of 17
An improved analysis of GW150914 using a fully spin-precessing waveform model
The LIGO Scientific Collaboration and The Virgo Collaboration
This paper presents updated estimates of source parameters for GW150914, a binary black-hole
coalescence event detected by the Laser Interferometer Gravitational-wave Observatory (LIGO)
on September 14, 2015 [1]. Reference [2] presented parameter estimation of the source using
a 13-dimensional, phenomenological precessing-spin model (precessing IMRPhenom) and a 11-
dimensional nonprecessing effective-one-body (EOB) model calibrated to numerical-relativity sim-
ulations, which forces spin alignment (nonprecessing EOBNR). Here we present new results that
include a 15-dimensional precessing-spin waveform model (precessing EOBNR) developed within
the EOB formalism. We find good agreement with the parameters estimated previously [2], and
we quote updated component masses of 35
+5
3
M
and 30
+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 esti-
mate
<
0
.
75 at 90% probability. Reference [2] estimated the systematic parameter-extraction errors
due to waveform-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, suggesting that these systematic errors are smaller than previously quoted.
PACS numbers: 04.80.Nn, 04.25.dg, 95.85.Sz, 97.80.–d
I. INTRODUCTION
The detection of the first gravitational-wave (GW)
transient, GW150914 , by the Laser Interferometer
Gravitational-wave Observatory (LIGO) on Septem-
ber 14, 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 ob-
servation, 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
+5
4
M
and 29
+4
4
M
, at a luminosity distance
of 410
+160
180
Mpc [2]. Throughout this paper, we quote pa-
rameter estimates as the median of their posterior prob-
ability density, together with the width of the 90% sym-
metric credible interval.
The GW signal emitted by a binary black hole (BBH)
depends on 15 independent parameters: the BH masses,
the BH spin vectors, the line of sight to the detector
(parametrized by two angles), the sky location of the bi-
nary (parametrized by two angles), the polarization an-
gle of the GW, the luminosity distance of the binary,
and the time of arrival of the GW at the detector. The
task of extracting all 15 parameters from interferomet-
ric detector data relies on efficient Bayesian inference al-
gorithms and on the availability of accurate theoretical
predictions of the GW signal. State-of-the-art numerical-
relativity (NR) simulations [3–8] can generate very accu-
rate BBH waveforms over a large region of parameter
space; however, this region does not yet include (i) bi-
nary configurations that have both large dimensionless
spins (
>
0
.
5), exterme mass ratios (
>
1
/
3) and many
GW cycles (
40–60), except for a few cases [8–10]; nor
does it include (ii) systems undergoing significant spin-
induced precession of the orbital plane. In practice, since
parameter estimation requires very many waveform eval-
uations that span a large region of the parameter space, a
purely NR approach is unfeasible. Therefore, great effort
has been devoted to building semi-analytical waveform
models that are more efficient computationally than full-
fledged NR, while still sufficiently accurate for the pur-
poses of extracting unbiased physical information from
the data.
The first parameter-estimation study of GW150914
[2] used two such models: an effective-one-body (EOB,
[11, 12]) model that restricts spins to be aligned with the
orbital angular momentum [13], and a phenomenological
model that includes spin-precession effects governed by
four effective spin parameters [14]. Here we present up-
dated parameter estimates using a
fully spin-precessing
EOB model [15, 16], which is parametrized by the full set
of BBH properties listed above, including all six BH-spin
degrees of freedom, and which reflects additional phys-
ical 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 yet been collected
by the time the study was finalized.
The main result of our analysis is that the two pre-
cessing models (phenomenological and EOB) are broadly
consistent, more so than the precessing phenomenological
and nonprecessing EOB models compared in Ref. [2]. In
that 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 dif-
ferenced to characterize systematic errors due to wave-
form 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
arXiv:1606.01210v1 [gr-qc] 3 Jun 2016
2
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
combina-
tion 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 num-
ber 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 cred-
ible intervals) due to the finite number of samples. We
apply the same analysis to the precessing phenomenolog-
ical and nonprecessing EOB models, and to their com-
binations. 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 are affected. Last, as
a further test on 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
=
c
= 1.
II. MODELING ORBITAL PRECESSION IN
BBH WAVEFORM MODELS
Astrophysical stellar-mass BHs are known to possess
significant intrinsic spins, which can engender large ef-
fects 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 fundamental chirping structure of emitted GWs in
a manner dependent on the relative angular geometry
of binary and observatory [17]. 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 [18]: aligned spins suggest that the two BHs were
born from an undisturbed binary star in which both com-
ponents successively 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 ac-
curacy. These include finite PN order, calibration to a
limited number of NR simulations, rotation to precessing
frames, and more. Thus, being able to compare parame-
ter estimates performed with different waveform models,
derived under different assumptions and approximations
(e.g., in time- vs. frequency-domain formulations), be-
comes desirable to assess the systematic biases due to
waveform mismodeling. While observing consistent re-
sults does not guarantee the absence of systematics (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 re-
sult matched the finding that the BH spins were approx-
imately aligned in GW150914, or that precession effects
were too weak to be detected, because of the small num-
ber of GW cycles and of the (putative) face-on/face-off
presentation of the binary. Nevertheless, it may be ar-
gued that the conclusion of consistency remained suspect,
because only one model in the analysis carried informa-
tion about the effects of precession; conversely, the esti-
mates of mismodeling systematics 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 lim-
itations, and sets up a more robust framework to assess
systematic biases in future detections where spin effects
play a larger role. In the rest of this section, we dis-
cuss the features and formulation of the fully precessing
EOBNR model. The reader not interested in these tech-
nical details (and in the Bayesian-inference setup of Sec.
III) may proceed directly to Sec. IV.
The precessing EOBNR model (henceforth, “pre-
cessing EOBNR”) used here describes inspiral–merger–
ringdown (IMR) waveforms for coalescing, quasicircu-
lar BH binaries with mass ratio 0
.
01
q
m
2
/m
1
1, dimensionless BH spin magnitudes 0
χ
1
,
2
|
S
1
,
2
|
/m
2
1
,
2
0
.
99, and arbitrary BH spin orientations.
1
We denote with
m
1
,
2
the masses of the component ob-
jects in the binary, and with
S
1
,
2
their spin vectors.
The fundamental idea of EOB models consists in map-
ping the conservative dynamics of a binary to that of a
spinning particle that moves in a deformed Kerr space-
time [11, 12, 19–24], where the magnitude of the defor-
mation is proportional to the mass ratio of the binary.
This mapping can be seen as a resummation of post-
Newtonian (PN) formulas [25] with the aim of extending
their validity to the strong-field regime. As for dissipa-
tive 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
1
In LIGO Algorithm Library (LAL), as well as in technical pub-
lications, the precessing EOBNR model that we use is called
SEOBNRv3
.
3
are themselves resummations of PN expressions [26–28],
and are functionals of the orbital dynamics. The ring-
down signal is described by a linear superposition of the
quasinormal modes [29–31] of the remnant BH.
EOB models can be tuned to NR by introducing ad-
justable parameters at high, unknown PN orders. For
the precessing EOB model used in this work, the rele-
vant calibration to NR was carried out in Ref. [13] against
38 NR simulations of nonprecessing-spins systems from
Ref. [32], 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 [33, 34]. Prescriptions for
the onset and spectrum of ringdown for precessing BBHs
were first given in Ref. [15], and significantly improved
in Ref. [16].
In the model, the BH spin vectors precess according to
d
S
1
,
2
d
t
=
∂H
EOB
S
1
,
2
×
S
1
,
2
;
(1)
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 New-
tonian orbital angular momentum
L
N
μ
r
×
̇
r
, where
μ
m
1
m
2
/
(
m
1
+
m
2
) and
r
is the relative BH separa-
tion. One defines a (noninertial) precessing frame whose
z
-axis is aligned with
L
N
(
t
), and whose
x
- and
y
-axes
obey the mimimum-rotation prescription of Ref. [35, 36].
In this frame, the waveform amplitude and phase modu-
lations induced by precession are minimized, as pointed
out in several studies [35–39].
Thus, the construction of a precessing EOB waveform
consists of the following steps: (i) compute orbital dy-
namics 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 [13]; (iii) rotate the wave-
forms 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) ro-
tate the waveforms to the inertial frame of the observer.
A phenomenological precessing-spins IMR model
(henceforth, “precessing IMRPhenom”) was proposed in
Refs. [40, 41].
2
These waveforms are generated in the
frequency domain by rotating nonprecessing phenomeno-
logical waveforms [42] from a precessing frame to the in-
ertial 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
2
In LAL this precessing model is called
IMRPhenomPv2
.
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 mod-
eled with a single spin parameter (a function of the two
BH spins), and depends also on the initial phase of the bi-
nary in the orbital plane. Thus, this model has only four
independent parameters to describe the six spin degrees
of freedom, which is justified by the analysis of dominant
spin effects performed in Ref. [40].
While both precessing EOBNR and IMRPhenom mod-
els 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 frequen-
cies are computed in BH perturbation theory.
By contrast, precessing IMRPhenom generates the
merger–ringdown signal directly in the precessing
frame.
3. The IMRPhenom precessing-frame waveforms con-
tain only the dominant (2
,
±
2) modes, while pre-
cessing EOBNR includes also (2
,
±
1) modes in the
precessing frame, although these are not calibrated
to NR.
4. In IMRPhenom, the frequency-domain rotation of
the GW modes from the precessing frame to the in-
ertial 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 straightfor-
ward.
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 ap-
proximated by its 2PN nonspinning expression; the
evolution of frequency is approximated as adia-
batic; and the PN formulas that regulate the be-
havior of the Euler angles at high frequencies are re-
summed partially. By contrast, precessing EOBNR
defines these Euler angles on the basis of the com-
pletely general motion of
L
N
(
t
); this motion is a
direct consequence of the EOB dynamics, and as
such it is sensitive to the full precessional dynam-
ics of the six spin components.
4
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 nonprecessing model;
a fortiori
we should expect
similar results between two precessing models. Indeed,
the GW150914 binary is most likely face-off or face-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 approxi-
mants were tested against a large catalog of NR simula-
tions [13, 42, 43], finding a high degree of accuracy in the
GW150914 parameter region. However, it is important
to bare 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 af-
fected by systematics.
Since the generation of precessing EOBNR waveforms
(at least in the current implementation in LAL) is
a rather time-consuming process,
3
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 (2
,
±
2) inertial-frame modes, since in that
case the marginalization can be performed analytically.
We have determined that the impact of this approxi-
mation is negligible by conducting a partial parameter-
estimation study where we do not marginalize over the
arrival time and phase. We can understand this physi-
cally for GW150914 because in a nearly face-on/face-off
binary the (2
,
±
1) observer-frame modes are significantly
sub-dominant compared to (2
,
±
2) modes.
4
III. BAYESIAN INFERENCE ANALYSIS
For each waveform model under consideration, we
estimate the posterior probability density [44, 45] 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
3
Generating a nonspinning, equal-mass binary black-hole wave-
form for a total mass of 70 M
from a starting frequency of
20 Hz is about a factor of 20 slower for precessing EOBNR than
for precessing IMRPhenom.
4
By construction, the precessing IMRPhenom inertial-frame po-
larizations
h
+
,
×
depend on the arrival time and phase exactly as
they would in a model that includes only (2
,
±
2) inertial-frame
modes. Thus, although precessing IMRPhenom does include
(2
,
±
1) inertial-frame modes, the analytical marginalization that
we just discussed is exact.
nested sampling [46] to sample the posterior density
p
(
θ
|
model, data) as a function of the parameter vector
θ
:
p
(
θ
|
model, data)
∝L
(data
|
θ
)
×
p
(
θ
)
.
(2)
To obtain the likelihood
L
(data
|
θ
), we first generate the
GW polarizations
h
+
(
θ
intrinsic
) and
h
×
(
θ
intrinsic
) accord-
ing to the waveform model. We then combine the polar-
izations into the LIGO detector responses
h
1
,
2
by way of
the detector antenna patterns:
h
k
(
θ
) =
h
+
(
θ
intrinsic
)
F
(+)
k
(
θ
extrinsic
)
+
h
×
(
θ
intrinsic
)
F
(
×
)
k
(
θ
extrinsic
)
.
(3)
Finally, we compute the likelihood as the sampling dis-
tribution of the residuals (i.e., the detector data
d
k
mi-
nus the GW response
h
k
(
θ
)), under the assumption that
these are distributed as Gaussian noise characterized by
the power spectral density (PSD) of nearby data [46]:
L
(data
|
θ
)
exp
1
2
k
=1
,
2
h
k
(
θ
)
d
k
|
h
k
(
θ
)
d
k
,
(4)
where
〈·|·〉
denotes the noise-weighted inner product [47].
The prior probability density
p
(
θ
) follows the choices
of Ref. [2]. In particular, we assume uniform mass priors
m
1
,
2
[10
,
80] M
, with the constraint
m
2
m
1
, and
uniform spin-amplitude priors
a
1
,
2
=
|
S
1
,
2
|
/m
2
1
,
2
[0
,
1],
with spin directions distributed uniformly on the two-
sphere; and we assume that sources are distributed uni-
formly 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 mo-
mentum,
θ
LS
1
,
2
) are defined at a reference GW frequency
f
ref
= 20 Hz.
Following [2], we marginalize over the uncertainty in
the calibration of LIGO data [48]. This broadens the pos-
teriors but reduces calibration biases. For the precessing
EOBNR analysis, we also marginalize over the time of
arrival and reference phase of the GW signal, following
the prescription of Ref. [49].
To assess whether the data is
informative
with regard
to a source parameter (i.e., where it
updates
the prior sig-
nificantly), we perform a Kolmogorov–Smirnov (KS) test.
Given an empirical distribution (in our case, the Monte
Carlo posterior samples) and a probability distribution
(in our case, the prior), the KS test measures the maxi-
mum deviation between the two cumulative distributions
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 sig-
nificance. The outcomes of our KS tests are only state-
ments about how much the posteriors deviate from the
respective priors, but they do not tell us anything about
the astrophysical relevance of 90% credible intervals.
5
IV. RESULTS
The first question that we address is whether param-
eter estimates performed using the two precessing mod-
els (precessing IMRPhenom and precessing EOBNR) are
compatible. In particular, we wish to compare 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 nom-
inal values of the medians and 5% and 95% quantiles
for the two models are listed in the “EOBNR” and “IM-
RPhenom” columns of Table I. However, it is unclear
a priori
whether any differences are due to the models
themselves, or to the imperfect sampling of the posteri-
ors in Markov Chain Monte Carlo runs. This is a con-
cern especially for the precessing EOBNR results, since
the slower speed of EOBNR waveform generation means
that shorter chains are available for parameter estima-
tion. 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 con-
sisting of 27,000 approximately independent samples, ob-
tained by downsampling the original MCMC run by a fac-
tor equal to the largest autocorrelation length measured
for the parameters of interest (those of Table I). We gen-
erate 1,000 Bayesian-bootstrap weighted resamplings [50]
of the equal-weight population,
5
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 1,000 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 2,700 approximately in-
dependent samples, obtained by selecting every 11th
sample in the original MCMC run. Again we gener-
ate 1,000 Bayesian-bootstrap resamplings, compute sum-
mary statistics on each, and measure their variation.
However, to improve the representativeness of this anal-
ysis given the smaller number of samples in play, we use
9 additional equal-weight populations, obtained by se-
lecting every (11 +
i
)-th sample in the original MCMC
run, for
i
= 1
,...,
9. For each of the 1,000 Bayesian-
bootstrap resamplings, we first choose randomly among
the 10 equal-weight populations.
Monte Carlo errors are expected to shrink as the in-
verse square root of the number of samples; this is indeed
what we observe, with precessing EOBNR finite-sample
errors
(27,000
/
2,700)
1
/
2
3 times larger than for pre-
cessing IMRPhenom. Table I and Figure 1 present the
5
For
n
samples, this involves generating 1,000 realizations of
weights according to the (
n
1)-variate Dirichlet distribution.
results of this study for several key physical parameters
of the source of GW150914. We display with darker col-
ors 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-
mismodeling errors in its fiducial parameter estimates,
Ref. [2] cited quantiles for combined posteriors ob-
tained by averaging 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 prior probability). We repeat the same pro-
cedure 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 combina-
tion due to 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 precess-
ing 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
magnitudes
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 1). The final spin
presented in Table I and Figure 1 was obtained including
the contribution from the in-plane spin components to
the final spin [51]; previous publications [1, 2] just use the
contribution from the aligned components of the spins,
which remains sufficient for the final mass computation.
Just using the aligned components does not change the
precessing EOBNR result, but gives a precessing IMR-
Phenom result of 0
.
66
+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 di-
mensionless 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 proba-
bility mass (i.e., histogramming the prior would result
in a uniform shading). It is apparent that the data
disfavor large spins aligned or antialigned with the or-
bital angular momentum, consistently with precessing
IMRPhenom results. Because precessing EOBNR favors
smaller values of the dimensionless spin magnitudes, the
plot is darker toward its center than its counterpart in
Ref. [2]. In agreement with that paper, our analysis
does not support strong statements on the orientation
of the BH spins with respect to the orbital angular mo-
6
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
+4
.
3
4
.
1
70
.
9
+4
.
0
3
.
9
71
.
3
+4
.
3
4
.
1
Detector-frame chirp mass
M
/
M
30
.
9
+2
.
0
1
.
9
30
.
6
+1
.
8
1
.
8
30
.
8
+1
.
9
1
.
8
Detector-frame primary mass
m
1
/
M
38
.
9
+5
.
1
3
.
7
38
.
5
+5
.
6
3
.
6
38
.
7
+5
.
3
3
.
7
Detector-frame secondary mass
m
2
/
M
32
.
7
+3
.
6
4
.
8
32
.
2
+3
.
6
4
.
8
32
.
5
+3
.
7
4
.
8
Detector-frame final mass
M
f
/
M
68
.
3
+3
.
8
3
.
7
67
.
6
+3
.
6
3
.
5
68
.
0
+3
.
8
3
.
6
Source-frame total mass
M
source
/
M
65
.
6
+4
.
1
3
.
8
65
.
0
+4
.
0
3
.
6
65
.
3
+4
.
1
3
.
7
Source-frame chirp mass
M
source
/
M
28
.
3
+1
.
8
1
.
7
28
.
1
+1
.
7
1
.
6
28
.
2
+1
.
8
1
.
7
Source-frame primary mass
m
source
1
/
M
35
.
6
+4
.
8
3
.
4
35
.
3
+5
.
2
3
.
4
35
.
4
+5
.
0
3
.
4
Source-frame secondary mass
m
source
2
/
M
30
.
0
+3
.
3
4
.
4
29
.
6
+3
.
3
4
.
3
29
.
8
+3
.
3
4
.
3
Source-frame final mass
M
source
f
/
M
62
.
5
+3
.
7
3
.
4
62
.
0
+3
.
7
3
.
3
62
.
2
+3
.
7
3
.
4
Mass ratio
q
0
.
84
+0
.
14
0
.
20
0
.
84
+0
.
14
0
.
20
0
.
84
+0
.
14
0
.
20
Effective inspiral spin parameter
χ
eff
0
.
02
+0
.
14
0
.
16
0
.
05
+0
.
13
0
.
15
0
.
04
+0
.
14
0
.
16
Effective precession spin parameter
χ
p
0
.
28
+0
.
38
0
.
21
0
.
35
+0
.
45
0
.
27
0
.
31
+0
.
44
0
.
23
Dimensionless primary spin magnitude
a
1
0
.
22
+0
.
43
0
.
20
0
.
32
+0
.
53
0
.
29
0
.
26
+0
.
52
0
.
24
Dimensionless secondary spin magnitude
a
2
0
.
29
+0
.
52
0
.
27
0
.
34
+0
.
54
0
.
31
0
.
32
+0
.
54
0
.
29
Final spin
a
f
0
.
68
+0
.
05
0
.
05
0
.
68
+0
.
06
0
.
06
0
.
68
+0
.
05
0
.
06
Luminosity distance
D
L
/
Mpc
440
+160
180
440
+150
180
440
+160
180
Source redshift
z
0
.
094
+0
.
032
0
.
037
0
.
093
+0
.
029
0
.
036
0
.
093
+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
mentum. The spin opening angles (the
tilts
), defined by
cos(
θ
LS
1
,
2
) = (
S
1
,
2
·
ˆ
L
N
)
/
|
S
1
,
2
|
, are distributed broadly.
However, the KS test described at the end of Sec. III
does indicate some deviation between priors and poste-
riors, with
p
-values much smaller than 0.05 for cos(
θ
LS
1
)
and cos(
θ
LS
2
).
Posterior histograms: effective spin parameters.
In
Fig. 5, we show the posteriors of the effective spin com-
binations
χ
eff
[19, 52–54] and
χ
p
[40] defined by
χ
eff
=
c
G
(
S
1
m
1
+
S
2
m
2
)
·
ˆ
L
N
M
,
(5)
χ
p
=
c
B
1
Gm
2
1
max(
B
1
S
1
,B
2
S
2
)
,
(6)
where
S
i
is the component of the spin perpendicular
to the orbital angular momentum
L
N
,
M
is the total
observed mass,
B
1
= 2 + 3
q/
2 and
B
2
= 2 + 3
/
(2
q
), and
i
=
{
1
,
2
}
.
While
χ
eff
combines the projections of the BH spins
onto the orbital angular momentum,
χ
p
depends on their
in-plane components, and thus relates to precessional ef-
fects. Both models have credible intervals for
χ
eff
that
contain the value 0, and deviate from the prior signifi-
cantly. The data provides little information about pre-
cession, but show a slightly stronger preference for lower
values of
χ
p
than expressed by our priors; the deviation is
more pronounced for precessing EOBNR. The 90% credi-
ble intervals contain the value 0, and extend up to about
0.7 and 0.8 for precessing EOBNR and precessing IM-
RPhenom, respectively. Thus, precessing EOBNR pro-
vides a tighter upper bound.
Posterior histograms: other spin angles.
To explore
other possible differences between the two precessing
models, we now consider spin parameters that were not
reported in Ref. [2]. In particular, we compute posteri-
ors for
θ
12
, the opening angle between the spin vectors,
and
φ
12
, the opening angle between the in-plane projec-
tions of the spins. The prior on cos
θ
12
is uniform in
[
1
,
1], while the prior on
φ
12
is uniform in [0
,
2
π
]. We
show these posteriors in Fig. 6. The
θ
12
posteriors de-
viate appreciably from the prior, and are rather similar.
By contrast, comparing the opening angle between spin
projections onto orbital plane,
φ
12
, we find that the pre-
cessing EOBNR posterior deviates significantly from the
prior (with KS
p
-value
[0
.
0077
,
0
.
075]), while the pre-
7
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.
M/
M
M
source
/
M
M
/
M
M
source
/
M
m
1
/
M
m
source
1
/
M
m
2
/
M
m
source
2
/
M
M
f
/
M
M
source
f
/
M
q
χ
eff
χ
p
a
1
a
2
a
f
z
D
L
/
Mpc
θ
LS
1
θ
LS
2
φ
12
cessing IMRPhenom posterior does not (with KS
p
-value
[0
.
30
,
0
.
60]). This is as it should be, since in precess-
ing IMRPhenom binaries with identical projection of the
total spin on the orbital plane have identical waveforms.
Although the KS
p
-values suggest that the data provide
information about
θ
12
and
φ
12
beyond the prior, we note
that the 90% confidence intervals for both of these pa-
rameters cover approximately 90% of their valid ranges,
and are indistinguishable for each waveform model.
Spin evolution
All the source parameters discussed
above are measured at a reference frequency of 20 Hz.
Exploiting the capability of precessing EOBNR of evolv-
ing the BH spin vectors in the time domain, we may
address the question of estimating values for the spin
parameters at the time of the merger. To do so, we
randomly sample 1,000 distinct configurations from the
precessing EOBNR posteriors, and we evolve them to
the maximum EOB orbital frequency (a proxy for the
merger in the model). We then produce histograms of
the evolved values of
χ
eff
and
χ
p
. We find little if any
change between 20 Hz and the merger. Indeed, a KS test
suggests that the original and evolved distributions are
very consistent, with
p
-values close to 1.
Comparison with numerical relativity
The precess-
ing EOBNR waveform model has been tested against
numerical relativity (NR) waveforms using simulations
from the SXS catalog [15, 16, 32].
We can provide
a targeted cross-check on the accuracy of precessing