# Identifying modified theories of gravity using binary black-hole ringdowns

Costantino Pacilio<EMAIL_ADDRESS>Dipartimento di Fisica “G.
Occhialini”, Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3,
20126 Milano, Italy  INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3,
20126 Milano, Italy  Swetha Bhagwat<EMAIL_ADDRESS>Institute for
Gravitational Wave Astronomy $\&$ School of Physics and Astronomy, University
of Birmingham, Edgbaston, Birmingham B15 2TT, UK

###### Abstract

Black-hole spectroscopy, that is, measuring the characteristic frequencies and
damping times of different modes in a black-hole ringdown, is a powerful probe
for testing deviations from the general theory of relativity (GR). In this
work, we present a comprehensive study on its ability to identify deviations
from the spectrum of a Kerr black hole in GR. Specifically, we investigate the
performance of black hole spectroscopy on a diverse set of theoretically
motivated as well as phenomenologically modified spectra. We find that while
the signal-to-noise ratio $\rho_{\rm RD}$ in the ringdown required to identify
a modification to the GR Kerr black hole spectrum depends on the details of
the modifications, a modification that introduces $\sim 1\%$ shift in the
fundamental mode frequencies can typically be distinguished with $\rho_{\rm
RD}\in[150,500]$. This range of $\rho_{\rm RD}$ is feasible with the next-
generation detectors, showing a promising science case for black hole
spectroscopy.

##  I Introduction

Gravitational waves (GWs) with characteristic frequencies and damping times
are radiated as the distorted black hole (BH) formed during a binary BH merger
relaxes into its final stable state. This signal is called the ringdown and
comprises of a linear superposition of the spectral modes of the BH, known as
the quasi-normal modes (QNMs). We can obtain the QNMs by solving the BH
perturbation equations Teukolsky (1972); Press and Teukolsky (1973); Teukolsky
and Press (1974) and the ringdown signal can be used to validate dynamics in
linear strong field regime. The QNM spectra of a perturbed BH in the general
theory of relativity (GR) are obtained by solving Teukolsky’s equation Pani
(2013) and under the Kerr hypothesis Cosimo (2011); Teukolsky (2015) i.e.,
that the remnant BH in binary BH coalescence relaxes to a Kerr BH.

BH spectroscopy Dreyer  _et al._ (2004), defined as measuring QNM spectra from
ringdown signals, allows to put forth consistency tests of the joint
hypotheses that

  1. 1.

the asymptotic equilibrium state of the remnant is described by the Kerr
metric a.k.a., the Kerr hypothesis and

  2. 2.

the dynamics of the perturbed Kerr BH is governed by Teukolsky’s equation,
i.e., (linearized) GR dynamics.

Further, BH spectroscopy can observationally validate the no-hair theorem
obeyed by BHs in GR; it demands that all aspects of a Kerr spacetime,
including its QNM spectrum, be fully characterized by just two parameters.
Often and most naturally, the two parameters are chosen as the mass $M_{f}$
and spin $\chi_{f}$ of the BH. If more than two QNM parameters are measured,
it can be verified that different pairs of QNM parameters solve for the same
$M_{f}$ and $\chi_{f}$. This allows us to perform a null test of the no-hair
theorem.

There has been much focus in the literature on the feasibility of measuring
the subdominant QNM modes and performing null tests to validate the underlying
theory of gravity as GR with BH spectroscopy Bhagwat  _et al._ (2022, );
Maggiore  _et al._ (2020); Berti  _et al._ (2016); Maselli  _et al._ (2017);
Carullo  _et al._ (2019); Bhagwat and Pacilio (2021); Ghosh  _et al._ (2021);
Bhagwat  _et al._ (2020a); Ota and Chirenti (2022); Baibhav and Berti (2019);
Baibhav  _et al._ (2020); Brito  _et al._ (2018). In this study, we
concentrate on a complementary aspect and investigate the ability of BH
spectroscopy to identify deviations from GR when the spectrum is not described
by the GR Kerr QNMs. We perform a comprehensive study utilizing the publicly
available QNM spectra in various modified theories as well as two
phenomenological modifications and assess the performance of BH spectroscopy
to differentiate them from a GR Kerr BH spectra. We then investigate the
signal-to-noise ratio $\rho_{\rm RD}$ in the ringdown at which different
modified theories of gravity can be distinguished from GR using BH
spectroscopy. We find that the required $\rho_{\rm RD}$ depends on the details
of the QNM spectra in a given theory and on their degeneracies with the GR
Kerr BH spectra in the mass-spin space. However, at a broad level, we observe
that $\rho_{\rm RD}\geq 150$ is required to confidently identify modified
theories that produce $\leq 1\%$ deviation in the dominant mode from GR using
BH spectroscopy (c.f., Bhagwat  _et al._ (2022, ) for a detailed study on
expected $\rho_{\rm RD}$ for measurability of QNM parameters with the next-
generation GW detectors).

The remainder of this paper is organized as follows. In Section II we outline
the conceptual structure adopted for this study to test the no-hair theorem
using BH spectroscopy. In Section III, we detail the modified QNM spectra used
in this study. Then, in Section IV we summarize the setup and implementation
used to perform this study. This is followed by the results in Section V and a
discussion of their implications in Section VI.

##  II Testing the no-hair hypothesis with BH spectroscopy

The ringdown waveform observed at asymptotic infinity can be approximated as a
linear superposition of a countably infinite set of (complex) QNMs with

$\omega_{lmn}=2\pi f_{lmn}-i/\tau_{lmn}.$ (1)

Here $f_{lmn}$ and $\tau_{lmn}$ are the characteristic frequencies and damping
times of the spectral modes. $(l,m,n)$ index the mode’s angular, azimuthal and
overtone numbers. As in any perturbation theory, the excitation amplitude of
the modes depend on the initial perturbation conditions; for a binary BH
merger these are set largely during the plunge-merger phase. A quasi-circular
merger excites $(2,2,0)$ dominantly, and depending on the initial binary BH’s
mass ratio and spins, the most prominent subdominant angular modes can be
$\\{(3,3,0),(2,1,0),(4,4,0)\\}$ Kamaretsos  _et al._ (2012a); Gossan  _et al._
(2012); Kamaretsos  _et al._ (2012b); London  _et al._ (2014); Borhanian  _et
al._ (2020); Magaña Zertuche  _et al._ (2022); Jiménez Forteza  _et al._
(2020); Forteza  _et al._ (2023); Borhanian  _et al._ (2020).

To outline our setup, let us consider a ringdown where more than two QNM
parameters are measurable, and a case where we have identified its QNM
indices. The minimum $\rho_{\rm RD}$ required for this has been investigated
in studies such as Berti  _et al._ (2007); Bhagwat  _et al._ (2020b, 2018);
London  _et al._ (2014). If the underlying theory of gravity is GR and if the
Kerr hypothesis holds, we can invert any pair of QNM parameters, preferably
the frequency and damping time of the dominant mode, to infer the mass and
spin of the BH –

$\\{f_{220},\tau_{220}\\}\to\\{{M_{f}^{\rm Kerr}},{\rm\chi_{f}^{\rm
Kerr}}\\}.$ (2)

From this mass and spin estimate, we can compute the full set of QNM spectra
of the Kerr BH. Let $f_{lmn}^{\rm(infer)}$ be the inferred subdominant mode
frequency

$\\{{M_{f}^{\rm Kerr}},{\chi_{f}^{\rm Kerr}}\\}\to f_{lmn}^{\rm(infer)}\,.$
(3)

Here we use the superscript (infer) to differentiate $f_{lmn}^{\rm(infer)}$
from $f_{lmn}^{\rm(meas)}$ which are the frequencies measured from the
ringdown signal. While a similar argument holds for QNM damping times,
subdominant mode damping times are poorly measured Gossan  _et al._ (2012);
Berti  _et al._ (2006a); Bhagwat  _et al._ (2022) and therefore, we focus on
tests using solely the subdominant mode frequencies.

A null test can be performed by checking if the relative difference between
the inferred and the measured quantity is compatible with zero. We define the
relative difference as

$\delta
f_{lmn}=\frac{f_{lmn}^{\rm(meas)}-f_{lmn}^{\rm(infer)}}{f_{lmn}^{\rm(infer)}}\,.$
(4)

We can infer steps (2)-(4) through a convenient reparametrization of the
waveform during the parameter estimation. We briefly summarize this and point
the reader to a detailed treatment in Isi and Farr (2021).

A generic modified QNM spectrum can be phenomenologically written as

$\displaystyle f_{lmn}=f_{lmn}^{\rm Kerr}(M_{f},\chi_{f})(1+\delta
f_{lmn})\,,$ (5a) $\displaystyle\tau_{lmn}=\tau_{lmn}^{\rm
Kerr}(M_{f},\chi_{f})(1+\delta\tau_{lmn})\,,$ (5b)

where $\\{M_{f},\chi_{f}\\}$ are the true values of the final mass and spin,
and $\\{\delta f_{lmn},\delta\tau_{lmn}\\}$ are the relative shifts of the
spectrum w.r.t. the QNM spectra of a GR BH. $\\{\delta
f_{lmn},\delta\tau_{lmn}\\}$ can be non-trivial functions of
$\\{M_{f},\chi_{f}\\}$ and of the physical parameters of the modified theory
such as the additional coupling constants or charges. At this stage of setting
up the formalism, we do not differentiate between a modification to the
underlying theory of gravity and a modification in the nature of the compact
object.

Now, notice that $\\{\delta f_{220},\delta\tau_{220}\\}$ are redundant
parameters because we can always find a pair
$\\{\tilde{M}_{f},\tilde{\chi}_{f}\\}$ of effective final mass and spin that
satisfy

$\displaystyle f_{220}=f_{220}^{\rm Kerr}(\tilde{M}_{f},\tilde{\chi}_{f})\,,$
(6a) $\displaystyle\tau_{220}=\tau_{220}^{\rm
Kerr}(\tilde{M}_{f},\tilde{\chi}_{f})\,.$ (6b)

The subdominant modes can be re-expressed as

$\displaystyle f_{lmn}=f_{lmn}^{\rm
Kerr}(\tilde{M}_{f},\tilde{\chi}_{f})(1+\tilde{\delta}f_{lmn})\,,$ (7a)
$\displaystyle\tau_{lmn}=\tau_{lmn}^{\rm
Kerr}(\tilde{M}_{f},\tilde{\chi}_{f})(1+\tilde{\delta}\tau_{lmn})\,,$ (7b)

for $(lmn)\neq(220)$. Further, the effective shifts
$\\{\tilde{\delta}f_{lmn},\tilde{\delta}\tau_{lmn}\\}$ satisfy

$\displaystyle f_{lmn}^{\rm Kerr}(M_{f},\chi_{f})\left(1+\delta
f_{lmn}\right)=f_{lmn}^{\rm
Kerr}(\tilde{M}_{f},\tilde{\chi}_{f})(1+\tilde{\delta}f_{lmn})\,,$ (8a)
$\displaystyle\tau_{lmn}^{\rm
Kerr}(M_{f},\chi_{f})\left(1+\delta\tau_{lmn}\right)=\tau_{lmn}^{\rm
Kerr}(\tilde{M}_{f},\tilde{\chi}_{f})(1+\tilde{\delta}\tau_{lmn})\,.$ (8b)

For the QNM spectrum of a Kerr BH in GR, $\\{\delta
f_{lmn},\delta\tau_{lmn}\\}$ vanish; therefore
$\\{\tilde{M}_{f},\tilde{\chi_{f}}\\}=\\{M_{f},\chi_{f}\\}$ and
$\\{\tilde{\delta}f_{lmn},\tilde{\delta}\tau_{lmn}\\}$ vanish. We set up our
framework to identify departure from the GR Kerr BH QNM spectrum by
constraining the effective shifts away from zero.

Note that the mass and spin appearing in Eq.s (2)-(3) are not the true values
$\\{M_{f},\chi_{f}\\}$ but rather the effective values
$\\{\tilde{M}_{f},\tilde{\chi}_{f}\\}$. We emphasise that we can only measure
the _effective_ final mass and spin, and not the true values corresponding to
the BHs. While developing a framework for observational test, the effective
(measured) parameters deviations
$\\{\tilde{\delta}f_{lmn},\tilde{\delta}\tau_{lmn}\\}$ are the instrumental
variables 111Note also that if one is not interested in testing the no-hair
theorem, the signals can be analyzed by assuming the Kerr BH spectrum (i.e.,
setting all $\\{\delta f_{lmn},\delta\tau_{lmn}\\}$ to zero) and recovering
posterior estimates of the mass and spin. The posteriors so obtained will
generally differ from the posteriors of
$\\{\tilde{M}_{f},\tilde{\chi}_{f}\\}$. This means that
$\\{\tilde{M}_{f},\tilde{\chi}_{f}\\}$ obtained here cannot be used to gauge
the performance of tests like the Inspiral-merger-ringdown test Ghosh  _et
al._ (2018).. Similarly, the magnitudes of $\\{\delta
f_{lmn},\delta\tau_{lmn}\\}$ are not directly accessible in BH spectroscopy
and we can only estimate
$\\{\tilde{\delta}f_{lmn},\tilde{\delta}\tau_{lmn}\\}$. In Section III, we
inspect modified QNM spectra and show that
$\\{\tilde{\delta}f_{lmn},\tilde{\delta}\tau_{lmn}\\}$ can be significantly
different from $\\{\delta f_{lmn},\delta\tau_{lmn}\\}$.

##  III Modified QNM spectra

In this study, we quantify the ability of BH spectroscopy to constrain
$\tilde{\delta}f_{lmn}$ away from zero for various class of modifications to
the Kerr BH spectrum (c.f., Tattersall and Ferreira (2019); Bao  _et al._
(2019); Carullo  _et al._ (2022); Silva  _et al._ (2023) for other works on BH
spectroscopy in the context of modified theories of gravity). Given the lack
of a best-candidate theory for modified gravity and the fact that QNM spectra
in modified theories are available in a very few theories Ferrari  _et al._
(2001); Konoplya (2002); Brito  _et al._ (2013a, b); Babichev  _et al._
(2016); Molina  _et al._ (2010); Pani and Cardoso (2009); Blázquez-Salcedo
_et al._ (2016, 2017); Brito and Pacilio (2018); Pierini and Gualtieri (2022);
Dias  _et al._ (2015); Wagle  _et al._ (2022); Cano  _et al._ (2020, 2022);
Tattersall and Ferreira (2018), of which even fewer theories have QNMs
computed at a beyond-leading order in BH spins Dias  _et al._ (2015); Pierini
and Gualtieri (2022), we consider both publicly available modified spectra and
phenomenologically modified spectra.

The spectra are chosen to encompass a variety of modifications to stress-test
the ability of spectroscopy to distinguish them from a GR Kerr spectrum. We
don’t concern ourselves with the physical plausibility of these modifications.
Below we describe the modifications to the GR Kerr BH spectrum used in this
study:

  * •

EdGB: The Einstein-dilaton-Gauss-Bonnet theory Mignemi and Stewart (1993);
Kanti  _et al._ (1996) is a modified theory of gravity that introduces a
dilaton scalar field that is non-minimally coupled to higher orders of the
curvature, specifically to the Gauss-Bonnet invariant. The BHs in EdGB have a
scalar hair as they are endowed with a monopole scalar charge. However, this
charge is not an independent parameter but it is a “secondary hair” Herdeiro
and Radu (2015), i.e., it is completely determined by the mass and spin of the
BH and by the coupling constants of the theory. The QNMs of EdGB BHs at the
next-to-leading order in the spin are derived in Pierini and Gualtieri (2022).
The numerical approximations in Pierini and Gualtieri (2022) restrict the
validity of the spectrum to $\zeta_{\rm EdGB}\lesssim 0.4$, where $\zeta_{\rm
EdGB}=\alpha_{\rm EdGB}/M_{f}^{2}$ and here $\alpha_{\rm EdGB}$ is the
coupling constant of the theory. Further, the final spins $\chi_{f}\gtrsim
0.3$ can be potentially outside the range of validity of the
$\mathcal{O}(\chi_{f}^{2})$ expansion. To mitigate these effects, we consider
the Padé resummed version of the spectrum provided in Pierini and Gualtieri
(2022). Note that the EdGB QNMs break isospectrality Chandrasekhar (1998)
between axial and polar sectors due to the non-minimal coupling of the scalar
field. In this work, we choose to not include the axial sector in the spectrum
and model it as given by the polar sector only.

  * •

Kerr-Newman: The Kerr-Newman spectrum for GW perturbation is derived in Pani
_et al._ (2013a, b) at first order in the final spin expansion (c.f., Mark
_et al._ (2015) for a perturbative expansion in the electric charge). There is
strong numerical evidence that the Kerr-Newman spectrum is isospectral, which
is confirmed by the full non-perturbative analysis in Dias  _et al._ (2015).
Therefore, unlike in the EdGB, there is no ambiguity in choosing polar or
axial sectors in its modified spectrum. Note that a Kerr-Newman BH becomes
extremal at charge-to-mass ratio $Q=(1-\chi_{f}^{2})^{1/2}$ but we only
consider values of $Q$ away from this limit.

  * •

Horndeski: The Horndeski action gives a general scalar-tensor gravity with
second order equations of motion Horndeski (1974). The Horndeski field
equations admit standard GR BH solutions under various conditions Motohashi
and Minamitsuji (2018). Linear perturbations around slowly rotating Kerr BHs
were studied in Tattersall and Ferreira (2018) for the sub-class of Horndeski
theories in which GW propagates at the speed of light. They show that the
equations are reduced to a massive scalar perturbation with an effective mass
parameter $\mu$. Although the spectrum does not correspond to perturbations in
GW sector, in principle, it can be sourced by the GW sector, and we expect
imprints of these frequencies in the GW signals Evstafyeva  _et al._ (2022).
In this work, we only look at the QNMs in scalar sector presented in Eq.s
(34)-(35) of Tattersall and Ferreira (2018). Note that this spectrum reduces
to the scalar perturbations of the Kerr BH in the limit $\mu\to 0$; therefore,
to augment our battery of modifications, we linearly re-scale it to recover
the gravitational GR Kerr BH spectrum in the limit $\mu\to 0$ and promote it
as yet another modified spectrum. We remind that for this work, we are
interested in studying the performance of BH spectroscopy to distinguish a
non-Kerr GR spectrum and do not aim to put bounds on any given modified
theory/spectrum in particular.

  * •

dCS: In dynamical Chern-Simons (dCS) theory Alexander and Yunes (2009), a
scalar field is non-minimally coupled to the higher-curvature Pontryagin
invariant, resulting in a breakdown of parity symmetry. Rotating BHs in dCS
have a secondary hair in the form of a monopole scalar charge Yagi  _et al._
(2012). The QNM spectrum of dCS BHs were computed in Wagle  _et al._ (2022) at
the leading order in the spin and at second order in the non-minimal coupling
constant of the scalar field $\alpha_{\rm dCS}$ (see also Srivastava  _et al._
(2021)). In the following we will use the dimensionless coupling $\zeta_{\rm
dCS}=\alpha^{2}_{\rm dCS}/M_{f}^{4}$. Here, we are extrapolating the spectrum
in Wagle  _et al._ (2022) beyond the small spin approximation but this is not
a critical concern for our study. Also, the non-minimal coupling of the scalar
field breaks isospectrality and therefore, similar to EdGB we consider the
polar sector of the dCS spectrum.

  * •

Delta: We generate an ad-hoc phenomenological spectrum by modifying the
frequencies of all modes by a constant relative shift, $f_{lmn}=f_{lmn}^{\rm
Kerr}(1+\Delta)$. We also choose to leave all damping times unchanged.

  * •

Delta220: We modify only the the frequency of the dominant mode
$f_{220}=f_{220}^{\rm Kerr}(1+\Delta_{220})$ and leave all other mode
frequencies and damping times unchanged.

The above scenarios are distinct modifications to the GR Kerr BH spectra where
the no-hair hypothesis can be violated. While Kerr-Newman BHs deviate from the
Kerr background due to the presence of a “primary hair” (the electric charge),
in the case of EdGB and dCS BHs the background possesses a “secondary hair”
(the monopole scalar charge). Further in the Horndeski BHs we consider here,
the background coincides with Kerr. Moreover, note that in all the theories
here, the deviations from GR appear also at the level of the field equations.

The Delta and Delta220 are simplistic ad-hoc modification schemes. We study
them as they are easy to implement and to interpret the performance of BH
spectroscopy and as a benchmark. We opt for modifications in the frequencies
as deviations in the damping times are more difficult to constrain. Note that,
in a realistic scenario, we typically expect all or at least a subset of
frequencies and damping times to be modified and it is unlikely that all modes
are modified by the same amount. Nonetheless, as shown in Fig. 4, they behave
as simpler representatives of the more complex spectra considered in this
work.

For the EdGB, Kerr-Newman, and dCS spectra, we opt to impose consistency with
the Kerr BH spectrum by linear rescaling, similar to the Horndeski case
described above. If we take the limit of $\zeta_{\rm EdGB}\to 0$ for the EdGB
spectrum in Pierini and Gualtieri (2022), we do not recover the GR Kerr BH
QNMs (which would be the case if the EdGB QNMs could be computed non-
perturbatively) because the spectrum is derived at
$\mathcal{O}(\chi_{f}^{2})$. Additionally, each spectrum is derived within its
own set of approximations, and therefore, they return a different
approximation to the GR Kerr BH spectrum in the limit of the vanishing
deviation parameters.

We provide a procedure for imposing consistency across the spectra by
redefining the spectra. We illustrate our procedure on EdGB below –

$\begin{split}&f_{lmn}^{\rm EdGB}(M_{f},\chi_{f},\zeta_{\rm EdGB})\\\
&=f_{lmn}^{\rm Kerr}(M_{f},\chi_{f})\left(\frac{\hat{f}_{lmn}^{\rm
EdGB}(M_{f},\chi_{f},\zeta_{\rm EdGB})}{\hat{f}_{lmn}^{\rm
EdGB}(M_{f},\chi_{f},0)}\right)\end{split}$ (9)

We enforce the GR Kerr BH spectrum in the limit $\zeta_{\rm EdGB}\to 0$. Here,
the hat denotes the expression of EdGB spectrum presented in Pierini and
Gualtieri (2022). Then, we defined the relative shifts as –

$\delta f_{lmn}=\frac{\hat{f}_{lmn}^{\rm EdGB}(M_{f},\chi_{f},\zeta_{\rm
EdGB})}{\hat{f}_{lmn}^{\rm EdGB}(M_{f},\chi_{f},0)}-1$ (10)

and parametrized

$f_{lmn}^{\rm EdGB}=f_{lmn}^{\rm Kerr}(1+\delta f_{lmn})\,.$ (11)

The definition (10) preserves the values of the shifts given by the
traditional parametrization of the spectrum. We also repeat the same procedure
for the damping times. The procedure is extended to the other spectra
considered above, by replacing $\zeta_{\rm EdGB}$ with $Q,\mu,\zeta_{\rm
dCS},\Delta$ and $\Delta_{220}$ respectively.

Here, we re-emphasis that this procedure is only necessary because the QNM
spectra in these modified theories are calculated perturbatively to a limited
order in the final spin of the BH. In the absence of the exact spectra, we use
Eq. (11) as a fiducial definition for all the modified spectra considered
here.

Finally, in Fig. 1 we plot the relative deviations $\delta f_{lmn}$ and
effective deviations $\tilde{\delta}f_{lmn}$ in the QNM frequencies for the
modified spectra listed above. We remind that it is the effective deviations
$\tilde{\delta}f_{lmn}$ that are measured when performing BH spectroscopy.
Interestingly, we find that for some spectra even when the actual spectrum has
deviations at a percent level, the measurable effective spectrum deviates from
GR Kerr BH at a much smaller sub-percent level. This is particularly evident
for the Delta spectrum where all the true frequencies are shifted by the same
amount but the measurable deviations turn out to be much smaller. We also see
this in the Kerr-Newman spectrum.

Spectrum | $\alpha_{0.01}$ |  $\tilde{M}_{f}$ ($M_{\odot}$) | $\tilde{\chi}_{f}$ | $\tilde{\delta}f_{330}$ | $\tilde{\delta}f_{210}$ | $\tilde{\delta}\tau_{330}$ | $\tilde{\delta}\tau_{210}$
---|---|---|---|---|---|---|---
$q=1.4\,,\quad\chi_{f}=0.67$ |  |
EdGB | $0.28$ | $71.10$ | $0.68$ | $-5.1\times 10^{-3}$ | $9.7\times 10^{-3}$ | $-1.2\times 10^{-2}$ | $-1.6\times 10^{-2}$
Kerr-Newman | $0.25$ | $69.71$ | $0.68$ | $-1.6\times 10^{-5}$ | $6.6\times 10^{-5}$ | $3.9\times 10^{-4}$ | $1.9\times 10^{-4}$
Horndeski | $0.16$ | $70.38$ | $0.69$ | $-4.3\times 10^{-3}$ | $1.1\times 10^{-2}$ | $-8.3\times 10^{-3}$ | $9.9\times 10^{-3}$
dCS | $0.055$ | $71.40$ | $0.68$ | $9.7\times 10^{-3}$ | $1.3\times 10^{-2}$ | $-3.7\times 10^{-2}$ | $1.2\times 10^{-2}$
Delta | $0.01$ | $69.76$ | $0.68$ | $4.2\times 10^{-4}$ | $2.8\times 10^{-3}$ | $-4.7\times 10^{-4}$ | $6.4\times 10^{-4}$
Delta220 | $0.01$ | $69.76$ | $0.68$ | $-9.5\times 10^{-3}$ | $-7.1\times 10^{-3}$ | $-4.7\times 10^{-4}$ | $6.4\times 10^{-4}$
$q=3\,,\quad\chi_{f}=0.54$ |  |
EdGB | $0.31$ | $70.54$ | $0.54$ | $-6.4\times 10^{-3}$ | $4.8\times 10^{-3}$ | $-7.6\times 10^{-3}$ | $-9.3\times 10^{-3}$
Kerr-Newman | $0.26$ | $69.75$ | $0.55$ | $-5.7\times 10^{-5}$ | $4.5\times 10^{-4}$ | $3.9\times 10^{-4}$ | $-2.7\times 10^{-5}$
z Horndeski | $0.15$ | $70.57$ | $0.57$ | $-4.3\times 10^{-3}$ | $1.1\times 10^{-2}$ | $-8.6\times 10^{-3}$ | $6.9\times 10^{-3}$
dCS | $0.058$ | $72.30$ | $0.58$ | $1.0\times 10^{-2}$ | $1.8\times 10^{-2}$ | $-5.7\times 10^{-2}$ | $-3.1\times 10^{-2}$
Delta | $0.01$ | $69.82$ | $0.55$ | $3.6\times 10^{-4}$ | $3.1\times 10^{-3}$ | $-4.5\times 10^{-4}$ | $4.0\times 10^{-4}$
Delta220 | $0.01$ | $69.82$ | $0.55$ | $-9.5\times 10^{-3}$ | $-6.8\times 10^{-3}$ | $-4.5\times 10^{-4}$ | $4.0\times 10^{-4}$
$q=5\,,\quad\chi_{f}=0.42$ |  |
EdGB | $0.34$ | $70.01$ | $0.40$ | $-7.5\times 10^{-3}$ | $5.2\times 10^{-5}$ | $-4.7\times 10^{-3}$ | $-4.7\times 10^{-3}$
Kerr-Newman | $0.27$ | $69.77$ | $0.43$ | $-1.1\times 10^{-4}$ | $8.2\times 10^{-4}$ | $4.7\times 10^{-4}$ | $-9.3\times 10^{-5}$
Horndeski | $0.14$ | $70.71$ | $0.46$ | $-4.3\times 10^{-3}$ | $1.1\times 10^{-2}$ | $-8.6\times 10^{-3}$ | $4.8\times 10^{-3}$
dCS | $0.063$ | $76.26$ | $0.55$ | $1.2\times 10^{-2}$ | $4.2\times 10^{-2}$ | $-1.3\times 10^{-1}$ | $-9.6\times 10^{-2}$
Delta | $0.01$ | $69.86$ | $0.43$ | $2.9\times 10^{-4}$ | $3.5\times 10^{-3}$ | $-3.7\times 10^{-4}$ | $2.8\times 10^{-4}$
Delta220 | $0.01$ | $69.86$ | $0.43$ | $-9.6\times 10^{-3}$ | $-6.5\times 10^{-3}$ | $-3.7\times 10^{-4}$ | $2.8\times 10^{-4}$
Table 1: Values $\alpha_{0.01}$ of the deviation parameter $\alpha$ inducing a
$1\%$ shift in $f_{220}$, alongside the effective measurable final mass and
spin $\\{\tilde{M}_{f},\tilde{\chi}_{f}\\}$ as well as the effective shifts
$\\{\tilde{\delta}f_{lmn},\tilde{\delta}\tau_{lmn}\\}$ of the subdominant
modes, for the different spectra considered in this work. Here $\alpha$ is a
collective name for the additional theory parameter. It represents
$\\{\zeta_{\rm EdGB},Q,\mu,\zeta_{\rm dCS},\Delta,\Delta_{220}\\}$ depending
on the spectrum.

For each spectrum and mass ratio, we quantify these differences in Tab. 1. We
present the values of the deviation parameters $\alpha$ such that the
deviation in the dominant mode frequency is at $1\%$ level i.e., $|\delta
f_{220}|=0.01$. In Sec. V, using the values in Tab. 1, we study the ability of
BH spectroscopy to constrain $\tilde{\delta}f_{lmn}$ away from zero i.e.,
exclude the GR Kerr BH spectrum.

Figure 1: Rows 1-4: Relative deviations $\delta f_{lmn}$ (left) and relative
effective deviations $\tilde{\delta}f_{lmn}$ (right) from the GR Kerr BH
spectrum, as induced by the different modified spectra discusses in Sec. III:
EdGB, Kerr-Newman, Horndeski and dCS, respectively. Last row: Relative
effective deviations $\tilde{\delta}f_{lmn}$ induces by the Delta spectrum
(left) and the Delta220 spectrum (right).

##  IV Methods and implementations

We construct ringdowns that comprise of
$(l,m,n)\in\\{(2,2,0),(3,3,0),(2,1,0)\\}$ modes with the modified QNM spectra
described in the previous Section 222We do not consider overtones for
simplicity as their measurablity and physical interpretation is still debated
even when assuming that the underlying theory of gravity is GR Bhagwat  _et
al._ (2020b); Cotesta  _et al._ (2022); Jiménez Forteza  _et al._ (2020);
Dhani (2021); Isi and Farr (2022); Cheung  _et al._ (2023); Mitman  _et al._
(2023).. We remind that the coupling parameters of the theory or the extra
charges that modify the QNM spectra are chosen such that the dominant-mode
frequency $f_{220}$ differs from the GR Kerr value $f_{220}^{\rm Kerr}$ by
$1\%$. This is a heuristic choice but studying the effect of $1\%$ deviation
in $f_{220}$ is a reasonable goal from the data analysis perspective as the
next-generation detectors is expected to measure $f_{220}$ with sub-percent
accuracy Maselli  _et al._ (2017); Shi  _et al._ (2019); Bhagwat  _et al._
(2022). For Delta and Delta220 modifications, the sign of the deviation is
chosen to be positive i.e., fractional frequency shift of $+0.01$.

We fix the final mass $M_{f}=70M_{\odot}$ in the detector frame. To compute
$\rho_{\rm RD}$, we consider events with extrinsic parameters listed in Tab.
2. These choices are compatible with the GW150914 event, with the only
exception of the inclination angle; the posteriors distribution for the
inclination angle of GW150914 prefers a face-on/face-off orientation for which
the subdominant modes excitation is suppressed significantly. Instead, we use
a more optimal inclination angle for BH spectroscopy and set $\iota=\pi/3$.
For each modified QNM scenario, we study mass ratio $q\in\\{1.4,3,5\\}$. The
final spin of the remnant and the mode excitation amplitudes
$\mathcal{A}_{lmn}$ are computed from the $q$ using the numerical fits
provided in Hofmann  _et al._ (2016) and Forteza  _et al._ (2023),
respectively, by assuming non-spinning progenitors, while the absolute
amplitude scale $\mathcal{A}_{220}$ is set by Eq. (5) in Gossan  _et al._
(2012)333Here we emphasize that depending on the modified theory, these
assumptions may not hold to differing extents. Amplitudes are governed
dominantly by the plunge-merger dynamics and we typically need a fully
numerical simulation to infer them. However, we do not have numerical
simulations for most of the theories considered here; thus, we do not have
knowledge of the amplitudes of mode excitations in these theories. Moreover,
there can be differences in the energy and momentum radiated and therefore, in
the expressions for the remnant mass and spin. Since we are only interested in
providing order of magnitude estimates, we use the GR amplitudes and remnant
properties as an approximate proxy for practicality.. In particular,
$\chi_{f}\approx\\{0.67,0.54,0.52\\}$ for the three values of $q$ chosen here.
The phases $\phi_{lmn}$ do not significantly affect the recovery of the QNM
frequencies and damping times (c.f., Bhagwat  _et al._ (2020b); Berti  _et
al._ (2006a)) and therefore for this study, we set it to $0$.

The ringdown waveform is modeled as $h=h_{+}+ih_{\times}$, with

$\displaystyle
h_{+}=\sum_{l,m>0,n}\mathcal{A}_{lmn}~{}{{}_{-2}}\tensor{Y}{{}_{lm,+}}{}(\iota)e^{-t/\tau_{lmn}}\cos\Phi_{lmn}\,,$
(12a) $\displaystyle
h_{\times}=\sum_{l,m>0,n}\mathcal{A}_{lmn}~{}{{}_{-2}}\tensor{Y}{{}_{lm,\times}}{}(\iota)e^{-t/\tau_{lmn}}\sin\Phi_{lmn}\,,$
(12b)

where $\Phi_{lmn}=2\pi f_{lmn}t+\phi_{lmn}$, $\mathcal{A}_{lmn}$ and
$\phi_{lmn}$ are the (real) excitation amplitudes and phases of the modes, and
the plus and cross spherical harmonics are defined by444A global phase in the
definition of the spherical harmonics can be reabsorbed in the definition of
the phases $\phi_{lmn}$ of the ringdown modes.

$\displaystyle{{}_{-2}}\tensor{Y}{{}_{lm,+}}{}(\iota)={{}_{-2}}\tensor{Y}{{}_{lm}}(\iota,0)+(-1)^{l}{{}_{-2}}\tensor{Y}{{}_{lm}}(\iota,0)\,,$
(13a)
$\displaystyle{{}_{-2}}\tensor{Y}{{}_{lm,+}}{}(\iota)={{}_{-2}}\tensor{Y}{{}_{lm}}(\iota,0)-(-1)^{l}{{}_{-2}}\tensor{Y}{{}_{lm}}(\iota,0)\,.$
(13b)

In writing the expressions (12a)-(12b), we assume a non-precessing quasi-
circular binary progenitor. For these systems equatorial reflection symmetries
gives
$\mathcal{A}_{l-mn}e^{i\phi_{l-mn}}=(-1)^{l}\mathcal{A}_{lmn}e^{-i\phi_{lmn}}$
and we simultaneously sum over $\pm m$ using the symmetry relation
$\omega_{l-mn}=-\omega^{*}_{lmn}$555We use $\omega_{lmn}$ to denote the
prograde modes of the spectrum. The QNM solutions also contain a set of
retrograde modes, but in the numerical simulations it is seen that these modes
are not excited significantly London  _et al._ (2014). See also Dhani (2021);
Finch and Moore (2021) on the role of retrograde modes in describing the
ringdown waveform.. We assume a circularly polarized ringdown as the numerical
simulations favour it (c.f., Borhanian  _et al._ (2020) for a details). We
also use spherical harmonics instead of the more natural spheroidal harmonics
basis function for ringdown; this is a fairly standard approximation which is
known to introduces substantial errors only in high spin limits Berti  _et
al._ (2006b); Berti and Klein (2014).

To predict the statistical uncertainty in the measurement of
$\tilde{\delta}f_{lmn}$, we employ a Fisher matrix formalism following our
work in Bhagwat  _et al._ (2022). We use the power spectral density (PSD) of
the next-generation ground-based detector – the Einstein Telescope, assuming a
triangular geometry, located in Sardinia Gossan  _et al._ (2022) and operating
at the ET-D sensitivity Hild  _et al._ (2011). However, note that all the
qualitative statements and ball-park numbers reported in this study should
hold for any detector for a given $\rho_{\rm RD}$. Our study is not heavily
influenced by the choice of the detector as long as the modifications in the
spectra are chosen to have $1\%$ deviation of $f_{220}$. Note that while the
results in this study will be approximately similar for the next space-based
detector LISA, LISA will be sensitive to supermassive binary BH signals and
therefore, the theory-specific parameters i.e., coupling constants or extra
charges, that lead to $1\%$ deviation of $f_{220}$ will be different from the
choice made here. However, we expect theory-parameters similar to the choice
made here for the Cosmic Explorer, as it is sensitive to stellar mass ranges
similar to the Einstein Telescope Berti  _et al._ (2016).

For computing the Fisher covariance matrix, we parametrize the waveform as
Eq.s (6a)-(6b) and (7a)-(7b), i.e., using effective parameters
$\\{\tilde{M}_{f},\tilde{\chi}_{f},\tilde{\delta}f_{lmn},\tilde{\delta}\tau_{lmn}\\}$
instead of the traditionally used $\\{f_{lmn},\tau_{lmn}\\}$. Next, for a
choice of theory parameters that produces a modified spectra such that
$f_{220}$ deviates by $1\%$ w.r.t. GR Kerr BH, we map the true values
$\\{M_{f},\chi_{f}\\}$ to the measured values
$\\{\tilde{M}_{f},\tilde{\chi}_{f}\\}$ using Eq.s (2) and (8a)-(8b). Note that
once we compute the values of $\\{\tilde{M}_{f},\tilde{\chi}_{f}\\}$,
deviation parameters for all the subdominant modes get fixed. We construct the
covariance matrix for a set of $4N$ parameters, where $N$ is the number of
modes as –

$\bm{\theta}=\\{\tilde{M}_{f},\tilde{\chi}_{f},\tilde{\delta}f_{lmn},\tilde{\delta}\tau_{lmn},\log_{10}\mathcal{A}_{lmn},\phi_{lmn}\\}\,.$
(14)

For the dominant mode, we estimate $\\{\tilde{M}_{f},\tilde{\chi}_{f}\\}$, and
not $\\{\tilde{\delta}f_{220},\tilde{\delta}\tau_{220}\\}$.

Next, we quantify the departure of the modified QNM spectra from GR Kerr QNM
spectra using $Q_{\rm GR}$ as a measure, where $Q_{\rm GR}$ is defined as the
quantile at which the posterior distribution of $\tilde{\delta}f_{lmn}$
excludes zero. Higher the value of $Q_{\rm GR}$, more confidently we can
exclude the null hypothesis that the ringdown contains QNMs corresponding to a
GR Kerr BH. $Q_{\rm GR}$ can either be defined mode-wise over a marginalized
1-d probability distribution $P(\tilde{\delta}f_{lmn})$ or defined on a joint
probability distribution of all the measured modes. Here, we use $Q_{\rm GR}$
defined on the 1-d posterior distributions corresponding to – a)
$\tilde{\delta}f_{330}$ : $Q_{\rm GR}^{330}$, b) $\tilde{\delta}f_{210}$ :
$Q_{G\rm R}^{210}$ and c) on a joint posterior distribution of
$\tilde{\delta}f_{330}$ and $\tilde{\delta}f_{210}$ : $Q_{\rm GR}^{\rm 2D}$.

In the Fisher matrix formalism, by construction, $P(\tilde{\delta}f_{lmn})$ is
a Gaussian distribution centered at the true value of $\tilde{\delta}f_{lmn}$.
From this, $Q^{lmn}_{\rm GR}$ can be easily computed as

$Q^{lmn}_{\rm GR}={\rm erf}\left(p/\sqrt{2}\right)\,,$ (15)

where the $p$-value is given as

$p=\frac{|\tilde{\delta}f_{lmn}|}{\sigma}\,.$ (16)

Further, Eq. (16) can be generalized to a 2-dimensional case of the joint
posterior $P(\tilde{\delta}f_{330},\tilde{\delta}f_{210})$. In the cases of
2-D posteriors –

$Q_{\rm GR}^{\rm
2D}=1-\exp\left(-\frac{1}{2}\vec{\mu}^{T}\cdot\mathbf{\Sigma}^{-1}\cdot\vec{\mu}\right)$
(17)

where $\vec{\mu}$ is the vector corresponding to the true values of
$\\{\tilde{\delta}f_{330},\tilde{\delta}f_{210}\\}$ and $\mathbf{\Sigma}$ is
the covariance matrix.

##  V Results

ra | dec | $\psi$ | $\iota$ | $t_{\rm GPS}$ | $d_{L}({\rm Mpc})$
---|---|---|---|---|---
$1.16$ | $-1.19$ | $1.12$ | $\pi/3$ | $1126259462.423$ | $403$
Table 2: Right ascension, declination, polarization angle, inclination angle,
GPS time and luminosity distance for GW150914 that were used to compute the
SNR and the Fisher covariance matrix.

Before turning our attention to the modified spectra, we first look at the
measurability of a GR Kerr BH spectrum with the next generation ground-based
detectors. We used ET-D PSD to illustrate the expected orders of magnitudes
and trends. Specifically, we consider ringdowns corresponding to
$q\in\\{1.4,3,5\\}$ with $M_{f}=70M_{\odot}$ and the extrinsic parameters
displayed in Tab. 2. Tab. 3 shows $\rho_{\rm RD}$ and the expected measurement
uncertainties on the deviation parameters $\sigma(\tilde{\delta}f_{lmn})$ —
note that for a GR Kerr BH spectrum, $\tilde{\delta}f_{lmn}=\delta f_{lmn}$
and we use tilde here for consistency of notation. We report the normalized
uncertainties $\kappa\equiv\rho_{\rm RD}\sigma$, where $\rho_{\rm RD}$ is the
SNR in the ringdown and $\sigma$ is the expected statistical uncertainty in
the parameter computed with a Fisher information matrix framework 666Note, the
relative excitation amplitudes $\mathcal{A}_{lmn}/\mathcal{A}_{220}$ of the
subdominant modes increase monotonically with $q$ for non-spinning systems
Kamaretsos  _et al._ (2012a, b); Gossan  _et al._ (2012); Borhanian  _et al._
(2020); Forteza  _et al._ (2023). However, $\rho_{\rm RD}$ decreases with $q$
Gossan  _et al._ (2012). Therefore, while spectroscopy with high $q$ is
favourable for fixed $\rho_{\rm RD}$, it need not be the case for fixed
luminosity distance.. Further, we confirm that the uncertainty in the
measurement of the deviations in damping times
$\sigma(\tilde{\delta}\tau_{lmn})$ is large; therefore, we concentrate on the
subdominant mode frequencies.

$q$ | $\rho_{\rm RD}$ | $\kappa=\rho_{\rm RD}\,\sigma$
---|---|---
|  | $\tilde{M}_{f}\,(M_{\odot})$ | $\tilde{\chi}_{f}$ | $\tilde{\delta}f_{330}$ | $\tilde{\delta}f_{210}$ | $\tilde{\delta}\tau_{330}$ | $\tilde{\delta}\tau_{210}$
1.4 | 87 | 184.63 | 3.68 | 1.56 | 7.78 | 22.43 | 46.14
3 | 74 | 294.63 | 7.43 | 1.00 | 3.27 | 9.17 | 19.73
5 | 57 | 436.37 | 13.08 | 1.27 | 3.38 | 8.80 | 22.26
Table 3: $\rho_{\rm RD}$ and uncertainties $\sigma$ over mass, spin and the
deviations parameters for a Kerr BH spectrum, as measured by a triangular
configuration of ET detector with ET-D PSD. The ringdown corresponds to a BH
whose detector-frame final mass is $M_{f}=70M_{\odot}$ and the extrinsic
system parameters are enlisted in Tab. 2.

Figure 2: Projection of BH spectroscopy on the mass-spin plane for the Kerr-
Newman (top) and dCS (bottom) spectra. We set $\rho_{\rm RD}=300$ and $q=5$.
The bands correspond to the 90% credible intervals. The black and red markers
indicate the measurable and true values for mass and spin, respectively.

To appreciate the quantitative results, we begin with illustrating the
performance of BH spectroscopy for a) the best-case scenario, the dCS
spectrum, and b) the worst-case scenario, the Kerr-Newman spectrum in Fig. 2.
For both cases, we use ringdowns with $\rho_{\rm RD}=300$ and $q=5$.
Specifically, we show the projections of the 90% credibility bands of
$f_{220}$, $\tau_{220}$, $f_{330}$ and $f_{210}$ on the mass-spin plane. The
common region of intersection of $f_{220}$ and $\tau_{220}$ corresponds to the
measurable mass and spin $\\{\tilde{M}_{f},\tilde{\chi}_{f}\\}$ estimate
(indicated by a black marker), while the true value of
$\\{{M}_{f},{\chi}_{f}\\}$ (indicated by a red marker) lies outside the
intersection region. For the Kerr-Newman spectrum, all bands have a common
intersection region, and therefore, BH spectroscopy fails to detect deviations
from the GR BH Kerr QNMs. On the contrary, for the dCS spectrum, there is no
common intersection region and BH spectroscopy can be used to identify that
this spectrum is incompatible with GR at the 90% confidence level.

Next, looking at the values of $\\{\tilde{M}_{f},\tilde{\chi}_{f}\\}$ and of
$\tilde{\delta}f_{lmn}$ in Tab. 1, we observe that
$\\{\tilde{M}_{f},\tilde{\chi}_{f}\\}$ does not differs significantly from
$\\{M_{f},\chi_{f}\\}$ and $\tilde{\delta}f_{lmn}\ll 1$. Consequently, the
results reported in Tab. 3 for the GR Kerr BH spectra can be expected to
approximately hold for the modified QNM spectra too. This allows to derive a
back-of-the-envelope $\rho_{\rm RD}$ estimate required to distinguish various
modified spectra from the GR Kerr BH spectrum. Further, we can use Eq.s
(15)-(16) to find the $\rho_{\rm RD}$ necessary to exclude 0 from
$P(\tilde{\delta}f_{330})$ at $90\%$ confidence.For instance, in the case of
an EdGB spectrum, from (15) we observe that $Q^{lmn}_{\rm GR}=0.9$ corresponds
to $p\approx 1.64$. Inverting (16) for $\rho_{\rm RD}=\kappa/\sigma$ and using
the results in Tab.s 1 and 3, we get an approximate minimum value of
$\rho_{\rm RD}$ as –

$\rho^{0.9}_{\rm RD}\approx\begin{cases}502&(q=1.4)\\\ 256&(q=3)\\\
278&(q=5)\end{cases}$ (18)

If we repeat this exercise for the dCS spectrum, we find

$\rho^{0.9}_{\rm RD}\approx\begin{cases}264&(q=1.4)\\\ 164&(q=3)\\\
174&(q=5)\end{cases}$ (19)

Note that it is only for the sake of demonstrating a fast and easy approximate
estimation that we derive (18) using the values in Tab. 3, which assumed a GR
Kerr BH spectrum. However, there is no fundamental obstruction to being fully
consistent by applying Eq.s (15) and (17) using the covariance matrices from
the modified spectra.

In Fig. 3, we depict marginalized 2-d posterior estimates of
$\tilde{\delta}f_{330}$ and $\tilde{\delta}f_{210}$ for all the modified
ringdowns studied. Here again, we use ringdowns with $\rho_{\rm RD}=300$ to
illustrate $Q_{\rm GR}$ as a measure to distinguish a modified QNM spectrum
from the GR Kerr BH spectrum. When the posterior distributions are compatible
with $(0,0)$ (indicated with a black dot), the QNM spectra for the $f_{330}$
and $f_{210}$ are compatible with the GR Kerr BH spectrum. Note also that the
shape of the contours changes with $q$ (and that the contours do not trivially
shrink with changing in $q$). This foreshadows the non-trivial dependence of
$Q_{\rm GR}$ on $q$ which will be further emphasised in fig 4. We can identify
the modifications to various level of confidence – $Q_{\rm GR}^{\rm
2D}=\\{0.33,0.85,0.97\\}$ for Horndeski, $Q_{\rm GR}^{\rm
2D}=\\{0.39,0.91,0.96\\}$ for EdGB, $Q_{\rm GR}^{\rm 2D}=\\{0.85,0.99,1\\}$
for dCS for $q=\\{1.4,3,5\\}$ respectively. We can contrast this to the case
of Kerr-Newman where we do not expect to detect any deviations – we have
$Q_{\rm GR}^{\rm 2D}=\mathcal{O}(10^{-6})$ for $q=1.4$, $\mathcal{O}(10^{-3})$
for $q=3$ and $\mathcal{O}(10^{-2})$ for $q=5$. We will see in Fig. 4 that
this is true even at very high $\rho_{\rm RD}$.

Figure 3: Density plots of $\\{\tilde{\delta}f_{330},\tilde{\delta}f_{210}\\}$
for the modified spectra considered in Sec. III and different mass ratios.
Solid and dashed lines indicate the $90\%$ and $50\%$ probability contours
respectively. We assume a true injected mass $M_{f}=70M_{\odot}$ and vanishing
initial spins (see Tab. 1 for the values of the corresponding effective
measurable masses and spins). We set to $\rho_{\rm RD}=300$ to clearly display
the different trends of the spectra in distinguishing deviations from GR. The
null hypothesis under test is denoted by a black marker.

Figure 4: Values of the GR quantile $Q_{\rm GR}$ as a function of the
$\rho_{\rm RD}$. We consider quantiles for the marginalised single parameter
posteriors $P(\tilde{\delta}f_{330})$ (leftmost column) and
$P(\tilde{\delta}f_{210})$ (central column) and for the 2D joint posterior
$P(\tilde{\delta}f_{330},\tilde{\delta}f_{210})$ (rightmost column). The
quantiles are computed using the exact expressions (15) and (17).

Finally, in Fig. 4, we present our main results on $\rho_{\rm RD}$ required to
identify deviations from the GR Kerr QNM spectrum for the various modification
schemes. The first two columns show $Q_{\rm GR}$ computed using a 1-d
posterior distribution of $f_{330}$ and $f_{210}$, respectively, and the
rightmost column corresponds to the 2-d joint posterior distribution on
$f_{330}$ and $f_{210}$. Again, we study systems corresponding to three mass
ratios: in the first row, we study the near-equal-mass scenario of $q=1.4$; in
the second row, we study $q=3$ and in the last row, we study $q=5$.

Let us examine the $\rho_{\rm RD}$ at which $Q_{\rm GR}\geq 0.9$ for all the
modified QNMs considered here. We observe that $\rho_{\rm RD}$ required to
distinguish the modified QNMs from GR Kerr BH spectrum depends acutely on the
details of the modifications. Modification of the spectra, such as that
predicted by Kerr-Newman and Delta, cannot be confidently differentiated from
the GR spectrum, even when $\rho_{\rm RD}=10^{3}$. This happens because these
modifications can be approximated by the GR by suitably selecting the values
of $\tilde{M}_{f}$ and $\tilde{\chi}_{f}$. In other words, these modified
spectra are highly degenerate with the GR Kerr BH QNM in the mass-spin space.
This can be quantitatively seen in Fig. 1. For instance,
$\tilde{\delta}f_{lmn}$ for these modified spectra is an order of magnitude
smaller than $\tilde{\delta}f_{lmn}$ for other modifications such as dCS or
EdGB. In contrast to this, modifications predicted by the EdGB and Horndeski
theories can be distinguished confidently from the GR at a $\rho_{\rm
RD}\in[250,10^{3}]$. Furthermore, the dCS or Delta220-like QNM spectra could
be confidently identified with an even lower $\rho_{\rm RD}\in[250,400]$.

We highlight two non-trivial trends from Fig. 4 that we observe.

  * •

While all of them are valid measures to identify modifications in a QNM
spectrum, we find that the ability to distinguish a modified theory from GR
using 1-d $Q_{\rm GR}^{330}$, $Q_{\rm GR}^{210}$ and 2-d joint $Q_{\rm
GR}^{\rm 2D}$ are fairly different and depends on theory. While modifications
in the spectra predicted by dCS and Delta220 benefit by using $Q_{\rm
GR}^{330}$, Kerr-Newmann and Delta are more distinguishable using $Q_{\rm
GR}^{210}$. Furthermore, certain types of modifications will only manifest in
a detectable fashion in one of the $Q_{\rm GR}$ measures. For example, EdGB
spectra with $\rho_{\rm RD}\leq 10^{3}$ cannot be confidently distinguished if
we used $Q_{\rm GR}^{210}$ while it can be using $Q_{\rm GR}^{330}$ or $Q_{\rm
GR}^{\rm 2D}$.

  * •

The _distinguishability_ of a given modified spectrum does not have a
monotonic behaviour with $q$. This occurs because $q$ dictates both the mode
excitation amplitudes and the final spin of the remnant BH, and the interplay
between the two produces the non-monotonic trend we see here. For instance, at
a given $\rho_{\rm RD}$, $Q_{\rm GR}^{330}$ for dCS increases monotonically
with $q$, while $Q_{\rm GR}^{330}$ for EdGB and Horndeski spectra seem to
perform better for the case of $q=3$ than for $q=1.4$ or $5$. It is also worth
noting that the performance of $Q_{\rm GR}^{330},Q_{\rm GR}^{210}$ and $Q_{\rm
GR}^{\rm 2D}$ for different theory can differ with $q$.

##  VI Discussion and conclusion

We performed a comprehensive study on the ability of BH spectroscopy to
distinguish a modified spectra from a GR Kerr BH QNMs. We studied theory-
motivated modifications for QNM spectra that were publicly available –
specifically, EdGB, Kerr-Newman, Horndeski and dCS theory, and for two
phenomenologically modified QNM spectra – Delta and Delta220. To investigate
$\rho_{\rm RD}$ necessary to distinguish these modified spectra from a GR Kerr
BH spectrum, we assessed the performance of BH spectroscopy using a Fisher
information matrix formalism. The ringdowns with the modified spectra are
generated such that in each case $f_{220}$ deviates from the corresponding GR
Kerr QNM by $1\%$.

First, we re-iterate that we can only measure the effective deviation
parameters $\tilde{\delta}f_{lmn}$ and not the absolute deviation $\delta
f_{lmn}$ as the mass and the spin estimation can be chosen suitably to
compensate for the deviations of the QNMs. Further, we show that in many
theories $\tilde{\delta}f_{lmn}$ significantly differs from $\delta f_{lmn}$.
Therefore, to study the ability BH spectroscopy to distinguish modified
spectra, the framework must be setup using the measurable effective deviation
parameters $\tilde{\delta}f_{lmn}$ .

We found that $\rho_{\rm RD}$ necessary to distinguish a modified spectrum
from the GR Kerr one depends on the details of the modification and on the
mass ratio. Further, we used three different measures that quantify the amount
by which: a) the 1-d posterior estimate of $\tilde{\delta}f_{330}$ excludes 0
a.k.a. $Q_{\rm GR}^{330}$, b) the 1-d posterior estimate of
$\tilde{\delta}f_{210}$ excludes 0 a.k.a. $Q_{\rm GR}^{210}$ and c) the 2-d
joint posterior estimation of $\tilde{\delta}f_{330}-\tilde{\delta}f_{210}$
excludes (0,0) a.k.a. $Q_{\rm GR}^{\rm 2D}$. Depending on the spectrum and the
magnitudes of $\tilde{\delta}f_{lmn}$, the performance of $Q_{\rm
GR}^{330},Q_{\rm GR}^{210}$ and $Q_{\rm GR}^{\rm 2D}$ can vary significantly.
Roughly, we find that a $\rho_{\rm RD}\geq 150$ is required to identify
deviations from GR at a 90% credibility level. This range of $\rho_{\rm RD}$
is attainable with the next generation detectors and BH spectroscopy will be a
powerful tool to constrain GR as well as for identifying modifications to Kerr
BH spectrum.

In previous works Bhagwat  _et al._ (2022, ), we studied the measurability of
QNM parameters, that is, the statistical uncertainty with which a QNM mode
parameter can be estimated from a signal to assess the landscape of BH
spectroscopy with a next-generation detector. However, we note here that the
subdominant QNM mode with the best measurability may not always be the optimal
mode for distinguishing a modified QNM spectrum from a GR Kerr BH QNM
spectrum. The best mode to identify a departure from GR Kerr spectrum depends
on the interplay between measurablity and the details of modification of the
spectra. For instance, measurablity of $f_{330}$ is better than $f_{210}$ in a
non-spinning binary BH ringdown. However, if the compact object is a Kerr-
Newman BH instead of a Kerr BH, $f_{330}$ measured in the ringdown would be
compatible with the Kerr BH even for ringdown with $\rho_{\rm RD}\sim 10^{3}$.
The modifications in the spectra in this case can be observed predominantly by
looking at $f_{210}$. Therefore, using a setup that uses information in all
measurable QNM mode parameters is optimal while looking for departure from the
GR Kerr spectrum in a ringdown signal. Greater the number of QNM mode
parameters measured, greater the chances that we can distinguish a modified
QNM spectra from the GR Kerr spectra.

Furthermore, among the modified QNM spectra we have studied, we see the
theories separate out into two groups :

  * •

EdGB, Horndeski, dCS, Delta220: For these spectra, a GR Kerr ringdown can be
excluded at $90\%$ confidence level with a SNR $\rho^{0.9}_{\rm
RD}\in[150,500]$. The deviations manifest more prominently in the $(3,3,0)$
mode and the $(2,1,0)$ mode offers a relatively poor constraints.

  * •

Kerr-Newman, Delta: here the deviations manifest almost exclusively in the
$(2,1,0)$ mode, with $Q_{\rm GR}^{210}$ returning the most performative no-
hair test. However, the $\rho_{\rm RD}$ required to exclude 0 at 90%
confidence is so high that these deviations are indistinguishable, even at
$\rho_{\rm RD}=1000$.

Finally, works such as this are essential for gauging the potential of
ringdown-based tests of gravity; but unfortunately, we are restricted by
publicly available modified QMN spectra. While it is difficult to solve the
QNM spectra in modified theories of gravity, to adequately prepare for test of
GR with next-generation detectors including designing the implementation of BH
spectroscopy, we feel that the field would benefit substantially from
investing effort in this direction.

###### Acknowledgements.

We thank L. Pierini for clarifications about the work in Pierini and Gualtieri
(2022). We thank E. Berti, L. Gualtieri, A. Maselli and P. Pani for fruitful
discussions. C.P. is supported by European Union’s H2020 ERC Starting Grant
No. 945155–GWmining and by Cariplo Foundation Grant No. 2021-0555. 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.

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