GW170817: Implications for the Stochastic Gravitational-Wave Background
from Compact Binary Coalescences
The LIGO Scientific Collaboration and The Virgo Collaboration
(Dated: October 16, 2017)
The LIGO Scientific and Virgo Collaborations have announced the first detection of gravita-
tional waves from the coalescence of two neutron stars. The merger rate of binary neutron stars
estimated from this event suggests that distant, unresolvable binary neutron stars create a signifi-
cant astrophysical stochastic gravitational-wave background. The binary neutron star background
will add to the background from binary black holes, increasing the amplitude of the total astro-
physical background relative to previous expectations. In the Advanced LIGO-Virgo frequency
band most sensitive to stochastic backgrounds (near 25 Hz), we predict a total astrophysical back-
ground with amplitude Ω
GW
(
f
= 25 Hz) = 1
.
8
+2
.
7
−
1
.
3
×
10
−
9
with 90% confidence, compared with
Ω
GW
(
f
= 25 Hz) = 1
.
1
+1
.
2
−
0
.
7
×
10
−
9
from binary black holes alone. Assuming the most probable rate
for compact binary mergers, we find that the total background may be detectable with a signal-
to-noise-ratio of 3 after 40 months of total observation time, based on the expected timeline for
Advanced LIGO and Virgo to reach their design sensitivity.
Introduction
— On 17 August 2017, the Laser Inter-
ferometer Gravitational-Wave Observatory (LIGO) [1]
Scientific and Virgo [2] Collaborations detected a new
gravitational-wave source: the coalescence of two neu-
tron stars [3]. This event, GW170817, comes almost
two years after GW150914, the first direct detection of
gravitational waves from the merger of two black holes
[4]. In total, four high confidence detections and one sub-
threshold candidate from binary black hole merger events
have been reported: GW150914 [4], GW151226 [5],
LVT151012 [6], GW170104 [7] and GW170814 [8]. The
last of these events was reported as a three-detector ob-
servation by the two Advanced LIGO detectors located
in Hanford, WA and Livingston, LA in the United States
and the Advanced Virgo detector, located in Cascina,
Italy.
In addition to loud and nearby events that are de-
tectable as individual sources, there is also a population
of unresolved events at greater distances. The superpo-
sition of these sources will contribute an astrophysical
stochastic background, which is discernible from detec-
tor noise by cross-correlating the data streams from two
or more detectors [9, 10]. For a survey of the expected
signature of astrophysical backgrounds, see [11–17]. A
recent review of data analysis for stochastic backgrounds
is given in [18].
Following the first detection of gravitational waves
from GW150914, the LIGO and Virgo Collaborations
calculated the expected stochastic background from a bi-
nary black hole (BBH) population with similar masses
to the components of GW150914 [19]. These calcula-
tions were updated at the end of the first Advanced
LIGO observing run (O1) [20] to take into account
the two other black hole merger events observed during
the run: GW151226, a high-confidence detection; and
LVT151012, an event of lower significance. The results
indicated that the background from black hole mergers
would likely be detectable by Advanced LIGO and Ad-
vanced Virgo after a few years of their operation at de-
sign sensitivity. In the most optimistic case, a detection
is possible even before instrumental design sensitivity is
reached.
In this paper, we calculate the contribution to the
background from a binary neutron star (BNS) popula-
tion using the measured BNS merger rate derived from
GW170817 [3]. Also, we update the previous predic-
tions of the stochastic background taking into account
the most recent published statements about the rate of
BBH mergers [7]. We find the contributions to the back-
ground from BBH and BNS mergers to be of similar mag-
nitude. As a consequence, an astrophysical background
(including contributions from BNS and BBH mergers)
may be detected earlier than previously anticipated.
Background from compact binary mergers
— The
energy-density spectrum of a gravitational wave back-
ground can be described by the dimensionless quantity
Ω
GW
(
f
) =
f
ρ
c
dρ
GW
df
,
(1)
which represents the fractional contribution of gravita-
tional waves to the critical energy density of the Universe
[9]. Here
dρ
GW
is the energy density in the frequency
interval
f
to
f
+
df
,
ρ
c
= 3
H
2
0
c
2
/
(8
πG
) is the critical en-
ergy density of the Universe, and the Hubble parameter
H
0
= 67
.
9
±
0
.
55 km s
−
1
Mpc
−
1
is taken from Planck
[21].
In order to model the background from all the binary
mergers in the Universe, we follow a similar approach
to [19]. A population of binary merger events may be
characterized by a set of average source parameters
θ
(such as component masses or spins). If such a popula-
tion merges at a rate
R
m
(
z
;
θ
) per unit comoving volume
per unit source time at a given redshift
z
, then the total
gravitational-wave energy density spectrum from all the
2
sources is given by (see, e.g. [11, 19])
Ω
GW
(
f,θ
) =
f
ρ
c
H
0
∫
z
max
0
dz
R
m
(
z
;
θ
)
dE
GW
(
f
s
;
θ
)
/df
s
(1 +
z
)
E
(Ω
M
,
Ω
Λ
,z
)
.
(2)
Here
dE
GW
(
f
s
,θ
)
/df
s
is the energy spectrum emitted
by a single source evaluated in terms of the source fre-
quency
f
s
= (1 +
z
)
f
. The function
E
(Ω
M
,
Ω
Λ
,z
) =
√
Ω
M
(1 +
z
)
3
+ Ω
Λ
accounts for the dependence of co-
moving volume on redshift assuming the best-fit cosmol-
ogy from Planck [21], where Ω
M
= 1
−
Ω
Λ
= 0
.
3065. We
choose to cut off the redshift integral at
z
max
= 10. Red-
shifts larger than
z
= 5 contribute little to the integral
because of the [(1 +
z
)
E
(
z
)]
−
1
factor in Eq. 2, as well as
the small number of stars formed at such high redshift,
see for example [11–17, 22].
The energy spectrum
dE
GW
/df
s
is determined from
the strain waveform of the binary system. The domi-
nant contribution to the background comes from the in-
spiral phase of the binary merger, for which
dE/df
s
∝
M
5
/
3
c
f
−
1
/
3
, where
M
c
= (
m
1
m
2
)
3
/
5
/
(
m
1
+
m
2
)
1
/
5
is the
chirp mass for a binary system with component masses
m
1
and
m
2
. In the BNS case we only consider the in-
spiral phase, since neutron stars merge at
∼
2 kHz, well
above the sensitive band of stochastic searches. We intro-
duce a frequency cutoff at the innermost stable circular
orbit. For BBH events, we include the merger and ring-
down phases using the waveforms from [12, 23] with the
modifications from [24].
The merger rate
R
m
(
z
;
θ
) is given by
R
m
(
z
;
θ
) =
∫
t
max
t
min
R
f
(
z
f
;
θ
)
p
(
t
d
;
θ
)
dt
d
,
(3)
where
t
d
is the time delay between formation and merger
of a binary,
p
(
t
d
;
θ
) is the time delay distribution given
parameters
θ
,
z
f
is the redshift at the formation time
t
f
=
t
(
z
)
−
t
d
, and
t
(
z
) is the age of the Universe
at merger. We assume that the binary formation rate
R
f
(
z
f
;
θ
) scales with the star formation rate. For the
BNS background, we make similar assumptions to those
used in [19], which are outlined in what follows below.
We adopt the star formation model of [25], which pro-
duces very similar results as compared to the model de-
scribed by [26]. We assume a time delay distribution
p
(
t
d
)
∝
1
/t
d
, for
t
min
< t
d
< t
max
. Here
t
min
is the
minimum delay time between the binary formation and
merger. We assume
t
min
= 20 Myr [27]. The maximum
time delay
t
max
is set to the Hubble time [28–36]. We
also need to consider the distribution of the component
masses to calculate Ω
GW
. We assume that each mass of
the binary is drawn from uniform distribution ranging
from 1 to 2
M
. The value of
R
m
at
z
= 0 is normalized
to the median BNS merger rate implied by GW170817,
which is 1540
+3200
−
1220
Gpc
−
3
yr
−
1
[3].
The calculation of the BBH background is similar, with
the following differences. We assume
t
min
= 50 Myr for
the minimum time delay [19, 36]. Massive black holes
are formed preferentially in low-metallicity environments.
For binary systems where at least one black hole has a
mass larger than 30
M
, we therefore re-weight the star
formation rate
R
f
(
z
) by the fraction of stars with metal-
licities
Z
≤
Z
/
2. Following [19], we adopt the mean
metallicity-redshift relation of [26], with appropriate scal-
ings to account for local observations [25, 37]. It is also
important to specify the mass distribution. We use a
power-law distribution of the primary (i.e., larger mass)
component
p
(
m
1
)
∝
m
−
2
.
35
1
and a uniform distribution
of the secondary [6, 7]. In addition, we require that the
component masses take values in the range 5
−
95
M
with
m
1
+
m
2
<
100
M
and
m
2
< m
1
, in agreement
with the observations of BBHs to date [7].
For the
rate of BBH mergers, we use the most recent published
result associated with the power-law mass distribution
103
+110
−
63
Gpc
−
3
yr
−
1
[7, 38]. As shown in [20], using a
flat-log mass distribution instead of the power-law only
affects Ω
GW
(
f
) at frequencies above 100 Hz, which has
very little impact on the detectability of the stochastic
background with LIGO and Virgo. Frequencies below
100 Hz contribute to more than 99% of the sensitivity of
the stochastic search [20].
Predictions and detectability
— A stochastic back-
ground of gravitational-waves introduces a correlated sig-
nal in networks of terrestrial detectors. This signal is ex-
pected to be much weaker than the detector noise, but
can be distinguished from noise by cross-correlating the
strain data from two or more detectors. For a network
of
n
detectors, assuming an isotropic, unpolarized, Gaus-
sian, and stationary background, the optimal signal-to-
noise ratio (SNR) of a cross-correlation search is given
by
SNR =
3
H
2
0
10
π
2
√
2
T
∫
∞
0
df
n
∑
i
=1
∑
j>i
γ
2
ij
(
f
)Ω
2
GW
(
f
)
f
6
P
i
(
f
)
P
j
(
f
)
1
/
2
,
(4)
in which
i,j
run over detector pairs,
P
i
(
f
) and
P
j
(
f
) are
the one-sided strain noise power spectral densities of the
two detectors, and
γ
ij
(
f
) is the normalized isotropic over-
lap reduction function between the pair [9, 19]. While the
cross correlation search is not optimal for non-Gaussian
backgrounds, Eq. 4 gives the correct expression for the
cross-correlation signal-to-noise ratio irrespective of the
Gaussianity of the background [27, 39].
On the left hand panel of Fig. 1, we show the estimates
on the background energy density Ω
GW
(
f
) for the BNS
and BBH merger populations described in the previous
section (red and green curves, respectively). The total
(combined) background from BBH and BNS mergers is
also plotted (solid blue curve) along with the 90% cred-
ible Poisson uncertainties in the local rate (indicated by
the grey shaded region). Considering this uncertainty,
we predict Ω
tot
GW
(
f
= 25 Hz) = 1
.
8
+2
.
7
−
1
.
3
×
10
−
9
.