# Biases in parameter estimation from overlapping gravitational-wave signals
in the third generation detector era

Anuradha Samajdar1,2,3 Justin Janquart1,2 Chris Van Den Broeck1,2 Tim
Dietrich4,5 1Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands
2Institute for Gravitational and Subatomic Physics (GRASP), Utrecht
University, Princetonplein 1, 3584 CC Utrecht, The Netherlands  3Department of
Physics, University of Milano – Bicocca, Piazza della Scienza 3, 20126 Milano,
Italy  4Institut für Physik und Astronomie, Universität Potsdam, Haus 28,
Karl-Liebknecht-Str. 24/25, 14476, Potsdam, Germany  5Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam
14476, Germany

###### Abstract

In the past few years, the detection of gravitational waves from compact
binary coalescences with the Advanced LIGO and Advanced Virgo detectors has
become routine. Future observatories will detect even larger numbers of
gravitational-wave signals, which will also spend a longer time in the
detectors’ sensitive band. This will eventually lead to overlapping signals,
especially in the case of Einstein Telescope (ET) and Cosmic Explorer (CE).
Using realistic distributions for the merger rate as a function of redshift as
well as for component masses in binary neutron star and binary black hole
coalescences, we map out how often signal overlaps of various types will occur
in an ET-CE network over the course of a year. We find that a binary neutron
star signal will typically have tens of overlapping binary black hole and
binary neutron star signals. Moreover, it will happen up to tens of thousands
of times per year that two signals will have their end times within seconds of
each other. In order to understand to what extent this would lead to
measurement biases with current parameter estimation methodology, we perform
injection studies with overlapping signals from binary black hole and/or
binary neutron star coalescences. Varying the signal-to-noise ratios, the
durations of overlap, and the kinds of overlapping signals, we find that in
most scenarios the intrinsic parameters can be recovered with negligible bias.
However, biases do occur for a short binary black hole or a quieter binary
neutron star signal overlapping with a long and louder binary neutron star
event when the merger times are sufficiently close. Hence our studies show
where improvements are required to ensure reliable estimation of source
parameters for all detected compact binary signals as we go from second-
generation to third-generation detectors.

##  I Introduction

The direct observation of gravitational waves (GWs) Abbott  _et al._ (2016a)
has had a tremendous impact in fundamental physics Abbott  _et al._ (2016b,
2019a, 2019b, 2019c), astrophysics Abbott  _et al._ (2016c, 2019d, 2020a,
2017a, 2018a, 2020b, 2020c, 2020d, 2020e, 2020f), and cosmology Abbott  _et
al._ (2017b); Soares-Santos  _et al._ (2019); Palmese  _et al._ (2020), and
starting from the observation of the binary neutron star (BNS) signal GW170817
Abbott  _et al._ (2017a) has opened a new era in multi-messenger astronomy
with GWs Soares-Santos  _et al._ (2017); Cowperthwaite  _et al._ (2017);
Abbott  _et al._ (2017c); Albert  _et al._ (2017). The third observing run
(O3) of Advanced LIGO Aasi  _et al._ (2015) and Advanced Virgo Acernese  _et
al._ (2015) ended in March 2020, and together these interferometers have found
more than 50 GW candidates gra , with 39 candidates observed during the first
half of O3 Abbott  _et al._ (2020a). The detector sensitivities will be
improved further, and the frequency with which signals are observed is
expected to keep increasing in coming years. In particular, in the transition
to the envisaged third generation (3G) era, with Einstein Telescope (ET)
Punturo  _et al._ (2010a); Hild  _et al._ (2011a) and Cosmic Explorer (CE)
Reitze  _et al._ (2019); Abbott  _et al._ (2017d); Regimbau  _et al._ (2017),
the detection rate will go up steeply, and signals will also spend much longer
times in the detectors’ sensitive band Sathyaprakash  _et al._ (2012). As
first pointed out in Regimbau and Hughes (2009) and studied further in this
paper, the probability of overlapping signals will then become signficant.

In view of this, it will be important to assess to what extent the science
goals of 3G detectors (see e.g. Sathyaprakash  _et al._ (2010); Van Den Broeck
(2010); Zhao  _et al._ (2011); Punturo  _et al._ (2010b); Sathyaprakash  _et
al._ (2011, 2012); Van Den Broeck (2014); Maggiore  _et al._ (2020);
Sathyaprakash  _et al._ (2019a); Vitale  _et al._ (2019); Ng  _et al._ (2020))
may be affected by signals overlapping with each other. Apart from science
with signals from compact binary coalescences (CBCs), this includes searches
for primordial backgrounds, since the subtraction of “foreground” CBC sources
Cutler and Harms (2006); Harms  _et al._ (2008); Regimbau  _et al._ (2017);
Sachdev  _et al._ (2020); Sharma and Harms (2020); Biscoveanu  _et al._ (2020)
will rely on our ability to characterize them individually. As shown in
Regimbau  _et al._ (2012); Meacher  _et al._ (2016), even using current data
analysis techniques, the _detection rates_ of individual CBC sources would
likely not be significantly impacted by the occurrence of overlapping signals.
However, a study of the effect on _parameter estimation_ had not yet been
performed.

Earlier works Vitale (2016); Vitale and Evans (2017); Vitale and Whittle
(2018) have studied parameter estimation for single sources in the 3G era.
Here we take the first step in assessing possible biases in the recovery of
parameters characterizing a GW signal when signals from different sources are
simultaneously present in the detectors’ sensitive band. Before doing this, we
map out how often signal overlaps of various types will occur in a network of
two CEs and one ET over the course of a year, assuming realistic distributions
for merger rate as a function of redshift and for component masses in binary
neutron star and binary black hole (BBH) coalescences. We find that a typical
BNS signal will be overlapped by tens of BBH signals. Moreover, BBH or BNS
signals whose mergers occur within seconds from each other will be quite
common. Since these are the cases for which we can expect the largest
parameter estimation biases to occur, we focus on them in setting up
simulations whereby signals are added to synthetic data from the ET-CE
network, and analyzed using current state-of-the-art parameter estimation
techniques. We explore various scenarios of signals from different kinds of
sources overlapping: two BBH signals, two BNS signals, and a BBH signal with a
BNS. For our simulations we choose signal parameters consistent with what has
been observed as being representative of each kind of source: parameter values
similar to the ones of GW170817 Abbott  _et al._ (2019e) to represent a BNS,
similar to the ones of GW150914 Abbott  _et al._ (2016d) to represent a high-
mass BBH, and similar to the ones of GW151226 Abbott  _et al._ (2016e) for a
lower-mass BBH. We find that in most cases, the intrinsic parameters can be
recovered with negligible bias. However, if the merger times of the two
signals are sufficiently close, considerable biases can occur when a short BBH
signal or a quieter BNS signal overlaps with a louder BNS signal. Though our
study should be considered exploratory, it already points to where
improvements over current parameter estimation pipelines will be needed the
most.

This paper is structured as follows. In Sec. II we obtain detection rate
estimates for signals in the 3G era, from which we calculate overlap rates. In
Sec. III we lay out the settings and methods we use for parameter estimation.
Parameter estimation results for various scenarios are shown in Sec. IV. A
summary and conclusions are presented in Sec. V, where we also give
recommendations for future improvements of parameter estimation techniques.

##  II Overlap rate estimates

###  II.1 Methodology

Before looking at the impact of overlapping signals on parameter estimation
for the individual ones, we want to address the question of how frequently
such overlaps will occur, depending on the type. Previous characterizations of
the overlap probabilities for 3G detectors were based on the _duty cycle_ ,
which is defined as the ratio of the typical duration of a particular type of
event (BNS or BBH) to the average time interval between two successive events
of that type, assuming some fixed canonical values of the component masses for
each type Regimbau and Hughes (2009). However, here we also want to allow for
overlaps of mixed types, and for a range of component masses (and hence signal
durations) within a given type, so as to arrive at a detailed assessment of
overlap rates. Therefore, what we will do is to assume particular merger rates
as function of redshift for BBH and BNS, as well as component mass
distributions, and on the basis of these create simulated “catalogs” of
signals in the detectors. This will allow us to make quantitative statements
regarding BNS signals overlapping with other BNS signals and with BBHs, and
the same for overlaps of BBH with BBH events, in a much more detailed and
realistic fashion.111Since neutron star-black hole (NS-BH) rates are less
certain (see e.g. Abbott  _et al._ (2020d); Hoang  _et al._ (2020)), we will
not consider them here, but we expect general conclusions regarding parameter
estimation to largely carry over when signal durations are similar.

We start by estimating the number of individual BBH and BNS coalescences that
happen in a given volume, up to a maximum redshift which is chosen to be
$z_{\rm max}=30$ for BBH events and $z_{\rm max}=6$ for BNS events Regimbau
and Hughes (2009); Regimbau  _et al._ (2012); Sathyaprakash  _et al._ (2012);
Safarzadeh  _et al._ (2019). For this we need the intrinsic merger rate
density for the events as a function of redshift. We will assume that the
compact binaries originate from stellar populations, and adopt the merger rate
estimates of Belcynski et al. Belczynski  _et al._ (2017) with Oguri’s
analytical fit Oguri (2018)222Strictly speaking this merger rate distribution
refers to BBH mergers. However, when computing the merger rate density (see
e.g. Regimbau  _et al._ (2012); Belczynski  _et al._ (2017); Safarzadeh  _et
al._ (2019)), one assumes a time delay distribution (e.g. $P(t_{\rm d})\propto
1/t_{\rm d}$), with a minimum time delay that is higher for BBH than for BNS.
Using the distribution of Belczynski  _et al._ (2017) for both BNS and BBH
(with some overall rescaling) then implies that we will underestimate the BNS
merger rate density Safarzadeh  _et al._ (2019) and hence the frequency of
overlaps involving BNS signals., whose general expression is

$R_{\rm GW}(z)=\frac{a_{1}e^{a_{2}z}}{e^{a_{3}z}+a_{4}}.$ (1)

Here $R_{\rm GW}$ is expressed in ${\rm Gpc^{-3}\,yr^{-1}}$, and the
coefficients $a_{i}$, $i=1,\ldots,4$ depend on the star populations that are
considered; see Fig. 1. For our purposes, we consider the combination of
population I and II stars for BNS, and populations I, II, and III for BBH, as
the contribution of the latter type of stars is important only at redshifts of
$\gtrsim 4$. However, these relations are rescaled to match the local merger
rate estimates obtained observationally by LIGO and Virgo so far; see Abbott
_et al._ (2020g). In this work, we focus on the lowest, the median, and the
highest local rate for each type of event. For BNS, the lowest, median, and
highest local rates are, respectively, $80\,{\rm Gpc^{-3}\,yr^{-1}}$,
$320\,{\rm Gpc^{-3}\,yr^{-1}}$, and $810\,{\rm Gpc^{-3}\,yr^{-1}}$, which are
obtained by changing the value of $a_{1}$ to 2480, 9920, and 25110,
respectively. On the other hand, for the BBH events, we apply a multiplicative
constant to the sum of the population I and II and the population III rates,
equal to 0.0709, 0.112, and 0.178 for the lowest, median, and highest local
rates, which are $15.1\,{\rm Gpc^{-3}\,yr^{-1}}$, $23.8\,{\rm
Gpc^{-3}\,yr^{-1}}$, and $37.9\,{\rm Gpc^{-3}\,yr^{-1}}$, respectively.

Figure 1: The BBH merger rate density according to Oguri’s fit Oguri (2018)
for population I, II, and III stars, as well as the total rate, when all the
star populations are accounted for.

An intrinsic merger rate density $R_{\rm GW}(z)$ is then converted to an
_observed_ merger rate density as function of redshift by multiplying by the
differential comoving volume Regimbau and Hughes (2009):

$R^{\rm obs}_{\rm GW}(z)=R_{\rm GW}(z)\,\frac{dV_{c}}{dz}(z).$ (2)

To obtain $dV_{c}/dz$, we assume the _Planck13_ cosmological model Ade  _et
al._ (2014) of _Astropy_ Robitaille  _et al._ (2013); Price-Whelan  _et al._
(2018).

As a next step, we simulate the population of systems by constructing a
“catalog”, and determine which events are actually detected. For the BBH
population, we assume that the masses follow the “ _power law + peak_ ”
distribution presented in Ref. Abbott  _et al._ (2020g) for the primary
component mass, and the corresponding power law distribution for the mass
ratio, through which we sample the secondary mass Abbott  _et al._ (2020g).
For BNS events we distribute component masses uniformly, where for the primary
mass $m_{1}\in[1,2.5]\,M_{\odot}$, and for the secondary mass
$m_{2}\in[1\,M_{\odot},m_{1}]$. Events are distributed over comoving distance
$D$ according to $R_{\rm GW}(z)$, converting between $D$ and $z$ using the
above mentioned cosmology and cutting off at the maximum redshifts $z_{\rm
max}$ stated above. Sky positions and unit normals to the orbital plane are
taken to be uniform on the sphere.

In this work we assume a network of two CEs located at the LIGO Hanford and
Livingston sites, and one ET located at the Virgo site. For each event we
calculate the optimal signal-to-noise ratios (SNRs) in the three
observatories, which are added in quadrature to obtain a network SNR. In
computing SNRs we only consider the inspiral part of binary coalescence, so
that in the stationary phase approximation Sathyaprakash and Dhurandhar (1991)
and for a single interferometer Kastha  _et al._ (2020)

$\displaystyle{\rm SNR}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{5}{6}}\frac{1}{\pi^{2/3}}\frac{c}{D(1+z)^{1/6}}\bigg{(}\frac{G\mathcal{M}}{c^{3}}\bigg{)}^{5/6}$
(3) $\displaystyle\times\,\,g(\theta,\phi,\psi,\iota)\,\sqrt{I(M)}.$

Here $\mathcal{M}=(m_{1}\,m_{2})^{3/5}/(m_{1}+m_{2})^{1/5}$ is the chirp mass
in the source frame. The geometric factor is given by

$\begin{split}g(\theta,\phi,\psi,\iota)=\bigg{(}F_{+}^{2}&(\theta,\phi,\psi)(1+\cos(\iota)^{2})^{2}\\\
&+4F_{\times}^{2}(\theta,\phi,\psi)\cos(\iota)^{2}\bigg{)}^{1/2},\end{split}$
(4)

where $F_{+,\times}$ are the beam pattern functions in terms of sky position
$(\theta,\phi)$ and polarization angle $\psi$, while $\iota$ is the
inclination angle. We take Einstein Telescope to consist of three detectors
with $60^{\circ}$ opening angle, arranged in a triangle with sides of 10 km
Freise  _et al._ (2009), and add the corresponding SNRs in quadrature; for
Cosmic Explorer we assume a single L-shaped detector of 40 km arm length
Reitze  _et al._ (2019); Abbott  _et al._ (2017d). Finally,

$I(M)=\int_{f_{\rm low}}^{f_{\rm high}}\frac{f^{-7/3}}{S_{h}(f)}df.$ (5)

Here $f_{\rm low}$ is a low-frequency cut-off that depends on the observatory;
we set $f_{\rm low}=5$ Hz for both ET and CE, though lower values may be
achieved in the case of ET Hild  _et al._ (2011b, 2010). For $f_{\rm high}$ we
use the frequency of the innermost stable circular orbit:

$f_{\rm
high}(m_{1},m_{2},z)=\frac{1}{1+z}\frac{1}{6\pi\sqrt{6}}\frac{c^{3}}{GM},$ (6)

where $M=m_{1}+m_{2}$ is the total mass. We take the noise power spectral
density (PSD) $S_{h}(f)$ to be ET-D in the case of Einstein Telescope Punturo
_et al._ (2010a); Hild  _et al._ (2011a); for the projected PSD of Cosmic
Explorer, see Reitze  _et al._ (2019); Abbott  _et al._ (2017d).

The network SNR, denoted $\mbox{SNR}_{\rm net}$, is defined as

$\mbox{SNR}_{\rm net}^{2}=\sum_{i=1}^{3}\mbox{SNR}_{i}^{2},$ (7)

where the sum is over the two CE and the one (triangular) ET observatories. We
consider an event as detectable if the network SNR is above 13.85
($=\sqrt{3}\times 8$), without imposing SNR thresholds in individual
observatories. For the BNS and BBH mass ranges considered here, this means
that detection rates will mainly be driven by the CEs, but we note that ET
will have an advantage at higher masses Sathyaprakash  _et al._ (2019b).

Finally, signals will be present in a detector for a duration given by

$\displaystyle\tau=2.18\,\bigg{(}\frac{1.21\,M_{\odot}}{\mathcal{M}}\bigg{)}^{5/3}\bigg{[}\bigg{(}\frac{100\,\mbox{Hz}}{f_{\rm
low}}\bigg{)}^{8/3}-\bigg{(}\frac{100\,\mbox{Hz}}{f_{\rm
high}}\bigg{)}^{8/3}\bigg{]}\,\mbox{s}.$

Simulated catalogs of events happening over the course of a year are
constructed as follows. The year is split into a grid in which each cell
corresponds to one second, and merger times are drawn from a uniform
distribution over these cells. For a given type of event (BNS or BBH), one
associates to each merger time a mass pair, redshift, sky position, and
orientation of the orbital plane drawn from the corresponding distributions,
as well as a signal duration computed from Eq. (II.1). Doing this for the
three choices of local merger rate, and in each case putting together the BNSs
and BBHs, catalogs of events are obtained. Finally, within each catalog, it is
assessed which events will be detectable with the ET-CE network according to
the criteria spelled out above, leading to an overview of what we may expect
to be contained in one year’s worth of data. In particular, we can check how
often and in what way events tend to overlap, depending on their types.

| # of detections | $\mbox{SNR}_{\rm net}$ |  # with $\mbox{SNR}_{\rm net}>250$ |  # with $\mbox{SNR}_{\rm net}>100$ |  # with $\mbox{SNR}_{\rm net}>50$ |  # with $\mbox{SNR}_{\rm net}>20$
---|---|---|---|---|---|---
BBH |  |  |  |  |  |
Low rate | 53756 | $81.1_{-57.3}^{+94.2}$ | 3069 (5%) | 20605 (35%) | 40063 (68%) | 52239 (89%)
Median rate | 85725 | $81.3_{-57.5}^{+93.9}$ | 4972 (5%) | 33148 (39%) | 63958 (75%) | 83333 (97%)
High rate | 137225 | $81.5_{-57.4}^{+94.2}$ | 7860 (6%) | 53419 (39%) | 102766 (75%) | 133460 (97%)
BNS |  |  |  |  |  |
Low rate | 98898 | $19.2_{-4.9}^{+22.1}$ | 17 (0.017%) | 298 (0.30%) | 2712 (2.7%) | 44350 (48%)
Median rate | 396793 | $19.1_{-4.8}^{+22.0}$ | 73 (0.018%) | 1257 (0.32%) | 10659 (2.7%) | 177296 (45%)
High rate | 1004525 | $19.1_{-4.8}^{+22.1}$ | 196 (0.020%) | 3255 (0.32%) | 27135 (2.7%) | 448610 (45%)
Table 1: The number of events detected by a network of two CEs and one ET in
one year of simulated data, the median network SNRs and their 90% spreads, and
the detection numbers and percentages (in brackets) for different choices of
minimum network SNR.

###  II.2 Overlap estimates

The three different local merger rates give the following typical numbers of
events happening over one year, prior to imposing detectability thresholds:
$\sim 59000$, $93000$, $148000$ BBH events, and $286000$, $1145000$, $2900000$
BNS events for the low, median, and high local rate, respectively. The network
of two CEs and one ET will detect 93% of BBHs and 35% of BNSs. The number of
detected signals is shown in Table 1 for the three local rates, along with
median and 90% spreads on SNRs, and a breakdown of detections according to
their loudness.

Within our simulated catalogs of events, we can look at the numbers of
detected signals that overlap depending on the types. We focus on two
quantities: (i) the number of seconds in a year where at least two detected
signals have their merger, and (ii) the typical number of mergers that happen
during the time a given signal is in a detector’s sensitivity band.

The numbers of seconds in a year that have at least two mergers taking place
is given in Table 2; clearly this will happen frequently over the course of a
year. Indeed, we find that even more than two mergers can occur within the
same second. The proportion of detected signals merging together with at least
one other goes up with increasing local merger rate, potentially reaching
thousands per year.

Rate | BBH mergers > 1 | BNS mergers > 1 | Any mergers > 1
---|---|---|---
Low rate | 48 | 310 | 750
Median rate | 127 | 2412 | 7347
High rate | 303 | 15581 | 20149
Table 2:  The number of seconds in a year with at least two mergers occurring,
depending on their types.

In addition to the scenario where different compact binary mergers happen in
the same second, we investigate the typical number of mergers that will happen
over the entire duration of a BNS event while it is in band, depending on
their type; see Table 3 and Fig. 2. Because BNS events are in the detector
band for a long time (several hours for $f_{\rm low}=5$ Hz), quite a number of
such overlaps will indeed occur. If one does the same for BBHs, one finds that
either zero or one BBH or BNS merger (at 90% confidence) will happen in its
duration; this is due to BBH events being shorter (the median duration being
$\sim 45$ seconds).

Before moving on to parameter estimation issues, let us briefly look at other
future GW observatories that are being planned or considered. Constructing
simulated catalogs of detectable sources in the same way as above, and
focusing on the high local merger rate, we find that over the course of a
year, Advanced LIGO+ Abbott  _et al._ (2018b) will typically have no events
merging within the same second, and only a few occurrences of a BBH merging in
the duration of a BNS (assuming $f_{\rm low}=15$ Hz). For Voyager Adhikari
_et al._ (2020) we find $\mathcal{O}(1)$ instances of two events merging
within the same second, and BNS signals will typically have at most one other
signal’s merger in their duration (for $f_{\rm low}=10$ Hz). These numbers
refer to signals detectable with a single interferometer (with SNR threshold
8) rather than with a network of them, but it will be clear that overlapping
signals are going to become an important consideration mainly in the 3G era.

Rate | Number of | Number of | Number of
---|---|---|---
| BBH mergers | BNS mergers | any type
Low rate | $8_{-5}^{+10}$ | $16_{-8}^{+16}$ | $25_{-12}^{+23}$
Median rate | $13_{-7}^{+14}$ | $62_{-27}^{+58}$ | $76_{-33}^{+77}$
High rate | $21_{-11}^{+21}$ | $157_{-66}^{+144}$ | $178_{-75}^{+164}$
Table 3: Typical numbers of compact binary mergers happening during the time a
BNS signal is in band. Figure 2: Fraction of detected BNS mergers with a given
number of compact binary mergers (blue), BBH mergers (red), and BNS mergers
(green) taking place while the BNS signal is in band.

##  III Parameter estimation setup

Having established that third-generation detectors will see a considerable
number of overlapping signals whose mergers occur very close in time, we want
to find out what this will imply for parameter estimation. To this end, we
simulate BBH and BNS signals in a network consisting of one ET and two CE
observatories as in the previous section, assuming stationary, Gaussian noise
following the PSDs used above.

Run | BBH-BBH | BBH-BNS | BNS-BNS
---|---|---|---
Low rate | $5$ | $57$ | $416$
Median rate | $11$ | $304$ | $6752$
High rate | $15$ | $1594$ | $41306$
Table 4: Number of pairs of binary coalescence events with both SNRs between
15 and 30, and such that their mergers occur within 2 seconds or less from
each other.

Since we expect parameter estimation biases to be more pronounced when SNRs of
overlapping signals are similar to each other, and on the low side, we focus
on network SNRs roughly between 15 and 30. We consider overlapping events
whose merger times either coincide (as a proxy for merger within the same
second), or are separated by 2 seconds, again because these are the types of
scenarios where biases will likely be the largest. The number of overlaps from
the previous section that satisfy these criteria is given in Table 4, for
different local merger rates; we see that they will be fairly common.

In our parameter estimation studies, for definiteness we take the BBH events
to have masses similar to those of either GW150914 (a higher-mass, shorter-
duration signal) or GW151226 (a lower-mass, longer-duration event), while for
BNSs we take the masses to be similar to those of GW170817. Overlapping
signals are given different injected sky locations. All analyses are done with
3 different noise realizations. For each example of overlapping signals,
parameter estimation is also done on the individual signals, for the same
noise realizations, in order to assess what biases occur. Fig. 3 provides an
overview of the various overlap scenarios that will be considered in the rest
of this paper, in terms of masses and SNRs.

To reduce computational cost, we focus on non-spinning sources. A BBH signal
is then characterized by parameters
$\vec{\theta}=\\{m_{1},m_{2},\alpha,\delta,\iota,\psi,D_{\rm
L},t_{c},\varphi_{c}\\}$, where $m_{1}$, $m_{2}$ are the component masses,
$(\alpha,\delta)$ specifies the sky position in terms of right ascension and
declination, $\iota$ and $\psi$ are respectively the inclination and
polarization angles which specify the orientation of the orbital plane with
respect to the line of sight, $D_{\rm L}$ is the luminosity distance, and
$t_{c}$ and $\varphi_{c}$ are respectively the time and phase at coalescence.
BNS signals have two additional parameters $(\Lambda_{1},\Lambda_{2})$,
corresponding to the (dimensionless) tidal deformabilities Flanagan and
Hinderer (2008); Vines  _et al._ (2011); Damour  _et al._ (2012); Del Pozzo
_et al._ (2013); Agathos  _et al._ (2015).

In this work we focus specifically on potential biases in _intrinsic_
parameters. For BBHs, results will be shown for the total mass $M=m_{1}+m_{2}$
and mass ratio $q=m_{2}/m_{1}$ (with the convention $m_{2}\leq m_{1}$). For
BNSs, we show chirp mass $\mathcal{M}$ instead of total mass, since that
parameter is usually the best-determined one for long signals. As the
individual tidal deformabilities tend to be poorly measurable for the SNRs
considered here, we will be showing results for a parameter $\tilde{\Lambda}$
defined as Wade  _et al._ (2014)

$\tilde{\Lambda}=\frac{16}{13}\sum_{i=1,2}\Lambda_{i}\frac{m_{i}^{4}}{M^{4}}\left(12-11\frac{m_{i}}{M}\right),$
(9)

since this is how tidal deformabilities enter the waveform phase to leading
(5PN) order Flanagan and Hinderer (2008).

Figure 3: Individual waveforms and the overlap scenarios considered in our
simulations. All signals are injected in 3 different simulated noise
realizations for a third-generation detector network. Signals are either
overlapped using the same end time (blue waveforms), or 2 seconds earlier than
the “primary” signal’s end-time (orange waveforms). Top three panels: BNS
signals (top) with an SNR of 30, 20, or 15 being overlapped with either a
high-mass BBH signal (middle; GW150914-like) or a low-mass BBH signal (bottom;
GW151226-like). Middle panels: Overlapping waveforms in the case of two BBH
signals. The higher-mass BBH signal (bottom; GW150914-like) is overlapped with
the lower-mass BBH signal (top; GW151226-like). Bottom panels: Overlapping
waveforms in the case of two BNS signals.

In the Bayesian framework, all information about the parameters of interest is
encoded in the posterior _probability density function_ (PDF), given by Bayes’
theorem Veitch and Vecchio (2010):

$p(\vec{\theta}|\mathcal{H}_{s},d)=\frac{p(d|\vec{\theta},\mathcal{H}_{s})\,p(\vec{\theta}|\mathcal{H}_{s})}{p(d|\mathcal{H}_{s})},$
(10)

where $\vec{\theta}$ is the set of parameter values and $\mathcal{H}_{s}$ is
the hypothesis that a GW signal depending on the parameters $\vec{\theta}$ is
present in the data $d$. For parameter estimation purposes, the factor
$p(d|\mathcal{H}_{s})$, called the _evidence_ for the hypothesis
$\mathcal{H}_{s}$, is effectively set by the requirement that PDFs are
normalized. Assuming the noise to be Gaussian, the _likelihood_
$p(d|\vec{\theta},\mathcal{H}_{s})$ of obtaining data $d(t)$ given the
presence of a signal $h(t)$ is determined by the proportionality

$p(d|\vec{\theta},\mathcal{H}_{s})\propto\exp{\left[-\frac{1}{2}(d-h(\vec{\theta})|d-h(\vec{\theta}))\right]},$
(11)

where the noise-weighted inner product $(\,\cdot\,|\,\cdot\,)$ is defined as

$(a|b)=4\Re\int_{f_{\rm low}}^{f_{\rm
high}}\frac{\tilde{a}^{\ast}(f)\,\tilde{b}(f)}{S_{h}(f)}\,df.$ (12)

Here a tilde refers to the Fourier transform, and $S_{h}(f)$ is the PSD, as in
the previous section. Due to computational limitations, in our parameter
estimation studies we use a lower frequency cut-off of $f_{\mathrm{low}}=23$
Hz. Since both ET and CE will be sensitive down to lower frequencies than
that, we expect that our choice will lead to conservative estimates of
parameter estimation biases, as the same signal will in reality accumulate
more SNR when it is visible in the detector already from a lower frequency.

Our choices for the _prior probability density_
$p(\vec{\theta}|\mathcal{H}_{s})$ in Eq. (10) are similar to what has been
used for the analyses of real data when BBH or BNS signals were present with
masses similar to the ones specified in Fig. 3. In all cases we sample
uniformly in component masses. For the GW150914-like signals, we do this in
the range $m_{1},m_{2}\in[10,80]\,M_{\odot}$. For analyzing the GW151226-like
signals, the component mass range is $m_{1},m_{2}\in[3,54.4]\,M_{\odot}$, and
in addition we restrict chirp mass to $\mathcal{M}\in[5,20]\,M_{\odot}$ and
mass ratio $q$ to the range $[0.05,1]$. For BNSs we sample component masses in
the range $m_{1},m_{2}\in[1,2]\,M_{\odot}$, restricting
$\mathcal{M}\in[0.7,2]\,M_{\odot}$, while tidal deformabilities are sampled
uniformly in the range $\Lambda_{1},\Lambda_{2}\in[0,5000]$. When we show PDFs
for the derived quantity $\tilde{\Lambda}$, they will have been reweighted
with the prior probability distribution of this parameter induced by the flat
priors on component masses and $\Lambda_{1}$, $\Lambda_{2}$, such as to
effectively have a uniform prior on $\tilde{\Lambda}$.

To sample the likelihood function in Eq. (11), we use the LALInference library
Veitch  _et al._ (2015), and specifically the lalinference_mcmc algorithm. The
waveforms we use for the BNS and BBH signals are IMRPhenomD_NRTidalv2 Dietrich
_et al._ (2017, 2019a, 2019b) and IMRPhenomD Husa  _et al._ (2016); Khan  _et
al._ (2016) respectively, both computed with the waveform library
LALSimulation. To inject the signals and add noise to them, we use standard
tools available within the LALSimulation package. All these codes are openly
accessible in LALSuite LIGO Scientific Collaboration (2018).

Before performing parameter estimation, we verify the detectability of the
individual signals in the overlap scenarios of Fig. 3 using the PyCBC software
package Biwer  _et al._ (2019). We inject overlapping signals in noise
generated from the PSD and check that the individual signals show up as
triggers with masses that are consistent between detectors, at a network SNR
above a threshold of 8. This turns out to be true for all the cases
considered, except for two BBH signals merging at the same time. In the latter
case we still have triggers in individual detectors, but with masses differing
by up to $\sim 5M_{\odot}$. Using the SNRs in single detectors as detection
statistics, detection is still achieved. For all scenarios, the end times of
individual signals tend to be identified with a precision of a few
milliseconds Van Den Broeck and Sengupta (2007); when subsequently performing
parameter estimation, we use a prior range for end time that is centered on
the true end time, leaving an interval of 0.1 s on either side.

For parameter estimation, all simulations are done with three different noise
realizations. In the next section, results are shown for one of those; for the
other two noise realizations, see Appendix A.

As usual, the one-dimensional PDF $p(\lambda|\mathcal{H}_{s},d)$ for a
particular parameter $\lambda$ is obtained from the joint PDF
$p(\vec{\theta}|\mathcal{H}_{s},d)$ by integrating out all other parameters.
In assessing the effect on parameter estimation of signals that overlap in
various ways, we will frequently be comparing one-dimensional PDFs for the
same parameter in different situations. A convenient way of quantifying the
difference between two distributions $p_{1}(\lambda)$ and $p_{2}(\lambda)$ is
by means of the _Kolmogorov-Smirnov (KS) statistic_ Kolmogorov (1933); Smirnov
(1948). Let $P_{1}(\lambda)$, $P_{2}(\lambda)$ be the associated _cumulative_
distributions; then the KS statistic is just the largest distance between
these two:

$\mbox{KS}=\mbox{sup}_{\lambda}|P_{1}(\lambda)-P_{2}(\lambda)|.$ (13)

By construction, this yields a number between 0 and 1; if the KS statistic is
close to zero, then the distributions $p_{1}(\lambda)$ and $p_{2}(\lambda)$
will be considered close to each other.

##  IV Results

Figure 4: Posterior PDFs showing estimation of intrinsic parameters when the BNS signal has SNR = 30 (top row), SNR = 20 (middle row), and SNR = 15 (bottom row). Results are shown for the cases when the GW150914-like signal ends at the same time as the BNS signal (GW150914-tc), when it ends 2 seconds earlier (GW150914-tc-2), when the GW151226-like signal ends at the same time as the BNS (GW151226-tc), when it ends 2 seconds earlier (GW151226-tc-2), and finally when the injected signal is only the BNS (BNS). The true values of the parameters are indicated by vertical, black lines.  BBH overlapped | BNS (SNR = 30) | BNS (SNR = 20) | BNS (SNR = 15)
---|---|---|---
| $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$
GW150914-tc | 0.0112 | 0.00915 | 0.0277 | 0.0162 | 0.0204 | 0.0275 | 0.0297 | 0.0323 | 0.00947
GW150914-tc-2 | 0.0320 | 0.0389 | 0.0168 | 0.0235 | 0.0273 | 0.0331 | 0.0704 | 0.0840 | 0.0218
GW151226-tc | 0.00754 | 0.00748 | 0.0113 | 0.0123 | 0.0139 | 0.0173 | 0.0403 | 0.0516 | 0.0305
GW151226-tc-2 | 0.0187 | 0.0220 | 0.0309 | 0.0227 | 0.0233 | 0.0259 | 0.0521 | 0.0513 | 0.0159
Table 5: Values of the KS statistic comparing PDFs for BNS parameters (columns) in the BNS+BBH overlap scenarios (rows) with the corresponding PDFs when there is no overlapping BBH signal. The small numbers indicate absence of significant bias. The numbers shown here correspond to the PDFs in Fig. 4. Figure 5: Posterior PDFs for total mass and mass ratio, for the GW150914-like signal (top panel) and the GW151226-like signal (bottom panel) when they are respectively being overlapped with a BNS signal of SNR = 30 (solid lines), SNR = 20 (dashed lines), and SNR = 15 (dotted lines). The overlaps are being done when the BBH and the BNS end at the same time (tc), and when the BBH ends 2 seconds before the BNS (tc-2). Finally, posterior PDFs for the two BBH signals by themselves are shown as green, dashed-dotted lines (BBH). The injected parameter values are indicated by black, vertical lines. BNS overlapped | GW150914-tc | GW150914-tc-2 | GW151226-tc | GW151226-tc-2
---|---|---|---|---
| $M$ | $q$ | $M$ | $q$ | $M$ | $q$ | $M$ | $q$
BNS (SNR = 15) | – | – | 0.0504 | 0.0807 | 0.00933 | 0.0117 | 0.0687 | 0.0657
BNS (SNR = 20) | – | – | 0.0427 | 0.0698 | 0.0107 | 0.0106 | 0.0727 | 0.0700
BNS (SNR = 30) | – | – | 0.0379 | 0.0673 | 0.0187 | 0.183 | 0.0819 | 0.0793
Table 6: Values of the KS statistic comparing PDFs for BBH parameters
(columns) in the BNS+BBH overlap scenarios (rows) with the corresponding PDFs
when there is no overlapping BNS signal. In the case of a GW150914-like signal
merging at the same time as a BNS, the sampler fails to find the signal, but
other scenarios are not so problematic. For GW151226, the slightly higher
values for the tc-2 case compared to the tc case are likely due to the signals
being placed in a slightly different part of the noise stream (two seconds
earlier) from the BBH-only cases that are used for comparison. The numbers
shown here correspond to the PDFs in Fig. 5.

###  IV.1 Overlap of a BNS signal with a BBH signal

First we look at the results of parameter estimation for the overlap of a BNS
signal with a BBH, either ending at the same time or with the BBH signal
ending 2 seconds earlier than the BNS. This is the scenario shown in the top
panels of Fig. 3. We perform parameter estimation first on the BNS and then on
the BBH, with priors as specified in the previous section.

####  IV.1.1 BNS recovery

Fig. 4 shows posterior probability distributions for intrinsic parameters
characterizing the BNS signal, for 3 different SNRs of the BNS, and the
different overlap scenarios. The PDFs tend to widen with decreasing SNR, as
expected. We see that estimation of the mass parameters are essentially
unaffected, regardless of the type of overlapping BBH signal (GW150914-like or
GW151226-like) or of its merger time (identical to that of the BNS, or 2
seconds earlier). For a given SNR of the BNS, the PDFs for the tidal parameter
$\tilde{\Lambda}$ differ slightly more between the overlap scenarios. However,
we note that most of the information on tides enters the signal at high
frequencies, where the detectors are less sensitive; and in fact, as shown in
Appendix A (Fig. 8), differences in the underlying noise realization tend to
have a larger effect on the measurement of $\tilde{\Lambda}$ than overlapping
signals.

We conclude that an overlapping BBH signal does not have much impact on the
estimation of the BNS parameters, even if the BBH merger time is arbitrarily
close to that of the BNS. This is corroborated by the KS statistics in Table
5, which compare PDFs for the various overlap scenarios with the corresponding
PDFs in the absence of overlapping signals. It is reasonable to assume that
placing a BBH signal even earlier in the BNS would also have had little
impact.

####  IV.1.2 BBH recovery

Figure 5 shows parameter estimation on the BBHs when the SNR of the BNS signal
is varied from 30, to 20, to 15. Table 6 has the corresponding KS statistics
comparing with PDFs obtained in the absence of overlap. Again results are
shown for a particular noise realization; see Fig. 9 in Appendix A for two
other noise realizations. We see that when the BBH signal has a time of
coalescence 2 seconds earlier than the BNS (tc-2 in the figure), the signal is
well recovered. However, when the BBH signal and the BNS signal end at the
same instant of time, the BBH recovery deteriorates, and in the case of the
GW150914-like signal, the sampling process in fact fails to find the signal.
For the GW151226-like signal, while the estimates are offset from their true
values, there is some measurability of the signal when the times of
coalescence of the BBH and BNS are the same. The different outcomes between
the GW150914-like and GW151226-like injection are likely due to the short
duration of the GW150914-like signal, effectively leading to a distortion of
the entire signal when the merger happens at the same instant as the BNS
merger. By contrast, the much longer inspiral of the GW151226-like signal
implies many more wave cycles for the parameter estimation algorithm to latch
on to. Finally, as the SNR over the underlying BNS signal is varied (keeping
the SNR of the BBH signal the same), the PDFs for the BBH show essentially no
change. Placing a BBH signal only 2 seconds before the BNS merger causes the
BBH to be recovered without appreciable biases, so it is reasonable to assume
that placing a BBH signal still earlier in the BNS inspiral would also have
little effect on its recovery.

###  IV.2 Overlap of 2 BBH signals

Figure 6: Posterior PDFs for total mass and mass ratio when a GW150914-like signal and a GW151226-like signal are being overlapped at the same trigger times (tc) and when the trigger time of the GW150914-like BBH ends 2 seconds earlier (tc-2), compared with parameter estimation in the absence of overlap (BBH). The top panel shows the recovery of the GW150914-like signal and the bottom one that of the GW151226-like signal. Black vertical lines indicate the true values of the parameters. GW150914-tc | GW150914-tc-2 | GW151226-tc | GW151226-tc-2
---|---|---|---
$M$ | $q$ | $M$ | $q$ | $M$ | $q$ | $M$ | $q$
0.0195 | 0.0109 | 0.162 | 0.103 | 0.0446 | 0.0478 | 0.0844 | 0.127
Table 7: Values of the KS statistic comparing PDFs for BBH parameters in the
BBH+BBH overlap scenarios with the corresponding PDFs without an overlapping
signal. The slightly higher values for the tc-2 cases are likely due to the
signals being in a slightly different part of the noise stream (two seconds
earlier) from the BBH-only cases used for comparison. However, in all cases
there is no significant bias. The numbers shown here correspond to the PDFs in
Fig. 6.

The scenario being analyzed here is the one in the middle panels of Fig. 3.
Fig. 6 shows the posterior PDFs on total mass $M$ and mass ratio $q$ when two
BBH signals of different masses are being overlapped, compared with parameter
estimation on the same signals in situations where there is no overlap (BBH).
The corresponding KS statistic values are given in Table 7. We find the
results to be consistent within statistical fluctuations. Here too, the
signals are overlapped once with the same coalescence times (tc), and once
with one of the signals, GW150914, ending 2 seconds earlier (tc-2). The SNRs
of the two signals, GW150914-like, and GW151226-like, are 30 and 15,
respectively. As can be seen in the Figure, the two BBH signals’ parameters
can be extracted without any biases even when they end simultaneously. Again
see Appendix A for other noise realizations, with the same conclusion.

###  IV.3 Overlap of 2 BNS signals

Figure 7: Posterior PDFs showing recovery on chirp mass, mass ratio and tidal deformability $\tilde{\Lambda}$ when two BNSs, referred to as BNS1 and BNS2, are being overlapped at the same time of coalescence (tc) and when BNS2 ends 2 seconds earlier than BNS1 (tc-2). These are compared with results in the absence of overlap (BNS). The top panel is for the recovery of BNS1 and the bottom one for the recovery of BNS2. The solid black vertical lines indicate the injected values of the source being recovered each time. We note that when the times of coalescence of the two BNSs are the same, the parameter estimates recovered are those of BNS1, whose injected values are also shown in the bottom panel, as dashed vertical black lines.  BNS1 (tc) | BNS1 (tc-2) | BNS2 (tc) | BNS2 (tc-2)
---|---|---|---
$\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$
0.269 | 0.270 | 0.202 | 0.0309 | 0.0216 | 0.0129 | 1.0 | 0.955 | 0.0762 | 0.384 | 0.0951 | 0.368
Table 8: Values of the KS statistic comparing PDFs for BNS parameters in the
BNS+BNS overlap scenarios with the corresponding PDFs without an overlapping
signal; see also Fig. 7. We see that the numbers are higher for both BNSs when
they end at the same time; in fact, the measured parameters for BNS2 are those
of BNS1. However, when the BNSs merge 2 seconds apart, the values are much
lower, showing that the biases largely disappear. The numbers shown here
correspond to the PDFs in Fig. 7.

Finally, we analyze the simulations in the bottom panels of Fig. 3. Figure 7
shows the recovery of BNS parameters for each BNS signal when two BNS signals
are being overlapped, again with either the same coalescence times and when
one of the BNSs (henceforth BNS2) ends 2 seconds earlier than the other BNS
signal (henceforth BNS1). For KS statistic values comparing PDFs with the
corresponding non-overlapping cases, see Table 8. BNS1 and BNS2 respectively
have SNRs of 30 and 20, and component masses
$(m_{1},m_{2})=(1.68,1.13)\,M_{\odot}$ and
$(m_{1},m_{2})=(1.38,1.37)\,M_{\odot}$. These particular choices cause both
signals to have very similar chirp masses. Given these masses, their tidal
deformabilities, $\tilde{\Lambda}=303$ for BNS1 and $\tilde{\Lambda}=292$ for
BNS2, follow the equation of state APR4; these were the simulated signals used
for investigating systematics in the measurements on GW170817 in Ref. Abbott
_et al._ (2019e).

In Fig. 7, the top panel shows the posterior PDFs on chirp mass, mass ratio,
and tidal deformability for BNS1 when BNS2 ends at the same time (tc) and when
BNS2 ends 2 seconds earlier (tc-2), together with the case where only BNS1 is
present in the data (BNS). The bottom panels show the same, but for the
recovery of BNS2. When the two signals end at the same time, the parameters
characterizing BNS1 are being recovered, which likely happens because of the
higher SNR of BNS1. As the tidal deformabilities of the two sources are so
close, the PDFs for $\tilde{\Lambda}$ look similar in all cases. However, also
looking at the mass parameters, parameter estimation is rather robust when the
signals end 2 seconds apart.

##  V Conclusions

Given regular improvements in the sensitivity of gravitational-wave detectors
and especially the planned construction of the next generation of
interferometers, it will become increasingly likely that individually
detectable gravitational-wave signals will end up overlapping in the data. In
this paper we (i) assessed how often different types of overlap will happen in
ET and CE, and (ii) tried to quantify the impact this would have on parameter
estimation with current data analysis techniques.

To address the question of the nature and frequency of different overlap
scenarios, for each of three possible local merger rates, we constructed a
“catalog” of signals in ET and CE, enabling a more in-depth study of overlaps
than in previous works. We showed that there will be a significant number of
signals for which the merger happens within the same second, varying from tens
to thousands depending on the local merger rate. Additionally, the substantial
increase in the duration of BNS events due to the improved low frequency
sensitivity of third generation observatories will lead to the occurrence of
up to tens of other signals overlapping with a given BNS.

Motivated by these results, we performed the first detailed Bayesian analysis
study on possible biases that may arise in future as detection rates become
higher and overlapping signals start to occur. We focused on overlapping
signals for which the end times were close to each other, so that in
particular there is overlap at times where both signal amplitudes are high; it
is this type of situation where we expect parameter estimation biases to be
the most pronounced. Specifically, merger times were taken to be either the
same (as a proxy for being arbitrarily close to each other), or separated by 2
seconds. Our preliminary conclusions (based on a limited number of
investigations) are as follows:

  * •

When BBH signals are overlapping with a BNS signal of similar SNR, parameter
estimation on the BNS is hardly affected, even with the merger time of the BBH
arbitrarily close to that of the BNS. Presumably this is due to the much
larger number of BNS wave cycles in band compared to the BBH.

  * •

However, in the same scenario, parameter estimation on the BBH can be subject
to significant biases if the BBH is high-mass, so that its signal is short.
That said, the problem largely disappears when the BNS and BBH merger times
are separated by 2 seconds, or when the BBH is low-mass.

  * •

When two BBHs with sufficiently dissimilar masses overlap with close-by merger
times, parameter estimation on either of the signals will not be much
affected.

  * •

When two BNS signals overlap with close-by merger times, parameter estimation
will recover the louder signal reasonably well. With a 2 second separation of
merger times, good-quality parameter estimation can already be done on the two
signals separately.

These results suggest that current parameter estimation techniques will, in
several types of situations of interest, already perform reasonably well in
the 3G era when applied to overlapping signals, even when the individual
signals have similar SNRs, and even when the SNRs are on the low side given
the projected distribution for these observatories. Nevertheless, a number of
questions remain. What happens when SNRs are gradually increased? Related to
this is the choice of lower cut-off frequency; to what extent will parameter
estimation improve as one goes to $f_{\rm low}=5$ Hz or even lower, so that
signals have a much larger number of wave cycles in the detector’s sensitive
band? Though not the focus here, at higher SNRs the use of currently available
waveform approximants to analyze BNS signals in 3G detectors would lead to
biases in the estimation of $\tilde{\Lambda}$ even in the absence of overlap
Samajdar and Dietrich (2018), also motivating further research in waveform
modeling. Spins were not included in our study, but it would be of interest to
see their effect: large precessing spins will complicate parameter estimation
in the case of BBHs, while for BNSs, having access to the spin-induced
quadrupole moment can aid in determining tidal deformabilities Samajdar and
Dietrich (2019). Finally, what happens when overlaps involve (much) more than
two signals, e.g. a long BNS signal overlapping with a large number of BBH
signals? These questions are left for future work.

In order to make optimal scientific use of the capabilities of 3G detectors,
it will be appropriate to develop Bayesian parameter estimation techniques for
which the likelihood function assumes multiple signals to be present in a
given stretch of data, e.g. replacing Eq. (11) by

$\displaystyle p(d|\\{\vec{\theta}_{i}\\},\mathcal{H}_{s})$
$\displaystyle\propto\exp{\left[-\frac{1}{2}\bigg{(}d-\sum_{i=1}^{N}h(\vec{\theta_{i}})\bigg{|}d-\sum_{i=1}^{N}h(\vec{\theta}_{i})\bigg{)}\right]},$
(14)

with $N$ the number of signals found by a detection pipeline, and
$\vec{\theta}_{i}$, $i=1,\ldots,N$ the associated parameters. Additionally,
one could let $N$ itself be a parameter to be sampled over, thus allowing for
an a priori unknown number of signals in the given stretch of data. In all
this, it may be possible to borrow from techniques developed in the context of
somewhat related problems in GW data analysis, such as the characterization of
the large number of (in this case near-monochromatic) signals from galactic
white dwarf binaries in the space-based LISA Crowder and Cornish (2004);
Cornish and Crowder (2005); Cornish and Porter (2007); Crowder and Cornish
(2007); Littenberg (2011); Robson and Cornish (2017); Littenberg  _et al._
(2020), BNSs in BBO Cutler and Harms (2006), or supermassive black hole
binaries in pulsar timing searches Petiteau  _et al._ (2013).

###### Acknowledgements.

We are grateful to Elia Pizzati, Surabhi Sachdev, Anuradha Gupta, and
Bangalore Sathyaprakash for sharing and discussing their results on a related
study. A.S., J.J, and C.V.D.B are supported by the research programme of the
Netherlands Organisation for Scientific Research (NWO). The authors are
grateful for computational resources provided by the LIGO Laboratory and
supported by the National Science Foundation Grants No. PHY-0757058 and No.
PHY-0823459. 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.

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##  Appendix A Parameter estimation for different noise realizations

We have performed all our simulations in three different noise realizations.
To avoid plots getting too busy, in Sec. IV we only showed results for one of
these; here we also give them for the other two noise realizations.

In the case of a BNS overlapping with a BBH, the measurements on the BNS are
shown in Fig. 8 and those on the BBH in Fig. 9. The corresponding KS values
are given in Tables 9 and 10, respectively. For measurements on the mass
parameters of the BNS, we find that the results are consistent between noise
realizations. For the tidal parameter $\tilde{\Lambda}$, the PDFs differ
somewhat more; compare the right columns in the two panels of Fig. 8. This is
likely because most of the information on tides enters the signal at higher
frequencies, where the variance of the noise is larger; hence the measurement
of $\tilde{\Lambda}$ will be more affected by the noise realization than the
mass measurements, especially when SNRs are not high. Indeed, though not shown
here explicitly, KS statistics for $\tilde{\Lambda}$ between different noise
realizations, but for the same overlap situation, tend to be significantly
larger than within the same noise realization but for different overlaps. For
parameter estimation on the BBH, there are differences in the PDFs for the
masses when the BBH merger time coincides with that of the BNS, but not so
much if it occurs 2 seconds earlier.

In the case of two overlapping BBH signals, parameter estimation results are
shown in Fig. 10, and KS statistics in Table 11. The results are quite robust
under a change of noise realization.

Finally, the case of two overlapping BNSs with different noise realizations is
shown in Fig. 11, and KS statistics in Table 12. As in the case of a BNS
overlapping with a BBH, the PDFs for the masses are not much affected by
differences in noise, but the ones for $\tilde{\Lambda}$ are more susceptible.

Figure 8: Posterior PDFs for BNS parameters when a BNS and BBH signal are being overlapped; same as Fig. 4 when injections are done in two other noise realizations (left and right panels). BBH overlapped | BNS (SNR = 30) | BNS (SNR = 20) | BNS (SNR = 15)
---|---|---|---
_Noise realization 2_ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$
GW150914-tc | 0.0267 | 0.0248 | 0.0224 | 0.0106 | 0.0141 | 0.0169 | 0.0146 | 0.0211 | 0.0290
GW150914-tc-2 | 0.0287 | 0.0282 | 0.0338 | 0.00601 | 0.0108 | 0.0486 | 0.0263 | 0.0308 | 0.0137
GW151226-tc | 0.0125 | 0.0141 | 0.0421 | 0.0376 | 0.0471 | 0.0723 | 0.0155 | 0.0152 | 0.0333
GW151226-tc-2 | 0.0337 | 0.0346 | 0.0815 | 0.0244 | 0.0258 | 0.0179 | 0.0113 | 0.0108 | 0.00923
_Noise realization 3_ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$
GW150914-tc | 0.0140 | 0.0143 | 0.0251 | 0.0236 | 0.0298 | 0.0481 | 0.0114 | 0.0255 | 0.0378
GW150914-tc-2 | 0.0296 | 0.0396 | 0.0255 | 0.0272 | 0.0218 | 0.0125 | 0.0186 | 0.0125 | 0.0299
GW151226-tc | 0.0135 | 0.0161 | 0.0347 | 0.0215 | 0.0312 | 0.0412 | 0.00750 | 0.00868 | 0.0239
GW151226-tc-2 | 0.0142 | 0.0140 | 0.0334 | 0.0109 | 0.00833 | 0.0310 | 0.0223 | 0.0292 | 0.0169
Table 9: Values of the KS statistic comparing PDFs for BNS parameters
(columns) in the BNS+BBH overlap scenarios (rows) with the corresponding PDFs
when there is no overlapping BBH signal, when injections are done in two other
noise realizations. The numbers shown correspond to the PDFs in Fig. 8, noise
realisation 2 corresponding to the left panel and noise realisation 3 to the
right panel.

Figure 9: Posterior PDFs for BBH parameters when a BNS and BBH signal are being overlapped; same as Fig. 5 when injections are done in two other noise realizations. BNS overlapped | GW150914-tc | GW150914-tc-2 | GW151226-tc | GW151226-tc-2
---|---|---|---|---
_Noise realization 2_ | $M$ | $q$ | $M$ | $q$ | $M$ | $q$ | $M$ | $q$
BNS (SNR = 15) | – | – | 0.0134 | 0.011 | 0.00832 | 0.00890 | 0.411 | 0.398
BNS (SNR = 20) | – | – | 0.0104 | 0.0109 | 0.0169 | 0.0172 | 0.390 | 0.377
BNS (SNR = 30) | – | – | 0.0100 | 0.0113 | 0.0140 | 0.0146 | 0.367 | 0.357
_Noise realization 3_ | $M$ | $q$ | $M$ | $q$ | $M$ | $q$ | $M$ | $q$
BNS (SNR = 15) | – | – | 0.0168 | 0.0100 | 0.0140 | 0.0142 | 0.318 | 0.131
BNS (SNR = 20) | – | – | 0.0189 | 0.0131 | 0.0132 | 0.0137 | 0.322 | 0.315
BNS (SNR = 30) | – | – | 0.0287 | 0.295 | 0.0136 | 0.0130 | 0.334 | 0.327
Table 10: Values of the KS statistic comparing PDFs for BBH parameters
(columns) in the BNS+BBH overlap scenarios (rows) with the corresponding PDFs
when there is no overlapping BNS signal, when injections are done in two other
noise realizations. As before, when the GW150914-like signal ends at the same
time as a BNS, it is not found by the sampling algorithm, but other scenarios
are less problematic. The numbers shown correspond to the PDFs in Fig. 9,
noise realisation 2 corresponding to the left panel and noise realisation 3 to
the right panel.

Figure 10: Posterior PDFs for BBH parameters when a BNS and BBH signal are being overlapped; same as Fig. 6 when injections are done in two other noise realizations. GW150914-tc | GW150914-tc-2 | GW151226-tc | GW151226-tc-2
---|---|---|---
$M$ | $q$ | $M$ | $q$ | $M$ | $q$ | $M$ | $q$
0.0195 | 0.00854 | 0.163 | 0.0395 | 0.0299 | 0.0309 | 0.0417 | 0.0746
$M$ | $q$ | $M$ | $q$ | $M$ | $q$ | $M$ | $q$
0.0291 | 0.0110 | 0.188 | 0.0625 | 0.0477 | 0.0497 | 0.0225 | 0.0440
Table 11: Values of the KS statistic comparing PDFs for BBH parameters in the
BBH+BBH overlap scenarios with the corresponding PDFs for the BBH-only case,
when injections are done in two other noise realizations. The numbers shown
correspond to the PDFs in Fig. 10, the upper row corresponding to the left
panel and the lower row to the right panel.

Figure 11: Posterior PDFs when two BNS signals are being overlapped; same as Fig. 7 when injections are done in two other noise realizations. BNS1 (tc) | BNS1 (tc-2) | BNS2 (tc) | BNS2 (tc-2)
---|---|---|---
$\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$
0.316 | 0.282 | 0.0743 | 0.0385 | 0.0339 | 0.0325 | 1.0 | 0.936 | 0.382 | 0.0271 | 0.0858 | 0.248
$\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$ | $\mathcal{M}$ | $q$ | $\tilde{\Lambda}$
0.278 | 0.257 | 0.123 | 0.0475 | 0.0381 | 0.0630 | 1.0 | 0.902 | 0.341 | 0.226 | 0.101 | 0.128
Table 12: Values of the KS statistic comparing PDFs for BNS parameters in the
BNS+BNS overlap scenarios with the corresponding PDFs for the BNS-only case,
when injections are done in two other noise realizations. The numbers shown
correspond to the PDFs in Fig. 11, the upper row corresponding to the left
panel and lower row to the right panel.