of 19
Improved Analysis of GW150914 Using a Fully Spin-Precessing Waveform Model
B. P. Abbott
etal.
*
(LIGO Scientific Collaboration and Virgo Collaboration)
(Received 4 June 2016; revised manuscript received 30 July 2016; published 21 October 2016)
This paper presents updated estimates of source
parameters for GW150914, a binary black-hole
coalescence event detected by the Laser Interfero
meter Gravitational-wave Observatory (LIGO) in
2015 [Abbott
et al.
Phys.Rev.Lett.
116
, 061102 (2016).]. Abbott
et al.
[Phys. Rev. Lett.
116
, 241102
(2016).] presented parameter estimation of the
source using a 13-dimensional, phenomenological
precessing-spin model (precessing IMRPhenom) and a
n 11-dimensional nonprecessing effective-one-
body (EOB) model calibrated to numerical-relati
vity simulations, which forces spin alignment
(nonprecessing EOBNR). Here, we present new resu
lts that include a 15-dimensional precessing-
spin waveform model (precessing EOBNR) developed within the EOB formalism. We find good
agreement with the parameters e
stimated previously [Abbott
et al.
Phys. Rev. Lett.
116
, 241102
(2016).], and we quote updated component masses of
35
þ
5
3
M
and
30
þ
3
4
M
(where errors correspond
to 90% symmetric credible intervals). We also present
slightly tighter constraints on the dimensionless
spin magnitudes of the two black holes, with a primary spin estimate
<
0
.
65
and a secondary spin
estimate
<
0
.
75
at 90% probability. Abbott
et al.
[Phys. Rev. Lett.
116
, 241102 (2016).] estimated the
systematic parameter-extraction errors due to wave
form-model uncertainty by combining the posterior
probability densities of precessing IMRPhenom and nonprecessing EOBNR. Here, we find that the two
precessing-spin models are in closer agreement, sugg
esting that these systematic errors are smaller
than previously quoted.
DOI:
10.1103/PhysRevX.6.041014
Subject Areas: Astrophysics, Gravitation
I. INTRODUCTION
The detection of the first gravitational-wave (GW)
transient, GW150914, by the Laser Interferometer
Gravitational-wave Observatory in 2015
[1]
marked the
beginning of a new kind of astronomy, fundamentally
different from electromagnetic or particle astronomy.
GW150914 was analyzed using the most accurate signal
models available at the time of observation, which were
developed under the assumption that general relativity is
the correct theory of gravity. The analysis concluded that
GW150914 was generated by the coalescence of two black
holes (BHs) of rest-frame masses
36
þ
5
4
M
and
29
þ
4
4
M
,at
a luminosity distance of
410
þ
160
180
Mpc
[2]
. Throughout this
paper, we quote parameter estimates as the median of their
posterior probability density, together with the width of the
90% symmetric credible interval.
The GW signal emitted by a binary black hole (BBH)
depends on 15 independent parameters: the BH masses and
the BH spin vectors (the intrinsic parameters); the incli-
nation and the phase of the observer in the orbital plane, the
sky location of the binary (parametrized by two angles, the
right ascension and declination), the polarization angle of
the GW, the luminosity distance of the binary, and the
time of arrival of the GW at the detector (all of which are
known as extrinsic parameters). The task of extracting all
15 parameters from interferometric detector data relies on
efficient Bayesian inference algorithms and on the avail-
ability of accurate theoretical predictions of the GW signal.
State-of-the-art numerical-relativity (NR) simulations
[3
8]
can generate very accurate BBH waveforms over a large
region of parameter space; however, this region does not
yet include (i) binary configurations that have large
dimensionless spins (
>
0
.
5
), extreme mass ratios (
<
1
=
3
),
and many GW cycles (
40
60
), except for a few cases
[8
10]
; nor does it include (ii) systems undergoing sig-
nificant spin-induced precession of the orbital plane. In
practice, parameter estimation requires very many wave-
form evaluations that span a large region of parameter
space, and a purely NR approach is possible if one coarsely
discretizes the intrinsic parameters, as has been done for
GW150914
[11]
, or constructs interpolants (surrogates)
across NR simulations
[12]
. However, a continuous sam-
pling of the intrinsic parameter space, even outside regions
where NR runs are available, is unfeasible.
The first parameter-estimation study of GW150914
[2]
used two such models: an effective-one-body
(EOB, Refs.
[13,14]
) model that restricts spins to be
aligned with the orbital angular momentum
[15]
, and a
phenomenological model that includes spin-precession
*
Full author list given at the end of the article.
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 3.0 License
. Further distri-
bution of this work must maintain attribution to the author(s) and
the published article
s title, journal citation, and DOI.
PHYSICAL REVIEW X
6,
041014 (2016)
2160-3308
=
16
=
6(4)
=
041014(19)
041014-1
Published by the American Physical Society
effects governed by four effective spin parameters
[16]
.
Here, we present updated parameter estimates using a fully
spin-precessing EOB model
[17,18]
, which is parametrized
by the full set of BBH properties listed above, including all
six BH-spin degrees of freedom, and which reflects addi-
tional physical effects described in Sec.
II
. The inclusion of
these effects motivates us to repeat the Bayesian analysis of
GW150914 with precessing EOB waveforms. This model
was not used in Ref.
[2]
because it requires costly time-
domain integration for each set of BBH parameters; thus,
not enough Monte Carlo samples had been collected by the
time the study was finalized
[19]
.
The main result of our analysis is that the two
precessing models (phenomenological and EOB) are
broadly consistent, showing largely overlapping 90%
credible intervals for all m
easured binary parameters,
more so than the precessing phenomenological and non-
precessing EOB models compared in Ref.
[2]
.Inthat
study, the parameter estimates obtained with those two
models were combined with equal weights to provide the
fiducial values quoted in Ref.
[1]
, and they were differ-
enced to characterize systematic errors due to waveform
mismodeling. Because the two precessing models yield
closer results, we are now able to report smaller combined
credible intervals, as well as smaller estimated systematic
errors. Nevertheless, the combined medians cited as
fiducial estimates in Ref.
[1]
change only slightly. In
addition, we find that some of the intrinsic parameters that
affect BBH evolution, such as the in-plane combination of
BH spins that governs precession, are constrained better
using the precessing EOB model.
Because precessing-EOB waveforms are so computa-
tionally expensive to generate, we cannot match the number
of Monte Carlo samples used in Ref.
[2]
. Thus, we carry out
a careful statistical analysis to assess the errors of our
summary statistics (posterior medians and credible inter-
vals) due to the finite number of samples. We apply the
same analysis to the precessing phenomenological and
nonprecessing EOB models, and to their combinations.
Although finite-sample errors are a factor of a few larger for
the precessing EOB model than for the other two, they
remain much smaller than the credible intervals, so none of
our conclusions is affected. Last, as a further test of the
accuracy and consistency of the two precessing models, we
use them to estimate the known parameters of a
GW150914-like NR waveform injected into LIGO data.
The resulting posteriors are similar to those found for
GW150914.
This article is organized as follows. In Sec.
II
, we discuss
the modeling of spin effects in the BBH waveforms used in
this paper. In Sec.
III
, we describe our analysis. We present
our results in Sec.
IV
and our conclusions in Sec.
V
.
Throughout the article, we adopt geometrized units,
with
G
¼
c
¼
1
.
II. MODELING ORBITAL PRECESSION
IN BBH WAVEFORM MODELS
Astrophysical stellar-mass BHs are known to possess
significant intrinsic spins, which can engender large effects
in the late phase of BBH coalescences: they affect the
evolution of orbital frequency, and (if the BH spins are not
aligned with the orbital angular momentum) they induce
the precession of the orbital plane, modulating the funda-
mental chirping structure of emitted GWs in a manner
dependent on the relative angular geometry of binary and
observatory
[20]
. While measuring BH spins is interesting
in its own right, the degree of their alignment and the
resulting degree of precession hold precious clues to the
astrophysical origin of stellar-mass BBHs
[21]
: Aligned
spins suggest that the two BHs were born from an
undisturbed binary star in which both components succes-
sively collapsed to BHs; nonaligned spins point to an origin
from capture events and three-body interactions in dense
stellar environments.
Clearly, the accurate modeling of BH-spin effects is
crucial to BBH parameter-estimation studies. Now, even
state-of-the-art semianalytical waveform models still rely
on a set of approximations that necessarily limit their
accuracy. These include finite post-Newtonian (PN) order,
calibration to a limited number of NR simulations, rotation
to precessing frames, and more. Thus, being able to
compare parameter estimates performed with different
waveform models, derived under different assumptions
and approximations (e.g., in time- vs frequency-domain
formulations), becomes desirable to assess the systematic
biases due to waveform mismodeling. While observing
consistent results does not guarantee the absence of
systematic errors (after all, multiple models could be wrong
in the same way), the fact that we do not observe
inconsistencies does increase our confidence in the models.
Such a comparison was performed in the original
parameter-estimation study of GW150914
[2]
, showing
consistency between the precessing phenomenological
model and the aligned-spin EOBNR model. This result
matched the finding that the BH spins were approximately
aligned in GW150914, or that precession effects were too
weak to be detected, because of the small number of GW
cycles and of the (putative) face-on/face-off presentation of
the binary. Nevertheless, it may be argued that the con-
clusion of consistency remained suspect because only one
model in the analysis carried information about the effects
of precession; conversely, the estimates of mismodeling
systematic errors performed in Ref.
[2]
were likely
increased by the fact that the nonprecessing model would
be biased by what little precession may be present in the
signal.
The analysis presented in this article, which relies on two
precessing-spin waveform families, removes both limita-
tions and sets up a more robust framework to assess
systematic biases in future detections where spin effects
B. P. ABBOTT
et al.
PHYS. REV. X
6,
041014 (2016)
041014-2
play a larger role. In the rest of this section, we discuss the
features and formulation of the fully precessing EOBNR
model. The reader not interested in these technical details
(and in the Bayesian-inference setup of Sec.
III
)may
proceed directly to Sec.
IV
.
The precessing EOBNR model (henceforth,
precessing
EOBNR
) used here can generate inspiral-merger-ring-
down (IMR) waveforms for coalescing, quasicircular BH
binaries with mass ratio
0
.
01
q
m
2
=m
1
1
, dimen-
sionless BH spin magnitudes
0
χ
1
;
2
j
S
1
;
2
j
=m
2
1
;
2
0
.
99
, and arbitrary BH spin orientations
[22]
. We denote
with
m
1
;
2
the masses of the component objects in the binary
and with
S
1
;
2
their spin vectors. Note that the model was
calibrated only to 38 nonspinning NR simulations that span
a smaller portion of the parameter space than defined
above, but it was not calibrated to any precessing NR
waveform (see below for more details).
The fundamental idea of EOB models consists in mapping
the conservative dynamics of a binary to that of a spinning
particle that moves in a deformed Kerr spacetime
[13,14,
23
28]
, where the magnitude of the deformation is propor-
tional to the mass ratio of the binary. This mapping can be
seen as a resummation of PN formulas
[29]
with the aim of
extending their validity to the strong-field regime. As for
dissipative effects, EOB models equate the loss of energy to
the GW luminosity, which is expressed as a sum of squared
amplitudes of the multipolar waveform modes. In the
nonprecessing limit, the inspiral-plunge waveform modes
are themselves resummations of PN expressions
[30
32]
and
are functionals of the orbital dynamics. The ringdown signal
is described by a linear superposition of the quasinormal
modes
[33
35]
of the remnant BH.
EOB models can be tuned to NR by introducing
adjustable parameters at high, unknown PN orders. For
the precessing EOB model used in this work, the relevant
calibration to NR was carried out in Ref.
[15]
against 38
NR simulations of nonprecessing-spin systems from
Ref.
[36]
, with mass ratios up to
1
=
8
and spin magnitudes
up to almost extremal for equal-mass BBHs and up to 0.5
for unequal-mass BBHs.
Furthermore, information from inspiral, merger, and
ringdown waveforms in the test-particle limit were also
included in the EOBNR model
[37,38]
. Prescriptions for
the onset and spectrum of ringdown for precessing BBHs
were first given in Ref.
[17]
and significantly improved
in Ref.
[18]
.
In the model, the BH spin vectors precess according to
d
S
1
;
2
d
t
¼
H
EOB
S
1
;
2
×
S
1
;
2
;
ð
1
Þ
when the BH spins are oriented generically, the orbital
plane precesses with respect to an inertial observer.
The orientation of the orbital plane is tracked by the
Newtonian orbital angular momentum
L
N
μ
r
×
_
r
, where
μ
m
1
m
2
=
ð
m
1
þ
m
2
Þ
and
r
is the relative BH separation.
One defines a (noninertial) precessing frame whose
z
axis
is aligned with
L
N
ð
t
Þ
, and whose
x
and
y
axes obey the
minimum-rotation prescription of Refs.
[39,40]
. In this
frame, the waveform amplitude and phase modulations
induced by precession are minimized, as pointed out in
several studies
[39
43]
.
Thus, the construction of a precessing EOB waveform
consists of the following steps: (i) Compute orbital dynam-
ics numerically, by solving Hamilton
s equation for the
EOB Hamiltonian, subject to energy loss, up until the light-
ring (or photon-orbit) crossing; (ii) generate inspiral-plunge
waveforms in the precessing frame as if the system were
not precessing
[15]
; (iii) rotate the waveforms to the inertial
frame aligned with the direction of the remnant spin;
(iv) generate the ringdown signal, and connect it smoothly
to the inspiral-plunge signal; (v) rotate the waveforms to the
inertial frame of the observer.
A phenomenological precessing-spin IMR model
(henceforth,
precessing IMRPhenom
) was proposed in
Refs.
[16,44,45]
. These waveforms are generated in the
frequency domain by rotating nonprecessing phenomeno-
logical waveforms
[46]
from a precessing frame to the
inertial frame of the observer, according to PN formulas
that describe precession in terms of Euler angles. The
underlying nonprecessing waveforms depend on the BH
masses and on the two projections of the spins on the
Newtonian angular momentum, with the spin of the BH
formed through merger adjusted to also take into account
the effect of the in-plane spin components. The influence of
the in-plane spin components on the precession is modeled
with a single-spin parameter (a function of the two BH
spins) and also depends on the initial phase of the binary in
the orbital plane. Thus, this model only has four indepen-
dent parameters to describe the 6 spin degrees of freedom,
which is justified by the analysis of dominant spin effects
performed in Ref.
[44]
.
While both precessing EOBNR and IMRPhenom models
describe spin effects, there are important differences in
how they account for precession, which is the main focus of
this paper.
(1) In precessing IMRPhenom, the precessing-frame
inspiral-plunge waveforms are strictly nonprecess-
ing waveforms, while for precessing EOBNR, some
precessional effects are included (such as spin-spin
frequency and amplitude modulations) since the
orbital dynamics that enters the nonprecessing ex-
pressions for the GW modes is fully precessing.
(2) The precessing EOBNR merger-ringdown signal is
generated in the inertial frame oriented along the
total angular momentum of the remnant
the very
frame where quasinormal mode frequencies are
computed in BH perturbation theory. By contrast,
precessing IMRPhenom generates the merger-
ringdown signal directly in the precessing frame.
IMPROVED ANALYSIS OF GW150914 USING A FULLY
...
PHYS. REV. X
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(3) The IMRPhenom precessing-frame waveforms con-
tain only the dominant
ð
2
;

2
Þ
modes
[47]
, while
precessing EOBNR also includes
ð
2
;

1
Þ
modes in
the precessing frame, although these are not cali-
brated to NR.
(4) In IMRPhenom, the frequency-domain rotation of
the GW modes from the precessing frame to the
inertial frame is based on approximate formulas (i.e.,
on the stationary-phase approximation), while pre-
cessing EOBNR computes the rotations fully in the
time domain, where the formulas are straight-
forward.
(5) In precessing IMRPhenom, the frequency-domain
formulas for the Euler angles that parametrize the
precession of the orbital plane with respect to a fixed
inertial frame involve several approximations: In-
plane spin components are orbit averaged; the
magnitude of the orbital angular momentum is
approximated by its 2PN nonspinning expression;
the evolution of frequency is approximated as
adiabatic; and the PN formulas that regulate the
behavior of the Euler angles at high frequencies
are partially resummed. By contrast, precessing
EOBNR defines these Euler angles on the basis
of the completely general motion of
L
N
ð
t
Þ
; this
motion is a direct consequence of the EOB dynam-
ics, and as such, it is sensitive to the full precessional
dynamics of the six spin components.
A priori
, it is not obvious that these approximations will
not impact parameter estimation for a generic BBH.
However, as far as GW150914 is concerned, Ref.
[2]
showed
broadly consistent results between a precessing and a non-
precessing model;
a fortiori
, we should expect similar results
between two precessing models. Indeed, the GW150914
binary is more probable to be face-off or face-on than edge-
on with respect to the line of sight to the detector, and the
component masses are almost equal
[2]
: Both conditions
imply that subdominant modes play a minor role.
The nonprecessing models that underlie both approx-
imants were tested against a large catalog of NR simu-
lations
[15,46,48]
, finding a high degree of accuracy in the
GW150914 parameter region. However, it is important to
bear in mind that these waveform models can differ from
NR outside the region in which they were calibrated, and
they do not account for all possible physical effects that are
relevant to generic BBHs, such us higher-order modes.
Finally, neither of the two precessing models has been
calibrated to any precessing NR simulation. Thus, we
cannot exclude that current precessing models are affected
by systematics. References
[17,18]
compared the precess-
ing EOBNR model to 70 NR runs with mild precession
(with mass ratios 1 to
1
=
5
, spin magnitudes up to 0.5,
generic spin orientations, and each about 15
20 orbital
cycles long) finding sky-location and polarization-averaged
overlaps typically above 97% without recalibration.
Since the generation of precessing EOBNR waveforms
[at least in the current implementation in the LIGO
Algorithm Library (LAL)] is a rather time-consuming
process (see
[19]
), when carrying out parameter-estimation
studies with this template family, we introduce a time-
saving approximation at the level of the likelihood function.
Namely, we marginalize over the arrival time and phase of
the signal as if the waveforms contained only
ð
2
;

2
Þ
inertial-frame modes since in that case the marginalization
can be performed analytically
[49]
. We have determined
that the impact of this approximation is negligible by
conducting a partial parameter-estimation study where we
do not marginalize over the arrival time and phase. We can
understand this physically for GW150914 because in a
nearly face-on/face-off binary, the
ð
2
;

1
Þ
observer-frame
modes are significantly subdominant compared to
ð
2
;

2
Þ
modes
[50]
.
III. BAYESIAN INFERENCE ANALYSIS
For each waveform model under consideration, we
estimate the posterior probability density
[51,52]
for the
BBH parameters, following the prescriptions of Ref.
[2]
.To
wit, we use the LAL implementation of parallel-tempering
Markov chain Monte Carlo and nested sampling
[49]
to
sample the posterior density
p
ð
θ
j
model
;
data
Þ
as a function
of the parameter vector
θ
:
p
ð
θ
j
model
;
data
Þ
L
ð
data
j
θ
Þ
×
p
ð
θ
Þ
:
ð
2
Þ
To obtain the likelihood
L
ð
data
j
θ
Þ
, we first generate the
GW polarizations
h
þ
ð
θ
intrinsic
Þ
and
h
×
ð
θ
intrinsic
Þ
according
to the waveform model. We then combine the polarizations
into the LIGO detector responses
h
1
;
2
by way of the
detector antenna patterns:
h
k
ð
θ
Þ¼
h
þ
ð
θ
intrinsic
Þ
F
ðþÞ
k
ð
θ
extrinsic
Þ
þ
h
×
ð
θ
intrinsic
Þ
F
ð
×
Þ
k
ð
θ
extrinsic
Þ
:
ð
3
Þ
Finally, we compute the likelihood as the sampling dis-
tribution of the residuals [i.e., the detector data
d
k
minus the
GW response
h
k
ð
θ
Þ
], under the assumption that these are
distributed as Gaussian noise characterized by the power
spectral density (PSD) of nearby data
[49]
:
L
ð
data
j
θ
Þ
exp

1
2
X
k
¼
1
;
2
h
h
k
ð
θ
Þ
d
k
j
h
k
ð
θ
Þ
d
k
i

;
ð
4
Þ
where
h
·
j
·
i
denotes the noise-weighted inner product
[53]
.
The prior probability density
p
ð
θ
Þ
follows the choices of
Ref.
[2]
. In particular, we assume uniform mass priors
m
1
;
2
½
10
;
80

M
, with the constraint
m
2
m
1
, and uni-
form spin-amplitude priors
a
1
;
2
¼j
S
1
;
2
j
=m
2
1
;
2
½
0
;
1

, with
spin directions distributed uniformly on the two-sphere;
B. P. ABBOTT
et al.
PHYS. REV. X
6,
041014 (2016)
041014-4
and we assume that sources are distributed uniformly in
Euclidian volume, with their orbital orientation distributed
uniformly on the two-sphere. All the binary parameters that
evolve during the inspiral (such as tilt angles between the
spins and the orbital angular momentum,
θ
LS
1
;
2
) are defined
at a reference GW frequency
f
ref
¼
20
Hz. Following
Ref.
[2]
, we marginalize over the uncertainty in the
calibration of LIGO data
[54]
. This broadens the posteriors
but reduces calibration biases.
To assess whether the data are
informative
with regard to
a source parameter (i.e., where it
updates
the prior
significantly), we perform a Kolmogorov-Smirnov (KS)
test. Given an empirical distribution (in our case, the
Monte Carlo posterior samples) and a probability distri-
bution (in our case, the prior), the KS test measures the
maximum deviation between the two cumulative distribu-
tions and associates a
p
-value to that: For samples
generated from the probability distribution against which
the test is performed, one expects a
p
-value around 0.5;
p
-
values smaller than 0.05 indicate that the samples come
from a different probability distribution with a high level of
significance
that is, there is only a 5% (or less) chance
that the two sets of samples come from the same distri-
bution. The outcomes of our KS tests are only statements
about how much the posteriors deviate from the respective
priors, but they do not tell us anything about the astro-
physical relevance of 90% credible intervals.
IV. RESULTS
The first question that we address is whether parameter
estimates derived using the two precessing models (pre-
cessing IMRPhenom and precessing EOBNR) are com-
patible. In particular, we compare posterior medians and
90% credible intervals (the summary statistics used in
Ref.
[2]
) for the parameters tabulated in Table I of Ref.
[2]
,
as well as additional spin parameters. The nominal values
of the medians and 5% and 95% quantiles for the two
models are listed in the
EOBNR
and
IMRPhenom
columns of Table
I
and Fig.
1
. However, it is unclear
a priori
whether any differences are due to the models
themselves or to the imperfect sampling of the posteriors in
Markov chain Monte Carlo runs. This is a concern
especially for the precessing EOBNR results since the
slower speed of EOBNR waveform generation means that
shorter chains are available for parameter estimation. To
gain trust in our comparisons, we characterize the
Monte Carlo error of the medians and quantiles by a
bootstrap analysis, as follows.
The Monte Carlo runs for the precessing IMRPhenom
model produced an equal-weight posterior sampling
TABLE I. Median values of source parameters of GW150914 as estimated with the two precessing waveform models and with an
equal-weight average of posteriors (in the
Overall
column). The models are described in the text. Subscripts and superscripts indicate
the range of the symmetric 90% credible intervals. When useful, we quote 90% credible bounds.
Precessing EOBNR
Precessing IMRPhenom
Overall
Detector-frame total mass
M=
M
71
.
6
þ
4
.
3
4
.
1
70
.
9
þ
4
.
0
3
.
9
71
.
3
þ
4
.
3
4
.
1
Detector-frame chirp mass
M
=
M
30
.
9
þ
2
.
0
1
.
9
30
.
6
þ
1
.
8
1
.
8
30
.
8
þ
1
.
9
1
.
8
Detector-frame primary mass
m
1
=
M
38
.
9
þ
5
.
1
3
.
7
38
.
5
þ
5
.
6
3
.
6
38
.
7
þ
5
.
3
3
.
7
Detector-frame secondary mass
m
2
=
M
32
.
7
þ
3
.
6
4
.
8
32
.
2
þ
3
.
6
4
.
8
32
.
5
þ
3
.
7
4
.
8
Detector-frame final mass
M
f
=
M
68
.
3
þ
3
.
8
3
.
7
67
.
6
þ
3
.
6
3
.
5
68
.
0
þ
3
.
8
3
.
6
Source-frame total mass
M
source
=
M
65
.
6
þ
4
.
1
3
.
8
65
.
0
þ
4
.
0
3
.
6
65
.
3
þ
4
.
1
3
.
7
Source-frame chirp mass
M
source
=
M
28
.
3
þ
1
.
8
1
.
7
28
.
1
þ
1
.
7
1
.
6
28
.
2
þ
1
.
8
1
.
7
Source-frame primary mass
m
source
1
=
M
35
.
6
þ
4
.
8
3
.
4
35
.
3
þ
5
.
2
3
.
4
35
.
4
þ
5
.
0
3
.
4
Source-frame secondary mass
m
source
2
=
M
30
.
0
þ
3
.
3
4
.
4
29
.
6
þ
3
.
3
4
.
3
29
.
8
þ
3
.
3
4
.
3
Source-frame final mass
M
source
f
=
M
62
.
5
þ
3
.
7
3
.
4
62
.
0
þ
3
.
7
3
.
3
62
.
2
þ
3
.
7
3
.
4
Mass ratio
q
0
.
84
þ
0
.
14
0
.
20
0
.
84
þ
0
.
14
0
.
20
0
.
84
þ
0
.
14
0
.
20
Effective inspiral spin parameter
χ
eff
0
.
02
þ
0
.
14
0
.
16
0
.
05
þ
0
.
13
0
.
15
0
.
04
þ
0
.
14
0
.
16
Effective precession spin parameter
χ
p
0
.
28
þ
0
.
38
0
.
21
0
.
35
þ
0
.
45
0
.
27
0
.
31
þ
0
.
44
0
.
23
Dimensionless primary spin magnitude
a
1
0
.
22
þ
0
.
43
0
.
20
0
.
32
þ
0
.
53
0
.
29
0
.
26
þ
0
.
52
0
.
24
Dimensionless secondary spin magnitude
a
2
0
.
29
þ
0
.
52
0
.
27
0
.
34
þ
0
.
54
0
.
31
0
.
32
þ
0
.
54
0
.
29
Final spin
a
f
0
.
68
þ
0
.
05
0
.
05
0
.
68
þ
0
.
06
0
.
06
0
.
68
þ
0
.
05
0
.
06
Luminosity distance
D
L
=
Mpc
440
þ
160
180
440
þ
150
180
440
þ
160
180
Source redshift
z
0
.
094
þ
0
.
032
0
.
037
0
.
093
þ
0
.
029
0
.
036
0
.
093
þ
0
.
030
0
.
036
Upper bound on primary spin magnitude
a
1
0.54
0.74
0.65
Upper bound on secondary spin magnitude
a
2
0.70
0.78
0.75
Lower bound on mass ratio
q
0.69
0.68
0.68
IMPROVED ANALYSIS OF GW150914 USING A FULLY
...
PHYS. REV. X
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041014-5
consisting of 27 000 approximately independent samples,
obtained by downsampling the original MCMC run by a
factor equal to the largest autocorrelation length measured
for the parameters of interest (those of Table
I
). We generate
1000 Bayesian-bootstrap weighted resamplings
[55]
of the
equal-weight population
[56]
, and for each, we compute
the weighted medians and quantiles. We characterize the
Monte Carlo error of these summary statistics as the 90%
symmetric interquantile interval across the 1000 realiza-
tions. For completeness, we apply the same analysis to the
45 000 samples of the nonprecessing EOBNR that were
employed in Ref.
[2]
.
The Monte Carlo runs for the precessing EOBNR model
produced a sampling of 2700 approximately independent
samples, obtained by selecting every 1100th sample in the
original MCMC run. Again, we generate 1000 Bayesian-
FIG. 1. Comparing nonprecessing EOBNR (light yellow, top), precessing IMRPhenom (light blue, middle), and precessing EOBNR
(light red, bottom) 90% credible intervals for select GW150914 source parameters. The darker intervals represent error estimates for
(from left to right) the 5%, 50%, and 95% quantiles, estimated by Bayesian bootstrapping.
B. P. ABBOTT
et al.
PHYS. REV. X
6,
041014 (2016)
041014-6
bootstrap resamplings, compute summary statistics on
each, and measure their variation. However, to improve
the representativeness of this analysis given the smaller
number of samples in play, we use nine additional equal-
weight populations, obtained by selecting every (
1100
þ
i
)
th sample in the original MCMC run, for
i
¼
1
;
...
;
9
.For
each of the 1000 Bayesian-bootstrap resamplings, we first
choose randomly among the ten equal-weight populations.
Monte Carlo errors are expected to shrink as the inverse
square root of the number of samples; this is indeed what
we observe, with precessing EOBNR finite-sample errors
about
ð
27 000
=
2700
Þ
1
=
2
3
times larger than for precess-
ing IMRPhenom. Table
I
and Fig.
1
present the results of
this study for several key physical parameters of the source
of GW150914. With darker colors, we display the finite-
sample error estimates on the position of the medians and
5% and 95% quantiles. Lighter colors represent the 90%
credible intervals.
Combined estimates.
To account for waveform-mis-
modeling errors in its fiducial parameter estimates, Ref.
[2]
cited quantiles for combined posteriors obtained by aver-
aging the posteriors for its two models (in Bayesian terms,
this corresponds to assuming that the observed GW signal
could have come from either model with equal posterior
probability). We repeat the same procedure for the two
precessing models, and we show the resulting estimates in
the column
Overall
of Table
I
. Quantiles are more
uncertain for the precessing combination because of the
larger finite-sampling error of precessing EOBNR.
Nevertheless, 90% credible intervals are slightly tighter
than cited in Ref.
[2]
. In the Appendix, we provide a
graphical representation of the combined estimates.
Posterior histograms: Masses and spin magnitudes.
We now discuss in some detail the salient features of
parameter posteriors. In Figs.
2
6
, we show the one-
dimensional marginalized posteriors for selected pairs of
parameters and 90% credible intervals (the dashed lines), as
obtained for the two precessing models, as well as the two-
dimensional probability density plots for the precessing
EOBNR model. In Fig.
2
, we show the posteriors for the
source-frame BH masses
m
1
;
2
: These are measured fairly
well, with statistical uncertainties around 10%. In Fig.
3
,
we show the posteriors for the dimensionless spin magni-
tudes
a
1
;
2
: The bound on
a
1
is about 20% more stringent for
precessing EOBNR. This is true even if we account for the
larger finite-sampling uncertainty in the precessing
EOBNR quantiles (see Table
I
). The final spin presented
in Table
I
and Fig.
1
was obtained, including the con-
tribution from the in-plane spin components to the final
spin
[57]
; previous publications
[1,2]
just use the contri-
bution from the aligned components of the spins, which
remains sufficient for the final mass computation. Just
using the aligned components does not change the pre-
cessing EOBNR result but gives a precessing IMRPhenom
result of
0
.
66
þ
0
.
04
0
.
06
.
Posterior histograms: Spin directions.
Figure
4
repro-
duces the disk plot of Ref.
[2]
for precessing EOBNR. In
this plot, the three-dimensional histograms of the dimen-
sionless spin vectors
S
1
;
2
=m
2
1
;
2
are projected onto a plane
perpendicular to the orbital plane; the bins are designed so
that each contains the same prior probability mass (i.e.,
histogramming the prior would result in a uniform shad-
ing). It is apparent that the data disfavor large spins aligned
or antialigned with the orbital angular momentum, con-
sistently with precessing IMRPhenom results. Because
precessing EOBNR favors smaller values of the dimen-
sionless spin magnitudes, the plot is darker towards its
FIG. 2. Posterior probability densities for the source-frame
component masses
m
source
1
and
m
source
2
, where
m
source
2
m
source
1
.
We show one-dimensional histograms for precessing EOBNR
(red) and precessing IMRPhenom (blue); the dashed vertical lines
mark the 90% credible intervals. The two-dimensional density
plot shows 50% and 90% credible regions plotted over a color-
coded posterior density function.
FIG. 3. Posterior probability densities for the dimensionless
spin magnitudes. (See Fig.
2
for details.)
IMPROVED ANALYSIS OF GW150914 USING A FULLY
...
PHYS. REV. X
6,
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