies that a more distant or higher bump obstructs
more the absorption effect of the black hole. The relative differences from
$\nu_{2}^{B}|_{\delta V=0}$ ($=0.0333$) are kept within $20\%$ in the current
parameter domain. The less sensitivity to the property of the modification
means that the dissipation numbers are stable.

Why does such destabilization occur for tidal Love numbers and does not for
dissipation numbers? The answer is as follows: a tidal response of a modified
system consists of that of the black hole and of the outer Gaussian bump. In
other words, the tidal Love and dissipation numbers in a composite system are
approximately determined by the linear combination of those of each component
of the system. A Gaussian bump in the Minkowski spacetime has non-zero Love
numbers and zero dissipation numbers (see Appendix A). In fact, the former
takes a value close to that of the Schwarzschild black hole with the same
Gaussian bump and shares the almost same dependence on the parameters of the
bump (see Figs. 9(a) and 9(b)). The latter has less impact on the dissipation
numbers of the black hole. The above result on the tidal Love number supports
the claim of Ref. Cardoso:2019upw .

We have checked that a negative $h$, i.e., a “dip”, leads to a positive tidal
Love number and a positive dissipation number, which is larger than
$\nu_{2}^{B}|_{\delta V=0}$ ($=0.0333$). They are more for a deeper Gaussian
dip. It is worth noting that the sign of the tidal Love number is governed
solely by that of $h$, which controls whether the Gaussian modification is
repulsive or attractive for scattering waves at sufficiently low frequencies.

The behavior of the dissipation number can be interpreted in terms of the
scattering. The nonzero dissipation number arises from the dissipation of
sufficiently low-frequency scattering waves. In the presence of a Gaussian
bump, the obstruction of absorption into the event horizon results in the
decrease of the dissipation number. A Gaussian dip, on the other hand, leads
to more attenuation of outgoing spherical waves, resulting in larger
dissipation numbers. Note that the dissipation number never changes its sign
while varying the height of the Gaussian bump because the modification plays a
role only in obstructing the absorption of waves into the black hole horizon.
The negative dissipation number means the existence of outgoing waves from the
horizon even though we imposed the ingoing-wave boundary condition,
contradicting physical intuition.

###  IV.3 Stability against combination of the potential deformation

Figure 6: The unperturbed potential $\delta V_{1}=\delta V_{2}=0$ (black line)
and deformed potential by the Gaussian bump and dip in Eq. (57) with
$h_{1}=-h_{2}=10^{-3}/M^{2}$, $\sigma_{1}=\sigma_{2}=M$, $a_{1}=20M$, and
$a_{2}=a_{1}+10M$ (orange line). Here, $V_{\rm RW}$ is the Regge-Wheeler
potential, $V_{\rm RW}=(1-2M/r)[\ell(\ell+1)/r^{2}-6M/r^{3}]$. The inset shows
the enlargement around the bump and dip.

We consider a system where a potential deformation comes from combination of
any two of Gaussian bump(s) and/or dip(s),

$\displaystyle\left(1-\frac{2M}{r}\right)\frac{d}{dr}\left[\left(1-\frac{2M}{r}\right)\frac{d\Phi}{dr}\right]$
(55)
$\displaystyle+\left\\{\omega^{2}-\left(1-\frac{2M}{r}\right)\left[\frac{\ell(\ell+1)}{r^{2}}-\frac{6M}{r^{3}}\right]-\delta
V_{1}-\delta V_{2}\right\\}\Phi=0,$ (56)

where

$\begin{split}\delta
V_{1}&=h_{1}\exp\left[-\dfrac{\left(r-a_{1}\right)^{2}}{2\sigma_{1}^{2}}\right],\\\
\delta
V_{2}&=h_{2}\exp\left[-\dfrac{\left(r-a_{2}\right)^{2}}{2\sigma_{2}^{2}}\right].\end{split}$
(57)

Here, we have assumed $|h_{1}|\ll 1/M^{2}$, $|h_{2}|\ll 1/M^{2}$, and that the
second modification $\delta V_{2}$ is located at a larger radius than the
first one $\delta V_{1}$, i.e., $a_{2}\geq a_{1}$. The explicit form of the
deformed potential is depicted in Fig. 6 for the quadrupolar mode $\ell=2$. An
example of such a bumpy deformation around the Regge-Wheeler potential, which
is caused by a local matter distribution, is presented in Appendix B (see Fig.
11).

(a) For various locations

(b) For various depths

Figure 7: The quadrupolar magnetic-type tidal Love numbers with the opposite
sign, i.e., $-\kappa_{2}^{B}$, for various locations $a_{2}$ and depths
$h_{2}$ of the Gaussian dip $\delta V_{2}$ in Eq. (57) with a fixed Gaussian
bump $\delta V_{1}$ (orange solid line) and the values of
$-(\kappa_{2}^{B}|_{\delta V_{2}=0}+\kappa_{2}^{B}|_{\delta V_{1}=0})$ (blue
dashed line), (Left) for the locations from $a_{2}=a_{1}(=20M)$ to
$a_{2}=a_{1}+10M$ with $h_{2}=-h_{1}=-10^{-3}/M^{2}$ and
$\sigma_{2}=\sigma_{1}=M$, and (Right) for the depth from
$-h_{2}=10^{-6}/M^{2}$ to $-h_{2}=10^{-2}/M^{2}$ with $a_{2}=a_{1}+5M$ and
$\sigma_{2}=\sigma_{1}=M$. The parameter set of $\delta V_{1}$ is the same as
those in Fig. 3(a).

(a) For various locations

(b) For various depths

Figure 8: The quadrupolar magnetic-type dissipation numbers $\nu_{2}^{B}$ for
various locations $a_{2}$ and depths $h_{2}$ of the Gaussian dip $\delta
V_{2}$ in Eq. (57) with a fixed Gaussian bump $\delta V_{1}$ (orange solid
line) and the values of $\nu_{2}^{B}|_{\delta V_{2}=0}+\nu_{2}^{B}|_{\delta
V_{1}=0}-\nu_{2}^{B}|_{\delta V_{1}=\delta V_{2}=0}$ (blue dashed line). The
last subtraction eliminates the overlapping of the contribution from the
purely black hole case, $\nu_{2}^{B}|_{\delta V_{1}=\delta V_{2}=0}=0.0333$,
in the combination of the first two components. The horizontal black solid
line corresponds to $\nu_{2}^{B}|_{\delta V_{1}=\delta V_{2}=0}=0.0333$. The
parameter set of $\delta V_{1}$ and $\delta V_{2}$ is, respectively, the same
as those in Fig. 3(a) and those in Figs. 7(a) and 7(b).

To be specific, we consider combination of a Gaussian bump $\delta V_{1}$ with
$h_{1}>0$ and dip $\delta V_{2}$ with $h_{2}<0$ in the following. For other
cases where the inner and outer modifications are the Gaussian dip and bump
respectively, or both are bumps or dips, the following results remain
qualitatively the same.

Figures 7(a) and 7(b), respectively, show the quadrupolar magnetic-type tidal
Love number with the opposite sign, i.e., $-\kappa_{2}^{B}$, as a function of
the location and depth of the Gaussian dip $\delta V_{2}$ with the Gaussian
bump $\delta V_{1}$ being fixed with the same parameter set as in Figs. 3(a)
and 3(b). Notice that the value in the presence of both $\delta V_{1}$ and
$\delta V_{2}$ (orange solid line) is almost the same as the linear
combination of those in case where each of the Gaussian bump $\delta V_{1}$ or
the Gaussian dip $\delta V_{2}$ is present, i.e., $-(\kappa_{2}^{B}|_{\delta
V_{2}=0}+\kappa_{2}^{B}|_{\delta V_{1}=0})$ (blue dashed line). This implies
that the tidal Love number in a composite system is mostly determined by the
linear combination of the Love number of each component in the system. It is
worth mentioning that in Fig. 7(b), the tidal Love number can vanish for a
specific modification.

Figures 8(a) and 8(b) present the quadrupolar magnetic-type dissipation number
$\nu_{2}^{B}$ as a function of the location $a_{2}$ and depth $h_{2}$ of the
Gaussian dip $\delta V_{2}$, respectively. The value of $\nu_{2}^{B}$ (orange
solid line) is almost the same as $\nu_{2}^{B}|_{\delta
V_{2}=0}+\nu_{2}^{B}|_{\delta V_{1}=0}-\nu_{2}^{B}|_{\delta V_{1}=\delta
V_{2}=0}$ (blue dashed line), where the last subtraction eliminates the
overlap of the contribution from the purely black hole case,
$\nu_{2}^{B}|_{\delta V_{1}=\delta V_{2}=0}=0.0333$, in the combination of the
first two components. This implies that the dissipation number is
approximately determined by the linear combination of the number of each
component in the system. Since its deviation from $\nu_{2}^{B}|_{\delta
V_{1}=\delta V_{2}=0}$ is small, we conclude that the dissipation numbers are
still stable even in the presence of another modification.

##  V Discussion

We discuss the astrophysical implication and theoretical application of the
results shown in the previous sections.

###  V.1 Astrophysical implications

In the previous sections, it is shown that a Schwarzschild black hole acquires
non-zero tidal Love numbers due to the presence of a Gaussian small
modification and their values are sensitive to the property of the
deformation. This means that, even if non-zero Love number is measured in
future gravitational-wave observations, we cannot immediately conclude
deviation of the underlying theory of gravity from General Relativity without
careful consideration on environmental effects. From another viewpoint, a non-
zero Love number allows us to catch a glimpse of the extreme property of
matter fields around a black hole through gravitational-wave observations. In
yet another context, the destabilization of the Love numbers may hinder
constraining the matter equation of state in neutron stars because the
destabilization occurs even for horizonless compact objects including a
neutron star as seen in Appendix C.

The dissipation numbers are stable for small modifications. Therefore, in the
context of a test of quantum corrections in the strong-field regime
Addazi:2018uhd ; Cardoso:2019rvt , quantifying the existence of the event
horizon is not spoiled even with deformation of the potential due to the
presence of a matter field.

In an inspiraling binary, the environmental effect causing a potential
deformation varies with time as the orbital separation decreases. Therefore,
the tidal response induced by the environment is not constant (see e.g., Ref.
DeLuca:2022xlz ). On the other hand, the tidal Love numbers arising from the
modification in theories of gravity remains constant. The tidal response
measured with gravitational-wave observations will be approximately determined
by the linear combination of the time-varying part and the constant part.
Thus, the extraction of the constant component from the time-varying tidal
response will be an important step in testing theories of gravity in the
strong-field regime.

###  V.2 Theoretical application: power-law correction to the effective
potential

Toward testing theories of gravity in the strong-gravity regime within linear
perturbation theory, “parametrized” formalism is expected to be useful
Cardoso:2019mqo ; McManus:2019ulj ; Volkel:2022aca :

$\left(1-\frac{r_{H}}{r}\right)\frac{d}{dr}\left[\left(1-\frac{r_{H}}{r}\right)\frac{d\Phi}{dr}\right]+\left\\{\omega^{2}-\left(1-\frac{r_{H}}{r}\right)\left[\frac{\ell(\ell+1)}{r^{2}}-\frac{3r_{H}}{r^{3}}\right]-\delta
V\right\\}\Phi=0,$ (58)

with

$\delta
V=\frac{1}{r_{H}^{2}}\left(1-\frac{r_{H}}{r}\right)\sum_{j=0}^{\infty}\alpha_{j}\left(\frac{r_{H}}{r}\right)^{j},$
(59)

where $|\alpha_{j}|\ll(1+1/j)^{j}(j+1)$ Cardoso:2019mqo . Choosing the
coefficient $\alpha_{j}$ appropriately, one can reconstruct the effective
potential for linear odd-parity gravitational perturbations around a static
and spherically symmetric black hole in a specific theory (see Ref.
Cardoso:2019mqo ). As another application, the parametrized formalism may also
be of use for modeling the deviation from a Schwarzschild background due to a
continuous matter distribution Barausse:2014tra .

The analysis in terms of the Gaussian bump or dip in Sec. IV gives various
suggestions for the properties of tidal Love and dissipation numbers in a
deformed system (58). First, the dissipation numbers take values close to
those in the purely Schwarzschild case at least for small $|\alpha_{j}|$.
Second, the Love and dissipation numbers for multiple corrections can be
interpreted as the linear combination of those of single power-law corrections
and those of the Schwarzschild black hole approximately. Third, the Love
number can vanish even with the corrections. Fourth, different parameter sets
of $\alpha_{j}$ and $j$ can give an identical Love number. For the Gaussian
bump, this degeneracy can be seen in Figs. 4(a) and 4(b). Finally, for
multiple corrections, a lower-order contribution of $j$ dominates over a
higher-order one in determining the Love numbers if the absolute value of the
coefficient of the lower-order one is larger than or comparable with those of
the higher-order one.

###### Acknowledgements.

The authors wish to express their cordial gratitude to Prof. Takahiro Tanaka,
the Leader of Innovative Area Grants-in-Aid for Scientific Research
“Gravitational wave physics and astronomy: Genesis”, for his continuous
interest and encouragement. The authors would also like to thank Vitor
Cardoso, Kazumi Kashiyama, and Shijun Yoshida for useful comments. The authors
acknowledge Chams Gharib Ali Barura, Shinji Mukohyama, Kazufumi Takahashi, and
Vicharit Yingcharoenrat for a lot of critical comments and fruitful
discussions on the analysis in Sec. III. This research is supported by Grants-
in-Aid for Scientific Research (TK and KO: 17H06360, HN: 17H06358, 21H01082,
21K03582, 23K03432, KO: 17H01102, 17H02869, 22H00149) from the Japan Society
for the Promotion of Science. TK thanks for support by VILLUM FONDEN (grant
no. 37766), by the Danish Research Foundation, and under the European Union’s
H2020 ERC Advanced Grant “Black holes: gravitational engines of discovery”
grant agreement no. Gravitas–101052587. KO acknowledges support from the
Amaldi Research Center funded by the MIUR program “Dipartimento di Eccellenza”
(CUP:B81I18001170001).

##  Appendix A Tidal response of a Gaussian bump in the Minkowski spacetime

(a) For various locations

(b) For various heights

Figure 9: The quadrupolar magnetic-type tidal Love numbers with the opposite
sign, i.e., $-\kappa_{2}^{B}$, for various locations $a$ and heights $h$ of
the Gaussian bump (54). The black and orange lines correspond to the values of
the Schwarzschild black hole with the Gaussian bump and those of the Gaussian
bump in the Minkowski spacetime, respectively. Here, $L$ is a length scale of
the Gaussian bump in the Minkowski spacetime. (Left) for the locations from
$a=20M=20L$ to $a=30M=30L$ with $h=10^{-3}/M^{2}=10^{-3}/L^{2}$ and
$\sigma=M=L$. (Right) for the heights from $h=10^{-6}/M^{2}=10^{-6}/L^{2}$ to
$h=10^{-2}/M^{2}=10^{-2}/L^{2}$ with $a=20M=20L$ and $\sigma=M=L$.

We discuss a tidal response of a Gaussian bump in the Minkowski spacetime. The
bump (54) is introduced in Eq. (53) with $M=0$:

$\frac{d^{2}\Phi}{dr^{2}}+\left[\omega^{2}-\frac{\ell(\ell+1)}{r^{2}}+\delta
V\right]\Phi=0.$ (60)

If $\delta V=0$, we have an analytic solution regular at the origin $r=0$,
i.e.,

$\Phi=r^{1/2}J_{\ell+1/2}\left(\omega r\right),$ (61)

where $J_{\ell+1/2}$ is the Bessel function of the first kind. We obtain the
response function from a numerical solution which is obtained by integrating
Eq. (60) from the origin to large distances under the boundary condition (61)
at $r=0$.

Figures 9(a) and 9(b) give the quadrupolar magnetic-type tidal Love numbers
for various location with the opposite sign, i.e., $-\kappa_{2}^{B}$, and
heights of the Gaussian bump, demonstrating that those of the bump in the
Minkowski spacetime and those in the Schwarzschild spacetime have close values
and share qualitatively the almost same behavior for the property of the
potential deformation. We have checked that the relative difference of them is
at most $10\%$. This implies that the tidal Love numbers of a Schwarzschild
black hole with a Gaussian bump are mostly determined by the property of the
bump.

It is found that the dissipation number $\nu_{2}^{B}$ of the Gaussian bump in
the Minkowski spacetime is a quite smaller value than unity, meaning its
vanishing. This implies that the Gaussian bump has less impact on the
dissipation numbers of a Schwarzschild black hole.

##  Appendix B Potential deformation by a local matter distribution

We here construct a static and spherically symmetric black hole solution with
an anisotropic matter field, and then derive an effective potential for an
odd-parity linear gravitational perturbation. In spherical polar coordinates
$(t,r,\theta,\varphi)$, a line element of a static and spherically symmetric
spacetime is given by

$ds^{2}=-A(r)dt^{2}+\left(1-\frac{2m(r)}{r}\right)^{-1}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right),$ (62)

where $A$ and $m$ are functions of the areal radius $r$. The function $m$ is
the Misner-Sharp mass PhysRev.136.B571 ; Hayward:1994bu . We introduce a
stationary system consisting of many gravitating masses which are assumed to
be anisotropic and have only tangential pressure in the angular directions
without the radial pressure. The energy-momentum tensor is given by

$T_{\mu}{}^{\nu}={\rm diag}[-\rho(r),0,P_{t}(r),P_{t}(r)].$ (63)

The Bianchi identity gives the relation between the local energy density
$\rho$ and the pressure $P_{t}$:

$P_{t}=\frac{\rho r}{4A}\frac{dA}{dr}.$ (64)

In addition, the $(r,r)$-component of the Einstein equation leads to a
constraint:

$\frac{r}{2}\frac{dA}{dr}=\frac{m}{r-2m}.$ (65)

Figure 10: The local energy density $\rho$ in Eq. (66) with $\epsilon_{m}=M$
and $a_{m}=20M$.

Now, we assume that the local energy density is given by

$\rho=\frac{2\epsilon_{m}}{Mr^{2}}\cosh^{-2}\left(\frac{r-a_{m}}{M}\right),$
(66)

where $\epsilon_{m}$ and $a_{m}$ determine the peak value of the energy
density and a location of the peak, respectively; $M$ is a length scale to be
interpreted as a mass of a black hole below. The distribution is depicted in
Fig. 10. From the $(t,t)$-component of the Einstein equation, the function $m$
in Eq. (62) takes the form,

$m=M+\epsilon_{m}\left[1+\tanh\left(\frac{r-a_{m}}{M}\right)\right].$ (67)

Equations (64) and (65) then determine the pressure $P_{t}$ and the metric
component $A$.

Figure 11: For the quadrupolar mode $\ell=2$, the Regge-Wheeler potential,
$V_{\rm RW}=(1-2M/r)[\ell(\ell+1)/r^{2}-6M/r^{3}]$ (black line) and the
potential $V_{m}$ in Eq. (70) (orange line) whose parameter set is the same as
that in Fig. 10. The inset shows the enlargement around the peak of the energy
density.

A linear gravitational perturbation to the geometry constructed above is
written as

$g_{\mu\nu}=g_{\mu\nu}^{(0)}+h_{\mu\nu},~{}~{}T_{\mu\nu}=T_{\mu\nu}^{(0)}+\delta
T_{\mu\nu},$ (68)

where the superscript $(0)$ denotes a background tensor field. As reviewed in
Sec. II, a symmetric tensor-field perturbation can be decomposed into two
independent components, i.e., the even- and odd-parity perturbations, because
of the parity invariance of the background. The linearized Einstein equation,
$\delta G_{\mu\nu}=8\pi\delta T_{\mu\nu}$, leads to a radial equation with the
harmonic decomposition, thereby obtaining the equation for the odd-parity
perturbation $\Phi$:

$\frac{d^{2}\Phi}{dR^{2}}+\left(\omega^{2}-V_{m}\right)\Phi=0,$ (69)

with

$V_{m}=A\left[\frac{\ell\left(\ell+1\right)}{r^{2}}-\frac{6m}{r^{3}}+\frac{1}{r^{2}}\frac{dm}{dr}\right],$
(70)

where $dR/dr=[A(1-2m/r)]^{1/2}$. The explicit form of $V_{m}$ is presented in
Fig. 11.

##  Appendix C Stability of the tidal response of a horizonless compact object

Figure 12: The dependence of the quadrupolar magnetic-type tidal Love number
with the opposite sign, i.e., $-\kappa_{2}^{B}$, on the radius at which the
reflective boundary condition (71) is imposed. The orange solid and blue
dashed lines correspond to the results without the Gaussian bump. The former
is the numerical result, while the latter is a model function,
$k_{2}^{B}=-0.0177/(1.0375+\ln\xi)$, whose logarithmic dependence on $\xi$ is
consistent with the analytical result in Ref. Cardoso:2017cfl . The black
solid line is the result with the Gaussian bump (54) whose parameter set is
the same as those in Fig. 3(a). The black solid line is almost constant,
$\kappa_{2}^{B}\simeq-96$, which is the almost same as the Love number of the
Schwarzschild black hole with the same Gaussian bump (see the orange line in
Fig. 3(a)).

We discuss stability of tidal Love and dissipation numbers when imposing a
reflective boundary condition, i.e., the Dirichlet boundary condition,

$\frac{A_{{\rm out},N}}{A_{{\rm in},N}}=-\xi^{-2i\omega r_{H}},$ (71)

on Eq. (29), at a radius $r_{0}:=r_{H}(1+\xi)~{}(0<\xi\ll 1)$. Figure 12 shows
that in the absence of the Gaussian bump (orange solid and blue dashed lines),
the Dirichlet boundary condition leads to a non-zero tidal Love number whose
absolute value increases for a larger boundary radius. The orange solid line
and the blue dashed line are, respectively, the numerical result and a model
function, $k_{2}^{B}=-0.0177/(1.0375+\ln\xi)$. The results of the logarithmic
dependence on $\xi$ and the negative Love numbers are consistent with the
analytical result in Ref. Cardoso:2017cfl . We have checked that the
dissipation numbers are quite smaller values than unity with or without the
Gaussian small bump.

In the presence of the Gaussian bump (black solid line), the quadrupolar
magnetic-type tidal Love number $\kappa_{2}^{B}$ takes approximately $-96$.
This value is less sensitive to the boundary radii and close to the tidal Love
number of a Schwarzschild black hole with the same Gaussian bump (see the
orange line in Fig. 3(a)), i.e., $\kappa_{2}^{B}\simeq-96$. This implies that
the property of the Gaussian bump mostly determines the value of the Love
numbers in the current system. On the other hand, the dissipation numbers may
tell the difference between black holes and horizonless objects.

##  Appendix D Useful formulas for special functions

The useful formulas used in the main text are summarized based on Ref.
NIST:DLMF  in the following. For the Gaussian hypergeometric functions,

$\begin{split}~{}_{2}F_{1}\left(a,b;c;s\right)=&\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}\left(1-s\right)^{-a-b+c}\\\
&\times\biggl{[}~{}_{2}F_{1}\left(-a+c,-b+c;-a-b+c+1;1-s\right)\\\
&~{}~{}~{}~{}+\left(1-s\right)^{a+b-c}\frac{\Gamma\left(-a-b+c\right)\Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b-c\right)\Gamma\left(-b+c\right)\Gamma\left(-a+c\right)}\\\
&~{}~{}~{}~{}~{}~{}\times{}_{2}F_{1}\left(a,b;a+b-c+1;1-s\right)\biggr{]},\end{split}$
(72)

and

$\lim_{s\to\infty}~{}_{2}F_{1}\left(~{},~{};~{};1-s\right)=1+\mathcal{O}(1/|s|).$
(73)

For the confluent hypergeometric functions,

$\lim_{s\to\infty}M\left(a,b,s\right)=\frac{\Gamma(b)}{\Gamma(a)}e^{s}s^{a-b}\left[1+\mathcal{O}(1/|s|)\right]+\frac{\Gamma(b)}{\Gamma(b-a)}(-s)^{-a}\left[1+\mathcal{O}(1/|s|)\right],$
(74)

and

$\lim_{s\to\infty}U\left(a,b,s\right)=s^{-a}\left[1+\mathcal{O}(1/|s|)\right].$
(75)

For the gamma function,

$\frac{\Gamma\left(-2\ell-1\right)}{\Gamma\left(-\ell+2\right)}=\left(-1\right)^{\ell+1}\frac{\Gamma\left(\ell-1\right)}{2\Gamma\left(2\ell+2\right)},$
(76)

and

$\frac{\Gamma\left(s\right)}{\Gamma\left(2s\right)}=\frac{\pi^{1/2}2^{-2s+1}}{\Gamma\left(s+1/2\right)}.$
(77)

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