# TESLA-X: An effective method to search for sub-threshold lensed
gravitational waves with a targeted population model

Alvin K. Y. Li<EMAIL_ADDRESS>LIGO Laboratory, California Institute of
Technology, Pasadena, CA 91125, USA  Juno C. L. Chan<EMAIL_ADDRESS>Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100
Copenhagen, Denmark  Heather Fong<EMAIL_ADDRESS>RESCEU, The University of
Tokyo, Tokyo, 113-0033, Japan  Graduate School of Science, The University of
Tokyo, Tokyo 113-0033, Japan  University of British Columbia, Vancouver
Campus, Vancouver, B.C. V6T 1Z4, 2329 West Mall, Canada  Aidan H.Y. Chong
Department of Physics, The Chinese University of Hong Kong, Hong Kong  Alan J.
Weinstein  LIGO Laboratory, California Institute of Technology, Pasadena, CA
91125, USA  Jose M. Ezquiaga  Niels Bohr International Academy, Niels Bohr
Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark

(August 28, 2024)

###### Abstract

Strong gravitational lensing can produce copies of gravitational-wave signals
from the same source with the same waveform morphologies but different
amplitudes and arrival times. Some of these strongly-lensed gravitational-wave
signals can be demagnified and become sub-threshold. We present TESLA-X, an
enhanced approach to the original GstLAL-based TargetEd Subthreshold Lensing
seArch (TESLA) method, for improving the detection efficiency of these
potential sub-threshold lensed signals. TESLA-X utilizes lensed injections to
generate a targeted population model and a targeted template bank. We compare
the performance of a full template bank search, TESLA, and TESLA-X methods via
a simulation campaign, and demonstrate the performance of TESLA-X in
recovering lensed injections, particularly targeting a mock event. Our results
show that the TESLA-X method achieves a maximum of $\sim 20\%$ higher search
sensitivity compared to the TESLA method within the sub-threshold regime,
presenting a step towards detecting the first lensed gravitational wave.
TESLA-X will be employed for the LIGO-Virgo-KAGRA’s collaboration-wide
analysis to search for lensing signatures in the fourth observing run.

##  I Introduction

Gravitational lensing refers to the phenomenon in which properties of
propagating waves are influenced by the distortion of spacetime caused by
massive intervening objects. The effects of gravitational lensing depends on
the mass properties of the lens Ohanian (1974); Thorne (1982); Deguchi and
Watson (1986); Wang  _et al._ (1996); Nakamura (1998); Takahashi and Nakamura
(2003). For massive lenses significantly larger than the gravitational-wave
wavelength, we can assume geometric optics, which results in repeated signals
with relative (de-)magnifications, time delays, and phase shifts Wang  _et
al._ (1996); Dai and Venumadhav (2017); Ezquiaga  _et al._ (2021); Ng  _et
al._ (2018); Li  _et al._ (2018); Oguri (2018); Smith  _et al._ (2018, 2017,
2019); Robertson  _et al._ (2020); Ryczanowski  _et al._ (2020); Ezquiaga  _et
al._ (2023). Lenses with sizes comparable to gravitational-wave wavelengths
can produce beating patterns through frequency-dependent wave optics effects,
formally known as microlensing Deguchi and Watson (1986); Nakamura (1998);
Takahashi and Nakamura (2003); Cao  _et al._ (2014, 2014); Lai  _et al._
(2018); Christian  _et al._ (2018); Dai  _et al._ (2018); Diego  _et al._
(2019); Çalışkan  _et al._ (2023a); Cheung  _et al._ (2021); Mishra  _et al._
(2021); Meena  _et al._ (2022); Cremonese  _et al._ (2021); Çalışkan  _et al._
(2023b).

The search for gravitationally-lensed gravitational waves is driven by various
scientific objectives, such as testing general relativity Ezquiaga and
Zumalacárregui (2020); Baker and Trodden (2017); Collett and Bacon (2017);
Goyal  _et al._ (2023); Ezquiaga  _et al._ (2021); Goyal  _et al._ (2023),
improving source localization accuracy Hannuksela  _et al._ (2020); Yu  _et
al._ (2020), characterizing lens models Lai  _et al._ (2018); Diego  _et al._
(2019); Oguri and Takahashi (2020); Liu  _et al._ (2023), and performing
precision cosmology Sereno  _et al._ (2011); Liao  _et al._ (2017); Cao  _et
al._ (2019); Hannuksela  _et al._ (2020) and statistical cosmology Xu  _et
al._ (2021).

The LIGO-Virgo-KAGRA (LVK) collaboration has conducted various analyses
searching for lensing signatures in the event catalogs of LVK’s observing runs
O1-O2 Hannuksela  _et al._ (2019) and O3 Abbott  _et al._ (2021a, 2023), but
no convincing evidence of lensing signatures has been found. Additionally,
other analyses have concluded that there are no confident lensing signatures
in the detected gravitational-wave events Li  _et al._ (2023); McIsaac  _et
al._ (2020); Dai  _et al._ (2020); Liu  _et al._ (2021); Ezquiaga  _et al._
(2023).

Strongly lensed gravitational waves are likely to be originating from sources
living in the higher redshift universe. These signals, without being lensed,
will only be barely detectable or even undetectable. Under strong lensing,
copies of gravitational waves from these high redshift sources will be
produced, some may be magnified and are detected as super-threshold
gravitational waves, but some may be potentially de-magnified with a reduced
signal-to-noise (SNR) ratio that is below detection threshold. These signals,
formally known as “sub-threshold” signals, are of particular interest, as a
significant proportion of strongly-lensed gravitational-wave events falls into
this category Wierda  _et al._ (2021). Retrieving these possible sub-threshold
lensed counterparts is a key step to boost the confidence for the first
detection of strongly-lensed gravitational waves.

Two targeted search methods Li  _et al._ (2023); McIsaac  _et al._ (2020) have
been developed and employed to recover potential sub-threshold lensed
counterparts to known super-threshold gravitational waves. The GstLAL-based
TargetEd Subthreshold Lensing seArch (TESLA) described in Li  _et al._ (2023)
strategically reduces the search parameter space to look for possible sub-
threshold lensed counterparts. However, as we will explain in the paper, the
traditional TESLA method can potentially lose signals due to an excessive loss
in signal-to-noise (SNR) ratio due to how the targeted template bank is
constructed. Additionally, TESLA’s efficiency in finding potential sub-
threshold lensed gravitational waves can be improved further if a targeted
population model built based on information from the target superthreshold
event is used. A proper targeted population model in the ranking statistics
can further reduce the influence of the noise background. Detailed information
on the TESLA method can be found in Li  _et al._ (2023).

In this paper, we present a novel method, TESLA-X, to search for strongly-
lensed sub-threshold gravitational-wave signals. TESLA-X employs a targeted
population model to enhance the search sensitivity for potential lensed sub-
threshold signals, and it addresses the excessive SNR loss issue by utilizing
a new targeted template bank construction method. The paper is organized as
follows: Section II provides a brief review of strong lensing of gravitational
waves. Section III outlines the workflow of the traditional TESLA method. In
Section IV, we give an overview on how the population prior term in the
likelihood ratio calculations for triggers from the GstLAL search pipeline is
calculated, and detail the TESLA-X methodology. Section V presents a
simulation campaign to evaluate the TESLA-X method. Finally, in Section VI, we
summarize our findings and discuss potential future work to further enhance
the search sensitivity of the TESLA-X pipeline.

##  II Strong lensing of gravitational waves

Massive objects, like galaxies and galaxy clusters, can produce distortion in
spacetime. General relativity predicts that when waves emitted from a source
pass by these massive objects (also known as _gravitational lenses_), they
will be deflected due to the curvature of spacetime. This effect is known as
_gravitational lensing_ , which is universal according to the equivalence
principle, i.e. it affects gravitational waves in the exact same way as
electromagnetic waves (EM waves). Depending on the spatial extent of the
potential well of gravitational lenses and the scales of wavelength, the
effect of gravitational lensing can be vastly different111In fact,
gravitational lensing can be sub-classified into strong lensing, weak lensing
and microlensing. Some literature also distinguishes millilensing as a sub-
type of gravitational lensing.. In this paper, we focus on _strong lensing_
with geometric optics, in which we assume that gravitational waves have a much
shorter wavelength that the spatial scale of the gravitational lenses.

Under the strong lensing hypothesis, strong gravitational lenses can produce
multiple copies of transient gravitational waves from the same source. These
repeated signals have identical waveforms (i.e. their underlying intrinsic
parameters $\vec{\theta}$, e.g. component masses and spins, are the same),
except that (I) their amplitudes may differ by some amplitude scaling factor
$\sqrt{\mu_{j}}$222Note that the amplitude scaling factor is achromatic under
the geometric optics assumption, and the amplification factor can be $>1$
(magnified) or $\leq 1$ (de-magnified)., (II) they are separated by a relative
arrival time delay $\Delta t_{j}$ with respect to the not-lensed signal’s
coalescence time, and (III) they have an additional Morse phase factor
$\Delta\phi_{j}$ applied to depending on the type of lensed signals333There
are three types of lensed signals, type I, II and III. They correspond to the
minimum, saddle point and maximum time-delay solution to hte lens equation
respectively., given by Ezquiaga and Zumalacárregui (2020); Dai  _et al._
(2020)

$\Delta\phi_{j}=-\frac{n_{j}\pi}{2},\,\,n_{j}=\begin{cases}0,\text{Type I
lensed signals}\\\ 1,\text{Type II lensed signals}\\\ 2,\text{Type III lensed
signals}\end{cases}.$ (1)

Mathematically, if we denote the not-lensed gravitational wave in the
frequency domain as $\tilde{h}^{\text{NL}}(f;\vec{\theta},\Delta t_{j}=0)$,
then the $j^{\text{th}}$ strongly-lensed counterpart waveform $\tilde{h}_{j}$
is

$\tilde{h}_{j}(f;\vec{\theta},\mu_{j},\Delta
t_{j},\Delta\phi_{j})=\sqrt{|\mu_{j}|}\tilde{h}^{\mathrm{NL}}(f;\vec{\theta},\Delta
t_{j})e^{\left(i\,\mathrm{sign}(f)\Delta\phi_{j}\right)}.$ (2)

Since the travelling paths of these repeated signals have been altered, these
signals will appear to be originating from sources at different sky locations.
However, the deviation in sky location due to strong lensing is in order of
arc-seconds. This is negligible compared to the uncertainty in sky
localization for gravitational waves, which is in the order of degrees Abbott
_et al._ (2018). Therefore, for the rest of this paper we will assume multiple
strongly-lensed gravitational-wave signals from the same source to have the
same sky location.

##  III The search for sub-threshold strongly-lensed gravitational waves:
Objectives and current approach

###  III.1 Current effort to search for strongly-lensed sub-threshold lensed
counterparts to super-threshold gravitational waves

When searching for gravitational waves in general, signals with sufficiently
high amplitudes that can be identified as gravitational waves from the noisy
data are known as _super-threshold_. In contrast, weaker gravitational waves
that are not distinguishable from noise are known as _sub-threshold_. As
discussed in section II, strong lensing can produce multiple copies of
gravitational waves coming from the same source with different amplification
factors $\sqrt{\mu_{j}}$. $\sqrt{\mu_{j}}$ can take on values smaller than
$1$, i.e. the repeated gravitational-wave signals are _de-magnified_ compared
to the not-lensed gravitational-wave signal and may become sub-threshold. It
is then natural to ask the question: If we assume that one of the identified
super-threshold gravitational waves detected so far is strongly lensed, can we
find its sub-threshold strongly lensed counterparts, if they exist?

There have been ongoing efforts to employ _matched-filtering based search
pipelines_ with modifications Li  _et al._ (2023); McIsaac  _et al._ (2020) to
search for strongly-lensed sub-threshold lensed counterparts to identified
super-threshold gravitational waves from LVK’s first three observing runs (O1,
O2 and O3) Hannuksela  _et al._ (2019); Abbott  _et al._ (2021a, 2023). In
this paper, we focus on the _TargetEd subthreshold Lensing seArch (TESLA)_
method Li  _et al._ (2023), which is built based on the GstLAL search pipeline
Messick  _et al._ (2017); Sachdev  _et al._ (2019); Hanna  _et al._ (2020);
Allen  _et al._ (2012); Cannon  _et al._ (2021); Godwin  _et al._ (2020); Chan
_et al._ (2020); Mukherjee  _et al._ (2021); Messick  _et al._ (2020). The
following sub-section gives a brief introduction to the traditional TESLA
method.

###  III.2 The traditional TESLA search method

Traditional TESLA is built based on the matched-filtering based pipeline
GstLAL. A detailed introduction to the GstLAL search pipeline can be found in
Li  _et al._ (2023); Messick  _et al._ (2017); Cannon  _et al._ (2021). In a
general search for gravitational waves from compact binaries coalescences, the
goal is to detect signals that span the entire parameter space to which the
detector network’s frequency bandwidth is sensitive. Hence, a large template
bank composed of over a million template waveforms is constructed and used. A
large number of candidates (also known as triggers), regardless of whether
they are noise or signals, is generated. In general, the more templates we use
in a search, the larger the trials factor, and hence the larger the noise
background will become/ Gravitational waves with weaker amplitudes, for
instance strongly-lensed sub-threshold lensed counterparts, can be buried in
the large noise background. Hence, it is necessary to tactically reduce the
noise background, while keeping the targeted foreground constant, in order to
uncover possible weaker gravitational waves.

Assuming a super-threshold gravitational wave is strongly lensed, we want to
search for its possible sub-threshold lensed counterparts. In section II we
explained that strongly-lensed gravitational waves from the same source should
have identical intrinsic parameters (e.g. component masses and spins), and
hence identical waveforms. For each super-threshold event (target), we obtain
the posterior probability distribution that gives the best estimates of the
source parameters of the target using the Bayesian parameter estimation
analyses described in Veitch  _et al._ (2015); Ashton  _et al._ (2019);
Romero-Shaw  _et al._ (2020); Ashton and Talbot (2021). We can then narrow
down the parameter search space for possible strongly-lensed sub-threshold
counterparts to regions that are consistent with the posterior parameter space
of the target. However, as explained in Li  _et al._ (2023), the posterior
parameter space of the target itself is insufficient because of noise
fluctuations in the data. Non-Gaussianity in the data can cause false signals
(especially those that have weaker amplitudes and are hence sub-threshold), to
be registered by templates that have very different parameters than those of
the posterior samples for the target.

TESLA accounts for both the signal sub-space and noise fluctuations in the
data when searching for possible strongly-lensed sub-threshold counterparts to
super-threshold gravitational waves. Here, we outline the steps for running a
TESLA analysis. For each super-threshold gravitational event (target), we
start with the posterior samples ordered in decreasing order of log likelihood
from the parameter estimation for the target. Since the amplitudes of possible
strongly-lensed sub-threshold gravitational waves are lower, they will be
registered as triggers in the search pipeline with lower signal-to-noise ratio
$\rho$ (SNR). For each posterior sample, we generate $1$ injection with the
exact same parameters (including luminosity distance to the source $D_{L}$ and
sky location, i.e. right ascension $\alpha$ and declination $\delta$) as the
sample, and $9$ extra injections with $D_{L}$ increased (and hence optimal SNR
$\rho_{\mathrm{opt}}$ decreased) to mimic the de-magnifying effect of strong
lensing on the amplitudes of possible sub-threshold lensed counterparts,
according to the inverse proportionality of $D_{L}$ and the optimal SNR
$\rho_{\mathrm{opt}}$:

$D_{L}\propto\frac{1}{\rho_{\mathrm{opt}}}.$ (3)

We require the latter $9$ injections to have detector SNR
$\rho_{\mathrm{det}}\geq 4$ in at least one detector as a constraint from the
GstLAL search pipeline444We do not lower the single detector SNR threshold in
the GstLAL search pipeline since this will lead to an exponential increase in
the number of both signal and noise triggers. As a future work, we will
explore the possibility to further lower the single detector SNR threshold..
The set of lensed injections represents the possible sub-threshold lensed
counterparts to the target that we are searching for. They are then injected
into real data, and we perform an injection campaign using GstLAL with a full
template bank to recover them. After the injection campaign, templates that
are able to find these injections555An injection is “found” if the FAR of the
associated trigger is $\leq$ $1$ in $30$ days. are kept to construct a
targeted template bank to search for possible strongly-lensed sub-threshold
counterparts to the target. The end product of TESLA is a ranked list of
possible lensed counterparts to the target in increasing order of FARs.
Readers are reminded that the FARs (or other ranking statistics) reported in
the search’s ranked list do not indicate how likely the signal candidate is a
lensed counterpart to the target, but only a priority label for follow-up
analysis.

For a detailed explanation of the TESLA method, please refer to Li  _et al._
(2023).

##  IV The upgraded TESLA-X search method

###  IV.1 The missing pieces in the traditional TESLA method: Non-optimal
template bank and population model

We have shown in Li  _et al._ (2023) that the traditional TESLA method succeed
in improving the search sensitivity to possible strongly-lensed sub-threshold
gravitational waves. However, there are two puzzle pieces missing from the
full picture.

First, in the original full template bank, the parameter ranges are informed
by the masses and spins of detectable sources (BNS, NSBH, BBH) and the
templates are distributed with a density that allows a maximum loss of $3\%$
SNR when a gravitational-wave signal is recovered (compared to the optimal
SNR666The optimal SNR is the SNR when the signal is cross-correlated with a
template with the exact same parameters as the signal.) Mukherjee  _et al._
(2021). However, when a reduced template bank is generated using traditional
TESLA, we discard templates that did not recover lensed injections for the
target super-threshold gravitational waves. This breaks the original
optimality of the full template bank, caused by the finite number of lensed
injections that are used to generate the reduced template bank, and strongly-
lensed sub-threshold signals whose optimal detector SNR is near threshold
(i.e. around $\rho_{\mathrm{opt}}\approx 4.12$) will not be detected.
Therefore, it is important that the optimality of the reduced template bank
must be conserved to maximize our search sensitivity towards possible
strongly-lensed sub-threshold gravitational waves.

Second, a population model was not set in traditional TESLA. In the analysis,
the absence of a population model carries an implicit assumption that every
template in the template bank has equal probability to recover a possible sub-
threshold counterpart. Since our goal is to detect sub-threshold counterparts
whose intrinsic parameters are identical to a super-threshold signal,
implementing a population model centred around the signal’s recovered
parameters will increase the pipeline’s sensitivity in detecting signals in
that region. The population model should also be dependent on SNR to account
for sub-threshold signals that are coloured by noise and, thus, could be
recovered by a template that is different from that recovered the super-
threshold signal.

###  IV.2 The likelihood ratio in the GstLAL search pipeline

To rank candidates, GstLAL assigns each trigger a likelihood ratio as its
ranking statistic, defined as Sachdev  _et al._ (2019); Cannon  _et al._
(2021); Messick  _et al._ (2017):

$\mathcal{L}=\frac{P(\vec{D}_{H},\vec{O},\vec{\rho},\vec{\xi}^{2},\left[\Delta\vec{t},\Delta\vec{\phi}\right]|\vec{\theta},\mathrm{signal})}{P(\vec{D}_{H},\vec{O},\vec{\rho},\vec{\xi}^{2},\left[\Delta\vec{t},\Delta\vec{\phi}\right]|\vec{\theta},\mathrm{noise})}\cdot\mathcal{L}(\theta),$
(4)

where the first term,
$P(...|\vec{\theta},\mathrm{signal})/P(...|\vec{\theta},\mathrm{noise})$, is
the ratio of the probability of obtaining the candidate event under the signal
model to that under the noise model. It depends on (1) the detectors $\vec{O}$
that are participating at the time of the candidate event, (2) the horizon
distances of the participating detectors $\vec{D}_{H}$, (3) the detector SNRs
$\vec{\rho}$ and (4) the auto-correlation based signal consistency test values
of the candidate event at each detector $\vec{\xi}^{2}$. If a candidate event
is found in coincidence at multiple detectors (i.e. the event is a
_coincident_ event, $\ln\mathcal{L}$ will also include (5) the relative time
delays $\Delta\vec{t}$ and (6) the relative phase delays $\Delta\vec{\phi}$ of
the candidate event between participating detectors777These relative time
delays and phase delays should not be confused with the relative time delays
and phase delays coming from strong lensing.

The second term in Equation 4,

$\mathcal{L}(\theta)=\frac{P(\vec{\theta}|\mathrm{signal})}{P(\vec{\theta}|\mathrm{noise})},$
(5)

is the ratio of the probability that a trigger is recovered by a template with
parameters $\vec{\theta}$, given the trigger is an astrophysical signal or
noise. Because this ratio does not depend on the data observed by the
detectors and depends on input parameters given prior to performing the
search, it has been written as a separate term.

The method of calculating the numerator and denominator are described in
detail in Fong (2018) and Cannon  _et al._ (2015), respectively. While the
derivations are not repeated here, we wish to highlight that information about
the population model is enfolded into the numerator; therefore, it is
$P(\vec{\theta}|\mathrm{signal})$ that allows the pipeline to perform targeted
source searches. In the next section, we describe the changes that were made
to improve the strongly-lensed gravitational-wave search.

###  IV.3 A better reduced template bank, and a lensing-based targeted
population model

Traditional TESLA aims to effectively reduce the noise background to uncover
possible strongly-lensed counterparts to the target super-threshold
gravitational wave. We reduce the number of templates, and hence trials
factors, by keeping only templates that can cover the parameter space where
possible lensed counterparts may live in. By doing an injection campaign,
traditional TESLA locates a subset of templates from the full template bank to
construct a reduced template bank to search for possible lensed counterparts
to a given target super-threshold gravitational wave. However, as explained at
the beginning of IV, there are two problems in traditional TESLA, both which
lead to a decrease in the search sensitivity of TESLA.

We now propose two improvements to the TESLA method that will resolve these
problems: (1) constructing a better reduced template bank, and (2) generating
a targeted population model.

We recall that the sole purpose of the injection campaign is to determine the
search parameter subspace that we should focus on to look for possible
strongly-lensed sub-threshold counterparts to the target super-threshold
gravitational waves. The injection campaign is always imperfect because we can
only do limited number of injections with discretized SNRs, i.e. we cannot
cover the full spectrum of SNRs that possible lensed sub-threshold signals can
have. Therefore, the templates that are able to recovered lensed injections in
the injection campaign can only serve as a pointer as to the region of search
parameter space we should be targeting, i.e. a search parameter subspace where
possible lensed sub-threshold counterparts to the target super-threshold
gravitational waves may live in. We should hence make use of the injection
campaign results to _define a boundary_ within which templates live should be
used collectively to construct the reduced template bank to search for
possible lensed counterparts to the target, regardless of whether the
templates are able to recover injections from the injection campaign. The
proposed new reduced template bank does not discard templates within the
narrowed-down search parameter space defined by the boundary, and hence it
still maintains the optimality guaranteed by the original full template bank
within the limited search space. In turn, any possible lensed counterparts
that live within the boundary will be recovered with an SNR loss less than
$3\%$, preventing any loss of potential signals due to excessive SNR loss.

Under the strong lensing hypothesis, possible lensed counterparts, regardless
of being super-threshold or sub-threshold, should have the exact same
parameters as the target super-threshold gravitational waves. Therefore, the
population of possible lensed counterparts to the target gravitational waves
should have a higher probability of lying near the signal subspace defined by
the posterior parameter probability distribution of the super-threshold event,
and a lower probability to be living far away from that. When we perform the
injection campaign in traditional TESLA, we can obtain both the recovered
injections’ SNRs and the corresponding templates that recover the injections.
Generally speaking, sub-threshold signals which have lower optimal SNRs are
affected by noise fluctuations more substantially and hence can be recovered
by templates with parameters different from those of the target super-
threshold event. That said, the templates that are closer to the signal
subspace defined by the posterior parameter probability distribution of the
target are associated with recovered injections with higher SNR values.
Therefore, we can make use of the results from the injection campaign to
construct a _Gaussian Kernel Density Estimation_ (KDE) function
$\text{KDE}(\vec{\gamma})$ using the distribution of the templates that
recovered the lensed injections, with the corresponding injections’ recovered
SNRs as weights. The KDE is a function of source parameters $\vec{\gamma}$
which, to match the parameters of the template bank, are set to be the masses
and spins of the source. When weighted with the recovered SNRs, regions that
are closer to the signal subspace that the target super-threshold
gravitational wave lies in will give a higher value for the KDE function,
which matches our expectation that lensed counterparts to the target should
have similar parameters as the target. The KDE function can then be redefined
as the probability density function that describes the “source population” for
our possible strongly-lensed sub-threshold counterparts to the target
gravitational wave. Using this as our population model, we then follow the
steps detailed in Fong (2018) to calculate $P(\vec{\theta}|\mathrm{signal})$,
i.e. the probability that a signal is recovered by a template with parameters
$\vec{\theta}$. Notably, this probability is dependent on SNR, such that the
search does not overly penalize templates that are far away from the signal
subspace for weak sub-threshold signals. These methods instill a correct
population model to the new TESLA method compared to the incorrect uniform
population model in traditional TESLA, and in turn improve the search
sensitivity.

###  IV.4 The flow of the TESLA-X search method

We now describe in details the actual flow of the proposed TESLA-X search
method.

Figure 1:  Workflow of TESLA-X. Elements in grey are the same procedures as in
traditional TESLA. Colored elements (color online) are new procedures in
TESLA-X.

Figure 1 presents the workflow of the TESLA-X method. As usual, we start with
a target super-threshold gravitational-wave event that is found during a
general search with GstLAL using a full template bank. Using the posterior
samples obtained by Bayesian parameter estimation of the target event, we
generate, for each sample, one injection that has the same optimal SNR as the
sample, and nine additional injections with smaller optimal SNRs by increasing
their effective distances, requiring that their SNRs in at least one detector
have to be $\geq 4$ to ensure they can be registered as a trigger during the
matched-filtering process in the search. These injections will represent
possible sub-threshold lensed counterparts to the target super-threshold
event. We then inject these simulated lensed injections into real data, and
use the GstLAL search pipeline to try and recover them with a full template
bank. Up to this stage, TESLA-X is still identical to the traditional TESLA
method.

After the injection campaign, we now retrieve a set of templates that are able
to recover the simulated lensed injections, and the associated observed SNRs
of the recovered injections. We can plot the retrieved templates on the chirp
mass $\mathcal{M}_{c}$ \- effective spin $\chi_{\text{eff}}$ space, with the
colors of the points being the associated recovered injections’ observed SNRs
$\rho_{\text{obs}}$. The top left panel of Figure 2 shows an example cartoon
that plots the templates on the $\mathcal{M}_{c}\mathrm{-}\chi_{\text{eff}}$
space, with the color being the recovered injections’ network SNRs.

Figure 2:  A cartoon depicting how we construct a reduced template bank from
the injection results following the Traditional TESLA method (top right) and
the proposed TESLA-X method (bottom left and right).

Traditional TESLA keeps only templates that are able to recover the lensed
injections and use them to construct the reduced bank, hereby known as “TESLA
BANK” (top right panel of Figure 2). For TESLA-X, we construct a Gaussian
Kernel Density Estimation (KDE) function $f(\vec{\gamma})$ for the templates
that recover the injections, with the corresponding recovered SNRs as the
weights, where $\gamma$ is the set of template parameters we are considering,
for example in Figure 2 $\vec{\gamma}$ will be chirp mass $\mathcal{M}_{c}$
and effective spin $\chi_{\text{eff}}$. Using the KDE function, we can then
evaluate the probability density for each template point $\vec{\theta}$ in the
full template bank. We then use “matplotlib.pyplot.contour” with the default
settings of “matplotlib.ticker.MaxNLocator” to determine the characteristic
contour levels from the results (see bottom left panel of Figure 2) for
example). Then, using the lowest contour as the boundary, TESLA-X keeps _all_
the templates within the boundary and use them to construst the reduced
template bank, hereby known as “TESLA-X BANK”. As we have explained in the
previous section, TESLA-X BANK is expected to give better search sensitivity
than TESLA BANK because it conserves the optimality of the original full
template bank, and hence we will not lose signals due to excessive loss in
SNR.

Because the Gaussian KDE function $f(\vec{\gamma})$ is essentially a
probability density function (PDF) that describes the expected distribution of
possible (sub-threshold) lensed counterparts to the target super-threshold
gravitational wave, we can use it as the astrophysical source population prior
to construct a targeted population model using the same methodology described
in the previous section and in Fong (2018). Figure 3 shows an example cartoon
to illustrate the idea.

Figure 3:  A cartoon depicting how we use the injection campaign results to
construct a targeted population model in TESLA-X.

Once the reduced bank and the targeted population model have been constructed,
we again use GstLAL to search through all possible data with the targeted bank
and the targeted population model to look for potential sub-threshold lensed
counterparts to the target super-threshold event. GstLAL will then output a
list of candidate events that are ranked by their assigned ranking statistics,
which include both the FARs and the likelihood ratios $\mathcal{L}$. We have
to stress again that the assigned FARs to the candidate events here are not
quantifying how likely they are to be lensed counterparts to the target event,
but they are measures to distinguish noise triggers (false alarms) to real
astrophysical signals, whether or not they are lensed counterparts to a target
event. The ranking statistics here should only be used as a priority ranking
for follow-up analysis to decide how likely each candidate event is a lensed
counterpart to the target event. As a future work, we will explore the idea of
evaluating for each trigger an additional lensing likelihood ratio that
including additional lensing information as a separate ranking statistic. At
that stage, the lensing likelihood may have the potential to reveal how likely
a trigger is a lensed counterpart to the target event.

##  V Method verification and results

In Li  _et al._ (2023), we carry out a full-scale simulation campaign, where
we prepare mock data with a pair of super-threshold and sub-threshold lensed
signals injected, and apply the TESLA method to recover the two signals. We
have proved that the TESLA method is able to recovered the injected sub-
threhsold lensed signal, if they exist. TESLA-X and TESLA have essentially the
same working principle, but the only differences are how we construct the
reduced template bank and the population model. Therefore, we conduct a
simulation campaign solely to compare the search effectiveness and sensitivity
towards possible sub-threshold lensed gravitational-wave signals using TESLA
and the proposed TESLA-X method.

###  V.1 Mock data generation

First, we generate a 9.15-day long data stream with Gaussian noise recolored
with O4 characteristic power spectral densities for LIGO Hanford, LIGO
Livingston and Virgo detector888The exact data for O4 characteristic power
spectral densities for LIGO Hanford, LIGO Livingston and Virgo detector can be
found at https://dcc.ligo.org/T2200043-v3/public., with the assumptions that
there is no detector downtime999A detector is considered “down” if it is not
in observing mode., and no times are vetoed. We then inject a super-threshold
gravitational wave generated using the IMRPhenomXPHMpseudoFourPN waveform
Pratten  _et al._ (2021) into the mock data. Details about the source
parameters of the injected gravitational-wave are listed in Table 1.

Properties | Injected super-threshold signal
---|---
UTC time | May 08 2022 $11:06:00$
GPS time | $1336043178.397$
Distance (Mpc) | $2858.18$
Primary mass $m_{1}^{\text{det}}$ | $70.08M_{\odot}$
Secondary mass $m_{2}^{\text{det}}$ | $38.83M_{\odot}$
Dimensionless spins |  $\chi_{1x}=0.182$, $\chi_{1y}=0.182$, $\chi_{1z}=-0.0363$,
|  $\chi_{2x}=-0.113$, $\chi_{2y}=0.132$, $\chi_{2z}=0.116$,
Right ascension $\alpha$ | $2.811$
Declination $\delta$ | $0.819$
Inclination $\iota$ | $2.513$
Polarization $\Psi$ | $1.187$
Waveform | IMRPhenomXPHMpseudoFourPN
Table 1: Information of the injected super-threshold gravitational-wave signal
MS220508a in the simulation campaign. All properties reported here are
measured in the detector frame.

In later parts of this paper, we may refer to the super-threshold signal as
MS220508a.

###  V.2 Performing a general search

We then use GstLAL to perform a search over the mock data stream following the
same settings used to search for gravitational waves within O3 data in GWTC-3
Abbott  _et al._ (2021b). We use the same general template bank as described
in Li  _et al._ (2023) for the general search. As expected, the search
recovers MS220508a with the highest ranking statistics (FAR$=2.972\times
10^{-35}$ Hz, rank 1) among all other triggers. We then apply a Bayesian
inference library for gravitational-wave astronomy Bilby Ashton  _et al._
(2019)101010For reference, we used the default MCMC approach to do the
sampling. to perform parameter estimation (PE) for MS220508a, which outputs a
set of posterior samples required for applying the TESLA and TESLA-X search
pipeline.

###  V.3 Performing an injection campaign

Using the posterior samples from the PE for MS220508a, we generated $24272$
simulated lensed injections following equation 3. These simulated signals are
injected into the mock data and we perform another search using GstLAL with
the general tempalte bank and try recovering them. The results of the
injection campaign are then used for performing the TESLA and TESLA-X
analyses.

In the injection campaign, the general template bank recovers $17769$ out of
$24272$ injections (See the column “General” in Table 2), ringing up a total
of $1008$ templates111111Note that the same template may be rung up multiple
times by different injections, and hence the number of rung-up templates can
be fewer than the number of found injections.. Figure 4 plots the rung-up
templates on the chirp mass $\mathcal{M}_{c}$ \- effective spin
$\chi_{\text{eff}}$ space, with the colors of the markers being the network
SNRs of the associated injections. We can see that the results match our
expectation: Templates with parameters closer to the true parameters of
MS220508a are associated with injections with higher recovered SNRs (i.e.
super-threshold). Those that have parameters much different that the true
parameters of MS220508a are in general associated with injections with lower
SNRs (i.e. sub-threshold).

Figure 4:  Rung-up templates by the simulated lensed injections of MS220508a
in the injection campaign, plotted on the chirp mass $\mathcal{M}_{c}$ \-
effective spin $\chi_{\text{eff}}$ space. The colors of the markers represent
the network SNRs of the associated injections.

###  V.4 Generating the TESLA and TESLA-X reduced template banks

We keep all templates that are rung-up during the injection campaign, and use
them to generate a TESLA targeted template bank. Figure 5 shows the templates
in the full template bank (in grey) and those in the TESLA template bank (in
orange).

Figure 5:  (Color online) The templates in the full template bank (in grey)
and in the TESLA reduced template bank (in orange), plotted in the
$\mathcal{M}_{c}$ \- $\chi_{\text{eff}}$ space.

For TESLA-X, we first use the rung-up templates in the injection campaign to
construct a Gaussian Kernel Density Estimation (KDE) function
$\text{KDE}(\vec{\gamma})$ with the associated injections’ network SNRs as the
weights. $\gamma$ represents the template parameters used to label the
templates. Here, $\vec{\gamma}=\\{\mathcal{M}_{c},\chi_{\text{eff}}\\}$. Then,
we evaluate the probability density at each template point in the full
template bank, then plot the results on a contour map. Figure 6 shows the
contour map for the Gaussian KDE function estimated for MS220508a.

Figure 6:  (Color online) The contour map of the Gaussian Kernel Density
Estimation function obtained for MS220508a in the $\mathcal{M}_{c}$ \-
$\chi_{\text{eff}}$ space. The colors represent the probability density
estimated by the KDE function.

We then take the lowest contour level (For the case of MS220508a, the lowest
contour level has a value of $0.015$) as a boundary, and use all the templates
that fall within this boundary to construct a TESLA-X targeted template bank.
Figure 7 shows the templates in the full template bank (in grey) and in the
TESLA-X reduced template bank (in red) in the $\mathcal{M}_{c}$ \-
$\chi_{\text{eff}}$ space. The yellow curve marks the lowest contour level we
obtained for the estimated Gaussian KDE function we showed in Figure 6. There
are $3928$ templates in the TESLA-X reduced template bank.

Figure 7:  (Color online) The templates in the full template bank (in grey)
and in the TESLA-X reduced template bank (in red), plotted in the
$\mathcal{M}_{c}$ \- $\chi_{\text{eff}}$ space. The TESLA-X template bank is
constructed by keeping templates that fall within the lowest contour level
from the Gaussian KDE function shown in Figure 6, i.e. $0.015$, which is
plotted as a yellow curve in this figure. For easy comparison, we also plot
the templates in the TESLA reduced template bank in the same figure (in blue).

###  V.5 Constructing the targeted population model for the TESLA-X reduced
template bank

Next, we use the estimated KDE function for MS220508a and the TESLA-X reduced
template bank to construct a targeted population model, following the same
steps detailed in IV and Fong (2018). Note that for practical reasons, the
Gaussian KDE function $\text{KDE}(\vec{\gamma})$ is re-calculated with
$\vec{\gamma}$ chosen to be the component masses and spins
$m_{1},m_{2},\chi_{1},\chi_{2}$ instead of $\mathcal{M}_{c}$ and
$\chi_{\text{eff}}$.

###  V.6 Performing a re-filtering to recover the lensed injections with the
TESLA and TESLA-X methods

Finally, we perform two searches, one using the TESLA reduced template bank,
and one using the TESLA-X reduced template bank together with the targeted
population model, over the same stretch of mock data to try to recover the
same set of lensed injections used in the injection campaign to compare the
performance of the TESLA and TESLA-X methods. Table 2 summarizes the findings.
As expected, the TESLA method succeeds in recovering more injections that the
full template bank, with an increase of $2.41\%$. However, we can see that the
proposed TESLA-X method actually out-performs the TESLA method. In particular,
it recovers even more lensed injections that the TESLA method, with an
increase of $4.0\%$.

Injections | General | TESLA | TESLA-X
---|---|---|---
Total | $24272$ | $24272$ | $24272$
Found | $17769$ | $18198$ | $18484$
Found $\%$ change | - | $+2.41\%$ | $+4.0\%$
Table 2: Number of injections found during the search of mock data using the
general template bank, TESLA method and TESLA-X method respectively. Figure 8:
(Color online) (Top panel) The sensitive Volume-Time v.s. FAR for
MS220508a-alike signals using the full tempalte bank (black), TESLA template
bank (red) and TESLA-X template bank (blue) respectively. The shaded band for
each curves represent the corresponding $1$-sigma region. (Bottom panel) The
corresponding percentage changes in sensitive Volume-Time v.s. FAR for the
different banks. We note that both the TESLA and TESLA-X bank improves in
terms of sensitivity towards MS220508a-alike (lensed) signals, but the curve
representing the TESLA-X bank (blue) is above that of the TESLA bank (red),
meaning that the TESLA-X method outperforms the TESLA method. In particular,
notice that the improvement is most significant in the region with FAR $\geq
10^{-8}\text{ Hz}$, which represents the “sub-threshold” region. This further
proves that TESLA-X method is a better method that the traditional TESLA
method.

Furthermore, if we look at the sensitive Volume-Time v.s. FAR for
MS220508a-alike signals using the full tempalte bank, TESLA template bank and
TESLA-X template bank respectively (see Figure 8), we can see that the TESLA-X
method gives a bigger improvement in terms of sensitivity towards
MS220508a-alike (lensed) signals than the traditional TESLA method. In
particular, we note that the improvement is most significant in the region
with FAR $\geq 10^{-8}\text{ Hz}$, which represents the “sub-threshold”
region. This means that TESLA-X outperforms TESLA in finding possible sub-
threshold strongly lensed gravitational waves.

##  VI Concluding Remarks

The detectability of strongly lensed gravitational wave (GW) signals can be
compromised as they may experience demagnification, falling below the
detection threshold. In a previous study, we introduced the TESLA method Li
_et al._ (2023) to extract strongly lensed sub-threshold GW signals by
constructing a target template bank based on the information from detected
events. In this article, we present an enhanced approach called TESLA-X, which
aims to improve the search sensitivity for strongly lensed sub-threshold GW
signals.

The TESLA-X methodology begins with the generation of simulated lensed
injections using posterior samples obtained from parameter estimation. These
injections are then subjected to an injection campaign using GstLAL, with a
threshold set at a FAR of $3.385\times 10^{-7}$ Hz, ensuring statistical
significance. Subsequently, the TESLA-X method utilizes the retrieved
injections to create a targeted population model and a densely distributed
template bank. Finally, the data is reprocessed to identify potential sub-
threshold lensed candidates by employing the new population model and template
bank.

The targeted population model is constructed through several steps. Initially,
a Gaussian kernel density estimation (KDE) function is applied to the source
parameters $\vec{\gamma}$ of the retrieved injections. This KDE function is
then transformed into a probability density function, representing the
parameter space of our potential strong-lensed sub-threshold GW signals Fong
(2018). Additionally, we generate the targeted template bank by defining a
boundary within the retrieved injections. This process involves discarding any
template from the original full template bank, resulting in a densely
distributed template bank that specifically targets a smaller region. Compared
to the TESLA method, this refined template bank ensures the recovery of any
possible lensed counterparts with less than a $3\%$ SNR loss. It is important
to note that the FARs and likelihood values ($\mathcal{L}$) do not provide
information about the likelihood of being lensed counterparts to the target
event. Instead, they serve as metrics to differentiate between noise triggers
and astrophysical signals.

In this study, we conducted a simulation campaign to evaluate and compare the
search sensitivity of three different methods: the full template bank search,
the TESLA method, and the TESLA-X method. Our analysis utilized a collection
of mock data, where Gaussian noise was introduced along with a significant GW
event labeled as MS220508a. Our findings demonstrate that the TESLA-X method
outperforms the other approaches by successfully recovering a greater number
of lensed injections specifically targeting MS220508a. Furthermore, we
observed a maximum of $\sim 20\%$ increase in the search sensitivity through
an examination of the Volume-Time versus FAR plot. An increase in volume-time
means that we are able to see signals coming from further sources, and
equivalently we are able to see weaker signals according to equation 3. These
results substantiate the superior performance of the TESLA-X method in
enhancing the search capabilities for gravitational lensing phenomena. TESLA-X
still carries the same problem as TESLA in terms of high computational
cost121212Running TESLA/TESLA-X for a single super-threshold event can take
$\mathcal{O}(\mathrm{weeks})$. for running the injection campaign for each
target super-threshold gravitational-wave event. As future work, we will
explore ways to improving the computational efficiency of TESLA-X. We will
also look into possible parameters other than chirp mass and effective spin to
represent the population that we are interested in.

###### Acknowledgements.

The authors acknowledge the generous support from the National Science
Foundation in the United States. AKYL would like to gratefully acknowledge the
sup- port from the National Science Foundation through the Grants NSF
PHY-1912594 and NSF PHY-2207758. JCLC acknowledges support from the Villum
Investigator program supported by the VILLUM Foundation (grant no. VIL37766)
and the DNRF Chair program (grant no. DNRF162) by the Danish National Research
Foundation. HF gratefully acknowledges the support of the Canadian Institute
for Theoretical Astrophysics (CITA) National Fellowship and the Japan Society
for the Promotion of Science (JSPS) Postdoctoral Fellowships for Research in
Japan. JME is supported by the European Union’s Horizon 2020 research and
innovation program under the Marie Sklodowska-Curie grant agreement No. 847523
INTERACTIONS, and by VILLUM FONDEN (grant no. 53101 and 37766). The authors
are also grateful for computational resources provided by the LIGO Laboratory
and supported by National Science Foundation Grants PHY-0757058 and
PHY-0823459. This research has made use of data, software and/or web tools
obtained from the Gravitational Wave Open Science Center (https://www.gw-
openscience.org) Vallisneri  _et al._ (2015), a service of LIGO Laboratory,
the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO was
constructed by the California Institute of Technology and Massachusetts
Institute of Technology with funding from the National Science Foundation and
operates under cooperative agreement PHY-0757058. HF is supported by a CITA
National Fellowship. Virgo is funded by the French Centre National de
Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica
Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and
Hungarian institutes. This paper carries LIGO Document Number LIGO-P1900059.

## References

  * Ohanian (1974) H. Ohanian, Int. J. Theor. Phys. 9, 425 (1974).
  * Thorne (1982) K. Thorne, in  _Les Houches Summer School on Gravitational Radiation_ (1982) pp. 1–57.
  * Deguchi and Watson (1986) S. Deguchi and W. D. Watson, Phys. Rev. D 34, 1708 (1986).
  * Wang  _et al._ (1996) Y. Wang, A. Stebbins, and E. L. Turner, Phys. Rev. Lett. 77, 2875 (1996), arXiv:astro-ph/9605140 .
  * Nakamura (1998) T. T. Nakamura, Phys. Rev. Lett. 80, 1138 (1998).
  * Takahashi and Nakamura (2003) R. Takahashi and T. Nakamura, Astrophys. J. 595, 1039 (2003), arXiv:astro-ph/0305055 .
  * Dai and Venumadhav (2017) L. Dai and T. Venumadhav, (2017), arXiv:1702.04724 [gr-qc] .
  * Ezquiaga  _et al._ (2021) J. M. Ezquiaga, D. E. Holz, W. Hu, M. Lagos, and R. M. Wald, Phys. Rev. D 103, 6 (2021), arXiv:2008.12814 [gr-qc] .
  * Ng  _et al._ (2018) K. K. Ng, K. W. Wong, T. Broadhurst, and T. G. Li, Phys. Rev. D 97, 023012 (2018), arXiv:1703.06319 [astro-ph.CO] .
  * Li  _et al._ (2018) S.-S. Li, S. Mao, Y. Zhao, and Y. Lu, Mon. Not. Roy. Astron. Soc. 476, 2220 (2018), arXiv:1802.05089 [astro-ph.CO] .
  * Oguri (2018) M. Oguri, Mon. Not. Roy. Astron. Soc. 480, 3842 (2018), arXiv:1807.02584 [astro-ph.CO] .
  * Smith  _et al._ (2018) G. P. Smith, M. Jauzac, J. Veitch, W. M. Farr, R. Massey, and J. Richard, Mon. Not. Roy. Astron. Soc. 475, 3823 (2018), arXiv:1707.03412 [astro-ph.HE] .
  * Smith  _et al._ (2017) G. Smith _et al._ , IAU Symp. 338, 98 (2017), arXiv:1803.07851 [astro-ph.CO] .
  * Smith  _et al._ (2019) G. P. Smith, A. Robertson, M. Bianconi, and M. Jauzac (LSST), (2019), arXiv:1902.05140 [astro-ph.HE] .
  * Robertson  _et al._ (2020) A. Robertson, G. P. Smith, R. Massey, V. Eke, M. Jauzac, M. Bianconi, and D. Ryczanowski,  (2020), 10.1093/mnras/staa1429, arXiv:2002.01479 [astro-ph.CO] .
  * Ryczanowski  _et al._ (2020) D. Ryczanowski, G. P. Smith, M. Bianconi, R. Massey, A. Robertson, and M. Jauzac, Mon. Not. Roy. Astron. Soc. 495, 1666 (2020), arXiv:2005.02296 [astro-ph.GA] .
  * Ezquiaga  _et al._ (2023) J. M. Ezquiaga, W. Hu, and R. K. L. Lo, (2023), arXiv:2308.06616 [astro-ph.CO] .
  * Cao  _et al._ (2014) Z. Cao, L.-F. Li, and Y. Wang, Phys. Rev. D 90, 062003 (2014).
  * Lai  _et al._ (2018) K.-H. Lai, O. A. Hannuksela, A. Herrera-Martín, J. M. Diego, T. Broadhurst, and T. G. Li, Phys. Rev. D 98, 083005 (2018), arXiv:1801.07840 [gr-qc] .
  * Christian  _et al._ (2018) P. Christian, S. Vitale, and A. Loeb, Phys. Rev. D 98, 103022 (2018), arXiv:1802.02586 [astro-ph.HE] .
  * Dai  _et al._ (2018) L. Dai, S.-S. Li, B. Zackay, S. Mao, and Y. Lu, Phys. Rev. D 98, 104029 (2018), arXiv:1810.00003 [gr-qc] .
  * Diego  _et al._ (2019) J. Diego, O. Hannuksela, P. Kelly, T. Broadhurst, K. Kim, T. Li, G. Smoot, and G. Pagano, Astron. Astrophys. 627, A130 (2019), arXiv:1903.04513 [astro-ph.CO] .
  * Çalışkan  _et al._ (2023a) M. Çalışkan, L. Ji, R. Cotesta, E. Berti, M. Kamionkowski, and S. Marsat, Phys. Rev. D 107, 043029 (2023a), arXiv:2206.02803 [astro-ph.CO] .
  * Cheung  _et al._ (2021) M. H. Y. Cheung, J. Gais, O. A. Hannuksela, and T. G. F. Li, Mon. Not. Roy. Astron. Soc. 503, 3326 (2021), arXiv:2012.07800 [astro-ph.HE] .
  * Mishra  _et al._ (2021) A. Mishra, A. K. Meena, A. More, S. Bose, and J. S. Bagla, arXiv e-prints , arXiv:2102.03946 (2021), arXiv:2102.03946 [astro-ph.CO] .
  * Meena  _et al._ (2022) A. K. Meena, A. Mishra, A. More, S. Bose, and J. S. Bagla, Mon. Not. Roy. Astron. Soc. 517, 872 (2022), arXiv:2205.05409 [astro-ph.GA] .
  * Cremonese  _et al._ (2021) P. Cremonese, J. M. Ezquiaga, and V. Salzano, Phys. Rev. D 104, 023503 (2021), arXiv:2104.07055 [astro-ph.CO] .
  * Çalışkan  _et al._ (2023b) M. Çalışkan, N. Anil Kumar, L. Ji, J. M. Ezquiaga, R. Cotesta, E. Berti, and M. Kamionkowski, (2023b), arXiv:2307.06990 [astro-ph.CO] .
  * Ezquiaga and Zumalacárregui (2020) J. M. Ezquiaga and M. Zumalacárregui, Phys. Rev. D 102, 124048 (2020), arXiv:2009.12187 [gr-qc] .
  * Baker and Trodden (2017) T. Baker and M. Trodden, Phys. Rev. D 95, 063512 (2017), arXiv:1612.02004 [astro-ph.CO] .
  * Collett and Bacon (2017) T. E. Collett and D. Bacon, Phys. Rev. Lett. 118, 091101 (2017), arXiv:1602.05882 [astro-ph.HE] .
  * Goyal  _et al._ (2023) S. Goyal, A. Vijaykumar, J. M. Ezquiaga, and M. Zumalacarregui, “Probing lens-induced gravitational-wave birefringence as a test of general relativity,” (2023), arXiv:2301.04826 [gr-qc] .
  * Hannuksela  _et al._ (2020) O. A. Hannuksela, T. E. Collett, M. Çalışkan, and T. G. F. Li, Mon. Not. Roy. Astron. Soc. 498, 3395 (2020), arXiv:2004.13811 [astro-ph.HE] .
  * Yu  _et al._ (2020) H. Yu, P. Zhang, and F.-Y. Wang, Mon. Not. Roy. Astron. Soc. 497, 204 (2020), arXiv:2007.00828 [astro-ph.CO] .
  * Oguri and Takahashi (2020) M. Oguri and R. Takahashi, Astrophys. J. 901, 58 (2020), arXiv:2007.01936 [astro-ph.CO] .
  * Liu  _et al._ (2023) A. Liu, I. C. F. Wong, S. H. W. Leong, A. More, O. A. Hannuksela, and T. G. F. Li, (2023), arXiv:2302.09870 [gr-qc] .
  * Sereno  _et al._ (2011) M. Sereno, P. Jetzer, A. Sesana, and M. Volonteri, Mon. Not. Roy. Astron. Soc. 415, 2773 (2011), arXiv:1104.1977 [astro-ph.CO] .
  * Liao  _et al._ (2017) K. Liao, X.-L. Fan, X.-H. Ding, M. Biesiada, and Z.-H. Zhu, Nature Commun. 8, 1148 (2017), [Erratum: Nature Commun. 8, 2136 (2017)], arXiv:1703.04151 [astro-ph.CO] .
  * Cao  _et al._ (2019) S. Cao, J. Qi, Z. Cao, M. Biesiada, J. Li, Y. Pan, and Z.-H. Zhu, Sci. Rep. 9, 11608 (2019), arXiv:1910.10365 [astro-ph.CO] .
  * Xu  _et al._ (2021) F. Xu, J. M. Ezquiaga, and D. E. Holz, (2021), arXiv:2105.14390 [astro-ph.CO] .
  * Hannuksela  _et al._ (2019) O. Hannuksela, K. Haris, K. Ng, S. Kumar, A. Mehta, D. Keitel, T. Li, and P. Ajith, Astrophys. J. Lett. 874, L2 (2019), arXiv:1901.02674 [gr-qc] .
  * Abbott  _et al._ (2021a) R. Abbott _et al._ (LIGO Scientific, Virgo), (2021a), arXiv:2105.06384 [gr-qc] .
  * Abbott  _et al._ (2023) R. Abbott _et al._ (LIGO Scientific, VIRGO, KAGRA), (2023), arXiv:2304.08393 [gr-qc] .
  * Li  _et al._ (2023) A. K. Y. Li, R. K. L. Lo, S. Sachdev, J. C. L. Chan, E. T. Lin, T. G. F. Li, and A. J. Weinstein (LIGO Scientific, Virgo), Phys. Rev. D 107, 123014 (2023), arXiv:1904.06020 [gr-qc] .
  * McIsaac  _et al._ (2020) C. McIsaac, D. Keitel, T. Collett, I. Harry, S. Mozzon, O. Edy, and D. Bacon, Phys. Rev. D 102, 084031 (2020), arXiv:1912.05389 [gr-qc] .
  * Dai  _et al._ (2020) L. Dai, B. Zackay, T. Venumadhav, J. Roulet, and M. Zaldarriaga, (2020), arXiv:2007.12709 [astro-ph.HE] .
  * Liu  _et al._ (2021) X. Liu, I. M. Hernandez, and J. Creighton, Astrophys. J. 908, 97 (2021), arXiv:2009.06539 [astro-ph.HE] .
  * Wierda  _et al._ (2021) A. R. A. C. Wierda, E. Wempe, O. A. Hannuksela, L. V. E. Koopmans, and C. Van Den Broeck, (2021), arXiv:2106.06303 [astro-ph.HE] .
  * Abbott  _et al._ (2018) B. P. Abbott _et al._ (KAGRA, LIGO Scientific, Virgo, VIRGO), Living Rev. Rel. 21, 3 (2018), arXiv:1304.0670 [gr-qc] .
  * Messick  _et al._ (2017) C. Messick, K. Blackburn, P. Brady, P. Brockill, K. Cannon, R. Cariou, S. Caudill, S. J. Chamberlin, J. D. E. Creighton, R. Everett, C. Hanna, D. Keppel, R. N. Lang, T. G. F. Li, D. Meacher, A. Nielsen, C. Pankow, S. Privitera, H. Qi, S. Sachdev, L. Sadeghian, L. Singer, E. G. Thomas, L. Wade, M. Wade, A. Weinstein, and K. Wiesner, Phys. Rev. D 95, 042001 (2017), arXiv:1604.04324 [astro-ph.IM] .
  * Sachdev  _et al._ (2019) S. Sachdev _et al._ , (2019), arXiv:1901.08580 [gr-qc] .
  * Hanna  _et al._ (2020) C. Hanna _et al._ , Phys. Rev. D 101, 022003 (2020), arXiv:1901.02227 [gr-qc] .
  * Allen  _et al._ (2012) B. Allen, W. G. Anderson, P. R. Brady, D. A. Brown, and J. D. E. Creighton, Phys. Rev. D 85, 122006 (2012), arXiv:gr-qc/0509116 .
  * Cannon  _et al._ (2021) K. Cannon, S. Caudill, C. Chan, B. Cousins, J. D. E. Creighton, B. Ewing, H. Fong, P. Godwin, C. Hanna, S. Hooper, R. Huxford, R. Magee, D. Meacher, C. Messick, S. Morisaki, D. Mukherjee, H. Ohta, A. Pace, S. Privitera, I. de Ruiter, S. Sachdev, L. Singer, D. Singh, R. Tapia, L. Tsukada, D. Tsuna, T. Tsutsui, K. Ueno, A. Viets, L. Wade, and M. Wade, SoftwareX 14, 100680 (2021), arXiv:2010.05082 [astro-ph.IM] .
  * Godwin  _et al._ (2020) P. Godwin _et al._ , (2020), arXiv:2010.15282 [gr-qc] .
  * Chan  _et al._ (2020) C. Chan _et al._ , (2020), arXiv:2009.03025 [astro-ph.IM] .
  * Mukherjee  _et al._ (2021) D. Mukherjee, S. Caudill, R. Magee, C. Messick, S. Privitera, S. Sachdev, K. Blackburn, P. Brady, P. Brockill, K. Cannon, S. J. Chamberlin, D. Chatterjee, J. D. E. Creighton, H. Fong, P. Godwin, C. Hanna, S. Kapadia, R. N. Lang, T. G. F. Li, R. K. L. Lo, D. Meacher, A. Pace, L. Sadeghian, L. Tsukada, L. Wade, M. Wade, A. Weinstein, and L. Xiao, Phys. Rev. D 103, 084047 (2021), arXiv:1812.05121 [astro-ph.IM] .
  * Messick  _et al._ (2020) C. Messick, S. Sachdev, K. Cannon, S. Caudill, C. Chan, J. D. E. Creighton, R. Everett, B. Ewing, H. Fong, P. Godwin, C. Hanna, R. Huxford, S. Kapadia, A. K. Y. Li, R. K. L. Lo, R. Magee, D. Meacher, S. R. Mohite, D. Mukherjee, A. Nishizawa, H. Ohta, A. Pace, A. Reza, M. Shikauchi, L. Singer, D. Singh, J. Rana SK, L. Tsukada, D. Tsuna, T. Tsutsui, K. Ueno, and A. Zimmerman, arXiv e-prints , arXiv:2011.02457 (2020), arXiv:2011.02457 [astro-ph.IM] .
  * Veitch  _et al._ (2015) J. Veitch _et al._ , Phys. Rev. D 91, 042003 (2015), arXiv:1409.7215 [gr-qc] .
  * Ashton  _et al._ (2019) G. Ashton _et al._ , Astrophys. J. Suppl. Ser. 241, 27 (2019), arXiv:1811.02042 [astro-ph.IM] .
  * Romero-Shaw  _et al._ (2020) I. M. Romero-Shaw _et al._ , Mon. Not. Roy. Astron. Soc. 499, 3295 (2020), arXiv:2006.00714 [astro-ph.IM] .
  * Ashton and Talbot (2021) G. Ashton and C. Talbot, Mon. Not. Roy. Astron. Soc. 507, 2037 (2021), arXiv:2106.08730 [gr-qc] .
  * Fong (2018) H. K. Y. Fong,  _From Simulations to Signals: Analyzing Gravitational Waves from Compact Binary Coalescences_ , Ph.D. thesis, Department of Physics, University of Toronto (2018).
  * Cannon  _et al._ (2015) K. Cannon, C. Hanna, and J. Peoples, arXiv e-prints , arXiv:1504.04632 (2015), arXiv:1504.04632 [astro-ph.IM] .
  * Pratten  _et al._ (2021) G. Pratten _et al._ , Phys. Rev. D 103, 104056 (2021), arXiv:2004.06503 [gr-qc] .
  * Abbott  _et al._ (2021b) R. Abbott _et al._ (LIGO Scientific, VIRGO, KAGRA), (2021b), arXiv:2111.03606 [gr-qc] .
  * Vallisneri  _et al._ (2015) M. Vallisneri, J. Kanner, R. Williams, A. Weinstein, and B. Stephens, in  _Journal of Physics Conference Series_, Journal of Physics Conference Series, Vol. 610 (2015) p. 012021, arXiv:1410.4839 [gr-qc] .