# Landscape of stellar-mass black-hole spectroscopy
with third-generation gravitational-wave detectors

Swetha Bhagwat<EMAIL_ADDRESS>Institute for Gravitational Wave
Astronomy $\&$ School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham B15 2TT, UK  Costantino Pacilio
<EMAIL_ADDRESS>Dipartimento di Fisica “G. Occhialini”,
Università degli Studi di Milano-Bicocca, Piazza della Scienza 3, 20126
Milano, Italy  INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126
Milano, Italy  Paolo Pani<EMAIL_ADDRESS>Dipartimento di Fisica,
Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy  INFN,
Sezione di Roma, Piazzale Aldo Moro 2, 00185, Roma, Italy  Michela Mapelli
<EMAIL_ADDRESS>Physics and Astronomy Department Galileo Galilei,
University of Padova, Vicolo dell’Osservatorio 3, I–35122, Padova, Italy  INFN
- Padova, Via Marzolo 8, I–35131 Padova, Italy  INAF - Osservatorio
Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy

###### Abstract

Gravitational-wave black-hole spectroscopy provides a unique opportunity to
test the strong-field regime of gravity and the nature of the final object
formed in the aftermath of a merger. Here we investigate the prospects for
black-hole spectroscopy with third-generation gravitational-wave detectors, in
particular the Einstein Telescope in different configurations, possibly in
combination with Cosmic Explorer. Using a state-of-the-art population model
for stellar-origin binary black holes informed by LIGO-Virgo-KAGRA data, we
compute the average number of expected events for precision black-hole
spectroscopy using a Fisher-matrix analysis. We find that Einstein Telescope
will measure two independent quasinormal modes within ${\cal O}(1)\%$ (resp.
${\cal O}(10)\%$) relative uncertainty for at least ${\cal O}(1)$ (resp.
${\cal O}(500)$) events per year, with similar performances in the case of a
single triangular configuration or two L-shaped detectors with same arm
length. A 15-km arm-length configuration would improve rates by roughly a
factor of two relative to a 10-km arm-length configuration. When operating in
synergy with Cosmic Explorer the rates will improve significantly, reaching
few-percent accuracy for ${\cal O}(100)$ events per year.

††preprint: ET-0106A-23

##  I Introduction

When a distorted black hole (BH) forms in a binary BH merger, it relaxes
emitting gravitational waves (GWs). Sufficiently later after the merger, the
evolution of the remnant spacetime can be described by perturbation theory and
the GW signal emitted during this phase is called the ringdown Chandrasekhar
and Detweiler (1975); Vishveshwara (1970).

At asymptotic infinity, the ringdown signal can be analytically approximated
as a linear superposition of countably infinite damped sinusoids corresponding
to a characteristic frequency and damping time spectra called the quasinormal
modes (QNMs) (see Kokkotas and Schmidt (1999); Berti  _et al._ (2009);
Konoplya and Zhidenko (2011) for some reviews). BH spectroscopy Berti  _et
al._ (2006); Gossan  _et al._ (2012); Dreyer  _et al._ (2004) –namely the
measurement of the frequencies and damping times in a ringdown signal – allows
us to perform multiple null tests of the General theory of Relativity (GR) and
of the nature of the BHs Nollert (1999); Berti  _et al._ (2018); Li  _et al._
(2012); Maselli  _et al._ (2020); Völkel and Barausse (2020); Carullo  _et
al._ (2018); Cardoso and Pani (2019).

GR predicts that the QNM spectrum of the remnant BH can be fully parameterized
by its mass and (dimensionless) spin. The measurement of the parameters of a
single complex QNM can be inverted to obtain an estimate of the mass and spin
of the BH, whereas with the measurement of additional QNMs one can perform
consistency tests for the Kerr nature of the source, including tests of the
no-hair and area theorems Carter (1971); Isi  _et al._ (2021).

Even in those cases in which the remnant is a Kerr BH, and therefore the QNM
spectrum is consistent with the GR prediction, the ringdown amplitudes and
phases can be used to test GR and the nature of the progenitor binary Forteza
_et al._ (2023).

Since the onset of GW observations in 2015, significant effort has been
devoted to measuring the ringdown in the LIGO-Virgo-KAGRA merger events. The
fundamental QNM has been measured for several events Abbott  _et al._ (2021a),
most notably in GW150914 Abbott  _et al._ (2016); Ghosh  _et al._ (2021,
2016); Abbott  _et al._ (2019, 2021b, 2021a). The inferred values of the final
mass and spin from the ringdown and those estimated from the inspiral-merger
part of the signals Abbott  _et al._ (2021a) have been found to be consistent.
There has been no unambiguous and confident measurements of any secondary QNM
parameters in the events detected thus far, although exciting hints of
overtones have been found in GW150914 Isi  _et al._ (2019); Finch and Moore
(2022) and the measurement of a secondary angular QNM has been reported in
GW190814, GW190412, and GW190521 Capano and Nitz (2020); Capano  _et al._
(2021, 2022). However, BH spectroscopy with overtones has potential
limitations related to the resolvability of the QNMs Bhagwat  _et al._ (2020);
Ota and Chirenti (2020), the number of overtones that need to be included for
an unbiased parameter recovery Isi  _et al._ (2019); Giesler  _et al._ (2019),
the physical interpretation of the measurement including the risk of over-
fitting Cotesta  _et al._ (2022); Baibhav  _et al._ (2023), and sensitivity to
the choice of the start time of the ringdown Cotesta  _et al._ (2022); Bhagwat
_et al._ (2020); Jiménez Forteza  _et al._ (2020); Ota and Chirenti (2020);
Nee  _et al._ (2023). Furthermore, merger events detected by current
interferometers are expected to have a low signal-to-noise ratio (SNR) in the
secondary angular mode Capano  _et al._ (2021); Berti  _et al._ (2016),
limiting the constraining power of such tests.

Next-generation GW detectors have the potential to perform _precision BH
spectroscopy_ , allowing for sub-percentage accuracy tests. In an earlier
paper Bhagwat  _et al._ (2022), we discussed the landscape of BH spectroscopy
using supermassive BH binary mergers detectable with the future space mission
LISA Amaro-Seoane  _et al._ (2017) and with third-generation ground-based
detectors such as the Einstein Telescope (ET) Punturo  _et al._ (2010);
Maggiore  _et al._ (2020); Reitze  _et al._ (2019); Kalogera  _et al._ (2021);
Hild  _et al._ (2011). In this study, we present a similar analysis of the
landscape of BH spectroscopy for the stellar-mass binary BH (BBH) population –
the prime candidates for ET Hild  _et al._ (2011); Maggiore  _et al._ (2020);
Kalogera  _et al._ (2021). We study the prospects of BH spectroscopy for both
individual events and for population studies using a state-of-the-art
population model that is informed by the LIGO-Virgo-KAGRA observations Mapelli
(2016); Périgois  _et al._ (2023). We consider two of the most relevant
configurations adopted in the recent analysis on the science impact of
different ET designs Branchesi  _et al._ (2023). Lastly, we also study the
prospects for joint ringdown measurement with ET and the Cosmic Explorer (CE)
Abbott  _et al._ (2017); Essick  _et al._ (2017) which is an American
initiative for a third-generation ground-based GW detector.

The remainder of this paper is organized as follows. In Sec. II, we outline
our setup for BH spectroscopy. Then, in Sec. III, we study BH spectroscopy for
golden events, i.e., loud events similar to GW150914. In Sec. IV, we discuss
the properties of the stellar-mass BBH population in the context of ringdown
tests, and present the results of ringdown population studies with ET. Next in
Sec. V, we study the advantages of a joint detection with ET and CE. Finally,
we conclude our work with a discussion and future research directions in Sec.
VI.

##  II Analysis setup

Within linear perturbation theory111See Refs. Cheung  _et al._ (2022); Mitman
_et al._ (2022); Lagos and Hui (2022); Kehagias  _et al._ (2023); Kehagias and
Riotto (2023); Baibhav  _et al._ (2023) for recent studies about next-to-
leading order ringdown effects. Note that the latter are particularly relevant
for the 440 mode (which can be sourced by the fundamental 220 mode at the
quadratic order), whereas the 220, 330, and 210 modes are less contaminated by
nonlinear effects. , the ringdown signals can be analytically modelled as
$h(t)=h_{+}(t)+ih_{\times}(t)$, where

$\begin{split}&h_{+}(t)=\sum_{lmn}{\cal A}_{lmn}\cos\left(2\pi
f_{lmn}t+\phi_{lmn}\right)e^{-t/\tau_{lmn}}\,\mathcal{Y}^{lm}_{+}(\iota)\,,\\\
&h_{\times}(t)=\sum_{lmn}{\cal A}_{lmn}\sin\left(2\pi
f_{lmn}t+\phi_{lmn}\right)e^{-t/\tau_{lmn}}\,\mathcal{Y}^{lm}_{\times}(\iota).\end{split}$
(1)

where $\iota$ is the remnant spin inclination angle and
$\\{f_{lmn},\tau_{lmn},\mathcal{A}_{lmn},\phi_{lmn}\\}$ are the frequency,
damping time, amplitude, and phase of the $(lmn)$ QNM, respectively. The
integers $(lmn)$ refer to the multipolar, azimuthal, and overtone index,
respectively, where $n=0$ corresponds to the fundamental tone and
$(lmn)=(220)$ is the dominant mode in a quasicircular coalescence. The
excitation amplitudes roughly scale as ${\cal A}_{lmn}\sim M_{f}/d_{L}$, where
$M_{f}$ is the mass of the remnant BH and $d_{L}$ is the luminosity distance
from the source. The $\\{+,\times\\}$ polarizations should be defined using a
spin-weighted spheroidal harmonics basis. However, for simplicity we
approximate it with their spherical harmonic counterpart, which is a good
approximation for moderately spinning remnants (see Refs. London and Fauchon-
Jones (2019); Cook (2020); Baibhav  _et al._ (2023) for related discussion).
The polarization basis can be written as Kamaretsos  _et al._ (2012a); Berti
_et al._ (2007)

$\mathcal{Y}_{+,\times}(\iota)=\tensor{[}_{-2}]{\mathcal{Y}}{{}^{lm}}(\iota,0)\pm(-1)^{l}\tensor{[}_{-2}]{\mathcal{Y}}{{}^{l-m}}(\iota,0)\,.$
(2)

In this work, we consider only BH spectroscopy with angular QNMs.

In this work, we consider the performance of BH spectroscopy with angular
QNMs, since they have longer damping times and they can be reliably extracted
sufficiently after the peak, where linear perturbation theory is accurate,
unlike the case of overtones Bhagwat  _et al._ (2020); Ota and Chirenti
(2020); Jiménez Forteza  _et al._ (2020); Baibhav  _et al._ (2023); Nee  _et
al._ (2023).

Using the above template, we infer the statistical errors associated to the
measurements of the ringdown parameters in a Fisher information matrix
framework. In particular, we use the same analysis setup developed in Bhagwat
_et al._ (2022) (to which we refer the readers for further technical details).
The setup is based on a fully numerical evaluation of the Fisher matrix.

We find it convenient to parameterize the frequencies and damping times as

$f_{lmn}=f_{lmn}^{\rm Kerr}(1+\delta
f_{lmn})\,,\quad\tau_{lmn}=\tau_{lmn}^{\rm Kerr}(1+\delta\tau_{lmn})\,,$ (3)

where $f_{lmn}^{\rm Kerr}$ and $\tau_{lmn}^{\rm Kerr}$ are the GR-predicted
frequencies and damping times of a remnant Kerr BH; these are functions of the
final mass $M_{f}$ and the final spin $\chi_{f}$. The (dimensionless)
deviation parameters $\delta f_{lmn}$ and $\delta\tau_{lmn}$ quantify the
departure of the measured spectrum from the GR prediction. For GR to pass a
null-hypothesis test with a certain level of confidence, the inferred
posterior distributions of $\delta f_{lmn}$ and $\delta\tau_{lmn}$ must
contain zero within a given confidence level. Note that, in the absence of
extra information coming from the inspiral, $\delta f_{220}$ and
$\delta\tau_{220}$ are degenerate with $M_{f}$ and $\chi_{f}$ Isi and Farr
(2021); Pacilio and Bhagwat (2023). Therefore, without loss of generality we
use ($M_{f},\chi_{f}$) in place of ($\delta f_{220},\delta\tau_{220}$), and we
use $\delta f_{lmn}$ and $\delta\tau_{lmn}$ for any other mode. Overall, the
parameters of our Fisher matrix are

$\\{M_{f},\chi_{f},\mathcal{A}_{220},\phi_{220},\delta
f_{lmn},\delta\tau_{lmn},\mathcal{A}_{lmn},\phi_{lmn}\\}\,,$ (4)

where $(lmn)\neq(220)$. Although different angular QNMs are almost orthogonal
and can be considered separately Berti  _et al._ (2006), in our analysis we
have included the 220, 330, 440, and 210 QNMs in the template at the same
time. Overall, the template has $16$ parameters.

The injected amplitudes $\mathcal{A}_{lmn}$ are obtained from the properties
of the progenitor binary (and therefore assuming GR). The injected amplitude
$\mathcal{A}_{220}$ of the fundamental mode is taken from Gossan  _et al._
(2012), while the amplitudes of the other modes relative to
$\mathcal{A}_{220}$ are obtained using the recent fits in Ref. Forteza  _et
al._ (2023). We also inject relative amplitudes $\phi_{lmn}-\phi_{220}$
consistently with the fits in Forteza  _et al._ (2023), while $\phi_{220}$ is
randomly sampled within $[0,\pi]$. The injected values of the final spin
$\chi_{f}$ are derived from the mass ratio $q$ using the numerical fits in
Hofmann  _et al._ (2016). We stress that the fits of Refs. Forteza  _et al._
(2023); Hofmann  _et al._ (2016) are only needed to select realistic injected
values, but $M_{f}$, $\chi_{f}$, and the QNM amplitudes and phases are free
parameters of the waveform. We assume a GR signal and inject $\delta
f_{lmn}=0=\delta\tau_{lmn}$.

$q$ | $\rho_{\rm RD}$ | $\sigma(M_{f})/M_{f}$ | $\sigma(\chi_{f})$ | $\sigma(\delta f_{330})$ | $\sigma(\delta f_{210})$ | $\sigma(\delta f_{440})$ | $\sigma(\delta\tau_{330})$ | $\sigma(\delta\tau_{210})$ | $\sigma(\delta\tau_{440})$ | $\sigma(\mathcal{A}_{330})/\mathcal{A}_{330}$ | $\sigma(\mathcal{A}_{210})/\mathcal{A}_{210}$ | $\sigma(\mathcal{A}_{440})/\mathcal{A}_{440}$
---|---|---|---|---|---|---|---|---|---|---|---|---
1.2 | 84 | 0.03 | 0.04 | 0.03 | 0.18 | 0.03 | 0.60 | 1.08 | 0.63 | 0.48 | 1.86 | 0.46
2 | 85 | 0.04 | 0.06 | 0.01 | 0.05 | 0.03 | 0.23 | 0.32 | 0.64 | 0.17 | 0.53 | 0.48
5 | 64 | 0.08 | 0.17 | 0.02 | 0.06 | 0.04 | 0.29 | 0.32 | 0.51 | 0.23 | 0.41 | 0.48
Table 1: Estimated $1$-$\sigma$ errors on the QNM parameters for a
GW150914-like system with $M_{f}=70M_{\odot}$ at luminosity distance
$d_{L}=450\,{\rm Mpc}$ with $({\rm
ra,dec},\psi,\iota)=(1.16,-1.19,1.12,\pi/3)$ and occurring at a geocenter GPS
time $t_{\rm GPS}=1126259466.43$. The reported ringdown SNR, $\rho_{\rm RD}$,
refers to a triangular ET detector placed in Sardinia with ET-D sensitivity.
Within the assumptions of the Fisher matrix, absolute errors scale inversely
with the SNR.

##  III BH Spectroscopy with ET for Golden events

In this section, we investigate the performance in BH spectroscopy on isolated
“golden” (i.e., loud and favorable) events using ET. We consider a
GW150914-like systems with $M_{f}=70M_{\odot}$ at a luminosity distance
$d_{L}=450\,{\rm Mpc}$. We place the event at the sky position
${\rm(ra,dec)}=(1.16,-1.19)$ with polariziation $\psi=1.12$, and assume it
occurs at a geocenter GPS time $t_{\rm GPS}=1126259466.43$. In order to ensure
that the subdominant modes are not suppressed, we choose an inclination angle
$\iota=\pi/3$.

We inject the mode amplitudes and phases as explained above. Since the mass
ratio $q$ has a leading effect on the amplitude excitation factors compared to
the binary spins London and Fauchon-Jones (2019); Kamaretsos  _et al._
(2012b); Gossan  _et al._ (2012); Forteza  _et al._ (2023), here we consider
three different values of $q\in\\{1.2,2,5\\}$ for nonspinning systems.

We tabulate the results of our Fisher matrix analysis in Table 1. Since the
results have a simple scaling with the SNR within the approximation of the
Fisher-matrix analysis, in this section we consider ET in a triangular
configuration adopting the standard ET-D sensitivity curve Hild  _et al._
(2011). We see that ET ringdown signals will have about one order of magnitude
higher SNR compared to the same events detected by LIGO-Virgo. This roughly
translates into an order-of-magnitude improvement on the measurement errors in
the ringdown parameters. Note that for the same final mass and luminosity
distance, the ringdown SNR ${\rho_{\rm RD}}$ decreases significantly with $q$;
this is unfortunate for BH spectroscopy because the excitation of the
subdominant angular modes is suppressed for almost equal-mass binaries. At the
same time, for a fixed SNR, the uncertainty of the QNM parameters tends to
decrease with increasing $q$ as the amplitude ratios of the subdominant modes
increase. Thus, as $q$ increases there is an interplay between decreasing
ringdown SNR and increasing $\mathcal{A}_{lmn}$ that produces the trend in
Table 1.

Overall, we see that ET can measure the subdominant QNM frequencies within a
few percent accuracy, with $\delta f_{330}$ always being the best constrained
subdominant parameter. For systems with $q\geq 2$, ET will measure
$\tau_{330}$ and $\tau_{210}$ within $20\%$ to $30\%$ accuracy, while
measuring $\tau_{440}$ is more difficult, with about $50\%$ to $65\%$
uncertainties in its recovery. Finally, although the recovery of amplitude
ratios is not directly relevant for performing BH spectroscopy, it can be used
to quantify the confidence level in the detection of a secondary mode (if
$\sigma(\mathcal{A}_{lmn})/\mathcal{A}_{lmn}\ll 1$) and to perform other tests
of GR, such as the recently-proposed amplitude-phase consistency test Forteza
_et al._ (2023) and merger-ringdown test Bhagwat and Pacilio (2021). Thus, we
report that, for $q\geq 2$, ET can measure $\mathcal{A}_{330}$ with $\approx
20\%$ accuracy, whereas it can measure $\mathcal{A}_{210}$ and
$\mathcal{A}_{440}$ with $\approx 50\%$ accuracy.

##  IV BH Spectroscopy with ET with a stellar mass BBH population

Having discussed the prospect for BH spectroscopy with single golden events,
in this section we move to the prospects for BH spectroscopy using a realistic
stellar-mass BBH population.

###  IV.1 Stellar-mass BBH population model

We use the BBH population described in Mapelli  _et al._ (2022). In
particular, we generated a population of merger signals by assuming that the
BBHs observed by the LIGO–Virgo–KAGRA Collaboration come from a mixture of the
dynamical and isolated channel. The isolated channel consists of BBHs that
evolve from unperturbed massive binary stars Bethe and Brown (1998);
Belczynski  _et al._ (2002). We evolved our massive binary systems with the
population-synthesis code mobse Mapelli  _et al._ (2017); Giacobbo  _et al._
(2018), which implements an up-to-date model for stellar winds (Giacobbo and
Mapelli, 2018), and a formalism for the main binary evolution processes
(Hurley  _et al._ , 2002). We describe the outcome of core-collapse supernovae
with the rapid model by Fryer  _et al._ (2012), which enforces a mass gap
between $2\,M_{\odot}$ and $5\,M_{\odot}$ Özel  _et al._ (2010); Farr  _et
al._ (2011). We also account for (pulsational) pair instability, as described
in Mapelli  _et al._ (2020). In our models, pair instability produces a mass
gap between $\approx{60\,M_{\odot}}$ and $\approx{120\,M_{\odot}}$ M⊙
(Giacobbo  _et al._ , 2018). This gap is partially filled by dynamically
formed binaries described below. We generate the dimensionless spin magnitudes
$\chi{}$ of the isolated BBHs from a Maxwellian distribution with root-mean-
square parameter $\sigma_{\chi}=0.1$ and truncated at $\chi=1$. This is a toy
model that allows us to reproduce the main features of BBHs observed by
LIGO–Virgo–KAGRA after the third observing run (Abbott  _et al._ , 2021). We
choose a toy model for spin magnitudes, because of the large uncertainties
still affecting astrophysical models and hampering their predictive power
(Périgois  _et al._ , 2023). We assume that binary evolution processes align
the spins of the progenitor stars with the orbital angular momentum of the
binary system: only the supernova explosion can produce a misalignment between
the BH spin and the orbital angular momentum (Rodriguez  _et al._ , 2016a).
This results in a preference for aligned spins in our isolated BBHs.

According to the dynamical channel, BBHs assemble in dense stellar clusters by
three-body encounters and dynamical exchanges (e.g., Portegies Zwart and
McMillan (2000); Banerjee  _et al._ (2010); Mapelli (2016); Rodriguez  _et
al._ (2016b); Fragione and Kocsis (2018); Kremer  _et al._ (2020); Banerjee
(2021)). We model three different astrophysical populations of star clusters:
nuclear, globular, and young clusters. Nuclear clusters lie at the center of
their host galaxies and can be very massive ($\approx{10^{7}}\,M_{\odot}$,
Neumayer  _et al._ (2020)). Globular clusters are massive
($\approx{10^{4-6}\,M_{\odot}}$, Harris (2010)) and mostly formed in the early
Universe ($z\sim{2-4}$, VandenBerg  _et al._ (2013)), while young clusters are
less massive than the other two families ($\leq{10^{5}}$M⊙) and are one of the
most common birthplaces of massive stars in the local Universe (Portegies
Zwart  _et al._ , 2010). We model the star formation history of nuclear,
globular, and young clusters as described in Mapelli  _et al._ (2022). In our
model, dynamically assembled BBHs in nuclear, globular, and young clusters can
undergo hierarchical mergers (Miller and Hamilton, 2002; Gerosa and Berti,
2017; Fishbach  _et al._ , 2017; Rodriguez  _et al._ , 2019; Antonini  _et
al._ , 2019, 2022; Gerosa and Fishbach, 2021): if the remnant of the merger of
two stellar-origin BHs is retained inside its parent cluster despite the
gravitational recoil (Merritt  _et al._ , 2004; Campanelli  _et al._ , 2007),
it can pair up again with another BH and lead to a second-generation (or nth-
generation) merger.

We generate the masses of first-generation BHs from the mobse population
synthesis code, i.e., the same code we use for the isolated binaries, for
consistency. We also randomly draw the spins of first-generation BHs from the
same Maxwellian distribution as we described for isolated BBHs. The masses and
the spins of second-generation BHs are obtained with fitting formulas to
numerical relativity simulations (Jiménez-Forteza  _et al._ , 2017). This
means that the spin magnitudes of our second-generation BHs peak at
$\chi{}\sim{0.7-0.8}$. Finally, the spins of both first-generation and second-
generation dynamical BHs are isotropically oriented over the sphere,
accounting for the effect of dynamical encounters Rodriguez  _et al._ (2016a).

With this set up, we obtain a primary BH mass distribution (Fig. 6 of Mapelli
_et al._ (2022)) and local BBH merger rate density ($R\approx{31}$ Gpc-3 yr-1)
that lie within the 90% credible intervals inferred by the LIGO–Virgo–KAGRA
Collaboration after the third observing run. We use this catalog because it is
grounded on state-of-the-art astrophysical models and matches the main
observed features. We refer to Mapelli  _et al._ (2022) for more details on
our simulations. Out of these simulations, we randomly extract a sub-sample
comprising all BBHs that we expect to merge in a time span of 10 years, from
the local Universe out to redshift $z=14$. This yields a final catalog of
1181195 BBHs.

###  IV.2 Population distributions in the context of BH spectroscopy

Before discussing our results it is useful to investigate some general
properties of our stellar-mass population model in the context of BH
spectroscopy.

Figure 1: Distributions of of remnant detector-frame mass $M_{f}\,(1+z)$ (top
left) and dimensionless spin $\chi_{f}$ (top right), the corresponding QNM
frequencies $f_{lmn}^{\rm Kerr}$ (middle left) and damping times
$\tau_{lmn}^{\rm Kerr}$ (middle right), and mass ratio $q$ (bottom left), and
symmetric spin $\chi_{s}$ and anti-symmetric spin $\chi_{a}$ components
(bottom right) of the progenitors. All histograms are not normalized and show
the actual number of events for the full 10-yr catalog.

The top panel in Fig. 1 shows the distributions of the detector-frame masses
and of the final spin of the BH remnant BHs. These distributions can be mapped
into the characteristic frequencies and damping times of the QNMs assuming a
Kerr BH, as presented in the middle panels of Fig. 1 for various subdominant
modes. If detector detuning is viable for next-generation ground-based
detectors, then the middle-left panel can be used to infer the frequency
ranges that optimize the detection of various subdominant modes. In the
middle-right panel, we see that all modes have a very similar damping time
distribution, peaking at approximately $10\,{\rm ms}$. These ranges can
provide informed priors in a Bayesian parameter estimation.

A notable property of the catalog in the context of BH spectroscopy is the
distribution of the binary mass ratio and of the spins. The asymmetry in the
progenitor binary systems is given by the mass ratio $q$ and the initial BH
spins. Note that the greater the asymmetry in the progenitor BBH, the higher
the subdominant mode excitation London  _et al._ (2014); Kamaretsos  _et al._
(2012b); Gossan  _et al._ (2012); Jiménez Forteza  _et al._ (2020); Forteza
_et al._ (2023). In the bottom panels of Fig. 1 we show the inverse cumulative
distribution, $1-{\rm CDF}$, of $q$ and the probability distribution of the
spin combinations $\chi_{s,a}=(m_{1}\chi_{1,z}\pm
m_{2}\chi_{2,z})/(m_{1}+m_{2})$ for a 10-year catalog. Although the
distribution of $q$ peaks at equal mass binaries, we see that there is a
significant number of events with $q\neq 1$. For these systems the subdominant
QNM excitation is nonnegligible. For instance, we expect $\sim 10^{3}$
event/yr with $q\geq 3$ and $\sim 10^{2}$ events/yr with $q\geq 6$.
Furthermore, even for $q\approx 1$, the subdominant QNM excitations can be
triggered by the spins of the progenitor BBHs. We see that $\chi_{a}$ peaks
close to zero with support in the range $\pm 0.2$, whereas $\chi_{s}$ favors a
positive value with a peak around 0.18.

Figure 2: Scatter plot of the amplitude ratio
$\mathcal{A}_{lmn}/\mathcal{A}_{220}$ on the $q$-$\chi_{\rm eff}$ plane for
the entire $10\,{\rm yr}$ catalog. Here $\chi_{\rm
eff}=(m_{1}\chi_{1}+m_{2}\chi_{2})/(m_{1}+m_{2})$ with $m_{i}$ denoting the
progenitor masses and $\chi_{i}$ progenitor spin. In each subplot, the color
bar displays $\mathcal{A}_{lmn}/\mathcal{A}_{220}$ of the $330$ mode (top),
the $210$ mode (center) and the $440$ mode (bottom).

Using the fits222Note that the fits in Forteza  _et al._ (2023) assume aligned
spins and, therefore, we used the spin components along the z-direction,
$\chi_{1,z}$ and $\chi_{2,z}$, to compute the injected values of the amplitude
ratios ${\cal A}_{lmn}/{\cal A}_{220}$. of Forteza  _et al._ (2023), we can
map these distributions into that of the amplitude ratio
$\mathcal{A}_{lmn}/\mathcal{A}_{220}$, as in the scattered plot in Fig. 2,
where the color bar shows the magnitude of the mode excitation. For 330 and
440 the spin contribution is much less important than that for 210 Forteza
_et al._ (2023). Indeed, the amplitude ratio for 210 spans a larger range
owing to its stronger dependence on the progenitor spins. Note that these
distributions of the amplitude ratios of the various subdominant modes can be
used to choose informed priors when performing Bayesian parameter estimation.

Finally, in Fig. 3, we present the (unnormalized) probability distribution
(left) and the inverse cumulative distribution (right) of the amplitude
ratios. We see that $\sim 20\%$ and $\sim 10\%$ of the events have
$\mathcal{A}_{330}\sim 0.1$ and $\mathcal{A}_{221}\sim 0.1$, respectively.
While the amplitude distributions of $330$ and $210$ are comparable, the peak
of the amplitude ratio of $440$ is very sharp at $\mathcal{A}_{440}\sim 0.04$
so that it is very unlikely to find events with larger $\mathcal{A}_{440}$.

Figure 3: Unnormalized probability distributions (left) and inverse cumulative
distributions (right) of the amplitude ratios
$\mathcal{A}_{lmn}/\mathcal{A}_{220}$ for the excitation modes considered in
this work.

###  IV.3 Landscape of BH spectroscopy with ET

In this section we present the landscape of BH spectroscopy for ringdowns of
stellar mass BH mergers with ET. We consider the most relevant configurations
adopted in the recent Branchesi  _et al._ (2023). We focus on the ‘hybrid’
cryogenic configuration comprising of a high-frequency (HF) and a low-
frequency (LF) instruments (which updates the standard ET-D curve and was
labelled ‘HFLF’ in Branchesi  _et al._ (2023)). We did not find any
significant difference when the low-frequency (LF) instrument is absent. This
is due to the fact that the ringdown signal is almost monochromatic and, in
our catalog, there exists a negligible fraction of events with
$f_{220}<20\,{\rm Hz}$ (see Fig. 1), where the contribution of the LF
instrument improves the sensitivity curve. While we found that the detector
geometry (either triangular or L-shaped) does not significantly impact the
performances of BH spectroscopy, the detector arm-length is more relevant.
Therefore, we present the result for two representative configurations: 1) a
single, 10-km long triangle-shaped interferometer (labeled as
‘$\Delta$-10km’), and 2) two, 15-km long L-shaped interferometers (labeled as
$2{\rm L}$-15km), see Ref. Branchesi  _et al._ (2023) for further details on
the configurations. We anticipate that the performances of a 15-km long
configuration (either triangular or 2L-shaped) are roughly a factor of two
better.

Figure 4: Left: Distribution of the number of events $N_{\rm det}$ with the redshift $z$ in the full 10-yr catalog, for small ($12$), high ($50$) and golden ($100$) ${\rho_{\rm RD}}$ thresholds. Right: Distribution of $N_{\rm det}$ with the mass ratio $q$. Configuration |  ${\rho_{\rm RD}}\geq 12$ ${\rm yr}^{-1}$ |  ${\rho_{\rm RD}}\geq 50$ ${\rm yr}^{-1}$ |  ${\rho_{\rm RD}}\geq 100$ ${\rm yr}^{-1}$ | max(${\rho_{\rm RD}}$)
---|---|---|---|---
$\Delta$-10km | $4594\pm 61$ | $28\pm 7$ | $3\pm 1$ | 1134
2L-15km | $10071\pm 88$ | $70\pm 9$ | $7\pm 3$ | 1262
Table 2: Detection rates per year with ${\rho_{\rm RD}}$ equal to or larger
than a small (12), high ($50$) and golden ($100$) threshold, respectively, for
two representative ET configurations. The last column indicates the maximum
${\rho_{\rm RD}}$ in the whole 10yr catalog.

As shown in the distribution of the detected events in Fig. 4 and from Table
2, ET will detect $\sim 10^{5}$ BBH mergers per year, of which $\sim 4600$
(resp. $\sim 10000$) events/yr have ${\rho_{\rm RD}}\geq 12$ in the
$\Delta$-10km (resp. 2L-15km) configuration. For comparison, the GW150914
signal, which had one of the loudest ringdown signal detected by LIGO, had
${\rho_{\rm RD}}\sim 8$. Also, we anticipate the detection of a handful of
events/yr with ${\rho_{\rm RD}}\geq 100$. As shown in Sec. III, these golden
events allow us to perform precision tests with unprecedented accuracy. We
summarize the median event rates and their corresponding errors across the
full 10-yr catalog in Table 2, and in Fig. 5 we show the inverse cumulative
distribution of the ringdown SNR. As it is expected, the event rate
uncertainties scale approximately as Poisson counting errors,
$\sigma(N)\approx\sqrt{N}$, and depending on the particular realization of the
catalog one can have individual signal-to-noise ratios as large as $\rho_{\rm
RD}\sim 1000$; these constitute promising candidates to identify spectra from
modified gravity theories Pacilio and Bhagwat (2023).

Figure 5: Inverse cumulative distribution of ${\rho_{\rm RD}}$ per year for
the ET detector configurations considered in this work. The shaded gray band
indicates the region where ${\rho_{\rm RD}}<12$. See also Table 2.

Figure 6:  Cumulative distributions of the uncertainties in measuring various
ringdown quantities (events per year for the ET configurations considered in
this work). Top panels: $\sigma\left(M_{f}\right)/M_{f}$ (left) and
$\sigma\left(\chi_{f}\right)$ (right). Second to fourth row: cumulative
distributions for the error on the QNM frequency, $\sigma\left(\delta
f_{lmn}\right)$ (left) and QNM damping time,
$\sigma\left(\delta\tau_{lmn}\right)$ (right). The colored bands indicate the
uncertainties on the $y$-axis computed across a 10-year catalog. The shaded
vertical gray band indicates the region $\sigma(\delta f_{lmn})>1$ or
$\sigma(\delta\tau_{lmn})>1$ where the constraints on $\delta f_{lmn}$ or
$\delta\tau_{lmn}$ are uninformative.

Our primary results for prospects of BH spectroscopy are summarized in Fig. 6
where we show the cumulative distributions of the uncertainties in
measurements of the QNM frequencies and damping times333Here and in Fig. 9 we
compute uncertainties only for the events that meet an SNR threshold for
parameter estimation, conventionally set to $\rho_{\rm RD}\geq 12$.. The first
row corresponds to the final mass and spin of the BH; these are degenerate
with $\delta f_{220}$ and $\delta\tau_{220}$. Although this does not have a
direct consequence on prospects of BH spectroscopy, it gives an estimate of
the performance of consistency tests informed by the inspiral part of the
signal. Note that ET will measure the final mass within fractional errors
$\leq 0.1$ for $\sim 400$ (resp. $\sim 1000$) event/yr in the $\Delta$-10km
(resp. 2L-15km) configuration. Similarly, for the final spin, we the absolute
errors are expected to be $\leq 0.1$ for $\sim 200$-$400$ event/yr.

From the second to forth rows of Fig. 6 we show the cumulative distribution
for the uncertainties in measurements of the QNM frequencies (left panels) and
damping times (right panels) for some of the promising subdominant modes.
Comparing the left and the right panels, we confirm that the deviation in the
subdominant mode frequencies can be measured more accurately than their
corresponding damping times. We expect between a few hundred and a thousand of
events per year to allow for $\leq 10\%$ uncertainty in $f_{330}$ and a
handful of events per year with an uncertainty $\leq 2\%$. In addition, a few
event/yr allow for the measurement of $f_{210}$ with an uncertainty $\leq
10\%$, and between a few hundred and a thousand events/yr allow for the
measurement of $f_{440}$ with $\leq 10\%$ uncertainty.

Note that, while $\mathcal{A}_{440}$ is not very high, the $440$ mode can
compete with the performance of the $330$ mode due to statistical abundance of
the systems in which it is excited.

##  V Prospects for combined CE-ET BH spectroscopy

In this section we study the prospect of BH spectroscopy assuming a combined
detection by ET and a single CE detector with 40-km arm length. As we shall
see, we find that the recovery of QNM parameters is significantly improved in
this case.

To facilitate the comparison, we will show exactly the same plots presented in
Sec. IV.3. Figures 7 and 8 are the analog of Figs. 4 and 8, respectively,
whereas Table 3 is the analog of Table 2. One of the most striking features is
that the number of events per year with ringdown SNR ${\rho_{\rm RD}}\geq 100$
increases by roughly a factor 3 detected up to redshift $z\approx 6$.
Furthermore, a handful of events will have ringdown SNR of a few hundreds.

Figure 7: Same as Fig. 4 for a network of ET operating in synergy with a single CE detector. Configuration |  ${\rho_{\rm RD}}\geq 12$ ${\rm yr}^{-1}$ |  ${\rho_{\rm RD}}\geq 50$ ${\rm yr}^{-1}$ |  ${\rho_{\rm RD}}\geq 100$ ${\rm yr}^{-1}$ | max(${\rho_{\rm RD}}$)
---|---|---|---|---
$\Delta$-10km+CE | $17174\pm 115$ | $161\pm 14$ | $13\pm 5$ | 1508
2L-15km+CE | $22144\pm 122$ | $246\pm 16$ | $18\pm 7$ | 1607
Table 3: Same as Table 2 but for ET operating in synergy with a single CE
detector. Figure 8: Same as Fig. 5 but for ET operating in synergy with a
single CE detector.

Finally, our main results for spectroscopy are again summarized in a single
plot (Fig. 9, which is the analog of Fig. 6). We see that $\sim 10$ events per
year will allow for fractional uncertainty $\sigma(M_{f})/M_{f}\leq 0.02$ (to
be compared with $\sim 1$ event/yr for ET alone at this level of accuracy) and
that few thousands event/yr will allow for $\sigma(M_{f})/M_{f}\leq 10^{-1}$
(to be compared with $\sim 400$-$1000$ event/yr with ET alone). Similar
improvements are found for the measurement of the final spin. Finally, the
combined CE-ET measurements of the subdominant QNMs yield $\sim 2000$ event/yr
with uncertainty $\sigma(\delta f_{lmn})\leq 0.1$, and $\sim 10$ events with
$\sigma(\delta f_{lmn})\leq 0.02$ for both the 330 and the 440 modes. The
accuracy in measuring $f_{210}$ performs worse and we predict $\sigma(\delta
f_{lmn})\leq 0.3$ for $\sim 100$ events/yr. Looking at the right panels in
Fig. 9, we see that the best combined CE-ET measurement of the damping time is
for $\tau_{330}$ with $\sim 10$ event/yr allowing for
$\sigma(\delta\tau_{330})\leq\ 0.3$.

Figure 9:  Same as Fig. 6 but for ET operating in synergy with a single CE
detector.

##  VI Discussion and Conclusions

In this paper, we have investigated the prospects for BH spectroscopy with the
ET detector in different configurations, possibly in combination with CE. We
estimated both projected bounds with isolated golden merger events and rates
of accurate measurements using a state-of-the-art population model for
stellar-origin BBHs informed by LIGO-Virgo-KAGRA data. Our results highlight
the importance of longer detector arms and combined CE-ET detections for what
concerns ringdown tests.

Our analysis was intended to explore broadly the landscape of ringdown tests
with third-generation interferometers, considering different detector
configurations and networks, and including different subdominant modes. As
such, it can be extended in various ways if one wishes to perform a more
detailed and focused analysis. An obvious extension is to perform a Bayesian
inference, possibly including the priors on the QNM amplitudes and phases as
discussed in this work. Furthermore, given the large number of ringdown
signals expected in the ET-CE era, a natural extension is to explore the
possibility of stacking multiple signals to improve the accuracy of ringdown
tests Yang  _et al._ (2017); Berti  _et al._ (2018). For what concerns
possible improvements on the ringdown modeling, it would be relevant to
include mode-mixing due to the expansion in spheroidal harmonics and possibly
also overtones and quadratic effects, along the lines of the recent analysis
in Ref. Baibhav  _et al._ (2023). In this context, it is also relevant to note
that, given the stellar-mass BBH population favored by current GW data, the
330 mode is the optimal angular mode for BH spectroscopy with ET. This is
fortunate, because the 330 mode is expected to be less contaminated by QNM
mixing at the quadratic level relative to the 440 mode (which shows only
slightly worse measurement accuracy). Indeed, angular-momentum sum rules imply
that, within GR, the 440 mode is sourced by the dominant 220+220 mode Cheung
_et al._ (2022); Mitman  _et al._ (2022); Lagos and Hui (2022), whereas the
330 mode could be sourced by the 220+210 mode, whose contribution is
suppressed due to the smaller excitation of the 210. Therefore, standard
(linear) ringdown tests with the 330 mode should be reliable also for very
loud events as those expected in the third-generation era.

###### Acknowledgements.

We thank Emanuele Berti and Francesco Iacovelli for useful conversations. The
research leading to these results has been conceived and developed within the
ET Observational Science Board (OSB). Numerical calculations have been made
possible through a CINECA-INFN agreement, providing access to resources on
MARCONI at CINECA. S.B. would like to acknowledge the UKRI Stephen Hawking
Fellowship funded by the Engineering and Physical Sciences Research Council
(EPSRC) with grant reference number EP/W005727 for support during this
project. C.P. is supported by European Union’s H2020 ERC Starting Grant No.
945155–GWmining and by Cariplo Foundation Grant No. 2021-0555. MM acknowledges
financial support from the European Research Council for the ERC Consolidator
grant DEMOBLACK, under contract no. 770017. P.P. acknowledge financial support
provided under the European Union’s H2020 ERC, Starting Grant agreement no.
DarkGRA–757480 and under the MIUR PRIN programme, and support from the Amaldi
Research Center funded by the MIUR program “Dipartimento di Eccellenza” (CUP:
B81I18001170001). This work was supported by the EU Horizon 2020 Research and
Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No.
101007855.

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