# Wafer-Scale Integration of Freestanding Photonic Devices with Color Centers
in Silicon Carbide

Sridhar Majety 111Corresponding author<EMAIL_ADDRESS>Department of
Electrical and Computer Engineering, University of California, Davis, CA
95616, USA  Victoria A. Norman  Department of Electrical and Computer
Engineering, University of California, Davis, CA 95616, USA  Department of
Physics and Astronomy, University of California, Davis, CA 95616, USA  Pranta
Saha  Department of Electrical and Computer Engineering, University of
California, Davis, CA 95616, USA  Alex H. Rubin  Department of Electrical and
Computer Engineering, University of California, Davis, CA 95616, USA
Department of Physics and Astronomy, University of California, Davis, CA
95616, USA  Scott Dhuey  The Molecular Foundry, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Marina Radulaski  Department of
Electrical and Computer Engineering, University of California, Davis, CA
95616, USA

###### Abstract

Color center platforms have been at the forefront of quantum nanophotonics for
applications in quantum networking, computing, and sensing. However, large-
scale deployment of this technology has been stifled by a lack of ability to
integrate photonic devices at scale while maintaining the properties of
quantum emitters. We address this challenge in silicon carbide which has both
commercially available wafer-scale substrates and is a host to color centers
with desirable optical and spin properties. Using ion beam etching at an
angle, we develop a 5-inch wafer process for the fabrication of triangular
cross-section photonic devices in bulk 4H-SiC. The developed process has a
variability in etch rate and etch angle of 5.4% and 2.9%, respectively.
Furthermore, the integrated color centers maintain their optical properties
after the etch, thus achieving the nanofabrication goal of wafer-scale
nanofabrication in quantum-grade silicon carbide.

##  1 Introduction

Nanophotonics has gained prominence over the last few decades for applications
in classical fields of communications, nonlinear optics, sensing and displays,
as well as the quantum field of information processing (QIP). Solid-state
emitters with optically addressable spin states have been the frontrunners for
scalable QIP implementation due to their capability of creating entanglement
between electron spin and photon, and subsequently, the emitted photon can be
used to distribute entanglement over long distances. When such an emitter,
with the ability to generate indistinguishable spin-entangled photons, is
integrated into a nanophotonic structure, it can reach the efficiencies in
emission and collection of photons [1, 2], necessary to implement QIP
protocols. Earlier efforts in this direction, using quantum dots and the
nitrogen vacancy color center in diamond, that demonstrated groundbreaking
effects of quantum nanophotonics, struggled to reach scalability due to either
the lack of scalable nanofabrication of the lack of spectral uniformity among
emitters. Silicon carbide (SiC) promises to bridge these gaps. SiC is a well
astablished host of color centers that combine long spin coherence times [3],
excellent brightness [4], room-temperature spin manipulation [5], ample
nuclear spins [6], ultra-narrow inhomogeneous broadening [7], and emission
wavelengths in the near-infrared and telecom bands [2]. Most importantly, SiC
is available as a wafer-scale substrate, and its nanofabrication processes are
CMOS compatible, making it a suitable platform for large-scale quantum
nanophotonic hardware.

Suspended photonics offer the best optical confinement due to their highest
refractive index contrast with the surrounding media, air (n = 1). Such
suspended photonics are mainly fabricated through heteroepitaxial growth of
the photonic device layer on a sacrificial layer and selective etching of this
sacrificial layer upon fabrication of the photonic devices to release them
from the bulk. The ability to grow silicon on silicon dioxide in a scalable
way has accelerated the field of large-scale silicon photonics [8]. Similar
efforts have been made to grow 3C-SiC on silicon [9, 10, 11], but such a SiC
layer develops defects up to thickness of a few microns due to the large
difference in the lattice constants of the two materials [12]. This has
encouraged researchers to develop unconventional methods such as the Smart-Cut
method [13], photoelectrochemical etch [14], and wafer bonding and thinning
[15]. Each of these methods has limitations in achieving scalability, such as
implantation damages (Smart-Cut method), requirement of specific doping
profiles (photoelectrochemical etch), and uniformity over wafer-scale (wafer
bonding and thinning).

Color centers are best defined in bulk. Angle etching method is a bulk
processing technique to fabricate suspended photonic devices, first
demonstrated in diamond [16] and later in SiC [17, 18]. These angle etching
demonstrations were Faraday cage-assisted, in which the ions in the reactive
ion etch chamber are directed at an angle by the Faraday cage to etch
underneath the mask, producing suspended photonic devices with triangular
cross-sections. Finite-difference time-domain (FDTD) modeling of triangular
cross-section photonics has shown suitability for color center integration and
QIP hardware applications [19, 18, 20, 21, 22]. However, Faraday cage-assisted
angle etching is not scalable and requires a separate Faraday cage design for
each fabrication etch angle.

Ion Beam etching (IBE) was used to achieve angle etching on a chip-scale in
bulk diamond [23]. In this method, ions from an ion source are accelerated
towards the substrate, and the substrate holder can be tilted to etch
underneath the mask to create an undercut. Color centers integrated into
triangular photonic devices in diamond fabricated using IBE were used to make
several key QIP demonstrations such as nanocavities with cooperativities
exceeding 100 [24], $>$99% single-shot readout and spin initialization
fidelities [24], cavity-coupled two-qubit registers with seconds-long
coherence times [25, 26, 27], generation of single photons with highly tunable
temporal wave packets and high spectral purity useful for generation of
cluster states [28], $>$ 90% outcoupling efficiencies from emitter-cavity
system into fiber [24, 29], memory enhanced quantum communication [24], strain
control for emitter tuning [30, 31], and quantum frequency conversion to the
telecom band with high indistinguishability [32]. As high purity 4H-SiC wafers
are commercially available and with several key QIP demonstrations missing,
developing a scalable angle-etch process in 4H-SiC using IBE is imperative.

In this work, we examine a novel method for fabricating suspended photonic
devices with triangular cross-sections on a wafer-scale in bulk 4H-SiC using
reactive ion beam etching (RIBE), a variation of IBE. We develop a RIBE
process to etch a variety of suspended photonic structures for a range of
angles. Next, we demonstrate the wafer-scale nature of the etch process for a
5-inch diameter wafer, which is the first such wafer-scale process
demonstrated for etching suspended structures in any bulk material. Finally,
we use photoluminesence measurements to confirm that the optical properties of
the color centers integrated into the fabricated photonic structures were
preserved, which is key to building QIP hardware with color centers as quantum
emitters.

##  2 Wafer-scale etching of 4H-SiC

Ion beam etching is a physical milling process in which inert gases such as
Argon are ionized in an RF ion source, and these ions are extracted as a
collimated beam through a series of grids and accelerated towards the
substrate to etch the target substrate. Variations of this process, reactive
ion beam etching (RIBE) and chemically assisted ion beam etching (CAIBE) have
a chemical etch component and are particularly useful for selective etching or
etching materials that are tolerant to milling. In RIBE, inert gases in the
ion source are substituted by reactive gases (SF6 and O2 for SiC), and the
reactive gas ion beam accelerated towards the target substrate generates the
etch. On the other hand, in CAIBE, inert gases are used in the ion source and
reactive gases are separately introduced into a gas ring close to the target
substrate. The accelerating inert gas ion beam, upon striking the reactive gas
molecules close to the target substrate produces reactive gas ions that
generate the etch. While both of these variations generates desirable etch
outcomes, RIBE offers superior etch mask selectivity and wafer-scale
uniformity [33], whereas CAIBE requires low maintainance for the ion source
[34].

The triangular cross-section photonic devices were nanofabricated using the
process flow as shown in Figure 1a (details in Methods). We developed a RIBE
process involving SF6 and O2 gas chemistry, while maintaining the SF6 to O2
flowrates at a ratio 4:1 (etch process parameters in Methods). The undercut in
these photonic structures was achieved by tilting the substrate holder (Figure
1b), to direct the ions underneath the nickel hard mask layer and rotation of
substrate holder was used to ensure uniform etch in all directions. The etch
angle ($\alpha$), defined as the half angle at the apex of the triangular
cross-section, varies linearly with the tilt angle of the substrate holder
($\varphi$), as shown in Figure 1c. The realized etch angle of the fabricated
devices deviates from the expected etch angle, determined by the tilt angle,
by 15∘ due to secondary fabrication processes. Figure 1d-f shows the SEM
images of the cross-section of a 1 $\mu$m waveguides etched at tilt angles
30∘, 45∘, and 60∘ respectively. The devices etched at a tilt angle of 60∘
(Figure 1f) is the highest etch angle (45∘) reported for triangular cross-
section devices fabricated in 4H-SiC, with an undercut of $>$ 1$\mu$m.

Figure 1: a) The process flow for the nanofabrication of triangular cross-
section photonics in 4H-SiC. b) Schematic of the Reactive Ion Beam Etching
(RIBE) with a tilting substrate holder, used to nanofabricate suspended
structures with triangular cross-section in bulk 4H-SiC. c) Variation of the
etch-angle of triangular structures ($\alpha$) as a function of the tilt angle
of the substrate holder ($\varphi$). d-f) SEM images of the triangular cross-
section profile of a 1 $\mu$m wide waveguide, showing etch angles for tilt
angles d) 30∘, e) 45∘, and f) 60∘.

We tested the wafer-scale nature of the RIBE process by placing a total of
eight patterned 4H-SiC samples on a 6-inch silicon substrate holder at
locations as follows: one sample each at the center, on the circumferences of
1-, 2- and 5-inch wafers, and four samples on the circumferences of a 4-inch
wafer, as shown in Figure 2a. All the eight samples were etched simultaneously
using the RIBE process developed at a tilt angle of 45∘ for a total of 70
minutes. The SEM images of the cross-section profiles of a 1 $\mu$m waveguide
on samples placed at the center and on the circumferences of 4- and 5-inch
wafers are shown in the insets of Figure 2a. We find that both the etch rate
and etch angle decreases linearly along the radius of the wafer, as shown in
Figure 2b-c. The etch rate and etch angle change linearly by 12.3% and 5%,
respectively, from the center to the circumference of a 5-inch wafer. The
variation in etch rate and etch angle among the four samples on the
circumference of a 4-inch wafer was 1.2% and 0.6% respectively. This RIBE
process has etch rate of $(25.9\pm 1.4)$ nm/min, etch angle of $(27\pm
0.8)^{\circ}$, and etch selectivity of $>$ 8:1. To our knowledge, this is the
first demonstration of a wafer-scale etch process for nanofabricating
suspended photonic structures in bulk 4H-SiC.

Figure 2: a) Illustration showing the placement of 4H-SiC samples on a 6-inch
substrate holder. The positioning of the 4H-SiC samples represent
circumferences of wafers up to 5 inches in diameter. (Insets) 4H-SiC samples
on the substrate holder and the cross-sectional profiles of the nanofabricated
1 $\mu$m waveguides from some of the illustrated chips. b-c) The achieved b)
etch rate and c) etch angle ($\alpha$) uniformity as a function of distance
from the center of the substrate holder, showing 5.4% and 2.9% variability,
respectively.

The RIBE etch performs uniformly for a range of neighboring active and passive
nanophotonic devices, as shown in Figure 3. We measured the emission
properties of the nitrogen vacancy centers integrated in the fabricated
triangular photonic devices using our 3-in-1 integrated cryogenic system [35].
The photoluminescence spectrum of a triangular cross-section ring resonator at
7 K is shown in Figure 4. The spectrum features zero-phonon lines (ZPLs)
corresponding to the naturally occuring divacancy centers (red) and the
implanted nitrogen-vacancy centers (green), indicating that the color center
optical properties are preserved upon integration into triangular cross-
section photonic devices fabricated using the developed RIBE process.

Figure 3: SEM images of the nanofabricated SiC photonic devices using the
wafer-scale angle-etch RIBE process: a) waveguides with notch couplers, b)
microring resonator, c) photonic crystal cavity, d) racetrack resonator, e)
fish-bone grating coupler, and f) microdisk resonator. SEM images in a-d) are
obtained with the nickel metal hard mask on top of the devices, while e-f)
have the nickel mask removed.

##  3 Discussion

The data in previous sections show results of what we believe is the first
wafer-scale angle etch process using ion beam etching to fabricate triangular
cross-section photonics in quantum-grade bulk 4H-SiC. Importantly, the etching
process does not interfere with the optical properties of the photonic
integrated color centers. This approach can be used to fabricate devices with
a wide range of etch angles. Over a 5-inch wafer diameter, the variation in
the etch rate and the etch angle is 5.4% and 2.9%, respectively, which can be
countered by adjustments in the lithographic mask. In comparison, silicon
carbide on insulator (SiCOI) approach quotes 1$\mu$m variation thickness
across a 4-inch wafer [15], which could potentially triple the device
thickness across the wafer if fabricated in a unified process. Previous
modeling results have shown that variation in etch angles of triangular
photonic crystal cavities shifts the resonant wavelength of the cavity without
significantly affecting the quality factor and the mode volume [19]. The
change in etch angle from the center to 2.5 inch away from the center results
in a 1.2% change in the resonant wavelength, and this can be compensated by
including a variation of 1.6% in the lattice constant within the mask design
to counter this resonant wavelength shift. For triangular waveguides, a
variation of 2.9% in the etch angle does not significantly change the coupling
efficiencies of the color center emission or the single-mode nature of these
waveguides [19, 18]. The variation in the etch rate does not play a
significant role in the triangular device fabrication because it does not
affect the shape of the device, but only the height of the undercut region.

Figure 4: Photoluminescence spectrum of an angle-etched 4H-SiC ring resonator
with integrated color centers at $T=7$ K, showing ZPLs from divacancy
($\mathrm{VV^{0}}$) centers (1070 - 1130 nm) and NV centers (1170 - 1245 nm),
marked with red and green arrows respectively. (Inset) SEM image of the
triangular cross-section ring resonator fabricated using the optimized wafer-
scale RIBE process, blue dot indicates the spot where photoluminescence
spectra were obtained from.

##  4 Methods

Prior to the lithographic process, we generated nitrogen vacancy centers in
4H-SiC through commercial (CuttingEdge Ions, LLC) implantation using 14N+ ions
at an energy of 65 keV and a dose of 1 $\times$ 1014 cm-2 [36], resulting in a
peak nitrogen concentration at a depth of $\sim$ 100 nm, as calculated by
stopping and range of ions in matter (SRIM) simulations. This was followed by
an annealing step in a 1-inch Lindberg Blue tube furnace at 1050 ∘C in
nitrogen atmosphere for 60 minutes [36], to activate the implants. The process
flow used for nanofabrication of triangular photonic devices in bulk 4H-SiC is
shown in Figure 1a. First the 4H-SiC samples were spin coated with a 350 nm
electron beam resist PMMA layer, and photonic patterns were transferred into
this layer using a 100 keV electron-beam lithography (ii-iii). Then e-beam
evaporation was used to deposit a 5 nm titanium adhesion layer and a 120 nm
nickel hard mask layer, and lift-off was used to transfer the photonic
patterns to this metal layer (iv-v). Triangular cross-section photonic devices
were etched in 4H-SiC using the Intlvac Nanoquest II ion beam etching tool and
finally the metal layers were removed by wet etching the samples in Transene’s
nickel and titanium etchants (v-vi).

The SiC chips with Ni patterns were etched using RIBE in the Intlvac Nanoquest
II, equipped with a 16 cm diameter Kaufman & Robinson RF ion source and a
tilting and rotating stage. The optimized RIBE process parameters are: beam
voltage = 300 V, accelerating voltage = 120 V, beam current = 110 mA, RF power
= 150 W, Ar/SF6/O2 flow rates = 10/10/2.5 sccm, pressure = 0.24 mTorr,
substrate temperature = 20 ∘C, etch time = 2 etch cycles of 35 minutes each
with 2 minutes cooling time between the two etch cycles. Argon gas is added to
the ion source along with SF6 and O2 to maintain the stability of beam
parameters.

##  5 Data availability

Data underlying the results presented in this article are available from the
authors upon reasonable request.

##  6 Acknowledgments

The authors are thankful for the valuable discussions with Marko Loncar, Bart
Machielse and Haig Atikian. We acknowledge support from NSF CAREER (Award
2047564) and AFOSR Young Investigator Program (Award FA9550-23-1-0266). Work
at the Molecular Foundry was supported by the Office of Science, Office of
Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-
AC02-05CH11231. Part of this study was carried out at the UC Davis Center for
Nano and Micro Manufacturing (CNM2). The authors would like to acknowledge the
valuable support provided by the CNM2 staff members Ryan Anderson and Vishal
Narang.

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# Greybody Factors Imprinted on Black Hole Ringdowns.
II. Merging Binary Black Holes

Kazumasa Okabayashi1<EMAIL_ADDRESS>Naritaka
Oshita1,2,3<EMAIL_ADDRESS>1Center for Gravitational
Physics, Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan  2The Hakubi Center
for Advanced Research, Kyoto University, Yoshida Ushinomiyacho, Sakyo-ku,
Kyoto 606-8501, Japan  3RIKEN iTHEMS, Wako, Saitama, 351-0198, Japan

###### Abstract

The spectral amplitude of the merger-ringdown gravitational wave (GW) emitted
by a comparable mass-ratio black hole merger is modeled by the greybody factor
of the remnant black hole. Our model does not include fitting parameters
except for a single overall spectral amplitude. We perform the mass-spin
inference from the SXS data without introducing fitting parameters and without
tuning the data range of each SXS template. Also, we find that the exponential
damping in the ringdown spectral amplitude can be modeled well with the
exponential damping in the greybody factor at high frequencies. Based on the
findings, we propose a conjecture that the light ring of the remnant black
hole, which sources the ringdown, forms as early as during the merger stage.
We discuss the formation of the light ring in the static binary solution as a
first step towards the understanding of how the separation of merging black
holes may affect the formation of the light ring.

††preprint: YITP-24-35, RIKEN-iTHEMS-Report-24

##  I Introduction

Modeling of gravitational-wave (GW) ringdown has been actively studied so far
as it is important to test gravity and to probe new physics in strong gravity
regimes. The most standard ringdown model is the superposed multiple
quasinormal (QN) modes. Each QN mode has a complex frequency
$\omega_{lmn}=f_{lmn}-i(\tau_{lmn})^{-1}$, where $f_{lm}$ and $\tau_{lm}$ are
real values and the subscripts $(l,m,n)$ represent the multipole mode $(l,m)$
and the overtone number $n$. The complex frequency $\omega_{lmn}$ is unique to
the spin and mass of a remnant black hole (for a review of black hole QN
modes, see e.g. Ref. [1]). Soon after the merger of two progenitor black
holes, the system is described by the Kerr solution with small perturbations.
Indeed, it was proposed that the excitation of QN modes can be seen around the
strain peak of GW signal for comparable mass-ratio mergers [2]. Then it was
recognized [3, 4, 5, 6, 7] that non-linearities, i.e., the second-order
perturbations, are also important to precisely model merger-ringdown waveforms
up to higher multipole moments as was pioneered by Ioka and Nakano in Ref.
[3]. Although ringdown and the excitation of QN modes are well understood,
there is an unavoidable issue so-called overfitting problem (see e.g., Refs.
[8, 9]) if the ringdown starts around the strain peak and several QN modes,
including the quadratic QN modes, should be taken into account in the model.
It is caused by having many fitting parameters in the ringdown model. Also,
the time-shift problem [10, 11, 12] is another main issue in the model of
superposed QN modes as we have to guess the start time of ringdown to perform
the data analysis for ringdown. To avoid these problems, one of the authors
has recently proposed a model alternative to the superposed QN modes [13], by
which the spectral amplitude of ringdown can be modeled only with the
reflectivity of the black hole light ring ${\cal R}_{lm}(\omega)$ or the black
hole greybody factor, which is a function of mode frequency $\omega$.

Figure 1:  A schematic picture describing the conjectured scenario that the
remnant light ring forms as early as in the merger phase of a BBH system.

The proposed greybody factor model for ringdown does not use individual QN-
mode frequencies and does not include fitting parameters except for the
overall amplitude. The reflectivity ${\cal R}_{lm}$ is represented by the
greybody factor $\Gamma_{lm}$, i.e., transmissivity of the light ring, as
${\cal R}_{lm}=\sqrt{1-\Gamma_{lm}}$. This model has some limitations, e.g.,
it works when the source term has its small dependence on $\omega$ at
$\omega\gtrsim f_{lm}\coloneqq f_{lm0}$, but the number of fitting parameters
is significantly reduced. The frequency region relevant to ringdown, i.e.,
$\omega\gtrsim f_{lm}(M,a)$, depends only on the remnant parameters, mass $M$
and spin $a$. On the other hand, the data analysis of ringdown in the time
domain involves an uncertainty in the start time of ringdown. In this sense,
modeling of ringdown with the greybody factor has some advantages in the
extraction of remnant parameters and in the test of the no-hair theorem of
black holes without the uncertainty in the range of time-domain data and
without overfitting that may affect data analysis. Also, the QNM filtering
[14, 15] is recently proposed as a technique to erase the excitation of QN
modes. It is an important technique as it also does not require fitting
parameters and only requires properly adjusting the start time of ringdown. We
should note that in the frequency domain, the amplitude of the merger-ringdown
part includes a small amount of the inspiral GW, which can be contamination in
the greybody factor model.

The ringdown model based on the greybody factor $\Gamma_{lm}$ was investigated
only for extreme mass-ratio mergers [13]. In this paper, we investigate the
feasibility of modeling ringdown with greybody factors for the case of
comparable mass-ratio mergers in a phenomenological manner. To this end, we
fit the reflectivity of a Kerr black hole ${\cal R}_{22}$ with the SXS’s
spectral amplitude of ringdown for the quadrupole moment. We will also fit the
greybody factor model with the SXS templates with various mass ratios and
remnant spin parameters to see the limitations of the model. We then
demonstrate that the best-fit remnant parameters obtained with the greybody
factor model are well consistent with the true remnant parameters.

We here consider GW signals sourced by comparable mass-ratio binary black hole
(BBH) mergers involving a highly non-linear phase. On the other hand, our
phenomenological model implies that the spectral amplitude of the merger-
ringdown phase can be modeled by the greybody factor based on the linear
perturbation theory. For the consistency between the two views, we conjecture
that the light ring forms at a very early stage of the merger (see Figure 1).
It is similar to the formation of the outermost common horizon in the merger
phase [16, 17, 18]. Although there are some relevant proposals in the
definition of the light ring in more general cases [19, 20, 21, 22, 23, 24,
25, 26], it is difficult to define the formation of the light ring in the
dynamical and less-symmetric spacetime. To get insight into the formation of
the light ring during a merger, we then consider a very simple solution, which
is static but still less-symmetric one, the Majumdar-Papapetrou (MP) solution
[27, 28]. We discuss a scenario that the remnant light ring forms as early as
the separation of the two equal-mass black holes is of the order of the
horizon size. Our discussion on the formation of the light ring in the
simplest MP solution might shed light on how the separation of merging two
black holes can affect the formation of the light ring. Note that a conclusive
statement on the dynamical formation of the light ring requires a careful
study of dynamical BBH solutions.

Our paper is organized as follows. In Sec. II.1, we review how we can model
the spectral amplitude of GW signals sourced by BBH mergers. We also explain
how we obtain the spectral amplitude of GW waveforms from the SXS templates.
In Sec. II.2, we introduce the greybody factor model for our merger-ringdown
phase based on the previous proposal [13] and explain the difference between
the previous model [13] and ours. In Sec. II.3, we carefully study how well
our model matches with the SXS templates at higher frequencies relevant to
merger-ringdown signals, i.e., $\omega\gtrsim f_{22}$. In Sec. II.4, we
compute the mismatch between a SXS template and our model for various assumed
remnant parameters and show that the remnant parameters leading to the least
value of the mismatch are consistent with the true remnant values. In Sec. III
, with a simple analytic solution, the MP solution, we discuss how the light
ring of the remnant black hole forms and how the size of the light ring
approaches that of the remnant one with respect to the separation of the
binary. In Sec. IV, we conclude and discuss the validity and limitations of
our model and possible extensions of our work. We use the natural units $c=1$
and $G=1$ throughout the manuscript.

##  II Modeling ringdown for a BBH merger by greybody factors

###  II.1 spectral amplitude of GWs from a BBH: a brief review

The merger-ringdown GW waveforms sourced by comparable mass-ratio mergers can
be obtained by numerical relativity. We here use SXS waveform templates [29]
to check our greybody factor model for merger-ringdown amplitudes. The SXS
collaboration provides GW waveforms in normalized strain $rh_{lm}$ for each
multipole moment. The strain data is decomposed into the real and imaginary
parts $\text{Re/Im}(h_{lm})\equiv h^{\rm(Re/Im)}_{lm}$ with various resolution
levels. We here use SXS data extracted at $r/M_{0}=100$ with their highest
available resolution, where the sum of the two Christodoulou masses $M_{0}$ is
defined at $t=0$.

A BBH merger, with the masses of the progenitors $M_{1}$ and $M_{2}$, emits a
GW signal consisting of three phases: inspiral, merger, and ringdown (Figure
2). Note that Figure 2 shows the spectral amplitude for the real part of the
strain amplitude, i.e., $rh_{22}^{\rm(Re)}$, for SXS:BBH:0305. The spectral
amplitude is computed by the following Fourier transform in the data range of
$t_{i}\leq t\leq t_{f}$:

$\tilde{h}_{22}^{\rm(Re/Im)}(\omega)=\int_{t_{i}}^{t_{f}}dte^{-i\omega
t}h_{22}^{\rm(Re/Im)}(t),$ (1)

where we set $t_{i}=100$ and $t_{f}$ is set to the maximum time available in
the SXS template. Also, we normalize the frequency in the unit of $2M=1$
throughout the manuscript. The values of a remnant mass $M$ and a remnant spin
$j$ are available in the SXS catalog. In the following, we often omit the
superscript (Re/Im). The inspiral phase contributes to the low-frequency part
of GW spectral amplitude. Based on the adiabatic analysis, an inspiral GW
waveform is approximated as

$h_{22}\sim\frac{\mu}{r}\left(\frac{1}{5}\frac{t_{c}-t}{\mu}\right)^{-1/4}\cos\left[{-2\left(\frac{1}{5}\frac{t_{c}-t}{\mu}\right)^{5/8}+\delta}\right],$
(2)

where ${\mu}\equiv(M_{1}M_{2})^{3/5}/(M_{1}+M_{2})^{1/5}$, $t_{c}$ is the time
of coalescence and $\delta$ is a phase. The amplitude of this in the frequency
domain is also analytically obtained by using the stationary phase
approximation as

$|\tilde{h}_{22}|\sim\frac{\mu^{5/6}}{r}\omega^{-7/6}.$ (3)

One can see that the spectral amplitude indeed follows
$|\tilde{h}_{22}|\propto\omega^{-7/6}$ at lower frequencies (blue dot-dashed
line in Figure 2). On the other hand, the contribution of the merger and
ringdown phase comes at higher frequencies, where there are two characteristic
features: a cut-off frequency $\omega\simeq
f_{22}\coloneqq\text{Re}(\omega_{220})$ (grey solid line in Figure 2) and an
exponential damping of spectral amplitude at $\omega\gtrsim f_{22}$. Our main
argument is that those two features in the quadrupole spectral amplitude for
BBH mergers can be modeled by the greybody factor $\Gamma_{22}$ at
$\omega\gtrsim f_{22}$, i.e., merger and ringdown part in the GW spectrum (red
dashed line in Figure 2). In the next section, we will briefly review the
greybody factor $\Gamma_{lm}$ and how the greybody factor model works for
merger-ringdown waveforms sourced by comparable mass-ratio BBH mergers.

Figure 2: GW spectral amplitude $|\tilde{h}_{22}^{\rm(Re)}|$ for SXS:BBH:0305.
We set $t_{i}=100$ and $t_{f}=4457$ in (1). The amplitude of the inspiral
phase can be approximated by $\omega^{-7/6}$ (blue dot-dashed) based on the
stationary phase approximation. Merger and ringdown phases correspond to the
high-frequency domain at $\omega\gtrsim f_{22}$. Our model based on the
greybody factor $\Gamma_{22}$ is shown with the red dashed line.

###  II.2 a ringdown model based on greybody factors

The black hole greybody factor is one of the important quantities
characterizing black holes. Given the mass, spin and charge of a black hole,
the greybody factor $\Gamma_{lm}(\omega)$ is uniquely determined like QN
modes. It quantifies the transmissivity of the black hole’s geometry for a
mode frequency $\omega$ with a multipole moment $(l,m)$. As the greybody
factor is obtained from the analysis of the scattering amplitude in the black
hole’s geometry, let us consider the Sasaki-Nakamura (SN) equation:

$\left[\frac{d^{2}}{dr^{\ast}{}^{2}}-F_{lm}\frac{d}{dr^{\ast}}-U_{lm}\right]X_{lm}=T_{lm},$
(4)

where $F_{lm}$ and $U_{lm}$ are given in the original paper [30] and $T_{lm}$
is the source term in the SN formalism. We here introduced a new variable
$r^{\ast}$ which is defined by

$r^{\ast}\equiv
r+\frac{2M}{r_{+}-r_{-}}\left[r_{+}\ln\left(\frac{r-r_{+}}{2M}\right)-r_{-}\ln\left(\frac{r-r_{-}}{2M}\right)\right].$
(5)

Note that $r^{\ast}\to-\infty$ and $r^{\ast}\to+\infty$ corresponds to the
horizon limit $r\to r_{+}$ and the far limit $r\to\infty$, respectively. As
the asymptotic behaviour of $F_{lm}$ and $U_{lm}$ in the SN equation is

$F_{lm}\to 0\ \text{for}\ r^{\ast}\to\pm\infty,$ (6)

and

$\displaystyle U_{lm}\to\begin{cases}-\omega^{2}\ &\text{for}\
r^{\ast}\to\infty,\\\ -k_{\rm H}^{2}\ &\text{for}\
r^{\ast}\to-\infty,\end{cases}$ (7)

the asymptotic form of the homogeneous solution to the SN equation
$X_{lm}^{\rm(hom)}$ is given by the linear combination of the two independent
solutions, $X_{lm}^{\rm(in)}$ and $X_{lm}^{\rm(out)}$:

$\displaystyle X_{lm}^{\rm(in)}=\begin{cases}A_{\rm
in}^{(l,m)}(\omega)e^{-i\omega r^{\ast}}+A_{\rm out}^{(l,m)}(\omega)e^{i\omega
r^{\ast}}&\text{for}\ r^{\ast}\to\infty,\\\ e^{-ik_{\rm
H}r^{\ast}}&\text{for}\ r^{\ast}\to-\infty,\end{cases}$ (8)

and

$\displaystyle X_{lm}^{\rm(out)}=\begin{cases}e^{i\omega r^{\ast}}&\text{for}\
r^{\ast}\to\infty,\\\ B_{\rm out}^{(l,m)}(\omega)e^{ik_{\rm H}r^{\ast}}+B_{\rm
in}^{(l,m)}(\omega)e^{-ik_{\rm H}r^{\ast}}&\text{for}\
r^{\ast}\to-\infty,\end{cases}$ (9)

where $k_{\rm H}\equiv\omega-m\Omega_{\rm H}$, $\Omega_{\rm H}\equiv j/2r_{+}$
and $A_{\rm in/out}^{(l,m)}$ and $B_{\rm in/out}^{(l,m)}$ are coefficients
depending on $\omega$. The greybody factor is identical to the absorption
probability of an incoming mode $\omega$. Given the coefficients $A_{\rm
in}^{(l,m)}$ and $A_{\rm out}^{(l,m)}$, the greybody factor is given by

$\Gamma_{lm}=\left|\frac{C}{c_{0}}\right|^{2}\left|\frac{A_{\rm out}}{A_{\rm
in}}\right|^{2},$ (10)

where $|C|^{2}$ and $c_{0}$ [31, 32, 33] are

$\displaystyle|C|^{2}$
$\displaystyle\equiv\lambda^{4}+4\lambda^{3}+\lambda^{2}(-40a^{2}\omega^{2}+40am\omega+4)+48a\lambda\omega(a\omega+m)+144\omega^{2}(a^{4}\omega^{2}-2a^{3}m\omega+a^{2}m^{2}+M^{2}),$
(11) $\displaystyle c_{0}$
$\displaystyle\equiv\lambda(\lambda+2)-12a\omega(a\omega-m)-i12M\omega.$ (12)

In this paper, as shown in Figure 2, we introduce a phenomenological model for
the merger-ringdown spectral amplitudes with the quadrupole moment:

$|\tilde{h}_{22}^{\rm(Re/Im)}(\omega)|=C_{\rm
amp}\sqrt{1-\Gamma_{22}(\omega)},\ \text{for}\ \omega\gtrsim f_{22},$ (13)

where a constant $C_{\rm amp}$ is an overall amplitude. For comparable mass-
ratio mergers, the greybody factor model indeed works well for various spin
parameters as is shown in Figure 3.

Figure 3: Comparison between the greybody factor model (red dashed) and the
spectral amplitude of GW from the SXS catalog (black solid). The vertical thin
lines indicate the value of $2Mf_{22}$. Mass ratio is in the range of $1\leq
M_{1}/M_{2}\leq 2$.

In the following sections, we will demonstrate that the spectral amplitude of
the merger and ringdown phase for $(l,m)=(2,2)$ can be modeled by a universal
quantity, $\Gamma_{22}(\omega)$. We also demonstrate that our greybody factor
model works to infer the remnant parameters.

###  II.3 exponential damping of the merger-ringdown spectral amplitude

We here study how well our model fits to the SXS data at higher frequencies
where the spectral amplitude is exponentially damped with $e^{-(\omega-
f_{22})/(2T^{\rm(GW)})}$. A constant $1/T^{\rm(GW)}$ is the strength of the
damping in the spectral amplitude. Our argument is that the damping amplitude
can be modeled by $C_{\rm amp}\sqrt{1-\Gamma_{22}(\omega)}$, which has a
damping $\sim e^{-(\omega-f_{22})/(2T)}$ at $\omega\gtrsim f_{22}$. Here
$C_{\rm amp}$ is an overall amplitude. We will see how $T^{\rm(GW)}$ extracted
from SXS templates is consistent with the value of $T$ in the greybody
factor.111A similar analysis for extreme mass-ratio mergers has been performed
in Ref. [34] and [13] by one of the authors.

Figure 4: (a) Exponent parameter $T^{\rm(GW)}$ of the damping amplitude
$|\tilde{h}_{22}|\sim e^{-\omega/(2T^{\rm(GW)})}$ at $\omega\geq\alpha f_{22}$
is shown with respect to the remnant spin $j$ (black markers). The constant
$\alpha$ determines the cut-off frequency, and how we determine $\alpha$ is
described in Appendix A. The mass ratio is in the range of $1\leq
M_{1}/M_{2}\leq 2$. The red solid line indicates the exponent of the damping
$T$ in $\sqrt{1-\Gamma_{22}}\propto e^{-\omega/(2T)}$ in $\omega\gtrsim
f_{22}$. (b) Exponent parameter $T^{\rm(GW)}$ are shown with respect to the
mass ratio $M_{1}/M_{2}$ (black markers). The mass ratio is in the range of
$1\leq M_{1}/M_{2}\lesssim 4$ and the spin parameter is in the range of
$0.6\leq j\leq 0.7$. The red band indicates the range of $T(j=0.7)\leq T\leq
T(j=0.6)$. In (a) and (b), we use 55 and 70 SXS templates, respectively, and
extract $T^{\rm(GW)}$ independently from each polarization $h_{+}$ and
$h_{\times}$.

The fitting function for the exponent parameter $T=T(j)$ with $2M=1$ is given
in Ref. [13]. We show the comparison between $T$ and $T^{\rm(GW)}$ in Figure
4. Figure 4-(a) shows the values of $T$ and $T_{\rm GW}$ with respect to the
remnant spin, and the mass ratio is restricted to a comparable mass ratio of
$1\leq M_{1}/M_{2}\leq 2$. On the other hand, in Figure 4-(b), the comparison
is performed with respect to the mass ratio of $1\leq M_{1}/M_{2}\lesssim 4$,
and the remnant spin is restricted to the typical values of $0.6\leq j\leq
0.7$. We normalize the spectral amplitude of SXS data with the scale of $2M=1$
and extract the exponent parameter $T^{\rm(GW)}$ by using the
NonlinearModelFit in the Mathematica. The error is estimated by Mathematica’s
option ParameterErrors which gives the standard errors for parameter
estimates.

From Figure 4-(a), we find that the exponent parameter $T^{\rm(GW)}$ extracted
from the SXS data is consistent with the exponent of the damping in
$\sqrt{1-\Gamma_{22}(\omega)}$ in the wide range of the spin parameter. From
Figure 4-(b), on the other hand, we find that the exponent of the damping in
the reflectivity agrees with $T^{\rm(GW)}$, especially for comparable mass
ratios $M_{1}/M_{2}\simeq 1$. For larger mass ratios, the exponent of
$T^{\rm(GW)}$ tends to be larger than $T$. This may be caused by the sourcing
effect of a small companion black hole. Indeed, in the previous work done by
one of the authors [13], one needs an extra factor of $1/\omega^{3}$ in the
model for the ringdown spectral amplitude for extreme mass ratios.

Our model could be applied to another model for the merger-ringdown amplitude
proposed in Refs. [35, 36]. They model the exponential damping in the
amplitude with $\tau_{lm0}\times\gamma_{3}$ where $\gamma_{3}$ is a free
parameter of order unity [35, 36]. We here do not use extra free parameters
except for the overall amplitude $C_{\rm amp}$.

###  II.4 mass-spin measurement

In this section, we show how our phenomenological model for the merger-
ringdown amplitude works to infer the remnant mass and spin from SXS’s
waveforms.

Figure 5: Mismatch between our model $C_{\rm amp}{\cal R}$ and the spectral
amplitude of the SXS’s waveforms, SXS:BBH:0305 ($j=0.6921$), SXS:BBH:0540
($j=0.8321$), and SXS:BBH:1124 ($j=0.9507$). The white lines indicate the true
value of the remnant quantities and the remnant mass $M_{\rm true}$ is
normalized with $M_{\rm true}=0.5$. We set $\omega_{i}=0.85\times
f_{22}(M,a)/(2M)$ and $\omega_{f}=1.7$ for all the three SXS templates.

To this end, we compute mismatch ${\cal M}$ between the SXS data,
$|\tilde{h}(\omega)|$, and our model for the merger-ringdown amplitude $C_{\rm
amp}{\cal R}(M,j)=C_{\rm amp}\sqrt{1-\Gamma(M,j,\omega)}$. We then find the
best-fit remnant parameters ($M,j$) for which the mismatch takes the least
value. We estimate the mismatch with the following formula

${\cal M}(M,j)=\left|1-\frac{\braket{}{\tilde{h}}{C_{\rm amp}\times{\cal
R}}}{\sqrt{\braket{}{\tilde{h}}{|\tilde{h}|}\braket{C_{\rm amp}\times{\cal
R}}{C_{\rm amp}\times{\cal
R}}}}\right|=\left|1-\frac{\braket{}{\tilde{h}}{{\cal
R}}}{\sqrt{\braket{}{\tilde{h}}{|\tilde{h}|}\braket{{\cal R}}{{\cal
R}}}}\right|,$ (14)

where $\braket{a(\omega)}{b(\omega)}$ is

$\braket{a(\omega)}{b(\omega)}=\int_{\omega_{i}}^{\omega_{f}}d\omega
a(\omega)b^{\ast}(\omega),$ (15)

and $\omega_{i}$ and $\omega_{f}$ determine the data range in the frequency
domain used to compute the mismatch. Our results are shown in Figure 5. We
here take the spectral data at $\omega\geq\omega_{i}=0.85\times
f_{22}(M,a)/(2M)$ throughout the mass-spin inference we performed for the
three SXS templates. This is one of the very important advantages of working
in the frequency domain. That is, there is less uncertainty in the range of
data in the frequency domain although in the time-domain analysis, we have
uncertainty originating from the start time of ringdown. Also, in our
analysis, there is no dependence of the mismatch ${\cal M}$ on the overall
amplitude $C_{\rm amp}$ when we consider only one multipole mode. We can see
that the true remnant values (white lines in Figure 5) are consistent with the
best-fit parameters where ${\cal M}$ takes the least value.

##  III Discussion: formation of the light ring in the static BBH

One may wonder why the greybody factor, obtained in the liner perturbation
regime, works to model the merger-ringdown waveform that may include more or
less nonlinear effects. As a possible scenario, we would argue that even if
the interior of the outermost common horizon is highly non-linear, its
exterior instantaneously relaxes to the Kerr geometry with perturbations, and
the remnant light ring forms there (see Figure 1). This conjecture may be
relevant to the previous work in Refs. [37, 38] where the authors provide some
supporting evidence by using both the post-Newtonian and black hole
perturbation techniques to describe GW emission sourced by a BBH merger
involving nonlinear collision. The relaxation of a merging BBH is complicated
as it is less symmetric and highly dynamical. There is no unique consensus on
the definition of the light ring in the dynamical spacetime although it has
been discussed how one can define it [19, 20, 21, 22, 23, 24, 25, 26].

As a first step towards the verification of our conjecture, we here consider a
binary system that is less symmetric but is static, i.e., has no dynamics.
Such a system can be described by the Majumdar-Papapetrou solution [27, 28].
For the case of less-symmetric and dynamical collisions, we need both
numerical relativity and a reasonable definition of the light ring applicable
even to dynamical systems, which is beyond the scope of our purpose of this
paper. We are interested in how the separation of two merging black holes may
affect the formation of the light ring. More concretely, we here consider the
MP solution with the two extreme black holes of mass $M/2$ and control the
separation of the binary denoted by $d$. When $d=0$, it leads to the remnant
black hole of mass $M$ which has spherical symmetry, and the light ring is
recovered. When $d\gg M$, it has no light ring enclosing the binary. At which
separation does the remnant light ring form? How does the formed light ring
approach the spherical form with respect to the separation $d$? The former
question is relevant to the formation during the merger phase and the latter
one would be relevant to the relaxation process of the light ring. It would be
useful to discuss the formation of the light ring from this point of view
although most of the facts relevant to the MP solution introduced here are
known [39, 40, 41, 42]. However, we should note that the two factors, the
separation and dynamics of a BBH, may not be independent of each other in a
realistic binary system.

The MP spacetime is a solution of the Einstein-Maxwell system, and its line
element $ds^{2}$ and the electromagnetic potential $A=A_{\mu}dx^{\mu}$ are
given by

$\displaystyle ds^{2}$
$\displaystyle=-U^{-2}dt^{2}+U^{2}(dx^{2}+dy^{2}+dz^{2}),$ (16) $\displaystyle
A$ $\displaystyle=U^{-1}dt,$ (17)

where

$\displaystyle U$ $\displaystyle\coloneqq
1+\sum_{i=1}^{N}\frac{M_{i}}{r_{i}},$ (18) $\displaystyle r_{i}$
$\displaystyle\coloneqq\sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}+(z-z_{i})^{2}}.$ (19)

The metric describes a system of $N$ extremely charged “particles”, and the
$i$-th particle is located at $(x,y,z)=(x_{i},y_{i},z_{i})$. Each mass and
charge is denoted by $M_{i}$ which is a positive real value.

In the case of $N=1$ that describes the remnant black hole, we adopt the
spherical coordinates $(r,\theta,\phi)$, in which the particle with mass
$M_{1}=M$ is located at the origin. Then, the metric is written as

$ds^{2}=-U^{-2}dt^{2}+U^{2}(dr^{2}+r^{2}d\Omega^{2}),$ (20)

where

$U=1+\frac{M}{r}.$ (21)

The metric reduces to a more familiar form of the extreme Reissner-Nordström
metric through the coordinate transformation: $r\rightarrow r-M$. Note that in
the coordinate in Eq. (20), the event horizon is located at $r=0$. Indeed, the
area at $r=0$ is not zero but $4\pi M^{2}$. This value is the same as the
horizon area of the extreme Reissner-Nordström black hole with mass $M$.
Regarding the geodesic equation, if the null particle’s energy and angular
momentum associated with the Killing vectors are denoted by $E$ and $L$,
respectively, then the radial component of the null geodesic equation reduces
to

$\frac{d^{2}r}{d\lambda^{2}}+V^{(1)}_{\rm eff}(r)=\frac{1}{b^{2}},$ (22)

where $\lambda$ is an affine parameter, $b\coloneqq L/E$ is the impact
parameter, and the effective potential $V^{(1)}_{\rm eff}(r)$ is given by

$V^{(1)}_{\rm eff}(r)=\frac{1}{r^{2}U^{4}}.$ (23)

By solving the equation $dV^{(1)}_{\rm eff}/dr=0$ outside the horizon, we
obtain the circular orbit with its radius $r=M$. The second derivative of the
effective potential $d^{2}V^{(1)}_{\rm eff}/dr^{2}$ has a negative sign at
this radius, and we find that the circular orbit is unstable. Due to the
spherical symmetry of the spacetime, the null geodesic tangent to the unstable
circular orbit generates the photon sphere with the radius $r=M$.

In the case of $N=2$ that describes a static BBH, the metric describes two
extremely charged particles, and the spacetime is static due to the balance of
the gravitational force and the electric force for each particle. We adopt the
cylindrical coordinates $(\rho,\phi,z)$, in which the extremely charged
particles are located at $z=\pm d$ on the $z$-axis. If the particles have the
identical mass $M_{1}=M_{2}=M/2$, then the metric is written as

$ds^{2}=-U^{-2}dt^{2}+U^{2}(d\rho^{2}+\rho^{2}d\phi^{2}+dz^{2}),$ (24)

where

$U=1+\frac{M}{2\sqrt{\rho^{2}+(z-d)^{2}}}+\frac{M}{2\sqrt{\rho^{2}+(z+d)^{2}}}.$
(25)

The spacetime has the apparent coordinate singularities at $z=\pm d$, and the
analytical continuation is possible at each point [43]. Actually, the points
at $z=\pm d$ are the event horizons, and the horizon area at each point is the
same as the extreme Reissner-Nordström black hole with mass $M/2$. To see this
statement, we move to the spherical coordinates $(R,\theta,\phi)$, in which
the singularity at $z=d$ is located at the origin. The coordinate
transformation is given by

$R\sin\theta=\rho,\quad R\cos\theta=z-d,$ (26)

and then, the metric becomes

$ds^{2}=-U^{-2}dt^{2}+U^{2}dr^{2}+U^{2}r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (27)

where

$U=1+\frac{M}{2R}+\frac{M}{2\sqrt{R^{2}+4dR\cos\theta+4d^{2}}}.$ (28)

Hence, we can evaluate the area at the origin ${\cal A}$ as

${\cal A}=\left.\int U^{2}r^{2}d\Omega\right|_{R=0}=\pi M^{2},$ (29)

which is identical to the area of the extreme Reissnor-Nordström black hole
with the mass of $M/2$. The area at $z=-d$ can be evaluated in the same way
and is identical to the area at $z=d$.

Figure 6:  Three-dimensional plots of closed orbits on the equatorial and
meridian plane in Majumdar-Papapetrou spacetime. We set the separation as
$d=M$ (left), $d=d_{\rm crit}$ (center), and $d=10^{-2}M$ (right). On the
meridian plane $(x=0)$, there are three types of closed orbits: an orbit
enclosing two black holes (blue solid), an 8-shaped orbit (brown solid), and
two orbits enclosing each black hole (green dashed). In the limit
$d\rightarrow 0$, the blue orbit converges to the photon sphere of the extreme
Reissner-Nordström black hole while the other orbits on the meridian plane
shrink to the origin. The top-right panel shows an enlarged view of the right
panel. On the equatorial plane $(z=0)$, the dashed blue line depicts an
unstable circular orbit. Since an unstable circular orbit on the equatorial
plane does not exist for $d>d_{\rm crit}$, there is no circular orbit in the
left figure ($d=M$). The dashed blue line on the equatorial plane converges to
the photon sphere of the extreme Reissner-Nordström black hole.  Figure 7: The
effective potential $V_{\rm eff}^{(2)}(\rho)$ of a null geodesic on the
equatorial plane for $d=M$ (dashed), $d=d_{\rm crit}$ (solid), and
$d=10^{-2}M$ (dot-dashed). For $d=d_{\rm crit}$, the effective potential has
an extreme value, indicated with a red marker, where we have the circular
photon orbit. For $d<d_{\rm crit}$, the effective potential has two extremal
values. One corresponds to a stable circular orbit and the other, i.e., the
maxima of the potential (red marker) corresponds to an unstable circular
orbit. If we take the limit $d\rightarrow 0$, the red point approaches
$\rho=M$ which corresponds to the radius of the photon sphere in the extreme
Reissner-Nordström black hole spacetime. For $d>d_{\rm crit}$ (e.g., dashed
line), there is no circular orbit on the equatorial plane.  Figure 8: The
difference between the size of the outer photon sphere on the equatorial plane
for $d>0$ and that for $d=0$ (i.e. Reissnor-Nordström black hole spacetime),
quantified by $1-\rho_{\rm out}/M$, is shown with respect to the separation
$d/M$. The blue solid line shows the analytical solution given by Eq. (32) and
the red dot-dashed line shows $(5/2)d^{2}/M^{2}$ (see Eq. (34)). The deviation
converges to zero with $\propto d^{2}/M^{2}$.

In the cylindrical coordinates of Eq.(24), we consider the two types of closed
null geodesics: (i) one on the equatorial plane ($z=0$) and (ii) another one
is on the meridian plane ($\phi={\rm const.}$). On the equatorial plane, there
are two circular orbits if half of the coordinate distance between black holes
$d$ is smaller than $d_{\rm crit}=\frac{M}{3}\sqrt{\frac{2}{3}}$. The outer
orbit approaches the circular orbit in the extreme Reissnor-Nordström black
hole spacetime, whose radius is $M$, as $d\rightarrow 0$ while the inner orbit
shrinks to the origin as $d\rightarrow 0$. On the meridian plane, there are
three types of closed orbits [39, 40, 41]. One is the orbit enclosing the two
black holes, the second one is the 8-shaped orbit, and the third ones are the
orbits enclosing each black hole. The orbit enclosing the two black holes
approaches the circular orbit in the extreme Reissnor-Nordström black hole
spacetime as $d\rightarrow 0$ while the others shrink to the origin as
$d\rightarrow 0$. These closed null orbits on the equatorial plane and the
meridian plane in terms of the coordinate distance $d$ are shown in Figure 6.
In the following, we analyze how the closed null orbits on the equatorial
plane and the meridian plane approach the circular orbit in the extreme
Reissnor-Nordström black hole spacetime in terms of the coordinate distance
$d$ instead of the proper distance. This is because the proper distance
between the two black holes on the same time slice diverges due to their
extreme charge.

On the equatorial plane, the geodesic equation is separable, and the radial
component of the null geodesic equation reduces to

$\frac{d^{2}\rho}{d\lambda^{2}}+V^{(2)}_{\rm eff}(\rho)=\frac{1}{b^{2}},$ (30)

where the effective potential is given by

$V^{(2)}_{\rm eff}(\rho)=\frac{1}{\rho^{2}U^{4}}.$ (31)

A physical circular orbit is given by a real root of $dV^{(2)}_{\rm
eff}/d\rho=0$. There is no circular orbit if the separation $2d$ is larger
than the critical value $2d_{\rm crit}=\frac{2M}{3}\sqrt{\frac{2}{3}}$. On the
other hand, for $d<d_{\rm crit}$, we find two circular orbits with the outer
radius $\rho_{\rm out}$ and the inner radius $\rho_{\rm in}$. Each radius has
an analytical expression [41, 42]:

$\displaystyle\frac{\rho_{\rm out}}{M}$
$\displaystyle=\sqrt{\frac{1}{9}\bigg{(}1+2\cos\bigg{[}\frac{1}{3}\cos^{-1}\bigg{(}1-\frac{27d^{2}}{M^{2}}\bigg{)}\bigg{]}\bigg{)}^{2}-\frac{d^{2}}{M^{2}}},$
(32) $\displaystyle\frac{\rho_{\rm in}}{M}$
$\displaystyle=\sqrt{\frac{1}{9}\bigg{(}1-2\sin\bigg{[}\frac{\pi}{6}-\frac{1}{3}\cos^{-1}\bigg{(}1-\frac{27d^{2}}{M^{2}}\bigg{)}\bigg{]}\bigg{)}^{2}-\frac{d^{2}}{M^{2}}}.$
(33)

This implies that the remnant light ring begins to form as early as the
separation of the two black holes is $\sim\mathcal{O}(M)$. We can check the
stability in the radial direction by evaluating the sign of the second
derivative of the effective potential. We then find that the outer one of the
two circular orbits is unstable and the inner one is stable, as is shown in
Figure 7. To analyze how the closed null orbits on the equatorial plane
approach the photon sphere of the extreme Reissnor-Nordström black hole
spacetime in the limit of $d\to 0$, we expand $1-\rho_{\rm out}/M$ in terms of
$d/M$ and get

$1-\frac{\rho_{\rm
out}}{M}=\frac{5}{2}\frac{d^{2}}{M^{2}}+\mathcal{O}\left(\frac{d^{4}}{M^{4}}\right).$
(34)

This shows that the outer circular orbit in the equatorial plane converges to
the remnant photon sphere with the deviation of ${\cal O}(d^{2}/M^{2})$. It
means that, at least on the equatorial plane, the common light ring starts to
form as early as the separation of the order of $M$ as is shown in Figure 8,
and it approaches the size of the remnant light ring with the deviation of
${\cal O}(d^{2}/M^{2})$. The similar behavior in the deviation is found for
the light ring on the meridian plane:

$\frac{r(\theta)}{M}=\frac{5(3+\cos
2\theta)}{12}\frac{d^{2}}{M^{2}}+\mathcal{O}\left(\frac{d^{4}}{M^{4}}\right),$
(35)

where $r(\theta)$ is the radius of the light ring and is defined by the
coordinate transformation $\rho=r\sin\theta$ and $z=r\cos\theta$. Hence, the
size of the light ring on the meridian plane also approaches the remnant light
ring with the deviation ${\cal O}(d^{2}/M^{2})$. The derivation of Eq. (35) is
given in Appendix B.

##  IV Conclusion

In this paper, we proposed a phenomenological model of ringdown amplitude for
binary black hole (BBH) mergers. Our model is based on the greybody factor
$\Gamma_{lm}(\omega)$, quantifying the absorption nature of a Kerr black hole,
and we argue that for the quadrupole moment, the spectral amplitude of
gravitational wave (GW) ringdown $|\tilde{h}_{22}(\omega)|$ can be modeled by
the greybody factor with $|\tilde{h}_{22}(\omega)|\sim C_{\rm
amp}\sqrt{1-\Gamma_{22}(\omega)}$ for $\omega\gtrsim f_{22}$ (see Figure 2),
where $f_{22}$ is the real part of the fundamental quasinormal (QN) mode
frequency. This phenomenological model is independent of another ringdown
model, i.e., superposed QN modes, which may involve many fitting parameters
and lead to the overfitting. Our model has an advantage in reducing fitting
parameters, i.e., it has the remnant mass $M$, spin $j$, and an overall
amplitude only. Also, it is known that there is an ambiguity in the start time
of ringdown and it propagates to uncertainty in the time-domain data analysis.
However, our model works in the frequency domain and the frequency region
relevant to merger and ringdown comes at $\omega\gtrsim f_{22}(M,j)$,
depending on the remnant parameters, which is less ambiguous compared with the
time-domain analysis of ringdown.

To show the validity and limitation of our model, we studied the consistency
between our model and the spectral amplitude of GW waveforms provided in the
SXS catalog. We confirmed that the exponent of the exponential damping of GW
spectral amplitude at high frequencies $\omega\gtrsim f_{22}$ is well
consistent with that of the reflectivity $\sqrt{1-\Gamma_{22}(\omega)}$ (see
Figures 3 and 4). The two exponents are well-matched with each other,
especially for comparable mass ratios. We also demonstrate the mass-spin
inference of the remnant black hole by computing the mismatch ${\cal M}$
between the greybody factor model and the SXS data. We then found that the
remnant parameters leading to the least value of ${\cal M}$ are consistent
with the true remnant values (see Figure 5).

Our model is relevant to another model for the merger-ringdown amplitude
proposed in Refs. [35, 36]. In our model, the exponential damping in the
amplitude is given by the damping in the greybody factor, but their model uses
the damping of the fundamental QN mode, $\tau_{lm0}$, with an extra free
parameter $\gamma_{3}\in[1.25,1.36]$ [35, 36]. It would be an interesting
direction to apply our greybody factor model to construct a more accurate
phenomenological model with a smaller number of parameters to contribute to
the existing model.

The greybody factor used in our phenomenological model is nothing but the
transmissivity of the light ring of the remnant Kerr black hole. In the merger
phase of a comparable mass-ratio BBH, the spacetime would be non-linearly
disturbed. The light ring of the remnant Kerr black hole should form soon
after or even during the merger. Otherwise, it is difficult to interpret how
our model works even for comparable mass-ratio mergers. We then consider the
Majumdar-Papapetrou (MP) solution as a simple analytic model and investigate
the relation between the separation distance of two charged black holes $d$
and the size of the light ring. We found that the light ring on the equatorial
plane forms at the critical distance of $d/M=d_{\rm
crit}/M\coloneqq\sqrt{2/3}/3\simeq 0.27$ and the radius of the light ring
$r_{\rm e}$ approaches to that of the remnant extremal Reisner-Nordström black
hole $r_{\rm rem}$ with $\Delta\propto d^{2}$, where $\Delta\coloneqq|r_{\rm
e}-r_{\rm rem}|$. A similar behavior is found for the outermost closed null
orbit on the meridian plane. Note that in the paper, we have investigated the
light ring only on the two planes, equatorial and meridian planes. It makes
sense that the merger-ringdown part of the GW spectrum can be modeled by the
greybody factor if the light ring forms at the early stage of the merger
phase, similar to the light ring on the equatorial plane of the MP solution,
and if the spacetime can be described by the perturbation of the Kerr solution
except for the interior of the outermost common horizon. This statement may be
relevant to the analysis performed in Refs. [37, 38]. They demonstrated that
black hole perturbation can describe the non-linear aspects of a binary merger
accessible to observers far from the collision.

Our discussion is based on the MP solution to see the formation of the light
ring in an analytic way. However, the MP solution is a static solution and has
no dynamics in it. What we could shed light on with this model is the relation
between the separation distance of two black holes and the formation of the
light ring. Defining the light ring in a dynamical spacetime would be
challenging but should be important as GW ringdown is sourced by the light
ring.

###### Acknowledgements.

N. O. is thankful for the valuable discussions with Niayesh Afshordi, Yanbei
Chen, and Sizheng Ma. K. O. is supported by JSPS KAKENHI Grant No. JP23KJ1162.
N. O. is supported by the Grant-in-Aid for Scientific Research (KAKENHI)
project for FY2023 (23K13111) and by the Hakubi project at Kyoto University.

##  Appendix A Extraction of $T^{\rm(GW)}$ from the SXS data

We here describe the methodology to extract the value of $T^{\rm(GW)}$ which
is the exponent parameter of the exponential damping in the GW spectral
amplitude at $\omega\gtrsim f_{22}$. We first perform the Fourier transform
with the formula of (1) in the frequency range of $10^{-3}\leq\omega\leq 1$
divided into $10^{3}$ bins. We then obtain the spectral amplitude as is shown
in Figure 3. We then perform the non-linear model fitting to fit the function
$\tilde{C}_{\rm amp}\exp{(-\omega/T^{\rm(GW)})}$ that has two fitting
parameters, $\tilde{C}_{\rm amp}$ and $T^{\rm(GW)}$, with the absolute square
of the spectral amplitude in the frequency range of

$\alpha f_{22}\leq\omega\leq 1.$ (36)

The value of the constant $\alpha$ we take is shown in Table 1.

spin range | cut off parameter ($\alpha$)
---|---
$0.85\leq j$ | 1.02
$0.65\leq j<0.85$ | 1.05
$j<0.65$ | 1.1
Table 1: The cut off parameter $\alpha$ we set in our analysis.

##  Appendix B The null orbit on the meridian plane in the MP solution

On the meridian plane, the geodesic equation is not separable, and we need to
numerically solve it. The null geodesic equations for $\rho$ and $z$
components are written as

$\displaystyle\ddot{\rho}-\frac{b^{2}(U+\rho\partial_{\rho}U)}{\rho^{3}U^{5}}+\frac{2\dot{\rho}\dot{z}\partial_{z}U-(1+\dot{z}^{2}-\dot{\rho}^{2})\partial_{\rho}U}{U}=0,$
(37)
$\displaystyle\ddot{z}-\frac{b^{2}\partial_{z}U}{\rho^{2}U^{5}}+\frac{2\dot{\rho}\dot{z}\partial_{\rho}U-(1-\dot{z}^{2}+\dot{\rho}^{2})\partial_{z}U}{U}=0.$
(38)

where $b$ is the impact parameter defined by $b=L/E$, and the dot means the
derivative with respect to the affine parameter $\lambda$. To consider closed
null geodesics on the meridian plane, we choose the impact parameter as $b=0$
since $L$ is the angular momentum associated with the Killing vectors and
written as $L=\rho^{2}U^{2}\dot{\phi}$. Thus, we consider the following
geodesic equation to find a closed null orbit,

$\displaystyle\ddot{\rho}+\frac{2\dot{\rho}\dot{z}\partial_{z}U-(1+\dot{z}^{2}-\dot{\rho}^{2})\partial_{\rho}U}{U}=0,$
(39)
$\displaystyle\ddot{z}+\frac{2\dot{\rho}\dot{z}\partial_{\rho}U-(1-\dot{z}^{2}+\dot{\rho}^{2})\partial_{z}U}{U}=0.$
(40)

In addition to these equations, we have the null condition:

$\dot{\rho}^{2}+\dot{z}^{2}=1.$ (41)

If we start to solve the geodesic equations from the $z$-axis at $\lambda=0$,
the initial conditions are chosen as $(\rho,z)|_{\lambda=0}=(0,z_{0})$,
$\dot{z}(0)=0$ where $z_{0}$ is a coordinate value of the $z$-axis. The
initial condition for the derivative of $\rho$ is determined by the null
condition Eq. (41), and $\dot{\rho}(0)=1$ is obtained. To obtain the closed
orbit, we solve the geodesic equations by using the shooting method with the
parameter $z_{0}$. As a result, three types of closed orbits are obtained: the
orbit enclosing the two black holes, the 8-shaped orbit, and the orbit
enclosing the individual black hole [39, 40, 41]. Notice that each orbit
always exists and is unstable as far as we have numerically confirmed. Figure
9 shows several examples of the three closed orbits. The outermost closed
orbit (blue solid) converges to the circular orbit on the photon sphere in the
remnant extreme Reissner-Nordström spacetime (red dotted) and the others
(brown solid and green dashed) converge to the origin as the separation
distance $d$ decreases. Hence, the outermost closed orbit is relevant to the
remnant photon sphere.

Figure 9:  The closed null orbits on the meridian plane is shown for $d=M$
(left), $d=d_{\rm crit}$ (center), and $d=10^{-2}M$ (right). As a reference,
the circular orbit of the extreme Reissner-Nordström solution is depicted as a
red dashed line. In the limit of $d/M\to 0$, the 8-shaped orbit and the orbits
enclosing an individual hole shrink to the origin. However, as seen in the
top-right figure, these orbits do not disappear even for smaller values of
$d/M$ since the spacetime is static. On the other hand, the light ring
enclosing the two holes converges to the remnant light ring.

To analyze how the outermost closed null orbit on the meridian plane
approaches the circular orbit in the extreme Reissnor-Nordström black hole
spacetime, we focus on null geodesics on the hypersurface described by
$\phi={\rm const.}$ Hereafter, this hypersurface is denoted by
$\Sigma_{\phi}$. The induced metric on $\Sigma_{\phi}$ is given by

$ds^{2}=-U^{-2}dt^{2}+U^{2}(d\rho^{2}+dz^{2}).$ (42)

To study the outermost closed null orbit, we adopt the spherical coordinate
$(t,r,\theta)$ through the coordinate transformation on $\Sigma_{\phi}$:
$\rho=r\sin\theta$ and $z=r\cos\theta$. Then, the induced metric is written as

$ds^{2}=-\tilde{U}^{-2}(r,\theta)dt^{2}+\tilde{U}^{2}(r,\theta)(dr^{2}+r^{2}d\theta^{2}),$
(43)

where

$\tilde{U}(r,\theta)\coloneqq
1+\frac{M}{2\sqrt{r^{2}-2\,d\,r\cos\theta+d^{2}}}+\frac{M}{2\sqrt{r^{2}+2\,d\,r\cos\theta+d^{2}}}.$
(44)

The geodesic equations on the hypersurface $\Sigma_{\phi}$ are given by

$\displaystyle\ddot{r}-\frac{1}{\tilde{U}}\Big{[}r\dot{\theta}^{2}\tilde{U}-2\dot{r}\dot{\theta}\tilde{U}_{,\theta}+(1-\dot{r}^{2}+r^{2}\dot{\theta}^{2})\tilde{U}_{,r}\Big{]}=0,$
(45)
$\displaystyle\ddot{\theta}-\frac{1}{r^{2}\tilde{U}}\Big{[}-2r\dot{r}\dot{\theta}\tilde{U}-2r^{2}\dot{r}\dot{\theta}\tilde{U}_{,r}+(1+\dot{r}^{2}-r^{2}\dot{\theta}^{2})\tilde{U}_{,\theta}\Big{]}=0,$
(46)

and the null condition is given by

$\dot{r}^{2}+r^{2}\dot{\theta}^{2}=1.$ (47)

Since we focus on the outermost closed null orbit, it is useful to use the
coordinate $\theta$ as a parameter of the orbit instead of the affine
parameter $\lambda$.

Figure 10: The difference between the size of the photon sphere on the
meridian plane for $d>0$ and that for $d=0$ (i.e. Reissnor-Nordström black
hole spacetime), quantified by $1-r(\theta)/M$, is shown with respect to the
separation $d/M$. We show the approximated values
$1-r(\theta)/M\simeq(5/12)(3+\cos 2\theta)(d^{2}/M^{2})$ for $\theta=0$
(solid) and $\theta=\pi/2$ (dashed). The red markers show the values obtained
numerically which agrees with the approximated formula in Eq. (35).

After rewriting the above equations, we obtain the following equation:

$r^{\prime\prime}-\frac{2r^{\prime 2}}{r}-r+2(r^{2}+r^{\prime
2})\left[\frac{r^{\prime}}{r^{2}}\frac{U_{,\theta}}{U}-\frac{U_{,r}}{U}\right]=0,$
(48)

where the prime indicates the derivative with respect to $\theta$. The
relation between $\lambda$ and $\theta$ is determined by the null condition
Eq. (47). Obtaining the analytical solution of the closed orbit from the
geodesic equation is difficult in general, but we can iteratively solve the
geodesic equation Eq. (48) for the sufficiently small coordinate distance
$d/M$. Since we know that the outermost closed orbit approaches the circular
orbit with the radius $M$ from the numerical result, we can expand the
function $r(\theta)$ by using dimensionless functions
$\tilde{r}_{i}(\theta)~{}(i=1,2,3)$ as

$\frac{r(\theta)}{M}=1+\tilde{r}_{1}(\theta)\frac{d}{M}+\tilde{r}_{2}(\theta)\frac{d^{2}}{M^{2}}+\tilde{r}_{3}(\theta)\frac{d^{3}}{M^{3}}+\mathcal{O}\left(\frac{d^{4}}{M^{4}}\right).$
(49)

Substituting Eq. (49) into the geodesic equation Eq. (48), we obtain
differential equations at each order of $d/M$. At the first order and the
third order, we obtain the solutions
$\tilde{r}_{1}(\theta)=\tilde{r}_{3}(\theta)=0$ by imposing the periodicity of
a closed orbit, i.e., $r(0)=r(2\pi)$. At the second order, the above equation
Eq. (48) can be expanded as

$\frac{d^{2}\tilde{r}_{2}}{d\theta^{2}}-\frac{\tilde{r}_{2}}{2}-\frac{5(1-3\cos^{2}\theta)}{4}=0.$
(50)

This can be solved analytically, and the solution is given by,

$\tilde{r}_{2}=\frac{5(3+\cos 2\theta)}{12}.$ (51)

As a result, the radius $r(\theta)$ of the outermost closed null orbit is
given by

$\frac{r(\theta)}{M}=1+\frac{5(3+\cos
2\theta)}{12}\frac{d^{2}}{M^{2}}+\mathcal{O}\left(\frac{d^{4}}{M^{4}}\right),$
(52)

for $d/M\ll 1$. This shows that the outermost circular orbit enclosing two
black holes on the meridian plane converges to the remnant photon sphere of
the extreme Reissnor-Nordström spacetime with the deviation of ${\cal
O}(d^{2}/M^{2})$. As shown in Figure 10, the numerical results of the radius
agree with the approximated solution Eq. (52) in the limit of $d/M\to 0$. This
does not change our statement concluded from the analysis on the equatorial
plane, i.e., that the common light ring may form as early as the separation of
the two black holes is $\sim{\cal O}(M)$.

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Y_{3}+Y_{3}\circ d(\alpha).\end{split}$ (220)

Again, by applying Proposition 2.1, we obtain that the identity above holds
for all $\beta$ in $dom(d_{min})$.
Then, in $L^{q,p}$-cohomology, $T_{f}$ does not depend on the choice of the
smooth Lipschitz approximation of $f$.
Point 4. Let us consider $f^{\prime}$ and $F^{\prime}$ two differentiable maps
which are Lipschitz-homotopic to $f$ and $F$. Let $H$ be a smooth Lipschitz
homotopy between $f^{\prime}$ and $F^{\prime}$. Then we can conclude as the
previous point.
Point 5. Thanks to the previous points, we can consider $f$ and $g$ as smooth
maps. Let us define the vector bundles $pr_{S}:g^{*}T^{\sigma}M\longrightarrow
S$, $pr_{S2}:(f\circ g)^{*}T^{\delta}N\longrightarrow S$. Moreover denote by
$\overline{g}:g^{*}T^{\sigma}M\longrightarrow M$ the map defined as
$\overline{g}:=g\circ pr_{S}$. We know that

$(f\circ\overline{g})^{*}TN=pr_{S2}^{*}g^{*}(TM)=pr_{S}^{*}(f\circ
g)^{*}TN=(f\circ g)^{*}TN\oplus g^{*}TM$ (221)

as vector bundles over $S$. Then, we obtain the following diagram

$\textstyle{(f\circ\overline{g})^{*}T^{\delta}N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{PR_{S}}$$\scriptstyle{PR_{S2}}$$\scriptstyle{P_{g}}$$\textstyle{(f\circ
g)^{*}T^{\delta}N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{S2}}$$\textstyle{g^{*}T^{\sigma}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{S}}$$\scriptstyle{p_{g}}$$\textstyle{f^{*}T^{\delta}N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{M}}$$\scriptstyle{p_{f}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{N}$
(222)

where $PR_{S}$, $PR_{S2}$ and $P_{g}$ are the bundle maps induced by $pr_{S}$,
$pr_{S2}$ and by $p_{g}$. In particular

$pr_{S2}\circ PR_{S}=pr_{S}\circ PR_{S2}.$ (223)

Observe that on $(f\circ\overline{g})^{*}TN=pr_{S2}^{*}g^{*}(TM)$ there is the
Sasaki metric $g_{1}$ induced by the Sasaki metric on $(f\circ g)^{*}TN$, the
pullback bundle metric of $g^{*}(TM)$ induced by the metric on $M$ and the
pullback connection of $\nabla^{LC}_{M}$.
Moreover if we consider $(f\circ\overline{g})^{*}TN=pr_{S}^{*}(f\circ
g)^{*}TN$, then we can define the Sasaki metric $g_{2}$ induced by the Sasaki
metric on $g^{*}(TM)$, the pullback bundle metric of $(f\circ g)^{*}TN$
induced by the metric on $N$ and the pullback connection of $\nabla^{LC}_{N}$.
In order to prove this equality between metrics, let us fix some normal
coordinates $\\{x^{i}\\}$ around a point $s$ in $S$, $\\{y^{j}\\}$ around
$g(s)$ and $\\{z^{k}\\}$ around $f\circ g(s)$. Then we consider on
$(f\circ\overline{g})^{*}TN$ the fibered coordinates
$\\{x^{i},\mu^{j},\sigma^{k}\\}$ related to the coordinates $\\{x^{i}\\}$, the
frame $\\{\frac{\partial}{\partial y^{j}}\\}$ and the frame
$\\{\frac{\partial}{\partial z^{k}}\\}$. Then by the formula 89, we obtain
that the matrices related to $g_{1}$ and $g_{2}$ on a point $(0,\mu,\sigma)$
are both the identity. So $g_{1}=g_{2}$.
This means that, with respect to the metric $g_{1}$, the maps $PR_{S}$,
$PR_{S_{2}}$ are R.N.-Lipschitz. Moreover, since $P_{g}\sim\overline{g}$, we
also have that $P_{g}$ is R.N.-Lipschitz. Finally, because of Proposition 3.4,
we also have that $PR_{S\star}$ and $PR_{S2\star}$ are both
$\mathcal{L}^{*}$-bounded oprators.
Consider the submersions $p_{f}:f^{*}T^{\delta}N\longrightarrow N$,
$p_{g}:g^{*}T^{\sigma}M\longrightarrow M$, $p_{f\circ g}:(f\circ
g)^{*}T^{\delta}N\longrightarrow N$ related to $f$, $g$, and $f\circ g$. Then,
as a consequence of the previous Proposition, we obtain for every $\alpha$ in
$\Omega^{*}_{c}(N)$

$(P_{g})^{*}\circ p_{f}^{*}-\overline{p}^{*}_{f\circ g}(\alpha)=d\circ
K_{2}+K_{2}\circ d(\alpha).$ (224)

Let us denote by $\omega$ and $\omega^{\prime}$ some Thom forms of $TN$ and of
$TM$ with uniformly bounded pointwise norms and support contained in
$T^{\delta}N$ and $T^{\sigma}M$.
Let us denote by $FG:(f\circ g)^{*}T^{\delta}N\longrightarrow T^{\delta}N$ the
bundle map induced by $f\circ g$. Moreover we also denote by
$F:f^{*}T^{\delta}N\longrightarrow T^{\delta}N$ and
$G:g^{*}T^{\sigma}M\longrightarrow T^{\sigma}M$ the bundle maps induced by $f$
and $g$.
Then

$\begin{split}T_{f\circ g}&=pr_{S2\star}\circ e_{(FG)^{*}\omega}\circ
p_{f\circ g}^{*}\\\ &=pr_{S2\star}\circ id_{(f\circ g)^{*}T^{\delta}N}\circ
e_{(FG)^{*}\omega}\circ p_{f\circ g}^{*}\\\ &=pr_{S2\star}\circ
PR_{S,\star}\circ e_{PR_{S2}^{*}G^{*}\omega^{\prime}}\circ PR_{S}^{*}\circ
e_{(FG)^{*}\omega}\circ p_{f\circ g}^{*}\\\ &=pr_{S2\star}\circ
PR_{S,\star}\circ e_{PR_{S2}^{*}G^{*}\omega^{\prime}}\circ
e_{PR_{S}^{*}(FG)^{*}\omega}\circ PR_{S}^{*}\circ p_{f\circ
g}^{*},\end{split}$ (225)

Observe that $p_{f\circ\overline{g}}=p_{f\circ g}\circ PR_{S}$, and so

$T_{f\circ g}=pr_{S2\star}\circ PR_{S,\star}\circ
e_{PR_{S2}^{*}G^{*}\omega^{\prime}}\circ e_{PR_{S}^{*}(FG)^{*}\omega}\circ
p_{f\circ\overline{g}}^{*}.$ (226)

Now we will focus on $T_{g}\circ T_{f}$. Observe that

$T_{g}\circ T_{f}=pr_{S\star}\circ e_{G^{*}\omega^{\prime}}\circ
p_{g}^{*}\circ pr_{M\star}\circ e_{F^{*}\omega}\circ p_{f}^{*}.$ (227)

It is possible to apply the Proposition VIII of Chapter 5 in Conn  to the
fiber bundles
$((f\circ\overline{g})^{*}T^{\delta}N,PR_{S2},g^{*}T^{\sigma}M,B^{\delta})$
and $(T^{\sigma}M,pr_{M},M,B^{\sigma})$ and the bundle morphism $P_{g}$
induced by $p_{g}$. We obtain that $p_{g}^{*}\circ
pr_{M\star}=PR_{S2,\star}\circ P_{g}^{*}$. Moreover $P_{g}^{*}\circ
e_{F^{*}\omega}=e_{P_{g}^{*}F^{*}\omega}\circ P_{g}^{*}$. This means

$\begin{split}T_{g}\circ T_{f}&=pr_{S\star}\circ e_{G^{*}\omega^{\prime}}\circ
p_{g}^{*}\circ pr_{M\star}\circ e_{F^{*}\omega}\circ p_{f}^{*}\\\
&=pr_{S\star}\circ e_{G^{*}\omega^{\prime}}\circ PR_{S2,\star}\circ
e_{P_{g}^{*}F^{*}\omega}\circ P_{g}^{*}\circ p_{f}^{*}\\\ &=pr_{S\star}\circ
PR_{S2,\star}\circ e_{PR_{S2}^{*}G^{*}\omega^{\prime}}\circ
e_{P_{g}^{*}F^{*}\omega}\circ P_{g}^{*}\circ p_{f}^{*}\end{split}$ (228)

Because of (223) we obtain $pr_{S\star}\circ PR_{S2,\star}=pr_{S2\star}\circ
PR_{S,\star}.$
Moreover $PR_{S}^{*}(FG)^{*}\omega$ is the pullback of $\omega$ along the
bundle map induced by $f\circ g\circ pr_{S2}$, but it can be also seen as the
pullback of $\omega$ along the bundle map induced by $f\circ g\circ
pr_{S}=f\circ\overline{g}$.
On the other hand $P_{g}^{*}F^{*}\omega$ is the pullback of $\omega$ along the
bundle map induced by $f\circ p_{g}$. Then, by Point 2 and since
$f\circ\overline{g}$ and $f\circ p_{g}$ are uniformly homotopy equivalent,
there is an $\mathcal{L}^{*}$-bounded operator $K$ such that

$e_{PR_{S}^{*}(FG)^{*}\omega}-e_{P_{g}^{*}F^{*}\omega}(\alpha)=d\circ K-K\circ
d(\alpha)$ (229)

and $K\beta$ has compact support if $\beta$ has compact support.
This means that, on $\Omega_{c}^{*}(N)$, because of Proposition 4.15, we have

$\begin{split}T_{f\circ g}-T_{g}\circ T_{f}&=pr_{S2\star}\circ
PR_{S,\star}\circ e_{PR_{S2}^{*}G^{*}\omega^{\prime}}\circ[dK-Kd]\circ
p_{f\circ\overline{g}}^{*}\\\ &+pr_{S2\star}\circ PR_{S,\star}\circ
e_{PR_{S2}^{*}\circ G^{*}\omega^{\prime}}\circ
e_{PR_{S}^{*}(FG)^{*}\omega}\circ[K_{2}\circ d+d\circ K_{2}]\\\ &=d\circ
Q-Q\circ d+d\circ W+W\circ d\end{split}$ (230)

where $Q:=pr_{S2\star}\circ PR_{S,\star}\circ
e_{PR_{S2}^{*}G^{*}\omega^{\prime}}\circ K\circ p_{f\circ\overline{g}}^{*}$
and $W:=pr_{S2\star}\circ PR_{S,\star}\circ
e_{PR_{S2}^{*}G^{*}\omega^{\prime}}\circ e_{PR_{S}^{*}(FG)^{*}\omega}\circ
K_{2}.$ Observe that $Q$ and $W$ are $\mathcal{L}^{*}$-bounded operators.
Moreover $W(\Omega_{c}^{*}(N))\subseteq\Omega^{*}_{c}(S)$ and
$Q(\Omega_{c}^{*}(N))\subseteq\Omega^{*}_{c}(S)$. Because of Proposition 2.1,
$W(dom(d_{p,q}))\subseteq dom(d_{p,q})$ and $Q(dom(d_{p,q}))\subseteq
dom(d_{p,q})$ and so, in $L^{q,p}$-cohomology,

$T_{f\circ g}=T_{g}\circ T_{f}.$ (231)

Point 6. In order to prove this statement we have to observe that, by Lemma
3.5, $p_{f}=p_{id}\circ F$. Let us consider a form
$\alpha\in\Omega_{c}^{*}(N)$:

$T_{f}\alpha=pr_{M\star}\circ e_{\omega}\circ F^{*}\circ
p_{id}^{*}\alpha=pr_{M\star}\circ F^{*}\circ e_{\omega}\circ
p_{id}^{*}\alpha.$ (232)

Now, thanks to the Proposition VIII of Chapter 5 of Conn ,

$\begin{split}T_{f}\alpha&=pr_{M\star}\circ F^{*}\circ e_{\omega}\circ
p_{id}^{*}\alpha\\\ &=f^{*}\circ pr_{N\star}\circ e_{\omega}\circ
p_{id}^{*}\alpha=f^{*}\circ T_{id_{N}}\alpha\end{split}$ (233)

Since $f$ is R.-N.-Lipschitz, $f^{*}$ is $\mathcal{L}^{*}$-bounded. Then (233)
implies $T_{f}=f^{*}\circ T_{id_{N}}$. So on $dom(d_{min})$ the following
holds

$f^{*}-T_{f}=f^{*}\circ(1-T_{id_{N}})=f^{*}\circ(d\circ Y_{1}+Y_{1}\circ d).$
(234)

Observe that $f^{*}(\Omega_{c}^{*}(N))\subseteq\Omega_{c}^{*}(M)$ since $f$ is
proper. Moreover $f^{*}d\alpha=df^{*}\alpha$ for all smooth form $\alpha$.
Then on $dom(d_{min})$

$f^{*}-T_{f}=f^{*}\circ(d\circ Y_{1}+Y_{1}\circ d)=d\circ W+W\circ d$ (235)

where $W=f^{*}\circ Y_{1}.$ And so in $L^{q,p}$-cohomology

$f^{*}=T_{f}.$ (236)

In order to conclude the proof we have to show that all the identities above
also hold in reduced $L^{q,p}$-cohomology and in $L^{q,p}$-quotient
cohomology. To this end observe that if $Q=dZ\pm Zd$ as operator on $dom(d)$,
and $Z$ is an $\mathcal{L}^{*}$-bounded operator then, in reduced $L^{q,p}$
cohomology, $Q$ is the null operator. This is a consequence of Corollary 2.2.

######  Corollary 4.18.

Let $(M,g)$ and $(N,h)$ be two oriented manifolds of bounded geometry. Let
$f:(M,g)\longrightarrow(N,h)$ be a uniform homotopy equivalence. Then the
(reduced or not) $L^{q,p}$-cohomologies and the $L^{q,p}$-quotient cohomology
groups are isomorphic.

Proof: Since $f$ and its homotopy inverse $g$ are uniform homotopy
equivalences, then they are uniform map. Moreover also $g\circ f$ and $f\circ
g$ are uniformly homotopic to the identities. Then

$1_{H^{*}_{q,p}(M)}=T_{id_{M}}=T_{g\circ f}=T_{f}\circ T_{g}\mbox{ and
}1_{H^{*}_{q,p}(N)}=T_{id_{N}}=T_{f\circ g}=T_{g}\circ T_{f}.$ (237)

The reduced case and the quotient cohomology case can be proved exactly with
the same argument.

###  4.7 Consequence: uniform homotopy invariance of $L^{2}$-index of
signature operator

A consequence of this work is the uniform homotopy invariance, for manifolds
of bounded geometry, of the index of the $L^{2}$-version of the signature
operator $d+d^{*}$ defined by Bei in page 20 of Bei3 . Fix $(M,g)$ an oriented
manifold of bounded geometry and let $dim(M)=4k$ for some positive $k$ in
$\mathbb{N}$. Let us consider

$\begin{split}\langle\cdot,\cdot:\rangle_{M}&\overline{H}^{i}_{2}(M,\mathbb{R})\times\overline{H}^{i}_{2}(M,\mathbb{R})\longrightarrow\mathbb{R}\\\
&([\eta],[\omega])\longrightarrow\int_{M}\eta\wedge\omega\end{split}$ (238)

where $\overline{H}^{i}_{2}(M,\mathbb{R})$ denotes the $i$-th group of reduced
$L^{2}$-cohomology defined by using forms with value in $\mathbb{R}$. This is
a well-defined and non-degenerate pairing: the proof is exactly the same given
by Bei in Proposition 4.1 of Bei3 . In particular, if $i=2k$ we obtain a
symmetric bilinear form. Then, if $\overline{H}^{2n}_{2}(M,\mathbb{R})$ is
finite dimensional, we can denote by $\sigma_{M}$ the signature of the
pairing.
Let us consider the signature operator $d_{M}+d^{*}_{M}$ where
$d_{M}:=d_{min,M}=d_{max,M}$ and let $d^{*}_{M}$ be the adjoint of $d_{M}$. In
page 20 of Bei3 , the index of the signature operator
$ind((d_{M}+d_{M}^{*})^{+})$ is defined and, in Theorem 4.2, the author proves

$\sigma_{M}=ind((d_{M}+d_{M}^{*})^{+}).$ (239)

######  Proposition 4.19.

Let $(M,g)$ and $(N,h)$ be two oriented manifolds of bounded geometry and let
$dim(M)=dim(N)=4k$ for some positive $k$ in $\mathbb{N}$. Let
$f:(M,g)\longrightarrow(N,h)$ be a uniform homotopy equivalence which
preserves the orientations and assume that
$\overline{H}^{2k}_{2}(M,\mathbb{R})\cong\overline{H}^{2k}_{2}(N,\mathbb{R})$
are finite-dimensional. Then

$ind((d_{M}+d_{M}^{*})^{+})=ind((d_{N}+d_{N}^{*})^{+})$ (240)

Proof: In order to prove the statement it is sufficient to prove that $T_{f}$
induces an isometry between $\overline{H}^{2n}_{2}(M,\mathbb{R})$ and
$\overline{H}^{2n}_{2}(N,\mathbb{R})$ which respects the bilinear form defined
in (238).
Let us consider $[\eta]$ and $[\omega]$ two classes in
$\overline{H}^{2n}_{2}(N,\mathbb{R})$. Thanks to Theorem 12.7 of Gol , we can
suppose that $\eta$ and $\omega$ are smooth forms. Observe that, since the
operator $Y_{1}$ in point 1 of Proposition 4.17 is $\mathcal{L}^{*}$-bounded,
$T_{id}\eta=\eta+d\alpha$ and $T_{id}\omega=\omega+d\beta$ where $\alpha$ and
$\beta$ are smooth forms. We obtain that

$\begin{split}\langle[T_{f}\eta],[T_{f}\omega]\rangle_{M}&=\int_{M}T_{f}\eta\wedge
T_{f}\omega\\\ &=\int_{M}f^{*}(T_{id}\eta)\wedge
f^{*}(T_{id}\omega)=\int_{M}f^{*}(T_{id}\eta\wedge T_{id}\omega)\\\
&=deg(f)\cdot\int_{N}T_{id}\eta\wedge
T_{id}\omega=\int_{N}(\eta+d\alpha)\wedge(\omega+d\beta)\\\
&=\int_{N}\eta\wedge\omega+\int_{N}d(\alpha\wedge\omega)+\int_{N}d(\eta\wedge\beta)+\int_{N}d(\alpha\wedge
d\beta)\\\
&=\int_{N}\eta\wedge\omega+0=\langle[\eta],[\omega]\rangle_{N}.\end{split}$
(241)

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