draft
Properties of the binary black hole merger GW150914
The LIGO Scientific Collaboration and The Virgo Collaboration
(compiled 12 February 2016)
On September 14, 2015, the Laser Interferometer Gravitational-wave Observatory (LIGO) detected a
gravitational-wave transient (GW150914); we characterise the properties of the source and its parameters.
The data around the time of the event were analysed coherently across the LIGO network using a suite
of accurate waveform models that describe gravitational waves from a compact binary system in general
relativity. GW150914 was produced by a nearly equal mass binary black hole of masses
36
+5
−
4
M
and
29
+4
−
4
M
(for each parameter we report the median value and the range of the
90%
credible interval). The
dimensionless spin magnitude of the more massive black hole is bound to be
<
0
.
7
(at 90% probability).
The luminosity distance to the source is
410
+160
−
180
Mpc
, corresponding to a redshift
0
.
09
+0
.
03
−
0
.
04
assuming
standard cosmology. The source location is constrained to an annulus section of
590 deg
2
, primarily in
the southern hemisphere. The binary merges into a black hole of mass
62
+4
−
4
M
and spin
0
.
67
+0
.
05
−
0
.
07
. This
black hole is significantly more massive than any other known in the stellar-mass regime.
PACS numbers: 04.80.Nn, 04.25.dg, 95.85.Sz, 97.80.–d
Introduction—
In Ref. [1] we reported the detec-
tion of gravitational waves (GWs), observed on Septem-
ber 14, 2015 at 09:50:45 UTC by the twin instruments of
the Laser Interferometer Gravitational-wave Observatory
(LIGO) located at Hanford, Washington, and Livingston,
Louisiana, in the USA [2, 3]. The transient signal, named
GW150914, was detected with a false-alarm-probability of
<
2
×
10
−
7
and has been associated with the merger of a
binary system of black holes (BHs).
Here we discuss the properties of this source and its in-
ferred parameters. The results are based on a complete
analysis of the data surrounding this event. The only in-
formation from the search-stage is the time of arrival of the
signal. Crucially, this analysis differs from the search in
the following fundamental ways: it is coherent across the
LIGO network, it uses waveform models that include the
full richness of the physics introduced by BH spins, and
it covers the full multidimensional parameter space of the
considered models with a fine (stochastic) sampling; we
also account for uncertainty in the calibration of the mea-
sured strain.
In general relativity, two objects in orbit slowly spiral
together due to the loss of energy and momentum through
gravitational radiation [4, 5]. This is in contrast to New-
tonian gravity where bodies can follow closed, elliptical
orbits [6, 7]. As the binary shrinks, the frequency and am-
plitude of the emitted GWs increase. Eventually the two
objects merge. If these objects are BHs, they form a sin-
gle perturbed BH that radiates GWs at a constant frequency
and exponentially damped amplitude as it settles to its final
state [8, 9].
An isolated BH is described by only its mass and spin,
since we expect the electric charge of astrophysical BHs to
be negligible [10–13]. Merging binary black holes (BBHs)
are therefore relatively simple systems. The two BHs are
described by eight intrinsic parameters: the masses
m
1
,
2
and spins
S
1
,
2
(magnitude and orientation) of the individ-
ual BHs. For a BH of mass
m
, the spin can be at most
Gm
2
/c
; hence it is conventional to quote the dimension-
less spin magnitude
a
=
c
|
S
|
/
(
Gm
2
)
≤
1
. Nine ad-
ditional parameters are needed to fully describe the binary:
the location (luminosity distance
D
L
, right ascension
α
and
declination
δ
); orientation (the binary’s orbital inclination
ι
and polarization
ψ
); time
t
c
and phase
φ
c
of coalescence,
and the eccentricity (two parameters) of the system.
Radiation reaction is efficient in circularising orbits [14]
before the signal enters the sensitive frequency band of the
instruments. In our analysis, we assume circular orbits (we
therefore do not include the eccentricity parameters), and
we find no evidence for residual eccentricity, see the Dis-
cussion. Under the approximation of a circular orbit, dom-
inant emission from the binary occurs at twice the orbital
frequency [15].
The gravitational waveform observed for GW150914
comprises
∼
10
cycles during the inspiral phase from
30 Hz
, followed by the merger and ringdown. The proper-
ties of the binary affect the phase and amplitude evolution
of the observed GWs allowing us to to measure the source
parameters. Here we briefly summarise these signatures,
and provide an insight into our ability to characterise the
properties of GW150914 before we present the details of
the Results; for methodological studies, we refer the reader
to [16–21] and references therein.
In general relativity, gravitational radiation is fully de-
scribed by two independent, and time-dependent polariza-
tions,
h
+
and
h
×
. Each instrument
k
measures the strain
h
k
=
F
(+)
k
h
+
+
F
(
×
)
k
h
×
,
(1)
a linear combination of the polarisations weighted by the
antenna beam patterns
F
(+
,
×
)
k
(
α,δ,ψ
)
, which depend on
the source location in the sky and the polarisation of the
waves [22, 23]. During the inspiral and at the leading order,
the GW polarizations can be expressed as
h
+
(
t
) =
A
GW
(
t
)
(
1 + cos
2
ι
)
cos
φ
GW
(
t
)
,
(2a)
h
×
(
t
) =
−
2
A
GW
(
t
) cos
ι
sin
φ
GW
(
t
)
,
(2b)
arXiv:1602.03840v1 [gr-qc] 11 Feb 2016
draft
2
where
A
GW
(
t
)
and
φ
GW
(
t
)
are the GW amplitude and
phase, respectively. For a binary viewed face-on, GWs are
circularly polarized, whereas for a binary observed edge-
on, GWs are linearly polarized.
During
the
inspiral,
the
phase
evolution
φ
GW
(
t
;
m
1
,
2
,
S
1
,
2
)
can
be
computed
using
post-
Newtonian (PN) theory, which is a perturbative expansion
in powers of the orbital velocity
v/c
[24]. For GW150914,
v/c
is in the range
≈
0
.
2
–
0
.
5
in the LIGO sensitivity
band. At the leading order, the phase evolution is driven
by a particular combination of the two masses, commonly
called the chirp mass [25],
M
=
(
m
1
m
2
)
3
/
5
M
1
/
5
'
c
3
G
[
5
96
π
−
8
/
3
f
−
11
/
3
̇
f
]
3
/
5
,
(3)
where
f
is the GW frequency,
̇
f
is its time derivative and
M
=
m
1
+
m
2
is the total mass. Additional parameters
enter at each of the following PN orders. First, the mass
ratio,
q
=
m
2
/m
1
≤
1
, and the BH spin components
parallel to the orbital angular momentum vector
L
affect
the phase evolution. The full degrees of freedom of the
spins enter at higher orders. Thus, from the inspiral, we
expect to measure the chirp mass best and only place weak
constraints on the mass ratio and (the components parallel
to
L
of) the spins of the BHs [18, 26].
Spins are responsible for an additional characteristic ef-
fect: if misaligned with respect to
L
, they cause the bi-
nary’s orbital plane to precess around the almost-constant
direction of the total angular momentum of the binary,
J
=
L
+
S
1
+
S
2
. This leaves characteristic amplitude
and phase modulations in the observed strain [27], as
ψ
and
ι
become time-dependent. The size of these modula-
tions depends crucially on the viewing angle of the source.
As the BHs get closer to each other and their veloci-
ties increase, the accuracy of the PN expansion degrades,
and eventually the full solution of Einstein’s equations is
needed to accurately describe the binary evolution. This is
accomplished using numerical relativity (NR) which, after
the initial breakthrough [28–30], has been improved con-
tinuously to achieve the sophistication of modeling needed
for our purposes. The details of the merger and ringdown
are primarily governed by the mass and spin of the final
BH. In particular, the final mass and spin determine the
(constant) frequency and decay time of the BH’s ringdown
to its final state [31]. The late stage of the coalescence al-
lows us to measure the total mass which, combined with
the measurement of the chirp mass and mass-ratio from the
early inspiral, yields estimates of the individual component
masses for the binary.
The observed frequency of the signal is redshifted by a
factor of
(1 +
z
)
, where
z
is the cosmological redshift.
There is no intrinsic mass or length scale in vacuum gen-
eral relativity, and the dimensionless quantity that incorpo-
rates frequency is
fGm/c
3
. Consequently, a redshifting
of frequency is indistinguishable from a rescaling of the
masses by the same factor [32, 33]. We therefore measure
redshifted masses
m
, which are related to source frame
masses by
m
= (1 +
z
)
m
source
. However, the GW am-
plitude
A
GW
, Eq. (2), also scales linearly with the mass
and is inversely proportional to the comoving distance in
an expanding universe. This implies that
A
GW
∝
1
/D
L
and from the GW signal alone we can directly measure the
luminosity distance, but not the redshift.
The observed time delay, and the need for the regis-
tered signal at the two sites to be consistent in amplitude
and phase, allow us to localize the source to a ring on the
sky [34, 35]. Where there is no precession, changing the
viewing angle of the system simply changes the observed
waveform by an overall amplitude and phase. Furthermore,
the two polarizations are the same up to overall amplitude
and phase. Thus, for systems with minimal precession, the
distance, binary orientation, phase at coalescence and sky
location of the source change the overall amplitude and
phase of the source in each detector, but they do not change
the signal morphology. Phase and amplitude consistency
allow us to untangle some of the geometry of the source. If
the binary is precessing, the GW amplitude and phase have
a complicated dependency on the orientation of the binary,
which provides additional information.
Our ability to characterise GW150914 as the signature
of a binary system of compact objects, as we have outlined
above, is dependent on the finite signal-to-noise ratio of the
signal and the specific properties of the underlying source.
These properties described in detail below, and the inferred
parameters for GW150914 are summarised in Table I and
Figures 1–6.
Method—
Full information about the properties of the
source is provided by the probability density function
(PDF)
p
(
~
θ
|
~
d
)
of the unknown parameters, given the two
data-streams from the instruments
~
d
.
The posterior PDF is computed through a straightfor-
ward application of Bayes’ theorem [36, 37]. It is propor-
tional to the product of the likelihood of the data given the
parameters
L
(
~
d
|
~
θ
)
, and the prior PDF on the parameters
p
(
~
θ
)
before we consider the data. From the (marginalised)
posterior PDF, shown in Figures 1–5 for selected param-
eters, we then construct credible intervals for the parame-
ters, reported in Table I.
In addition, we can compute the evidence
Z
for the
model under consideration. The evidence (also known as
marginal likelihood) is the average of the likelihood under
the prior on the unknown parameters for a specific model
choice.
At the detector output we record the data
d
k
(
t
) =
n
k
(
t
) +
h
M
k
(
t
;
~
θ
)
, where
n
k
is the noise, and
h
M
k
is the
measured strain, which differs from the physical strain
h
k
from Eq. (1) as a result of the detectors’ calibration [38].
In the frequency domain, we model the effect of calibra-
draft
3
tion uncertainty by considering
̃
h
M
k
(
f
;
~
θ
) =
̃
h
k
(
f
;
~
θ
)
[
1 +
δA
k
(
f
;
~
θ
)
]
×
exp
[
iδφ
k
(
f
;
~
θ
)
]
,
(4)
where
δA
k
(
f
;
~
θ
)
and
δφ
k
(
f
;
~
θ
)
are the frequency-
dependent amplitude and phase calibration-error functions,
respectively. These calibration-error functions are mod-
elled using a cubic spline polynomial, with five nodes per
spline model placed uniformly in
ln
f
[39].
We have analyzed the data at the time of this event using
a
coherent
analysis. Under the assumption of stationary,
Gaussian noise uncorrelated in each detector [40], the like-
lihood function for the LIGO network is [17, 41]:
L
(
~
d
|
~
θ
)
∝
exp
[
−
1
2
∑
k
=1
,
2
〈
h
M
k
(
~
θ
)
−
d
k
∣
∣
∣
h
M
k
(
~
θ
)
−
d
k
〉
]
,
(5)
where
〈·|·〉
is the noise-weighted inner product [17]. We
model the noise as a stationary Gaussian process of zero
mean and known variance, which is estimated from the
power spectrum computed using up to
1024 s
of data adja-
cent to, but not containing, the GW signal [41].
The source properties are encoded into the two polar-
izations
h
+
and
h
×
. Here we focus on the case in which
they originate from a compact binary coalescence; we use
model waveforms (described below) that are based on solv-
ing Einstein’s equations for the inspiral and merger of two
BHs.
The computation of marginalized PDFs and the model
evidences require the evaluation of multi-dimensional in-
tegrals. This is addressed by using a suite of Bayesian
parameter-estimation and model-selection algorithms tai-
lored to this problem [41]. We verify the results by us-
ing two
independent
stochastic sampling engines based on
Markov-chain Monte Carlo [42, 43] and nested sampling
[44, 45] techniques.
1
In addition to this analysis, we consider a model which is
not derived from a particular physical scenario and makes
minimal assumptions about
h
+
,
×
. In this case we com-
pute directly the posterior
p
(
~
h
|
~
d
)
by reconstructing
h
+
,
×
using a linear combination of elliptically polarized sine–
Gaussian wavelets whose amplitudes are assumed to be
consistent with a uniform source distribution [46], see Fig-
ure 6. The number of wavelets in the linear combination
is not fixed a priori but is optimized via Bayesian model
selection. This analysis directly infers the PDF of the GW
strain given the data,
p
(
~
h
|
~
d
)
.
1
The marginalized PDFs and model evidences are computed using the
LALInference
package of the LIGO Algorithm Library (LAL) software
suite available from www.lsc-group.phys.uwm.edu/lal.
BBH waveform models—
For the modelled analysis of
binary coalescences, an accurate waveform prediction for
the gravitational radiation
h
+
,
×
is essential. As a conse-
quence of the complexity of solving the two body prob-
lem in general relativity, several techniques have to be
combined to describe all stages of the binary coalescence.
While the early inspiral is well described by the analyti-
cal PN expansion [47], which relies on small velocities and
weak gravitational fields, the strong-field merger stage can
only be solved in full generality by large-scale NR sim-
ulations [28–30]. Since these pioneering works, numer-
ous improvements have enabled numerical simulations of
BBHs with sufficient accuracy for the applications consid-
ered here and for the region of parameter space of relevance
to GW150914 (see, e.g. [48–50]). Tremendous progress
has also been made in the past decade to combine analytical
and numerical approaches, and now several accurate wave-
form models are available, and they are able to describe the
entire coalescence for a large variety of possible configura-
tions [49, 51–57]. Extending and improving such models
is an active area of research, and none of the current mod-
els can capture all possible physical effects (eccentricity,
higher order gravitational modes in the presence of spins,
etc.) for all conceivable binary systems. We discuss the
current state of the art below.
In the Introduction, we outlined how the binary parame-
ters affect the observable GW signal, and now we discuss in
the BH spins greater detail. There are two main effects that
the BH spins
S
1
and
S
2
have on the phase and amplitude
evolution of the GW signal. The spin projections along
the direction of the orbital angular momentum
ˆ
L
affect the
inspiral rate of the binary. In particular, spin components
aligned (antialigned) with
ˆ
L
increase (decrease) the num-
ber of orbits from any given separation to merger with re-
spect to the nonspinning case [47, 58]. Given the limited
signal-to-noise ratio of the observed signal, it is difficult to
untangle the full degrees of freedom of the individual BH’s
spins, see e.g., [59, 60]. However, some spin information
is encoded in a dominant spin effect. Several possible
1
-
dimensional parametrizations of this effect can be found
in the literature [18, 61, 62]; here, we use a simple mass-
weighted linear combination of the spins [63, 64]
χ
eff
=
c
G
(
S
1
m
1
+
S
2
m
2
)
·
ˆ
L
M
,
(6)
which takes values between
−
1
(both BHs have maximal
spins antialigned with respect to the orbital angular mo-
mentum) and
+1
(maximal aligned spins).
Having described the effect of the two spin components
aligned with the orbital angular momentum, four in-plane
spin components remain. These lead to precession of the
spins and the orbital plane, which in turn introduces mod-
ulations in the strain amplitude and phase as measured at
the detectors. At leading order in the PN expansion, the
draft
4
equations that describe precession in a BBH are [27]
̇
L
=
G
c
2
r
3
(
B
1
S
1
⊥
+
B
2
S
2
⊥
)
×
L
,
(7)
̇
S
i
=
G
c
2
r
3
B
i
L
×
S
i
,
(8)
where
S
i
⊥
is the component of the spin perpendicular
to
L
;
r
is the orbital separation;
B
1
= 2 + 3
q/
2
and
B
2
= 2 + 3
/
(2
q
)
, and
i
=
{
1
,
2
}
. It follows from (7)
and (8) that
ˆ
L
and
ˆ
S
i
precess around the almost-constant
direction of the total angular momentum
J
. For a nearly
equal-mass binary (as we find is the case for GW150914),
the precession angular frequency can be approximated by
Ω
p
≈
7
GJ/
(
c
2
r
3
)
, and the total angular momentum is
dominated during the inspiral by the orbital contribution,
J
≈
L
. Additional, higher order spin-spin interactions can
also contribute significantly to precession effects for some
comparable-mass binaries [65, 66].
The in-plane spin components rotate within the orbital
plane at different velocities. Because of nutation of the or-
bital plane, the magnitude of the in-plane spin components
oscillates around a mean value, but those oscillations are
typically small. To first approximation, one can quantify
the level of precession in a binary by averaging over the
relative in-plane spin orientation. This is achieved by the
following effective precession spin parameter [67]
χ
p
=
c
B
1
Gm
2
1
max(
B
1
S
1
⊥
,B
2
S
2
⊥
)
>
0
,
(9)
where
χ
p
= 0
corresponds to an aligned-spin (non-
precessing) system, and
χ
p
= 1
to a binary with the max-
imum level of precession. Although the definition of
χ
p
is
motivated by configurations that undergo many precession
cycles during inspiral, we find that it is also a good approx-
imate indicator of the in-plane spin contribution for the late
inspiral and merger of GW150914.
For the analysis of GW150914, we consider waveform
models constructed within two frameworks capable of ac-
curately describing the GW signal in the parameter space of
interest. The effective-one-body (EOB) formalism [68–72]
combines perturbative results from the weak-field PN ap-
proximation with strong-field effects from the test-particle
limit. These results are resummed in a suitable Hamil-
tonian, radiation-reaction force and gravitational polariza-
tions, and are improved by calibrating (unknown) higher-
order PN terms to NR simulations. Henceforth, we use
“EOBNR” to indicate waveforms within this formalism.
Waveform models that include spin effects have been de-
veloped, for both double non-precessing spins [52, 57] (11
independent parameters) and double precessing spins [53]
(15 independent parameters). Here, we report results us-
ing the non-precessing model [52] tuned to NR simula-
tions [73]. This model is formulated as a set of differ-
ential equations that are computationally too expensive to
solve for the millions of likelihood evaluations required
for the analysis. Therefore, a frequency-domain reduced-
order model [74, 75] was implemented that faithfully rep-
resents the original model with an accuracy that is better
than the statistical uncertainty caused by the instruments’
noise.
2
Bayesian analyses that use the double precessing
spin model [53] are more time consuming and are not yet
finalized. The results will be fully reported in a future pub-
lication.
An alternative inspiral–merger–ringdown phenomeno-
logical formalism [76–78] is based on extending
frequency-domain PN expressions and hybridizing PN
and EOB with NR waveforms. Henceforth, we use “IMR-
Phenom” to indicate waveforms within this formalism.
Several waveform models that include aligned-spin effects
have been constructed within this approach [56, 63, 64],
and here we employ the most recent model based on a
fitting of untuned EOB waveforms [52] and NR hybrids
[55, 56] of non-precessing systems. To include leading-
order precession effects, aligned-spin waveforms are
rotated into precessing waveforms [79]. Although this
model captures some two-spin effects, it was principally
designed to accurately model the waveforms with respect
to an effective spin parameter similar to
χ
eff
above. The
dominant precession effects are introduced through a
lower-dimensional effective spin description [67, 80],
motivated by the same physical arguments as the definition
of
χ
p
. This provides us with an effective-precessing-spin
model [80] with 13 independent parameters.
3
All models we use are restricted to non-eccentric inspi-
rals. They also include only the dominant spherical har-
monic modes in the non-precessing limit.
Choice of priors—
We analyse coherently
T
= 8
s of
data with a uniform prior on
t
c
of width of
±
0
.
1 s
, centred
on the time reported by the online analysis [1, 81], and a
uniform prior in
[0
,
2
π
]
for
φ
c
. We consider the frequency
region between
20 Hz
, below which the sensitivity of the
instruments significantly degrades (see panel (b) of Figure
3 in Ref. [1]), and
1024 Hz
, a safe value for the highest fre-
quency contribution to radiation from binaries in the mass
range considered here.
Given the lack of any additional astrophysical con-
straints on the source at hand, our prior choices on the pa-
rameters are uninformative. We assume sources uniformly
distributed in volume, and the orbital orientation priors are
uniform on the
2
-sphere. We use uniform priors in
m
1
,
2
∈
[10
,
80] M
, with the constraint that
m
2
≤
m
1
. We use a
uniform prior in the spin magnitudes
a
1
,
2
∈
[0
,
1]
. For an-
gles subject to change due to precession effects we give val-
ues at a reference gravitational-wave frequency
f
ref
= 20
2
In LAL, as well as in technical publications, the aligned, precessing and
reduced-order EOBNR models are called
SEOBNRv2
,
SEOBNRv3
and
SEOBNRv2
ROM
DoubleSpin
, respectively.
3
In LAL, as well as in some technical publications, the model used is called
IMRPhenomPv2
.
draft
5
Hz. The priors on spin orientation for the precessing model
is uniform on the
2
-sphere. For the non-precessing model,
the prior on the spin magnitudes may be interpreted as the
dimensionless spin projection onto
ˆ
L
having a uniform dis-
tribution
[
−
1
,
1]
. This range includes binaries where the
two spins are strongly antialigned relative to one another.
Many such antialigned-spin comparable-mass systems are
unstable to large-angle precession well before entering our
sensitive band [82, 83] and could not have formed from an
asymptotically spin antialigned binary. We could exclude
those systems if we believe the binary is not precessing.
However, we do not make this assumption here and instead
accept that the models can only extract limited spin infor-
mation about a more general, precessing binary.
We also need to specify the prior ranges for the
amplitude and phase error functions
δA
k
(
f
;
~
θ
)
and
δφ
k
(
f
;
~
θ
)
. The calibration during the time of observa-
tion of GW150914 is characterised by a
1
-
σ
statistical
uncertainty of no more than
10%
in amplitude and
10
◦
in phase [1, 38]. We use zero-mean Gaussian priors on
the values of the spline at each node with widths corre-
sponding to the uncertainties quoted above [39]. Calibra-
tion uncertainties therefore add
10
parameters per instru-
ment to the model used in the analysis. For validation pur-
poses we also considered an independent method that as-
sumes frequency-independent calibration errors [84], and
obtained consistent results.
Results—
The results of the analysis using binary coa-
lescence waveforms are posterior PDFs for the parameters
describing the GW signal and the model evidence. A sum-
mary is provided in Table I. For the model evidence, we
quote (the logarithm of) the Bayes factor
B
s
/
n
=
Z
/
Z
n
,
which is the ratio of the evidence for a coherent signal hy-
pothesis divided by that for (Gaussian) noise [45]. At the
leading order, the Bayes factor and the optimal signal-to-
noise ratio
ρ
= [
∑
k
〈
h
M
k
|
h
M
k
〉
]
1
/
2
are related by
ln
B
s
/
n
≈
ρ
2
/
2
[85].
Before discussing parameter estimates in detail, we
consider how the inference is affected by the choice of
compact-binary waveform model. From Table I, we see
that the posterior estimates for each parameter are broadly
consistent across the two models, despite the fact that they
are based on different analytical approaches and that they
include different aspects of BBH spin dynamics. The mod-
els’ log Bayes factors,
288
.
7
±
0
.
2
and
290
.
1
±
0
.
2
, are also
comparable for both models: the data do not allow us to
conclusively prefer one model over the other [88]. There-
fore, we use both for the Overall column in Table I. We
combine the posterior samples of both distributions with
equal weight, in effect marginalising over our choice of
waveform model. These averaged results give our best es-
timate for the parameters describing GW150914.
In Table I, we also indicate how sensitive our results are
to our choice of waveform. For each parameter, we give
systematic errors on the boundaries of the
90%
credible
25
30
35
40
45
50
m
source
1
/
M
̄
20
25
30
35
m
source
2
/
M
̄
Overall
IMRPhenom
EOBNR
FIG. 1. Posterior PDFs for the source-frame component masses
m
source
1
and
m
source
2
, where
m
source
2
≤
m
source
1
.
In the
1
-dimensional marginalised distributions we show the Overall
(solid black), IMRPhenom (blue) and EOBNR (red) PDFs; the
dashed vertical lines mark the
90%
credible interval for the Over-
all PDF. The
2
-dimensional plot shows the contours of the
50%
and
90%
credible regions plotted over a colour-coded posterior
density function.
intervals due to the uncertainty in the waveform models
considered in the analysis; the quoted values are the
90%
range of a normal distribution estimated from the variance
of results from the different models.
4
Assuming normally
distributed error is the least constraining choice [89] and
gives a conservative estimate. The uncertainty from wave-
form modelling is less significant than statistical uncer-
tainty; therefore, we are confident that the results are ro-
bust against this potential systematic error. We consider
this point in detail later in the paper.
The analysis presented here yields an optimal coherent
signal-to-noise ratio of
ρ
= 25
.
1
+1
.
7
−
1
.
7
. This value is higher
than the one reported by the search [1, 3] because it is ob-
tained using a finer sampling of (a larger) parameter space.
GW150914’s source corresponds to a stellar-mass BBH
with individual source-frame masses
m
source
1
= 36
+5
−
4
M
and
m
source
2
= 29
+4
−
4
M
, as shown in Table I and Figure 1.
4
If
X
were an edge of a credible interval, we quote systematic uncertainty
±
1
.
64
σ
sys
using the estimate
σ
2
sys
= [(
X
EOBNR
−
X
Overall
)
2
+
(
X
IMRPhenom
−
X
Overall
)
2
]
/
2
. For parameters with bounded ranges,
like the spins, the normal distributions should be truncated. However, for
transparency, we still quote the
90%
range of the uncut distributions. These
numbers provide estimates of the order of magnitude of the potential sys-
tematic error.
draft
6
TABLE I. Summary of the parameters that characterise GW150914. For model parameters we report the median value as well as
the range of the symmetric
90%
credible interval [86]; where useful, we also quote
90%
credible bounds. For the logarithm of the
Bayes factor for a signal compared to Gaussian noise we report the mean and its
90%
standard error from
4
parallel runs with a nested
sampling algorithm [45]. The source redshift and source-frame masses assume standard cosmology [87]. The spin-aligned EOBNR
and precessing IMRPhenom waveform models are described in the text. Results for the effective precession spin parameter
χ
p
used in
the IMRPhenom model are not shown as we effectively recover the prior; we constrain
χ
p
<
0
.
81
at
90%
probability, see left panel of
Figure 5. The Overall results are computed by averaging the posteriors for the two models. For the Overall results we quote both the
90%
credible interval or bound and an estimate for the
90%
range of systematic error on this determined from the variance between
waveform models.
EOBNR
IMRPhenom
Overall
Detector-frame total mass
M/
M
70
.
3
+5
.
3
−
4
.
8
70
.
7
+3
.
8
−
4
.
0
70
.
5
+4
.
6
±
0
.
9
−
4
.
5
±
1
.
0
Detector-frame chirp mass
M
/
M
30
.
2
+2
.
5
−
1
.
9
30
.
5
+1
.
7
−
1
.
8
30
.
3
+2
.
1
±
0
.
4
−
1
.
9
±
0
.
4
Detector-frame primary mass
m
1
/
M
39
.
4
+5
.
5
−
4
.
9
38
.
3
+5
.
5
−
3
.
5
38
.
8
+5
.
6
±
0
.
9
−
4
.
1
±
0
.
3
Detector-frame secondary mass
m
2
/
M
30
.
9
+4
.
8
−
4
.
4
32
.
2
+3
.
6
−
5
.
0
31
.
6
+4
.
2
±
0
.
1
−
4
.
9
±
0
.
6
Detector-frame final mass
M
f
/
M
67
.
1
+4
.
6
−
4
.
4
67
.
4
+3
.
4
−
3
.
6
67
.
3
+4
.
1
±
0
.
8
−
4
.
0
±
0
.
9
Source-frame total mass
M
source
/
M
65
.
0
+5
.
0
−
4
.
4
64
.
6
+4
.
1
−
3
.
5
64
.
8
+4
.
6
±
1
.
0
−
3
.
9
±
0
.
5
Source-frame chirp mass
M
source
/
M
27
.
9
+2
.
3
−
1
.
8
27
.
9
+1
.
8
−
1
.
6
27
.
9
+2
.
1
±
0
.
4
−
1
.
7
±
0
.
2
Source-frame primary mass
m
source
1
/
M
36
.
3
+5
.
3
−
4
.
5
35
.
1
+5
.
2
−
3
.
3
35
.
7
+5
.
4
±
1
.
1
−
3
.
8
±
0
.
0
Source-frame secondary mass
m
source
2
/
M
28
.
6
+4
.
4
−
4
.
2
29
.
5
+3
.
3
−
4
.
5
29
.
1
+3
.
8
±
0
.
2
−
4
.
4
±
0
.
5
Source-fame final mass
M
source
f
/
M
62
.
0
+4
.
4
−
4
.
0
61
.
6
+3
.
7
−
3
.
1
61
.
8
+4
.
2
±
0
.
9
−
3
.
5
±
0
.
4
Mass ratio
q
0
.
79
+0
.
18
−
0
.
19
0
.
84
+0
.
14
−
0
.
21
0
.
82
+0
.
16
±
0
.
01
−
0
.
21
±
0
.
03
Effective inspiral spin parameter
χ
eff
−
0
.
09
+0
.
19
−
0
.
17
−
0
.
03
+0
.
14
−
0
.
15
−
0
.
06
+0
.
17
±
0
.
01
−
0
.
18
±
0
.
07
Dimensionless primary spin magnitude
a
1
0
.
32
+0
.
45
−
0
.
28
0
.
31
+0
.
51
−
0
.
27
0
.
31
+0
.
48
±
0
.
04
−
0
.
28
±
0
.
01
Dimensionless secondary spin magnitude
a
2
0
.
57
+0
.
40
−
0
.
51
0
.
39
+0
.
50
−
0
.
34
0
.
46
+0
.
48
±
0
.
07
−
0
.
42
±
0
.
01
Final spin
a
f
0
.
67
+0
.
06
−
0
.
08
0
.
67
+0
.
05
−
0
.
05
0
.
67
+0
.
05
±
0
.
00
−
0
.
07
±
0
.
03
Luminosity distance
D
L
/
Mpc
390
+170
−
180
440
+140
−
180
410
+160
±
20
−
180
±
40
Source redshift
z
0
.
083
+0
.
033
−
0
.
036
0
.
093
+0
.
028
−
0
.
036
0
.
088
+0
.
031
±
0
.
004
−
0
.
038
±
0
.
009
Upper bound on primary spin magnitude
a
1
0
.
65
0
.
71
0
.
69
±
0
.
05
Upper bound on secondary spin magnitude
a
2
0
.
93
0
.
81
0
.
88
±
0
.
10
Lower bound on mass ratio
q
0
.
64
0
.
67
0
.
65
±
0
.
03
Log Bayes factor
ln
B
s
/
n
288
.
7
±
0
.
2
290
.
1
±
0
.
2
—
The two BHs are nearly equal mass. We bound the mass
ratio to the range
0
.
65
≤
q
≤
1
with
90%
probability.
For comparison, the highest observed neutron star mass is
2
.
01
±
0
.
04 M
[90], and the conservative upper-limit for
the mass of a stable neutron star is
3 M
[91, 92]. The
masses inferred from GW150914 are an order of magni-
tude larger than these values, which implies that these two
compact objects of GW150914 are BHs, unless exotic al-
ternatives, e.g., boson stars [93], do exist. This result estab-
lishes the presence of stellar-mass BBHs in the Universe. It
also proves that BBHs formed in Nature can merge within
an Hubble time [94].
To convert the masses measured in the detector frame to
physical source-frame masses, we required the redshift of
the source. As discussed in the Introduction, GW obser-
vations are directly sensitive to the luminosity distance to a
source, but not the redshift [95]. We find that GW150914 is
at
D
L
= 410
+160
−
180
Mpc
. Assuming a flat
Λ
CDM cosmol-
ogy with Hubble parameter
H
0
= 67
.
9 km s
−
1
Mpc
−
1
and matter density parameter
Ω
m
= 0
.
306
[87], the in-
ferred luminosity distance corresponds to a redshift of
z
=
0
.
09
+0
.
03
−
0
.
04
.
The luminosity distance is strongly correlated to the in-
clination of the orbital plane with respect to the line of
sight [17]. For precessing systems, the orientation of the
orbital plane is time-dependent. We therefore describe the
source inclination by
θ
JN
, the angle between the total an-
gular momentum (which typically is approximately con-
stant throughout the inspiral) and the line of sight, and we
quote its value at a reference gravitational-wave frequency
f
ref
= 20
Hz. The posterior PDF shows that an orientation
of the total orbital angular momentum of the BBH strongly
draft
7
0
◦
30
◦
60
◦
90
◦
120
◦
150
◦
180
◦
θ
JN
0
200
400
600
800
D
L
/
Mpc
Overall
IMRPhenom
EOBNR
FIG. 2. Posterior PDFs for the source luminosity distance
D
L
and
the binary inclination
θ
JN
. In the
1
-dimensional marginalised
distributions we show the Overall (solid black), IMRPhenom
(blue) and EOBNR (red) PDFs; the dashed vertical lines mark the
90%
credible interval for the Overall PDF. The
2
-dimensional
plot shows the contours of the
50%
and
90%
credible regions
plotted over a colour-coded PDF.
misaligned to the line of sight is disfavoured; the probabil-
ity that
45
◦
< θ
JN
<
135
◦
is
0
.
35
.
The masses and spins of the BHs in a (circular) binary
are the only parameters needed to determine the final mass
and spin of the BH that is produced at the end of the
merger. Appropriate relations are embedded intrinsically
in the waveform models used in the analysis, but they do
not give direct access to the parameters of the remnant BH.
However, applying the fitting formula calibrated to non-
precessing NR simulations provided in [96] to the posterior
for the component masses and spins [97], we infer the mass
and spin of the remnant BH to be
M
source
f
= 62
+4
−
4
M
,
and
a
f
= 0
.
67
+0
.
05
−
0
.
07
, as shown in Figure 3 and Table I.
These results are fully consistent with those obtained us-
ing an independent non-precessing fit [55]. The systematic
uncertainties of the fit are much smaller than the statistical
uncertainties. The value of the final spin is a consequence
of conservation of angular momentum in which the total
angular momentum of the system (which for a nearly equal
mass binary, such as GW150914’s source, is dominated by
the orbital angular momentum) is converted partially into
the spin of the remnant black hole and partially radiated
away in GWs during the merger. Therefore, the final spin
is more precisely determined than either of the spins of the
binary’s BHs.
The calculation of the final mass also provides an esti-
50
55
60
65
70
M
source
f
/
M
̄
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
a
f
Overall
IMRPhenom
EOBNR
FIG. 3. PDFs for the source-frame mass and spin of the rem-
nant BH produced by the coalescence of the binary. In the
1
-dimensional marginalised distributions we show the Overall
(solid black), IMRPhenom (blue) and EOBNR (red) PDFs; the
dashed vertical lines mark the
90%
credible interval for the Over-
all PDF. The
2
-dimensional plot shows the contours of the
50%
and
90%
credible regions plotted over a colour-coded PDF.
mate of the total energy emitted in GWs. GW150914 ra-
diated a total of
3
.
0
+0
.
5
−
0
.
5
M
c
2
in GWs, the majority of
which was at frequencies in LIGO’s sensitive band. These
values are fully consistent with those given in the literature
for NR simulations of similar binaries [98, 99]. The ener-
getics of a BBH merger can be estimated at the order of
magnitude level using simple Newtonian arguments. The
total energy of a binary system at separation
r
is given by
E
≈
(
m
1
+
m
2
)
c
2
−
Gm
1
m
2
/
(2
r
)
. For an equal-mass
system, and assuming the inspiral phase to end at about
r
≈
5
GM/c
2
, then around
2
–
3%
of the initial total energy
of the system is emitted as GWs. Only a fully general rela-
tivistic treatment of the system can accurately describe the
physical process during the final strong-field phase of the
coalescence. This indicates that a comparable amount of
energy is emitted during the merger portion of GW150914,
leading to
≈
5%
of the total energy emitted.
We further infer the peak GW luminosity achieved dur-
ing the merger phase by applying to the posteriors a sep-
arate fit to non-precessing NR simulations [100]. The
source reached a maximum instantaneous GW luminosity
of
3
.
6
+0
.
5
−
0
.
4
×
10
56
erg s
−
1
= 200
+30
−
20
M
c
2
/
s
. Here, the
uncertainties include an estimate for the systematic error
of the fit as obtained by comparison with a separate set
of precessing NR simulations, in addition to the dominant
statistical contribution. An order-of-magnitude estimate of
the luminosity corroborates this result. For the dominant