# Quadratic Quasi-Normal Modes of a Schwarzschild Black Hole

Bruno Bucciotti<EMAIL_ADDRESS>Leonardo Juliano
<EMAIL_ADDRESS>Scuola Normale Superiore, Piazza dei Cavalieri 7,
56126, Pisa, Italy  INFN Sezione di Pisa, Largo Pontecorvo 3, 56127 Pisa,
Italy  Adrien Kuntz<EMAIL_ADDRESS>SISSA, Via Bonomea 265, 34136
Trieste, Italy and INFN Sezione di Trieste  IFPU - Institute for Fundamental
Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy  Enrico
Trincherini<EMAIL_ADDRESS>Scuola Normale Superiore, Piazza dei
Cavalieri 7, 56126, Pisa, Italy  INFN Sezione di Pisa, Largo Pontecorvo 3,
56127 Pisa, Italy

(August 31, 2024)

###### Abstract

Quadratic quasi-normal modes, generated at second order in black hole
perturbation theory, are a promising target for testing gravity in the
nonlinear regime with next-generation gravitational wave detectors. While
their frequencies have long been known, their amplitudes remain poorly
studied. We introduce regular variables and compute amplitudes for
Schwarzschild black holes with the Leaver algorithm. We find a nonlinear ratio
$\mathcal{R}\simeq 0.154e^{-0.068i}$ for the most excited $\ell=4$ mode,
matching results from Numerical Relativity. We also predict new low-frequency
$\ell=2$ quadratic modes.

Introduction Observations of binary black hole (BH) mergers hold promise for
testing General Relativity (GR) in the strong gravity regime with
unprecedented precision and perhaps even finding evidence of novel phenomena
in the gravitational domain. While the precise moment of the actual merger can
only be described using Numerical Relativity (NR), analytic perturbation
theory plays a fundamental role at earlier and later times. In this letter, we
will focus on the final phase of the merger process, known as the ringdown,
specifically examining the scenario where the black hole formed in the process
has zero angular momentum.

Linear perturbations around the spacetime geometry resulting from the merger
are described by a superposition of a discrete set of damped sinusoids, known
as quasinormal modes (QNMs) [1]. They are characterized by complex frequencies
determined in GR solely by the mass and angular momentum of the black hole.
This is because, at linear order and in the limit of zero spin, the dynamics
of the two physical degrees of freedom, corresponding to the two gravitational
wave polarizations, can be described by two “master scalar” variables
$\Psi_{\pm}^{(1)}$ that obey independent wave equations in a potential
$V_{\pm}$:

$\Big{(}\Box-V_{\pm}\Big{)}\Psi_{\pm}^{(1)}=0\;.$ (1)

The equations are decoupled because the two perturbations transform oppositely
under parity (hence the subscript $\pm$), which is a symmetry of the
Schwarzschild geometry. The resulting frequencies $\omega_{\ell mn}$, labeled
by two angular harmonic numbers $(\ell,m)$ and an overtone number $n$, can be
utilized for accurate gravity testing [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. On the other hand, the amplitudes
of these modes depend on the initial conditions of the merger and should be
fitted against data or Numerical Relativity simulations [22, 23, 24, 25, 26,
27, 7, 28, 29].

Since the perturbations decay exponentially, it is not surprising that linear
Black Hole Perturbation Theory (BHPT) [30, 31] can effectively model the
ringdown after a time roughly equivalent to the QNM decay timescale. However,
during the initial phase of the ringdown, just after the peak of the strain,
it becomes essential to account for nonlinear effects. This is particularly
relevant because the first few cycles encompass most of the observable signal,
providing the best opportunities for precise measurements. Indeed, recent
studies have shown that second-order BHPT may be necessary to accurately
describe the signals obtained from NR [32, 33, 34, 35, 29, 36, 37, 38].

To second order in perturbation theory, the dynamics is governed by the same
wave equation for the second order master scalars $\Psi_{\pm}^{(2)}$, but now
with a source term $S^{(2)}$ quadratic in the linear perturbations,

$\Big{(}\Box-V_{\pm}\Big{)}\Psi_{\pm}^{(2)}=S^{(2)}\;.$ (2)

As a consequence, there exists a set of quadratic QNMs (QQNMs) generated by
the product of two linear modes. For each $(\ell,m)$ the waveform contains
additional modes whose frequencies can be trivially obtained from the product
of two monochromatic first-order waves as
$\omega^{\mathrm{Q}}=\omega_{\ell_{1}m_{1}n_{1}}+\omega_{\ell_{2}m_{2}n_{2}}$
or
$\omega^{\mathrm{Q}}=\omega_{\ell_{1}m_{1}n_{1}}-(\omega_{\ell_{2}m_{2}n_{2}})^{*}$
[39], where $*$ denotes complex conjugation. A crucial feature of QQNMs is
that, unlike the linear ones, their amplitude is also entirely determined by
GR. This essential property arises from the fact that these modes solve a non-
homogeneous equation (2). Despite this fact, to date, only a few specific
second-order amplitudes have been computed, either by fitting NR simulations
[32, 33, 34, 35, 29, 36, 37, 38] or by integrating BHPT equations [40, 41, 42,
43]. Since at least the loudest QQNM will be observable by next-generation
interferometers [44], there is an increased need for precise computation of
these amplitudes.

This letter aims to provide the amplitude of all low multipole QQNMs for a
Schwarzschild BH. One of the main obstacles in solving the second-order
equation can be traced back to the redundancy that inevitably arises when
describing the two degrees of freedom (DOF) of a massless spin-2 particle in
terms of the metric tensor. Conceptually, the procedure should follow what is
done at linear order and appears straightforward: one first fixes the gauge
redundancy associated with diffeomorphism invariance. Secondly, one selects
two master scalar variables $\Psi_{\pm}^{(2)}$, expressed in terms of the
metric components, which capture the dynamics of the physical DOF. Once the
equation for each $\Psi_{\pm}^{(2)}$ is solved with the appropriate boundary
conditions, it becomes possible to reconstruct the full metric and transition
to an outgoing radiation gauge, where the amplitude of the physical
gravitational wave observable at infinity can be computed.

Regrettably, as noted in the literature [45, 46, 47, 48, 49, 50, 40, 41, 43],
when the choice of a second-order master scalar is equivalent to the first-
order one, quadratic perturbations exhibit a divergent behavior at infinity or
the black hole horizon. Although this divergence will eventually become
immaterial, it signifies a poor choice of variables, making the extraction of
the subleading physical effect extremely challenging. To circumvent this
problem, we will demonstrate how to define two new master scalars for
arbitrary $(\ell,m)$ that are regular everywhere. With these variables, the
resulting equations, along with appropriate QNM boundary conditions, can be
integrated using the Leaver algorithm [51].

Our main results are shown in fig. 1, where one can directly read the
amplitude of QQNMs generated by several dominant linear modes. One of our most
surprising findings is the existence of QQNMs with an appreciable amplitude in
the $\ell=2$ multipole, a point we will elaborate on later. Additionally, in
fig. 2 we illustrate the impact of quadratic modes on a typical ringdown
waveform for a GW190521-like merger. From the figure, it is evident that the
quadratic oscillations occur at twice the frequency of the linear modes, and
the fact that the quadratic amplitude is proportional to the square of the
linear one. Notice that, for the purpose of this figure, we have extrapolated
our results to the spinning case by simply neglecting the spin dependence of
the nonlinear ratio $\mathcal{R}$. This assumption is approximately supported
by NR simulations [29, 37]. While we explicitly present in this article a set
of amplitudes for the first few QQNMs, our complete results are available
online [52]. In the following, we will use units in which $G=c=1$.

Second-order BHPT

Figure 1: Nonlinear ratio of amplitudes $|\mathcal{R}|$ for different mode
numbers. The $x$ axis represents the mode numbers $\ell mn\mathfrak{m}$ of the
first linear parent mode, while the second linear parent mode is represented
with different markers. Filled symbols represents the combination of two
normal modes with $m>0$
($\omega^{\mathrm{Q}}=\omega_{\ell_{1}n_{1}+}+\omega_{\ell_{2}n_{2}+}$), while
empty ones represents a normal mode with $m>0$ combined with a mirror mode
with $m<0$
($\omega^{\mathrm{Q}}=\omega_{\ell_{1}n_{1}+}-\big{(}\omega_{\ell_{2}n_{2}+}\big{)}^{*}$).
A missing marker means that $|\mathcal{R}|<2.5\times 10^{-4}$.

We expand the spacetime metric as $g_{\mu\nu}=\bar{g}_{\mu\nu}+\varepsilon
h^{(1)}_{\mu\nu}+\varepsilon^{2}h^{(2)}_{\mu\nu}$, where $\varepsilon$ is a
small bookkeeping parameter for the perturbation order (physically
corresponding to the small magnitude of linear QNMs (LQNMs)) and
$\bar{g}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=-f(r)\mathrm{d}t^{2}+f(r)^{-1}\mathrm{d}r^{2}+r^{2}\big{(}\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\big{)}$
is the Schwarzschild metric, with $f(r)=1-2M/r$. Up to second order,
Einstein’s equations read

$\displaystyle G^{(1)}_{\mu\nu}\big{[}h^{(1)}\big{]}$ $\displaystyle=0\;,$ (3)
$\displaystyle G^{(1)}_{\mu\nu}\big{[}h^{(2)}\big{]}$
$\displaystyle=-G^{(2)}_{\mu\nu}\big{[}h^{(1)},h^{(1)}\big{]}\;,$ (4)

where we schematically denoted by $G^{(1)}_{\mu\nu}\left[\cdot\right]$ the
part of the Einstein tensor linear in perturbations and by
$G^{(2)}_{\mu\nu}\big{[}\cdot,\cdot\big{]}$ its part bilinear in
perturbations, suppressing indices on $h_{\mu\nu}$ for clarity. Thus, the LQNM
$h^{(1)}$ will provide a source term for the QQNM $h^{(2)}$; notice that the
same differential operator acts similarly on $h^{(1)}$ and $h^{(2)}$ on the
left-hand side of these equations. Spherical symmetry of the background allows
us to split spacetime between a “$t-r$ plane” $\mathcal{M}$ (indices
$a,b,\dots$ and metric $\bar{g}_{ab}=\mathrm{diag}\left(-f,f^{-1}\right)$) and
the 2-sphere $\mathcal{S}$ (indices $A,B,\dots$ and metric
$\Omega_{AB}=\mathrm{diag}(1,\sin^{2}\theta)$). We decompose metric
perturbations according to

$\displaystyle h_{ab}^{(i)}$ $\displaystyle=\sum_{\omega,\ell,m}e^{-i\omega
t}h_{ab,\ell m\omega}^{(i)}Y^{\ell m}\;,$ (5) $\displaystyle h_{aB}^{(i)}$
$\displaystyle=\sum_{\omega,\ell,m}e^{-i\omega t}h_{a+,\ell
m\omega}^{(i)}Y_{B}^{\ell m}+h_{a-,\ell m\omega}^{(i)}X_{B}^{\ell m}\;,$ (6)
$\displaystyle h_{AB}^{(i)}$ $\displaystyle=\sum_{\omega,\ell,m}e^{-i\omega
t}h_{\circ,\ell m\omega}^{(i)}\Omega_{AB}Y^{\ell m}$ $\displaystyle+h_{+,\ell
m\omega}^{(i)}Y_{AB}^{\ell m}+h_{-,\ell m\omega}^{(i)}X_{AB}^{\ell m}\;,$ (7)

where $i=1,2$ is the order of perturbation, the sum goes over all QNM
frequencies $\omega$ and mode numbers $\ell,m$, and
$Y,Y_{A},X_{A},Y_{AB},X_{AB}$ are a set of tensor spherical harmonics defined
in app. A. To streamline the discussion, we will omit the indices $\ell
m\omega$ from variables in the following. We will focus on perturbations with
$\ell\geq 2$, as according to the peeling theorem [53, 54, 55], these are the
only ones that carry radiation to infinity. Following the approach outlined in
[56], we define a gauge-invariant variable $\tilde{h}_{\mu\nu}$, decomposed in
spherical harmonics as in (5)-(7), which coincides with the metric
perturbation $h_{\mu\nu}$ in Regge-Wheeler-Zerilli (RWZ) gauge where
$h_{\pm}=h_{a+}=0$. Detailed expressions are provided in app. A. On the other
hand, we want to extract the QNM amplitudes at large distances and in the
physical Transverse-Traceless (TT) gauge, where the (real) $\mathfrak{h}_{+}$
and $\mathfrak{h}_{\times}$ components of the metric perturbation
$h_{\mu\nu}=h_{\mu\nu}^{(1)}+\varepsilon h_{\mu\nu}^{(2)}$ are, up to
$\mathcal{O}(r^{-2})$:

$\mathfrak{h}_{+}-i\mathfrak{h}_{\times}=\frac{M}{r}\sum_{\ell
m\mathcal{N}}\mathcal{A}_{\ell
m\mathcal{N}}e^{i\omega_{\ell\mathcal{N}}(r_{*}-t)}\vphantom{|}_{-2}Y^{\ell
m}(\theta,\phi)\;.$ (8)

In this equation, which applies to both linear and quadratic order,
$\vphantom{e}{}_{-2}Y$ are the spin-weighted spherical harmonics,
$r_{*}=r+2M\log(r/2M-1)$ is the tortoise coordinate, $\mathcal{A}_{\ell
m\mathcal{N}}$ and $\omega_{\ell\mathcal{N}}$ are the amplitudes and
frequencies of QNMs, and $\mathcal{N}$ is an additional mode number defined as
follows. For linear modes, $\mathcal{N}=(n,\mathfrak{m})$ is composed of the
overtone number $n$ and of the mirror modes $\mathfrak{m}=\pm$ . A mirror mode
has a frequency related to standard modes by complex conjugation:
$\omega_{\ell n-}=-\omega_{\ell n+}^{*}$. Note that for a Schwarzschild BH,
QNMs frequencies do not depend on the mode number $m$; we will use the
convention that standard $\mathfrak{m}=+$ modes have positive real part. For
quadratic modes,
$\mathcal{N}=(\ell_{1},m_{1},n_{1},\mathfrak{m}_{1})\times(\ell_{2},m_{2},n_{2},\mathfrak{m}_{2})$
contains the mode numbers of the product of linear modes which generate them
[39]. Finally, we will assume (as is common in the literature [26, 27, 44,
57]) that the strain (11) has a circular polarization pattern. This implies
that mirror modes with $m<0$ have the same amplitude as standard modes with
$m^{\prime}=-m>0$ up to a phase. This assumption is typically well satisfied
in practise [26] 111Notice that mirror modes are related to but different than
retrograde modes introduced in [57, 29], whose amplitude are suppressed in a
typical merger [3, 25, 33, 58]..

By relating $\tilde{h}$ to $\mathfrak{h}_{+}$ and $\mathfrak{h}_{\times}$, it
can be checked that $\tilde{h}_{ab}^{(i)}=\mathcal{O}(r)$,
$\tilde{h}_{a-}^{(i)}=\mathcal{O}(r)$,
$\tilde{h}_{\circ}^{(i)}=\mathcal{O}(r^{2})$ for $r\rightarrow\infty$
regardless of the gauge choice (since $\tilde{h}$ is gauge-invariant). The
precise relationships between asymptotic amplitudes of the various variables
will be provided in a forthcoming publication [59]. We now define modified RWZ
variables as

$\displaystyle\Psi_{\mathrm{-}}^{(i)}$
$\displaystyle=\frac{r}{\lambda}\bigg{[}D_{r}\tilde{h}_{t-}^{(i)}-D_{t}\tilde{h}_{r-}^{(i)}-\frac{2}{r}\tilde{h}_{t-}^{(i)}\bigg{]}+g^{(i)}_{\mathrm{-}}(r)\;,$
(9) $\displaystyle\Psi_{\mathrm{+}}^{(i)}$
$\displaystyle=\frac{2r}{\ell(\ell+1)}\Bigg{[}\frac{\tilde{h}_{\circ}^{(i)}}{r^{2}}+\frac{1}{\lambda+3M/r}\Bigg{(}f^{2}(r)\tilde{h}_{rr}^{(i)}$
$\displaystyle-
rf(r)D_{r}\bigg{(}\frac{\tilde{h}_{\circ}^{(i)}}{r^{2}}\bigg{)}\Bigg{)}\Bigg{]}+g^{(i)}_{\mathrm{+}}(r)\;,$
(10)

where $D$ denotes the covariant derivative on $\mathcal{M}$,
$\lambda=(\ell+2)(\ell-1)/2$, and $g_{\pm}^{(i)}$ is a _regularizing_ term
present at second order only (i.e., $g_{\pm}^{(1)}=0$) enforcing the correct
asymptotic limits. Indeed, one can check that, were that term not present, the
RWZ variables would exhibit more divergent behaviors at second order compared
to the first-order QNM [45, 46, 47, 48, 49, 50, 40]. Our definition ensures
that

$\Psi_{\pm}^{(i)}\mathrel{\mathop{\simeq}\limits_{{r\rightarrow\infty}}}\mathcal{B}^{(i)}_{\pm}e^{i\omega(r_{*}-t)}\;,\;\Psi_{\pm}^{(i)}\mathrel{\mathop{\simeq}\limits_{{r\rightarrow
2M}}}\mathcal{C}^{(i)}_{\pm}e^{-i\omega(r_{*}+t)}\;,$ (11)

where $\mathcal{B}^{(i)}_{\pm}$, $\mathcal{C}^{(i)}_{\pm}$ are two constants.
At second order, a specific QQNM is generated by the product of two LQNMs
$\Psi_{1}^{(1)}$ and $\Psi_{2}^{(1)}$ (for brevity we include the parity of
first-order modes in the set of suppressed indices). We thus take

$g^{(2)}_{\pm}=\left(a_{\pm,2}r^{2}+a_{\pm,1}r\right)\Psi^{(1)}_{1}\Psi^{(1)}_{2}$
(12)

where $a_{\pm,1},a_{\pm,2}$ are chosen to cancel the divergences of the
unregulated master scalars. Another possibility would have been to work with
“unregularized” variables; however, for numerical accuracy we have chosen here
to analytically subtract these spurious terms. We have verified that they do
not contribute to the asymptotic waveform in TT gauge. Details and explicit
expressions for $g_{\pm}^{(2)}$ will be given in the companion article [59].
Eqs. (3)-(4) then translate in the RWZ equations with a source term

$\frac{\mathrm{d}^{2}\Psi_{\pm}^{(i)}}{\mathrm{d}r_{*}^{2}}+\big{(}\omega^{2}-V_{\pm}\big{)}\Psi_{\pm}^{(i)}=S_{\pm}^{(i)}\;,$
(13)

where $V_{\pm}$ are the usual RWZ potentials given in app. A and the source is
zero at first order, $S_{\pm}^{(1)}=0$. Our choice of regulator for
$\Psi^{(2)}_{\pm}$ ensures that the left-hand side of eq. (13) is the same at
first and second order. On the other hand, the explicit source at second order
is regulator-dependent and we get

$\displaystyle
S^{(2)}_{\pm}=\mathcal{F}_{1}(r)\Psi_{1}^{(1)}\Psi_{2}^{(1)}+\mathcal{F}_{2}(r)(\Psi_{1}^{(1)})^{\prime}\Psi_{2}^{(1)}$
$\displaystyle+\mathcal{F}_{3}(r)\Psi_{1}^{(1)}(\Psi_{2}^{(1)})^{\prime}+\mathcal{F}_{4}(r)(\Psi_{1}^{(1)})^{\prime}(\Psi_{2}^{(1)})^{\prime}\;,$
(14)

where ${}^{\prime}=\mathrm{d}/\mathrm{d}r$, and the four functions
$\mathcal{F}$ depend on $\ell m$, the parities of second and first-order modes
as well as
$\mathcal{N}=(\ell_{1},m_{1},n_{1},\mathfrak{m}_{1})\times(\ell_{2},m_{2},n_{2},\mathfrak{m}_{2})$.
We will give their explicit expression in the companion article [59]. We have
verified that the asymptotic behavior of LQNMs (11) imposes that
$S_{\pm}^{(2)}\propto r^{-2}$ for $r\rightarrow\infty$ and
$S_{\pm}^{(2)}\propto 1-2M/r$ for $r\rightarrow 2M$, which is consistent with
eq. (13) at second order once we impose QNM boundary conditions on
$\Psi_{\pm}^{(2)}$. This serves as a confirmation of the consistency of our
computations.

Checks and results

Figure 2:  Waveform polarization $|h_{+}|$ in eq. (8) including only linear
modes, only quadratic modes, and both of them for a GW190521-like merger.
LQNMs amplitudes are taken from the fits in [29] with progenitor parameters
from [60]; we include the $(220)$, $(221)$, $(210)$, $(330)$ and $(440)$
LQNMs. All quadratic modes amplitudes are added using results from this
article. The starting time is chosen to be $t_{0}=10M$ after the peak of the
waveform to ensure QNM regime.

Second-order RWZ equations were investigated in several other works, see e.g.
[40, 41, 42, 48, 49, 50, 56, 43, 45, 46, 47, 61]. However, we cannot directly
compare our expression for the source term in (14) to their computations since
our definitions of the master scalars in (9)-(10) differ. What is
unequivocally defined, however, is the asymptotic amplitude of quadratic modes
in the GW strain (8), which is our main result and that we will now provide.

We numerically solve eq. (13) with QNM boundary conditions (11). We employ the
adaptation of the Leaver algorithm to QQNMs described in [40], which provides
the ratio of QQNMs to LQNMs amplitudes at infinity
$\mathcal{B}^{(2)}_{\pm}/\big{(}\mathcal{B}^{(1)}_{1}\mathcal{B}^{(1)}_{2}\big{)}$
with negligible numerical error. Subsequently, we invert eqs. (9)-(10) to
reconstruct the gauge-invariant variable $\tilde{h}$ from the RWZ scalars,
which finally provides us the physical amplitude of QNMs in the TT gauge as
per eq. (8). Rotational symmetry imposes that the only dependence of
$\mathcal{A}_{\ell m\mathcal{N}}^{(2)}$ on $m,m_{1},m_{2}$ is contained in the
3j-symbol $\begin{pmatrix}\ell&\ell_{1}&\ell_{2}\\\
-m&m_{1}&m_{2}\end{pmatrix}$ which we factor out from our numerical
computations. We give in the companion csv file [52] the ratios
$\mathcal{R}=\mathcal{A}_{\ell
m\mathcal{N}}^{(2)}/\big{(}\mathcal{A}_{\ell_{1}m_{1}\mathcal{N}_{1}}^{(1)}\mathcal{A}_{\ell_{2}m_{2}\mathcal{N}_{2}}^{(1)}\big{)}$
for all $2\leq\ell,\ell_{1},\ell_{2}\leq 7$ respecting the Clebsch-Gordan
rules $|\ell_{1}-\ell_{2}|\leq\ell\leq\ell_{1}+\ell_{2}$. We also include the
cases where one of the linear amplitudes is an overtone $n=1$ or a mirror mode
$\mathfrak{m}=-$.

Our results for some of the low-lying modes are displayed in fig. 1. In
particular, we find $\mathcal{R}\simeq 0.154e^{-0.068i}$ for
$(220+)\times(220+)\rightarrow(44)$ which is the most excited nonlinear mode
in NR simulations [33, 32, 34, 35, 29, 37]. This is $\sim 10$% higher than the
semi-analytical calculation on a Kerr background given in [43]. However, our
result is in perfect agreement with the fits to NR simulations provided in
[37]222We refer here to their results based on full NR simulations, which is
still $\sim 10$% lower than their time evolution of second-order perturbation
equations. Based on this observation we conjecture that the fits to second-
order perturbation theory in [37] contains some source of systematic
uncertainties, as was suggested in that article.. Our phase seem also to be in
good agreement with [37]. Another QQNM of interest is
$(330+)\times(220+)\rightarrow(55)$ for which we find $\mathcal{R}\simeq
0.417e^{-0.081i}$, which is a $\sim 7$% difference in amplitude from the fits
provided in [29]. The phase is however half of the value provided in [29]:
this difference may be due to the extrapolation of their data to zero spin.
Finally we find $\mathcal{R}\simeq 0.194e^{-0.084i}$ for
$(220+)\times(210+)\rightarrow(43)$, a QQNM not yet mentioned in the NR
literature.

Let us now describe other peculiarities of our results. First, one has
$|\mathcal{R}|<1$ for all of the ratios displayed in fig. 1; this was not
granted since perturbation theory only requires
$\mathcal{A}^{(2)}\ll\mathcal{A}^{(1)}$. Second, it appears that combinations
of a standard $\mathfrak{m}=+$ with a mirror $\mathfrak{m}=-$ linear modes
give a non-negligible ratio in the $\ell=2$ sector; we find for example
$\mathcal{R}\simeq 0.169e^{-0.194i}$ for
$(330+)\times(2(\raisebox{0.75pt}{-}1)0-)\rightarrow(22)$ and
$\mathcal{R}\simeq 0.104e^{-1.58i}$ for
$(220+)\times(2(\raisebox{0.75pt}{-}1)0-)\rightarrow(21)$. These QQNMs weren’t
mentioned in previous works using fits to NR [33, 32, 34, 35, 29, 37]. This
may be due to the fact that: a) They have a small or even vanishing real part
of the frequency due to the relation
$\omega^{\mathrm{Q}}=\omega_{\ell_{1}n_{1}+}-(\omega_{\ell_{2}n_{2}+})^{*}$,
making them potentially more challenging to incorporate into fitting
algorithms. b) Their ratio $\mathcal{R}$ is non-negligible mostly for odd
$m_{1}$ or $m_{2}$, whose parent modes amplitudes are suppressed in an equal-
mass merger [62, 29]. Notice however that, since standard modes with $m>0$
have a similar amplitude to mirror modes with $m<0$ [57, 26], the actual
amplitude of these QQNMs – proportional to the ratio times the amplitudes of
parent modes – should not deviate significantly from the dominant QQNM in the
$\ell=5$ multipole observed in NR simulations. Third, the combination of
fundamental $n=0$ and overtone $n=1$ can also present a non-negligible ratio
$\mathcal{R}$, although these modes are expected to be more challenging to
detect due to their shorter decay time.

Conclusions In this letter we have provided the first ab initio computation of
all low multipole QQNMs amplitudes using perturbation theory on a
Schwarzschild BH background. Our findings should facilitate a straightforward
integration of second-order nonlinearities into ringdown models, as QQNMs
frequencies and amplitudes are now fully determined by their parent LQNMs. In
addition to providing more accurate models for fitting to data or NR, our work
enables to test GR by measuring the nonlinear ratio $\mathcal{R}$ in data and
comparing it with our prediction. We leave these aspects, as well as the
extension of our results to Kerr backgrounds, to future investigations.

###### Acknowledgements.

A. Kuntz acknowledges support from the European Union’s H2020 ERC Consolidator
Grant “GRavity from Astrophysical to Microscopic Scales” (Grant No.
GRAMS-815673), the PRIN 2022 grant “GUVIRP - Gravity tests in the UltraViolet
and InfraRed with Pulsar timing”, and the EU Horizon 2020 Research and
Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No.
101007855. This project made use of the Black Hole Perturbation Toolkit [63].
We would like to thank Enrico Barausse and Luca Santoni for discussions.

##  Appendix A Technical reference

Spherical harmonics. Following the literature, we define tensor spherical
harmonics as

$\displaystyle Y_{A}^{\ell m}=D_{A}Y^{\ell m}\;,$ (15) $\displaystyle
X_{A}^{\ell m}=-\epsilon_{A}^{\;\;C}D_{C}Y^{\ell m}\;,$ (16) $\displaystyle
Y_{AB}^{\ell
m}=\left(D_{A}D_{B}+\frac{1}{2}\ell(\ell+1)\Omega_{AB}\right)Y^{\ell m}\;,$
(17) $\displaystyle X_{AB}^{\ell
m}=-\epsilon_{(A}^{\;\;\;\;C}D_{B)}D_{C}Y^{\ell m}\;,$ (18)

where $Y^{\ell m}$ are the standard spherical harmonics, $D_{A}$ and
$\epsilon_{AB}$ are the covariant derivative and the Levi-Civita symbol on the
unit sphere respectively.

Gauge-invariant perturbations. Following [56], we define

$\displaystyle\tilde{h}_{ab}^{(i)}$
$\displaystyle=h_{ab}^{(i)}+H_{ab}^{(i)}+2D_{(a}\zeta^{(i)}{}_{b)}\;,$ (19)
$\displaystyle\tilde{h}_{a-}^{(i)}$
$\displaystyle=h_{a-}^{(i)}+H_{a-}^{(i)}+r^{2}D_{a}Z_{-}^{(i)}\;,$ (20)
$\displaystyle\tilde{h}_{\circ}^{(i)}$
$\displaystyle=h_{\circ}^{(i)}+H_{\circ}^{(i)}+2rf(r)\zeta^{(i)}_{r}-\ell(\ell+1)r^{2}Z_{+}^{(i)}\;,$
(21)

where $Z_{\pm}^{(i)}=-(h_{\pm}^{(i)}+H_{\pm}^{(i)})/(2r^{2})$ and
$\zeta_{a}^{(i)}=-h_{a+}^{(i)}-H_{a+}^{(i)}-r^{2}D_{a}Z_{+}^{(i)}$. The
various $H^{(i)}$ components can be found by decomposing a vector
$H^{(i)}_{\mu\nu}$ in spherical harmonics in the same way as eqs (5)-(7),
where $H^{(1)}_{\mu\nu}=0$ and
$H^{(2)}_{\mu\nu}=\mathcal{L}_{\xi}\big{(}h_{\mu\nu}^{(1)}+\mathcal{L}_{\xi}\bar{g}_{\mu\nu}/2\big{)}$,
where $\mathcal{L}$ is the Lie derivative,
$\xi_{\mu}=(\zeta^{(1)}_{a},Z^{(1)}_{A})$ and
$Z_{A}^{(1)}=Z_{+}^{(1)}Y_{A}+Z_{-}^{(1)}X_{A}$.

RWZ potentials. They are given by

$\displaystyle V_{-}$
$\displaystyle=f(r)\bigg{(}\frac{\ell(\ell+1)}{r^{2}}-\frac{6M}{r^{3}}\bigg{)}\;,$
$\displaystyle V_{+}$
$\displaystyle=f(r)\frac{2\lambda^{2}(\lambda+1)r^{3}+6\lambda^{2}Mr^{2}+18\lambda
M^{2}r+18M^{3}}{r^{3}(\lambda r+3M)^{2}}\;,$ (22)

where $\lambda=(\ell+2)(\ell-1)/2$.

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