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A search for the isotropic stochastic background using data from Advanced LIGO’s
second observing run
The LIGO Scientific Collaboration and The Virgo Collaboration
(Dated: April 11, 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.
Introduction
— A superposition of gravitational waves
from many astrophysical and cosmological sources cre-
ates a stochastic gravitational-wave background. Sources
which may contribute to the stochastic background in-
clude compact binary coalescences [1–8], core collapse
supernovae [9–14], neutron stars [15–24], stellar core col-
lapse [25, 26], cosmic strings [27–31], primordial black
holes [32–34], superradiance of axion clouds around black
holes [35–37], and gravitational waves produced during
inflation [38–46]. A particularly promising source is the
stochastic background from compact binary coalescences,
especially in light of the detections of one binary neutron
star and ten binary black hole mergers [47–54] by the
Advanced LIGO Detector, installed in the Laser Interfer-
ometer Gravitational-wave Observatory (LIGO) [55], and
by Advanced Virgo [56] so far. Measurements of the rate
of binary black hole and binary neutron star mergers im-
ply that the stochastic background may be large enough
to detect with the Advanced LIGO-Virgo detector net-
work [57, 58]. The stochastic background is expected to
be dominated by compact binaries at redshifts inaccessi-
ble to direct searches for gravitational-wave events [59].
Additionally, a detection of the stochastic background
would enable a model-independent test of general relativ-
ity by discerning the polarization of gravitational waves
[60, 61]. Because general relativity predicts only two ten-
sor polarizations for gravitational waves, any detection of
alternative polarizations would imply a modification to
our current understanding of gravity [62–64]. For recent
reviews on relevant data analysis methods, see [65, 66].
In this manuscript, 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 de-
tectors [67, 68]. We first review the stochastic search
methodology, then describe the data and data quality
cuts. As we do not find evidence for the stochastic back-
ground, we place upper limits on the possible amplitude
of an isotropic stochastic background, as well as limits
on the presence of alternative gravitational-wave polar-
izations. Upper limits on anisotropic stochastic back-
grounds are given in a companion publication to this
one [69]. We then give updated forecasts of the sensi-
tivities of future stochastic searches and discuss the im-
plications 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 [70], and discuss mitigation strategies that are
being pursued for future observing runs.
Method
— The isotropic stochastic background can be
described in terms of the energy density per logarithmic
frequency interval
GW
(
f
) =
f
ρ
c
d
ρ
GW
d
f
,
(1)
where d
ρ
GW
is the energy density in gravitational waves
in the frequency interval from
f
to
f
+ d
f
, and
ρ
c
=
3
H
2
0
c
2
/
(8
πG
) 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
= 67
.
9 kms
1
Mpc
1
[71].
We use the optimal search for a stationary, Gaussian,
unpolarized, and isotropic stochastic background, which
is the cross-correlation search [65, 66, 72, 73] (however,
see [74]). For two detectors, we define a cross-correlation
statistic
ˆ
C
(
f
) in every frequency bin
ˆ
C
(
f
) =
2
T
Re[ ̃
s
?
1
(
f
) ̃
s
2
(
f
)]
γ
T
(
f
)
S
0
(
f
)
,
(2)
where ̃
s
i
(
f
) is the Fourier transform of the strain time
series in detector
i
=
{
1
,
2
}
,
T
is the segment duration
arXiv:1903.02886v2 [gr-qc] 10 Apr 2019
2
used to compute the Fourier transform, and
S
0
(
f
) is the
spectral shape for an Ω
GW
= const background given by
S
0
(
f
) =
3
H
2
0
10
π
2
f
3
.
(3)
The quantity
γ
T
(
f
) is the normalized overlap reduction
function for tensor (T) polarizations [72], which encodes
the geometry of the detectors and acts as a transfer func-
tion between strain cross power and Ω
GW
(
f
). Equation
(2) has been normalized so that the expectation value of
ˆ
C
(
f
) is equal to the energy density in each frequency bin
ˆ
C
(
f
)
= Ω
GW
(
f
)
.
(4)
In the limit where the gravitational-wave strain ampli-
tude is small compared to instrumental noise, the vari-
ance of
ˆ
C
(
f
) is approximately given by
σ
2
(
f
)
1
2
T
f
P
1
(
f
)
P
2
(
f
)
γ
2
T
(
f
)
S
2
0
(
f
)
,
(5)
where
P
1
,
2
(
f
) are the one-sided noise power spectral den-
sities of the two detectors and ∆
f
is the frequency reso-
lution, 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
ref
=
k
w
(
f
k
)
1
ˆ
C
(
f
k
)
σ
2
(
f
k
)
k
w
(
f
k
)
2
σ
2
(
f
k
)
,
σ
2
=
k
w
(
f
k
)
2
σ
2
(
f
k
)
,
(6)
where the optimal weights for spectral shape Ω
GW
(
f
) are
given by
w
(
f
) =
GW
(
f
ref
)
GW
(
f
)
.
(7)
The broadband estimators are normalized so that
ˆ
ref
= Ω
GW
(
f
ref
).
By appropriate choices of the
weights
w
(
f
), one may construct an optimal search for
stochastic backgrounds with arbitrary spectral shapes, or
for stochastic backgrounds with scalar and vector polar-
izations.
Many models of the stochastic background can be ap-
proximated as a power laws [73, 75],
GW
(
f
) = Ω
ref
(
f
f
ref
)
α
,
(8)
with a spectral index
α
and an amplitude Ω
ref
at a refer-
ence frequency
f
ref
. As in the search in Advanced LIGO’s
first observing run (O1) [67], we will take
f
ref
= 25 Hz,
which is a convenient choice in the most sensitive part
of the frequency band. While we will seek to generi-
cally 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 back-
ground from compact binaries is well-approximated by
a power law with
α
= 2
/
3 [76]. Slow roll inflation and
cosmic string models can be described with
α
= 0 [77].
Finally, following previous analyses [67], we use
α
= 3 as
an approximate value to stand in for a variety of astro-
physical models with positive slopes, such as unresolved
supernovae [11–14].
Data
— We analyze data from Advanced LIGO’s sec-
ond observing run, which took place from 16:00:00 UTC
on 30 November, 2016 to 22:00:00 UTC on 25 August,
2017. We cross correlate the strain data measured by the
two Advanced LIGO detectors, located in Hanford, WA
and Livingston, LA in the United States [55]. Linearly
coupled noise has been removed from the strain time
series at Hanford and Livingston using Wiener filtering
[78, 79], see also [80–82]. By comparing coherence spec-
tra and narrowband estimators formed with and without
Wiener filtering, we additionally verified that this noise
subtraction scheme does not introduce correlated arti-
facts into the Hanford and Livingston data.
Virgo does not have a significant impact on the sensi-
tivity of the stochastic search in O2, because of the larger
detector noise, the fact that less than one month of coin-
cident integration time is available, and because the over-
lap 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 anal-
ysis.
The raw strain data are recorded at 16384 Hz. We
first downsample the strain time series to 4096 Hz, and
apply a 16th-order high-pass Butterworth filter with knee
frequency of 11 Hz to avoid spectral leakage from the
noise power spectrum below 20 Hz. Next we apply a
Fourier transform to segments with a duration of 192 s,
using 50% overlapping Hann windows, then we coarse-
grain six frequency bins to obtain a frequency resolution
of 1/32 Hz. As in [67], 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 re-
move non-Gaussian features of the data. We remove
times when the detectors are known to be unsuitable
for science results [83] and times associated with known
gravitational-wave events [54]. We also remove times
where the noise is non-stationary, following the proce-
dure described in the supplement of [68] (see also [67]).
These cuts remove 16% of the coincident time which is
in principle suitable for data analysis, leading to a coin-
cident livetime of 99 days.
In the frequency domain, we remove narrowband co-
herent lines that are determined to have instrumental
or environmental causes, using the methods described
in [84]. These cuts remove 15% of the total observing
3
20
30
40
50
60
70
80
90
100
f
(Hz)
1
.
5
1
.
0
0
.
5
0
.
0
0
.
5
1
.
0
1
.
5
ˆ
C
(
f
)
×
10
5
FIG. 1. The cross-correlation spectrum
ˆ
C
(
f
) measured be-
tween Advanced LIGO’s Hanford and Livingston detectors
during its second observing run. The estimator is normalized
so that
ˆ
C
(
f
)
= Ω
GW
(
f
) for tensor-polarized gravitational
waves. The black traces mark the
±
2
σ
uncertainties on the
measured cross-correlations. Coherent lines that were identi-
fied 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
(
f
).
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 co-
herent 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 [85].
O2 Results
— In Figure 1, we plot the observed cross-
correlation spectrum
ˆ
C
(
f
) and uncertainty
σ
(
f
) obtained
from Advanced LIGO’s O2 run. We only plot the spec-
trum up to 100 Hz to focus on the most sensitive part of
the frequency band. These data are also publicly avail-
able on the webpage [85], 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
(
f
)
(
f
) of 4.2. With a to-
tal of 46227 (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 assum-
ing power laws with
α
= 0, 2
/
3, and 3. Given the un-
certainties, 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
α
ˆ
ref
(O2)
ˆ
ref
(O1)
O2 Sensitive band
0
(2
.
2
±
2
.
2)
×
10
8
(4
.
4
±
6
.
0)
×
10
8
20-81.9 Hz
2/3
(2
.
0
±
1
.
6)
×
10
8
(3
.
5
±
4
.
4)
×
10
8
20-95.2 Hz
3
(3
.
5
±
2
.
8)
×
10
9
(3
.
7
±
6
.
6)
×
10
9
20-301 Hz
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 [67]. 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 [67] (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.
background.
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 [60, 61, 75], applied to the cross-correlation spectrum
obtained by combining the results from O1 given in [67],
with those from O2 which are described above (please see
the Technical Supplement 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 uncertainties [86].
In O2 we assume 2.6% and 3.85% amplitude uncertain-
ties in Hanford and Livingston, respectively [87, 88]. In
O1, the calibration uncertainty for Hanford was 11.8%
and for Livingston was 13.4% [67]. Phase calibration un-
certainty is negligible.
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
=
0, 2
/
3, and 3.
When adopting a uniform amplitude prior and fixing
α
= 0, we obtain an upper limit of Ω
ref
<
6
.
0
×
10
8
, im-
proving 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 com-
pared 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 stochas-
tic background model of
α
= 2
/
3, we place a limit of
ref
<
4
.
8
×
10
8
, and for
α
= 3, Ω
ref
<
7
.
9
×
10
9
. Fi-
nally, when we marginalize 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 Technical Supplement.
Implications for compact binary background
— In Fig-
ure 3 we show the prediction of the astrophysical stochas-
tic background from binary black holes (BBH) and bi-
4
Uniform prior
Log-uniform prior
α
O1+O2
O1
O1+O2
O1
0
6
.
0
×
10
8
1
.
7
×
10
7
3
.
5
×
10
8
6
.
4
×
10
8
2/3
4
.
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
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) [67]. We show results for two priors, one which is
uniform in Ω
ref
, and one which is uniform in the logarithm of Ω
ref
.
10
13
10
11
10
9
10
7
10
5
ref
8
6
4
2
0
2
4
6
8
α
68%
95%
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 proba-
bility centered around log Ω
ref
=
8 and
α
= 2. This is not
statistically significant, 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 Technical Supple-
ment.
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
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, then marginalize over the spectral
indices and two amplitudes for the three different polarization
modes, as described in the main text.
nary neutron stars (BNS), along with its statistical un-
certainty 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) bi-
naries 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 [58], but we have up-
dated the mass distributions and rates to be consistent
with the most recent results given in [54, 89]. For NSBH,
we use the same evolution with redshift as BNS. As in
[53], for BBH we include inspiral, merger and ringdown
contributions computed in [90], while for NSBH and BNS
we use only the inspiral part of the waveform. For the
BBH mass distribution, we assume a power law in the
primary mass
p
(
m
1
)
m
2
.
3
1
with the secondary mass
drawn from a uniform distribution, subject to the con-
straints 5
M
m
2
m
1
50
M
. In Ref. [54], rate
estimates were computed by two pipelines, PyCBC [91]
and GstLAL [92]. We use the merger rate measured by
GstLAL,
R
local
= 56
+44
27
Gpc
3
yr
1
[54], because it gives
a more conservative (smaller) rate estimate. Using the
methods described in [58], the inferred amplitude of the
stochastic background is Ω
BBH
(25 Hz) = 5
.
3
+4
.
2
2
.
5
×
10
10
.
For the BNS mass distribution, following the analysis
in [54], 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
= 920
+2220
790
Gpc
3
yr
1
[54]. From these inputs,
we predict Ω
BNS
(25 Hz) = 3
.
6
+8
.
4
3
.
1
×
10
10
. Combining
the BBH and BNS results yields a prediction for the total
SGWB of Ω
BBH+BNS
(25 Hz) = 8
.
9
+12
.
6
5
.
6
×
10
10
. This
value is about a factor of 2 smaller the one in [58], 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 up-
per limit on the rate from GstLAL [54]. The upper limit
from NSBH is Ω
NSBH
(25 Hz) = 9
.
1
×
10
10
. We show the
sum of the upper limit of Ω
NSBH
(
f
), with the 90% upper
limit on Ω
BBH+BNS
(
f
), as a dotted line in Figure 3.
We also show the power-law-integrated curves (PI
curves) [93] of the O1 and O2 isotropic background
searches. A power-law stochastic background that is
tangent to a PI curve is detectable with SNR = 2 by
the given search. We additionally show a projected PI
curve based on operating Advanced LIGO and Advanced
5
10
1
10
2
10
3
f
(Hz)
10
10
10
9
10
8
10
7
10
6
10
5
GW
O1 (2
σ
)
O1+O2 (2
σ
)
Design (2
σ
)
BBH+BNS+NSBH
BBH+BNS
(Median)
BBH+BNS
(Poisson)
FIG. 3. Sensitivity curves for O1, combined O1+O2, and de-
sign sensitivity. A power law stochastic background which
lies tangent to one of these curves is detectable with 2
σ
sig-
nificance. We have used the Advanced LIGO design sensitiv-
ity given in [94], 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 stochas-
tic background, combining BBH and BNS, using the model
described in [58] with updated mass distributions and rates
from [54, 89], and the gray box is the Poisson error region.
The dotted gray line is the sum of the upper limit for the
BBH+BNS backgrounds with the upper limit on the NSBH
background.
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 [94], which incorporates improved measurements
of coating thermal noise relative to the one assumed in
[57]. This updated curve introduces additional broad-
band noise at low frequencies relative to previous esti-
mates. As a result, the updated design-sensitivity PI
curve is less sensitive than the one shown in [57].
Implications for cosmic string models
— Cosmic
strings [95, 96] are linear topological defects which are
expected to be generically produced within the context
of Grand Unified Theories [97]. The dynamics of a cos-
mic string network is driven by the formation of loops
and the emission of gravitational waves [98, 99]. 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 [100, 101], for
which the string thickness is zero and the intercommu-
tation probability equals unity. Gravitational waves will
allow us to constrain the string tension
Gμ/c
2
, where
μ
denotes the mass per unit length. This dimension-
less parameter is the single quantity that characterizes a
Nambu-Goto string network.
We will consider two analytic models of cosmic string
loop distributions [102, 103]. The former [102] 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 [103]
is based on a different numerical simulation [104], and
gives the distribution of non-self intersecting loops at a
given time [105].
The corresponding limits found by combining O1 and
O2 data are
Gμ/c
2
1
.
1
×
10
6
for the model of [102]
and
Gμ/c
2
2
.
1
×
10
14
for the model of [103]. The
Advanced LIGO constraints are stronger for the model
of [103] because the predicted spectrum is larger at 100
Hz for that model. This can be compared with the pulsar
timing limits,
Gμ/c
2
1
.
6
×
10
11
and
Gμ/c
2
6
.
2
×
10
12
, respectively [106].
Test of General Relativity
— Alternative theories of
gravity generically predict the presence of vector or scalar
gravitational-wave polarizations in addition to the stan-
dard 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 [60, 61].
When allowing for the presence of alternative
gravitational-wave polarizations, the expectation value
of the cross-correlation statistic becomes
ˆ
C
(
f
)
=
A
β
A
(
f
)Ω
A
GW
(
f
) =
A
β
A
(
f
)Ω
A
ref
(
f
f
ref
)
α
A
,
(9)
where
β
A
=
γ
A
(
f
)
T
(
f
), and
A
labels the polarization,
A
=
{
T,V,S
}
. The functions
γ
T
(
f
),
γ
V
(
f
), and
γ
S
(
f
)
are the overlap reduction functions for tensor, vector, and
scalar polarizations [60]. Because these overlap reduction
functions are distinct, the spectral shape of
ˆ
C
(
f
) enables
us to infer the polarization content of the stochastic back-
ground. While we use the notation Ω
A
GW
(
f
) in analogy
with the GR case, in a general modification of gravity,
the quantities Ω
T
GW
(
f
), Ω
V
GW
(
f
), and Ω
S
GW
(
f
) are best
understood as a measurement of the two-point correla-
tion statistics of different components of the stochastic
background rather than energy densities [107].
Following Refs. [60, 61], we compute two Bayesian
odds: odds
O
s
n
for the presence of a stochastic signal
of any polarization(s) versus Gaussian noise, and odds
O
ngr
gr
between a hypothesis allowing for vector and scalar
modes and a hypothesis restricting to standard tensor
polarizations. Using the combined O1 and O2 measure-
ments, we find log
O
s
n
=
0
.
64 and log
O
ngr
gr
=
0
.
45,
consistent with Gaussian noise. Given the non-detection
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 Technical Supplement. These limits are shown in
Table III, again for both choices of amplitude prior.
Estimate of correlated magnetic noise
— Coherent
noise between gravitational-wave interferometers may be
6
introduced by terrestrial sources such as Schumann res-
onances, which are global electromagnetic modes of the
cavity formed by the Earth’s surface and ionosphere [70].
These fields have very long coherence lengths [108] and
can magnetically couple to the gravitational-wave chan-
nel and lead to broadband noise that is coherent between
different gravitational-wave detectors. As the detectors
become more sensitive, eventually this source of corre-
lated noise may become visible to the cross-correlation
search, and, if not treated carefully, will bias the analy-
sis by appearing as an apparent stochastic background.
Unlike the lines and combs discussed in [84], we cannot
simply remove affected frequency bins from the analysis
because Schumann noise is broadband.
Here, we estimate the level of correlated electromag-
netic noise (from Schumann resonances or other sources)
in O2 following [67, 109, 110]. We first measure the
cross power spectral density
M
12
(
f
) between two Bart-
ington Model MAG-03MC magnetometers [111] installed
at Hanford and Livingston. We then estimate the trans-
fer function
T
i
(
f
) (
i
=
{
1
,
2
}
) between the magnetome-
ter channel and the gravitational-wave channel at each
site, as described in [112]. Finally, we combine these re-
sults 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
(
f
)
mag
(
f
) =
|
T
1
(
f
)
||
T
2
(
f
)
|
Re[
M
12
(
f
)]
γ
T
(
f
)
S
0
(
f
)
.
(10)
We show Ω
mag
(
f
) in Figure 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 cou-
pling was reduced by approximately an order of magni-
tude, as indicated by the dotted and dot-dashed curves in
Fig. 4. Additionally, work is ongoing to develop Wiener
filtering to subtract Schumann noise [108, 110, 113], and
to develop a parameter estimation framework to mea-
sure or place upper limits on the level of magnetic con-
tamination [114]. This work will take advantage of low
noise LEMI-120 magnetometers [115] that were recently
installed at both LIGO sites, as described in the Techni-
cal Supplement.
Conclusions
— We have presented the results of a
cross-correlation search for the isotropic stochastic back-
ground using data from Advanced LIGO’s first and sec-
ond observing runs. While we did not find evidence for
the stochastic background, we obtain the most sensitive
upper limits to date in the
20-100 Hz frequency band.
We have also placed improved upper limits on the exis-
10
1
10
2
f
(Hz)
10
19
10
16
10
13
10
10
10
7
10
4
GW
O1+O2 PI (2
σ
)
Design PI (2
σ
)
mag
(O1 TF)
mag
Fit (O1 TF)
mag
(O2 TF)
mag
Fit (O2 TF)
FIG. 4. Conservative estimate of correlated magnetic noise.
We assume a conservative transfer function (TF) based on
measurements as described in the text. The first Schumann
resonance at 8 Hz is visible, 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 har-
monics have been removed, as in the cross-correlation analy-
sis. The two trend lines show power law fits to the magne-
tometer 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 neg-
ligible in O2. However magnetic contamination may be an
issue in future observing runs.
tence of a stochastic background from vector and scalar-
polarized gravitational waves.
While we have targeted an isotropic, stationary, and
Gaussian background, other search techniques can probe
backgrounds that violate one or more of these assump-
tions. Upper limits on an anisotropic gravitational-wave
background from O1 were presented in [116]. Further-
more, non-Gaussian searches targeting the compact bi-
nary stochastic background are currently being devel-
oped [117–120]. A successful detection of the stochastic
background by any of these approaches would offer a new
probe of the gravitational-wave sky.
Acknowledgments
— The authors gratefully acknowl-
edge 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 construc-
tion and operation of the GEO600 detector. Additional
support for Advanced LIGO was provided by the Aus-
tralian Research Council. The authors gratefully ac-
knowledge the Italian Istituto Nazionale di Fisica Nucle-
are (INFN), the French Centre National de la Recherche
Scientifique (CNRS) and the Foundation for Fundamen-
7
tal Research on Matter supported by the Netherlands
Organisation for Scientific Research, for the construc-
tion and operation of the Virgo detector and the cre-
ation 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 In-
dustrial Research of India, the Department of Science
and Technology, India, the Science & Engineering Re-
search Board (SERB), India, the Ministry of Human
Resource Development, India, the Spanish Agencia Es-
tatal de Investigaci ́on, the Vicepresid`encia i Conselle-
ria d’Innovaci ́o, Recerca i Turisme and the Conselleria
d’Educaci ́o i Universitat del Govern de les Illes Balears,
the Conselleria d’Educaci ́o, Investigaci ́o, Cultura i Es-
port de la Generalitat Valenciana, the National Science
Centre of Poland, the Swiss National Science Foundation
(SNSF), the Russian Foundation for Basic Research, the
Russian Science Foundation, the European Commission,
the European Regional Development Funds (ERDF), the
Royal Society, the Scottish Funding Council, the Scot-
tish Universities Physics Alliance, the Hungarian Scien-
tific Research Fund (OTKA), the Lyon Institute of Ori-
gins (LIO), the Paris
ˆ
Ile-de-France Region, the National
Research, Development and Innovation Office Hungary
(NKFIH), 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, Innova-
tions, and Communications, the International Center for
Theoretical Physics South American Institute for Fun-
damental 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 Tech-
nology (MOST), Taiwan and the Kavli Foundation. The
authors gratefully acknowledge the support of the NSF,
STFC, MPS, INFN, CNRS and the State of Niedersach-
sen/Germany for provision of computational resources.
This article has been assigned the document number
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