# Imprint of black hole area quantization and Hawking radiation on inspiraling
binary

Sayak Datta<EMAIL_ADDRESS>Inter-University Centre for Astronomy and
Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India  Max-Planck-
Institut für Gravitationsphysik (Albert-Einstein-Institut), D-30167 Hannover,
Germany  Leibniz Universität Hannover, D-30167 Hannover, Germany  Khun Sang
Phukon<EMAIL_ADDRESS>Nikhef - National Institute for Subatomic
Physics, Science Park, 1098 XG Amsterdam, Netherlands  Institute for High-
Energy Physics, University of Amsterdam, Science Park, 1098 XG Amsterdam,
Netherlands  Institute for Gravitational and Subatomic Physics, Utrecht
University, Princetonplein 1, 3584 CC Utrecht, The Netherlands

###### Abstract

We study the potential of gravitational wave astronomy to observe the quantum
aspects of black holes. According to Bekenstein’s quantization, we find that
black hole area discretization can have observable imprints on the
gravitational wave signal from an inspiraling binary black hole. We study the
impact of quantization on tidal heating. We model the absorption lines and
compute gravitational wave flux due to tidal heating in such a case. By
including the quantization we find the dephasing of the gravitational wave, to
our knowledge it has never been done before. We discuss the observability of
the phenomena in different parameter ranges of the binary. We show that in the
inspiral, it leads to vanishing tidal heating for the high spin values.
Therefore measuring non-zero tidal heating can rule out area quantization. We
also argue that if area quantization is present in nature then our current
modeling with reflectivity can possibly probe the Hawking radiation which may
bring important information regarding the quantum nature of gravity.

##  I Introduction

Since the observation of GW190514 Abbott  _et al._ (2019a) gravitational wave
(GW) astronomy has grown to be one of the dominant contributors to
astrophysical observations. These observations have led to an unprecedented
probe of strong gravity Abbott  _et al._ (2019b). Properties of vacuum
spacetime, propagation of GW, violation of Lorentz invariance have been tested
rigorously, which has resulted in stringent bounds on the mass of the graviton
and violations of Lorentz invariance Abbott  _et al._ (2016a, b, 2017). We
also have begun to investigate the properties of these dark compact objects,
both in inspiraling binary as well as in the merger.

To resolve the information-loss paradox, Planck scale modifications of black
hole (BH) horizons and BH structure have been proposed Lunin and Mathur
(2002); Almheiri  _et al._ (2013). Other exotic compact object (ECOs) have
also been proposed in the literature Mazur and Mottola (2004); Liebling and
Palenzuela (2012). To probe the nature of the compact objects in binary,
several tests have been proposed too Cardoso  _et al._ (2016a, b); Maselli
_et al._ (2018); Tsang  _et al._ (2020); Abedi  _et al._ (2017); Westerweck
_et al._ (2018); Cardoso and Pani (2019); Tsang  _et al._ (2020); Abedi  _et
al._ (2017); Westerweck  _et al._ (2018); Cardoso  _et al._ (2017); Sennett
_et al._ (2017); Brustein and Sherf (2020, 2021); Krishnendu  _et al._ (2017);
Datta and Bose (2019); Datta (2021).

In General Relativity, the horizon of the classical BHs (CBHs) are perfect
absorbers Thorne  _et al._ (1986); Damour (1982); Poisson (2009); Cardoso and
Pani (2013). This is due to the causal structure of the geometry of CBH. This
null surface, which is the defining feature of a CBH, is a one-way membrane.
Due to the nature of the horizon, the reflectivity of the CBH horizon is
considered to be zero Berti  _et al._ (2009). But in the case of the ECOs, it
is required to introduce non-zero reflectivity Cardoso  _et al._ (2019). This,
as a result, brings changes to the observable quantities. Having such scopes,
we have begun to investigate the nature of these compact objects in detail. In
light of this, it has also become possible to investigate the quantum gravity
corrections near horizon scales observationally.

In this work, we focus primarily on the possibility that the GWs emitted from
an inspiraling binary BH merger can carry imprints regarding the quantum
properties of the BHs. In some works, this question have been posed and have
been addressed primarily in the context of the postmerger Cardoso  _et al._
(2019); Laghi  _et al._ (2021); Agullo  _et al._ (2021). In this work, we will
focus on the possibility of observing these effects in the inspiral phase.

##  II Quantum Black Holes

It is expected that the quantum BHs (QBHs) have a discrete energy spectrum. As
a result, it has been put forward that they will behave in a similar manner as
the excited atoms Mukhanov (1986); Bekenstein (1974); Bekenstein and Mukhanov
(1995); Kothawala  _et al._ (2008); Davidson (2019); de Freitas Pacheco and
Silk (2020). Bekenstein had proposed the idea based on the adiabatic
invariance of BH area in classical GR and applying Bohr-Sommerfeld
quantization

$A_{N}=\alpha\ell_{P}^{2}N,$ (1)

where $\ell_{P}$ is the Planck length, N a positive integer, and $\alpha$ is a
non-negative number.

Bekenstein and Mukhanov had concluded that BHs must have a discrete spectrum
of mass and investigated the emission spectrum Bekenstein and Mukhanov (1995);
Agullo  _et al._ (2010). In loop quantum gravity too BH area has been shown to
be quantized Rovelli and Smolin (1995); Rovelli (1996); Ashtekar  _et al._
(1998, 2000); Agullo  _et al._ (2008). Hence, the area quantization can be
considered to be one of the possible signatures of quantum gravity.
Interestingly, as we will demonstrate, despite the area quantization being
related to the Planck scale, it can leave observable imprints on GWs.

In Ref. Agullo  _et al._ (2021) how the area quantization leads to an
absorption spectrum has been studied in detail. They have argued that the
“astrophysical BHs act as magnifying lenses,” in the sense that they bring the
Planck-scale discretization of the horizon within the realm of GW
observations. In Agullo  _et al._ (2021) various mechanisms where BH area
quantization may leave observable imprints in GWs has also been discussed. One
such mechanism is tidal heating. In the present work, we will investigate the
impact of area quantization on tidal heating in more details. Therefore, our
primary focus would be the inspiral phase of a binary.

##  III Effect of area quantization

Effect of area quantization for a spinning BH has been explicitly computed in
Ref. Agullo  _et al._ (2021); Chakraborty and Lochan (2019). As our work
depends on it, we will shortly review the results here. A spinning BH, which
is represented by a Kerr metric, is parametrized by its mass $(M)$ and angular
momentum $(J)$. Since the area of the horizon $(A)$ of the BH can completely
be determined in terms of $M$ and $J$, it can be used to represent $M$ in
terms of $A$ and $J$:

$M=\sqrt{\frac{A}{16\pi}+\frac{4\pi J^{2}}{A}}.$ (2)

Considering the area quantization and quantization of the angular momentum,
$J=\hbar j$ $(0\leq j\leq\alpha N/8\pi)$, Ref. Agullo  _et al._ (2021) found
that,

$M_{N,j}=\sqrt{\hbar}\sqrt{\frac{\alpha N}{16\pi}+\frac{4\pi
j^{2}}{N\alpha}}.$ (3)

This set of $M_{N,j}$ constitutes the mass spectrum of QBH for a set
$\\{N,j\\}$.

By focusing on the dominant GW mode $(l=2,\,m=2)$, which is most interesting
for astrophysical systems (in quasicircular binary inspiral), and considering
the transition $M_{N,j}\rightarrow M_{N+\Delta N,j+2}$ in Ref. Agullo  _et
al._ (2021) it is shown that the frequency at which a BH can have non-zero
absorption is,

$\omega_{n}=n\frac{\kappa\alpha}{8\pi}+2\Omega_{H},$ (4)

where $\kappa$ is the surface gravity and $\Omega_{H}$ is the angular velocity
of the horizon111When GW frequency does not match with the frequency in Eq.
(4), the absorption is more likely to be suppressed rather than being exactly
equal to zero. This is primarily because the particular details of the
frequency dependencies of the absorption should depend on the details of the
microscopic theory of quantum gravity, that one lacks). . Note, for higher
modes of GW with $l>2$, different transitions will also contribute. This will
bring out more interesting features. But throughout our current work, we will
assume that Eq. (4) is valid, and we will study the consequence of it in the
inspiral phase.

Assuming Hawking radiation as the decay channel, in Ref. Agullo  _et al._
(2021) the width $(\Gamma)$ of the lines have also been calculated. We use
these numerically calculated values of $\Gamma$, that has been provided to us
by the authors of Ref. Agullo  _et al._ (2021), to find an analytical fitting
function for $\Gamma$ as follows,

$M\Gamma_{E}=1.005e^{(-6.42+1.8\chi^{2}+1.9\chi^{12}-.1\chi^{14})}.$ (5)

In the upper panel of the Fig. 1 we plot the analytical function,
$\Gamma_{E}$, (red curve) and the numerical data, $\Gamma_{N}$ (blue dotted
curve) with respect to the dimensionless spin parameter $(\chi)$. In the lower
panel we plot the percantage fitting error defined as follows,

$|\Delta\Gamma|\equiv|\Gamma_{N}-\Gamma_{E}|,$ (6)

where $\Gamma_{N}$ represents the numerically evaluated values of $\Gamma$. We
find that for the entire range of $\chi$, error in fitting is less than $8\%$,
while for $\chi\leq.85$ it is less than $2\%$. Error is highest for $\chi=.9$,
i.e $\sim 8\%$. We will use this analytic expression throughout our work.

Figure 1: We plot the analytical function $\Gamma_{E}$ (red curve) and the
numerical data $\Gamma_{N}$ (blue dotted curve) with respect to the
dimensionless spin parameter $(\chi)$ in the upper panel. In lower panel, we
plot the error in the fitting. We find that for entire range of $\chi$ error
is less than $8\%$, while for $\chi\leq.85$ it is less than $2\%$. Error is
highest for $\chi=.9$, i.e $\sim 8\%$.

##  IV modelling the absorption: tidal heating

A CBH absorbs radiation of frequency $\omega>m\Omega_{H}$ (where m is the
azimuthal number of the wave and $\Omega_{H}$ is the angular velocity of the
horizon) but amplifies radiation of smaller frequency, due to superradiance
Brito  _et al._ (2015). When CBHs are in inspiraling binary, they expirence
the other component’s tidal field during the inspiral phase. If the bodies are
(at least partially) absorbing, then these tides backreact on the orbit. As a
result, energy and angular momentum is transferred from their spin into the
orbit. This phenomena is called tidal heating Hartle (1973); Hughes (2001);
Poisson and Will (1953). This effect is due to the dissipative nature of the
CBH horizons. Modification of physics near horizon scale, as a result, can
modify the TH Datta  _et al._ (2020a); Datta (2020). For this reason, the TH
contribution in an inspiraling binary can deviate significantly if the quantum
effects near the horizon are taken into account, such as area quantization.

As has been discussed earlier, a QBH can absorb (emit) GW only in some
discrete frequency values, unlike CBH where the process is continuous across
the frequency band. Therefore in the presence of area quantization tidal
heating should contribute only to those frequency values that are allowed by
Eq. (4). This can be modeled by assuming that the BHs have a frequency-
dependent reflectivity $\mathcal{R}(f)$. For this purpose, in this section we
will try to model the reflectivity of a QBH.

The primary focus of our current work is to investigate what is the
implication of such an effect in the tidal heating of a QBH in a binary, which
was partially addressed in Ref. Agullo  _et al._ (2021). In Ref. Datta (2020);
Datta  _et al._ (2020a) it has been established how the tidal heating in ECOs
can get modified. It has been established that this can be done by defining
the Horizon parameter $(H)$ and multiplying the TH flux of CBH with $H$
Maselli  _et al._ (2018); Datta and Bose (2019); Sherf (2021). For horizonless
bodies, $H=0$ and for CBH, $H=1$, and “anything” in-between can be modeled
assuming $0<H<1$. In Ref. Datta (2020) it has been established that
$H=1-|\mathcal{R}|^{2}$ at $\mathcal{O}(\epsilon^{0})$. Where the position of
the reflective surface $(r_{s})$ is defined in terms of $\epsilon$ and the
position of the CBH horizon $(r_{H})$, as $r_{s}=r_{H}(1+\epsilon)$. In the
current work, this process can be continued assuming $\epsilon=0$ and the
reflectivity $\mathcal{R}$ is frequency-dependent. Rather than constructing
$\mathcal{R}(f)$, it is much easier to construct $\mathcal{H}(f)$ in the
current case. We will follow that route here. Once the frequency-dependent
Horizon parameter $(\mathcal{H}(f))$ is defined, frequency-dependent
reflectivity can be constructed by assuming
$\mathcal{H}(f)=1-|\mathcal{R}(f)|^{2}$. Note, in this method only
$|\mathcal{R}(f)|$ can be constructed and it will not be possible to construct
the phase of the reflectivity (which becomes relevant when $\epsilon\neq 0$
Chakraborty  _et al._ (2021)). Note, once the reflectivity is constructed,
irrespective of how, it can not only be used for TH computation but also for
any kind of study regarding QBH. In inspiral along with TH, excited resonace
modes can also contribute Maggio  _et al._ (2021); Sago and Tanaka (2021). To
study such behavior also the reflectivity constructed in this work can be
used.

Figure 2: We plot the horizon parameter $\mathcal{H}(f)$ w.r.t. the GW
frequency $f$, for different spin and $\alpha$. Black and blue plot represents
$\chi=.7$, where black curve has $\alpha=2$ and blue curve represents
$\alpha=4\pi$. The first line falls on top of each other in this case, since
$f_{0}$ depends only on the spin. With a lower value of $\alpha$, the lines in
the black curves are closer to each other compared to the blue curve
representing a larger $\alpha$ value. Similar phenomena happen for $\chi=.9$,
represented by the red curves. Note, the width of the lines is larger for a
higher spin.

We observe that for QBH, TH will contribute only at those GW frequencies $(f)$
where $f=\omega_{n}/2\pi$. At these GW frequencies $\mathcal{H}(f_{n})=1$
should be satisfied, where $2\pi f_{n}\equiv\omega_{n}$. Since there is a non-
zero line width $\Gamma$, TH should also contribute in the frequency range
$f_{n}-\Gamma/2<f<f_{n}+\Gamma/2$, following a pattern of absorption line
structure. Everywhere else $\mathcal{H}(f)=0$ should be satisfied. This can be
done by constructing an absorption line profile at
$f_{n}-\Gamma/2<f<f_{n}+\Gamma/2$ for each $n$ and add them. Therefore,
$\mathcal{H}(f)$ can be defined as follows:

$\displaystyle\mathcal{H}(f)=$
$\displaystyle\sum_{n=0}^{n_{max}}\mathcal{P}_{n}(f)$ (7)
$\displaystyle\mathcal{P}_{n}(f)=$
$\displaystyle\mathfrak{P}(f-f_{n},\frac{\Gamma}{2}),$ (8)

where $\mathfrak{P}(f-f_{n},\frac{\Gamma}{2})$ is a Hann window function
defined as follows,

$\begin{split}\mathfrak{P}(x,y)=\,&0,\qquad\qquad\qquad\qquad\qquad\,\,\,x\leq-y\\\
&.5+.5\cos(2\pi\frac{.5x}{y}),\,\,\,\,-y\leq x\leq y\\\
\,&0,\qquad\qquad\qquad\qquad\qquad\,\,\,x\geq y\end{split}$ (9)

$\displaystyle\omega_{n}=n\frac{\kappa\alpha}{8\pi}+2\Omega_{H},\,\,n_{max}=Ceiling[\frac{8\pi}{\kappa\alpha}(\omega_{contact}-\omega_{0})]$
$\displaystyle\kappa=\frac{\sqrt{1-\chi^{2}}}{2M(1+\sqrt{1-\chi^{2}})},\,\,\,\Omega_{H}=\frac{\chi}{2M(1+\sqrt{1-\chi^{2}})}.$

In Fig. 2 we plot $\mathcal{H}(f)$ for multiple parameter sets. Black and blue
plot represents $\chi=.7$, where black curve has $\alpha=2$ and blue curve
represents $\alpha=4\pi$. The first line falls on top of each other in this
case, since $f_{0}$ depends only on the spin. With a lower value of $\alpha$
in the black curve, the lines are very close to each other compared to the
blue curve with larger value of $\alpha$. Similar phenomena happens for
$\chi=.9$, which is represented by the red curve. Note that the width of the
lines are larger for a higher spin. This can be understood by comparing the
red and the black curve. This is also evident from Fig. 1.

In Ref Agullo  _et al._ (2021) critial values of $\alpha$, say
$\alpha_{crit}(\chi)$, below which lines start to overlap, has been calculated
as a function of spin. By taking sufficiently small $\alpha$, it is possible
to get $\mathcal{H}(f)\rightarrow 1$ for the entire frequency range where
$f>f_{0}$. Therefore, in frequency range $f>f_{0}$ it can mimic the result of
a CBH. This happens because for $\alpha<\alpha_{crit}$ the closely spaced
lines starts to overlap with each other. As a result, their contributions get
added up to give a “semi-uniform” value of $\mathcal{H}(f)$. Although for
small values of $\alpha$, $H\rightarrow 1$, it can not mimic CBH result
exactly. For CBH there is no lowest value of $f$ below which TH vanishes. But
for QBH if $f<f_{0}$, then TH vanishes, even for very small values of
$\alpha$. This phenomena is very different from CBH, as in CBH (unlike QBH) TH
will be non-zero not only for $f>f_{0}$ but also for $0<f<f_{0}$.

Figure 3: We plot the fractional flux due to tidal heating w.r.t. the flux at
infinity assuming the area quantization. As expected, the CBH flux envelopes
the QBH flux. It can have both the absorption lines as well as emission lines
which should be understood in the context of the superradiance of classical
BH.

Once $\mathcal{H}(f)$ is constructed, flux for TH of QBH in a binary can be
defined by multiplying $\mathcal{H}(f)$ to the TH flux for CBH, namely
$\mathcal{F}_{CBH}(f)$. Therefore, the flux for QBH, namely
$\mathcal{F}_{QBH}(f)\equiv\mathcal{H}(f)\mathcal{F}_{CBH}(f)$. Using this
definition, we express the total flux due to TH of a binary QBH in the
following manner,

$\mathcal{F}_{\rm
TH,QBH}=\mathcal{H}^{(1)}\mathcal{F}^{(1)}_{CBH}+\mathcal{H}^{(2)}\mathcal{F}^{(2)}_{CBH},$
(10)

where, $M_{i}$ and $\chi_{i}$ is the mass and the dimensionless spin of the
$i$th QBH, mass ratio $q=M_{2}/M_{1}$, $M=M_{1}+M_{2}$ and $\mathcal{H}^{(i)}$
is the horizon parameter, and $\mathcal{F}^{(i)}$ is flux of the $i$th body if
it were a CBH (expression can be found in Ref. Alvi (2001); Datta  _et al._
(2020b); Chatziioannou  _et al._ (2013, 2016)).

In Fig. 3 we have plotted an example of fractional flux due to TH of QBH with
respect to the flux at infinity. Due to the area quantization, TH flux has a
discretized structure that is determined by the area quantization parameter
$\alpha$. But as expected, the discretized structure follows the CBH envelope.
For all of the curves the components of the binaries are equally massive,i.e.
$q=1$. Individual mass of the components are $30M_{\odot}$. The red and the
black curve represents $\chi=.85$ and $\alpha=20,\,4\pi$ respectively. The
blue curve is for $\chi=.9$ and $\alpha=6$. Note, this plot is for
representation purpose only, as we will discuss later, there will be no TH for
these spin values in equal mass binary.

We deduce the GW phase involving tidal heating by using Eq. (2.7) of Ref.
Tichy  _et al._ (2000) (see Isoyama and Nakano (2018) for the details). We
find the phase shift due to the associated horizon absorption to be,

$\displaystyle\Psi_{\rm THQBH}=$
$\displaystyle{}\frac{3}{128\nu}\left(\frac{1}{v}\right)^{5}\left[-\frac{10}{9}v^{5}\Psi_{5}\left(3\log\left(v\right)+1\right)\right.$
(11) $\displaystyle-\frac{5}{168}v^{7}\Psi_{5}\left(952\nu+995\right)$
$\displaystyle\left.+\frac{5}{9}v^{8}\left(3\log\left(v\right)-1\right)(-4\Psi_{8}+\Psi_{5}\psi_{\text{SO}})\right]\,,$

where $\Psi_{5}=(\mathcal{H}^{(1)}A^{(1)}_{5}+\mathcal{H}^{(2)}A^{(2)}_{5})$
and $\Psi_{8}=(\mathcal{H}^{(1)}A^{(1)}_{8}+\mathcal{H}^{(2)}A^{(2)}_{8})$,
$v$ is the post-Newtonian velocity parameter, $\nu$ is the symmetric mass
ratio, $\hat{L}$ is the direction of the orbital angular momentum,
$\hat{S}_{i}$ is the direction of the $i-$th component’s spin,

$\begin{split}\psi_{\text{SO}}\equiv~{}&\frac{1}{6}\big{[}\big{(}-56\nu-73\sqrt{1-4\nu}+73\big{)}\big{(}\hat{L}.\hat{S}_{1}\big{)}\chi_{1}\\\
&+\big{(}-56\nu+73\sqrt{1-4\nu}+73\big{)}\big{(}\hat{L}.\hat{S}_{2}\big{)}\chi_{2}\big{]}\,.\end{split}$
(12)

$\displaystyle A^{(i)}_{5}\equiv$
$\displaystyle{}\left(\frac{M_{i}}{M}\right)^{3}\left(\hat{L}.\hat{S}_{i}\right)\chi_{i}\left(3\chi_{i}^{2}+1\right)\,,$
(13a) $\displaystyle A^{(i)}_{8}\equiv$ $\displaystyle{}~{}4\pi
A^{(i)}_{5}+\left(\frac{M_{i}}{M}\right)^{4}\left(3\chi_{i}^{2}+1\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\times\left(\sqrt{1-\chi_{i}^{2}}+1\right)\,.$
(13b)

These expressions will be used later to calculate the dephasing which will
lead us to understand the observability of the area quantization.

##  V Observability before plunge

In the previous section, we have discussed the impact of area quantization on
the inspiral phase and how they can be modeled. In this section, we will focus
more on the details of the observability of this phenomena.

Figure 4: We plot $f_{0}<f_{contact}$ region with respect to the individual
spins of the binary components for mass ratio $q=1$ in blue and $q=.42$ in
orange. The dashed lines show the boundary. Note, we are focusing only on the
observability of the first body’s quantum nature. This is why the plot is
asymmetric between $\chi_{1}$ and $\chi_{2}$ even for $q=1$. A similar kind of
result can be found if we focus on the observability of the second body’s
quantum nature.

###  V.1 Absorption frequency vs contact frequency

For a given spin $\chi$ and area separation $\alpha$, there is a lowest
absorption frequency, i.e. $\omega_{0}=2\Omega_{H}$. Only above this
frequency, TH will be present for QBH. The range of frequency covered in an
inspiral upto merging of the two bodies in a binary is bounded from above by
the frequency corresponding to the length scale of the radial surface position
of the bodies. In a binary, the closest reach in the inspiral is
$r_{H1}+r_{H2}$, where they touch each other. Using Kepler’s third law,
corresponding frequency to this length can be found as,

$f_{contact}=\frac{1}{\pi
M_{1}}\sqrt{\frac{1+q}{(1+\sqrt{1-\chi_{1}^{2}}+q+q\sqrt{1-\chi_{2}^{2}})^{3}}},$
(14)

where $q=M_{2}/M_{1}$.

Note that both of the frequencies $(f_{0},f_{contact})$ are functions of mass
and spin. Where $f_{0}$ is the lowest frequency of the TH lines for QBH and
$f_{contact}$ is the upper frequency before they merge, above which TH
phenomenon is not relevant. Hence, non-zero discretized structure of TH flux
can only be observed iff $f_{0}\leq f_{contact}$. In Fig. 4, we plot
$f_{0}<f_{contact}$ region with respect to the individual spins of the binary
components for mass ratio $q=1$ in blue and $q=.42$ in orange. The dashed
lines show the boundary. Note, we are focusing only on the observability of
the first body’s quantum nature (i.e. $f_{0}$ of the first body). This is why
the plot is asymmetric between $\chi_{1}$ and $\chi_{2}$, even for $q=1$. A
similar kind of result can be found if we focus on the observability of the
second body’s quantum nature.

From the Fig. 4, we notice that for a given value of $q$ there is a region
above the dashed curve where $f_{0}/f_{contact}<1$ is not satisfied. We find
that below $\chi_{1}=0.679667$, the condition is always satisfied. However,
above this spin value the allowed region satisfying $f_{0}<f_{contact}$
depends on both $q$ and $\chi_{2}$. But for $q<.384646$, for entire range of
$\chi_{1}$ and $\chi_{2}$ the condition $f_{0}<f_{contact}$ will be satisfied.
In these ranges where $f_{0}<f_{contact}$, non-zero TH will be present during
the inspiral. But the feature of it will be completely different from TH of
CBH, as discussed before. This, as a result, can be used as a signature of the
area quantization. In the ranges where $f_{0}>f_{contact}$ TH will be exactly
equal to zero for QBH. In the parameter range where TH is expected to be
absent for QBH, a model selection strategy with the Bayes factor can be used
to test for the presence of “classical TH” to distinguish between CBH and QBH.
It has been addressed in Ref. Datta  _et al._ (2020b). Hence, using TH as a
possible probe for area quantization seems viable.

Figure 5: We plot first body’s $f_{0}$ and $f_{1}$ for $\alpha=20,25$ along
with the contact frequencies for different parameter values of the binary.

To measure $\alpha$, it is necessary to measure at least one higher frequency
absorption line, namely $f_{1}$. This implies that $\alpha$ will be measurable
if $f_{1}<f_{contact}$. In Fig. 5, we plot first body’s $f_{0}$ and $f_{1}$
for $\alpha=20,25$ along with the contact frequencies for different parameter
values of the binary. For equal mass binaries $(q=1)$ (dashed red and green
line), $f_{0}$ is greater than $f_{contact}$ for higher spin of the first
body. This is in accordance with Fig. 4 and illustrates the argument described
in earlier paragraphs. In Fig. 5, we observe that for smaller $q$,
$f_{0}<f_{contact}$ for all spin values of the first body (dotted green and
red line).

Figure 6: We plot different contact frequencies of the binaries for
$\chi_{2}=.1,\,q=.3$ in red dashed curve, and $\chi_{2}=.95,\,q=10^{-5}$ in
green dashed curve. We also plot $f_{1}$ for $\alpha=30,\,110,\,16\pi$ in blue
black, and purple.

Similar behavior is shown by the $f_{1}$ curves (solid blue and purple
curves). But in this case the value of $\alpha$ becomes an important
parameter. By increasing $\alpha$, $f_{1}$ curves can be shifted upwards, as
has been illustrated by the blue curve ($\alpha=25$) having higher values than
the purple curve ($\alpha=20$). Therefore, in the near-equal mass binaries
beyond a maximum value of $\alpha$, say $\alpha_{max}$, $f_{1}>f_{contact}$
for all $\chi_{1}$. Hence, even if area quantization is present in nature, we
can only measure it using TH if and only if
$\alpha<\alpha_{max}(\chi_{1},\chi_{2},q)$. This can be understood by noting
that the red dashed curve intersects with the $\alpha=25$ (solid blue) curve
at a smaller $\chi_{1}$ value than the $\alpha=20$ (solid purple) curve. It
also implies that $\alpha_{max}$ will be a function of the binary parameters.

We calculate $\alpha_{max}$ by using $f_{1}=f_{contact}$. We find,

$\begin{split}\frac{\alpha_{max}}{32\pi}=&\frac{1+\sqrt{1-\chi_{1}^{2}}}{\sqrt{1-\chi_{1}^{2}}}\left[\frac{1+q}{(1+q+\sqrt{1-\chi_{1}^{2}}+q\sqrt{1-\chi_{2}^{2}})^{3}}\right]^{\frac{1}{2}}\\\
-&\frac{\chi_{1}}{2\sqrt{1-\chi_{1}^{2}}}.\end{split}$ (15)

In Fig. 10, we show the value of $\alpha_{max}$ for different binary
parameters. As can be seen in the plot in the upper left corner, for high
values of $\chi_{1}$ in equal mass binaries $(q=1)$, $\alpha_{max}<0$. Which
in accordance with the Fig. 5. It implies that in equal mass binaries there
will be no observable higher absorption lines if the components of the
binaries are spinning very fast. In equal mass binaries we find the maximum
observable $\alpha_{max}\sim 54$. If $\alpha$ is very high $(\mathcal{O}(10))$
in nature, then to measure it the spin of the corresponding body in the equal
mass binary requires to be small. In the figures for $M_{1}>M_{2}$ we restrict
ourselves with $\chi_{1}\leq.85$, primarily because beyond this value
$\alpha_{max}$ can take extremely large values, i.e. $\alpha_{max}>600$ for
$\chi_{1}\sim.998$.

Figure 7: The blue shaded region represents the parameter region that
satisfies $f_{1}(\chi_{1}=1)<f_{contact}(\chi_{1}=1)$. We find for $q\leq
0.384646$ this condition is satisfied for the entire range of $\chi_{2}$.

Note, for $M_{1}>M_{2}$, $\alpha_{max}$ reaches the minimal values in the mid
range of $\chi_{1}$ and grows after that. This can be understood from Fig. 6.
Dashed red and green curves are the $M_{1}f_{contact}$ for
$q=.3,\,\chi_{2}=.1$ and $q=10^{-5},\,\chi_{2}=.95$, respectively. Blue,
black, and purple curves represents $f_{1}$ for $\alpha=30,\,110,\,16\pi$
respectively. We find that the $M_{1}f_{contact}$ frequency curves can
intersect $M_{1}f_{1}$ curves at two points
$\chi_{1}=\chi_{low},\,\chi_{high}$. This creates an intermediate region of
$\chi$ where the $\alpha_{max}$ decreases then it increases again for high
spin values.

The reason for such behavior is connected with the different behavior of
$M_{1}f_{contact}$ and $M_{1}f_{1}$ as a function of $\chi_{1}$. $f_{contact}$
is a monotonically increasing function with respect to spin. But $M_{1}f_{1}$
can have a maxima for a range of $\alpha$. This is primarily because
$M_{1}f_{1}$ is fixed to the value of $1/2\pi$ when $\chi_{1}=1$ , which is
independent of $\alpha$. But $M_{1}f_{contact}$ at $\chi_{1}=1$, increases
with increasing $\chi_{2}$ and decreasing $q$.

If $f_{1}(\chi_{1}=1)=1/(2\pi M_{1})<f_{contact}(\chi_{1}=1)$ then even for
arbitrarily large values of $\alpha$ there will be at least one intersection
point between $f_{1}$ and $f_{contact}$ near $\chi_{1}=1$. In Fig. 6, this
phenomena is being exhibited by $\alpha=110$ curve. In Fig. 7, we have plotted
the region that satisfies $f_{1}(\chi_{1}=1)<f_{contact}(\chi_{1}=1)$. We find
for $q\leq 0.384646$ this condition is satisfied for the entire range of
$\chi_{2}$.

Note, in Fig. 6, $\alpha=110$ curve has a maxima while there are no maximas
present for lower values of $\alpha$ shown in Fig. 5. For a fixed $\alpha$ the
condition for extrema of $f_{1}$ is $f_{1}^{\prime}(\chi_{1})=0$, where prime
denotes the derivative with respect to $\chi_{1}$. This results as,

$\alpha=\frac{16\pi(1+\sqrt{1-\chi_{1}^{2}})}{\chi_{1}}.$ (16)

It is easy to verify that the right hand side of Eq. (16) has a minimum value
of $16\pi$. Hence for $\alpha<16\pi$ this condition will not be satisfied and
there will not be any extrema. Interestingly, in the literature the range of
expected values is $1<\alpha<30$ Cardoso  _et al._ (2019). Therefore in this
physical range maxima will never occur. Hence, Fig. 10 implies that for
$q\leq.1$ entire physical range can be probed. This can help us in putting
bound on $\alpha$ from observation.

Figure 8: We plot $f_{0}$ and ISCO frequency in EMRI. Above $\chi=.359405$,
$f_{\rm ISCO}<f_{0}$. As a result, above this spin $\delta\phi=0$ and below
this spin value $\delta\phi\sim\mathcal{O}(.1)$. For CBH
$\delta\phi\sim\mathcal{O}(10^{2})$ and for $\chi=.9$,
$\delta\phi\sim\mathcal{O}(10^{3})$ Datta  _et al._ (2020a).

###  V.2 Dephasing in the inspiral phase

Using Eq. (11), we calculate the dephasing in radian due to TH in QBH. A
useful estimator to describe the effects of $\alpha$ in the phase is the total
number of GW cycles $(\equiv N)$ that accumulates within a given frequency
band of the detectors. In terms of the frequency-domain phase $\Psi(f)$ this
can be defined as,

$N\equiv\frac{1}{2\pi}\int_{f_{min}}^{f_{max}}fdf\left(\frac{d^{2}\Psi(f)}{df^{2}}\right).$
(17)

By substituting Eq. (11) into Eq. (17) the relative number of GW cycles
$(\equiv\Delta N)$ that is contributed by this effect and accumulated within
the frequency band $f\in[f_{min},f_{max}]$ is calculated. We find only in case
of extreme mass ratio inspiral (EMRI) dephasing is reasonable. For EMRI we
take $f_{min}=.5\,{\rm mHz},\,\,f_{max}=f_{ISCO}$. We compute the dephasing
(in radian), $\delta\phi=2\pi\Delta N$, by taking into account of non-zero
$\alpha$. We find that above $\chi=.359405$, $\delta\phi=0$ and below it
$\delta\phi\sim\mathcal{O}(10^{-1})$ even for high values of $\alpha$ (in the
physical range).

Figure 9: The dephasing (in radian) from ISCO frequency to the light ring
frequency is calculated. In the upper panel, the dephasing for QBH is
demonstrated with $\alpha$. In the lower panel, the ratio of QBH and CBH
dephasing is demonstrated in the same frequency band.

This can be understood from Fig. 8. In Fig. 8 we plot $f_{0}$ and the
frequency corresponding to the inner most stable circular orbit (ISCO) of
EMRI, with respect to $\chi$. In the figure above $\chi=.359405$,
$f_{0}>f_{ISCO}$. In such cases there will be no tidal heating contribution
for QBH. This is a telltale signature, as for CBH in EMRI
$\delta\phi_{CBH}\sim\mathcal{O}(10^{2})$($\mathcal{O}(10^{3})$)for
$\chi=.3$(.9). Therefore, it can safely be said that absence of any TH is a
potential signature of area quantization. But unfortunately, if the bodys are
spinning very fast then except for the absence of TH it is not possible to
find any other feature of area quantization in the inspiral up to ISCO. As a
result, it can potentially lead to degeneracy between horizonless ECOs and
QBHs. But this degeneracy can get broken in the merger phase. This has been
discussed in the next section.

###  V.3 Dephasing in the merger phase

In the previous section, we demonstrated that the TH effect of QBH is
vanishingly small in the inspiral phase of a binary. As a result, the
dephasing due to TH in QBH is either vanishes or is vanishingly small.
However, our investigation suggests that $f_{0}$ is always smaller than
$f_{contact}$ in EMRI, implying it is possible to find signatures in the
merger and post-postmerger phase. To explore it, we again calculate dephasing
$(\delta\phi)$ for an EMRI with a $10^{6}M_{\odot}$ supermassive body and a
$10M_{\odot}$ companion. But this time we integrate Eq. (17) from ISCO
frequency to the frequency corresponding to the light ring (see Ref.
Taracchini  _et al._ (2013) for definition).

In the upper panle of Fig. 9 we plot the accumulated dephasing due to QBH
$(\delta\phi_{QBH})$ in the frequency band, w.r.t. $\alpha$. In lower panel we
plot the ratio of dephasing between QBH TH and CBH TH, namely
$(\delta\phi_{QBH}/\delta\phi_{class})$. As expected, the dephasing is higher
for higher spin values. We also find that with increasing $\alpha$, the
dephasing decreases but also stays roughly unchanged in a range of $\alpha$,
creating a stair-like structure. The stair structure originates due to the
different number of absorption lines present in the frequency band. With
increasing $\alpha$, the lines get more separated. Due to the increased
separation, if a line goes out of the frequency band then the dephasing drops.
Otherwise, it increases very slowly. The increase occurs due to the increase
in the strength of TH with the absorption lines shifting towards higher
frequency while staying in the frequency band. It implies that the
corresponding dephasing in EMRI can be used to measure $\alpha$ and $\Gamma$
in the merger phase. However, we have used a stationary phase approximation to
find this dephasing, which is unlikely to be applicable. Hence, it remains to
be seen what happens when realistic phasing is considered in the merger phase.
However, the figures establish that the merger phase can introduce significant
deviation from CBH contribution, resulting in the measurability of $\alpha$
and $\Gamma$.

##  VI Probing Hawking radiation

We have explicitly shown that the presence of area quantization affect the
tidal heating phenomena of a QBH in a binary. Since, tidal heating affects the
inspiral rate of a binary, imprints of area quantization is possible to find
in the emitted GW from the system. This phenomena manifests through two
parameters, quantization spacing $(\alpha)$ and the line width $(\Gamma)$. We
have explored in details the effect of $\alpha$ and $\Gamma$ in the GW.

Figure 10: We plot the $\alpha_{max}$ above which $f_{1}$ will not be
observable before the binary merges. Since the comparison arises due to the
$f_{contact}$, it depends on $\chi_{2}$ and the mass ratio
$q=\frac{M_{2}}{M_{1}}$. $\alpha_{max}$ has been plotted in the colour bar
w.r.t. $\chi_{1}$ and $\chi_{2}$.

The width $\Gamma$ of the energy levels is written as the inverse of a decay
rate, $\tau$ , as $\Gamma=\hbar/\tau$. This timescale is associated with the
spontaneous decay of the BH energy states due to Hawking radiation, and has
been estimated as $\tau\equiv-\frac{\langle\hbar\omega\rangle}{\dot{M}}$,
where $\langle\hbar\omega\rangle$ denotes the average frequency over all the
decay channels, and $\dot{M}$ is the power (check Ref. Agullo  _et al._ (2021)
for more details).

During observation it is possible to treat $\Gamma$ as an independent
parameter, rather than using Eq. (5). Since we will measure spin $\chi$ along
with $\Gamma$, the measured value of $\chi$ can be used to find a fit between
$\Gamma$ and $\chi$ using data from multiple observations. Once such fitting
function is available, it is possible to compare the observed fitting function
with the analytic expression in Eq. (5). Any deviation from Eq. (5) will
represent the limitation of the estimation of $\tau$. This deviation can arise
due to modification of Hawking radiation itself or due to some beyond standard
model channels not considered in Ref. Agullo  _et al._ (2021).

As a result, we will have observational information regarding Hawking
radiation as well as the nature of the quantum states of QBH. This therefore,
has the potential to shed light on the information loss paradox from the side
of the observation 222From inspiraling binary, upper bound on the Hawking
temperature has been found in Ref. Chung and Sakellariadou (2021). However
this value is very high with respect to the expected value of the Hawking
temperature of these systems.. Note, it is unlikely that TH can be used for
that purpose unless the spin of the bodies are low and the signal to noise
ratio (SNR) is very high. But probing Hawking radiation by measuring $\Gamma$,
as described in previous paragraph, will be valid even in merger and
postmerger phase.

##  VII Observability

In our work, we establish that due to the presence of area quantization,
effect of TH becomes vanishingly small in EMRI. Above $\chi\sim.36$ it results
in vanishing TH. Therefore, measuring no TH in EMRI will be an indication
towards area quantization, this aspect was proposed in Ref. Agullo  _et al._
(2021). We agree with their claim after a detailed investigation. However, as
TH is not exactly equal to zero below $\chi\sim.36$, with high SNR, $\alpha$
and $\Gamma$ can be measured. This will be investigated in recent future.
Another key point is, we have not included the effect of resonances which can
modify our conclusion.

Although for high spins, TH vanishes exactly below ISCO frequency, $f_{0}$ and
$f_{1}$ are less than contact frequency. This implies that the frequency
corresponding to the area quantization will be in the range of the merger and
postmerger phase frequencies. As a result, in the merger phase of an EMRI it
will lead to non-zero dephasing for any non-zero spin of the supermassive
body.

The vanishing of TH for high spins below ISCO implies that there will not be
any absorption of GW flux in the inspiral phase. As a result, this can lead to
ergo region instability Maggio  _et al._ (2017, 2019), which can have
observable consequences. This requires a detailed investigation. Note, that
the information of area quantization is solely captured by the first term in
Eq. (4), while the second term corresponds to angular momentum states of the
QBH. For $\ell=m=2$ mode the second term is similar to the $m\Omega_{H}$ that
connects the frequency at the far range to the frequency at the near horizon
range. This indicates that even for a different area quantization law only the
first term in Eq. (4) will get modified while the second term will stay intact
for $\ell=m=2$ mode (assuming Eq. (2) stays valid). In such a case for nonzero
$\chi$ there will always be a minimum frequency $\propto\Omega_{H}$ below
which TH vanishes. Therefore non-vanishing TH can be a strong indication of a
violation of any kind of area quantization.

##  VIII Discussion

In this work, we have explored the possibility of observing BH area
quantization using tidal heating. We have argued that it can be done by
assuming a frequency-dependent reflectivity of the QBH. We have explicitly
constructed a model for such reflectivity. Once implemented, we find as
expected, the tidal heating flux becomes quantized, unlike classical black
holes. Using the quantized structure of the tidal heating flux, we find the
expression of the phase of the QBH binary. Since in the presence of area
quantization, TH is absent below $f_{0}$, it will be hard to distinguish
between QBH and horizonless ECOs for $f<f_{0}$. To address it, we explored the
parameter range where $f_{0}<f_{contact}$ is satisfied. We find that below
mass ratio $q<.384646$, $f_{0}<f_{contact}$ is always satisfied. We have also
found the expression for $\alpha_{max}$ for a set of binary parameters. We
find that for equal mass binary observable $\alpha_{max}<54$. For binaries
with $q<.1$ the entire physical range of $\alpha$ is observable. By
calculating dephasing, we demonstrate that in EMRI the effect of area
quantization is very strong in the sense that the effect of TH vanishes for
higher spin values below ISCO. This is drastically different from CBH, where
TH can add several cycles in the phase. However, in the merger phase of an
EMRI area quantization leads to non-zero dephasing. We also discuss the
possibility of probing Hawking radiation with the measurement of $\Gamma$.

To model the lines we have used the Hann window function. They can also be
modeled with Lorentzian profile,

$\mathfrak{P}(\omega)\propto\frac{2}{\pi}\frac{\Gamma}{4(\omega-\omega_{n})^{2}+\Gamma^{2}}.$
(18)

Under such modeling also the conclusion stays the same (check Ref. Chakravarti
_et al._ (2021) too). However, there remain differences in the interpretation
of the different modeling. I.e., if the normalized Lorentzian or Gaussian
profiles are used then at the position of the absorption frequency $f_{i}$,
the maximum value of each profile can be larger than one. This, as a result,
is different from the Hann window function. In choosing the Hann function, we
kept the maximums of $\mathcal{H}$ at equal to 1. It implies that when
absorption is allowed, it will be like CBH at these frequencies. In Gaussian
or Lorentzian profile $\mathcal{H}(f_{i})>1$, which is different from CBH. It
requires an investigation from the theoretical side.

In Ref. Agullo  _et al._ (2021) it has been assumed that the QBH can emit at
any real frequency $\omega$ during the Hawking evaporation process. It will
not be true if the energy levels are quantized. It is because only such decay
channels will be allowed that end at an allowed energy level. If that is taken
into account while calculating the values of $\Gamma$, not only the values of
$\Gamma$ will decrease, but also it will depend on the quantization $\alpha$.

It is extraordinary that we are at a juncture when Planck scale physics can be
tested with the current and the upcoming detectors. We have shown that the
tidal heating in the inspiral phase of a binary will look quite different from
binary CBHs. The area quantization of QBHs can lead to peculiar features in
tidal heating flux, which for CBH is a 2.5 PN$\times\log v$ effect in the
leading order. The analysis presented here heavily depends on Bekenstein-
Mukhanov’s semi-heuristic arguments on quantum BHs. It is mind-boggling that
the quantum aspects of BHs are within the reach of observations. It is high
time to advance the quantum gravity frameworks so that concrete predictions
are possible to make.

## Acknowledgement

It is a pleasure to thank Adri$\acute{\rm a}$n del Rio and Vitor Cardoso for
providing us numerical data that has been used to find the analytical
expression of the line width. We thank Adri$\acute{\rm a}$n del Rio and Vitor
Cardoso for reading an earlier draft of the paper carefully and giving us
valuable suggestions. We thank Yash Bhargava, Bhaskar Biswas, and Soumak
Maitra for useful discussions. SD would like to thank University Grants
Commission (UGC), India, for financial support for a senior research
fellowship. KSP acknowledges the support of the Science and Engineering
Research Board (SERB), India, and the Netherlands Organisation for Scientific
Research (NWO). This work was done with partial support provided by the Tata
Trusts. This paper has been assigned LIGO Document Number LIGO-P2000115.

We would like to thank all of the essential workers who put their health at
their risk during the COVID-19 pandemic, without whom we would not have been
able to complete this work.

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