The Rate of Binary Black Hole Mergers Inferred from Advanced LIGO Observations Surrounding GW150914

Abstract: A transient gravitational-wave signal, GW150914, was identified in the twin Advanced LIGO detectors on September 14, 2015 at 09:50:45 UTC. To assess the implications of this discovery, the detectors remained in operation with unchanged configurations over a period of 39 d around the time of the signal. At the detection statistic threshold corresponding to that observed for GW150914, our search of the 16 days of simultaneous two-detector observational data is estimated to have a false alarm rate (FAR) of < 4.9 × 10^(−6) yr^(−1), yielding a p-value for GW150914 of < 2 × 10^(−7). Parameter estimation followup on this trigger identifies its source as a binary black hole (BBH) merger with component masses (m_1, m_2) = (36^(+5)_(−4), 29^(+4)_(−4)) M_⊙ at redshift z = 0.09^(+0.03)_(−0.04) (median and 90\% credible range). Here we report on the constraints these observations place on the rate of BBH coalescences. Considering only GW150914, assuming that all BBHs in the Universe have the same masses and spins as this event, imposing a search FAR threshold of 1 per 100 years, and assuming that the BBH merger rate is constant in the comoving frame, we infer a 90% credible range of merger rates between 2--53 Gpc^(−3) yr^(−1) (comoving frame). Incorporating all search triggers that pass a much lower threshold while accounting for the uncertainty in the astrophysical origin of each trigger, we estimate a higher rate, ranging from 13--600 Gpc^(−3) yr^(−1) depending on assumptions about the BBH mass distribution. All together, our various rate estimates fall in the conservative range 2--600 Gpc^(−3) yr^(−1).

Figures

Figure 3. Left panel: the median value and 90% credible interval for the expected number of highly significant events (FARs<1/century) as a function of surveyed time-volume in an observation (shown as a multiple of á ñVT 0). The expected range of values of á ñVT for the observations in O2 and O3 are shown as vertical bands. Right panel: the probability of observing N>0 (blue), N>10 (green), N>35 (red), and N>70 (purple) highly significant events, as a function of surveyed timevolume. The vertical line and bands show, from left to right, the expected sensitive time-volume for each of the O1 (dashed line), O2, and O3 observations.

Figure 2. Sensitivity of the inferred BBH coalescence rate to the assumed astrophysical distribution of BBH masses. The curves represent the posterior assuming that BBH masses are flat in ( ) ( )-m mlog log1 2 , as in Equation (18) (green; “Flat”), are exactly GW150914-like or LVT151012-like as described in Section 2 (blue; “Reference”), or are distributed as in Equation (19) (red; “Power Law”). The pycbc results are shown as solid lines, and the gstlal results are shown as dotted lines. Though the searches differ in their models of the astrophysical and terrestrial triggers, the rates inferred from each search are very similar. The posterior median rates and symmetric 90% credible level (CL) intervals are given in Table 1. Comparing to the total rate computed using the assumptions in Kim et al. (2003) in Section 2 we see that the rate can change by a factor of a few depending on the assumed BBH population. In spite of this seemingly large variation, all three rate posterior distributions are consistent within our statistical uncertainties.

Table 1 Rates of BBH Mergers Estimated under Various Assumptions

Figure 1. Posterior density on the rate of GW150914-like BBH inspirals, R1 (green); LVT151012-like BBH inspirals, R2 (red); and the inferred total rate, = +R R R1 2 (blue). The median and 90% credible levels are given in Table 1. Solid lines give the rate inferred from the pycbc trigger set, while dashed lines give the rate inferred from the gstlal trigger set.

Full text

 PUBLISHED VERSION
Abbott, BP ... Hollitt, SE ... Ottaway, DJ, LIGO Scientific Collaboration and Virgo
Collaboration
The rate of binary black hole mergers inferred from advanced LIGO
observations surrounding GW150914
The Astrophysical Journal, 2016; 833(1):L1-L8
© 2016. The American Astronomical Society. All rights reserved.
Originally published by IOP Publishing at: http://dx.doi.org/10.3847/2041-8205/833/1/L1
http://hdl.handle.net/2440/103644
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7 March, 2017

THE RATE OF BINARY BLACK HOLE MERGERS INFERRED FROM
ADVANCED LIGO OBSERVATIONS SURROUNDING GW150914
LIGO Scientific Collaboration and Virgo Collaboration
(See the Supplement, Abbott et al. 2016g, for the full list of authors.)
Received 2016 February 12; revised 2016 September 19; accepted 2016 September 19; published 2016 November 30
ABSTRACT
A transient gravitational-wave signal, GW150914, was identified in the twin Advanced LIGO detectors on 2015
September 2015 at 09:50:45 UTC. To assess the implications of this discovery, the detectors remained in operation with
unchanged configurations over a period of 39 days around the time of the signal. At the detection statistic threshold
corresponding to that observed for GW150914, our search of the 16 days of simultaneous two-detector observational
data is estimated to have a false-alarm rate (FAR) of
<
´
--
4.9 10 yr
61
, yielding a p-value for GW150914 of
<
´
-
210
7
. Parameter estimation follow-up on this trigger identifies its source as a binary black hole (BBH) merger
with component masses
(
)( )

=
-
+
-
+
mm M,36,29
12
4
5
4
4
at redshift
=
-
+
z
0.09
0.04
0.03
(media n and 90% credible range).
Here, we report on the constraints these observations place on the rate of BBH coalescences. Considering only
GW150914, assuming that all BBHs in the universe have the same masses and spins as this event, imposing a search
FAR threshold of 1 per 100 years, and assuming that the BBH merger rate is constant in the comoving frame, we infer a
90% credible range of merger rates between
–
--
2
53 Gpc yr
31
(comoving frame). Incorporating all search triggers that
pass a much lower threshold while accounting for the uncertainty in the astrophysical origin of each trigger, we estimate
a higher rate, ranging from
–
--
13 600 Gpc yr
31
depending on assumptions about the BBH mass distribution. All
together, our various rate estimates fall in the conservative range
–
--
2
600 Gpc yr
31
.
Key words: black holes – gravitational waves – stars: massive
Supporting material: data behind figures
1. INTRODUCTION
The first detection of a gravitational-wave (GW) signal in the
twin Advanced LIGO detectors on 2015 September 2015,
09:50:45 UTC was reported in Abbott et al. (2016d). This
transient signal is designated GW150914. To assess the
implications of this discovery, the detectors remained in
operation with unchanged configurations over a period of
39 days around the time of the signal. At the detection statistic
threshold corresponding to that observed for GW150914, the
false-alarm rate (FAR) of the search of the available 16 days of
coincident data is estimated to be
<
´
--
4.9 10 yr
61
, yielding a
p-value for GW150914 of
<
´
-
210
7
(Abbott et al. 2016c) .
GW150914 is consistent with a GW signal from the merger of
two black holes with masses
(
)( )

=
-
+
-
+
mm M,36,29
12
4
5
4
4
at
redshift
=
-
+
z
0.09
0.04
0.03
(Abbott et al. 2016e). Here and
throughout, we report posterior medians and 90% symmetric
credible intervals. In this Letter, we discuss inferences on the
rate of binary black hole (BBH) mergers from this detection
and the surrounding data. This Letter is accompanied by Abbott
et al. (2016g, hereafter the Supplement) containing supple-
mentary information on our methods and computations.
Previous estimates of the BBH merger rate based on
population modeling are reviewed in Abadie et al. (2010).The
range of rates given there spans more than three orders of
magnitude, from 0.1 to 300
--
Gpc yr
31
. The rate of BBH
mergers is a crucial output from BBH population models, but
theoretical uncertainty in the evolution of massive stellar binaries
and a lack of constraining electromagnetic observations produce
a wide range of rate estimates. Observations of GWs can tightly
constrain this rate with minimal modeling assumptions, and thus
provide useful input on the astrophysics of massive stellar
binaries. In the absence of detections until GW150914, the most
constraining rate upper limits from GW observations, as detailed
in Aasi et al. (2013), lie above the model predictions. Here, for
the first time, we report on GW observations that constrain the
model space of BBH merger rates.
It is possible to obtain a rough estimate of the BBH
coalescence rate from the GW150914 detection by setting a
low search FAR threshold that eliminates other search triggers
(Abbott et al. 2016c). The inferred rate will depend on the
detector sensitivity to the BBH population, which strongly
depends on BBH masses. However, our single detection leaves
a large uncertainty in the mass distribution of merging BBH
systems. Kim et al. (2003) faced a similar situation in deriving
binary neutron star merger rates from the small sample of
Galactic double neutron star systems. They argued that a good
rate estimate follows from an approach assuming each detected
system belongs to its own class, deriving merger rates for each
class independently, and then adding the rates over classes to
infer the overall merger rate. If we follow Kim et al. (2003),
assume that all BBH mergers in the universe have the same
source-frame masses and spins as GW150914, and set a
nominal threshold on the search FAR of one per century—
eliminating all triggers but the one associated with GW150914
—then the inferred posterior median rate and 90% credible
range is
=
-
+
--
R
14 Gpc yr
100
12
39
31
(see Section 2.2).
Merger rates inferred from a single highly significant trigger
are sensitive to the choice of threshold. Less significant search
triggers eliminated under the strict FAR threshold can also
provide information about the merger rate. For example,
thresholded at the significance of the second-most-significant
trigger (designated LVT151012), our search FAR is
-
0
.43 yr
1
,
yielding a p-value for this trigger of 0.02. This trigger cannot
confidently be claimed as a detection on the basis of such a p-
The Astrophysical Journal Letters, 833:L1 (8pp), 2016 December 10 doi:10.3847/2041-8205/833/1/L1
© 2016. The American Astronomical Society. All rights reserved.
1

value, but neither is it obviously consistent with a terrestrial
origin, i.e., a result of either instrumental or environmental effects
in the detector. Under the assumption that this trigger is
astrophysical in origin, parameter estimation (PE; Veitch
et al. 2015) indicates that its source is also a BBH merger with
source-frame masses
(
)( )

=
-
+
-
+
mm M,23,13
12
6
18
5
4
at redshift
=
-
+
z
0.21
0.09
0.09
(Abbott et al. 2016c). Based on two different
implementations of a matched-filter search, we find posterior
probabilities 0.84 and 0.91 that LVT151012 is of astrophysical
origin (see Section 2.1). This is the only trigger besides
GW150914 that has probability greater than 50% of being of
astrophysical origin. Farr et al. (2015) presented a method by
which a set of triggers of uncertain origin like this can be used to
produce a rate estimate that is more accurate than that produced
by considering only highly significant events.
The mixture model of Farr et al. (2015) used here is similar to
other models used to estimate rates in astrophysical contexts.
Loredo & Wasserman (1995, 1998b) used a similar foreground/
background mixture model to infer the rate and distribution of
gamma-ray bursts. A subsequent paper used similar models in a
cosmological context, as we do here (Loredo &
Wasserman 1998a). Guglielmetti et al. (2009) used the same
sort of formalism to model ROSAT images, and it has also found
use in analysis of surveys of trans-Neptunian objects (Gladman
et al. 1998; Petit et al. 2008). Kelly et al. (2009) address selection
effects in the presence of population models, an issue that also
appears in this work. In contrast to previous analyses, here we
operate in the background-dominated regime, setting a search
threshold where the FAR is relatively high so that we can be
confident that triggers of terrestrial (as opposed to astrophysical)
origin dominate near threshold (see Section 2.1).
Incorporating our uncertainty about the astrophysical origin
of all search triggers that could represent BBH signals (Abbott
et al. 2016c) using the Farr et al. (2015) method, assuming that
the BBH merger rate is constant in comoving volume and
source-frame time, and making various assumptions about the
mass distribution of merging BBH systems as described in
Sections 2.2 and 3, we derive merger rates that lie in the
range
–
--
13 600 Gpc yr
31
.
Our rate estimates are summarized in Table 1; see Section 2.2
for more information. Each row of Table 1 represents a
different assumption about the BBH mass distribution. The first
two columns give rates that correspond to two different search
algorithms (called pycbc and gstlal, described in the
Supplement) with different models of the astrophysical and
terrestrial trigger distributions. Because the rate posteriors from
the different searches are essentially identical (see Figures 1
and 2), the third column gives rates that provide a combined
estimate that results from an average of the posterior densities
from each search. Including the rate estimate with a strict
threshold that considers only the GW150914 trigger as
described in Section 2.2, all our rate estimates lie in the
conservative range
–
--
2
600 Gpc yr
31
.
137
All our rate estimates are consistent within their statistical
uncertainties, and these estimates are also consistent with the
broad range of rate predictions reviewed in Abadie et al. (2010)
with only the low end (<1
--
Gpc yr
31
) of rate predictions
being excluded. The astrophysical implications of the
GW150914 detection and these inferred rates are further
discussed in Abbott et al. (2016a) .
This Letter presents the results of our rate inference. For
methodological and other details of the analysis, see the
Supplement.
The results presented here depend on assumptions about the
masses, spins, and cosmological distribution of sources. As
GW detectors acquire additional data and their sensitivities
improve, we will be able to test these assumptions and deepen
our understanding of BBH formation and evolution in the
universe.
2. RATE INFERENCE
A rate estimate requires counting the number of signals in an
experiment and then estimating the sensitivity to a population
of sources to transform the count into an astrophysical rate.
Individually, the count of signals and the sensitivity will
depend on specific detection and trigger generation thresholds
imposed by the pipeline, but the estimated rates should not
depend strongly on such thresholds. We consider various
methods of counting signals, employ two distinct search
pipelines, and obtain a range of broadly consistent rate
estimates.
2.1. Counting Signals
Two independent pipelines searched the coincident data for
signals matching a compact binary coalescence (CBC; Abbott
et al. 2016c), each producing a set of coincident search triggers.
Both the pycbc pipeline ( Usman et al. 2016) and the gstlal
pipeline (Messick et al. 2016) perform matched-filter searches
for CBC signals using aligned-spin templates (Taracchini
et al. 2014; Pürrer 2016) when searching the BBH parts of the
CBC parameter space. In these searches, single-detector
triggers are recorded at maxima of the signal-to-noise ratio
(S/N) time series for each template (Allen et al. 2012);
coincident search triggers are formed when pairs of triggers,
one from each detector, occur in the same template with a time
difference of ±15 ms or less. Our data set here consists of the
set of coincident triggers returned by each search over the
16 days of coincident observations. See the Supplement for
more information about the generation of triggers.
The Farr et al. (2015) framework considers two classes of
coincident triggers: those whose origin is astrophysical and
those whose origin is terrestrial. Terrestrial triggers are the
result of either instrumental or environmental effects in the
detector. The two types of sources produce triggers with
different densities in the space of detection statistics, which we
denote as x. We consider all triggers above a threshold chosen
so that triggers of terrestrial origin dominate at the threshold.
Triggers appear above threshold in a Poisson process with
number density in detection space
() () ()=L +L
dN
dx
px px,1
1
1
0
0
where the subscripts “1” and “0” refer to the astrophysical and
terrestrial origin, Λ
1
and Λ
0
are the Poisson mean numbers of
137
Following submission but before acceptance of this Letter we identified a
mistake in our calculation of the sensitive spacetime volume for the “Flat ” and
“Power Law ” BBH populations (see Section 3) that reduced those volumes and
increased the corresponding rates by a factor of approximately two. Since the
upper limit of this rate range is driven by the rate estimates for the “Power
Law” population, the range given here increased when the mistake was
corrected. Previous versions of this paper posted to the arXiv, Abbott et al.
(2016d), Abbott et al. (2016a), and others used the mistaken rate range
–
--
2
400 Gpc yr
31
. The correction does not affect the astrophysical interpreta-
tion appearing in Abbott et al. (2016d) or Abbott et al. (2016a).
2
The Astrophysical Journal Letters, 833:L1 (8pp), 2016 December 10 Abbott et al.

triggers of astrophysical and terrestrial type, and p
1
and p
0
are
the (normalized ) density of triggers of astrophysical and
terrestrial origin over detection space. We estimate the
densities, p
0
and p
1
, of triggers of the two types empirically
as described in the Supplement and in Abbott et al. (2016c).
Here, we ignore the time of arrival of the triggers in our data
set, averaging the rates of each type of trigger and the
sensitivity of the detector to astrophysical signals over time.
We do this because it is difficult to estimate p
0
and p
1
over
short times and because we see no evidence of time variation in
p
0
and p
1
; for more details see the Supplement.
The parameter Λ
1
is the mean number of signals of
astrophysical origin above the chosen threshold; it is not the
mean number of signals confidently detected (see Section 4).
Under the assumptions we make here of a rate that is constant
in the comoving frame, Λ
1
is related to the astrophysical rate of
BBH coalescences R by
()L= á ñRVT,2
1
where
á
ñVT
is the time- and population-averaged spacetime
volume to which the detector is sensiti ve at the chosen s ea rch
thresh ol d, defined in E qu at ion ( 15). Because the astrophysical
rate enters the likelihood only in the combination
áñ
R
VT
,
which represents a dimensionless count, we first discuss
estimation of Λ in this section, and then discuss the
relationship between the posterior on Λ andontherateR in
Section 2.2.
The likelihood for a trigger set with detection statistics
{
∣}=¼xj M1, ,
j
is (Loredo & Wasserman 1995; Farr
et al. 2015)
({ ∣ }∣ )
[() ()] [ ]()


=¼ LL
= L +L -L -L
=
⎪
⎪
⎪
⎪
⎧
⎨
⎩
⎫
⎬
⎭
xj M
px px
1, , ,
exp . 3
j
j
M
jj
10
1
1
1
0
0
10
See the Supplement for a derivation of this likelihood function
for our Poisson mixture model. In each pipeline, the shape of
the astrophysical trigger distribution p
1
(x) is, to a very great
extent, universal (Schutz 2011; Chen & Holz 2014); that is, it
does not depend on the properties of the source (see
Supplement). It is this remarkable property that motivates this
approach to our analysis. In principle, the data from the LIGO
detector contain much more information than can be summar-
ized by a trigger with detection statistic x. For example, to
obtain information about the source associated to a trigger (if
any) we can follow it up with a separate PE analysis (Veitch
et al. 2015). Unfortunately, with only two likely astrophysical
sources,
138
the amount of information available about the
distribution of source properties is minimal. Since we cannot
eliminate the dominant astrophysical systematic of uncertainty
about the distribution of source parameters through a more
detailed analysis, here we adopt this simpler method. In the
future, as detections accumulate, we expect to transition to a
method of analysis that incorporates estimation of population
parameters. Here, we deal with the uncertainty in the
astrophysical population by estimating rates under several
different assumptions about the population; see Section 2.2.
We impose a prior on the Λ parameters of
() ()LL µ
LL
p ,
11
.4
10
10
See Section 4 of the Supplement for a discussion of our choice
of prior. The posterior on expected counts is proportional to the
product of the likelihood from Equation (3) and the prior from
Equation (4):
(∣{∣ })
[() ()]
[] ()

LL = ¼
µL+L
´-L-L
LL
=
⎪
⎪
⎪
⎪
⎧
⎨
⎩
⎫
⎬
⎭
pxjM
px px
,1,,
exp
1
.5
j
j
M
jj
10
1
1
1
0
0
10
10
Posterior distributions for Λ
0
and Λ
1
were obtained using a
Markov Chain Monte Carlo; details are given in the
Supplement, along with the resulting expected counts Λ
1
.
Using the posterior on the Λ
0
and Λ
1
, we can compute the
posterior probability that each particular trigger comes from an
astrophysical versus terrestrial source. The conditional prob-
ability that an event at detection statistic x comes from an
astrophysical source is given by (Guglielmetti et al. 2009; Farr
et al. 2015)
(∣ )
()
() ()
()LL=
L
L+L
Px
px
px px
,.6
101
1
1
1
1
0
0
Marginalizing over the posterior for the expected counts gives
(∣{ ∣ })
(∣ )
(∣{∣ }) ()
ò
=¼
ºLL LL
´LL =¼
Px xj M
ddPx
pxjM
1, ,
,
,1,,, 7
j
j
1
011 01
10
which is the posterior probability that an event at detection
statistic x is astrophysical in origin given the observed event set
(and associated count inference). In particular, we calculate the
posterior probability that LVT151012 is of astrophysical origin
to be 0.84 with the gstlal pipeline and 0.91 with the pycbc
pipeline. These probabilities, while not high enough to claim
LVT151012 as a second detection, are large enough to
motivate exploring a second class of BBHs in the Kim et al.
(2003) prescription.
It is more difficult to estimate the posterior probability that
GW150914 is of astrophysical origin because there are no
samples from the empirical background estimation in this
region, so the probability estimate is sensitive to how the
background density, p
0
, is analytically extended into this
region. We estimate that the probability of astrophysical origin
for GW150914 is larger than 1 –10
−6
.
Under the assumption that GW150914 and LVT151012
are astrophysical, posterior distributions for system parameters
can be derived (Veitch et al. 2015). Both triggers are
consistent with BBH merger sources with masses
(
)( )

=
-
+
-
+
mm M,36,29
12
4
5
4
4
at redshift
-
+
0
.09
0.04
0.03
(GW150914)
and
(
)( )

=
-
+
-
+
mm M,23,13
12
6
18
5
4
at redshift
-
+
0
.21
0.09
0.0
9
(LVT151012; Abbott et al. 2016c, 2016e). Following Kim
138
Actually three, as this article goes to press (Abbott et al. 2016f; The LIGO
Scientific Collaboration et al. 2016).
3
The Astrophysical Journal Letters, 833:L1 (8pp), 2016 December 10 Abbott et al.

et al. (2003), we consider the second event, if astrophysical, to
be a separate class of BBH from GW150914.
We can incorporate a second class of BBH merger into our
mixture model in a straightforward way. Let there be two
classes of BBH mergers: type 1, which are GW150914-like,
and type 2, which are LVT151012-like. Our trigger set then
consists of triggers of type 1, triggers of type 2, and triggers of
terrestrial origin, denoted as type 0. The distribution of triggers
over detection statistic, x, now follows an inhomogeneous
Poisson process with three terms:
() () () ()=L +L +L
dN
dx
px px px,8
1
1
2
2
0
0
where Λ
1
, Λ
2
, and Λ
0
are the mean number of triggers of each
type in the data set and p
1
, p
2
, and p
0
are the probability
densities for triggers of each type over the detection statistic.
The shape of the distribution of S/Ns is independent of the
event properties, so p
1
=p
2
(Schutz 2011; Chen & Holz 2014);
we cannot distinguish BBH classes based only on their
detection statistic distributions, but rather require PE.
When an eventʼs parameters are known to come from a
certain class i, under the astrophysical origin assumption, then
the trigger rate becomes
() () ()=L +L
dN
dx
px p x,9
i
i
i
0
0
i.e., we permit the event to belong to either its astrophysical
class or to an terrestrial source, but not to the other
astrophysical class. The Poisson likelihood for the set of M
triggers,
{
∣}=¼xj M1, ,
j
, exceeding our detection statistic
threshold is similar to Equation (3), but we now account for the
distinct classification of GW150914 and LVT151012 based on
PE:
({ ∣ }∣ )
[ () ()][ () ()]
[() () ()]
[] ()


=¼ LLL
=L +L L +L
´ L +L +L
´-L-L-L
=
⎪
⎪
⎪
⎪
⎧
⎨
⎩
⎫
⎬
⎭
xj M
px px px px
px px px
1, , , ,
exp . 10
j
j
M
jjj
12 0
1
1
10
0
12
2
20
0
2
3
1
1
2
2
0
0
12 0
The first two terms in this product are the rates of the form of
Equation (9) for the GW150914 and LVT151012 triggers,
whose class, if not terrestrial, is known; the remaining terms in
the product over coincident triggers represent the other events,
whose class is not known.
As above, the expected counts of type 1 and 2 triggers are
related to the astrophysical rates of the corresponding events by
()L= á ñRVT,11
ii i
where
á
ñVT
i
is the time- and population-averaged spacetime
volume to which the detector is sensitive for event class i,
defined in Equation (15) under the population assumption in
Equation (16).
We impose a prior for the total astrophysical and terrestrial
expected counts:
() ()LLL µ
L+L L
p ,,
11
.12
12 0
120
This prior is chosen to match the prior in Equation (4).Itis
adequate for the two events that we are analyzing here, but
should be modified if a large number of events are being
analyzed in this formalism; with N categories of foreground
event under this prior, the expected number of total counts
becomes
+
N
1
2
. The posterior on expected counts given the
trigger set is proportional to the product of likelihood,
Equation (10), and prior, Equation (12):
(∣{})
({}∣)()()
LLL
µLLLLLL
px
xp
,,
,, ,, 13
j
j
12 0
12 0 12 0
We again use Markov Chain Monte Carlo samplers to obtain
resulting expected counts for Λ
1
, Λ
2
,and
LºL +L
12
.These
parameters represent the Poisson mean number of events of type
1 (GW150914-like),type2(LVT151012-like), and both types
over the observation, above a very low detection statistic
threshold. The estimates, which are given in the Supplement,are
consistent with one event of astrophysical origin (GW150914) at
very high probability, a further trigger (LVT151012) with high
probability, and possibly several more of each type in the set of
triggers at lower significance. In the next subsection, we will
describe how to turn these expected counts of events into
astrophysical rates.
2.2. Rates
The crucial element in the step from expected counts to rates
is to determine the sensitivity of the search. Search sensitivity is
described by the selection function, which gives, as a function
of source parameters, the probability of generating a trigger
above the chosen threshold. Here we assume that events are
uniformly distributed in comoving volume and source time and
describe the distribution of the other parameters (masses, spins,
orientation angles, etc., here denoted by θ) for events of type i
by a distribution function s
i
(θ). Because the shape of the
distribution p
1
(x) is universal (Schutz 2011; Chen &
Holz 2014), the source population enters the likelihood only
through the search sensitivity; this situation differs from
previous astrophysical rate calculations (Loredo &
Wasserman 1995, 1998a, 1998b; Gladman et al. 1998), where
information about the source properties is contained in each
trigger. Under these assumptions, a count at a chosen threshold
Λ
i
is related to an astrophysical rate R
i
by
()L= á ñRVT,14
ii i
where
() ( ) ( )
ò
qqqáñ=
+
VT T dz d
dV
dz z
sfz
1
1
,15
i
c
i
(see the Supplement). Here, R
i
is the spacetime rate density in
the comoving frame,
()qfz
0
,1
is the selection function,
T is the total observation time in the observer frame, and V
c
(z)
is the comoving volume contained within a sphere out to
redshift z (Hogg 1999).
139
In other words, the posterior on R
i
is
obtained by substituting Equation (14) into Equation (13).We
need to know (or assume) s
i
, the population distribution for
events of type i, before we can turn expected counts into rates.
139
Throughout this Letter, we use the “TT+lowP+lensing+ext” cosmologi-
cal parameters from Table 4 of Planck Collaboration et al. (2016).
4
The Astrophysical Journal Letters, 833:L1 (8pp), 2016 December 10 Abbott et al.

Frequently asked questions

Q1. What are the contributions in "The rate of binary black hole mergers inferred from advanced ligo observations surrounding gw150914" ?

At the detection statistic threshold corresponding to that observed for GW150914, their search of the 16 days of simultaneous two-detector observational data is estimated to have a false-alarm rate ( FAR ) of < ́ 4. 9 10 yr 6 1, yielding a p-value for GW150914 of < ́ 2 10 7. Parameter estimation follow-up on this trigger identifies its source as a binary black hole ( BBH ) merger with component masses ( ) ( )  = m m M, 36, 29 1 2 4 4 at redshift = z 0. 09 0. 04 0. 03 ( median and 90 % credible range ). Here, the authors report on the constraints these observations place on the rate of BBH coalescences. 

Q2. What have the authors stated for future works in "The rate of binary black hole mergers inferred from advanced ligo observations surrounding gw150914" ?

There is, unsurprisingly, a wide range of reasonable possibilities for the number of highly significant events in future observations. The power chosen here is the same as the Salpeter initial mass function ( Salpeter 1955 ), but this should not be understood to suggest that the distribution of the more massive BH in a binary would follow the IMF ; the initial mass–final mass relation for massive stars is complicated and nonlinear ( Fryer & Kalogera 2001 ; Dominik et al. 

Q3. What is the assumption of Kim et al. (2003)?

The Kim et al. (2003) assumption is that the population follows the observed sources:( ) ( ) ( )q d q q= -s , 16i iwhere δ is the Dirac delta function and θi are the parameters of source type i. 

Q4. What is the definition of a rate estimate?

A rate estimate requires counting the number of signals in an experiment and then estimating the sensitivity to a population of sources to transform the count into an astrophysical rate. 

Q5. What is the posterior on expected counts given the trigger set?

The posterior on expected counts given the trigger set is proportional to the product of likelihood, Equation (10), and prior, Equation (12):( ∣{ }) ({ }∣ ) ( ) ( ) L L L µ L L L L L L p x x p , , , , , , 13 jj1 2 01 2 0 1 2 0The authors again use Markov Chain Monte Carlo samplers to obtain resulting expected counts for Λ1, Λ2, and L º L + L1 2. 

Q6. What is the posterior probability of an event coming from an astrophysical source?

61 0 1 1 11 1 0 0Marginalizing over the posterior for the expected counts gives( ∣{ ∣ })( ∣ ) ( ∣{ ∣ }) ( ) ò= ¼º L L L L´ L L = ¼P x x j Md d P xp x j M1, ,,, 1, , , 7jj10 1 1 0 11 0which is the posterior probability that an event at detection statistic x is astrophysical in origin given the observed event set (and associated count inference). 

Q7. What is the relationship between the posterior on and the rate R?

Because the astrophysical rate enters the likelihood only in the combination á ñR VT , which represents a dimensionless count, the authors first discuss estimation of Λ in this section, and then discuss the relationship between the posterior on Λ and on the rate R in Section 2.2. 

Q8. How many sources are there in the GW150914 search?

GW150914 is unusually significant; only ∼8% of the astrophysical distribution of sources appearing in their search with a threshold at FARs of one per century will be more significant than GW150914. 

Q9. What is the likelihood of a trigger set with detection statistics?

The likelihood for a trigger set with detection statistics { ∣ }= ¼x j M1, ,j is (Loredo & Wasserman 1995; Farr et al. 2015)({ ∣ }∣ )[ ( ) ( )] [ ] ( )= ¼ L L= L + L -L - L =⎪⎪⎪⎪ ⎧ ⎨ ⎩ ⎫ ⎬ ⎭x j Mp x p x1, , ,exp .