Search for the isotropic stochastic background using data from Advanced
LIGO
’
s second observing run
B. P. Abbott
etal.
*
(LIGO Scientific and Virgo Collaboration)
(Received 14 April 2019; published 4 September 2019)
The stochastic gravitational-wave background is a superposition of sources that are either too weak or
too numerous to detect individually. In this study, we present the results from a cross-correlation analysis on
data from Advanced LIGO
’
s second observing run (O2), which we combine with the results of the first
observing run (O1). We do not find evidence for a stochastic background, so we place upper limits on the
normalized energy density in gravitational waves at the 95% credible level of
Ω
GW
<
6
.
0
×
10
−
8
for a
frequency-independent (flat) background and
Ω
GW
<
4
.
8
×
10
−
8
at 25 Hz for a background of compact
binary coalescences. The upper limit improves over the O1 result by a factor of 2.8. Additionally, we place
upper limits on the energy density in an isotropic background of scalar- and vector-polarized gravitational
waves, and we discuss the implication of these results for models of compact binaries and cosmic string
backgrounds. Finally, we present a conservative estimate of the correlated broadband noise due to the
magnetic Schumann resonances in O2, based on magnetometer measurements at both the LIGO Hanford
and LIGO Livingston observatories. We find that correlated noise is well below the O2 sensitivity.
DOI:
10.1103/PhysRevD.100.061101
I. INTRODUCTION
A superposition of gravitational waves from many astro-
physical and cosmological sources creates a stochastic
gravitational-wave background (SGWB). Sources which
may contribute to the stochastic background include com-
pact binary coalescences
[1
–
8]
, core collapse supernovae
[9
–
14]
, neutron stars
[15
–
24]
, stellar core collapse
[25,26]
,
cosmic strings
[27
–
31]
, primordial black holes
[32
–
34]
,
superradiance of axion clouds around black holes
[35
–
38]
,
and gravitational waves produced during inflation
[39
–
47]
.
A particularlypromisingsource isthestochasticbackground
from compact binary coalescences, especially in light of the
detections of one binary neutron star and ten binary black
hole mergers
[48
–
55]
by the Advanced LIGO detector,
installed in the Laser Interferometer Gravitational-wave
Observatory (LIGO)
[56]
, and by Advanced Virgo
[57]
so far. Measurements of the rate of binary black hole and
binary neutron star mergers imply that the stochastic back-
ground may be large enough to detect with the Advanced
LIGO-Virgo detector network
[58,59]
. The stochastic back-
ground is expected to be dominated by compact binaries at
redshifts inaccessible to direct searches for gravitational-
wave events
[60]
. Additionally, a detection of the stochastic
background would enable a model-independent test of
general relativity by discerning the polarization of gravita-
tional waves
[61,62]
. Because general relativity predicts
only two tensor polarizations for gravitational waves, any
detection of alternative polarizations would imply a modi-
fication to our current understanding of gravity
[63
–
65]
.For
recent reviews on relevant data analysis methods, see
Refs.
[66,67]
.
In this paper, we present a search for an isotropic
stochastic background using data from Advanced LIGO
’
s
second observing run (O2). As in previous LIGO and Virgo
analyses, this search is based on cross-correlating the strain
data between pairs of gravitational-wave detectors
[68,69]
.
We first review the stochastic search methodology and then
describe the data and data quality cuts. As we do not find
evidence for the stochastic background, we place upper
limits on the possible amplitude of an isotropic stochastic
background as well as limits on the presence of alternative
gravitational-wave polarizations. Upper limits on aniso-
tropic stochastic backgrounds are given in a publication
that is a companion to this one
[70]
. We then give updated
forecasts of the sensitivities of future stochastic searches and
discuss the implications of our current results for the
detection of the stochastic background from compact
binaries and cosmic strings. Finally, we present estimates
of the correlated noise in the LIGO detectors due to magnetic
Schumann resonances
[71]
and discuss mitigation strategies
that are being pursued for future observing runs.
II. METHOD
The isotropic stochastic background can be described in
terms of the energy density per logarithmic frequency
interval
Ω
GW
ð
f
Þ¼
f
ρ
c
d
ρ
GW
d
f
;
ð
1
Þ
*
Full author list given at end of the article.
PHYSICAL REVIEW D
100,
061101(R) (2019)
Rapid Communications
2470-0010
=
2019
=
100(6)
=
061101(16)
061101-1
© 2019 American Physical Society
where d
ρ
GW
is the energy density in gravitational waves
in the frequency interval from
f
to
f
þ
d
f
and
ρ
c
¼
3
H
2
0
c
2
=
ð
8
π
G
Þ
is the critical energy density required for
a spatially flat universe. Throughout this work, we will use
the value of the Hubble constant measured by the
Planck
satellite,
H
0
¼
67
.
9
km s
−
1
Mpc
−
1
[72]
.
We use the optimal search for a stationary, Gaussian,
unpolarized, and isotropic stochastic background, which is
the cross-correlation search
[66,67,73,74]
(however, see
Ref.
[75]
). For two detectors, we define a cross-correlation
statistic
ˆ
C
ð
f
Þ
in every frequency bin
ˆ
C
ð
f
Þ¼
2
T
Re
½
̃
s
⋆
1
ð
f
Þ
̃
s
2
ð
f
Þ
γ
T
ð
f
Þ
S
0
ð
f
Þ
;
ð
2
Þ
where
̃
s
i
ð
f
Þ
is the Fourier transform of the strain time series
in detector
i
¼f
1
;
2
g
,
T
is the segment duration used to
compute the Fourier transform, and
S
0
ð
f
Þ
is the spectral
shape for an
Ω
GW
¼
const background given by
S
0
ð
f
Þ¼
3
H
2
0
10
π
2
f
3
:
ð
3
Þ
The quantity
γ
T
ð
f
Þ
is the normalized overlap reduction
function for tensor (T) polarizations
[73]
, which encodes
the geometry of the detectors and acts as a transfer function
between strain cross-power and
Ω
GW
ð
f
Þ
. Equation
(2)
has
been normalized so that the expectation value of
ˆ
C
ð
f
Þ
is
equal to the energy density in each frequency bin
h
ˆ
C
ð
f
Þi ¼
Ω
GW
ð
f
Þ
:
ð
4
Þ
In the limit where the gravitational-wave strain amplitude is
small compared to instrumental noise, the variance of
ˆ
C
ð
f
Þ
is approximately given by
σ
2
ð
f
Þ
≈
1
2
T
Δ
f
P
1
ð
f
Þ
P
2
ð
f
Þ
γ
2
T
ð
f
Þ
S
2
0
ð
f
Þ
;
ð
5
Þ
where
P
1
;
2
ð
f
Þ
are the one-sided noise power spectral
densities of the two detectors and
Δ
f
is the frequency
resolution, which we take to be
1
=
32
Hz.
An optimal estimator can be constructed for a model of
any spectral shape by taking a weighted combination of the
cross-correlation statistics across different frequency bins
f
k
,
ˆ
Ω
ref
¼
P
k
w
ð
f
k
Þ
−
1
ˆ
C
ð
f
k
Þ
σ
−
2
ð
f
k
Þ
P
k
w
ð
f
k
Þ
−
2
σ
−
2
ð
f
k
Þ
;
σ
−
2
Ω
¼
X
k
w
ð
f
k
Þ
−
2
σ
−
2
ð
f
k
Þ
;
ð
6
Þ
where the optimal weights for spectral shape
Ω
GW
ð
f
Þ
are
given by
w
ð
f
Þ¼
Ω
GW
ð
f
ref
Þ
Ω
GW
ð
f
Þ
:
ð
7
Þ
The broadband estimators are normalized so that
h
ˆ
Ω
ref
i¼
Ω
GW
ð
f
ref
Þ
. By appropriate choices of the weights
w
ð
f
Þ
, one
may construct an optimal search for stochastic backgrounds
with arbitrary spectral shapes, or for stochastic back-
grounds with scalar and vector polarizations.
Many models of the stochastic background can be
approximated as a power laws
[74,76]
,
Ω
GW
ð
f
Þ¼
Ω
ref
f
f
ref
α
;
ð
8
Þ
with a spectral index
α
and an amplitude
Ω
ref
at a reference
frequency
f
ref
. As in the search in Advanced LIGO
’
s first
observing run (O1)
[68]
, we will take
f
ref
¼
25
Hz, which
is a convenient choice in the most sensitive part of the
frequency band. While we will seek to generically constrain
both
Ω
ref
and
α
from the data, we will also investigate
several specific spectral indices predicted for different
gravitational-wave sources. In the frequency band probed
by Advanced LIGO, the stochastic background from
compact binaries is well approximated by a power law
with
α
¼
2
=
3
[77]
. Slow roll inflation and cosmic string
models can be described with
α
¼
0
[78]
. Finally, follow-
ing previous analyses
[68]
, we use
α
¼
3
as an approximate
value to stand in for a variety of astrophysical models with
positive slopes, such as unresolved supernovae
[11
–
14]
.
III. DATA
We analyze data from Advanced LIGO
’
s second observ-
ing run, which took place from 16
∶
00:00 UTC on November
30, 2016 to 22
∶
00:00 UTC on August 25, 2017. We cross-
correlate the strain data measured by the two Advanced
LIGO detectors, located in Hanford, Washington, and
Livingston, Louisiana, in the United States
[56]
. Linearly
coupled noise has been removed from the strain time series
at Hanford and Livingston using Wiener filtering
[79,80]
;
see also Refs.
[81
–
83]
. By comparing coherence spectra and
narrowband estimators formed with and without Wiener
filtering, we additionally verified that this noise subtraction
scheme does not introduce correlated artifacts into the
Hanford and Livingston data.
Virgo does not have a significant impact on the sensitivity
of the stochastic search in O2 because of the larger detector
noise, the fact that less than one month of coincident
integration time is available, and that fact that the overlap
reduction function is smaller for the Hanford-Virgo and
Livingston-Virgo pairs than for Hanford-Livingston.
Therefore, we do not include Virgo data in the O2 analysis.
The raw strain data are recorded at 16,384 Hz. We first
downsample the strain time series to 4096 Hz and apply a
16th-order high-pass Butterworth filter with knee fre-
quency of 11 Hz to avoid spectral leakage from the noise
power spectrum below 20 Hz. Next, we apply a Fourier
B. P. ABBOTT
et al.
PHYS. REV. D
100,
061101 (2019)
061101-2
transform to segments with a duration of 192 s, using 50%
overlapping Hann windows, and then we coarse grain six
frequency bins to obtain a frequency resolution of
1
=
32
Hz.
As in Ref.
[68]
, we observe in the band 20
–
1726 Hz. The
maximum frequency of 1726 Hz is chosen to avoid aliasing
effects after downsampling the data.
Next, we apply a series of data quality cuts that remove
non-Gaussian features of the data. We remove times when the
detectors are known to be unsuitable for science results
[84]
and times associated with known gravitational-wave events
[55]
. We also remove times where the noise is nonstationary,
following the procedure described in the supplement of
Ref.
[69]
(see also Ref.
[68]
). These cuts remove 16% of
the coincident time, which is in principle suitable for data
analysis, leading to a coincident live time of 99 days.
In the frequency domain, we remove narrowband coher-
ent lines that are determined to have instrumental or
environmental causes, using the methods described in
Ref.
[85]
. These cuts remove 15% of the total observing
band, but only 4% of the band below 300 Hz, where the
isotropic search is most sensitive. The narrow frequency
binning of
1
=
32
Hz was needed to cut out a comb of
coherent lines found at integer frequencies. A list of notch
filters corresponding to lines which were removed from the
analysis is also available on the public data release page
[86]
.
IV. O2 RESULTS
In Fig.
1
, we plot the observed cross-correlation spec-
trum
ˆ
C
ð
f
Þ
and uncertainty
σ
ð
f
Þ
obtained from Advanced
LIGO
’
s O2 run. We only plot the spectrum up to 100 Hz to
focus on the most sensitive part of the frequency band.
These data are also publicly available on the webpage
[86]
and can be used to search for stochastic backgrounds of any
spectral shape.
We perform several tests that the cross-correlation
spectrum is consistent with uncorrelated Gaussian noise.
The
χ
2
per degree of freedom for the observed spectrum is
0.94. The loudest individual frequency bin is 51.53 Hz,
with a signal-to-noise ratio
C
ð
f
Þ
=
σ
ð
f
Þ
of 4.2. With a total
of 46,227 (un-notched) frequency bins, there is a 71%
probability that random Gaussian noise would yield an
equally loud bin.
In Table
I
, we list the broadband point estimates and
1
σ
uncertainties obtained from the O2 data when assuming
power laws with
α
¼
0
,
2
=
3
, and 3. Given the uncertainties,
uncorrelated Gaussian noise would produce point estimates
at least this large with probability 30%, 22%, and 21%,
respectively. We conclude there is not sufficient evidence to
claim detection of the stochastic background.
V. UPPER LIMITS ON ISOTROPIC STOCHASTIC
BACKGROUND
Since we do not find evidence for the stochastic back-
ground, we place upper limits on the amplitude
Ω
ref
.We
use the parameter estimation framework described in
Refs.
[61,62,76]
, applied to the cross-correlation spectrum
obtained by combining the results from O1 given in
Ref.
[68]
with those from O2 which are described above
(please see the Supplemental Material
[87]
for more
details). We present results assuming two priors, one which
is uniform in
Ω
ref
and one which is uniform in log
Ω
ref
.We
additionally marginalize over detector calibration uncer-
tainties
[88]
. In O2, we assume 2.6% and 3.85% amplitude
uncertainties in Hanford and Livingston, respectively
[89,90]
. In O1, the calibration uncertainty for Hanford
was 4.8% and for Livingston was 5.4%
[89]
. Phase
calibration uncertainty is negligible.
FIG. 1. The cross-correlation spectrum
ˆ
C
ð
f
Þ
measured between
Advanced LIGO
’
s Hanford and Livingston detectors during its
second observing run. The estimator is normalized so that
h
ˆ
C
ð
f
Þi ¼
Ω
GW
ð
f
Þ
for tensor-polarized gravitational waves.
The black traces mark the
1
σ
uncertainties on the measured
cross-correlations. Coherent lines that were identified to have an
instrumental cause have been removed from the spectrum. The
loss in sensitivity visible at approximately 64 Hz is due to a zero
in the tensor overlap reduction function
γ
T
ð
f
Þ
.
TABLE I. Point estimates and
1
σ
uncertainties for
Ω
ref
in O2,
for different power-law models, alongside the same quantities
measured in O1
[68]
. We also show the minimum contiguous
frequency band containing 99% of the sensitivity. For each power
law, the maximum of the frequency band is within 5% of the
value found in O1. The value of the Hubble constant used in this
paper is different than what was used in the O1 analysis
[68]
(
68
km s
−
1
Mpc
−
1
), which has led to some differences in the
numerical values of the point estimates and error bars that we
report for O1.
α
ˆ
Ω
ref
(O2)
ˆ
Ω
ref
(O1)
O2 sensitive band
0
ð
2
.
2
2
.
2
Þ
×
10
−
8
ð
4
.
4
6
.
0
Þ
×
10
−
8
20
–
81.9 Hz
2
=
3
ð
2
.
0
1
.
6
Þ
×
10
−
8
ð
3
.
5
4
.
4
Þ
×
10
−
8
20
–
95.2 Hz
3
ð
3
.
5
2
.
8
Þ
×
10
−
9
ð
3
.
7
6
.
6
Þ
×
10
−
9
20
–
301 Hz
SEARCH FOR THE ISOTROPIC STOCHASTIC BACKGROUND
...
PHYS. REV. D
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061101 (2019)
061101-3
Figure
2
shows the resulting posterior distribution in the
Ω
ref
vs
α
plane, along with 68% and 95% credibility
contours. Table
II
lists the marginalized 95% credible upper
limit on
Ω
ref
(for both choices of amplitude prior) as well as
the amplitude limits obtained when fixing
α
¼
0
,
2
=
3
, and 3.
When adopting a uniform amplitude prior and fixing
α
¼
0
, we obtain an upper limit of
Ω
ref
<
6
.
0
×
10
−
8
,
improving the previous O1 result by a factor of 2.8. The
1
σ
error bar is
2
.
2
×
10
−
8
, a factor of 2.7 times smaller than the
equivalent O1 uncertainty. This factor can be compared
with the factor of 2.1 that would be expected based on
increased observation time alone, indicating that the search
has benefited from improvements in detector noise between
O1 and O2. For the compact binary stochastic background
model of
α
¼
2
=
3
, we place a limit of
Ω
ref
<
4
.
8
×
10
−
8
,
and for
α
¼
3
,
Ω
ref
<
7
.
9
×
10
−
9
. Finally, when we mar-
ginalize over the power-law index
α
, we obtain the upper
limit
Ω
ref
<
1
.
1
×
10
−
7
. The prior for
α
is described in the
Supplemental Material
[87]
.
VI. IMPLICATIONS FOR COMPACT BINARY
BACKGROUND
In Fig.
3
, we show the prediction of the astrophysical
stochastic background from binary black holes (BBHs) and
binary neutron stars (BNSs), along with its statistical
uncertainty due to Poisson uncertainties in the local binary
merger rate. We plot the upper limit allowed from adding
the background from neutron star
–
black hole (NSBH)
binaries as a dotted line. We use the same binary formation
and evolution scenario to compute the stochastic back-
ground from BBH and BNS as in Ref.
[59]
, but we have
updated the mass distributions and rates to be consistent
with the most recent results given in Refs.
[55,91]
.For
NSBHs, we use the same evolution with redshift as BNSs.
As in Refs.
[54]
, for BBHs, we include inspiral, merger,
and ringdown contributions computed in Ref.
[92]
, while
for NSBH and BNSs, we use only the inspiral part of the
waveform. For the BBH mass distribution, we assume a
power law in the primary mass
p
ð
m
1
Þ
∝
m
−
2
.
3
1
with the
secondary mass drawn from a uniform distribution, subject
to the constraints
5
M
⊙
≤
m
2
≤
m
1
≤
50
M
⊙
. In Ref.
[55]
,
rate estimates were computed by two pipelines, PyCBC
[93]
and GstLAL
[94]
. We use the merger rate measured by
GstLAL,
R
local
¼
56
þ
44
−
27
Gpc
−
3
yr
−
1
[55]
, because it gives a
more conservative (smaller) rate estimate. Using the meth-
ods described in Ref.
[59]
, the inferred amplitude of the
stochastic background is
Ω
BBH
ð
25
Hz
Þ¼
5
.
3
þ
4
.
2
−
2
.
5
×
10
−
10
.
For the BNS mass distribution, following the analysis in
Ref.
[55]
, we take each component mass to be drawn from a
Gaussian distribution with a mean of
1
.
33
M
⊙
and a
standard deviation of
0
.
09
M
⊙
. We use the GstLAL rate
of
R
local
¼
920
þ
2220
−
790
Gpc
−
3
yr
−
1
[55]
. From these inputs,
we predict
Ω
BNS
ð
25
Hz
Þ¼
3
.
6
þ
8
.
4
−
3
.
1
×
10
−
10
. Combining
the BBH and BNS results yields a prediction for the total
SGWB of
Ω
BBH
þ
BNS
ð
25
Hz
Þ¼
8
.
9
þ
12
.
6
−
5
.
6
×
10
−
10
. This
value is about a factor of 2 smaller the one in Ref.
[59]
,
due in part to the decrease in the rate measured after
analyzing O1 and O2 data with the best available sensitivity
and data analysis techniques.
For NSBH, we assume a delta function mass distribu-
tion, where the neutron star has a mass of
1
.
4
M
⊙
and the
black hole has a mass of
10
M
⊙
, and we take the upper
limit on the rate from GstLAL
[55]
. The upper limit from
NSBH is
Ω
NSBH
ð
25
Hz
Þ¼
9
.
1
×
10
−
10
. We show the sum
of the upper limit of
Ω
NSBH
ð
f
Þ
, with the 90% upper limit on
Ω
BBH
þ
BNS
ð
f
Þ
, as a dotted line in Fig.
3
.
We also show the power law
–
integrated (PI) curves
[96]
of the O1 and O2 isotropic background searches. A power-
law stochastic background that is tangent to a PI curve is
detectable with SNR
¼
2
by the given search. We addi-
tionally show a projected PI curve based on operating
FIG. 2. Posterior distribution for the amplitude
Ω
ref
and slope
α
of the stochastic background, using a prior which is uniform in
the logarithm of
Ω
ref
, along with contours with 68% and
95% confidence level, using combined O1 and O2 data. There
is a small region of increased posterior probability centered
around log
Ω
ref
¼
−
8
and
α
¼
2
. This is not statistically signifi-
cant, and similar-size bumps have appeared in simulations of
Gaussian noise. An analogous plot with a prior uniform in
Ω
ref
can be found in the Supplemental Material
[87]
.
TABLE II. 95% credible upper limits on
Ω
ref
for different
power-law models (fixed
α
) as well as marginalizing over
α
, for
combined O1 and O2 data (current limits) and for O1 data
(previous limits)
[68]
. We show results for two priors, one which
is uniform in
Ω
ref
and one which is uniform in the logarithm
of
Ω
ref
.
Uniform prior
Log-uniform prior
α
O
1
þ
O
2
O1
O
1
þ
O
2
O1
0
6
.
0
×
10
−
8
1
.
7
×
10
−
7
3
.
5
×
10
−
8
6
.
4
×
10
−
8
2
=
34
.
8
×
10
−
8
1
.
3
×
10
−
7
3
.
0
×
10
−
8
5
.
1
×
10
−
8
3
7
.
9
×
10
−
9
1
.
7
×
10
−
8
5
.
1
×
10
−
9
6
.
7
×
10
−
9
Marg.
1
.
1
×
10
−
7
2
.
5
×
10
−
7
3
.
4
×
10
−
8
5
.
5
×
10
−
8
B. P. ABBOTT
et al.
PHYS. REV. D
100,
061101 (2019)
061101-4
Advanced LIGO and Advanced Virgo at design sensitivity
for 2 years, with 50% network duty cycle. By design
sensitivity, we refer to a noise curve which is determined by
fundamental noise sources. We use the Advanced LIGO
design sensitivity projection given in Ref.
[95]
, which
incorporates improved measurements of coating thermal
noise relative to the one assumed in Ref.
[58]
. This updated
curve introduces additional broadband noise at low fre-
quencies relative to previous estimates. As a result, the
updated design-sensitivity PI curve is less sensitive than the
one shown in Ref.
[58]
.
VII. IMPLICATIONS FOR COSMIC
STRING MODELS
Cosmic strings
[97,98]
are linear topological defects
which are expected to be generically produced within the
context of grand unified theories
[99]
. The dynamics of a
cosmic string network is driven by the formation of loops
and the emission of gravitational waves
[100,101]
.One
may therefore use the stochastic background in order to
constrain the parameters of a cosmic string network.
We will focus on Nambu-Goto strings
[102,103]
, for
which the string thickness is zero and the intercommutation
probability equals unity. Gravitational waves will allow us
to constrain the string tension
G
μ
=c
2
, where
μ
denotes the
mass per unit length. This dimensionless parameter is the
single quantity that characterizes a Nambu-Goto string
network.
We will consider two analytic models of cosmic string
loop distributions
[104,105]
. The former
[104]
gives the
distribution of string loops of given size at fixed time, under
the assumption that the momentum dependence of the loop
production function is weak. The latter
[105]
is based on a
different numerical simulation
[106]
and gives the distri-
bution of non
–
self intersecting loops at a given time
[107]
.
The corresponding limits found by combining O1 and O2
data are
G
μ
=c
2
≤
1
.
1
×
10
−
6
for the model of Ref.
[104]
and
G
μ
=c
2
≤
2
.
1
×
10
−
14
for the model of Ref.
[105]
. The
Advanced LIGO constraints are stronger for the model of
Ref.
[105]
because the predicted spectrum is larger at 100 Hz
for that model. This can be compared with the pulsar timing
limits,
G
μ
=c
2
≤
1
.
6
×
10
−
11
and
G
μ
=c
2
≤
6
.
2
×
10
−
12
,
respectively
[108]
.
VIII. TEST OF GENERAL RELATIVITY
Alternative theories of gravity generically predict the
presence of vector or scalar gravitational-wave polarizations
in addition to the standard tensor polarizations allowed in
general relativity. Detection of the stochastic background
would allow for direct measurement of its polarization
content, enabling new tests of general relativity
[61,62]
.
When allowing for the presence of alternative gravita-
tional-wave polarizations, the expectation value of the
cross-correlation statistic becomes
h
ˆ
C
ð
f
Þi ¼
X
A
β
A
ð
f
Þ
Ω
A
GW
ð
f
Þ¼
X
A
β
A
ð
f
Þ
Ω
A
ref
f
f
ref
α
A
;
ð
9
Þ
where
β
A
¼
γ
A
ð
f
Þ
=
γ
T
ð
f
Þ
and
A
labels the polarization,
A
¼f
T; V; S
g
. The functions
γ
T
ð
f
Þ
,
γ
V
ð
f
Þ
, and
γ
S
ð
f
Þ
are
the overlap reduction functions for tensor, vector, and scalar
polarizations
[61]
. Because these overlap reduction func-
tions are distinct, the spectral shape of
ˆ
C
ð
f
Þ
enables us to
infer the polarization content of the stochastic background.
While we use the notation
Ω
A
GW
ð
f
Þ
in analogy with the
general relativity (GR) case, in a general modification of
gravity, the quantities
Ω
T
GW
ð
f
Þ
,
Ω
V
GW
ð
f
Þ
, and
Ω
S
GW
ð
f
Þ
are
best understood as a measurement of the two-point corre-
lation statistics of different components of the stochastic
background rather than energy densities
[109]
.
Following Refs.
[61,62]
, we compute two Bayesian
odds: odds
O
S
N
for the presence of a stochastic signal of
any polarization(s) vs Gaussian noise and odds
O
NGR
GR
between a hypothesis allowing for vector and scalar modes
and a hypothesis restricting to standard tensor polariza-
tions. Using the combined O1 and O2 measurements, we
find log
O
S
N
¼
−
0
.
64
and log
O
NGR
GR
¼
−
0
.
45
, consistent
with Gaussian noise. Given the nondetection of any generic
stochastic background, we use Eq.
(9)
to place improved
upper limits on the tensor, vector, and scalar background
amplitudes, after marginalizing over all three spectral
indices, using the priors described in the Supplemental
FIG. 3. Sensitivity curves for O1, combined O
1
þ
O
2
, and
design sensitivity. A power law stochastic background which lies
tangent to one of these curves is detectable with
2
σ
significance.
We have used the Advanced LIGO design sensitivity given in
Ref.
[95]
, which incorporates improved measurements of coating
thermal noise. Design sensitivity assumes that the LIGO noise
curve is determined by fundamental noise sources only. The
purple line is the median total stochastic background, combining
BBHs and BNSs, using the model described in Ref.
[59]
with
updated mass distributions and rates from Refs.
[55,91]
, and the
gray box is the Poisson error region. The dotted gray line is
the sum of the upper limit for the BBH
þ
BNS backgrounds with
the upper limit on the NSBH background.
SEARCH FOR THE ISOTROPIC STOCHASTIC BACKGROUND
...
PHYS. REV. D
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061101 (2019)
061101-5
Material
[87]
. These limits are shown in Table
III
, again for
both choices of amplitude prior.
IX. ESTIMATE OF CORRELATED
MAGNETIC NOISE
Coherent noise between gravitational-wave interferom-
eters may be introduced by terrestrial sources such as
Schumann resonances, which are global electromagnetic
modes of the cavity formed by the Earth
’
s surface and
ionosphere
[71]
. These fields have very long coherence
lengths
[110]
and can magnetically couple to the gravita-
tional-wave channel and lead to broadband noise that is
coherent between different gravitational-wave detectors. As
the detectors become more sensitive, eventually this source
of correlated noise may become visible to the cross-
correlation search and, if not treated carefully, will bias
the analysis by appearing as an apparent stochastic back-
ground. Unlike the lines and combs discussed in Ref.
[85]
,
we cannot simply remove affected frequency bins from the
analysis because Schumann noise is broadband.
Here, we estimate the level of correlated electromagnetic
noise (from Schumann resonances or other sources) in O2
following Refs.
[68,111,112]
. We first measure the cross-
power spectral density
M
12
ð
f
Þ
between two Bartington
Model MAG-03MC magnetometers
[113]
installed at
Hanford and Livingston. We then estimate the transfer
function
T
i
ð
f
Þ
(
i
¼f
1
;
2
g
) between the magnetometer
channel and the gravitational-wave channel at each site,
as described in Ref.
[114]
. Finally, we combine these
results to produce an estimate for the amount of correlated
magnetic noise, which we express in terms of an effective
gravitational-wave energy density
Ω
mag
ð
f
Þ
,
Ω
mag
ð
f
Þ¼
j
T
1
ð
f
Þjj
T
2
ð
f
Þj
Re
½
M
12
ð
f
Þ
γ
T
ð
f
Þ
S
0
ð
f
Þ
:
ð
10
Þ
We show
Ω
mag
ð
f
Þ
in Fig.
4
, alongside the measured O1
+O2 PI curve and the projected design-sensitivity PI curve.
The trend for the magnetic noise lies significantly below the
O1+O2 PI curve, indicating that correlated magnetic noise
is more than an order of magnitude below the sensitivity
curve in O2, although it may be an issue for future runs.
Experimental improvements can mitigate this risk by
further reducing the coupling of correlated noise. From
O1 to O2, for instance, the magnetic coupling was reduced
by approximately an order of magnitude, as indicated by
the dotted and dot-dashed curves in Fig.
4
. Additionally,
work is ongoing to develop Wiener filtering to subtract
Schumann noise
[110,112,115]
and to develop a parameter
estimation framework to measure or place upper limits on
the level of magnetic contamination
[116]
. This work will
take advantage of low noise LEMI-120 magnetometers
[117]
that were recently installed at both LIGO sites, as
described in the Supplemental Material
[87]
.
X. CONCLUSIONS
We have presented the results of a cross-correlation
search for the isotropic stochastic background using data
from Advanced LIGO
’
s first and second observing runs.
While we did not find evidence for the stochastic back-
ground, we obtain the most sensitive upper limits to date in
the approximately 20
–
100 Hz frequency band. We have
also placed improved upper limits on the existence of a
stochastic background from vector and scalar-polarized
gravitational waves.
While the upper limits on the SGWB presented in this
work are the strongest direct limits in the frequency band of
current ground-based gravitational-wave detectors, other
TABLE III. Upper limits on different polarizations. To obtain
the upper limits, we assume a log uniform and a uniform prior on
the amplitude
Ω
ref
for each polarization, using combined O1 and
O2 data. We assume the presence of a tensor, vector, and scalar
backgrounds and then marginalize over the spectral indices and
two amplitudes for the three different polarization modes, as
described in the main text.
Polarization
Uniform prior
Log-uniform prior
Tensor
8
.
2
×
10
−
8
3
.
2
×
10
−
8
Vector
1
.
2
×
10
−
7
2
.
9
×
10
−
8
Scalar
4
.
2
×
10
−
7
6
.
1
×
10
−
8
FIG. 4. Conservative estimate of correlated magnetic noise. We
assume a conservative transfer function (TF) based on measure-
ments as described in the text. The first Schumann resonance at
8 Hz is visible, and higher harmonics are below the noise floor.
There is a zero of the overlap function at 64 Hz which leads to an
apparent feature in
Ω
mag
. Power line harmonics have been
removed, as in the cross-correlation analysis. The two trend
lines show power-law fits to the magnetometer spectra, scaled by
the O1 (purple dotted) and end-of-O2 (blue dot-dashed) transfer
functions. This demonstrates the effect of reducing the magnetic
coupling in O2. The trend for the noise budget lies well below the
solid black O2 PI curve, which indicates that correlated magnetic
noise is negligible in O2. However, magnetic contamination may
be an issue in future observing runs.
B. P. ABBOTT
et al.
PHYS. REV. D
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061101 (2019)
061101-6
observations place stronger constraints in other frequency
bands. The NANOGrav Collaboration has reported the
95% upper limit of
Ω
GW
<
7
.
4
×
10
−
10
at a frequency of
1
yr
−
1
after marginalizing over uncertainty in the Solar
system ephemeris
[118]
. Combining data from the
Planck
satellite and the BICEP2/Keck array constrains the tensor-
to-scalar ratio from the cosmic microwave background to
be
r<
0
.
064
at 95% confidence at comoving scales of
k
¼
0
.
002
Mpc
−
1
, corresponding to a gravitational-wave
frequency of
f
0
.
002
¼ð
2
π
Þ
−
1
ck
¼
3
.
1
×
10
−
18
Hz
[119]
,
assuming the single field slow roll consistency condition.
Using Eq.
(4)
of Ref.
[108]
, this can be converted into the
constraint
Ω
GW
ð
f
Þ
≤
3
.
2
×
10
−
16
×
ð
f=f
0
.
05
Þ
−
r=
8
½
16
=
9
þ
f
2
eq
=
ð
2
f
2
Þ
, where
f
eq
is the frequency of a gravitational
wave of which the wavelength was the size of the Universe
at matter-radiation equality and
f
0
.
05
is the pivot scale.
Combining constraints at different frequency ranges can
probe models which span many orders of magnitude in
frequency
[108,119]
.
While we have targeted an isotropic, stationary, and
Gaussian background, other search techniques can probe
backgrounds that violate one or more of these assumptions.
Upper limits on an anisotropic gravitational-wave back-
ground from O1 were presented in Ref.
[120]
. Furthermore,
non-Gaussian searches targeting the compact binary
stochastic background are currently being developed
[121
–
124]
. A successful detection of the stochastic back-
ground by any of these approaches would offer a new probe
of the gravitational-wave sky.
The supporting data for this paper are openly available
via the LIGO Document Control Center (DCC)
[86]
.
ACKNOWLEDGMENTS
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 European Gravitational
Observatory (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, 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; the Russian Foundation for Basic
Research; the Russian Science Foundation; the European
Commission; the European Regional Development Funds;
the Royal Society; the Scottish Funding Council; the
Scottish Universities Physics Alliance; the Hungarian
Scientific Research Fund; the Lyon Institute of Origins;
the Paris Île-de-France Region; the National Research,
Development and Innovation Office, Hungary; the National
Research Foundation of Korea; Industry Canada and the
Province of Ontario through the Ministry of Economic
Development and Innovation; the Natural Science
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; the
Research Grants Council of Hong Kong; the National
Natural Science Foundation of China; the Leverhulme
Trust, the Research Corporation; the Ministry of Science
and Technology, 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. This
article has been assigned the document number LIGO-
P1800258.
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