of 13
Search for Tensor, Vector, and Scalar Polarizations
in the Stochastic Gravitational-Wave Background
B. P. Abbott
etal.
*
(LIGO Scientific Collaboration and Virgo Collaboration)
(Received 1 March 2018; revised manuscript received 30 March 2018; published 16 May 2018)
The detection of gravitational waves with Advanced LIGO and Advanced Virgo has enabled novel tests
of general relativity, including direct study of the polarization of gravitational waves. While general
relativity allows for only two tensor gravitational-wave polarizations, general metric theories can
additionally predict two vector and two scalar polarizations. The polarization of gravitational waves is
encoded in the spectral shape of the stochastic gravitational-wave background, formed by the superposition
of cosmological and individually unresolved astrophysical sources. Using data recorded by Advanced
LIGO during its first observing run, we search for a stochastic background of generically polarized
gravitational waves. We find no evidence for a background of any polarization, and place the first direct
bounds on the contributions of vector and scalar polarizations to the stochastic background. Under log-
uniform priors for the energy in each polarization, we limit the energy densities of tensor, vector, and scalar
modes at 95% credibility to
Ω
T
0
<
5
.
58
×
10
8
,
Ω
V
0
<
6
.
35
×
10
8
, and
Ω
S
0
<
1
.
08
×
10
7
at a reference
frequency
f
0
¼
25
Hz.
DOI:
10.1103/PhysRevLett.120.201102
Introduction.
The direct detection of gravitational
waves offers novel opportun
ities to test general
relativity in previously unexplored regimes. Already,
the compact binary mergers
[1
5]
observed by
Advanced LIGO (the Laser Interferometer Gravitational
Wave Observatory)
[6,7]
and Advanced Virgo
[8]
have
enabled improved limits on the graviton mass, experi-
mental measurements of post-Newtonian parameters, and
inference of the speed of gravitational waves, among other
tests
[3,9
11]
.
Another central prediction of general relativity is
the existence of only two gravitational-wave polarizations:
the tensor plus and cross modes, with spatial strain
tensors
ˆ
e
þ
¼
0
B
@
100
0
10
000
1
C
A
ˆ
e
×
¼
0
B
@
010
100
000
1
C
A
ð
1
Þ
(assuming waves propagating in the
þ
ˆ
z
direction). Generic
metric theories of gravity, however, can allow for up to four
additional polarizations: the
x
and
y
vector modes and the
breathing and longitudinal scalar modes, with basis strain
tensors
[12
14]
ˆ
e
x
¼
0
B
@
001
000
100
1
C
A
ˆ
e
y
¼
0
B
@
000
001
010
1
C
A
ˆ
e
b
¼
0
B
@
100
010
000
1
C
A
ˆ
e
l
¼
0
B
@
000
000
001
1
C
A
:
ð
2
Þ
The observation of vector or scalar modes would be in
direct conflict with general relativity, and so the direct
measurement of gravitational-wave polarizations offers a
promising avenue by which to test theories of gravity
[14]
.
Recently, the Advanced LIGO-Virgo network has suc-
ceeded in making the first direct statement about the
polarization of gravitational waves. The gravitational-wave
signal GW170814, observed by both the Advanced LIGO
and Virgo detectors, significantly favored a model assum-
ing pure tensor polarization over models with pure vector or
scalar polarizations
[4,15]
. In general, however, the ability
of the Advanced LIGO-Virgo network to study the polari-
zation of gravitational-wave transients is limited by several
factors. First, the LIGO-Hanford and LIGO-Livingston
detectors are nearly co-oriented, preventing Advanced
LIGO from sensitively measuring more than a single
polarization mode
[4,9,10,15]
. Second, at least five detec-
tors are needed to fully characterize the five polarization
degrees of freedom accessible to quadrupole detectors.
Quadrupole detectors (those measuring differential arm
motion) have degenerate responses to breathing and
*
Full author list given at the end of the article.
PHYSICAL REVIEW LETTERS
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Editors' Suggestion
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=
18
=
120(20)
=
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201102-1
© 2018 American Physical Society
longitudinal modes, and can therefore measure only a
single linear combination of scalar breathing and longi-
tudinal polarizations
[14
17]
.
Beyond compact binary mergers, another target for
Advanced LIGO and Virgo is the stochastic gravita-
tional-wave background. An astrophysical stochastic back-
ground is expected to arise from the population of distant
compact binary mergers
[18
23]
, core-collapse supernovae
[24
26]
, and rapidly rotating neutron stars
[27
29]
.In
particular, the astrophysical background from compact
binary mergers is likely to be detected by LIGO and
Virgo at their design sensitivities
[23]
. A background of
cosmological origin may also be present, due to cosmic
strings
[30,31]
, inflation
[32
35]
, and phase transitions in
the early Universe
[32,33,36
38]
.
Long duration gravitational-wave sources, like the sto-
chastic background
[39
42]
or persistent signals from
rotating neutron stars
[43
45]
, offer a viable means of
searching for nonstandard gravitational-wave polarizations.
Unlike gravitational-wave transients, which sample only a
single point on the LIGO/Virgo antenna response patterns,
long-duration signals contain information about many
points on the antenna patterns. Long-duration signals,
therefore, enable the direct measurement of gravitational-
wave polarizations using the current generation of gravi-
tational-wave detectors, without the need for additional
detectors or an independent electromagnetic counterpart.
The stochastic background is thus a valuable laboratory for
polarization-based tests of general relativity
[42]
.
In this Letter, we present the first direct search for vector
and scalar polarizations in the stochastic gravitational-wave
background. We analyze data recorded during Advanced
LIGO
s first observing run (O1), which has previously been
searched for both isotropic and anisotropic backgrounds of
standard tensor polarizations
[46,47]
. First, we describe the
O1 data set and its initial processing. We then discuss the
stochastic analysis, including the construction of Bayesian
odds that indicate the nondetection of a generically polar-
ized stochastic background in our data. Finally, we present
upper limits on the joint contributions of tensor, vector, and
scalar polarizations to the stochastic gravitational-wave
background. Additional details and results are presented in
the Supplemental Material
[48]
, available online.
Data.
We search Advanced LIGO data for evidence of
a stochastic background, analyzing data recorded between
September 18, 2015 15
00 UTC and January 12, 2016
16
00 UTC during LIGO
s O1 observing run. We do not
include several days of O1 data recorded prior to September
18, but this has negligible impact on our results. We
exclude times containing the binary black hole signals
GW150914 and GW151226, as well as the signal candidate
LVT151012.
The initial data processing proceeds as in previous
analyses
[46,49]
. Time-domain strain measurements from
the LIGO-Hanford and LIGO-Livingston detectors are
down-sampled from 16 384 Hz to 4096 Hz and divided
into half-overlapping 192 s segments. Each time segment is
then Hann-windowed, Fourier transformed, and high-pass
filtered using a 16th order Butterworth filter with a knee
frequency of 11 Hz. Finally, the strain data are coarse-
grained to a frequency resolution of 0.03 125 Hz and
restricted to a frequency band from 20
1726 Hz. Within
each segment, we compute the LIGO-Hanford and LIGO-
Livingston strain auto-power spectral densities using
Welch
s method
[50]
.
Standard data quality cuts are performed in both the time
and frequency domains to mitigate the effects of non-
Gaussian instrumental and environmental noise
[46,47,51]
.
In the time domain, 35% of data is discarded due to
nonstationary detector noise, leaving 29.85 days of coinci-
dent observing time. In the frequency domain, an additional
21% of data is discarded to remove correlated narrow-band
features between LIGO-Hanford and LIGO-Livingston
[46,47,51]
. These narrow-band correlations are due to a
variety of sources, including injected calibration signals,
power mains, and GPS timing systems. To estimate
possible contamination due to terrestrial Schumann reso-
nances
[52
54]
, we additionally monitored coherences
between magnetometers installed at both detectors.
Schumann resonances were found to contribute negligibly
to the stochastic measurement
[46,51]
.
We assume conservative 4.8% and 5.4% calibration
uncertainties on the strain amplitude measured by LIGO-
Hanford and LIGO-Livingston, respectively
[55]
. Phase
calibration is a much smaller source of uncertainty and is
therefore neglected
[46,56]
. All results below are obtained
after marginalization over amplitude uncertainties; see the
Supplemental Material
[48]
for details.
Method.
To search for a generically polarized stochas-
tic background, we will apply the methodology presented
in Ref.
[42]
. This method is summarized below, and
additional details are discussed in the Supplemental
Material
[48]
.
The stochastic background may be detected in the form
of a correlated signal between pairs of gravitational-wave
detectors. We will assume that the stochastic background is
stationary, isotropic, and Gaussian. For simplicity, we also
assume that the background is uncorrelated between
polarization modes. Finally, we assume that the tensor
and vector contributions to the background are individually
unpolarized (with equal contributions, for instance, from
the tensor plus and cross modes). Certain theories may
violate one or more of these assumptions. For example, the
stochastic background is unlikely to remain strictly unpo-
larized in the presence of gravitational-wave birefringence,
as in Chern-Simons gravity
[57
59]
, while theories violat-
ing Lorentz invariance may yield a departure from isotropy
[60,61]
. The violation of one or more of our assumptions
would likely reduce our search
s sensitivity to the stochas-
tic background.
PHYSICAL REVIEW LETTERS
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201102 (2018)
201102-2
Given the above assumptions, the expected cross-corre-
lation between two detectors in the presence of a stochastic
background is of the form
[39
41,62]
h
̃
s
1
ð
f
Þ
̃
s

2
ð
f
0
Þi ¼
1
2
δ
ð
f
f
0
Þ
X
A
Γ
A
ð
f
Þ
S
A
h
ð
f
Þ
:
ð
3
Þ
Here,
S
A
h
ð
f
Þ
is the one-sided gravitational-wave strain
power spectral density of the net tensor (
A
¼
T
), vector
(
V
), and scalar (
S
) contributions to the stochastic back-
ground. The detectors
geometry is encoded in the overlap
reduction functions
Γ
A
ð
f
Þ
, defined
[39,42,62,63]
Γ
A
ð
f
Þ¼
1
8
π
X
a
A
Z
d
ˆ
nF
a
1
ð
ˆ
n
Þ
F
a
2
ð
ˆ
n
Þ
e
2
π
if
ˆ
n
·
Δ
x=c
:
ð
4
Þ
F
a
I
ð
ˆ
n
Þ
is the antenna response function of detector
I
to
signals of polarization
a
,
Δ
x
is the separation vector
between detectors, and
c
is the speed of light. The integral
is taken over all sky directions
ˆ
n
.
We will work not directly with
Γ
A
ð
f
Þ
, but rather with the
normalized
overlap reduction functions
γ
A
ð
f
Þ
Γ
A
ð
f
Þ
=
Γ
0
,
where the constant
Γ
0
is chosen such that
γ
T
ð
f
Þ¼
1
for co-
located and co-oriented detectors. For Advanced LIGO,
Γ
0
¼
1
=
5
, but in general its value will vary for other
experiments like LISA and pulsar timing arrays
[64]
. The
normalized overlap reduction functions for LIGO
s
Hanford-Livingston baseline are shown in Fig.
1
.
Because tensor, vector, and scalar modes each have distinct
overlap reduction functions, the shape of a measured cross-
correlation spectrum [Eq.
(3)
] will reflect the polarization
content of the stochastic background
[39,42]
. Of the three
curves in Fig.
1
, the scalar overlap reduction function is
smallest in magnitude. This reflects the fact that the
Advanced LIGO detectors have weaker geometrical
responses to scalar-polarized gravitational waves than to
tensor- and vector-polarized signals.
Conventionally, gravitational-wave backgrounds are par-
ameterized by their energy-density spectra
[62,64]
Ω
ð
f
Þ¼
1
ρ
c
d
ρ
GW
d
ln
f
;
ð
5
Þ
where
d
ρ
GW
is the energy density in gravitational waves per
logarithmic frequency interval
d
ln
f
. We normalize Eq.
(5)
by
ρ
c
¼
3
H
2
0
c
2
=
8
π
G
, the closure energy density of the
Universe. Here,
G
is Newton
s constant and
H
0
is the
Hubble constant; we take
H
0
¼
68
kms
1
Mpc
1
[65]
. The
precise relationship between
Ω
ð
f
Þ
and
S
h
ð
f
Þ
is theory
dependent. Under any theory obeying Isaacson
s formula
for the stress-energy of gravitational waves
[66]
, the
energy-density spectrum is related to
S
h
ð
f
Þ
by
[42,62,67]
Ω
ð
f
Þ¼
2
π
2
3
H
2
0
f
3
S
h
ð
f
Þ
:
ð
6
Þ
Equation
(6)
does not hold in general, however
[67]
.For
ease of comparison with previous studies, we will instead
take Eq.
(6)
as the
definition
of the canonical energy-
density spectra
Ω
A
ð
f
Þ
. The canonical energy-density spec-
tra can be directly identified with true energy densities
under any theory obeying Isaacson
s formula. For other
theories,
Ω
A
ð
f
Þ
can instead be understood simply as a
function of the detector-frame observable
S
A
h
ð
f
Þ
.
Within each 192 s time segment (indexed by
i
), we form
an estimator of the visible cross power between LIGO-
Hanford and LIGO-Livingston:
ˆ
C
i
ð
f
Þ¼
1
Δ
T
20
π
2
3
H
2
0
f
3
̃
s

1
;i
ð
f
Þ
̃
s
2
;i
ð
f
Þ
;
ð
7
Þ
normalized such that the estimator
s mean and variance are
[42]
h
ˆ
C
i
ð
f
Þi ¼
X
A
γ
A
ð
f
Þ
Ω
A
ð
f
Þð
8
Þ
and
σ
2
i
ð
f
Þ¼
1
2
Δ
Tdf

10
π
2
3
H
2
0

2
f
6
P
1
;i
ð
f
Þ
P
2
;i
ð
f
Þ
;
ð
9
Þ
respectively. Within Eqs.
(7)
and
(9)
,
Δ
T
is the segment
duration,
df
the frequency bin width, and
P
I;i
ð
f
Þ
is the
one-sided auto-power spectral density of detector
I
in time
segment
i
, defined by
h
̃
s

I;i
ð
f
Þ
̃
s
I;i
ð
f
0
Þi ¼
1
2
δ
ð
f
f
0
Þ
P
I;i
ð
f
Þ
:
ð
10
Þ
FIG. 1. Overlap reduction functions representing the Advanced
LIGO network
s sensitivity to stochastic backgrounds of tensor
(blue), vector (red), and scalar-polarized (green) gravitational
waves.
PHYSICAL REVIEW LETTERS
120,
201102 (2018)
201102-3
The normalization of
ˆ
C
ð
f
Þ
is chosen such that the con-
tribution from each polarization appears symmetrically in
Eq.
(8)
; this choice differs by a factor of
γ
T
ð
f
Þ
from the
point estimate
ˆ
Y
ð
f
Þ
typically used in stochastic analyses
[42,46,49]
. Finally, the cross-power estimators from each
segment are optimally combined via a weighted sum to
form a single cross-power spectrum for the O1 observing
run,
ˆ
C
ð
f
Þ¼
P
i
ˆ
C
i
ð
f
Þ
σ
2
i
ð
f
Þ
P
i
σ
2
i
ð
f
Þ
;
ð
11
Þ
with the corresponding variance
σ
2
ð
f
Þ¼
X
i
σ
2
i
ð
f
Þ
:
ð
12
Þ
Note that, unlike transient gravitational-wave searches,
searches for the stochastic background are well described
by Gaussian statistics due to the large number of time
segments contributing to the final cross-power spec-
trum
[68]
.
Given the measured cross-power spectrum
ˆ
C
ð
f
Þ
,we
compute Bayesian evidence for various hypotheses
describing the presence and polarization of a possible
stochastic signal within our data. Evidence is computed
using P
Y
M
ULTI
N
EST
[69]
, a Python interface to the nested
sampling code M
ULTI
N
EST
[70
74]
. We consider several
different hypotheses: (i) Gaussian noise (
N
): No stochastic
signal is present in our data, and the measured cross power
is due entirely to Gaussian noise. (ii) Signal (SIG): A
stochastic background of any polarization(s) is present.
(iii) Tensor-polarized (GR): The data contains a purely
tensor-polarized stochastic signal, consistent with general
relativity. (iv) Nonstandard polarizations (NGR): The data
contains a stochastic signal with vector and/or scalar
contributions. These evidences are combined to form
two Bayesian odds
[42]
: (1) Odds
O
SIG
N
for the presence
of a stochastic signal relative to pure noise, and (2) odds
O
NGR
GR
for the presence of nonstandard polarizations versus
ordinary tensor modes.
O
SIG
N
quantifies evidence for the
detection
of a generically polarized stochastic background,
and generally depends only on a background
s total power,
not its polarization content.
O
NGR
GR
indicates if the back-
ground
s polarization is inconsistent with general relativity.
In particular, the sensitivity of
O
NGR
GR
to nonstandard
polarizations is not significantly affected by the strength
of any tensor polarization which may also be present
[42]
.
See the Supplemental Material
[48]
for further details about
our hypotheses and odds ratio construction, including the
priors placed on these hypotheses and their parameters.
Results.
Using the cross power measured between
LIGO-Hanford and LIGO-Livingston during Advanced
LIGO
s O1 observing run, we obtain odds ln
O
SIG
N
¼
0
.
53
between signal and Gaussian noise hypotheses,
indicating a nondetection of the stochastic gravitational-
wave background. Additionally, we find ln
O
NGR
GR
¼
0
.
25
,
consistent with values expected in the presence of Gaussian
noise
[42]
. (We will use ln and log to denote base-
e
and
base-10 logarithms, respectively.).
Given our nondetection, we place upper limits on the
presence of tensor, vector, and scalar contributions to the
stochastic background. To simultaneously constrain the
properties of each polarization, we will restrict our analysis
to a model assuming the presence of tensor, vector, and
scalar-polarized signals (this is the TVS hypothesis in the
notation of the Supplemental Material
[48]
). Under this
hypothesis, we model the total canonical energy density of
the stochastic background as a sum of power laws:
Ω
ð
f
Þ¼
Ω
T
0

f
f
0

α
T
þ
Ω
V
0

f
f
0

α
V
þ
Ω
S
0

f
f
0

α
S
:
ð
13
Þ
Here,
Ω
A
0
is the amplitude of polarization
A
at a reference
frequency
f
0
, and
α
A
is the corresponding spectral index.
We take
f
0
¼
25
Hz
[46]
. Standard tensor-polarized sto-
chastic backgrounds are predicted to be well described by
power laws in the Advanced LIGO band. The expected
astrophysical background from compact binary mergers,
for instance, is well modeled by a power law with
α
T
¼
2
=
3
[18
20,75]
.
We will consider two different prior distributions for
the background amplitudes: a log-uniform prior between
10
13
Ω
A
0
10
5
and a uniform prior between
0
Ω
A
0
10
5
. The former (log-uniform) corresponds to
the prior adopted in Ref.
[42]
. The latter (uniform)
implicitly reproduces the maximum likelihood analysis
used in previous studies, and is included to allow direct
comparison to previous stochastic results
[46,49]
. The
upper amplitude bound (
10
5
) is consistent with limits
placed by Initial LIGO and Virgo
[49]
. In order to be
normalizable, the log-uniform prior requires a nonzero
lower bound; although parameter estimation results will
depend on the specific choice of lower bound, in general
this dependence is weak
[44]
. Our lower bound (
10
13
)is
chosen to encompass small energy densities well below the
reach of LIGO and Virgo at design sensitivity
[23,46]
.
Following Ref.
[42]
, we take our spectral index priors to
be
p
ð
α
A
Þ
1
j
α
A
j
=
α
MAX
for
j
α
A
j
α
MAX
and
p
ð
α
A
Þ¼
0
elsewhere. This prior preferentially weights flat energy-
density spectra, penalizing spectra which are more steeply
positively or negatively sloped in the Advanced LIGO
band. We conservatively choose
α
MAX
¼
8
, allowing for
energy-density spectra significantly steeper than back-
grounds predicted from known astrophysical sources (like
compact binary mergers).
We perform parameter estimation using posterior samples
obtainedby P
Y
M
ULTI
N
EST
. Figure
2
shows posteriors on the
tensor, vector, and scalar background amplitudes, under
each choice of amplitude prior. The dashed and dot-dashed
PHYSICAL REVIEW LETTERS
120,
201102 (2018)
201102-4
curves are proportional to the log-uniform and uniform
amplitude priors, respectively; each prior curve has been
renormalized by a constant factor to illustrate consistency
between our priors and posteriors at small
Ω
A
0
. We can now
place upper limits on the amplitude of each component at
f
0
¼
25
Hz. The 95% credible upper limits on the ampli-
tude of each polarization are listed in Table
I
for each choice
of prior (for convenience, we list limits in terms of both
log
Ω
A
0
and
Ω
A
0
). As no signal was detected, our posteriors on
the spectral indices
α
A
are dominated by our prior. Full
parameter estimation results, including posteriors on
α
A
, are
given in the Supplemental Material
[48]
.
Care should be taken when comparing these upper limits
to those obtained in previous analyses (e.g., Table I of
Ref.
[46]
). Three important distinctions should be kept in
mind. First, the amplitude posteriors in Fig.
2
(and hence
the limits in Table
I
) are obtained after marginalization over
spectral index. Previous analyses, on the other hand,
typically assume specific fixed spectral indices or present
exclusion curves in the
Ω
T
0
α
T
plane
[46]
. Second,
Bayesian upper limits may be strongly influenced by one
s
adopted prior. Uniform amplitude priors, for instance,
preferentially weight larger signals and hence yield larger
upper limits, while log-uniform priors support smaller
signal amplitudes, giving tighter limits. Finally, our results
are obtained under a specific signal hypothesis allowing
simultaneously for tensor, vector, and scalar polarizations.
These limits are not generically identical to those that
would be obtained if we allowed for tensor modes alone. In
the Supplemental Material
[48]
, we have tabulated upper
limits under a variety of signal hypotheses allowing for
each unique combination of gravitational-wave polariza-
tions (our results, though, do not vary considerably
between hypotheses). We have additionally verified that,
under the GR (tensor-only) hypothesis with delta-function
priors on the background
s spectral index, we recover upper
limits identical to results previously published in Ref.
[46]
.
Conclusion.
The direct measurement of gravitational-
wave polarizations may open the door to powerful new tests
of gravity. Such measurements largely depend only on the
geometry of a gravitational wave
s strain and its direction
of propagation, not on the details of any specific theory of
gravity. Recently, the Advanced LIGO-Virgo observation
of the binary black hole merger GW170814 has enabled the
first direct study of gravitational-wave polarizations
[4,15]
.
While LIGO and Virgo are limited in their ability to discern
the polarization of gravitational-wave transients, the future
construction of additional detectors, like KAGRA
[76,77]
Probability Density
FIG. 2. Posteriors on the tensor (left), vector (center), and scalar (right) stochastic background amplitudes at reference frequency
f
0
¼
25
Hz. Within each subplot, dark posteriors show results obtained assuming log-uniform priors (dashed curves) on
Ω
A
0
, while light
posteriors show results corresponding to uniform amplitude priors (dot-dashed curves). The prior curves shown here have been
renormalized by constant factors to illustrate consistency with the posteriors below our measured upper limits. These posteriors
correspond to the 95% credible upper limits listed in Table
I
. Relative to the log-uniform priors, the uniform amplitude priors
preferentially weight loud stochastic signals and therefore yield more conservative upper limits.
TABLE I. 95% credible upper limits on the log amplitudes of tensor, vector, and scalar modes in the stochastic background at
reference frequency
f
0
¼
25
Hz. We assume an energy-density spectrum in which all three modes are present, and present limits
following marginalization over the spectral index of each component [see Eq.
(13)
]. We show results for two different amplitude priors: a
log-uniform prior (
dp=d
log
Ω
0
1
; top row) and a uniform prior (
dp=d
Ω
0
1
; bottom row). Additional parameter estimation results
are shown in the Supplemental Material
[48]
.
Prior
log
Ω
T
0
log
Ω
V
0
log
Ω
S
0
Ω
T
0
Ω
V
0
Ω
S
0
Log uniform
7
.
25
7
.
20
6
.
96
5
.
58
×
10
8
6
.
35
×
10
8
1
.
08
×
10
7
Uniform
6
.
70
6
.
59
6
.
07
2
.
02
×
10
7
2
.
54
×
10
7
8
.
44
×
10
7
PHYSICAL REVIEW LETTERS
120,
201102 (2018)
201102-5
and LIGO-India
[78]
, will help to break existing degener-
acies and allow for increasingly precise polarization
measurements.
Long-duration signals offer further opportunities to
study gravitational-wave polarizations. Detections of con-
tinuous sources like rotating neutron stars
[44,45]
and the
stochastic background
[42]
will offer the ability to directly
measure and/or constrain gravitational-wave polarizations,
even in the absence of additional detectors. In this Letter,
we have conducted a search for a generically polarized
stochastic background of gravitational waves using data
from Advanced LIGO
s O1 observing run. Although we
find no evidence for the presence of a background (of any
polarization), we have succeeded in placing the first direct
upper limits (listed in Table
I
) on the contributions of vector
and scalar modes to the stochastic background.
The authors gratefully acknowledge the support of the
United States National Science Foundation (NSF) for the
construction and operation of the LIGO Laboratory and
Advanced LIGO as well as the Science and Technology
Facilities Council (STFC) of the United Kingdom, the Max-
Planck-Society (MPS), and the State of Niedersachsen,
Germany for support of the construction of Advanced
LIGO and construction and operation of the GEO600
detector. Additional support for Advanced LIGO was
provided by the Australian Research Council. The authors
gratefully acknowledge the Italian Istituto Nazionale di
Fisica Nucleare (INFN), the French Centre National de la
Recherche Scientifique (CNRS) and the Foundation for
Fundamental Research on Matter supported by the
Netherlands Organisation for Scientific Research, for the
construction and operation of the Virgo detector and the
creation and support of the EGO consortium. The authors
also gratefully acknowledge research support from these
agencies as well as by the Council of Scientific and
Industrial Research of India, the Department of Science
and Technology, India, the Science & Engineering Research
Board (SERB), India, the Ministry of Human Resource
Development, India, the Spanish Agencia Estatal
de Investigación, the Vicepresid`
encia i Conselleria
d
Innovació, Recerca i Turisme and the Conselleria
d
Educació i Universitat del Govern de les Illes Balears,
the Conselleria d
Educació, Investigació, Cultura i Esport de
la Generalitat Valenciana, the National Science Centre of
Poland, the Swiss National Science Foundation (SNSF), the
RussianFoundationfor Basic Research,the RussianScience
Foundation, the European Commission, the European
Regional Development Funds (ERDF), the Royal Society,
the Scottish Funding Council, the Scottish Universities
Physics Alliance, the Hungarian Scientific Research Fund
(OTKA), the Lyon Institute of Origins (LIO), the Paris Île-
de-France Region, the National Research, Development and
Innovation Office Hungary (NKFI), the National Research
Foundation of Korea, Industry Canada and the Province of
OntariothroughtheMinistry ofEconomicDevelopmentand
Innovation, the Natural Sciences and Engineering Research
Council Canada, the Canadian Institute for Advanced
Research, the Brazilian Ministry of Science, Technology,
Innovations, and Communications, the International Center
for Theoretical Physics South American Institute for
Fundamental Research (ICTP-SAIFR), the Research
Grants Council of Hong Kong, the National Natural
Science Foundation of China (NSFC), the Leverhulme
Trust, the Research Corporation, the Ministry of Science
and Technology (MOST), Taiwan and the Kavli Foundation.
The authors gratefully acknowledge the support of the NSF,
STFC, MPS, INFN, CNRS and the State of Niedersachsen,
Germany for provision of computational resources.
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1
R. Abbott,
1
T. D. Abbott,
2
F. Acernese,
3,4
K. Ackley,
5,6
C. Adams,
7
T. Adams,
8
P. Addesso,
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R. X. Adhikari,
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S. B. Anderson,
1
W. G. Anderson,
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27
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28
S. Appert,
1
K. Arai,
1
M. C. Araya,
1
J. S. Areeda,
29
N. Arnaud,
28,30
S. Ascenzi,
31,32
G. Ashton,
10
M. Ast,
33
S. M. Aston,
7
P. Astone,
34
D. V. Atallah,
35
P. Aufmuth,
22
C. Aulbert,
10
K. AultONeal,
36
C. Austin,
2
A. Avila-Alvarez,
29
S. Babak,
37
P. Bacon,
38
M. K. M. Bader,
14
S. Bae,
39
P. T. Baker,
40
F. Baldaccini,
41,42
G. Ballardin,
30
S. W. Ballmer,
43
S. Banagiri,
44
J. C. Barayoga,
1
S. E. Barclay,
45
B. C. Barish,
1
D. Barker,
46
K. Barkett,
47
F. Barone,
3,4
B. Barr,
45
L. Barsotti,
15
M. Barsuglia,
38
D. Barta,
48
J. Bartlett,
46
I. Bartos,
49,5
R. Bassiri,
50
A. Basti,
23,24
J. C. Batch,
46
M. Bawaj,
51,42
J. C. Bayley,
45
M. Bazzan,
52,53
B. B ́
ecsy,
54
C. Beer,
10
M. Bejger,
55
I. Belahcene,
28
A. S. Bell,
45
B. K. Berger,
1
G. Bergmann,
10
J. J. Bero,
56
C. P. L. Berry,
57
D. Bersanetti,
58
A. Bertolini,
14
J. Betzwieser,
7
S. Bhagwat,
43
R. Bhandare,
59
I. A. Bilenko,
60
G. Billingsley,
1
C. R. Billman,
5
J. Birch,
7
R. Birney,
61
O. Birnholtz,
10
S. Biscans,
1,15
S. Biscoveanu,
62,6
A. Bisht,
22
M. Bitossi,
30,24
C. Biwer,
43
M. A. Bizouard,
28
J. K. Blackburn,
1
J. Blackman,
47
C. D. Blair,
1,63
D. G. Blair,
63
R. M. Blair,
46
S. Bloemen,
64
O. Bock,
10
N. Bode,
10
M. Boer,
65
G. Bogaert,
65
A. Bohe,
37
F. Bondu,
66
E. Bonilla,
50
R. Bonnand,
8
B. A. Boom,
14
R. Bork,
1
V. Boschi,
30,24
S. Bose,
67,19
K. Bossie,
7
Y. Bouffanais,
38
A. Bozzi,
30
C. Bradaschia,
24
P. R. Brady,
21
M. Branchesi,
17,18
J. E. Brau,
68
T. Briant,
69
A. Brillet,
65
M. Brinkmann,
10
V. Brisson,
28
P. Brockill,
21
J. E. Broida,
70
A. F. Brooks,
1
D. A. Brown,
43
D. D. Brown,
71
S. Brunett,
1
C. C. Buchanan,
2
A. Buikema,
15
T. Bulik,
72
H. J. Bulten,
73,14
A. Buonanno,
37,74
D. Buskulic,
8
C. Buy,
38
R. L. Byer,
50
M. Cabero,
10
L. Cadonati,
75
G. Cagnoli,
26,76
C. Cahillane,
1
J. Calderón Bustillo,
75
T. A. Callister,
1
E. Calloni,
77,4
J. B. Camp,
78
M. Canepa,
79,58
P. Canizares,
64
K. C. Cannon,
80
H. Cao,
71
J. Cao,
81
C. D. Capano,
10
E. Capocasa,
38
F. Carbognani,
30
S. Caride,
82
M. F. Carney,
83
J. Casanueva Diaz,
28
C. Casentini,
31,32
S. Caudill,
21,14
M. Cavagli`
a,
11
F. Cavalier,
28
R. Cavalieri,
30
G. Cella,
24
C. B. Cepeda,
1
P. Cerdá-Durán,
84
G. Cerretani,
23,24
E. Cesarini,
85,32
S. J. Chamberlin,
62
M. Chan,
45
S. Chao,
86
P. Charlton,
87
E. Chase,
88
E. Chassande-Mottin,
38
D. Chatterjee,
21
B. D. Cheeseboro,
40
H. Y. Chen,
89
X. Chen,
63
Y. Chen,
47
H.-P. Cheng,
5
H. Chia,
5
A. Chincarini,
58
A. Chiummo,
30
T. Chmiel,
83
H. S. Cho,
90
M. Cho,
74
J. H. Chow,
25
N. Christensen,
70,65
Q. Chu,
63
A. J. K. Chua,
13
S. Chua,
69
A. K. W. Chung,
91
S. Chung,
63
G. Ciani,
5,52,53
R. Ciolfi,
92,93
C. E. Cirelli,
50
A. Cirone,
79,58
F. Clara,
46
J. A. Clark,
75
P. Clearwater,
94
F. Cleva,
65
C. Cocchieri,
11
E. Coccia,
17,18
P.-F. Cohadon,
69
D. Cohen,
28
A. Colla,
95,34
C. G. Collette,
96
L. R. Cominsky,
97
M. Constancio Jr.,
16
L. Conti,
53
S. J. Cooper,
57
P. Corban,
7
T. R. Corbitt,
2
I. Cordero-Carrión,
98
K. R. Corley,
49
N. Cornish,
99
A. Corsi,
82
S. Cortese,
30
C. A. Costa,
16
E. Coughlin,
70
M. W. Coughlin,
70,1
S. B. Coughlin,
88
J.-P. Coulon,
65
S. T. Countryman,
49
P. Couvares,
1
P. B. Covas,
100
E. E. Cowan,
75
D. M. Coward,
63
M. J. Cowart,
7
D. C. Coyne,
1
R. Coyne,
82
J. D. E. Creighton,
21
T. D. Creighton,
101
J. Cripe,
2
S. G. Crowder,
102
T. J. Cullen,
29,2
A. Cumming,
45
L. Cunningham,
45
E. Cuoco,
30
T. Dal Canton,
78
G. Dálya,
54
S. L. Danilishin,
22,10
S. D
Antonio,
32
K. Danzmann,
22,10
A. Dasgupta,
103
C. F. Da Silva Costa,
5
V. Dattilo,
30
I. Dave,
59
M. Davier,
28
D. Davis,
43
E. J. Daw,
104
B. Day,
75
S. De,
43
D. DeBra,
50
J. Degallaix,
26
M. De Laurentis,
17,4
S. Del ́
eglise,
69
W. Del Pozzo,
57,23,24
N. Demos,
15
T. Denker,
10
T. Dent,
10
R. De Pietri,
105,106
V. Dergachev,
37
R. De Rosa,
77,4
R. T. DeRosa,
7
C. De Rossi,
26,30
R. DeSalvo,
107
O. de Varona,
10
J. Devenson,
27
S. Dhurandhar,
19
M. C. Díaz,
101
L. Di Fiore,
4
M. Di Giovanni,
108,93
T. Di Girolamo,
49,77,4
A. Di Lieto,
23,24
S. Di Pace,
95,34
I. Di Palma,
95,34
F. Di Renzo,
23,24
Z. Doctor,
89
V. Dolique,
26
F. Donovan,
15
K. L. Dooley,
11
S. Doravari,
10
I. Dorrington,
35
R. Douglas,
45
M. Dovale Álvarez,
57
T. P. Downes,
21
M. Drago,
10
C. Dreissigacker,
10
J. C. Driggers,
46
Z. Du,
81
M. Ducrot,
8
P. Dupej,
45
S. E. Dwyer,
46
T. B. Edo,
104
M. C. Edwards,
70
A. Effler,
7
H.-B. Eggenstein,
37,10
P. Ehrens,
1
J. Eichholz,
1
S. S. Eikenberry,
5
R. A. Eisenstein,
15
R. C. Essick,
15
D. Estevez,
8
Z. B. Etienne,
40
T. Etzel,
1
M. Evans,
15
T. M. Evans,
7
M. Factourovich,
49
V. Fafone,
31,32,17
H. Fair,
43
S. Fairhurst,
35
X. Fan,
81
S. Farinon,
58
B. Farr,
89
W. M. Farr,
57
E. J. Fauchon-Jones,
35
M. Favata,
109
M. Fays,
35
C. Fee,
83
H. Fehrmann,
10
J. Feicht,
1
M. M. Fejer,
50
A. Fernandez-Galiana,
15
I. Ferrante,
23,24
E. C. Ferreira,
16
F. Ferrini,
30
F. Fidecaro,
23,24
D. Finstad,
43
I. Fiori,
30
D. Fiorucci,
38
M. Fishbach,
89
R. P. Fisher,
43
M. Fitz-Axen,
44
R. Flaminio,
26,110
M. Fletcher,
45
H. Fong,
111
J. A. Font,
84,112
P. W. F. Forsyth,
25
S. S. Forsyth,
75
J.-D. Fournier,
65
S. Frasca,
95,34
F. Frasconi,
24
Z. Frei,
54
A. Freise,
57
R. Frey,
68
V. Frey,
28
E. M. Fries,
1
P. Fritschel,
15
V. V. Frolov,
7
P. Fulda,
5
M. Fyffe,
7
H. Gabbard,
45
B. U. Gadre,
19
S. M. Gaebel,
57
J. R. Gair,
113
L. Gammaitoni,
41
M. R. Ganija,
71
S. G. Gaonkar,
19
C. Garcia-Quiros,
100
F. Garufi,
77,4
B. Gateley,
46
S. Gaudio,
36
G. Gaur,
114
V. Gayathri,
115
N. Gehrels,
78
,
G. Gemme,
58
E. Genin,
30
A. Gennai,
24
D. George,
12
J. George,
59
L. Gergely,
116
V. Germain,
8
S. Ghonge,
75
Abhirup Ghosh,
20
Archisman Ghosh,
20,14
S. Ghosh,
64,14,21
J. A. Giaime,
2,7
K. D. Giardina,
7
A. Giazotto,
24
,
K. Gill,
36
L. Glover,
107
E. Goetz,
117
R. Goetz,
5
S. Gomes,
35
B. Goncharov,
6
G. González,
2
J. M. Gonzalez Castro,
23,24
A. Gopakumar,
118
M. L. Gorodetsky,
60
S. E. Gossan,
1
M. Gosselin,
30
R. Gouaty,
8
A. Grado,
119,4
C. Graef,
45
M. Granata,
26
A. Grant,
45
S. Gras,
15
C. Gray,
46
G. Greco,
120,121
A. C. Green,
57
E. M. Gretarsson,
36
P. Groot,
64
H. Grote,
10
S. Grunewald,
37
P. Gruning,
28
G. M. Guidi,
120,121
X. Guo,
81
A. Gupta,
62
M. K. Gupta,
103
K. E. Gushwa,
1
E. K. Gustafson,
1
R. Gustafson,
117
PHYSICAL REVIEW LETTERS
120,
201102 (2018)
201102-8