# Constraints on the amplitude of gravitational wave echoes from black hole
ringdown using minimal assumptions

Andrea Miani  0000-0001-7737-3129<EMAIL_ADDRESS>Università di Trento,
Dipartimento di Fisica, I-38123 Povo, Trento, Italy  INFN, Trento Institute
for Fundamental Physics and Applications, I-38123 Povo, Trento, Italy  Claudia
Lazzaro  0000-0001-5993-3372 Università di Padova, Dipartimento di Fisica,
I-35121 Padova, Italy  INFN, Sezione di Padova, I-35131 Padova, Italy
Giovanni Andrea Prodi  0000-0001-5256-915X Università di Trento, Dipartimento
di Matematica, I-38123 Povo, Trento, Italy  INFN, Trento Institute for
Fundamental Physics and Applications, I-38123 Povo, Trento, Italy  Shubhanshu
Tiwari  0000-0003-1611-6625 Physik-Institut, University of Zurich,
Winterthurerstrasse 190, 8057 Zurich, Switzerland  Marco Drago
0000-0002-3738-2431 Università di Roma, Dipartimento di Fisica, I-00133 Roma,
Italy  INFN, Sezione di Roma, I-00133 Roma, Italy  Edoardo Milotti
0000-0001-7348-9765 Università di Trieste, Dipartimento di Fisica, I-34127
Trieste, Italy  INFN, Sezione di Trieste, I-34127 Trieste, Italy  Gabriele
Vedovato  0000-0001-7226-1320 Università di Padova, Dipartimento di Fisica,
I-35121 Padova, Italy  INFN, Sezione di Padova, I-35131 Padova, Italy

###### Abstract

Gravitational wave echoes may appear following a compact binary coalescence if
the remnant is an “exotic compact object” (ECO). ECOs are proposed
alternatives to the black holes of Einstein’s general relativity theory and
are predicted to possess reflective boundaries. This work reports a search for
gravitational wave transients (GWTs) of generic morphology occurring shortly
after ($\lesssim$1\text{\,}\mathrm{s}$$) binary black hole (BBH) mergers,
therefore targeting all gravitational wave echo models. We investigated the
times after the ringdown for the higher signal-to-noise ratio BBHs within the
public catalog GWTC-3 by the LIGO-Virgo-KAGRA collaborations (LVK). Our search
is based on the coherent WaveBurst pipeline, widely used in generic searches
for GWTs by the LVK, and deploys new methods to enhance its detection
performances at low signal-to-noise ratios. We employ Monte Carlo simulations
for estimating the detection efficiency of the search and determining the
statistical significance of candidates. We find no evidence of previously
undetected GWTs and our loudest candidates are morphologically consistent with
known instrumental noise disturbances. Finally, we set upper limits on the
amplitude of GW echoes for single BBH mergers.

††preprint: APS/123-QED

##  I Introduction

The LIGO [1] and Virgo [2] observatories have successfully detected about 90
gravitational wave transients (GWTs) in past observing runs [3, 4, 5], all
associated to compact binary coalescences (CBCs). More than $90\%$ of these
GWTs are identified as generated by the merger of binary black hole (BBH)
systems. Recently, this worldwide observatory has expanded to include the
KAGRA detector [6]. A new observing run is currently ongoing, and low latency
alerts of more CBC GWTs are being publicly released [7]. Investigating the
black hole (BH) nature through GW astronomy is therefore a very hot topic in
fundamental physics, especially in view of the so-called BH information
paradox [8]. The LIGO-Virgo-KAGRA collaboration (LVK) already published
several results of tests of the general relativity theory (GR) [9, 10, 11, 12,
13], exploiting the GWTs emitted by BBHs.

Several recent papers [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]
addressed the topic of exotic compact objects (ECOs) [26]: possible compact
objects (COs) alternative to the BHs predicted by Albert Einstein’s GR theory.
Examples of ECOs include wormholes [27], boson stars [28], gravastar [29], and
fuzzballs [30]. These ECO models are characterized by different astrophysical
properties, like their constituent “matter”, but they all share one physical
characteristic: Planck-scale modifications of the BH event horizon due to
quantum effects[18] or the presence of a surface of different nature [31, 17].
This feature would enable the emission of repeated GWTs occurring shortly
after the BBH merger time, echoes of the ECO remnant ringdown [14, 15, 32].

In this work, we report a systematic search for echo signals of generic
morphology occurring after the merger-ringdown phase of BBH GWTs [33]. The
detection performance of the method is demonstrated down to low signal-to-
noise (SNR) ratios, and the results are practically independent of the echo
signal morphology. We also provide upper limits on the strain amplitude at
earth of echoes for the loudest BBH GWTs in the LVK catalogs [3, 4, 5]. The
search is based on the coherent Wave Burst pipeline (cWB) [34, 35, 36, 37, 38]
widely used to search for generic short duration GWTs by LVK [39, 40, 41].

Section II provides a brief review of GW echo models and discusses the main
characteristics of the predicted echo signals. In section III we summarize the
data analysis method focusing on novel features: in particular, on the search
for weak post-merger-ringdown GWTs, the simulations with software signal
injections and the construction of the confidence belt on echoes’
$h_{\mathrm{rss}}$ [42] strain amplitude. Section IV reports the search
results including detection performances, checks of robustness to different
echo morphologies, and upper limits on the echoes’ $h_{\mathrm{rss}}$. A
comparative discussion with respect to four, previously published, echo
searches [24, 23, 21, 43] is reported in section V. Conclusions are drawn in
section VI.

Figure 1: Simulated inspiral-merger-ringdown-echoes GW transient signal vs.
time at one detector: the red line is the whitened reconstructed signal strain
$h(t)$; the grey line is the whitened data. The CBC signal peaks at the merger
time ($60.42\text{\,}\mathrm{s}$) and is followed by echoes, whose main
parameters are visualized.

##  II Gravitational Wave Echoes

Distinctive properties of an ECO remnant derives from the dynamics of its
inner barrier, interpreted as an effective surface, located above the would-be
BH event horizon (EH), $r_{\mathrm{eh}}$, [44, 16, 17] at a radius
$r_{\mathrm{in}}$ from the object’s center

$r_{\mathrm{in}}=r_{\mathrm{eh}}+\mathit{l}\,.$ (1)

Here, in eq.(1), $\mathit{l}$ is the length correction to the would-be BH
event horizon (EH), and it is theorised to be extremely small [17, 18, 15], of
the order of the Planck length ($l_{\mathrm{Planck}}\sim 2\cdot
10^{-35}$\mathrm{m}$$). The inner barrier couples to the outer one (i.e.
angular-momentum potential barrier) acting as a sort of cavity [19]. If the
remnant compact object of a CBC is not a GR BH, i.e. it is not fully
absorbing, the merger-ringdown can trigger multiple outer and inner barrier
excitations [14, 15]. This results in a train of pulses of outgoing GW
radiation of decreasing energy called echoes [45, 46].

Since the performance of our method is independent to details on the phase
evolution of the signal, we can safely test it under the simplifying
assumption of a non-rotating ECO remnant 111According to e.g. [15], echoes
from a spinning remnant show a much larger variety of morphologies, including
e.g. a redshift of the echo central frequency vs pulse order as well as
different relative contributions of the $+$ and $\times$ GW polarization
components at different pulses. However, the sensitivity of our search is
invariant when expressed in terms of the $h_{rss}$ component that is
detectable by the GW observatory for a given GWT. Of course, the
interpretation of our measurement in terms of the source model will be
different.. This assumption has also been adopted in many searches in the
literature [18, 19, 20, 21, 22, 23, 24]. A complete description of a possible
echo template for a spinning ECO remnant, as expected from a compact binary
coalescence, is provided in [15]. The main parameters characterizing the
models of echoes are [21, 22] (see also figure 1):

  * •

$\Delta t_{\mathrm{echo}}$ : the time separation between subsequent pulses as
measured by a distant observer. It corresponds to the round-trip travel time
of the space-time perturbation between the inner and outer barriers [17];

  * •

$t_{\mathrm{echo}}$ : the delay of the first echo pulse from the coalescence
time of the binary. In general $t_{\mathrm{echo}}\sim\Delta t_{\mathrm{echo}}$
apart from small effects related to the strong non-linearity close to the
merger time;

  * •

$\gamma$ : the attenuation per round-trip in terms of the GW amplitude ratio
between subsequent echo pulses ($0<\gamma<1$);

  * •

$A$ : amplitude ratio between the amplitude of the first echo and the one at
merger time ($0\leq A<1$).

Following [16, 17], the theoretical prediction for $\Delta t_{\mathrm{echo}}$
is clearly related to the space-time geometry outside the ECO remnant:

$\Delta t_{\mathrm{echo}}\sim
2\int_{r_{\mathrm{in}}}^{r_{\mathrm{out}}}\frac{1}{\sqrt{F(r)B(r)}}dr\,$ (2)

where in eq. 2, $F(r)$ and $1/B(r)$ 222The space-time geometry outside an ECO
remnant can be described with the
$ds^{2}=-F(r)dt^{2}+1/B(r)dr^{2}+r^{2}d\Omega^{2}$ metric. Such a metric is
used to generally describe a static CO with spherical symmetry and matter
localised only in the region $r<r_{shell}$. Following Birkhoff’s theorem, in
the region $r>r_{shell}$ the Schwarzschild metric holds:
$F(r)=B(r)=\left(1-\frac{2GM}{c^{2}r}\right)$. are the coefficients functions
for the time and radial component of the metric in a spherically symmetric
system, $r_{\mathrm{in}}$ is the radius of the inner barrier and
$r_{\mathrm{out}}$ the radius of the outer barrier. Eq. 2 takes into account
the effects of gravitational redshift and spatial curvature on the emission of
GW echoes. The resulting approximate expression for the time separation is
[16, 17]:

$\Delta t_{\mathrm{echo}}\approx
54\left(\frac{n}{4}\right)M_{30}\left[1-0.001\ln\left(\frac{l/l_{\mathrm{Planck}}}{M_{30}}\right)\right]\,$\mathrm{ms}$\,.$
(3)

Here, $n$ is a parameter of the order of the unity that takes into account the
structure of the ECO nature [17, 20], $M_{30}\equiv M/30M_{\odot}$ with $M$
standing for the final mass of the remnant. Therefore, a measurement of
$\Delta t_{\mathrm{echo}}$ would provide information over the theorized nature
of the ECO through the parameters $n$ and $\mathit{l}$, related to the
compactness of the ECO [14]. According to eq.(3), typical values for echoes
time separation are $\Delta t_{\mathrm{echo}}\in(30,400)\;$\mathrm{ms}$$ for
BBH mergers whose total mass ranges in $\in(10,100)\;M_{\odot}$, like most of
those detected during O1, O2 and O3 by the LV Collaborations [3, 4, 5].

###  II.1 Signal proxy for echoes

Our detection algorithm does not make use of signal templates, and for testing
its performance we can rely on loose signal proxies. The template we selected
to mimic echo signals $h_{\mathrm{echo}}(t)$ is a double sine-Gaussian (SGE)
pulse $h_{\mathrm{echo}}(t)=h_{\mathrm{SGE}}(t)+\gamma\cdot
h_{\mathrm{SGE}}(t-\Delta t_{\mathrm{echo}})$ with $h_{\mathrm{SGE}}(t)$ [47]:

$\displaystyle\begin{split}h_{+,\mathrm{SGE}}(t)&=h_{0}\,\frac{1+\cos^{2}(\iota)}{2}\,e^{-\frac{t^{2}}{\tau^{2}}}\,\sin(2\pi
f_{0}t+\phi_{0})\,\\\
h_{\times,\mathrm{SGE}}(t)&=h_{0}\,\cos(\iota)\,e^{-\frac{t^{2}}{\tau^{2}}}\,\cos(2\pi
f_{0}t+\phi_{0})\,,\end{split}$ (4)

In eq.(4), $h_{0}$ is the signal amplitude, $\iota$ is the inclination angle
of the source, $\tau$ the half-time duration of the pulse, $f_{0}$ and
$\phi_{0}$ its central frequency and phase respectively. The values we select
for these parameters are:

  * •

$h_{0}$ is defined as $h_{0}=A\cdot h_{\mathrm{max}}$, where
$h_{\mathrm{max}}$ is the GW amplitude at the merger. In our simulations, $A$
is randomly selected per each injection within a uniform distribution $0<A<1$
(see III.3).

  * •

$\gamma=0.5$ so that the second echo is contributing 1/3 of the injected SNR.
This is an intermediate condition on the concentration of the signal in time
and makes possible to study the reconstruction of a weaker echo, separately
from the first.

  * •

$\tau=$20\text{\,}\mathrm{ms}$$ and $f_{0}=$140\text{\,}\mathrm{Hz}$$, are
close to expectations for the typical mass range of BBH mergers in GWTC-3.

  * •

$\phi_{0}=0$. This is not impacting the results since the search method is
agnostic on the signal phase in each pulse.

  * •

$t_{\mathrm{echo}}=$300\text{\,}\mathrm{ms}$$ and $\Delta
t_{\mathrm{echo}}=$300\text{\,}\mathrm{ms}$$ are intermediate values for the
investigated BBH mergers (see section IV.1) according to eq. 3.

  * •

$\iota$ is set equal to the one of the injected BBH signal.

Furthermore, the sky location of the echo signal proxy is the same as the one
of the BBH GWT.

##  III Search methods

This section describes the methods developed to search for generic GWTs after
BBH mergers, such as echo signals. The analysis is based on cWB methods and
comprises Monte Carlo simulations to tune the search and interpret the results
in terms of gravitational wave echoes. We call this new analysis cWB echo
signal (ES) search.

###  III.1 Coherent WaveBurst

Coherent WaveBurst [38, 37] is a data analysis pipeline searching for generic
GWT signals in the data from the LVK GW detectors network [48, 49, 50].
Designed to operate without a specific waveform model, cWB first identifies
coincident excess power in the multi-resolution time-frequency (TF)
representations of the detectors’ strain data [36]. Then, for the selected
events, cWB reconstructs the source sky location and the signal waveform of
each GW candidate by means of a constrained maximum likelihood method [35].

To be robust against the non-stationary detector noise cWB employs signal-
independent vetoes, reducing the initial high rate of the excess power
triggers. The primary selection cut is on the network correlation coefficient
$c_{\mathrm{c}}$ [34], defined as:

$c_{\mathrm{c}}=\frac{E_{\mathrm{c}}}{E_{\mathrm{c}}+E_{\mathrm{null}}}\,,$
(5)

which is informative on the coherence of a signal among the detectors of the
network. Here, $E_{\mathrm{c}}$ and $E_{\mathrm{null}}$ [51, 35, 34] are the
coherent and the null energy of the signal. The algorithm also combines all
the data streams into one coherent statistic $\eta_{\mathrm{c}}$ [34], which
is used for ranking the detected events and is defined as:

$\eta_{\mathrm{c}}=\sqrt{\frac{c_{\mathrm{c}}\cdot E_{\mathrm{c}}}{N-1}}\,,$
(6)

with $N$ the number of detectors in the network. Typically, for a GW signal
$c_{\mathrm{c}}\sim 1$ while for instrumental glitches $c_{\mathrm{c}}\ll 1$.
By setting a threshold value on $c_{\mathrm{c}}$, it is possible to
reconstruct events with a lower or higher probability of being genuine GW
signals.

In the LVK analyses, different cWB searches are used depending on the target
GWT. In a previous work, cWB was used to investigate post-merger GW emission
in a configuration more sensitive to the chirping morphology of the CBCs
signals [52]. Currently, the most general cWB search is the all-sky burst
search [39, 40, 41], with a proven ability to detect the broadest variety of
GW signal morphologies. Our search method is based on this cWB instance, the
same version used in the LVK O3 analysis [41, 37], thus it is more agnostic
than [52]. The following subsections describe the peculiarities of cWB ES
search.

Figure 2: This plot shows the segmentation of the analyzed time following up
an event (red line). The pale blue opaque area is representative of the blind
time $\Delta t_{\mathrm{blind}}$, and the light blue transparent area after it
highlights the post-merger window (PMW).

###  III.2 Searching for echoes

Due to the expected nature of echoes, the cWB all-sky burst search is modified
to select more TF pixels with a low energy content and scattered over a wider
than usual time span (see appendix A). Triggering and final selection
thresholds are decreased, and to group different pulses (i.e. the BBH merger
and the echo-like signals) into a single event, we increase the maximum time
separation between disjoint clusters of pixels which define a single event.
Specifically, the $\eta_{\mathrm{c}}$ threshold is decreased from 5.0 to 3.5,
and the $T_{\mathrm{gap}}$ parameter [38] is increased up to
$2\text{\,}\mathrm{s}$. Also, the whitening [53] of the data is performed
using a TF map resolution which mitigates the leakage of the merger-ringdown
signal of the remnant into the subsequent TF pixels. Indeed, while the cWB
all-sky burst search performs the whitening in the TF map with the best
frequency resolution, typically $\Delta f=$1\text{\,}\mathrm{Hz}$$ and $\Delta
t=$0.5\text{\,}\mathrm{s}$$, here we adopt a better time resolution, using
pixels with a time width of $\Delta t=$0.125\text{\,}\mathrm{s}$$ and $\Delta
f=$4\text{\,}\mathrm{Hz}$$.

The search uses the BBH GWT as a trigger and focuses on a user-defined post-
merger time interval, called post-merger window (PMW), see figure 2. The PMW
starts at time $t_{\mathrm{start}}^{\scaleto{\mathrm{PMW}}{4pt}}$, defined as

$t_{\mathrm{start}}^{{\scaleto{\mathrm{PMW}}{4pt}}}=t_{\mathrm{coa}}+\Delta
t_{\mathrm{blind}}\,,$ (7)

where $t_{\mathrm{coa}}$ is the coalescence time of the BBH system and $\Delta
t_{\mathrm{blind}}$ a user-defined blind time. The blind time’s purpose is to
mask the ringdown of the BBH signal, and its impact on the analysis will be
discussed in IV.1. Limiting the ES search to a PMW allows to limit the noise
contribution in the post-merger without penalising the capability to detect
possible echo signals. We adopted two choices of PMW:

  * •

for $t_{\mathrm{echo}}\leq$200\text{\,}\mathrm{ms}$$ we use $\Delta
t_{\mathrm{blind}}=$50\text{\,}\mathrm{ms}$$ and $\Delta
t^{\scaleto{\mathrm{PMW}}{4pt}}=$300\text{\,}\mathrm{ms}$$;

  * •

for $t_{\mathrm{echo}}>$200\text{\,}\mathrm{ms}$$ we use $\Delta
t_{\mathrm{blind}}=$200\text{\,}\mathrm{ms}$$ and $\Delta
t^{\scaleto{\mathrm{PMW}}{4pt}}=$1\text{\,}\mathrm{s}$$.

Such choices are suitable to include the first $\sim$1-4 echo pulses according
to eq.(3).

Within the PMW, the main statistical parameters we compute are the network
correlation coefficient, $c_{\mathrm{c}}^{\scaleto{\mathrm{PMW}}{4pt}}$,
analogous to $c_{\mathrm{c}}$ (see eq.(5)), and the network signal to noise
ratio of the data, SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}$, defined as

$\text{SNR}^{\scaleto{\mathrm{PMW}}{4pt}}=\sqrt{\sum_{k}^{N}\sum_{j\in
J}(x_{\mathrm{k}}[j])^{2}}\,,$ (8)

where $J$ is the set of the TF pixels corresponding to times inside $\Delta
t^{\scaleto{\mathrm{PMW}}{4pt}}$, and $x_{\mathrm{k}}[j]$ are the whitened
reconstructed data.

While cWB can work with arbitrary detectors networks, the ES search deployed
here is run only over the two LIGO detectors network (H, Hanford, and L,
Livingston [1]). The motivation is two-fold. On one side, H and L detect most
of the GWTs’ SNR. Moreover, under O2 and O3 conditions, the detection
performance of minimally modeled searches for GWTs results more effective when
restricted to the LIGO network of almost co-aligned detectors [41]. This comes
from a complex balance between background rejection capabilities against
collection of GW information, as the target signal parameter space becomes
higher dimensional when searching also Virgo [2] due to the need of taking
into account both GW polarization components.

###  III.3 Monte Carlo estimators

The ES search follows a two-track scheme: the background (BKG) analysis, and
the signal (SIG) analysis. Both analyses are off-source experiments, meaning
that the data do not include the times corresponding to the detected GW
signals. The ES search is separately performed for each BBH GWT considered.

The background (BKG) analysis is used to estimate the noise statistics for the
null hypothesis in the PMW. We create a set of off-source software signal
injections over the data stream using waveform templates of the specific BBH
event under study. These templates are randomly selected from the CBC waveform
posterior samples [54, 55], provided by the Parameter Estimation (PE) methods
for the considered GW event (following the approximants used in [3, 4, 5]).
The signals are injected widely separated, i.e. one each
$600\text{\,}\mathrm{s}$, to avoid systematic interferences in the analysis.

The signal (SIG) analysis enables the measurement of the sensitivity of the ES
search to signals within the PMW. The injected BBH GWTs are the same as the
BKG analysis with, in addition, the injection of secondary signals after each
BBH merger according to the echo model of section II.1. Different morphologies
of secondary signals have been tested as well, see IV.1.

This double simulation scheme is depicted in figure 3. The data used for all
studies are real data available at the GW open science center of the LVK
collaboration, see [54, 55]. These two analyses allow us to study the
detection probability, DP, and the false alarm probability, FAP, as functions
of the reconstructed SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\mathrm{rec}}$.
Their definition is the following:

$\displaystyle\begin{split}\text{DP}&=\frac{\text{EV}_{\text{SIG}}(\text{SNR}^{\scaleto{\mathrm{PMW}}{4pt}}_{\mathrm{rec}}\geq
th_{\mathrm{snr}})}{\text{EV}}\\\
\text{FAP}&=\frac{\text{EV}_{\text{BKG}}(\text{SNR}^{\scaleto{\mathrm{PMW}}{4pt}}_{\mathrm{rec}}\geq
th_{\mathrm{snr}})}{\text{EV}}\,,\end{split}$ (9)

and here $\text{EV}_{\text{SIG}}$ and $\text{EV}_{\text{BKG}}$ are the number
of detected events above threshold in the PMW from the SIG and BKG
distributions, EV is the total number of injected signals, and
$th_{\mathrm{snr}}$ is the threshold value on
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\mathrm{rec}}$.

Figure 3: Flowchart of the echo signal (ES) search. Once the cWB all-sky burst
search detects a BBH event, the cWB ES search can be run as a follow-up. The
search runs two parallel studies on a common data selection: the background
(BKG) and the signal (SIG), and computes all the statistical estimators
described in section III.2. The BBH primary signal injections are randomly
picked from the PE samples distribution for that event.

###  III.4 Tuning the analysis internal thresholds

The cWB internal thresholds, described in III.2 and listed in appendix A, are
related to the energy content of a possible trigger, its energy per degree of
freedom, and its coherence within the detectors’ network. On the contrary,
they are agnostic to the signal morphology or spectral characterisation, so
they allow to address a very wide range of different statistical noise
conditions.

The tuning criteria of the internal thresholds are based on the receiver
operating characteristics (ROC) curves, which are built from the DP and FAP
measurements. The chosen configuration of the analysis is the one that
maximises the DP for low values of FAP, in the interval $0.5\%\leq$ FAP $\leq
5\%$. This region corresponds to the events which possess low to medium
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\mathrm{rec}}$, typically $4\leq$
SNR${}^{\scaleto{PMW}{4pt}}_{\mathrm{rec}}$ $\leq 8$. The tuning has been
extensively performed on the simulations related to the GW150914 event [56,
57, 58]. We have checked as well that the same setup is also providing the
best results for a few GWTs from O3, including GW190521 [59].

The list of tuned parameters is reported in appendix A.

Figure 4: Confidence belt for the echo’s injected amplitude
$h_{\mathrm{rss}}$, $h_{\mathrm{rss}}^{\mathrm{inj}}$, vs the reconstructed
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{rec}$ in the PMW for GW150914. The blue
region corresponds to $95\%$ coverage. The on-source $95\%$ confidence
interval in terms of the $h_{\mathrm{rss}}$ is set by the intersection between
the vertical line at the on-source value
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\scaleto{\mathrm{ON}}{4pt}}\,\sim\,0.38$
(red line) and the blue region. The y-axis values are in units of
$10^{-23}/\sqrt{$\mathrm{Hz}$}$. The scatter plot shows SIG simulation data
points but not BKG ones for readability. 95$\%$ of the BKG samples has an
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{rec}\leq 4$.

###  III.5 Inference of confidence intervals

Searches for GWTs of generic morphologies, such as cWB ES, directly measure
the energy or integrated squared amplitude of the candidate signal. Here,
results are presented in terms of $h_{\mathrm{rss}}$ [42] at earth

$h_{\mathrm{rss}}=\sqrt{\int_{t\in\mathrm{PMW}}(\mid h_{+}(t)\mid^{2}+\mid
h_{\times}(t)\mid^{2})dt}$ (10)

of signals consistent with the on-source data in the PMW.

The SIG simulations provide estimates of the conditional distributions of the
recovered SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\mathrm{rec}}$ as a function
of $h_{\mathrm{rss}}^{\mathrm{inj}}$, the $h_{\mathrm{rss}}$ of the injected
post-merger signal. We use the conditional distributions to build the
confidence belts [60] on $h_{\mathrm{rss}}^{\mathrm{inj}}$, as shown in figure
4. This is approximately achieved by introducing a binning in
$h_{\mathrm{rss}}^{\mathrm{inj}}$ which ensures a minimum number of samples,
hundreds per bin, and allows to target a confidence belt coverage of $95\%$.
The related cost is to perform specific SIG simulations with much higher
statistics than those needed for estimating the DP. For the special case of
$h_{\mathrm{rss}}^{\mathrm{inj}}=0$, the null hypothesis, we exploit the full
statistics of the BKG simulation. The confidence belt is then used to set the
$95\%$ confidence interval on the expected $h_{\mathrm{rss}}^{\mathrm{inj}}$
for signals inside the PMW that posses a SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}$
equal to the one measured on-source,
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\scaleto{\mathrm{ON}}{4pt}}$.

##  IV Results

List of analysed BBH events
---
Run - GW name | App. | $t_{\mathrm{coa}}$ |  SNRnet | $h_{\mathrm{rss}}^{50\%}\cdot\frac{10^{-23}}{\sqrt{$\mathrm{Hz}$}}$ |  $t_{\mathrm{echo}}$ [$\mathrm{ms}$] |  SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\scaleto{\mathrm{ON}}{4pt}}$ |  p-valueON
O1 - GW150914 | 1 | 1126259462.421 | 24.4 | $2.79\pm 0.02$ | $227^{+12}_{-11}$ | $0.4$ | $0.849\pm 0.006$
O1 - GW151012 | 1 | 1128678900.467 | 10.0 | $2.57\pm 0.03$ | $127^{+39}_{-14}$ | $0.1$ | $0.54\pm 0.01$
O1 - GW151226 | 1 | 1135136350.668 | 13.1 | $2.70\pm 0.03$ | $73^{+23}_{-5}$ | $0.2$ | $0.8\pm 0.1$
O2 - GW170104 | 1 | 1167559936.619 | 13.0 | $2.52\pm 0.01$ | $176^{+18}_{-14}$ | $0.4$ | $0.673\pm 0.006$
O2 - GW170608 | 1 | 1180922494.501 | 14.9 | $2.63\pm 0.01$ | $63^{+12}_{-2}$ | $2.2$ | $0.034\pm 0.003$
O2 - GW170729 | 1 | 1185389807.346 | 10.2 | $2.53\pm 0.01$ | $287^{+53}_{-37}$ | $2.5$ | $0.043\pm 0.003$
O2 - GW170809 | 1 | 1186302519.758 | 12.4 | $2.40\pm 0.02$ | $203^{+19}_{-14}$ | $\leq 0.1$ | $0.56\pm 0.02$
O2 - GW170814 | 1 | 1186741861.533 | 15.9 | $2.51\pm 0.02$ | $191^{+12}_{-9}$ | $1.5$ | $0.450\pm 0.007$
O2 - GW170823 | 1 | 1187529256.501 | 11.5 | $2.50\pm 0.02$ | $236^{+36}_{-27}$ | $\leq 0.1$ | $0.835\pm 0.006$
O3a - GW190408_181802 | 2 | 1238782700.279 | 14.7 | $1.82\pm 0.01$ | $147^{+14}_{-10}$ | $1.1$ | $0.155\pm 0.005$
O3a - GW190412 | 2 | 1239082262.165 | 18.9 | $1.82\pm 0.01$ | $134^{+14}_{-14}$ | $1.2$ | $0.273\pm 0.007$
O3a - GW190512_180714 | 2 | 1241719652.435 | 12.3 | $1.69\pm 0.02$ | $123^{+14}_{-12}$ | $\leq 0.1$ | $0.55\pm 0.02$
O3a - GW190513_205428 | 2 | 1241816086.800 | 12.3 | $1.83\pm 0.01$ | $185^{+29}_{-21}$ | $0.2$ | $0.882\pm 0.004$
O3a - GW190517_055101 | 2 | 1242107479.848 | 10.2 | $1.80\pm 0.02$ | $213^{+33}_{-32}$ | $\leq 0.1$ | $0.52\pm 0.01$
O3a - GW190519_153544 | 2 | 1242315362.418 | 12.0 | $1.84\pm 0.01$ | $365^{+45}_{-50}$ | $0.4$ | $0.302\pm 0.006$
O3a - GW190521 | 2 | 1242442967.471 | 14.4 | $1.74\pm 0.01$ | $568^{+133}_{-81}$ | $0.2$ | $0.858\pm 0.004$
O3a - GW190521_074359 | 2 | 1242459857.456 | 24.4 | $1.73\pm 0.01$ | $256^{+23}_{-16}$ | $0.5$ | $0.218\pm 0.006$
O3a - GW190602_175927 | 2 | 1243533585.093 | 12.1 | $1.98\pm 0.04$ | $402^{+64}_{-54}$ | $0.2$ | $0.838\pm 0.005$
O3a - GW190701_203306 | 2 | 1246048404.578 | 11.6 | $1.84\pm 0.01$ | $326^{+41}_{-32}$ | $6.4$ | $0.0019\pm 0.0007$
O3a - GW190706_222641 | 2 | 1246487219.361 | 12.3 | $1.82\pm 0.01$ | $358^{+66}_{-49}$ | $0.3$ | $0.100\pm 0.004$
O3a - GW190814 | 2 | 1249852257.009 | 22.2 | $1.82\pm 0.01$ | $91^{+4}_{-3}$ | $0.5$ | $0.718\pm 0.005$
O3a - GW190828_063405 | 2 | 1251009263.781 | 16.0 | $1.82\pm 0.01$ | $197^{+26}_{-15}$ | $1.0$ | $0.474\pm 0.006$
O3a - GW190915_235702 | 1 | 1252627040.693 | 13.1 | $1.88\pm 0.02$ | $205^{+26}_{-22}$ | $0.2$ | $0.720\pm 0.008$
O3a - GW190929_012149 | 1 | 1253755327.505 | 9.9 | $1.86\pm 0.02$ | $367^{+122}_{-92}$ | $0.1$ | $0.860\pm 0.005$
O3b - GW191109_010717 | 2 | 1257296855.783 | 17.3 | $1.85\pm 0.01$ | $387^{+65}_{-54}$ | $0.1$ | $0.715\pm 0.008$
O3b - GW191204_171526 | 2 | 1259514944.087 | 17.5 | $2.05\pm 0.05$ | $68^{+6}_{-4}$ | $0.3$ | $0.36\pm 0.02$
O3b - GW191215_223052 | 2 | 1260484270.995 | 11.2 | $1.69\pm 0.02$ | $148^{+18}_{-15}$ | $0.2$ | $0.50\pm 0.02$
O3b - GW191222_033537 | 2 | 1261020955.347 | 12.5 | $1.82\pm 0.01$ | $272^{+55}_{-36}$ | $\leq 0.1$ | $0.771\pm 0.007$
O3b - GW191230_180458 | 2 | 1261764316.898 | 14.4 | $1.77\pm 0.08$ | $296^{+61}_{-40}$ | $\leq 0.1$ | $0.31\pm 0.04$
O3b - GW200219_094415 | 2 | 1266140673.095 | 10.7 | $1.75\pm 0.07$ | $224^{+42}_{-28}$ | $\leq 0.1$ | $0.28\pm 0.05$
O3b - GW200224_222234 | 2 | 1266618172.381 | 20.0 | $1.74\pm 0.01$ | $247^{+24}_{-17}$ | $7.4$ | $0.0017\pm 0.0005$
O3b - GW200225_060421 | 2 | 1266645879.413 | 12.5 | $1.89\pm 0.04$ | $115^{+13}_{-10}$ | $0.7$ | $0.21\pm 0.02$
O3b - GW200311_115853 | 2 | 1267963151.380 | 17.8 | $1.80\pm 0.01$ | $212^{+17}_{-14}$ | $0.3$ | $0.561\pm 0.006$
Table 1: The analyzed BBH GWTs with SNR${}_{\mathrm{net}}\geq 10$ in the all-
sky cWB search [39, 40, 41], listed in chronological order. Columns report the
observing run; GWT name; waveform model used in BKG and SIG simulations (1 =
IMRPhenomPv2 approximant, 2 = SEOBNRv4PHM approximant, as the one used in [3,
4, 5]); GPS coalescence time $t_{\mathrm{coa}}$ to $\mathrm{ms}$ resolution
from the PE waveform posterior information available through GWOSC [54, 55];
network SNR as reconstructed by the cWB all-sky search for burst GWTs;
$h_{\mathrm{rss}}$ in PMW that ensures DP=$50\%$ at FAP=$5\%$; predicted
$t_{\mathrm{echo}}$ (following eq.(3) and assuming n=8, wormhole, and
$\mathit{l}/l_{\mathrm{Planck}}=2$); on-source SNR inside the PMW,
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\scaleto{\mathrm{ON}}{4pt}}$; p-value of
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\scaleto{\mathrm{ON}}{4pt}}$ with
statistical uncertainties. The analyzed PMW is [50,350]$\mathrm{ms}$ for GWTs
with $t_{\mathrm{echo}}\leq$200\text{\,}\mathrm{ms}$$, [0.2,1.2]$\mathrm{s}$
otherwise.

Using the search tuning described in III.4, we investigated a sub-set of 33
BBH events from the BBH detections from LVK collaboration [3, 4, 5]. The
subset comprises all the BBH events that possess a network SNR greater than 10
in the cWB search for generic GWTs [39, 40, 41] 333The network SNR recovered
by cWB is consistent to the one recovered by template searches for these loud
BBH events. The selection is motivated by the reasonable expectation that the
signal amplitude of echoes is such that $A\ll 1$, since no signals with
amplitude comparable to that of the merger have been observed after the
ringdown phase of any BBH GW emission. The list of investigated BBH events and
related main results is given in table 1.

Figure 5: Detection probability (DP) at FAP = $5\%$ as a function of
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\mathrm{inj}}$ for different
morphologies of simulated post-ringdown signals: a high mass
($80-80M_{\odot}$) BBH coalescence (blue), trains of two elliptically
polarised sine-Gaussian pulses as described in II.1 with different central
frequencies $f_{0}=80,140,200,400\,$\mathrm{Hz}$$ (orange, green, red, and
violet respectively) and a single pulse of white noise, WNB, of duration
$\sim$0.02\text{\,}\mathrm{s}$$, central frequency $150\text{\,}\mathrm{Hz}$
and bandwidth $100\text{\,}\mathrm{Hz}$ (brown). These results refer to
GW150914.

###  IV.1 Robustness of cWB ES search

The BKG simulations show that the statistical properties of the noise
background are weakly related to the choice of $\Delta t_{\mathrm{blind}}$
within the range $(0.05,0.4)\,$\mathrm{s}$$. Therefore, any $\Delta
t_{\mathrm{blind}}$ in this range can be freely selected for the cWB ES
search. Instead, the noise level starts to increase as $\Delta
t_{\mathrm{blind}}$ gets shorter due to some residual leakage from the primary
BBH GWT signal into the PMW. The duration of the PMW window, $\Delta
t^{\scaleto{PMW}{4pt}}$, affects as expected the mean
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}$ from the BKG analysis, the longer
$\Delta t^{\scaleto{PMW}{4pt}}$ the larger the noise in the PMW.

We also tested the robustness of the cWB ES search against variations of the
injected secondary signals in SIG analyses, see section III.3, for a few BBH
GWT cases. By changing the delay time $t_{\mathrm{eco}}$ and time separation
$\Delta t_{\mathrm{echo}}$ of the two pulses of the signal proxy defined in
section II.1, the detection probability at FAP=$5\%$ results unaffected as
long as both pulses occur inside the analyzed time window, PMW. Therefore, the
off-source results reported in this work can be considered valid as long as
$\Delta t^{\scaleto{PMW}{4pt}}$ and $\Delta t_{\mathrm{blind}}$ are included
in the tested ranges, [0.2.1.2]$\mathrm{s}$ and [0.05,0.2]$\mathrm{s}$
respectively, regardless of the choice $t_{\mathrm{eco}}=\Delta
t_{\mathrm{echo}}=0.3\,$\mathrm{s}$$ which we adopted in the SIG analyses of
all BBH GWTs.

Moreover, we checked the sensitivity of the cWB ES search to widely different
morphologies of post-ringdown signals, by performing additional SIG analyses.
Figure 5 shows the DP at FAP = $5\%$ as a function of the injected
SNR${}^{\scaleto{PMW}{4pt}}$ for different central frequencies of the SGE echo
signal proxy (see section II.1), for a single pulse made by a BBH merger
waveform and for a single burst of white noise. The resulting performances are
almost identical within uncertainties, which is an expected outcome due to the
general nature of the cWB search (see section III.1). The slight decrease in
performances when injecting white noise burst (WNB) signals in the PMW is
mostly related to their wider frequency band.

Figure 6: Plot of the detection probability as a function of
$h_{\mathrm{rss}}^{\mathrm{inj}}$ of the echo signal proxy for a selection of
BBH GWTs from each observing run of the LVK (O1, O2, and O3). The
$h_{\mathrm{rss}}^{\mathrm{inj}}$ units are
$10^{-23}\,\cdot\,\sqrt{$\mathrm{Hz}$}$. The temporal sensitivity improvement
achieved is visible by the shift of the curves to lower
$h_{\mathrm{rss}}^{\mathrm{inj}}$.

###  IV.2 Detection probability

We discuss here the detection probability measurements for the echo signal
proxy described in section II.1, with the requirement of FAP = $5\%$. Figure 6
shows the DP as a function of the $h_{\mathrm{rss}}$ injected inside the PMW
for a subset of GWTs from the three LVK observing runs (O1, O2, O3). The
visible improvement towards smaller $h_{\mathrm{rss}}$ comes from the temporal
enhancement of the detectors’ sensitivities. Between O1 and O2 observing runs,
the typical $h_{\mathrm{rss}}$ at $50\%$ DP decreases from $\sim 2.7\cdot
10^{-23}/\sqrt{$\mathrm{Hz}$}$ to $\sim 2.5\cdot
10^{-23}/\sqrt{$\mathrm{Hz}$}$. A more significant decrease in
$h_{\mathrm{rss}}$ at $50\%$ DP can be seen from O2 to O3, from average values
of $\sim 2.5\cdot 10^{-23}/\sqrt{$\mathrm{Hz}$}$ to $\sim 1.8\cdot
10^{-23}/\sqrt{$\mathrm{Hz}$}$, corresponding to an improvement of about
28$\%$. Column 6 of table 1 reports the resulting $h_{\mathrm{rss}}$ values
which ensure $50\%$ DP with FAP $5\%$ for all the studied GWTs.

The coherent WaveBurst ES search explores a significantly lower range of
$h_{\mathrm{rss}}$ values with respect to the cWB all-sky search for short-
duration bursts [41]. For the latter, the best results in terms of
$h_{\mathrm{rss}}$ values at DP=$50\%$ among the tested signal morphologies
has been achieved in O3 for a single pulse SGE, $Q=100$,
$f_{0}=$235\text{\,}\mathrm{Hz}$$, reaching $h_{\mathrm{rss}}=8\cdot
10^{-23}/\sqrt{$\mathrm{Hz}$}$ at a FAR of one per 100 years. Here instead,
with a more dispersed signal, the double pulse SGE, Q = 8.8,
$f_{0}=$140\text{\,}\mathrm{Hz}$$, the average $h_{\mathrm{rss}}$ values at
DP=$50\%$ in O3 reaches $\sim 1.9\cdot 10^{-23}/\sqrt{$\mathrm{Hz}$}$, but at
a much higher FAR of 2 per year, estimated by multiplying the FAP by the rate
of the investigated BBH GWTs.

###  IV.3 On-source p-value

The on-source (ON) data for each BBH GWT is analyzed using the same
configuration of the cWB ES search of the SIG and BKG analyses (see sec.
III.3). By comparing the ON results with their BKG distributions we can
estimate the p-value of
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\scaleto{\mathrm{ON}}{4pt}}$ per each
BBH GWT:

$\text{p-value}_{\text{ON}}=\frac{\text{EV}_{\text{BKG}}(\text{SNR}^{\mathrm{PMW}}_{\mathrm{rec}}\geq\text{SNR}^{\scaleto{\mathrm{PMW}}{4pt}}_{\scaleto{\mathrm{ON}}{4pt}})}{\text{EV}}\,,$
(11)

where SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\scaleto{\mathrm{ON}}{4pt}}$ is
the on-source reconstructed SNR inside the PMW, EV is the total number of BKG
instances and $\text{EV}_{\text{BKG}}$ is the number of BKG instances with
SNR${}^{\mathrm{PMW}}_{\mathrm{rec}}$ above the ON value. A low p-value points
to SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\scaleto{\mathrm{ON}}{4pt}}$ on the
high-energy tail of the SNR${}^{\mathrm{PMW}}_{\mathrm{rec}}$ distribution for
the null hypothesis. Columns 8 and 9 of table 1 list the
SNR${}^{\mathrm{PMW}}_{\mathrm{ON}}$ and p-value${}_{\text{ON}}$ per each BBH
GWT. Figure 7 reports the p-value for each investigated GWT, ranked from the
lowest to the highest. These estimates are based on the BKG analyses performed
over approximately one calendar month of data around each BBH GWT. We set an a
priori threshold on the false discovery rate [61], FDR $<\,0.1$, to select the
p-values hinting at a rejection of the null hypothesis. These cases are then
the object of deeper follow-up studies.

Figure 7: The plot shows the ordered p-values of
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\scaleto{\mathrm{ON}}{4pt}}$ for the
null hypothesis as measured by BKG analyses. P-values and their statistical
uncertainties are represented by the blue dots. The red dashed line
corresponds to the expected values for the null hypothesis (FDR$=50\%$). The
orange dashed line and orange filled area corresponds to a FDR =$10\%$, and to
the region FDR $<10\%$ respectively. This area is used to select echo
candidates.

Two GW events, GW190701 and GW200224, show an interesting
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{\scaleto{\mathrm{ON}}{4pt}}$ and their
p-values pass the a priori FDR threshold. In both cases, the morphological
information of the outliers reconstructed inside the PMW (see appendix B)
points to a dominant contribution by known instrumental disturbances in the
frequency range $(16,40)\,$\mathrm{Hz}$$ [62, 63]. These noise disturbances
are known to often occur as a train of more pulses with a quasi-regular time
separation. This feature is especially evident in our analysis of GW200224
(see appendix B.2) and can affect our p-values estimates, since it violates
the assumption of uniformly random occurrence times and of independence of
each noise pulse. Therefore, one can expect, at the very least, an
underestimation of the uncertainties of our p-values.

We checked for systematic errors in the p-values of GW190701 and GW200224 by
changing the off-source injection times of the BBH GWTs inside the BKG
analysis. In particular, we repeated the BKG analysis using only
$4096\text{\,}\mathrm{s}$ of data around the GWT time. The new local p-value
estimates are

$\displaystyle\text{GW190701:}\quad\text{p-value}_{\text{ON}}^{\mathrm{local}}=0.004\pm
0.002$ (12)
$\displaystyle\text{GW200224:}\quad\text{p-value}_{\text{ON}}^{\mathrm{local}}=0.007\pm
0.002\,,$ (13)

also reported in figure 11, in appendix B. In the case of GW200224, the
discrepancy between the estimates points to large systematic effects,
including a significant bias of the p-value, which weakens its initial
statistical significance. As for GW190701, the local p-value estimate is also
higher than the initial one, though it may still be compatible within the
stated statistical uncertainties.

Further statistical checks and more morphological tests on GW190701 and GW
200224 are reported in appendix B. Among these checks, the most important
observation is that the reconstructed frequency spectrum for both the
candidates does not match any expectation from echo models [18], so these
outliers cannot be considered plausible candidates for echoes. We conclude
that these two outliers are not suitable candidates for echo signals and are
very likely instrumental disturbances.

For all the other GWTs, our p-value estimates occur well above our FDR
threshold of attention, and their distribution is well described by the
empirical BKG model. Therefore, our work does not reject the null hypothesis,
confirming what was previously reported by different search methods:

  * •

the generic echo search of [24], which estimated p-values in the post-ringdown
of the GWTs detected in observing runs O1 and O2 [25] and in O3b [13];

  * •

the template-based searches [11, 12] on O1, O2 data, and [23, 21], which
provided p-value estimates for O1 GWTs plus GW170104.

We discuss the comparison of performances with the cWB ES search in section V.

Upper limits on echoes amplitude
---
GW name | $h_{\mathrm{rss}}^{\mathrm{MR}}\cdot\frac{10^{-23}}{\sqrt{Hz}}$ | $h_{\mathrm{rss}}^{\mathrm{UL}}\cdot\frac{10^{-23}}{\sqrt{Hz}}$ | $\frac{h_{\mathrm{rss}}^{\mathrm{UL}}}{h_{\mathrm{rss}}^{\mathrm{MR}}}$
GW150914 | 16.0 | 3.4 | 0.21
GW170104 | 11.5 | 2.2 | 0.20
GW170809 | 11.3 | 2.5 | 0.22
GW170814 | 12.1 | 2.5 | 0.21
GW170823 | 9.8 | 2.5 | 0.26
GW190408_181802 | 5.5 | 1.7 | 0.31
GW190412 | 4.2 | 1.3 | 0.31
GW190513_205428 | 8.1 | 1.4 | 0.17
GW190521 | 14.8 | 2.4 | 0.15
GW190521_074359 | 15.1 | 2.5 | 0.17
GW190814 | 2.0 | 1.5 | 0.75
GW190828_063405 | 15.0 | 1.6 | 0.11
GW191109_010717 | 6.2 | 2.1 | 0.34
GW200225_060421 | 5.1 | 2.8 | 0.55
GW200311_115853 | 7.9 | 2.0 | 0.25
GW200224_222234${}^{\text{\textdagger}}$ | 10.8 | 3.7 | 0.34
Table 2: List of the BBH GWTs selected for setting confidence intervals on the
echo’s amplitude. They are a subset of the loudest ones listed in table 1. The
columns report: the GWT name; the merger-ringdown $h_{\mathrm{rss}}$ of the
primary BBH GWT, $h_{\mathrm{rss}}^{\mathrm{MR}}$; the upper limit in terms of
$h_{\mathrm{rss}}$ of possible echo candidates inside the PMW,
$h_{\mathrm{rss}}^{\mathrm{UL}}$; the relative amplitude upper limit defined
as the ratio between the $h_{\mathrm{rss}}^{\mathrm{UL}}$ and
$h_{\mathrm{rss}}^{\mathrm{MR}}$.
†: this GWT event is affected by a loud instrumental glitch in the PMW (see
appendix B.2).

###  IV.4 Upper limits on $h_{\mathrm{rss}}$ of echoes

The confidence belt construction procedure requires SIG analyses with extended
statistics. Therefore, we prioritised the GWTs with a merger and ringdown (MR)
SNR${}_{\mathrm{MR}}\,\geq 7$, as reported in [13, 5], if detectable by the
cWB all-sky burst search. We also added to this list the outstanding GW event
GW190814 [64].

All confidence intervals result in upper limits on the $h_{\mathrm{rss}}$ of
the echo signals, $h_{\mathrm{rss}}^{\mathrm{UL}}$ (see table 2) with the
exception of GW200224 (see section IV.3, and appendix B.2).

Typical upper limits values are in the $h_{\mathrm{rss}}$ range $\sim 1\div
4\times 10^{-23}/\sqrt{Hz}$ at 95$\%$ coverage. The results in terms of
$h_{\mathrm{rss}}$ can be directly converted to GW strain amplitudes through
eq.(10), once a specific waveform of echo signal is assumed.

The ratios between $h_{\mathrm{rss}}^{\mathrm{UL}}$ and the merger-ringdown
$h_{\mathrm{rss}}$ of the primary BBH GWT, $h_{\mathrm{rss}}^{\mathrm{MR}}$,
are also reported in table 2. These ratios are our measured amplitude upper
limits in relative terms, though their connection to the echo’s $A$ parameter
(see section II) depends on the actual morphologies of echo models and of the
primary BBH GWT. In the approximation that the merger-ringdown and each echo
pulse share similar morphologies (e.g. similar central frequency and number of
cycles), then the reported $h_{\mathrm{rss}}$ ratios can be considered to be
equivalent to upper limits on $A$. They are conservative upper limits in case
more echo pulses are detected by cWB ES search within the PMW.

##  V Comparison with previous searches for echoes

Here we provide some comments on the performances of the cWB ES search with
respect to previously reported methods, being aware, however, that a full
comparison of performances would require additional coordinated simulations
which are computationally costly and beyond the scope of this paper. In
particular, we are not able to provide comparisons with LVK searches for
echoes reported in [11, 12, 13], because the published information on this
topic is not detailed enough. Instead, a partial comparison is feasible with a
few dedicated papers: we focus on a previous model-independent search using
simulated data [24], and on three template-based searches [23, 21, 43].

Model-independent search method by Tsang et al. [24]. This general search
method for echoes has been first tested on simulated LIGO Hanford and
Livingstone detector data assuming Gaussian noise [24], and then performed a
search using real data on GWTs detected in O1 and O2 [25] and in O3 [13]. In
simulated Gaussian noise, ref. [24] shows that echo signals are confidently
detectable above SNR $=12$. In addition, at SNR $=8$ the false alarm
probability of noise fluctuations misidentified as signals is at the level of
a few $\%$. The comparison with our cWB ES search can only be semi-
quantitative since no information about the detection efficiency as a function
of echo parameters is available in [24, 25]. We can point out that for the cWB
ES search on real HL data around GW150914, a signal delivering
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}=12$ would also ensure a very confident
detection, with a measured $100\%$ detection probability at false alarm
probability as low as our measurement limit, $0.1\%$. Moreover, with
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}=8$ at the selected false alarm
probability of $5\%$, the detection probability of cWB ES ranges from $95\%$
to $99\%$ depending on the statistics of noise outliers in different periods
of observation. This means that cWB ES achieves high detection performances
also at SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}=8$ in real noise. Moreover, our
off-source simulations clearly show that the data are not compliant with a
stationary Gaussian noise model in the low SNR range of interest in the
proximity of most BBH GWTs.

Model-dependent search method by Rico K. L. Lo et al. [23] This has been the
first model-dependent search that challenged the claim of an echo discovery
after GW150914 by Abedi [20]. Figure 4 from [23] shows that an A parameter
greater than 0.3 can be detected with a 5$\sigma$ threshold for the GW150914
emission in Gaussian noise and using Advanced LIGO design sensitivity. The
signal model’s parameters used in [23] for this result are very similar to the
ones used here: the only non-negligible differences are on $\gamma$ parameter
and number of pulses. In [23], three echo-like pulses have been injected with
$\gamma\sim$ 0.9, while here we injected two with $\gamma=0.5$. Detectability
of A $\sim 0.3$ is well in the ballpark of our method when using real data,
see last column from table 2. In particular, for GW150914, our search
constrains A below 0.21 with 95$\%$ confidence. Moreover, figure 4 shows that
cWB ES search can identify echo signals at 95$\%$ confidence when
SNR${}^{\scaleto{\mathrm{PMW}}{4pt}}_{rec}$ passes the 4 $\sigma$ threshold, a
performance which is comparable with what reported in table 3 from [23] 444In
table 3 of [23], the threshold on the detection statistics which corresponds
to 4 $\sigma$ in Gaussian noise, is delivering a 2 $\sigma$ confidence in real
GW150914 noise, similarly to our result..

Model-dependent search method by Westerweck et al. [21] This template-based
search has been deployed on real data analyzing four BBH GWTs (including
GW150914) and does not find violations of the null hypothesis. It estimates
the p-values of results by using different noise instantiations close to the
GWTs times, which is a similar method to our BKG analysis. Instead, the
sensitivity of this search is assessed by injecting echo waveforms on
simulated Gaussian noise which preserves the actual power spectral density of
the LIGO detectors at the GWTs detections. Figure 2 and 5 in [21] show that
peak amplitudes of echoes detach from the noise fluctuations starting from
$h_{\mathrm{p}}\simeq 2\cdot 10^{-22}$. In actual noise, our search achieves
$50\%$ detection probability with a false alarm probability of $5\%$ for a
peak amplitude of the assumed echo waveform $h_{\mathrm{p}}\sim 2.3\cdot
10^{-22}$ for GW150914, as estimated from our more general result in terms of
$h_{\mathrm{rss}50\%}$ (see Tab.1). Therefore, we conclude that the
sensitivity of the cWB ES search is at least competitive to that of this
template-based search on this specific echo model. We remark that the
implementation of the model-dependent search uses a template bank and requires
subtraction of the detected GWT trigger from data prior to matched filtering
for the template bank. Such steps add complexity with respect to the cWB ES
search.

Model-dependent analysis by Abedi. [43] Another systematic search for a
specific echo model has been very recently reported by Abedi [43]. This search
analyses 65 GWTs from the LVK catalog of compact binary coalescences. The
method assumes Gaussian noise close to each GW event. The main result reported
is an upper limit value to the echo amplitude, A, resulting to be A $\leq
0.42$ with a 90% credible interval, under the assumption that A is equal for
all analyzed events. In addition, the Bayes factor reported for GW190521
stands out as an outlier, suggesting a preference for post-merger echoes
rather than the null hypothesis. In our study, GW190521 shows an on-source
p-value equal to $0.569\pm 0.006$, suggesting that the data in the PMW are
compatible with noise. Moreover, our relative upper limit on the amplitude
$h_{\mathrm{rss}}$ ratio at 95% coverage is $0.23$ for GW190521 and is as low
as $0.13$ for the loudest GWTs.

##  VI Conclusions

This paper describes a search for secondary gravitational wave transients of
generic morphology which may occur shortly after the ringdown phase of a
primary signal from a Compact Binary Coalescence. The analysis method is
developed on top of the coherent WaveBurst pipeline: it uses the primary GWT
as a trigger and follows up the coherent response of the interferometric
gravitational wave detectors on a selectable time window, defined with respect
to the merger time.

The scientific motivation for this work is the search for gravitational wave
echoes after binary black hole mergers. Such echoes are expected if the final
remnant object is not a standard black hole from the general relativity
theory, either because the event horizon is not fully absorbing or because the
remnant is an exotic compact object larger than the would-be event horizon.
The detection performances of the current search are described in terms of
$h_{\mathrm{rss}}$ strain amplitude and are rather independent of the signal
waveform and spectra within a wide signal class. Therefore, as long as any
echo pulse occurs inside the selected time window, from 0.05 to 0.35
$\mathrm{s}$ or from 0.2 to 1.2 $\mathrm{s}$ after the merger, the reported
results can be interpreted in terms of any echoes’ model.

The analysis of the loudest 33 BBH mergers detected during the O1, O2, and O3
observing runs by the LIGO, Virgo, and KAGRA collaborations is consistent with
null results (see table 1), so no evidence of echo signals is found. This
search provides separate results for single BBH mergers. The off-source
characterization of the detection efficiency vs false alarm probability and
the estimation of p-values of candidates is performed using thousands of real
detector noise instantiations. Therefore, the results do not rely on an a
priori noise model and point out that the actual noise statistic is far from
Gaussian in most cases, even at low SNR. The search also provides a
morphological reconstruction of candidates and, for the first time, the
confidence intervals on the $h_{\mathrm{rss}}$ amplitude of gravitational wave
echoes. The latters turn out to be upper limits, typically ranging in the
interval $\sim 1\div 4\times 10^{-23}/\sqrt{Hz}$ in terms of
$h_{\mathrm{rss}}$ (see table 2).

The two loudest candidates found occur after GW190701 and GW200224. These
candidates are also the only ones featuring low enough p-values to require
further follow-up investigations. Their morphological reconstruction clearly
points to the dominating presence of known pulsating instrumental noise
disturbances at low frequencies, occurring in both the LIGO detectors, and
they are by far inconsistent with any published model of echoes. The pseudo-
regular cadence of these disturbances is the likely cause of a systematic
error in our initial p-values estimates.

To our knowledge, this search for echoes is delivering the highest sensitivity
to the possible presence of gravitational wave echoes occurring within a
selected post-merger time window, without relying on signal templates. Our a
posteriori use of morphological information to reject or accept candidates is
still a sub-optimal strategy. An a priori exploitation of loose morphological
priors of echo signals will likely improve the current method.

We plan to extend this method also to investigate the post-merger emission
after BNS (see e.g. [65]) and NSBH coalescence over a wider frequency range,
exploiting the entire spectral sensitivity of the LVK detectors. Remarkably,
this search can be adapted to study other science cases of interest in current
GW astronomy, which share the expectation of a weak GW feature close to the
coalescence time of the primary CBC GWT signal. Examples include investigation
of memory effects [66, 67, 68, 69, 70], precursors to highly eccentric BBHs
[71, 72, 73, 74, 75], or micro-lensing effects [76, 77].

###### Acknowledgements.

The authors would like to thank Andrea Maselli, Francesco Salemi, and Patrick
Sutton for their constructive inputs. We also acknowledge useful discussions
with Sophie Bini, Alessandro Martini, and Andrea Virtuoso. This research used
data, software, and web tools from the Gravitational Wave Open Science Center,
a service of LIGO Laboratory, the LIGO Scientific Collaboration, and the Virgo
Collaboration. This material is based upon work supported by NSF’s LIGO
Laboratory which is a major facility fully funded by the National Science
Foundation. Virgo is funded by the French Centre National de Recherche
Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN),
and the Dutch Nikhef, with contributions by Polish and Hungarian institutes.
The authors are grateful for computational resources provided by the LIGO
Laboratory and supported by National Science Foundation Grants PHY-0757058 and
PHY-0823459. Andrea Miani thankfully acknowledges the grant provided by the
EGO Consortium EGO-DIR-56-2021 and the University of Trento. Shubhanshu Tiwari
is supported by the Swiss National Science Foundation (SNSF) Ambizione Grant
Number: PZ00P2-202204.

##  Appendix A cWB all-sky burst search vs ES search

Table 3 lists cWB [37, 38] parameters (first column) and their threshold
values that can be tuned in a cWB search, comparing the configuration for the
cWB all-sky burst search [39, 40, 41] (second column) to that of the cWB ES
search (third column). The different tuning of $\eta_{\mathrm{c}}$, Tgap, and
SUBRHO thresholds is motivated by the need to grasp lower SNR triggers in the
search for echoes, while keeping under control the false alarms. Additional
configuration parameters are defined in the cWB ES search: the time width of
the PMW, $\Delta t^{\scaleto{\mathrm{PMW}}{4pt}}$; a blind time after the
coalescence time, $t_{\mathrm{blind}}$; the fraction of correlated energy in
the PMW, $c^{\scaleto{\mathrm{PMW}}{4pt}}_{\mathrm{c}}$.

configuration parameters
---
Parameters | All-Sky O3 search | ES search
bpp | 0.001 | 0.001
subnet | 0.5 | 0.5
$c_{\mathrm{c}}$ | 0.5 | 0.5
$\eta_{\mathrm{c}}$ | 5.0 | 3.5
$A_{\mathrm{core}}$ | 1.7 | 1.7
$T_{\mathrm{gap}}$ | 0.2 $\mathrm{s}$ | 2.0 $\mathrm{s}$
$F_{\mathrm{gap}}$ | 128.0 $\mathrm{Hz}$ | 128.0 $\mathrm{Hz}$
SUBRHO | 5.5 | 3.5
SUBNET | 0.1 | 0.1
PMW | not used | $\Delta t^{\scaleto{\mathrm{PMW}}{4pt}}=$1\text{\,}\mathrm{s}$$
$t_{\mathrm{blind}}$ | not used | $t_{\mathrm{blind}}\geq$40\text{\,}\mathrm{ms}$$
$c^{\scaleto{\mathrm{PMW}}{4pt}}_{\mathrm{c}}$ | not used | $c^{\scaleto{\mathrm{PMW}}{4pt}}_{\mathrm{c}}\geq 0.5$
Table 3: Comparison of the cWB main production thresholds between all-sky
burst search [41] (second column) and ES search (third column).

(a) GW190701 - L1 strain.

(b) GW200224 - L1 strain.

Figure 8: On top: Fig. 8(a) shows the strain amplitude waveform of GW190701
and its post-merger as function of time for L1 detector. On the bottom: Fig.
8(b) reports the strain amplitude of GW200224 and its post-merger as function
of time for L1 detector. In both plots the black vertical line marks the
coalescence time of the binary. For both the GW signals are clearly visible
the glitches providing the energy excesses when running the ES search.

##  Appendix B Followup of loudest candidates

From the analysis of the p-values of the BBH GWTs (see section IV.3), two
events are selected for deeper investigations since they are consistent with a
FDR $\leq 10\%$: GW190701 and GW200224. Estimating the p-values on a
different, more local set of noise instantiations results in higher p-values,
which points to some systematic bias in our estimating procedure.
Nevertheless, these two local p-values are still the only ones $\leq 1\%$,
further motivating the following deeper investigations on GW190701 and
GW200224.

The morphological study of the PMW on-source event allows to gather
information about the reconstructed SNR of the energy excess, its arrival
time, mean frequency, and the reconstructed waveform. Additional tests have
been deployed as well, like performing a single detector analysis of the on-
source morphology, with the subtraction of the primary BBH waveform. The
information of the morphological studies are then compared with the
theoretical expectation of echo models (see section II) and with the known
noise disturbances.

(a) L1 TF map.

(b) H1 TF map.

(c) L1 TF map - BBH sub.

(d) LH - Likelihood.

(e) L - Likelihood - BBH sub.

Figure 9: In this figure: plots 9(a) and 9(b) show the time-frequency map of
GW190701 event in L1 and H1 detector respectively. Plot 9(c) shows the event
in L1 detector once the best template of GW190701 between its posterior
samples is subtracted from the data. The white vertical line marks the
coalescence time of the BBH event. Plot 9(d) show the reconstructed maximum
likelihood of the event for the LH network, while plot 9(e) display the same
quantity but for a single detector search (L1) and after the best GW190701
template is subtracted from the data. The black vertical line marks the
coalescence time of the BBH event. Note that the color bars scale between L
and H detectors have a different range.

###  B.1 GW190701

Figure 8(a), shows the reconstructed strain signal waveform of GW190701 in L1
detector. Here, the BBH signal is the smallest bump on the left while the two
bumps on its right are the post-merger energy excesses. Among them, the most
interesting one is the second (at time $\sim$168.86\text{\,}\mathrm{s}$$)
since it is the one falling inside our PMW. The post-merger candidate shows
higher strain, and longer time duration, around
$\geq$100\text{\,}\mathrm{ms}$$, with respect to the BBH event, and no echo
models are consistent, to our knowledge, with these features.

The entire on-source event (BBH + PM signals) has an overall SNR content
around SNR $\sim 12.9$ with a $\rho\sim 4.8$, and $c_{\mathrm{c}}\sim 0.57$
that is an unusually low value for an event with such an SNR. Figures 9(a) and
9(b), the two TF maps of the event for each detector, show that in
L1-detector, after the BBH event, there are three post-merger energy excesses,
at times $\sim$168.64\text{\,}\mathrm{s}$$, $\sim$168.86\text{\,}\mathrm{s}$$,
and $169.07\text{\,}\mathrm{s}$, while in the post-merger of H1 there is only
one clear energy excess at $\sim$168.84\text{\,}\mathrm{s}$$. This energy
distribution asymmetry explains the low value of the correlation coefficient
$c_{\mathrm{c}}$ suggesting that a noise realisation is a preferred
explanation for such an observation since it does not match up with echo
signal predictions. Furthermore, in the bottom row of figure 9 there is the
network on-source likelihood 9(d) TF map. At time
$\sim$168.55\text{\,}\mathrm{s}$$ there is the chirping cluster of pixels
representing the GW190701 event, going from frequencies around
$\sim$40\text{\,}\mathrm{Hz}$$ up to $\sim$150\text{\,}\mathrm{Hz}$$, while in
the post-merger, at the time $\sim$168.84\text{\,}\mathrm{s}$$, is clearly
visible the energy excess. It has a central frequency around
$f_{0}\sim[30-40]\,$\mathrm{Hz}$$ which is not a frequency range expected for
echoes: they should possess similar frequencies or higher than the BBH merger
one [18]. Finally, figure 9(e) shows the on-source likelihood TF map after the
BBH subtraction for the single L1 detector configuration. A repetition of
similar pulses is visible both before and after GW190701 coalescence time
($\sim$168.55\text{\,}\mathrm{s}$$). This is again inconsistent with echo
models, and points to an accidental coincidence with noisy features polluting
L1 data.

(a) L1 TF map.

(b) H1 TF map.

(c) L1 TF map - BBH sub.

(d) LH - Likelihood.

(e) L - Likelihood - BBH sub.

Figure 10: In this figure: plots 10(a) and 10(b) show the time-frequency map
of GW200224 event in L1 and H1 detector respectively. Plot 10(c) shows the
event in L1 detector once the best template of GW200224 between its posterior
samples is subtracted from the data. The white vertical line marks the
coalescence time of the BBH event. Plot 10(d) show the reconstructed maximum
likelihood of the event for the LH network, while plot 10(e) display the same
quantity but for a single detector search (L1) and after the best GW200224
template is subtracted from the data. The black vertical line marks the
coalescence time of the BBH event. Note that the color bars scale between L
and H detectors have a different range.

###  B.2 GW200224

To study the post-merger on-source energy excess detected in GW200224 we
deploy the same strategy in GW190701. Figure 8(b) shows the on-source strain
waveform of the entire event, with the BBH being the small signal on the left.
The time duration of the PM signal, $\sim$400\text{\,}\mathrm{ms}$$ as well as
its time distance of $\sim$1\text{\,}\mathrm{s}$$ to the merger time of the
BBH do not match the theoretical predictions of echo signals. Following eq.(3)
$t_{\mathrm{echo}}$ is predicted to be $\sim$\text{\,}\mathrm{ms}$$ after
$t_{\mathrm{coa}}$. Moreover, the TF map of the on-source event, figure 10(a)
shows that the mean frequency of this PM excess of energy is around
$40\text{\,}\mathrm{Hz}$ well below the expected frequency values for echoes.

Figures 10(a) and 10(b), the TF maps of the event in L1 and H1 respectively,
shows that the PM signal is present only in L1 detector, while in H1 such high
energy excess is not reconstructed. Since the two LIGO detectors are nearly
aligned and are sensitive to the same GW’s polarisation, for real
astrophysical events such energy imbalance in the detectors is suspicious.

We proceed in subtracting to GW200224 on-source event the best PE model
describing that same BBH event, then on the subtracted data we run the single
detector ES search. The result is displayed in figure 10(e). Here, no
undetected energy excess other than the investigated one appears, suggesting
that we are not in a scenario similar to the single detector analysis of
GW190701. The energy outlier has a SNR $\sim 10.4$, while the overall SNR of
the BBH signal plus post-merger excess of energy is equal to $\sim 16.8$ (in
single detector mode).

Figure 11: Same plot of figure 7, where the green dots are the on-source
p-value obtained from the ES search carried on using only the
$4096\text{\,}\mathrm{s}$ around the main BBH event. The violet dots are the
on-source p-value for the ES search performed over the same standard analysis
data but applying a $32\text{\,}\mathrm{Hz}$ data mitigation plugin to
suppress some noisy contaminating the O3a and O3b observational periods.

###  B.3 cWB ES search with 32 Hz mitigation

The PMW on-source morphologies hint to a possible data pollution by a glitch
family identified in the frequency range
$\in\left(16,40\right)\,$\mathrm{Hz}$$ [62, 63]. Therefore, we repeated the ES
search for these two GWTs by including a specific single detector data filter
[78, 79] that estimates the power oscillations within the frequency range
$\in\left(16-40\right)\,$\mathrm{Hz}$$ and attenuates them. We label such
analysis as $32\text{\,}\mathrm{Hz}$-ES search, to differentiate it from the
standard ES search. The measured on-source null hypothesis p-values when the
noise around $32\text{\,}\mathrm{Hz}$ frequencies is mitigated result:

$\displaystyle\text{GW190701:}\quad\text{p-value}_{\text{ON}}^{$32\text{\,}\mathrm{Hz}$}=0.024\pm
0.002$ (14)
$\displaystyle\text{GW200224:}\quad\text{p-value}_{\text{ON}}^{$32\text{\,}\mathrm{Hz}$}=0.003\pm
0.001\,$ (15)

and they are plotted in figure 11 as the violet dots. This noise mitigation
rules out the post-merger event candidate GW190701, while for the PM of
GW200224 the p-value is still within the FRD$\leq 10\%$.

This study together with the morphological investigation of the PMW energy
excesses of GW190701 and GW200224 (see appendix B.1 and B.2), show that it is
reasonable to assume them to be non stationary noise feature polluting the
data and especially affecting L detector. These noise transients posses a
central frequency around $(30,40)$\mathrm{Hz}$$, and have a greater time
duration ($\sim$ hundred of $\mathrm{ms}$) with respect to the expected one
for echo signals ($\sim$ tens of $\mathrm{ms}$), so around one order of
magnitude bigger.

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