# Tidal-heating/viscous dissipation correspondence in black holes and viscous
compact objects

Yotam Sherf
Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel
<EMAIL_ADDRESS>
###### Abstract

The effect of energy absorption during the binary evolution of Exotic-compact-
objects (ECOs) is extensively studied. We review the underlying mechanism that
provides the energy dissipation in material objects - tidal friction. We show
that unlike typical astrophysical objects, where absorption due to viscosity
is negligible, in ECOs, absorption could potentially mimic the analogous
effect of black-holes (BHs) - tidal heating. We stand for their differences
and similarities in the context of energy dissipation during the inspiral.
Inspired by the membrane paradigm and recent studies, we demonstrate how
viscosity is a defining feature that quantifies how close is the ECO
absorption to that of a classical BH absorption. We show that for ECOs,
viscosity can induce significant modifications to the GW waveform, which in
some favorable scenarios of super-massive binaries of equal mass and spin,
enables the measurement of the ECO absorption in the future precision
gravitational-wave (GW) observations. Finally, we discuss the implications on
the ECO reflection coefficient and the relation to the universal viscosity to
volume entropy bound.

##  I Introduction

Inspired by current observations from LIGO TheLIGOScientific:2016src ;
TheLIGOScientific:2017qsa ; Abbott:2020uma  and future Laser Interferometer
Space Antenna (LISA) LISA ; AmaroSeoane:2012je , the era of gravitational-wave
(GW) spectroscopy entails an outstanding opportunity for strong gravity tests
with precision GW measurement and to examine the consistency of Einstein’s
General-Relativity (GR) with GW observations. One of the most tantalizing GR
predictions is the existence of BHs and their event horizon, a surface
characterized by the discontinuity of the causal structure and possess the
unique property of being a perfect absorber. Thus, any deviations aside from
the ultimate absorption support the possibility of the existence of ECOs and
the emergence of new physics Cardoso:2016rao ; Cardoso:2017cqb .

In GR, the emergence of new physics is expected to appear in the strong-field
regime, when approaching the UV cutoff. At this regime, modifications to GR
arise naturally, and additional quantum scales such as the string scale enters
Boulware:1985wk ; Zwiebach:1985uq . Furthermore, the existence of these
quantum motivated dark compact object is necessary for a consistent
description of evaporating BHs as well as other open questions
Braunstein:2009my ; Brustein:2013ena . Among the quantum inspired ECOs
alternatives are known BH polymers Brustein:2016msz ; Brustein:2017koc ,
fuzzballs Skenderis:2008qn , Firewalls Almheiri:2012rt ; Mathur:2009hf ,
gravastars Mazur:2001fv in addition to other quantum effects that were
recently studied Giddings:2017mym ; Cardoso:2019apo ; Wang:2019rcf ;
Brustein:2019twi ; Brustein:2020tpg ; Agullo:2020hxe , (see also
Cardoso:2019rvt for an extensive review). Moreover, recent GW observations do
not exclude the existence of dark compact objects LIGOScientific:2019fpa ;
Abbott:2020khf . Therefore, the corrections demonstrated by these beyond GR
models, especially those that modify the BH absorption, should be taken into
account when performing an ECO search.

During the inspiral evolution of BHs, and in particular at its early stages,
the GW emission is governed by the point mass orbital motion. At later stages,
the mutual tidal interaction of the companions becomes significant, this gives
rise to a dissipation of energy and angular momentum, a phenomenon known as
tidal heating Hawking:1972hy ; Hartle:1973zz ; Thorne:1984mz ; Poisson:1994yf
; Poisson:2004cw ; Alvi:2001mx . The effect of tidal heating in BHs can be
illustrated from a Newtonian dynamic perspective Thorne:1986iy ;
Poisson:2009di ; Cardoso:2012zn . Where one of the companions is an ECO,
characterized by an average kinematic viscosity $\nu_{{}_{ECO}}$, then
dissipation of energy and angular momentum are attributed to tidal work and
tidal torque exerted by the external tidal field on the body. The tidal
heating - Newtonian dynamics correspondence is best emphasized from a
thermodynamic picture Poisson:2009di . Where for a fluid star, energy is
dissipated into heat according to $\dot{Q}=\dot{W}-\Omega_{{}_{ECO}}\dot{J}$,
which means that heat generation is attributed to the tidal work ($\dot{W}$)
and to the angular momentum flow $(\Omega_{{}_{ECO}}\dot{J})$ where
$\Omega_{{}_{ECO}}$ is the body’s angular velocity. In the analogous BH
thermodynamic description we have
$(\kappa/8\pi)\dot{A}=\dot{M}-\Omega_{{}_{BH}}\dot{J}$, where
$T_{BH}=(\kappa/2\pi)$ is the BH temperature, $\kappa$ is its surface gravity
and $S_{BH}=A/4$ is the BH entropy. Then, from the second law of thermodynamic
$\delta Q=T\delta S$, we find that BHs dissipates energy in similar to a
Newtonian viscous body.

The uniqueness of the effect is revealed when observing the frequency content
of the heating formula. From Alvi:2001mx ; Poisson:2004cw tidal heating takes
$\dot{M}\propto\Omega(\Omega-\Omega_{{}_{BH}})$ and the rotational energy flow
$\Omega_{{}_{BH}}\dot{J}\propto\Omega_{{}_{BH}}(\Omega-\Omega_{{}_{BH}})$,
then dissipation into heat reads
$\dot{Q}\propto(\Omega-\Omega_{{}_{BH}})^{2}$, where $\Omega$ is the orbital
angular velocity. At low spins, $\Omega_{{}_{BH}}\ll\Omega$ heat generation is
governed by $\Omega$ and gets smaller as the body’s spin increase where exact
cancellation occurs for $\Omega_{{}_{BH}}=\Omega$, this threshold case is
known as tidal locking. At higher spins, $\Omega_{{}_{BH}}\gg\Omega$, which is
a typical situation at the inspiral phase,
$\dot{Q}\propto\Omega_{{}_{BH}}^{2}$, and the entire effect is dominated by
the object’s spin. Moreover, since $\dot{J}<0$ the body’s rotational energy
flows outward and angular momentum conservation implies that spin angular
momentum is removed to the orbit. The phenomena of energy extraction from the
rotating body is the underlying mechanism for superradiance Cardoso:2012zn ;
Brito:2015oca .

Here we study the viscous dissipation at the inspiral phase, which as
previously mentioned, since the inspiral time scale is longer than the binary
time scale $\Omega^{-1}\ll 1/R\sim\Omega_{{}_{ECO}}^{-1}$, viscous dissipation
is governed by $\Omega_{{}_{ECO}}$. Motivated by the membrane paradigm
Thorne:1986iy and the similarity shared by the tidal interaction of BHs and
Newtonian viscous bodies. We show that in some cases dissipation in ECOs is
comparable to that of a BHs and due to the large dephasing can produce an
observable imprint on the GW waveform as demonstrated in different context at
Datta:2019euh ; Datta:2019epe ; Datta:2020rvo ; Maselli:2017cmm , which
eventually could be detected with LISA.

##  II Tidal heating - viscous dissipation correspondence

In the current setup we consider the inspiral phase of a binary system, whose
companions are ECOs of comparable masses $M$ and radii $R$. The objects
revolve on a circular orbit of radius $b\gg M$. At this phase the companions
are tidally interacting, the mutual interaction is characterized to leading
order by the quadrupolar tidal field $\mathcal{E}_{ab}$, which for Newtonian
theory is defined in terms of the external Newtonian potential by
$\mathcal{E}_{ab}=\partial_{a}\partial_{b}U_{ext}$. As a result of the slowly
varying induced tidal field, the material body develops, to leading order,
quadrupole moment, given asymptotically in Cartesian coordinates by
$Q_{ab}=\int\rho d^{3}x(x_{i}x_{j}-\frac{1}{3}r^{2}\delta_{ij})$, in which
$\rho$ is the mass density. The strength of the response to the external tidal
field depends on the body composition and is quantified in terms of the tidal
Love number, which relates the induced quadrupole moment to the quadrupolar
tidal field by $Q_{ab}=-\frac{2}{3}k_{2}R^{5}\mathcal{E}_{ab}$. Here $k_{2}$
is the dimensionless $l=2$ electric tidal Love number Hinderer:2007mb ;
Damour:2009vw ; Binnington:2009bb and $R$ is the stars radius.

In the absence of viscosity, the tidal forces produced by each of the
companions raise a bulge in the object, the height of the bulge and its
position are determined by the external tidal field. The relation of the bulge
to the external tidal field is obtained by demanding that the potential energy
gained by the raised bulge $\Delta R$, is conserved with respect to the
external potential such that $(M/R)\Delta R\sim U_{ext}$, where $U_{ext}$ is
evaluated at the tides position. Then, expanding the Newtonian potential in
$b\gg R$ we get $U_{ext}\sim\mathcal{E}R^{2}$, where
$\mathcal{E}=M_{ext}/b^{3}$ is the magnitude of the tidal field. Eventually,
the height of the bulge takes the form $\Delta R(t)\sim\mathcal{E}(t)R^{3}$,
where the time dependence indicates that the object response to the applied
tidal field is instantaneous111 For a more detailed analysis regarding the
relation of $\Delta R$ to $\mathcal{E}$ see at Hartle:1974gy ; Damour:2009va
in the context of the shape Love number. However, when the body’s internal
properties are considered, the effect of viscosity might be non-negligible. In
this case, the response of the body to the exerted tidal force is not
instantaneous and it suffers a time lag, characterized by the viscous time
delay $\tau$. As a result, the tidal bulge is misaligned with the companion
position, an effect that displays a time delay in the induced quadrupole
moment Poisson:2009di , which for slowly-spinning becomes

$\displaystyle Q_{ij}(t)$
$\displaystyle=-\frac{2}{3}k_{2}R^{5}\mathcal{E}_{ij}(t-\tau)\leavevmode\nobreak\
,$ (.1)

where $\tau$ is the viscous time delay. For the case of our interest we assume
that the time scale associated with changes in the tidal field, given by
$\Omega^{-1}\sim\mathcal{E}^{-1/2}$, where $\Omega^{-1}$ is the orbital
evolution time scale, is long compared to the viscous time delay
$\tau\ll\Omega^{-1}$. Then, the tidal field can be expanded into

$\displaystyle
Q_{ij}(t)\approx-\frac{2}{3}k_{2}R^{5}\left(\mathcal{E}_{ij}-\tau\dot{\mathcal{E}}_{ij}\right).$
(.2)

In general, the dependence of the viscous time delay in the internal
properties is given by Poisson:2009di

$\displaystyle\tau\leavevmode\nobreak\ =\leavevmode\nobreak\
\alpha\nu\leavevmode\nobreak\ .$ (.3)

Where $\nu$ is the average kinematic viscosity and $\alpha$ is a dimensionless
coefficient that depends on the objects interior. For example, an isotropic
incompressible viscous body with compactness $C$ has
$\alpha=\alpha_{0}=\frac{19}{2C}$ Poisson:2009di . For anisotropic fluids
$\alpha=\alpha_{0}/\mathcal{A}$, where $\mathcal{A}$ is the anisotropy factor,
with $\mathcal{A}=1$ denoting isotropic fluid Glampedakis:2013jya . The
importance in considering ECOs composed of anisotropic matter is due to their
natural appearance at high densities, in addition to their ability to sustain
gravity above the Buchdahl bound $C>4/9$ Raposo:2018rjn , and up to the BH
limit $C=1/2$ Glampedakis:2013jya .

For the BH viscous delay, we follow the membrane paradigm Thorne:1986iy where
the BH horizon is treated as one way viscous membrane with an effective
kinematic viscosity of $\nu_{{}_{BH}}=M_{{}_{BH}}$. We find that for BHs,
where the only relevant scale is the BH scale $\tau_{{}_{BH}}=R_{{}_{BH}}$, so
$\alpha=2$. The result is also consistent with the calculation of the phase
lag of the bulge relative to the external companion position Hartle:1974gy ,
where for non-spinning BHs $\phi=4M\Omega$, then since $\phi=\omega\tau$, and
$\omega=2\Omega$ is twice the orbital frequency, we obtain
$\tau_{{}_{BH}}=R_{{}_{BH}}$. Besides, it is important to mention that
although, as pointed out in Raposo:2018xkf , horizon scale corrections leave
an indistinguishable imprint on the spacetime of ECOs in comparison to that of
BHs, the behaviour of the BH bulge is different than the ECO bulge. The
difference in the bulge behaviour is not related to the compactness of the
objects, nor how similar their spacetimes are, rather, it is related to the
definition of the event horizon - a globally defined surface, where no light
rays can escape from it to spatial infinity. Thus, no local physical
experiment can locate the event horizon Ashtekar:2004cn . Owing to this
property, the boundary conditions imposed on the event horizon are
teleological Thorne:1986iy . As a result, the bulge displays an a-causal
behaviour, which means that in contrast to material bodies, it leads the orbit
when $\Omega>\Omega_{{}_{BH}}$ and lags when $\Omega<\Omega_{{}_{BH}}$. This
a-causal behaviour guarantees the stability of the hole under perturbations,
as shown in Thorne:1986iy , where causal evolution leads to an exponential
expansion of the BH and eventually destroys it (see also Hawking:1971vc for a
similar discussion in the context of BHs area theorem). Moreover, the ECO
considered here are horizonless, and so their outer surface is locally defined
and smoothly connected to time-like geodesics.

Here we consider ECOs whose outer surface positioned around their photonsphere
$R\sim 3M$ 222In the next section we extend the results for Ultra-compact-
objects $R=2M(1+\epsilon)$, which requires full GR treatment.. In the presence
of viscosity, the work exerted on the star by the external tidal field is
partially converted into heat (tidal friction) and partially leads to angular
momentum flow (tidal torque exerted on the bulge). this is also reflected by
the energy balance formula $\dot{E}_{\nu}=\dot{Q}+\Omega_{{}_{ECO}}\dot{J}$
(we label $\dot{W}=\dot{E}_{\nu}$). Where in the case of slowly-spinning
objects ($\Omega\gg\Omega_{{}_{ECO}}$), most of the work exerted by the tidal
field is dissipated into heat by the interior viscosity. In Newtonian theory,
the work applied by the tidal field on a fluid element $dm$ is
$E_{\nu}=-\frac{1}{2}\int\mathcal{E}_{ab}x^{a}x^{b}dm$. For the rate of tidal
work, differentiation and integration by parts yields

$\displaystyle\dot{E}_{\nu}\leavevmode\nobreak\ =\leavevmode\nobreak\
\dfrac{1}{2}Q_{ij}\dot{\mathcal{E}}^{ij}\leavevmode\nobreak\ .$ (.4)

Then, substituting the induced quadrupole from Eq. (.2) and dismissing total
derivatives, viscous dissipation reads Hartle:1974gy ; Poisson:2009di 333The
calculations are performed in $M_{1}$ local asymptotic rest frame (LARF1).

$\displaystyle\dot{E}_{\nu}\leavevmode\nobreak\ =\leavevmode\nobreak\
\dfrac{1}{3C^{5}}k_{2}\tau
M^{5}\dot{\mathcal{E}}_{ij}\dot{\mathcal{E}}^{ij}\leavevmode\nobreak\ .$ (.5)

Where $C=M/R$ is the ECO compactness, $\dot{\mathcal{E}}=\Omega M_{ext}/b^{3}$
and $\Omega=\sqrt{M_{ext}/b^{3}}$ is the orbital frequency.

To proceed, we extend the relations of viscous bodies Eq. (.2)-(.5) to GR BHs.
First, mentioning that for non-spinning BHs, the rate of tidal heating takes
the form of Eq. (.4),
$\dot{M}=\frac{1}{2}Q_{ij}^{{}_{BH}}\dot{\mathcal{E}}_{ij}$ Thorne:1986iy ;
Poisson:2004cw , where the induced quadrupole moment of a mass $M$ BH is given
by

$\displaystyle Q_{ij}^{{}_{BH}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\dfrac{32}{45}M^{6}\dot{\mathcal{E}}_{ij}\leavevmode\nobreak\ ,$ (.6)

and is not proportional to the tidal field, as implied from the BHs no-hair
properties. For the rate of tidal heating, substituting $Q_{ij}^{{}_{BH}}$
into $\dot{M}$ immediately reads (we label $\dot{M}=\dot{E}_{{}_{BH}}$)

$\displaystyle\dot{E}_{{}_{BH}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\dfrac{16}{45}M^{6}\dot{\mathcal{E}}_{ij}\dot{\mathcal{E}}^{ij}\leavevmode\nobreak\
.$ (.7)

The similarity of these effects is revealed when the viscous dissipation is
given in terms of the tidal heating dissipation.

$\displaystyle\dot{E}_{\nu}\leavevmode\nobreak\ =\leavevmode\nobreak\
\gamma\dot{E}_{{}_{BH}}\leavevmode\nobreak\ .$ (.8)

Where $\gamma$ is defined as the dimensionless effective absorption
coefficient, that is given in terms of the viscous time delay $\tau$ Eq. (.3)
by

$\displaystyle\gamma\leavevmode\nobreak\ =\leavevmode\nobreak\
\dfrac{15\alpha}{32C^{5}}\dfrac{\nu}{M}k_{2}.$ (.9)

The effective absorption coefficient $\gamma$ quantifies how close is the body
absorption to that of a GR BH absorption, alternatively, how good is the body
as a BH mimicker. By definition, $0\leq\gamma\leq 1$, where $\gamma=0$
describes empty, nonviscous interior. On the contrary, $\gamma=1$ defines
objects whose dissipation matches the BH dissipation. In general, for ordinary
matter $\gamma\ll 1$, to show this we consider cold neutron stars ($T\approx
10^{6}$K) that posses large viscosities $\nu{{}_{NS}}\approx 3\times
10^{5}\frac{\text{m}^{2}}{\text{s}}$ LL1 ; LL2 (see Appendix for more
details). To estimate $\gamma$ from Eq. (.9), we take $\alpha_{0}$ and choose
typical neutron star parameters $R=10^{4}$m$,M=1.4M_{\odot}$ and tidal
deformability parameter $k_{2}=1/3$, the effective absorption for neutron
stars reads

$\displaystyle\gamma_{{}_{NS}}\leavevmode\nobreak\ $
$\displaystyle\leavevmode\nobreak\ \sim\leavevmode\nobreak\ 9\times
10^{-5}\times\left(\dfrac{\nu}{3\times
10^{5}\frac{\text{m}^{2}}{\text{s}}}\right)\left(\dfrac{1.4M_{\odot}}{M}\right)\leavevmode\nobreak\
.$ (.10)

Then, since $\gamma_{{}_{NS}}\ll 1$ we conclude that for typical matter,
viscous dissipation is negligible in comparison to the analogous effect in
BHs. However, this might not be the case for exotic matter, in which the
viscosity can be orders of magnitude larger than the value given in Eq. (.10).
Eventually, in such scenarios, the resulted effective absorption becomes
significant and approaches the BH absorption $\gamma\lesssim 1$.

The motivation in considering matter providing non-negligible absorption, and
hence large viscosity, is important when performing an ECO search, where any
departure from the complete absorption indicate on the existence of ECOs. The
existence of matter providing these properties is inspired from both classical
hydrodynamic and quantum mechanics perspective. Where large viscosity is
attributed to a matter whose viscosity is of order of the analogous horizon
viscosity Thorne:1986iy , where according to the membrane paradigm, from
$\nu_{{}_{BH}}=M$

$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\nu_{{}_{BH}}\leavevmode\nobreak\ \approx\leavevmode\nobreak\ 2.7\times
10^{13}\dfrac{m^{2}}{s}\left(\dfrac{M}{65M_{\odot}}\right)\leavevmode\nobreak\
.$ (.11)

An example for such matter can be found in non-rotating strongly magnetized
$(B=10^{15}\text{G})$ neutron stars, in which their viscous damping time
$\tau_{\nu}\sim R^{2}/\nu$ is compared to the Alfven time scale, then, for a
typical neutron star parameters we find $\nu\sim 4\times
10^{10}\text{m${}^{2}$/s}$ Shapiro:2000zh ; Yunes:2016jcc . For boson stars
444In most cases bosonic matter is assumed to have low viscosity., with radius
$R\sim 3M$ and mass $65M_{\odot}$, the damping time of the $l=2$ mode is
$\tau\sim 320$ms, which yields a viscosity of $\nu\sim 2\times 10^{10}$m2/s
Yunes:2016jcc . As for the quantum mechanical perspective, motivated by the
information paradox, where unitarization requires non-local effects, or exotic
matter that provides strong quantum effects that is outside of the standard
model Brustein:2013ena ; Brustein:2016msz ; Giddings:2017mym ;
Buoninfante:2019swn ; Giddings:2011ks ; Almheiri:2012rt . These theories
involves a new scale of order of the Planck scale (as in string theory), which
leads to deviation from the horizon and thus, display small corrections to the
would be BH entropy $S_{{}_{ECO}}=A/(4gl_{P}^{2})$, where $g\gtrsim 1$ is a
new dimensionless scale of order unity or more, as predicted in string theory
$g\sim l_{s}^{2}/l_{P}^{2}$, with $l_{s}$ being the string scale Dienes:1996du
. The link to viscosity is made by the requirement that the exotic matter
saturates the universal shear viscosity to entropy volume density (see also
Sec. V.1 for more details), $4\pi{\eta}/{s}=1$. Then since $\eta=\rho\nu$,
where $\rho$ is the mass density, we find that the ECO viscosity is suppressed
in comparison to the BH viscosity
$\nu_{{}_{ECO}}\sim\nu_{{}_{BH}}/g\leavevmode\nobreak\ $Thorne:1986iy ;
Brustein:2017koc ; Kovtun:2004de . The quantum perspective is discussed
separately in Sec. III.

At this stage, motivated by the vast BH alternatives, we turn to the
evaluation of the ECO effective absorption Eq. (.9). From the analogous BH
viscosity $\nu_{{}_{BH}}=M$, we get $\gamma_{{}_{ECO}}=\beta\nu/\nu_{BH}$.
Where $\beta$ depends on the ECO interior and can be found by specifying the
ECO equation-of-state (EOS). For example, we notice that for the previous
exotic alternatives suggested $\nu\sim 10^{-3}\nu_{{}_{BH}}$, and extremely
compact models with $C=1/3$ have $k_{2}\sim 10^{-2}$ and $\alpha\sim 10$ (see
examples for several NS models at Hinderer:2007mb ). This leads to
$\beta\nu\lesssim\nu_{{}_{BH}}$, which implies that viscous dissipation in
ECOs is comparable to horizon dissipation in an equal mass BHs. Here we do not
examine the specific details of the ECO interior neither the underlying
mechanism that provides large viscosity, rather, we are satisfied with knowing
that such scenarios are physically viable. Hence we define the ECO effective
viscosity $\beta\nu=\nu_{{}_{ECO}}$, and focus on scenarios where
$\nu_{{}_{ECO}}/\nu_{{}_{BH}}\lesssim 1$ in addition to their implications on
future GW observations. Then, we define the ECO effective absorption
coefficient

$\displaystyle\gamma_{{}_{ECO}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\dfrac{\nu_{{}_{ECO}}}{\nu_{{}_{BH}}}\leavevmode\nobreak\ .$ (.12)

Where $0\leq\gamma_{{}_{ECO}}\leq 1$, and is maximized when $\nu_{{}_{ECO}}$
approaches $\nu_{{}_{BH}}$, so
$\gamma_{{}_{ECO}}\rightarrow\gamma_{{}_{BH}}=1$.

The conclusion is that in general, viscous dissipation in ECOs can mimic the
effect of tidal heating in BHs, and as we show in details below, under some
favourable conditions can leave a significant imprints on the emitted GW
waveform during the inspiral phase as well at the ringdown phase in the form
of echoes when the ECO are very compact $R=2M(1+\epsilon)$, where $\epsilon\ll
1$.

The effect of spin  In the previous section, we focused on the absorption of
non-spinning ECOs. However, the inclusion of spin effects increases the energy
dissipation at 1.5 PN relative order and therefore is essential to the
detectability estimation of $\gamma_{{}_{ECO}}$.

The generalization of $\gamma_{{}_{ECO}}$ to the spinning case follows the
procedure given above, where now $\dot{E}_{{}_{ECO}}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \dot{E}_{TH}$ are amplified due to the effect
of rotation. The corresponding expressions for dissipation of energy in the
spinning case can be found at Poisson:2004cw ; Alvi:2001mx . Here for
simplicity we consider the case of a rigid rotation on equatorial orbits,
then, to leading order, we find that Eq. (.4) is valid and the relation given
in Eqs. (.8) holds. Hence, to leading order in the spin, tidal dissipation is
given by

$\displaystyle\dot{E}_{i}\leavevmode\nobreak\ =\leavevmode\nobreak\
\gamma_{i}\dfrac{32}{45}M^{6}\mathcal{E}^{2}(1+\sqrt{1-\chi^{2}})(1+3\chi^{2})\Omega(\Omega-\Omega_{{}_{i}})\leavevmode\nobreak\
.$ (.13)

Where the subscript $i=(\text{BH,\leavevmode\nobreak\ ECO})$, $\Omega_{i}$ is
the BH/ECO angular velocity, $\gamma_{{}_{BH}}=1$. In addition, from the
viscous dissipation of rotating stars given in Poisson:2009di we obtain the
spin-dependent ECO effective absorption

$\displaystyle\gamma_{{}_{ECO}}(\chi)\leavevmode\nobreak\
=\leavevmode\nobreak\
\dfrac{\nu_{{}_{ECO}}}{\nu_{{}_{BH}}}\dfrac{2}{(1+\sqrt{1-\chi^{2}})(1+3\chi^{2})}\leavevmode\nobreak\
.$ (.14)

We first note that at low spins we recover the non-rotating results Eqs.
(.5),(.7), and, from Berti:2004ny we also know that at low spins the ECO
metric can be approximated to that of a Kerr BH. Thus, the quadratic spin
modifications to $\gamma_{{}_{ECO}}$ can be regarded as a relativistic
corrections of the fluid velocity. For a viscous body, the velocity of a fluid
element positioned at the outer surface
$v_{{}_{R_{+}}}=R_{+}\Omega_{{}_{ECO}}\sim\chi$ (where
$\Omega_{i}=\chi/2R_{+}$), so spin effects brings to $\dot{E}_{{}_{ECO}}$
relativistic corrections of order $(v/c)^{2}$ Poisson:2009di . Eventually, we
conclude that the net effect of spin corrections is to reduce the non-rotating
ECO effective absorption at high spins ($\chi\gtrsim 0.85$) by not more than a
factor two, $\gamma_{{}_{ECO}}(\chi\gtrsim 0.85)\sim 2\gamma_{{}_{ECO}}$.

##  III Viscous dissipation in Ultra-Compact-Objects

In this section, we would like to extend the previous formalism to another
sub-family of ECOs - Ultra-compact-objects (UCOs), in which their outer
surface positioned at $R=2M(1+\epsilon)$ where $\epsilon\ll 1$ (see
classification at Cardoso:2019rvt ). Inspired by various quantum motivated
models (see introduction and discussion below Eq. (.11)) and especially by the
BH membrane paradigm, where the horizon is modeled as a one-way viscous
membrane with shear viscosity $\eta_{{}_{BH}}=1/(16\pi)$. We extend the
viscous description of BHs and generalize it to UCOs by considering horizon
scale corrections and non-negligible viscosity
$\eta_{{}_{UCO}}\lesssim\eta_{{}_{BH}}$.

The key point is the realization that the viscous dissipation of inspiraling
UCOs is similar to the tidal dissipation of BHs Datta:2019epe ,

Eqs. (.7),(.13), since as pointed out at Raposo:2018xkf the external geometry
of UCOs can be well approximated by the Schwarzschild geometry (or by the Kerr
geometry in the spinning case) up to some higher-order corrections in the
perturbative parameter $\epsilon$. Therefore, Eq. (.13) is valid in addition
to the correspondence Eq. (.8). The only difference is encoded in the UCO
effective absorption coefficient $\gamma_{{}_{UCO}}$, which in contrast to the
above reviewed ECOs, whose outer surface lies at $R\sim 3M$ so relativistic
correction are small, the surface of UCOs lies deep inside their photonsphere
and requires the full GR treatment. To proceed, we express Eq. (.13) in terms
of the UCO reflection coefficient $\mathcal{R}$ Datta:2019epe ,

$\displaystyle\dot{E}_{{}_{UCO}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\left(1-|\mathcal{R}|^{2}\right)\dot{E}_{TH}\leavevmode\nobreak\ ,$ (.1)

and yields the relation $|\mathcal{R}|^{2}=1-\gamma_{{}_{ECO}}$. This implies
that a perfect absorber with $\gamma_{{}_{UCO}}=1$ has $\mathcal{R}=0$ which
corresponds to $\eta_{{}_{UCO}}=\eta_{{}_{BH}}$, alternatively a perfect
reflector has $\mathcal{R}=1$ which corresponds to negligible absorption
$\eta_{{}_{UCO}}\ll\eta_{{}_{BH}}$.

In order to find the UCO viscosity dependent reflection coefficient, we follow
the derivation given in Maggio:2020jml and extend their results for the
inspiral phase where $M\omega\ll 1$. We consider the quadrupolar $l=2$
perturbations, sourced by the external tidal field. Gravitational perturbation
of the even-parity sector in the exterior Schwarzschild geometry are given by
a simple wave-like differential equation Regge:1957td ; Zerilli:1970se

$\displaystyle\dfrac{d^{2}\psi(x)}{dx^{2}}+\left(\omega^{2}-V\right)\psi(x)=0$
(.2)

where the even-parity potential is given by

$\displaystyle V(r)\leavevmode\nobreak\ =\leavevmode\nobreak\
f(r)\dfrac{24r^{2}(r+M)+18M^{2}(2r+M)}{r^{3}(2r+3M)^{2}}$ (.3)

here $x$ is the familiar tortoise coordinate $dx/dr=1/f(r)$ and $f(r)=1-2M/r$.
The scalar field $\psi$ obeys purely outgoing boundary conditions at infinity
and a superposition of an incoming and a reflected outgoing waves near the UCO
surface

$\displaystyle\psi(x)\leavevmode\nobreak\ \sim\leavevmode\nobreak\ e^{i\omega
x}\hskip 62.59596ptx\rightarrow\infty\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (.4) $\displaystyle\psi(x)\leavevmode\nobreak\
\sim\leavevmode\nobreak\ \mathcal{R}e^{i\omega x}+e^{-i\omega
x}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ x\rightarrow x(R)$ (.5)

To relate the properties of the membrane to the external geometry and in order
to compensate the discontinuity of the extrinsic curvature due to the viscous
membrane, we impose a junction condition on the extrinsic curvature555 We
assume that the extrinsic curvature of the interior is zero Thorne:1986iy .
Thorne:1986iy ; Maggio:2020jml

$\displaystyle Kh_{ab}-K_{ab}\leavevmode\nobreak\ =\leavevmode\nobreak\ 8\pi
T_{ab}\leavevmode\nobreak\ ,$ (.6)

where $K_{ab}$ is the extrinsic curvature about the 3-dimensional membrane
parametrized by the intrinsic coordinates $(t,\theta,\phi)$, $K=h_{ab}K^{ab}$
is the trace with respect to the induced metric $h_{ab}$ of the 3-dimensional
membrane. The stress-energy-momentum tensor (SEM) on the surface

$\displaystyle T_{ab}\leavevmode\nobreak\ =\leavevmode\nobreak\ \rho
u_{a}u_{b}+(p-{\Theta}\xi)\gamma_{ab}-2\eta_{{}_{UCO}}\sigma_{ab}$ (.7)

where $\rho,p,u_{a}$ are the density, pressure and the velocity of a fluid
element on the membrane. $\xi,\Theta$ are the bulk viscosity and the
expansion, $\gamma_{ab}=h_{ab}+u_{a}u_{b}$ is the projection metric and
$\sigma_{ab}$ is the shear tensor. As demonstrated in Maggio:2020jml (for
similar approach see Oshita:2019sat ) by imposing the junction condition Eq.
(.6) on the UCO surface, we get a boundary condition that relates the UCO
reflection coefficient to the membrane properties

$\displaystyle\dfrac{\psi^{\prime}(x)}{\psi}=-i\dfrac{\nu_{{}_{UCO}}}{\nu_{{}_{BH}}}\omega+G(R,\omega,\nu,\xi)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ x\rightarrow
x(R).$ (.8)

Where $G(R,\omega,\nu,\xi)$ is a function that depends on the membrane
properties and is given in the Appendix. We solve Eq. (.8) for the reflection
coefficient near the UCO surface by imposing the boundary condition Eq. (.5).
Then, to first order in $\epsilon\ll 1$ and for the low frequency tidal
perturbation $M\omega\ll 1$ that is related to the orbital frequency
$\Omega=\sqrt{M/b^{3}}$ by $\omega=2\Omega$, the UCO reflection coefficient
reads666The results agree with Maggio:2020jml for $\epsilon\ll 1$, where here
we also neglected the bulk viscosity since it is insignificant to the
absorption coefficient in the BH limit.

$\displaystyle|\mathcal{R}|^{2}\approx\left(\frac{1-\nu_{{}_{UCO}}/\nu_{{}_{BH}}}{1+\nu_{{}_{UCO}}/\nu_{{}_{BH}}}\right)^{2}\left(1-\frac{512(16\pi-3\nu_{{}_{UCO}}/\nu_{{}_{BH}})}{77\left(1-\nu_{{}_{UCO}}^{2}/\nu_{{}_{BH}}^{2}\right)}\epsilon\right)\leavevmode\nobreak\
,$ (.9)

Obviously, for $\nu_{{}_{ECO}}\ll 1$ absorption is negligible and the UCO
reflection is maximized $\mathcal{R}\approx 1$. On the other hand, for
$\nu_{{}_{UCO}}\approx\nu_{{}_{BH}}$ reflection is negligible $\mathcal{R}\ll
1$.

From the relation $|\mathcal{R}|^{2}=1-\gamma_{{}_{ECO}}$ and to leading
order, we obtain the UCO effective absorption coefficient

$\displaystyle\gamma_{{}_{UCO}}\leavevmode\nobreak\ =\leavevmode\nobreak\
\dfrac{4\nu_{{}_{UCO}}/\nu_{{}_{BH}}}{(1+\nu_{{}_{UCO}}/\nu_{{}_{BH}})^{2}}\leavevmode\nobreak\
.$ (.10)

As expected, in the absence of viscosity $\gamma_{{}_{ECO}}=0$, alternatively,
when $\nu_{{}_{UCO}}=\nu_{{}_{BH}}$ we recover the BH absorption
$\gamma_{{}_{UCO}}=1$. Furthermore, we notice that the effective absorption of
ECOs Eq. (.12) is recovered in the BH limit, where for
$\nu_{{}_{UCO}}/\nu_{{}_{BH}}\approx 1$ we find
$\gamma_{{}_{UCO}}\approx\nu_{{}_{UCO}}/\nu_{{}_{BH}}$.

Finally, as previously explained, the UCO viscous dissipation is obtained by
substituting $\gamma_{{}_{UCO}}$ into Eq. (.13). Moreover, we argue that
although the reflection coefficient is calculated for non-spinning UCOs, Eq.
(.10) is still a good approximation for rotating UCOs. This claim is justified
in the UCO corotating frame with angular velocity
$\Omega_{{}_{UCO}}=\chi/(2M(1+\sqrt{1-\chi^{2}}))$, where the frequency of the
external tidal perturbation is shifted to $\Omega-\Omega_{{}_{UCO}}$ such that
at high spins $\chi\lesssim 1$ the low-frequency approximation breaks.
However, from Eq. (.8) we notice that higher orders in the frequency are
coupled to $\epsilon$ through the function $G(\omega)$. Thus, since we are
particularly interested in quantum corrections to the horizon where
$\epsilon\ll 1$, which is parametrically orders of magnitude smaller than the
dimensionless spin parameter $\epsilon\ll\chi$. Higher-order spin corrections
are always suppressed by $\epsilon$ and are unimportant to the absorption
properties of the membrane.

##  IV  Detectability

Figure 1: The relative statistical error vs spin parameter for several values
of $\gamma_{{}_{ECO/UCO}}$ Eqs. (.12),(.10). We assume a binary of equal spin
and nearly equal masses $M_{1}/M_{2}=1.01$ with luminosity distance
$D_{l}=2$Gpc. Points below the horizontal dashed line correspond to detection
at better than $1\sigma$ confidence. The
$\gamma_{{}_{UCO}}=\left(0.3,0.5,0.7,0.9\right)$ corresponds to
$\nu_{{}_{UCO}}/\nu_{{}_{BH}}=\left(0.08,0.17,0.29,0.51\right)$.

In this section, we discuss the possibility of measuring the ECO effective
absorption $\gamma_{{}_{ECO}}$ in future LISA observations of supermassive,
highly-spinning equal-mass binaries. Which according to LISA , LISA could
track for a time period $T\sim 1$year and observe from the early stages of the
inspiral and up to the coalescence with high SNR ($\gtrsim 10^{3}$). We show
that for such binary systems, the statistical error due to the detector noise
enables to set constraints on $\gamma_{{}_{ECO}}$, which according to Eq.
(.12) constraints the properties of the exotic matter through
$\nu_{{}_{ECO}}$. When spin effects are considered, we find that the higher
the spin of the rotating body, the lower the relative statistical error. The
improvement in accuracy at large spins is mainly since tidal heating effects
of spinning BHs are leading to the non-spinning case at 1.5 PN relative order,
where the spinning (non-spinning) enters at 2.5 PN (4PN) relative to the
leading order GW phase term. Another important reason is due to the signal-to-
noise-ratio (SNR) sensitivity at the late stages of the inspiral, where the
orbital frequency is large
SNR${}^{2}\sim\int_{\min}^{\max}\frac{\tilde{h}(f)^{2}}{S_{n}(f)}df$. Where
$\tilde{h}(f)$ is the Fourier transform GW waveform and $S_{n}(f)$ is the
detector noise spectral density. We find this effect to be extremely important
for UCOs whose outer surface lies deep inside their photonsphere. We denote
$f_{\max}=f_{\text{ISCO}}$, and $f^{\text{Kerr}}_{{}_{\text{ISCO}}}$ is Kerr’s
ISCO which for certain spins ($\chi\gtrsim 0.7$) modifies $f_{\text{max}}$
substantially
$f^{\text{Kerr}}_{{}_{\text{ISCO}}}/f^{\chi=0}_{{}_{\text{ISCO}}}\approx 3$
Favata:2010ic .

The analysis given below follows the analysis given in Maselli:2017cmm , there
it is shown that highly-spinning super massive BHs characterized by maximal
absorption $\gamma_{BH}=1$ can be distinguished from ECOs with zero absorption
$\gamma_{{}_{ECO}}=0$, which means that the statistical error due to the
detector noise is small enough and enables the measurement of the absorption
coefficients with high certainty. However, here we consider the possibility
that dissipation effects in ECOs can mimic the effect of dissipation at the
horizon, as described by
$\dot{E}_{{}_{ECO}}=\gamma_{{}_{ECO}}\dot{E}_{{}_{BH}}$. Therefore, as
displayed in Eq. (.12), ECOs could, in principle, produce comparable effects
to the BH absorption, namely $\gamma_{{}_{ECO}}\lesssim 1$. The results are
displayed in Fig. 1.

To proceed, our aim is to estimate the statistical error in the measurement of
$\gamma_{{}_{ECO}}$. We apply a parameter estimation method based on the
Fisher information matrix that is accurate for large SNR events and described
by a Gaussian distribution. Where for a given detector signal
$s(t)=h(t,\theta^{i})+n$, where $n$ is the noise and $h(t,\theta^{i})$ is the
model signal that depends on the parameters $\theta^{i}$. The posterior
probability distribution can be approximated by

$\displaystyle p(\theta^{i}|s)\propto
e^{-\frac{1}{2}\Delta\theta^{i}\Delta\theta^{j}\Gamma_{ij}}\leavevmode\nobreak\
,$ (.11)

Where $\Gamma_{ij}$ is the Fisher matrix defined by

$\displaystyle\Gamma_{ij}=\left(\frac{\partial
h}{\partial\theta^{i}}\Big{|}\frac{\partial
h}{\partial\theta^{j}}\right)\leavevmode\nobreak\ ,$ (.12)

and the inner product $(\cdot|\cdot)$ is defined by

$\displaystyle(h_{1}|h_{2})\leavevmode\nobreak\ =\leavevmode\nobreak\
4\leavevmode\nobreak\
\text{Re}\int_{f_{min}}^{f_{max}}\frac{\tilde{h}_{1}(f)\tilde{h}_{2}^{*}(f)}{S_{n}(f)}df\leavevmode\nobreak\
.$ (.13)

Here $S_{n}(f)$ denotes LISAs noise spectral density LISA ; Cornish:2018dyw .
For the mass range considered below $M=10^{6}M_{\odot}$ the minimal frequency
is the entrance frequency to LISA’s observation band is denoted by,
$f_{min}\approx 10^{-5}$Hz which corresponds to observation time of about one
year Berti:2004bd . The maximal frequency for UCOs is taken to be the
frequency at the Kerr’s ISCO $f_{max}=f^{\text{Kerr}}_{{}_{\text{ISCO}}}$
Favata:2010ic , for ECOs we take the contact point. The model signal and the
true signal are parameterized by $\theta^{i}=(\text{ln}\leavevmode\nobreak\
\mathcal{A},\text{ln}\leavevmode\nobreak\
\mathcal{M},\text{ln}\leavevmode\nobreak\
\eta,\Psi_{c},t_{c},\chi_{1},\chi_{2},\gamma_{{}_{ECO/UCO}})$, whose arguments
are the amplitude $\mathcal{A}$, the chirp mass $\mathcal{M}=\eta^{3/5}M$, the
symmetric mass ratio $\eta=M_{1}M_{2}/M^{2}$, the phase $\Psi_{c}$, the time
at coalescence $t_{c}$, the dimensionless spin parameters $\chi_{1},\chi_{2}$
and the ECOs/UCOs dimensionless effective absorption coefficient given in Eqs.
(.12),(.10) respectively.

For this set of parameters, the root-mean-square error in measuring
$\gamma_{{}_{ECO/UCO}}$ is expressed through the inverse of the Fisher matrix

$\displaystyle\sigma_{\gamma}=\sqrt{\langle\left(\Delta\gamma\right)^{2}\rangle}=\sqrt{\left(\Gamma^{-1}\right)_{\gamma\gamma}}\leavevmode\nobreak\
.$ (.14)

For a binary inspiral, the Fourier transform of the signal is modelled by
$\tilde{h}(f,\theta^{i})=\mathcal{A}e^{i\Psi}$, where

$\displaystyle\Psi=\Psi_{PP}+\gamma_{i}\Psi_{TH}\leavevmode\nobreak\ ,$ (.15)

are the phases of the point-particle and tidal heating effects respectively
Datta:2020gem . $i=(\text{ECO,UCO})$ denotes the ECO/UCO effective absorption
Eqs. (.12),(.10). The phase is computed by solving the equation

$\displaystyle\dfrac{d^{2}\Psi}{df^{2}}\leavevmode\nobreak\
=\leavevmode\nobreak\ \dfrac{2\pi}{\dot{E}}\dfrac{dE}{df}\leavevmode\nobreak\
,$ (.16)

where $E=-M_{1}M_{2}/2b$ is the gravitational binding energy of the binary
system. The approximation method adopted here is the analytical “TaylorF2
approximant” Damour:2000zb ; Arun:2004hn ; Buonanno:2009zt . We include
correction terms to the GW phase in the form of spin-orbit, spin-spin and
cubic spin corrections up to 3.5PN order relative to the leading-order GW term
Isoyama:2017tbp ; Khan:2015jqa , and tidal heating correction term for
spinning objects to the leading 2.5 PN order relative to the leading-order GW
term Cardoso:2019rvt ; Isoyama:2017tbp . The amplitude is taken to leading PN
order and includes the sky-averaged prefactor Berti:2004bd .

##  V Discussion

###  V.1 Theoretical implications on black-hole mimickers

In this section we discuss, on a qualitative level, the possible
interpretations to a measurement of $\gamma_{{}_{UCO}}$ and its deviations in
BH mimickers. Furthermore, we show explicitly how stringy inspired quantum
corrections provides the deviation of the absorption coefficient from that of
a classical BH.

BHs-like objects are characterized by the BH compactness and an entropy that
matches the Bekenstein’s entropy $S_{{}_{BH}}=A/4$ 777These objects can have
Planckian corrections at the horizon scale, so $\epsilon=l_{p}$ and redshift
arguments provides the Bekenstein’s entropy $S_{{}_{ECO}}=1/\epsilon^{2}$
tHooft:1984kcu ; Brustein:2018ixz . This property is provided by Qunautm BHs
(QBHs) models considered in Bekenstein:1995ju ; Skenderis:2008qn ;
Almheiri:2012rt ; Brustein:2016msz . We can express the UCO absorption Eq.
(.10) in terms of their entropy (in units of $\hbar=k_{B}=1$),
$\rho\nu_{{}_{BH}}=\eta_{{}_{BH}}=s_{{}_{BH}}/(4\pi)$, where
$s_{{}_{BH}}=S_{{}_{BH}}/V$ is the entropy volume density. Then from Eq. (.10)
at large viscosities $\gamma_{{}_{UCO}}=\eta_{{}_{UCO}}/\eta_{{}_{BH}}$, the
absorption coefficient becomes

$\displaystyle\gamma_{{}_{UCO}}\leavevmode\nobreak\ =\leavevmode\nobreak\
4\pi\dfrac{\eta_{{}_{UCO}}}{s}\leavevmode\nobreak\ .$ (.17)

So for UCO posses BH entropy, constraints on the absorption coefficient
translates to constraints on the universal shear viscosity to entropy volume
density, known as the KSS bound Kovtun:2004de
$4\pi\frac{\eta_{{}_{UCO}}}{s}\geq 1$. Then, since the UCO absorption is
bounded from above ($\gamma_{{}_{UCO}}\leq 1$), the two conditions implies
$\gamma_{{}_{UCO}}=1$.

The conjectured KSS bound found to agree with common substances as water,
helium, and nitrogen. However, in strongly interacting systems such as quark-
gluon plasma, which appear at extreme densities and are relevant for our
understanding of the UCO interior, the ratio found to be close to the bound
and perhaps violating it Shuryak:2004cy . Therefore, it is requested to
explore the violation possibility ($\gamma_{{}_{UCO}}<1$) and its implications
on the UCO absorption.

Let us assume that the fluid interior is composed of strongly interacting
fluid in which, according to the fluid-gravity correspondence Rangamani:2009xk
their higher dimensional holographic dual is the stringy inspired $ADS_{5}$
black brane Gauss-Bonnet (GB) gravity. The Lagrangian density
$\mathcal{L}=R-2\Lambda+\frac{\lambda_{{GB}}}{2}L^{2}\left(R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)$,
where $L$ is the ADS radius and $\Lambda=-{6}/{L^{2}}$. $\lambda_{GB}$ is the
dimensionless parameter that controls the magnitude of the higher curvature
correction terms. In Brigante:2007nu ; Kats:2007mq , the bound was calculated
from the holographic gravity dual at large $N$ and large ’t Hooft coupling
$\lambda$ ($1\ll\lambda\ll N$), then to nonperturbative order in
$\lambda_{{GB}}$ the result is given by

$\displaystyle\gamma_{{}_{UCO}}\leavevmode\nobreak\ =\leavevmode\nobreak\
1-4\lambda_{GB}$ (.18)

Which can be expressed in terms of the reflection coefficient

$\displaystyle|\mathcal{R}|^{2}\leavevmode\nobreak\ =\leavevmode\nobreak\
4\lambda_{{GB}}$ (.19)

This demonstrates how quantum corrections in the form of higher derivative
terms modifies the BH absorption/reflection. The positive sign of
$\lambda_{GB}$ that is required in order to maintain causality
Brustein:2017iet is consistent with the positivity of $|\mathcal{R}|^{2}$ and
the UCO absorption bound $\gamma_{{}_{UCO}}<1$.

##  VI Summary

In this paper, we investigated the effect of energy dissipation in ECOs at the
inspiral phase. In particular, we were interested in whether and how can ECO
mimic the effect of tidal heating in BHs. Then, in analogy to the membrane
paradigm, by modelling ECOs as a Newtonian viscous body and UCOs as a viscous
membrane, we find the corresponding absorption coefficients. Then, the ECOs
effective absorption was generalized to the spinning case, according to Eq.
(.12).

The detectability of the ECO effective absorption $\gamma_{{}_{ECO}}$ in
future LISA observations was discussed. We performed a Fisher information
analysis to estimate the statistical error in measurement of
$\gamma_{{}_{ECO}}$ and to quantify how well can the ECO viscous interior
produce similar effects to that of a BH. The results show that for
supermassive, highly-spinning comparable-mass binaries, a relative absorption
of $\gamma_{{}_{UCO}}\gtrsim 0.5,\leavevmode\nobreak\ \gamma_{{}_{ECO}}\gtrsim
0.9$ could be measured for binary at a luminosity distance $D_{l}=2$Gpc of
$M=10^{6}M_{\odot}$ and spin $\chi\gtrsim 0.75$ with more than $1\sigma$
confidence, where at spins close to extremality $\chi\sim 0.95$,
$\gamma_{{}_{UCO}}$ can be measured with more than $3\sigma$ confidence. Our
results found to agree with Maselli:2017cmm . Nevertheless, a different
approach for calculating the reflection coefficient was recently studied in
Chen:2020htz , which by quantifying the linear response of the membrane to the
external tidal field, different boundary conditions are imposed on the
membrane.

## Acknowledgments

I thank Ramy Brustein, Yoav Zigdon and Sayak Datta for discussions and
comments on the manuscript and Kent Yagi for discussions. The research was
supported by the Israel Science Foundation grant no. 1294/16 and by the Negev
scholarship.

## Appendix

##  Appendix A Estimation of the neutron star shear viscosity

The estimated density for a typical neutron stars with $M\leavevmode\nobreak\
=\leavevmode\nobreak\ 1.4\times M_{\odot}$, $R\leavevmode\nobreak\
=\leavevmode\nobreak\ 10$km is

$\displaystyle\rho\leavevmode\nobreak\ =\leavevmode\nobreak\
\dfrac{3}{4\pi}\dfrac{M_{1}}{R_{1}^{3}}\leavevmode\nobreak\
\approx\leavevmode\nobreak\ 6.3\times
10^{17}\dfrac{kg}{m^{3}}\left(\dfrac{M}{1.4\times
M_{\odot}}\right)\left(\dfrac{10^{4}m}{R}\right)^{3}\leavevmode\nobreak\ .$
(A.1)

The estimated viscosity for ‘cold’ neutron star $T<10^{9}$K, where the
viscosity is governed by the electron scattering, is given by LL1 ; LL2

$\displaystyle\nu_{{}_{NS}}\leavevmode\nobreak\ \approx\leavevmode\nobreak\
3\times 10^{5}\dfrac{m^{2}}{s}\left(\dfrac{\rho}{6.3\times
10^{17}\frac{kg}{m^{3}}}\right)\left(\dfrac{10^{6}\text{K}}{T}\right)^{2}\leavevmode\nobreak\
.$ (A.2)

##  Appendix B The function $G(R,\omega,\nu)$

For completeness we bring the full form of the function $G(R,\omega,\nu)$ Eq.
(.8), for the case of our interest we consider zero bulk viscosity which found
to be negligible when $\epsilon\ll 1$. The function is taken from the Appendix
of Maggio:2020jml and defined by $G(R,\omega,\nu)=A/B$

$\begin{split}A\leavevmode\nobreak\ =\leavevmode\nobreak\
-&2\epsilon(49152\pi^{2}\eta w^{2}(\epsilon-1)^{5}(\eta(4\epsilon-5)-2)+\\\
&3(-4w^{2}(4\epsilon(\epsilon(4\epsilon-9)+6)+5)(\epsilon-1)^{2}+\\\
&4\epsilon(2\epsilon(15-8(\epsilon-1)\epsilon)-35)-7)-\\\ &512i\pi
w(\epsilon-1)^{3}\left(\eta(\epsilon(16\epsilon-5)-29)+8(\epsilon-1)^{2}\right))\\\
B\leavevmode\nobreak\ =\leavevmode\nobreak\
&4(7-4\epsilon)(\epsilon-1)^{2}(8w^{2}(2\epsilon(4\epsilon(12(\epsilon-4)\epsilon+71)-\\\
&181)+91)(\epsilon-1)^{3}+128i\pi\eta
w(8\epsilon(4\epsilon((\epsilon-4)\epsilon+6)-\\\
&17)+11)(\epsilon-1)^{3}-8\epsilon(\epsilon(2\epsilon+3)-3)-11)\leavevmode\nobreak\
.\end{split}$ (A.3)

Where $w=M\omega$ and $\omega=2\Omega$ is twice the orbital frequency.
$\epsilon=\Delta R/R=(R-2M)/2M$ is the deviation from the BH horizon. In
accordance with Eqs. (.8),(.9), since $G(\epsilon,w)\propto\epsilon$, in the
BH limit $\epsilon\rightarrow 0$ we get $\mathcal{R}=0,\leavevmode\nobreak\
\gamma_{{}_{UCO}}=\gamma_{{}_{BH}}=1$ as expected.

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