Source confusion from neutron star binaries in ground-based gravitational
wave detectors is minimal
Aaron D. Johnson ,
1
,*
Katerina Chatziioannou,
1,2
,
†
and Will M. Farr
3,4
,
‡
1
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
2
LIGO Laboratory, California Institute of Technology, Pasadena, California 91125, USA
3
Center for Computational Astrophysics, Flatiron Institute,
162 5th Avenue, New York, New York 10010, USA
4
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA
(Received 8 February 2024; accepted 7 March 2024; published 9 April 2024)
Upgrades beyond the current second generation of ground-based gravitational wave detectors will
allow them to observe tens of thousands neutron star
and black hole binaries. Given the typical minute-
to-hour duration of neutron star signals in the det
ector frequency band, a number of them will overlap in
the time-frequency plane, resulting in a
nonzero cross-correlation. We examine
“
source confusion
”
arising from overlapping signals whose time-frequency tracks cross. Adopting the median observed
merger rate of
100
Gpc
−
3
yr
−
1
, each neutron star binary signal overlaps with an average of 42(4)[0.5]
other signals when observed from 2(5)[10] Hz. The va
st majority of overlaps occur at low frequencies
where the inspiral evolution is slow: 91% of time-
frequency overlaps occur in band below 5 Hz. The
combined effect of overlapping signals does not sa
tisfy the central limit theorem and source confusion
cannot be treated as stationary, Gaussian noise: on av
erage 0.91(0.17)[0.05]
signals are present in a
single adaptive time-frequency bin centered at 2
(5)[10] Hz. We quantify source confusion under a
realistic neutron star binary population and find that
parameter uncertainty typically increases by less
than 1% unless there are overlapping signals wh
ose detector-frame chirp mass difference is
≲
0
.
01
M
⊙
and the overlap frequency is
≳
40
Hz. Out of
1
×
10
6
simulated signals, 0.14% fall within this region of
detector-frame chirp mass differences, but their o
verlap frequencies are typically lower than 40 Hz.
Source confusion for ground-based detectors, where
events overlap instantaneously, is significantly
milder than the equivalent Laser In
terferometer Space Antenna problem, where many classes of events
overlap for the lifetime of the mission.
DOI:
10.1103/PhysRevD.109.084015
I. INTRODUCTION
Planned improvements and upgrades of ground-based
gravitational wave (GW) detectors will expand both their
detection horizon and their sensitive frequency range
[1
–
3]
.
The expanded horizon leads to detection of binary neutron
stars (BNSs) and binary black holes (BBHs) to larger
distances, thus increasing the detection rate by orders of
magnitude. The increased bandwidth leads to observation
times that reach hours and minutes for BNSs and BBHs,
respectively. The combined outcome of these two effects is
that multiple signals will overlap in time and frequency in
the data streams, leading to source confusion. As discussed
in
[4,5]
and proven analytically in Appendix
C
, however,
the relevant condition is not whether two signals overlap in
time or frequency
only
, but rather whether they overlap
simultaneously
in both, i.e., if their time-frequency tracks
cross. We therefore define
“
overlapping signals
”
as those
whose time-frequency tracks cross, resulting in a nonzero
cross-correlation.
1
“
Signal confusion
”
is then the effect of
overlapping signals on inference.
Overlapping signals is not a new problem for GW
astronomy. The planned Laser Interferometer Space
Antenna (LISA) mission
[6]
will observe (among other
sources) tens of millions of Galactic white dwarf binaries,
thousands of which will be individually resolvable with the
rest contributing to the unresolvable Gaussian noise
[7]
.
However, the ground-based and LISA overlapping source
*
aaronj@caltech.edu
†
kchatziioannou@caltech.edu
‡
wfarr@flatironinstitute.org
1
The cross-correlation is defined as the noise-weighted inner
product between two signals, Eq.
(12)
. In Appendix
C
we
analytically prove under the stationary phase approximation
(SPA) that the integral is nonzero if and only if two signals overlap
in time and frequency simultaneously. This integral is sometimes
also referred to as the
“
overlap integral
”
leading to the confusing
definition: overlapping signals are those whose overlap is nonzero.
PHYSICAL REVIEW D
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=
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© 2024 American Physical Society
problems are not identical. The vast majority of LISA
’
s
white dwarf binaries have negligible frequency evolution
during the mission lifetime. As a result, two signals that
overlap in frequency at one time will continue doing so
practically indefinitely. BNSs as observed by ground-based
detectors, on the other hand, are transient sources with
strong frequency evolution. Two signals that overlap
temporally over a long time will only overlap in both time
and frequency
instantaneously
. Moreover, the frequency
evolution is faster at higher frequencies, suggesting that
most overlaps occur at low frequencies. While one LISA
white dwarf binary overlaps with another binary indefi-
nitely, one ground-based BNS overlaps with a large number
of BNSs each momentarily and preferentially at lower
frequencies. The latter resembles more the case of a single
massive BBH that overlaps with multiple white dwarf
binaries as it sweeps through the LISA frequency band
[8]
.
This general picture suggests that source confusion
is qualitatively different across detectors and astro-
physical sources. Through a Fisher formalism, Crowder
and Cornish
[9]
showed that inference accuracy for a single
white dwarf binary in LISA deteriorates
exponentially
with the number of overlapping sources. Moving up to
the decihertz range, Cutler and Harms
[4]
showed that BNS
source confusion in the Big Bang Observer instead grows
as the
square root
of the number of overlapping sources.
The scaling difference is exactly due to the fact that white
dwarf binaries overlap in the time-frequency domain over a
long time
[9]
, while frequency evolution makes BNS
overlaps momentary
[4]
. Each BNS time-frequency inter-
section happens at a random phase; it is therefore a random
walk that adds incoherently. A separate but related question
is whether overlapping signals add up to Gaussian noise.
Racine and Cutler
[10]
argued that the answer depends both
on the number of sources and on the type of signal we are
targeting on top of all other signals. The latter determines
how far in the tails of the noise distribution we have to go for
detection, i.e., to what
σ
level the central limit theorem has to
be satisfied. In the context of LISA, BBH signals are
“
sufficiently different
”
from white dwarf binaries that source
confusion can indeed be treated as Gaussian noise
[10]
.
Moving further up to the ground-based detector fre-
quency range, a comparison between the astrophysical rate
and observable duration reveals that multiple BNSs will be
simultaneously present in the data time series
[11
–
15]
.
Since signals, however, overlap mostly at low frequencies
and
“
separate
”
as they approach merger, current detection
techniques can identify them
[16
–
20]
and measure their
coalescence time to
O
ð
10
Þ
ms
[17]
. Ignoring the presence
of overlapping signals can lead to parameter biases for
two signals that merge sufficiently close
[5,13
–
15,21]
.
Considering more or louder signals
[22]
and going beyond
masses and aligned spins
[5,13
–
15,21]
would likely
increase biases. For example, inference of more subtle
effects such as spin precession
[15]
, tests of general
relativity
[18,22,23]
, and the neutron star (NS) equation
of state could be more severely affected by violations of
the assumption that the data are consistent with Gaussian
noise
[24,25]
. Quantitative conclusions about source con-
fusion are, however, complicated by the fact that the relevant
picture for ground-based detectors is time-frequency over-
laps of multiple signals, rather than temporal coincidences
between two signals as adopted in
[13
–
15]
. To emphasize
this distinction, we refer to overlapping signals as those
whose time-frequency tracks cross and coinciding signals as
those that exist simultaneously in the data stream.
In this study, we revisit overlapping signals in ground-
based detectors and quantify source confusion. We restrict
ourselves to BNSs that are expected to be the most
numerous and long-lasting binary source, thus resulting
in more overlapping signals. While BBH signals may suffer
from larger source confusion when they overlap, their
lower local event rate and short duration in band suggest
that this is more rare than BNSs, e.g.,
[13]
. Overall, we
expect that source confusion will depend on the astro-
physical population properties, astrophysical merger rate,
detector sensitivity (affecting the detected rates), and
detector low-frequency performance (affecting the signal
duration). We therefore consider different networks of
proposed detectors, astrophysical populations, and merger
rates as described in Sec.
II
. We address two questions.
(1) How much time-frequency overlap is there? In
Sec.
III
we simulate data with BNSs under different
astrophysical rates. We examine time-frequency
crossings and confirm the qualitative picture de-
scribed above. Under the median local rate of
100
Gpc
−
3
yr
−
1
[26]
, each BNS time-frequency
track crosses an average of 42 other BNS tracks
from 2 Hz. Because of the slow frequency evolution,
the majority of overlaps occurs at low frequencies:
91% in band below 5 Hz and very few above 20 Hz.
Splitting data into time-frequency bins that are
adapted to the signal morphology, each bin contains
on average (at most over 5 days of observation)
0.91(6) signals at 2 Hz, dropping to 0.05(3) signals
at 10 Hz. The low occupation number suggests that
the central limit theorem is not satisfied and BNS
source confusion in ground-based detectors is not
another source of Gaussian noise. Though not
quantified in this study, we expect this conclusion
to hold when further considering BBH and mixed
neutron star
–
black hole (NSBH) events given their
shorter duration and lower rates.
(2) What is the impact of overlapping signals on
parameter estimation? Again, with simulated data
we quantify the impact of overlapping signals on
parameter inference of a target BNS of interest. If the
overlapping signals were ignored altogether, param-
eter inference would be subject to systematic biases
[5,13
–
15,21,27]
.A
“
global fit
”
that simultaneously
JOHNSON, CHATZIIOANNOU, and FARR
PHYS. REV. D
109,
084015 (2024)
084015-2
analyzes all signals would mitigate such biases.
Despite its technical complications, progress in
the LISA
[28]
and ground-based
[29
–
33]
contexts
suggests that such solutions could be available in the
timescale of third-generation ground-based detec-
tors. As such, here we instead focus on the
statistical
uncertainty aspect of source confusion. Following
Crowder and Cornish
[9]
, we use the Fisher for-
malism as described in Sec.
IV
and compare stat-
istical uncertainty from data with only one signal
and data with multiple overlapping signals. In Sec.
V
we show that source confusion results in a sub-
percent increase in parameter uncertainty per signal
unless there exist overlapping signals with detector-
frame chirp masses
j
Δ
M
z
j¼j
M
z
2
−
M
z
1
j
≲
0
.
01
M
⊙
. Even when events with such similar masses
do overlap, parameter uncertainties increase by
≳
1%
only if the frequency of overlap is
≳
40
Hz. Out of
1
×
10
6
simulated signals, 0.14% fall within this
chirp mass threshold, but at frequencies lower than
40 Hz, implying that none have significant param-
eter uncertainty increases. Our results qualitatively
agree with those of Ref.
[21]
, generalized over BNS
populations and binary parameters.
Overall, we conclude that the confusion problem in
third-generation detectors, where signals usually overlap
instantaneously, will be a lot more mild than the LISA case,
where signals may overlap for the entirety of the mission
duration. By exploring the parameter uncertainty increase
and comparing to LISA calculations
[9]
, we quantify this
comparison and comment on the efficacy of LISA data
analysis strategies for third-generation detectors. Global fit
analyses that simultaneously model all data components,
including instrumental noise, astrophysical
=
cosmological
backgrounds, and transient signals, are likely to be suc-
cessful for third-generation detector data as well, hopefully
without loss of data
[34]
. We discuss these conclusions and
elaborate upon further work in Sec.
VI
.
II. DETECTOR NETWORK AND
ASTROPHYSICAL POPULATIONS
Source confusion depends both on the properties of the
detector network and on the astrophysical properties of the
signals. In Secs.
II A
and
II B
we describe the networks of
future detectors and astrophysical populations we consider,
respectively.
A. Detector networks
We consider several detectors whose location and
orientation are summarized in Table
IV
. Cosmic Explorer
(CE)
[2]
is envisioned as a 40 km
“
L
”
shaped detector.
Since the location remains to be determined, we set two CE
detectors at the current LIGO sites. We adopt its projected
low-frequency cutoff of 5 Hz. While a noise curve tuned to
low frequencies exists, we employ the standard noise curve
for the CE
[35,36]
, since the projected sensitivity remains the
same below 10 Hz where the majority of overlaps occur. The
Einstein Telescope (ET)
[3]
is designed with a triangular
shape and 10 km arms; we adopt the possible site of Sardinia
[37]
. Projected noise curves set the low-frequency sensitivity
cutoff at 1 Hz, however, here we adopt a cutoff of 2 Hz, as
the noise increases rapidly below this value. Design noise
curves for all detectors are shown in Fig.
1
.
We combine these detectors to form different networks:
(i) CE: a single CE detector in the location of LIGO
’
s
Hanford detector.
(ii) ET: the full triangular ET detector.
(iii) CE
þ
ET: two CE detectors at each of LIGO
’
s sites
and an ET detector.
B. Populations of neutron star binaries
We consider populations of quasicircular, spin-aligned
BNS inspirals and model the GW signal with the TaylorF2
[38]
waveform. Details about the waveform implementa-
tion and how we take Earth
’
s rotation into account are
given in Appendix
D
. Binary parameters (other than
redshift) are drawn from distributions that are summarized
in Table
I
. Since we assume uniformly distributed masses,
we adopt the corresponding local merger rates of 20, 100,
and
300
Gpc
−
3
yr
−
1
representing approximately the low,
median, and high inferred values
[26]
. Select results
are also presented for the extremely high rate of
1700
Gpc
−
3
yr
−
1
for reference.
We consider different redshift distributions computed as
follows. The source-frame merger rate density is
̇
n
ð
z
Þ
∝
Z
t
max
d
t
min
d
ψ
ð
z
f
ð
z; t
d
ÞÞ
P
ð
t
d
Þ
dt
d
;
ð
1
Þ
given a binary formation rate
ψ
ð
z
f
Þ
and a time delay
distribution
P
ð
t
d
Þ
between formation and merger. The
FIG. 1. Projected noise amplitude spectral densities (ASDs) for
the two detectors that form the different networks we consider.
Despite the nominal ET sensitivity going down to 1 Hz, we adopt
a low-frequency cutoff of 2 Hz due to the high ASD values below
that frequency. Outside of these frequency ranges, the ASD is
assumed to be infinite.
SOURCE CONFUSION FROM NEUTRON STAR BINARIES IN
...
PHYS. REV. D
109,
084015 (2024)
084015-3
constant of proportionality in Eq.
(1)
is determined by
matching
̇
n
ð
z
Þ
to the local merger rate. We assume that
binary formation follows the Madau-Dickinson
[41]
star
formation rate
ψ
ð
z
f
Þ
∼
ψ
SFR
ð
z
f
Þ
,
ψ
SFR
ð
z
f
;
α
;
β
;z
p
Þ¼
ð
1
þ
z
f
Þ
α
1
þ
1
þ
z
f
1
þ
z
p
α
þ
β
;
ð
2
Þ
with
α
¼
2
.
7
,
β
¼
2
.
9
, and
z
p
¼
1
.
9
. The delay time
between formation and merger is
P
ð
t
d
Þ
∝
t
−
1
d
. We adopt
minimum and maximum time delays of
t
min
d
¼
20
Myr and
a Hubble time
t
max
d
¼
t
H
¼
14
.
45
Gyr, respectively. The
mapping between redshift at formation
z
f
and merger
z
is
obtained by solving
t
d
−
½
t
L
ð
z
f
Þ
−
t
L
ð
z
Þ ¼
0
;
ð
3
Þ
where
t
L
ð
z
Þ¼
Z
z
0
dz
0
ð
1
þ
z
0
Þ
E
ð
z
0
Þ
ð
4
Þ
is the look-back time with
E
ð
z
Þ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ω
Λ
þ
Ω
M
ð
1
þ
z
Þ
3
q
;
ð
5
Þ
and
Ω
M
¼
0
.
3097
and
Ω
Λ
¼
0
.
6903
[42]
. From the rate
density of Eq.
(1)
, we obtain the rate in a redshift shell by
multiplying by the comoving volume element,
R
ð
z
Þ¼
̇
n
ð
z
Þ
dV
dz
:
ð
6
Þ
In the observer frame,
R
o
ð
z
Þ¼
R
ð
z
Þ
=
ð
1
þ
z
Þ
. The redshift
z
(equivalently, luminosity distance
d
L
) distribution is
P
ð
z
Þ¼
R
o
ð
z
Þ
R
∞
0
R
o
ð
z
0
Þ
dz
0
:
ð
7
Þ
Finally, the total number of BNS events is obtained by
integrating over redshift,
N
BNS
¼
Z
z
0
R
o
ð
z
Þ
dz
0
:
ð
8
Þ
Table
II
lists the total number of events and the average time
between them for different choices of the local merger rate.
Our results are broadly consistent with equivalent calcu-
lations using similar assumptions. Our numbers are similar
to those of
[12,43
–
45]
, but half of those obtained in
[18,46]
for equivalent local merger rates. All results are highly
dependent on the assumed event rates, with higher rates
leading to correspondingly higher numbers of overlaps
with more severe parameter estimation implications, and
vice versa. Unless otherwise noted, all subsequent results
incorporate a time delay.
III. THE PREVALENCE OF OVERLAPPING
SIGNALS
At each time, dozens of BNS signals are simultaneously
present in the detector data stream
[14]
. However, as
discussed further in Sec.
IV
and Appendix
C
, source
confusion is not driven by signals overlapping in time
(or frequency) alone, but by signals overlapping in time and
frequency simultaneously. In this section we study the
prevalence of overlapping signals and source confusion
through the time-frequency tracks of simulated signals,
Eq.
(D5)
.
2
Assuming each value for the local merger rate,
TABLE I. Population distributions for BNS parameters. We list
the component source-frame masses
m
1
,
m
2
; the component
spins along the orbital angular momentum
χ
1
,
χ
2
; the component
dimensionless tidal deformabilities
Λ
1
,
Λ
2
determined through a
fixed equation of state SFHo
[39]
that is consistent with current
observational constraints
[40]
; the declination
δ
and right
ascension
α
; the polarization angle
ψ
; the inclination
ι
; and
the time
t
c
and phase
φ
c
of coalescence.
Parameter
Prior
m
1
,
m
2
U
½
1
;
2
M
⊙
χ
1
,
χ
2
U
½
−
0
.
05
;
0
.
05
Λ
1
,
Λ
2
SFHo
[39]
cos
δ
U
½
−
1
;
1
α
U
½
0
;
2
π
ψ
U
½
0
;
π
cos
ι
U
½
−
1
;
1
t
c
U
½
0
;
3024000
s
φ
c
U
½
0
;
2
π
TABLE II. Simulated populations used throughout this study.
We vary the local merger rate between a low, median, high, and
very high value inferred in
[26]
and optionally include a delay
between formation and merger. The last two columns give the
total number of mergers in a year
N
BNS
and the average time
between successive events
Δ
t
c
.
Rate [Gpc
−
3
yr
−
1
]
Delay
N
BNS
h
Δ
t
c
i
(s)
20
Yes
28955
1090
100
Yes
144778
218
300
Yes
434336
73
1700
Yes
2461238
13
20
No
79694
396
100
No
398470
79
300
No
1195412
26
1700
No
6774004
5
2
A full calculation for the overlap is reserved for Sec.
V
.
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