# Addressing the challenges of detecting time-overlapping compact binary
coalescences

Philip Relton<EMAIL_ADDRESS>Gravity Exploration Institute, Cardiff
University, Cardiff CF24 3AA, United Kingdom  Andrea Virtuoso  Università di
Trieste, Dipartimento di Fisica, I-34127 Trieste, Italy; INFN Sezione di
Trieste, I-34127 Trieste, Italy  Sophie Bini  Università di Trento,
Dipartimento di Fisica, I-38123 Povo, Trento, Italy; INFN, TIFPA, I-38123
Povo, Trento, Italy  Vivien Raymond  Gravity Exploration Institute, Cardiff
University, Cardiff CF24 3AA, United Kingdom  Ian Harry  University of
Portsmouth, Portsmouth, PO1 3FX, United Kingdom  Marco Drago  Università di
Roma La Sapienza, I-00185 Roma, Italy; INFN, Sezione di Roma, I-00185 Roma,
Italy  Claudia Lazzaro  10 Università di Padova, Dipartimento di Fisica e
Astronomia, I-35131 Padova, Italy; INFN, Sezione di Padova, I-35131 Padova,
Italy  Andrea Miani  Università di Trento, Dipartimento di Fisica, I-38123
Povo, Trento, Italy; INFN, TIFPA, I-38123 Povo, Trento, Italy  Shubhanshu
Tiwari  Physik-Institut, University of Zurich, Winterthurerstrasse 190, 8057
Zurich, Switzerland

###### Abstract

Standard detection and analysis techniques for transient gravitational waves
make the assumption that detector data contains, at most, one signal at any
time. As detectors improve in sensitivity, this assumption will no longer be
valid. In this paper we examine how current search techniques for transient
gravitational waves will behave under the presence of more than one signal. We
perform searches on simulated data sets containing time-overlapping compact
binary coalescences. This includes a modelled, matched filter search (PyCBC),
and an unmodelled coherent search, coherent WaveBurst (cWB). Both of these
searches are used by the LIGO-Virgo-KAGRA collaboration [1]. We find that both
searches are capable of identifying both signals correctly when the signals
are dissimilar in merger time, $|\Delta t_{c}|\geq$1\text{\,}\mathrm{s}$$,
with PyCBC missing only $1\%$ of the overlapping binary black hole mergers it
was provided. Both pipelines can find signal pairings within the region
$|\Delta t_{c}|<$1\text{\,}\mathrm{s}$$. However, clustering routines in the
pipelines will cause only one of the two signals to be recovered, as such the
efficiency is reduced. Within this region, we find that cWB can identify both
signals. We also find that matched filter searches can be modified to provide
estimates of the correct parameters for each signal.

gravitational waves, signal detection, overlapping signals

##  I Introduction

In 2015 the Advanced Laser Interferometer Gravitational-Wave Observatory
(aLIGO) interferometers observed the merger of two black holes [2]. Since
then, there have been over 90 transient gravitational waves observed [1, 3].
The rate of these mergers is given by the rate of gravitational wave from
astrophysical sources and the observing range of the detector. As the current
network of detectors are improved [4, 5, 6, 7, 8], and future detectors such
as The Einstein Telescope (ET) [9] and Cosmic Explorer [10] begin
observations, this observing range will significantly increase. This will lead
to multiple transient signals being present in the detector concurrently, most
likely before the end of the decade [11], with most transient signals
overlapping another in the next generation of observatories.

Previous investigations into the case of time-overlapping compact binary
coalescences (CBCs), have focused on how parameter estimation techniques will
perform. Their findings indicate that the parameters of modelled signals
should be recoverable, unless the two signals overlap strongly with a small
relative merger time [11, 12, 13, 14, 15]. This is stronger if the signals
have similar signal-to-noise ratios (SNRs) [11], or chirp masses, mostly due
to similar frequencies in the signals [13]. In the case where overlapping
astrophysical signals are misidentified as a single signal, the data will be
passed on to these parameter estimation methods. As these methods aim to
minimise their residuals, it is possible that the overlap would not be
noticed. Any resulting oddities would likely be attributed to other causes
such as glitches, precession or eccentricity [16, 17, 18].

Current methods of gravitational wave detection rely upon the assumption that
only a single signal is present in the detector and, in most algorithms, that
this signal can be matched to modelled signal waveforms. As this assumption
will break for time-overlapping transients, we have investigated how current
search pipelines will behave in their presence.

Our investigation was performed by injecting pairs of overlapping CBCs into
the coloured-Gaussian noise of a three detector, LIGO-Virgo, network [19, 6].
We then performed an offline search of these data sets using a modelled
search, PyCBC) [20, 21, 22, 23, 24, 25], and an unmodelled search, cWB [26,
27]. We investigated how the parameters of the signal pairings affected
whether the signals were detected and how accurate the recovered parameters of
those signals were.

Section II outlines how the data were generated, and how each search works.
Section III covers the findings of how signal pairings were detected and the
effect of the overlap on recovered parameters. Section IV shows how these
search methods could lead to successful overlap identification and separation.
Section V contains a summary of our findings, followed by an appendix.

##  II Method

###  II.1 Overview

For this study we consider two types of CBCs, binary black hole mergers (BBH)
and binary neutron star mergers (BNS). Our main interest in including these
two forms of signal was to see how current search methods behave under the
presence of overlapping CBCs of different visible durations and frequency
profiles. We did not include neutron star-black hole signals, which should
have similar observable durations [28] to low mass BBH signals, as these
events are rare and less relevant to this study. Their presence in overlaps
should be considered in future studies. We also did not include tidal
deformability in the BNS signals, as it should have little effect on the
length and frequency of the signal [29]. Three separate investigations were
performed for this study, each examining one of three combinations of CBC
overlap: BBH+BBH, BNS+BNS and BNS+BBH.

###  II.2 Data generation

Each investigation in this study included generating ten days of coloured-
Gaussian noise for a three detector network. This length of time was chosen to
give a reasonable estimate of false alarm rate of significant signals and to
allow a reasonable number of events to be studied. The detectors were added as
two advanced LIGO detectors at design specification in the locations of LIGO-
Hanford and LIGO-Livingston [19, 30] and a third detector with an Advanced
Virgo sensitivity in the position of Virgo [6].

For each of the three studies mentioned in Section II.1 we made three
injection sets. The first two only included individual, non-overlapping
signals from the pairings, labelled as $\mathrm{SINGLES}_{\mathrm{A}}$ and
$\mathrm{SINGLES}_{\mathrm{B}}$. This provides comparison data to see whether
each signal is detectable alone. The final injection set, labelled as PAIRS,
contained both Signal A and Signal B as time-overlapping pairs.

Signals were injected into the data internally in each search pipeline. BBH
signals were generated using the SEOBNRv4PHM approximant [31] to allow for the
inclusion of spin precession and higher modes. BNS signals were generated with
the SEOBNRv4 [32] approximant; this only allows for aligned spin and, does not
include neutron star tidal deformability.

With the exception of merger time, the extrinsic parameters of both Signal A
and B were drawn from uniform distributions in sky locations, phase,
polarisation and binary inclination. Each signal was generated from
$9\text{\,}\mathrm{H}\mathrm{z}$, just below the noise cutoff of
$10\text{\,}\mathrm{H}\mathrm{z}$, to avoid discontinuities in the data.
Further details of intrinsic parameters, merger times and luminosity distances
are given in the subsections below.

This distribution is designed to be as close to a true astrophysical
distribution, but to also allow reasonable study of possible overlapping CBC
cases. As such our study is aimed at the methodology of the pipelines and not
at providing an astrophysical study of such events.

####  II.2.1 Masses and spins

The masses of the two objects in a binary were drawn in pairs. BBH mass
pairing were drawn from the most recent binary mass distribution estimation by
the LIGO-Virgo collaboration. The chosen mass distribution was the
PowerLaw+Peak distribution covering primary masses in the range
$[5,100]\,M_{\odot}$ and mass ratios in the range $[0.1,1]$ [33]. As the
detectors have some bias as to the events they will detect, we included a
biasing factor on the mass distribution. Using the power spectral density
(PSD) of the advanced LIGO detectors [30], and the inspiral-range PYTHON
package [34, 35], we estimated viewing ranges for each binary pairing in the
mass distribution grid. These distance estimates were used to generate volume
estimates and were then multiplied by the per volume merger rate to estimate a
true merger rate of each signal. This was then normalised to produce a
weighted distribution to bias the astrophysical merger rates, by mass, to the
observable merger rate. This produces binaries with more even mass ratios and
higher masses, as is expected in these detectors.

The neutron star (NS) binary mass range is still an area of active research
[33, 36], with observed and predicted maximum NS masses differing [37, 38]. As
such, for BNS mergers we defined a mass range of $[1.14,3]\,M_{\odot}$ and
drew uniformly across this. This is broadly consistent with both gravitational
wave and electromagnetic observations. No mass distribution biasing was
performed due to the smaller luminosity distances drawn for these binaries.

BBH spins were drawn from the spin distributions estimated by the
PowerLaw+Peak mass model from the most recent estimates by the LIGO-Virgo
collaboration [33]. This includes both aligned and in-plane spins with
azimuthal orientations between $[\frac{-\pi}{2},\frac{\pi}{2}]$, polar angles
between $[0,2\pi]$ and magnitudes ranging between $[0,0.99]$ in dimensionless
magnitude. Parameter estimation on time-overlapping signals has shown that
highly overlapping pairs can mimic precession effects in the waveform [11], so
it is interesting to consider how well searches will observe such events.
Spins were drawn independently of binary masses. BNS signals were given
aligned only spins with magnitude range $[-0.05,0.05]$, based on observations
of BNS systems [39, 16, 40].

####  II.2.2 Distances and times

The distance of events was drawn from a second order power law between
$[200,1300]\,$\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$ for BBH systems and
$[5,200]\,$\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$ for BNS systems. These
values were set to ensure a reasonable number of visible events in the
injection sets, without too many events being too quiet to observe,
$\mathrm{SNR}<5$, or overwhelmingly loud, $\mathrm{SNR}>50$. These ranges are
broadly consistent with observations in current detectors [1]. While redshift
does have an effect at these distances it will not change any conclusions
drawn from this study, and therefore was not included in these injections. For
consistency, all detector frame masses are identical to the source frame
masses.

The merger times of Signal A events were drawn uniformly across the ten days
of simulated data. To avoid overlaps of $N_{signals}>2$, these times were
spaced out within this data set such that no mergers were within $60$ seconds
of the start of when the next pairing reached
$10\text{\,}\mathrm{H}\mathrm{z}$. The merger times of Signal B events were
then selected by generating a waveform for each Signal A, calculating the time
of first visibility at $10\text{\,}\mathrm{H}\mathrm{z}$ and that of the
merger, and drawing a time uniformly between these values. To ensure
observations of close relative merger times, some pairings in the BNS+BNS and
BNS+BBH runs the times were drawn within two seconds of the merger. The BNS
studies with narrower merger time separations were performed as a separate ten
day segment, which does not affect the estimated false alarm rates in those
runs.

As lower mass signals, such as BNS inspirals, are in-band in the detector for
much longer than lower mass, BBH signals, this leads to many fewer signals in
the BNS study. However, we did not increase the length of the BNS studies
beyond ten days as this would have drastically increased the computational
expense.

###  II.3 Searches

####  II.3.1 PyCBC) search

PyCBC) is a matched-filter search pipeline for the detection of CBC signals in
a wide parameter space [20, 21, 22, 23, 24, 25, 3]. The triggers are generated
in coincidence with the network of detectors by correlating the data with
templates. The bank of templates used in this study is exactly the same as the
one used for PyCBC-broad in GWTC-3 [1]. In this work we have employed a
slightly modified version of PyCBC-broad which was used for the GWTC-3 catalog
[1]. These modifications were made in order to accommodate the complexity
induced in the signal space due to time overlap. In particular, for this work
the “clustering window” of the PyCBC) search is modified. This is discussed in
more detail in Section II.4.

Background estimates are generated by time shifting the data of the three
detectors by more than the time of flight between the detectors and a
background trigger list is obtained. This process is repeated until enough
events are acquired to measure a false alarm rate (FAR) of 1 per year
($1\text{\,}\mathrm{y}\mathrm{r}^{-1}$). The injections are then processed and
ranked according to the background. This process is same for both the
pipelines used in this study, however the detection statistics between the two
pipelines are very different.

PyCBC) uses detection statistics based on the matched-filter SNR, with
information about the event rate and the background rate incorporated as
described in [3, 41].

####  II.3.2 cWB search

cWB is an unmodelled analysis pipeline which detects and reconstructs
gravitational wave (GW) signals without assuming any waveform model [26, 27,
42]. cWB decomposes each interferometer data into a time-frequency (TF)
representation using Wilson-Daubechies-Meyer wavelets [43]. Each wavelet
amplitude is normalised by the corresponding detector amplitude spectral
density, cWB then selects those wavelets having energy above a fixed
threshold. Finally, clusters from different detectors are combined coherently
into a likelihood function, which is maximised with respect to the sky
location.

From the likelihood we define the statistical quantities to distinguish
between a possible GW signal and glitches coming from the noise. The first
being the coherent energy $E_{c}$, which represents the coherent contribution
of the likelihood by cross-correlating data from different detectors. Another
key statistic is the null, or residual, noise energy $E_{n}$, which is given
by subtracting the likelihood from the whitened data energy.

From coherent and null energy we can define two further quantities. The first
is the penalty, $\chi^{2}$, defined as the null energy divided by the number
of independent wavelet amplitudes used for describing the detected event. The
second is the correlation coefficient $c_{c}$:

$c_{c}\,=\,\frac{\left|E_{c}\right|}{\left|E_{c}\right|+E_{n}}$ (1)

This estimates the coherence of the data among different detectors: when
$c_{c}\simeq 1$ ($\left|E_{c}\right|\gg E_{n}$) there is a high coherence and
it is likely that there is an astrophysical signal, while when $c_{c}\simeq 0$
($E_{n}\gg\left|E_{c}\right|$) the data is incoherent and so is less likely to
contain an astrophysical signal.

The correlation coefficient and the penalty are used in post-production
analysis for recognising non-Gaussian noise transients, commonly referred to
as “glitches”, which could trigger the pipeline, even if they are not GW
events. In this work we applied the same cuts used for O3 analysis, these are
$c_{c}>0.7$ and $\log_{10}(\chi^{2})<0.2$ [1].

###  II.4 Finding signals

Once the searches were performed, trigger sets were recovered for each of the
three injection sets in each run. Injections were marked as found if there was
a trigger found in the trigger sets within a defined time separation at
merger. These separations differed between the two pipelines.

As a matched filter search, PyCBC) returns a time for each trigger that
corresponds to the visible end of the signal in the data. This was directly
matched to the injected signals merger time. If the injected signal had a
PyCBC) trigger with an end time in the range $\pm 0.1$ seconds then the
injection was counted as found by the pipeline.

A slight problem arises for signal pairings in which the mergers are within
$\pm 0.1$ seconds. PyCBC) often finds that several templates match the same
signal, particularly for significant triggers. To avoid this a clustering
window is specified for the run. This window, set to $\pm 1$ second for our
study, will reject all but the most significant trigger within the window. If
signals overlap in this region, then only one trigger will be returned for the
pairing. This situation is covered in more detail in Section III.2.

cWB’s approach of searching for regions of excess power requires a different
approach. The standard returned time is the mean time, weighted with energy,
but for this study it is more suitable to use the end time of the
reconstructed waveform as estimated by cWB. This allows for the reconstructed
waveforms to be counted correctly. Initial applications of the PyCBC) time
constraints led to a large number of triggered injections being rejected. A
window for signal finding of injected/reconstructed signal end time was set to
$\pm 2.5$ seconds. This time allows for the best recovery rate of cWB
triggers, without missing separate signal pairings.

##  III Results

###  III.1 Bias regions for overlapping signals

Previous studies of time-overlapping CBCs have focused on their effect on the
parameter estimation of the underlying signals [11, 12, 13, 14, 15]. By
considering the findings of these studies, we define three regions in which
the presence of a second signal affects the recovered parameters of the
primary signal:

Strong bias: This is the region in which the two signals most strongly affect
each other, recovered parameters will be significantly biased away from their
true values. Largely this is bound by the separation of merger times between
the signals. This boundary is not consistent between all studies, to be
conservative we define this as a merger time separation of
$|\Delta{t_{c}}|\leq$0.5\text{\,}\mathrm{s}$$ for BBH+BBH overlaps. However,
some studies [13, 15], indicate that the similar region for BNS+BNS overlaps
is closer at $|\Delta{t_{c}}|\leq$0.01\text{\,}\mathrm{s}$$. The literature
has not defined any such value for BNS+BBH overlaps, for this study we have
applied boundaries to be the same as BNS+BNS overlaps. This region is smaller,
likely due to the number of clean cycles in the later merging BNS, due to
their high merger frequency, unlike BBH mergers which merge at lower
frequency.

Weak bias: In this region signals are often recovered with slightly biased,
but broadly correct, parameters. Considering the literature we define this
region as $0.5<|\Delta{t_{c}}|\leq$2\text{\,}\mathrm{s}$$, with the lower
limit varying for overlaps containing a BNS merger.

Negligible bias: In this region the signals are dissimilar enough to not cause
any noticeable bias in the recovered parameters. Pairings in this region
should both be recovered. The bias in one signal, caused by the presence of
the other, should be small enough to not negatively impact the results. We
define this region to be for merger time separations of
$|\Delta{t_{c}}|>$2\text{\,}\mathrm{s}$$.

Parameter estimation studies also indicate that the bias will be negligible if
the ratio of the SNRs of the signals is particularly uneven, one signal being
at least greater than three times louder than the other [11]. Pairings in this
category are likely to fall into the negligible bias region, regardless of
merger time proximity. Some indication of relative chirp mass, and therefore
waveform frequency range, is likely to also have an effect [13]. If signals
differ significantly in chirp mass, $\mathcal{M}$, then the bias may be
smaller as the signals are dissimilar in frequency at merger. While we include
signal overlaps of differing chirp mass in the study, we do not examine how
this effect is present in trigger selection.

|  $N_{Overlaps}$ by region
---|---
Overlap configuration | Strong | Weak | Negligible
BBH+BBH | $5.6^{+9.1}_{-3.3}$ | $17.0^{+27.0}_{-9.8}$ | $13000.0^{+8300.0}_{-4900.0}$
BNS+BNS | $0.13^{+0.71}_{-0.12}$ | $26.0^{+140.0}_{-25.0}$ | $14000.0^{+22000.0}_{-11000.0}$
BNS+BBH | $0.11^{+0.18}_{-0.07}$ | $22.0^{+36.0}_{-13.0}$ | $13000.0^{+8300.0}_{-4800.0}$
Table 1: Estimated number of signals occurring for different CBC overlap
configurations in a year’s observations of the Einstein Telescope. The first
column, $N_{events}$ is an estimate of the total number of single signal
events, of that kind, that ET would expect to see in the year’s observation.
For the BNS+BBH row this is the expected number of BBHs, rather than the
number of BNSs. The other columns represent the predicted number of overlaps
of this kind in each bias region.

Table 1 shows estimated numbers of events falling within each of the bias
regions, defined above, over a year’s observation of ET [9]. These numbers
were estimated by calculating the visible volume for a variety of different
mass CBCs as described in [11], using the inspiral-range PYTHON package [34,
35], and a PSD for ET [44]. The viewing time for each signal, in the
negligible bias region, was estimated from $1\text{\,}\mathrm{H}\mathrm{z}$ to
the time of merger. Signals in the strong and weak bias regions were fixed to
the durations described above, merger time separations of
$0.1\text{\,}\mathrm{s}$ and $2\text{\,}\mathrm{s}$, respectively, for BBH
signals. For a more detailed explanation of this method see Section II-B of
[11].

From the numbers in Table 1 it is clear that, despite the strong and weak
regions being the most cause for concern, only a very small fraction of events
will fall in this region. By far the dominant case is the negligible bias
region, into which all signals in these detectors should fall.

###  III.2 Injection studies

####  III.2.1 BBH+BBH overlaps

BBH+BBH | Injected | PyCBC | cWB
---|---|---|---
SINGLES | PAIRS | SINGLES | PAIRS
Total | Counts | $13172$ | $10454$ | $9818$ | $6885$ | $5634$
Percentage | - | $79.37\%$ | $74.54\%$ | $52.27\%$ | $42.77\%$
FAR$<$1\text{\,}\mathrm{y}\mathrm{r}^{-1}$$ | Counts | $13172$ | $8436$ | $7947$ | $6883$ | $5310$
Percentage | - | $64.04\%$ | $60.33\%$ | $52.25\%$ | $40.31\%$
Table 2: Injected and recovered individual overlapping signals in different
injection sets and search pipelines. Here the two signals in a pairing are
both BBH mergers. The SINGLES column here is the union of the results from
both $\mathrm{SINGLES}_{\mathrm{A}}$ and $\mathrm{SINGLES}_{\mathrm{B}}$ data
sets. The FAR threshold of $<$1\text{\,}\mathrm{y}\mathrm{r}^{-1}$$ means
that, in a years observation, fewer than one event of this kind will occur due
to statistical fluctuations in the noise. This is a fairly typical cut for
assuring an event is astrophysical. The total column is for all events,
without the threshold.

Table 2 contains values of the number of injected and recovered signals in
each injection set in both PyCBC) and cWB. As can be seen, in PyCBC), the vast
majority, close to $80\%$, of injected non-overlapping signals were recovered
by any trigger. The other $20\%$ were largely missed due to the signals having
low network SNRs, $\lesssim 12$, or being in less sensitive sky locations for
the network. However, another $15\%$ are removed once a FAR threshold of
$<$1\text{\,}\mathrm{y}\mathrm{r}^{-1}$$ is applied, these could be true
astrophysical signals, but are rejected due to poor significance. This is an
artefact of the injected distribution to provide a reasonable number of
detectable events.

As expected [33], cWB is less sensitive, with respect to modelled searches,
for this range of masses. The missed signals here are largely low mass CBCs,
$\mathcal{M}\lesssim 15\,M_{\odot}$, in which the majority of the SNR comes
from the inspiral, which is difficult for cWB to recover. As expected, if the
injections are present in the template bank of the matched filter search, then
the unmodelled method will always be less optimal than the matched filter
searches. This is the case here since our injections have source parameters
which are well described by the template bank.

Figure 1: Bar charts for the found injections in both pipelines in the
different overlap bias regions. Each region has four bars, split into two for
each pipeline. These show the percentage injections in which the pipeline
found a trigger. The “unique” column shows the percentage of unique triggers,
i.e. if both signals are found by the same trigger, then they are only counted
once. The shaded segments show injections recovered at a FAR threshold of
$<$1\text{\,}\mathrm{y}\mathrm{r}^{-1}$$. The data here is for the BBH+BBH
injection sets.

Figure 1 shows bar charts for the percentage of injections that were found by
a trigger in each pipeline in the PAIRS injection set. As expected the
pipelines perform reasonably well in the negligible bias region, with the
majority of signals found in PyCBC). For both pipelines, in this region, the
number of injections found matches the findings in the SINGLES injections.
There is some decrease in the fraction successfully caught when moving to the
weak and strong bias regions.

Inside the strong bias region for PyCBC), and some of the weak region for cWB,
some of the injections are both found in the same trigger. This means that
they are counted twice. To show this we have added extra bars for each search
in which we have removed duplicated triggers here. This is an effect of the
criteria we have applied to decide if an injection is found. See Section II.4
for more details.

The strong bias region here shows a functional problem with the pipelines for
signals in this region. These numbers are slightly lower than they
theoretically should be, due to the clustering of triggers in the pipeline. In
PyCBC), as templates are matched to the data, numerous triggers are recorded
for each signal as multiple templates may match the signal to differing levels
of significance. To avoid all of these triggers being returned for a single
signal, a clustering window is set. This window, set to
$\pm$1\text{\,}\mathrm{s}$$ in our study, clusters these triggers and returns
a single trigger of highest significance.

In the case of overlapping signals within this $\pm$1\text{\,}\mathrm{s}$$
merger time separation window, only the most significant signal is returned.
As such, in the strong bias region and some of the weak bias region, many of
the signals are missed due to only the most significant being returned. In our
study the triggered signal is usually Signal A. Due to our method of drawing
of the merger time, Signal B always merges before Signal A. Therefore, in a
matched-filter search, Signal A is favoured as it is still providing power in
the data at this time. This is not necessarily the case with unmodelled
searches, such as cWB, see Section IV.1 for further details. An example of
this can be seen in Figure 2.

Figure 2: A stack plot showing the distribution of how injected signals were
found by relative merger time in the PyCBC) BBH+BBH PAIRS injection set. Blue
and red show the pairings in which only one signal was found, Signal A or
Signal B respectively. Green shows the the pairings in which both signals were
found, grey for entirely missed pairs. This plot covers both the strong and
weak bias regions. The time convention is $T_{B}-T_{A}$, this is always less
than $0$ due to the convention of drawing Signal B’s merger within the
observable duration of Signal A. The percentages in the caption refer to the
fraction of pairings in this region, $|\Delta
t_{c}|\leq$2\text{\,}\mathrm{s}$$, that fall in each category.

Both found triggers are only in the region outside the clustering window of
PyCBC), $|\Delta t_{c}|\leq$1\text{\,}\mathrm{s}$$111Three pairings have both
signals found inside the clustering window. These pairings have merger time
separations of approximately $-$0.999\text{\,}\mathrm{s}$$ and, in some
detectors have separations $|\Delta t_{c}|>$1\text{\,}\mathrm{s}$$. In the
region $|\Delta t_{c}|\leq$1\text{\,}\mathrm{s}$$, Signal A is clearly
favoured, as it is the later merging signal. Pairings in this region that are
found with Signal B are largely either very close in merger time or have
$\text{SNR}\textsubscript{B}>\text{SNR}\textsubscript{A}$. This is shown in
Figure 3, where the distribution of SNR ratio between signals in found
pairings is shown. In the region $<$1\text{\,}\mathrm{s}$$ here Signal A is
louder, and therefore favoured, and in the region $>$1\text{\,}\mathrm{s}$$
Signal B is louder and more likely to be found. Missed pairings here have
generally even SNR ratios due to them both having poor SNRs. These regions are
artefacts of the pipeline settings and as such remain the same for the longer
signals in the BNS+BNS and BNS+BBH runs outlined in Sections III.2.2 and
III.2.3.

Figure 3: A stack plot showing the distribution of how pairings were found as
a function of SNR ratio between the two signals. The ratio is defined to be
$\text{SNR}\textsubscript{B}/\text{SNR}\textsubscript{A}$. Colours are
consistent with those in Figure 2. Similarly, the data comes from pairings in
the strong and weak bias regions of the PyCBC) BBH+BBH PAIRS injection set.
The percentages in the caption refer to the fraction of pairings in this
region, $|\Delta t_{c}|\leq$2\text{\,}\mathrm{s}$$, that fall in each
category.

For Figures 2 and 3, we have only included PyCBC) results. This is due to the
large time window criteria we applied to count found cWB injections In this
region, most cWB pairings will be single signal found. However, most signals
with such close mergers will only be found as a single trigger in cWB. cWB
internally applies time windows on regions of excess power, under the
assumption that it would only ever be a single signal. In wider regions, such
as the negligible bias region, the search returns separate triggers for each
component signal in the pairing. This is akin to the clustering window of
PyCBC), however, when PyCBC) finds both injections by a single trigger it will
return a template for a single signal under the assumption that it is a single
signal, cWB returns the coherent power of one, or potentially both signals,
see Section IV.

There are some signals in the negligible bias region that are found in the
single signal injection sets, but missed in the overlapping runs. Generally,
these signals have relative merger times just beyond the two second negligible
bias region and have fairly uneven SNR-ratios. This follows the findings of
previous parameter estimation studies. In those studies, if one signal is much
louder than the other then samples are recovered matching the parameters of
the louder signal. In the case of match filter searches, the template bank
search will recover templates closer to the louder signal, the remaining power
from the weaker signal will be rejected as either noise, or excess power in
the inspiral of the louder signal.

The drop in efficiency due to the presence of time-overlapping signals can be
estimated by comparing the efficiency of the overlapping and non-overlapping
signals. For PyCBC) this drop off is approximately $4\%$ across the entire
run. In cWB this drop in efficiency is approximately $12\%$. The differences
in pipeline efficiency means that a direct comparison is not a simple matter,
we also report the relative efficiency drops as $6\%$ and $23\%$ for PyCBC)
and cWB respectively.

The errors on all efficiencies are less than $0.01\%$ and as such were not
included. All quoted efficiencies are for events found with a FAR threshold of
$<1$ per year.

Outside the clustering window of PyCBC), this fall in efficiency is
approximately $1\%$, while inside the clustering window the fall in efficiency
is $26\%$ due to the majority of paired signals being found by a single
trigger, or not at all. These regions are not directly comparable for cWB, in
these cases it will find both signals and return them as one trigger, however,
similar numbers would be $9\%$ and $32\%$ for outside and inside the $|\Delta
t_{c}|=$1\text{\,}\mathrm{s}$$ boundary.

####  III.2.2 BNS+BNS overlaps

BNS+BNS | Injected | PyCBC | cWB
---|---|---|---
SINGLES | PAIRS | SINGLES | PAIRS
Total | Counts | $2212$ | $2042$ | $1677$ | $554$ | $467$
Percentage | - | $92.31\%$ | $75.81\%$ | $25.05\%$ | $21.11\%$
FAR$<$1\text{\,}\mathrm{y}\mathrm{r}^{-1}$$ | Counts | $2212$ | $1854$ | $1544$ | $550$ | $461$
Percentage | - | $83.82\%$ | $69.80\%$ | $24.86\%$ | $20.84\%$
Table 3: Injected and recovered individual overlapping signals in different
injection sets and search pipelines. The SINGLES column here is the union of
the results from both $\mathrm{SINGLES}_{\mathrm{A}}$ and
$\mathrm{SINGLES}_{\mathrm{B}}$ data sets. Here the two signals in a pairing
are both BNSs. It should be noted that these values are not directly
comparable to those in Table 2, as the luminosity distances were arbitrarily
set for BBH and BNS injections such that most were visible in the detector. As
such we have a slightly higher proportion of BNS signals recovered in PyCBC
than we did for BBH signals.

Table 3 shows the results for the BNS+BNS overlap injection set. There are
fewer signals in these injection sets as a higher number of signals would have
caused $N_{signals}>2$ overlaps, which we do not consider in this study. As
expected, cWB finds a lower percentage of the injections, as it is not
designed to find longer, inspiral dominated signals such as binary neutron
stars. This can be seen in Figure 4 where cWB recovers only about $20\%$ of
injections with unique triggers. PyCBC) has a slightly higher efficiency here
than in the BBH+BBH run, however, this is most likely a consequence of the
injected luminosity distances of the signals rather than the design of the
search pipeline.

Due to these long durations and the uniform drawing of merger time separation,
the signals were drawn such that approximately half fell into the negligible
bias region with a further quarter falling in each of the weak and strong bias
regions. As the strong bias region of BNS+BNS events is, as predicted by
parameter estimation studies, $|\Delta t_{c}|\leq$0.01\text{\,}\mathrm{s}$$, a
very high percentage of injected pairs fall inside the PyCBC) clustering
window so are both found by the same trigger. This is apparent in the strong
region of Figure 4 where the unique bar is half that of the non-unique found.

Figure 4: Bar charts for the found injections in both pipelines in the
different overlap bias regions. Each region has four bars, split into two for
each pipeline. These show the percentage injections in which the pipeline
found a trigger. The “unique” column shows the percentage of unique triggers,
i.e. if both signals are found by the same trigger, then they are only counted
once. The shaded segments show injections recovered at a FAR threshold of
$<$1\text{\,}\mathrm{y}\mathrm{r}^{-1}$$. The data here is for the BNS+BNS
injection sets.

The efficiency drops between overlapping and non-overlapping BNS signals is
$14\%$ for PyCBC) and $4\%$ for cWB. The cWB values drop is small, compared to
PyCBC), as the majority of BNS signals found by cWB are very significant
regardless of overlap. The relative drop in these efficiencies are $17\%$ and
$16\%$ respectively.

####  III.2.3 BNS+BBH overlaps

BNS+BBH | Injected | PyCBC | cWB
---|---|---|---
SINGLES | PAIRS | SINGLES | PAIRS
Total | Counts | $2400$ | $2086$ | $1713$ | $928$ | $706$
Percentage | - | $86.92\%$ | $71.38\%$ | $38.67\%$ | $29.42\%$
FAR$<$1\text{\,}\mathrm{y}\mathrm{r}^{-1}$$ | Counts | $2400$ | $1782$ | $1519$ | $922$ | $673$
Percentage | - | $74.25\%$ | $63.29\%$ | $38.42\%$ | $28.04\%$
Table 4: Injected and recovered individual overlapping signals in different
injection sets and search pipelines. The SINGLES column here is the union of
the results from both $\mathrm{SINGLES}_{\mathrm{A}}$ and
$\mathrm{SINGLES}_{\mathrm{B}}$ data sets. Here the two signals in a pairing
are a BNS and a BBH. It should be noted that the comparison between PYCBC and
cWB for the SINGLES runs is challenging as cWB will perform significantly
differently for the BNS SINGLES run. See Table 3 for a more precise comparison
of single signal runs for these events.

As described in Section III.2.2 cWB is not tuned to low mass, inspiral
dominated signals, like BNSs. As such it does not perform well for the BNS
portion of the SINGLES injection sets. The efficiencies of the pipeline for
these injection sets highlights this. For cWB, the
$\mathrm{SINGLES}_{\mathrm{A}}$ run recovered $25\%$ of injections, while the
$\mathrm{SINGLES}_{\mathrm{B}}$ run recovered $52\%$, with $86\%$ and $82\%$
being the comparable numbers for PyCBC).

Table III.2.3 shows a significant falloff between the SINGLES and PAIRS
injection sets, compared to the BBH+BBH run. This, as in Section III.2.2, is
in large part due to a large percentage of strong bias pairs falling inside
the clustering windows and being found by the same trigger. This can be seen
in Figure 5 where, as in Figure 4 the strong bias region sees a significant
falloff from any to unique found injections.

Figure 5: Bar charts for the found injections in both pipelines in the
different overlap bias regions. Each region has four bars, split into two for
each pipeline. These show the percentage injections in which the pipeline
found a trigger. The “unique” column shows the percentage of unique triggers,
i.e. if both signals are found by the same trigger, then they are only counted
once. The shaded segments show injections recovered at a FAR threshold of
$<$1\text{\,}\mathrm{y}\mathrm{r}^{-1}$$. The data here is for the BNS+BBH
injection sets.

Here PyCBC) performs better for BNS+BBH overlaps than for BNS+BNS overlaps.
The same cannot be said for cWB, which would appear to perform more
consistently in the BNS+BNS run. Although this may be difficult to conclude
due to the low efficiency of that run.

The efficiency drop between overlapping and non-overlapping BNS signals is
$11\%$ for PyCBC) and $10\%$ for cWB. Care should be taken here in comparisons
between the pipelines, as PyCBC) has similar sensitivities to BBH and BNS
signals, whereas cWB will favour higher mass signals and is therefore more
sensitive to one half of the non-overlapping signals. The relative drops in
efficiency are $15\%$ and $26\%$ respectively.

###  III.3 Accuracy of recovered triggers

Figure 6: Distribution, across the overlap regions, of the fractional
difference between the recovered and injected chirp masses for PyCBC) triggers
in the PAIRS injection set. Also shown, in grey, is the same distribution for
recovered and injected in the SINGLES injection set. A perfect pipeline, with
an infinitely finely gridded template bank, would find injections with a delta
spike on zero in this plot. For direct comparison across the overlap regions,
we have included this plot as a function of regional density in Appendix VII.

Figure 6 contains histograms of the fractional chirp mass difference between
the triggers recovered, by PyCBC), in the PAIRS injection set and the true
injected values. It also includes a similar distribution for the SINGLES
injection set, shown in grey.

In most cases the recovered trigger in the PAIRS injections match what is
found in the single signal injections to a reasonable level,
$|\Delta\mathcal{M}|<1\,M_{\odot}$. This can be seen by comparing the
distribution of negligible bias region injections to the SINGLES distribution.
The weak and negligible regions match this distribution fairly well, with few
outliers. However, the strong bias region skews towards a high fractional
difference, up to $\sim 6$ times the value found in SINGLES. Therefore, we can
say that signal pairings in the strong bias region are affected in a similar
way in matched filter searches to matched filter based parameter estimation.

The skew of this distribution shows that incorrectly found signals tend to be
found with higher chirp mass templates than the injected signal. As this is
also true for the inaccurately found single signals, this is a feature of how
we calculate the fractional difference. The recovered chirp mass can never be
less than -1 as for a trigger to be recovered it must have a positive chirp
mass. Triggers with much larger values of fractional chirp mass are likely
extreme due to the lower density of templates at the high mass tail of the
template bank.

##  IV Overlap Identification and Separation

Figure 7: A plot of whitened strain, in the LIGO-Hanford detector, for a
single BBH injection. The blue lines shows the matched template found by
PyCBC) in the $\mathrm{SINGLES}_{\mathrm{A}}$ injection set. The red line is
the cWB reconstructed waveform from the same injection set. The green, dashed,
vertical line indicates the injected merger time. Figure 8: A plot of whitened
strain, in the LIGO-Hanford detector, for a BBH+BBH injection. The blue line
shows the matched templates found by PyCBC) in the PAIRS injection set. The
red line is the cWB reconstructed waveform from the PAIRS injection set. The
green, dashed, vertical lines indicates the injected merger times.

Figure 7 shows a comparison of the whitened strain of an injected BBH single
signal in simulated Hanford detector strain. This is accompanied by a whitened
version of the template found in the PyCBC) search and the reconstructed
waveform, produced by cWB. Here, outside the merger, the template found by
PyCBC) does not match the signal as well as the cWB reconstruction. This is
most noticeable at lower frequency, although the mergers are broadly similar.
This is due to the coarseness of the gridding in the PyCBC) template bank and
the lack of bank sensitivity to parameters such as spin and phase.

Figure 8 shows an equivalent plot involving the Signal A represented in Figure
7 overlapping with a Signal B. It can be seen that Signal B, merging first, is
found in cWB but not in PyCBC). Both signals are significant, with network
SNRs of 26 and 27. Despite this, due to the close mergers of the signals and
clustering, the template for Signal A is returned in the PyCBC) PAIRS run due
to its later merger time. The Signal B from this pairing was found by both
PyCBC) and cWB, as an individual signal, in the
$\mathrm{SINGLES}_{\mathrm{B}}$ injection set.

The phase of the cWB reconstruction for Signal A in Figure 8 is different
compared to that showing in Figure 7 as the algorithm is trying to fit both
signals to the same, coherent sky location. Indeed, as shown in Figures 9 and
10, in that case the likelihood maximisation is considerably affected by
Signal B, and so Signal A’s reconstruction is not optimal. This is discussed
in detail in Section IV.1.

###  IV.1 Separation through unmodelled algorithms

The cWB framework, as described in Section II.3.2, is designed to analyse a
single signal. The pipeline finds regions of excess power and then maximises
the likelihood with respect to one source location, regardless of the number
of present astrophysical signals. In the standard case of a single signal
trigger, the pipeline maximises the likelihood with respect to that signal,
and so almost all its energy is placed in the likelihood, while the null is
almost entirely noise, as shown in Figure 9.

For the case of two overlapping signals, the likelihood is then maximised
considering both signals, generally with one favoured over the other,
considering this signal as the “primary” and the other as the “secondary”. In
this case, the likelihood is largely maximised with respect to the primary
signal, though often with some contamination from the secondary signal,
depending on its energy. This means that the primary signal energy is almost
entirely found in the likelihood. The energy of the secondary signal is
divided between likelihood and null depending on its source location. If this
is considerably different with respect to that of the primary signal, then a
considerable amount of the energy of the secondary signal will remain in the
null. This is similar to the situation in Figures 7 and 8.

Figure 9 shows the time-frequency representation of the likelihood and null
energies of Signal A, in which likelihood has been properly maximised. The
signal is almost entirely caught in the likelihood. The remaining energy in
the null is likely excess noise. Figure 10 represents the same signal in an
overlapping pairing, as in Figure 8. In this case, the likelihood is maximised
mainly with respect to Signal B, here the primary signal. The null contains
the secondary, here Signal A, and is increased with respect to the single
detection. The likelihood has been maximised with respect to a different sky
location, and both the null and likelihood contain a non-negligible
contribution from each signal.

This contribution of the signal to the residual noise energy increases the
penalty, lowering the correlation coefficient. For this reason, triggers with
overlapping signals are penalised when applying post-production cuts, as it
can be seen in detection efficiencies reported in Tables 2, 3 and 4 and in
Figures 1, 4 and 5.

Figure 9: A spectrogram of likelihood and null energy for a single signal
event. The likelihood contains almost entirely signal energy, while the null
almost entirely noise. The data is the same as for the Single A signal in
Figure 7

Figure 10: A spectrogram of likelihood and null energy for two overlapping
signals. The likelihood has been maximised with respect to the primary signal,
so a relevant fraction of the energy associated with the secondary one remains
in the null. The data is the same as for the signals in Figure 8

.

These results suggest that, to produce an optimal analysis of overlapping
signals, a future development of cWB should allow for the estimation of
multiple likelihoods. If a TF map suggests the presence of two overlapping
signals, then the information about the way to analyse it can be obtained
firstly analysing the data with a likelihood following the primary signal. If
the null then contains an indication for another signal, then it could be used
to select the pixels related to this signal and try to maximise the likelihood
with respect to them only. It is important to remark that the TF
representation is able to disentangle, by itself, the two overlapping signals
only if these cover different TF pixels. In this case, the separate
maximisation of two different likelihoods, via two different sets of pixels,
would give the signal reconstruction as optimally as it is currently for
single signal triggers. It should be noted that in the unlikely case in which
the two signals come from almost the same source location the two likelihoods
would be almost the same.

If a pixel has contributions from both signals, then the TF separation this is
not possible. In that case there is still the possibility to exploit the
likelihood-null energy distribution for the involved overlapping pixels, if
the signals come from different sky locations. When the two signals cover the
same pixel the likelihood-null disentangling would be non-optimal as some
fraction of the secondary signal will fall into the likelihood of the primary,
as shown in Figure 10. This implies that we may not be able to fully recover
the secondary signal from the null. If the signals come from almost the same
source location, then there is no way to disentangle them in an unmodelled
way, at least in those TF pixels covered by both signals: the likelihood-null
approach fails as well because in that case also the null is almost the same
for both signals. All these considerations should be examined in future
studies.

###  IV.2 Separation through matched filter algorithms

When searching for signals in data, matched filter searches, such as PyCBC),
will match a template waveform to the data at time intervals in order to
create an SNR time series of the data. This time series is a map of the SNR of
that template with the data. When one of these templates closely matches a
signal, the time series will peak.

In a situation in which there are two signals in the data, this time series
should peak twice, once for each signal in the data. Figure 11 shows this for
two perfect templates against data with two signals injected. The blue line,
for Signal A’s template, shows a small peak around the merger of Signal B,
where the match is not perfect. It then peaks again, cleanly and more
significantly, around the merger of Signal A. The reverse is shown for a
perfect Signal B template in red.

Figure 11: A plot of SNR time series for two perfect templates against
simulated data containing both signals in the LIGO-Hanford detector. The blue
line shows the SNR of the Signal A template against the data, the red line
shows the equivalent for the Signal B template. The blue and red, dashed,
vertical lines show the injected merger times of Signal A and B respectively.
The signals are the same as those shown in Figure 8. The peak SNRs here are
smaller than those given in Section IV as they are single detector SNRs, not
network SNRs.

The signals and templates used in Figure 11 have very similar chirp masses,
and as such match reasonably well with each other’s templates. Conversely, if
the signals have very different chirp masses, then this plot will only show a
single peak for each signal. Although, there will likely be some non-
Gaussianity in the region of the non-ideal signals merger time.

PyCBC)’s clustering routines, as described in Section II.4, will ignore these
features and only return the maximum likelihood signal, in this case Signal A.
It should be possible to modify the clustering routines to check for multiple
peaks in a single template, or for single peaks in multiple templates. These
results can then be used to identify reasonable parameter ranges for each
signal in the overlap. This would allow for a reliable starting point for
multi-signal parameter estimation [14].

##  V Discussion

We showed here that both modelled, and unmodelled searches can detect
overlapping CBC signals in the negligible bias region of $|\Delta
t_{c}|>$2\text{\,}\mathrm{s}$$. Within the weak bias region, so long as the
signals are separated by more than the matched filter clustering window, here
$|\Delta t_{c}|>$1\text{\,}\mathrm{s}$$, the matched filter search should
recover both signals, provided one is not much louder than the other. We also
show that in the narrower weak and strong bias regions both pipelines can
successfully recover one or both signals in the overlap most of the time.

Inside the clustering window, matched filter pipelines do struggle to
successfully recover both signals due to internal methods designed to reduce
the number of false triggers returned. However, we believe that these searches
can be modified to find templates that best match each signal in the pairing.
This could provide reasonable best guesses for multi-signal parameter
estimation.

We showed that unmodelled searches can successfully recover both signals in
the pairing, both as a single trigger when the mergers are close, and as
separate triggers when far apart. cWB, when recovering signals via coherent
sky location, is able to disentangle overlapping signals, so long as they
cover different frequencies in the TF map. In the case in which the signals
are in the same TF pixels, the secondary signal should be at least partially
separable via the null energy map, if coming from a different sky location
than the first. This should be another indicator of the presence of a
secondary signal. It is possible that the second signal’s sky location may be
recoverable leading to further first guess parameters for multi-signal
parameter estimation.

We believe that, with modifications, the combination of these two forms of CBC
search pipelines should provide a reliable method for the detection,
identification and initial separation of time-overlapping CBCs.

##  VI Acknowledgements

We would like to thank Tim Dietrich, Stephen Fairhurst, Jonathan Thompson,
Alex Nitz, Edoardo Milotti, and Giovanni Prodi for useful discussions. We
would also like to thank the anonymous reviewer. We are grateful for
computational resources provided by Cardiff University and funded by an STFC
grant supporting UK Involvement in the Operation of Advanced LIGO. The authors
are grateful for computational resources provided by the LIGO Lab (CIT) and
supported by Natiwith the network of detectors by correlating the dataonal
Science Foundation Grants PHY-0757058 and PHY-0823459. This material is based
upon work supported by NSF’s LIGO Laboratory which is a major facility fully
funded by the National Science Foundation. Alongside previously mentioned
software packages we would also like to thank the following codes for their
use in this study: SCIPY [45], NUMPY [46], PANDAS [47, 48], MATPLOTLIB [49],
and GWPY [50]. This work has been allocated LIGO document number P2200229, and
Virgo document number VIR-0777A-22. PR is supported by STFC grant
ST/S505328/1. VR is supported by STFC grants ST/V001396/1 and ST/V00154X/1. IH
is supported by STFC grants ST/T000333/1 and ST/V005715/1. MD acknowledges the
support from the Amaldi Research Center funded by the MIUR program
’Dipartimento di Eccellenza’ (CUP:B81I18001170001) and the Sapienza School for
Advanced Studies (SSAS). ST is supported by Swiss National Science Foundation
(SNSF) Ambizione Grant Number: PZ00P2$\\_$202204\. AM acknowledges the support
of the European Gravitational Observatory under the convention EGO-DIR-63-2018

##  VII Appendix - Supplementary figure

The plot shown in Figure 12 is similar to that in Figure 6, however, the y
axis has been adjusted to show the normalised distributions, rather than the
number of triggers. This allows more direct comparison between the four
distributions.

Figure 12: Distribution, across the overlap regions, of the fractional
difference between the recovered and injected chirp masses for PyCBC) triggers
in the PAIRS injection set. Also shown, in grey, is the same distribution for
recovered and injected in the SINGLES injection set. A perfect pipeline, with
an infinitely finely gridded template bank, would find injections with a delta
spike on zero in this plot. Here the distribution is shown as a function of
the density of signals, rather than the signal numbers.

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