of 12
Constraining the
p
-Mode
g
-Mode Tidal Instability with GW170817
B. P. Abbott
etal.
*
(LIGO Scientific Collaboration and Virgo Collaboration)
(Received 13 September 2018; revised manuscript received 30 October 2018; published 13 February 2019)
We analyze the impact of a proposed tidal instability coupling
p
modes and
g
modes within neutron stars on
GW170817. This nonresonant instability transfers energy from the orbit of the binary to internal modes of the
stars,acceleratingthegravitational-wavedriven inspiral.Wemodeltheimpactofthisinstability onthephasing
of the gravitational wave signal using three parameters per star: an overall amplitude, a saturation frequency,
and a spectral index. Incorporating these additional parameters, we compute the Bayes factor (ln
B
pg
!
pg
)
comparing our
p
-
g
model to a standard one. We find that the observed signal is consistent with waveform
models that neglect
p
-
g
effects, with ln
B
pg
!
pg
¼
0
.
03
þ
0
.
70
0
.
58
(maximum
a posteriori
and90%credible region).By
injecting simulated signals that do not include
p
-
g
effects and recovering them with the
p
-
g
model, we show
that there is a
50%
probability of obtaining similar ln
B
pg
!
pg
even when
p
-
g
effects are absent. We find that the
p
-
g
amplitude for
1
.
4
M
neutron stars is constrained to less than a few tenths of the theoretical maximum,
with maxima
a posteriori
near one-tenth this maximum and
p
-
g
saturation frequency
70
Hz. This suggests
that there are less than a few hundred excited modes, assuming they all saturate by wave breaking. For
comparison, theoretical upper bounds suggest
10
3
modes saturate by wave breaking. Thus, the measured
constraints only rule out extreme values of the
p
-
g
parameters. They also imply that the instability dissipates
10
51
erg over the entire inspiral, i.e., less than a few percent of the energy radiated as gravitational waves.
DOI:
10.1103/PhysRevLett.122.061104
Introduction.
Detailed analysis of the gravitational-
wave (GW) signal received from the first binary neutron
star (NS) coalescence event (GW170817
[1]
) constrains the
tidal deformability of NSs and thus the equation of state
(EOS) above nuclear saturation density
[2
4]
. Studies of
NS tidal deformation typically focus on the linear, quasi-
static tidal bulge induced in each NS by its companion.
Such deformations modify the system
s binding energy and
GW luminosity and thereby alter its orbital dynamics. The
degree of deformation is often expressed in terms of the
tidal deformability
Λ
i
ð
R
i
=m
i
Þ
5
of each component
[5]
,
or a particular mass-weighted average thereof (
̃
Λ
)
[2]
. The
strong dependence on compactness
R=m
means that a
stiffer EOS, which has larger
R
for the same
m
, imprints
larger tidal signals than a softer EOS. Current analyses of
GW data from the LIGO
[6]
and Virgo
[7]
detectors favor a
soft EOS
[3,8]
. Specifically, Ref.
[2]
finds
̃
Λ
730
at the
90% credible level for all waveform models considered,
allowing for the components to spin rapidly. The pressure at
twice nuclear saturation density is also constrained to
P
¼
3
.
5
þ
2
.
7
1
.
7
×
10
34
dyn
=
cm
2
(median and 90% credible region)
[3]
assuming small component spins. In addition to GW
phasing, the EOS dependence of
̃
Λ
should correlate with
postmerger signals
[9]
, possible tidal disruptions, and
kilonova observations
[10]
. Observed light curves for the
kilonova suggest a lower bound of
̃
Λ
200
[11,12]
.
Although some dynamical tidal effects are incorporated
in these analyses (see, e.g., Refs.
[2,13]
), the impact of
several types of dynamical tidal effects are neglected
because they are believed to be small or have large
theoretical uncertainties. These effects arise because tidal
fields, in addition to raising a quasistatic bulge, excite
stellar normal modes. Three such excitation mechanisms
are (i) resonant linear excitation, (ii) resonant nonlinear
excitation, and (iii) nonresonant nonlinear excitation (see,
e.g., Ref.
[14]
). The first occurs when the GW frequency
(the oscillation frequency of the tidal field) sweeps through
a mode
s natural frequency (see, e.g., Refs.
[15
22]
).
However, since the GW frequency increases rapidly
during the late inspiral, the time spent near resonance is
too short to excite modes to large amplitudes. As a result,
for modes with natural frequencies within the sensitive
bands of ground-based GW detectors, the change in orbital
phasing is expected to be small (
ΔΨ
10
2
rad) unless the
stars are rapidly rotating
[17
19]
. The impact of resonant
nonlinear mode excitation (i.e., the parametric subharmonic
instability) is likewise limited by the swiftness of the
inspiral
[23]
.
The proposed
p
-
g
tidal instability is a nonresonant,
nonlinear instability in which the tidal bulge excites a
low-frequency buoyancy-supported
g
mode and a high-
frequency pressure-supported
p
mode
[23
26]
. It occurs in
the inner core of the NS, where the stratification is weak
and the shear due to the tidal bulge is especially susceptible
to instability. Unlike resonantly excited modes, an unstable
*
Full author list given at the end of the Letter.
PHYSICAL REVIEW LETTERS
122,
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0031-9007
=
19
=
122(6)
=
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© 2019 American Physical Society
p
-
g
pair continuously drains energy from the orbit once
excited, even after the orbital frequency changes signifi-
cantly. There are many potentially unstable
p
-
g
pairs, each
becoming unstable at a different frequency and growing at a
different rate. Although there is considerable uncertainty
about the number of unstable pairs, their exact growth rates,
and how they saturate, estimates suggest that the impact
could be measurable with current detectors
[27]
.
In this Letter, we investigate the possible impact of the
p
-
g
instability on GW170817 using the phenomenological
model developed in Ref.
[27]
. The model describes the
energy dissipated by the instability within each NS,
indexed by
i
, in terms of three parameters: (i) an overall
amplitude
A
i
, which is related to the number of modes
participating in the instability, their growth rates, and their
saturation energies, (ii) a frequency
f
i
corresponding to
when the instability saturates, and (iii) a spectral index
n
i
describing how the saturation energy evolves with fre-
quency. In the section
Phenomenological model,
we
describe our models in detail. In the section
Model
selection,
we compare the statistical evidence for models
that include the
p
-
g
instability relative to those that do not.
In the section
Parameter inference,
we investigate the
constraints on the
p
-
g
parameters from GW170817, and in
the section
Discussion,
we conclude.
Phenomenological model.
Following Ref.
[27]
,we
extend a post-Newtonian (PN) waveform by including a
parametrized model of the
p
-
g
instability. For the initial PN
model, we use the TaylorF2 frequency-domain approxim-
ant (see, e.g., Ref.
[28]
) terminated at the innermost stable
circular orbit, which includes the effects of linear tides (
̃
Λ
)
and spins aligned with the orbital angular momentum (the
impact of misaligned spins on
p
-
g
effects is not known).
Waveform systematics between different existing approx-
imants may be important for small
p
-
g
effects. However,
by comparing the waveform mismatches between several
other models (TaylorF2, SEOBNRT, PhenomDNRT, and
PhenomPNRT, see Ref.
[2]
), we find these systematics
induce waveform mismatches that correspond to
p
-
g
phenomenological amplitudes roughly an order of magni-
tude smaller than the upper limits set by our analysis (see
section
Parameter inference
). We expect the TaylorF2
approximant to be reasonably accurate and defer a com-
plete analysis of waveform systematics to future work.
Assuming the
p
-
g
effects are a perturbation to the
TaylorF2 approximant, we find that they modify the phase
in the frequency domain by
ΔΨ
ð
f
Þ¼
2
C
1
3
B
2
ð
3
n
1
Þð
4
n
1
Þ

Θ
1

f
f
ref

n
1
3
þð
1
Θ
1
Þ

f
1
f
ref

n
1
3

ð
4
n
1
Þ
ð
3
n
1
Þ

f
f
1

þð
1
2
Þ
;
ð
1
Þ
where
f
i
is the saturation frequency,
f
ref
100
Hz is
a reference frequency with no intrinsic significance,
C
i
¼
½
2
m
i
=
ð
m
1
þ
m
2
Þ
2
=
3
A
i
,
B
¼ð
32
=
5
Þð
G
M
π
f
ref
=c
3
Þ
5
=
3
,
M
¼ð
m
1
m
2
Þ
3
=
5
=
ð
m
1
þ
m
2
Þ
1
=
5
, and
Θ
i
¼
Θ
ð
f
f
i
Þ
where
Θ
is the Heaviside function. This approximant is
slightly different than that of Ref.
[27]
because they
incorrectly applied the saddle-point approximation to
obtain the frequency-domain waveform from time-domain
phasing
[29]
. This correction renders the
p
-
g
instability
slightly more difficult to measure than predicted in
Ref.
[27]
, although the observed behavior is qualitatively
similar. Specifically, we find that in order to achieve the
same
j
ΔΨ
j
,
A
i
needs to be larger than Ref.
[27]
found by a
factor of
ð
4
n
i
Þ
, although the precise factor also
depends on the other
p
-
g
parameters.
The
ΔΨ
expression contains three types of terms: a
constant term, a linear term
ð
1
Θ
i
Þ
f
, and a power-law
term
Θ
i
f
n
i
3
. The constant term corresponds to an
overall phase offset and is degenerate with the orbital
phase at coalescence. The linear term corresponds to a
change in the time of coalescence; because the
p
-
g
instability transfers energy from the orbit to stellar normal
modes, the binary inspirals faster than it would if the effect
was absent. The power-law term accounts for the com-
petition between the rate of
p
-
g
energy dissipation and the
rate of inspiral, both of which increase as
f
increases.
As argued in Ref.
[27]
, we expect
n
i
<
3
, which implies
that the phase shift accumulates primarily at frequencies
just above the
turn-on
(saturation) frequency
f
f
i
.
When
n
i
<
3
,
p
-
g
effects are most important at lower
frequencies whereas linear tides (
̃
Λ
) and spins (
χ
i
¼
cS
i
=Gm
2
i
, where
S
i
is the spin-angular momentum of each
component) have their largest impact at higher frequencies
(see, e.g., Ref.
[30]
). The priors placed on the latter qua-
ntities can, however, affect our inference of
p
-
g
parameters.
In order to account for a possible dependence on the
component masses (
m
i
), we parametrize our model using a
Taylor expansion in the
p
-
g
parameters around
m
i
¼
1
.
4
M
and sample from the posterior using the first
two coefficients. Our model computes
A
i
as
A
i
ð
m
i
Þ¼
A
0
þ

dA
dm




1
.
4
M

ð
m
i
1
.
4
M
Þ
;
ð
2
Þ
and uses
A
0
and
dA=dm
instead of
A
1
and
A
2
. The model
uses similar representations for
f
i
and
n
i
in terms of the
parameters
f
0
,
df=dm
,
n
0
, and
dn=dm
. We assume a
uniform prior on log
10
A
0
within
10
10
A
0
10
5
.
5
,a
uniform prior in
f
0
within
10
Hz
f
0
100
Hz, and a
uniform prior in
n
0
within
1
n
0
3
. The priors on the
first-order terms (
dA=dm; df=dm; dn=dm
) are the same as
those in Ref.
[27]
; when
m
1
m
2
, they imply
A
1
A
2
, etc.
We investigate GW170817 using data from several
different frequency bands and with different spin priors,
but unless otherwise noted we focus on results for data
PHYSICAL REVIEW LETTERS
122,
061104 (2019)
061104-2
above 30 Hz with
j
χ
i
j
0
.
89
. Throughout this Letter, results
from GW170817 were obtained using the same
data conditioning as Ref.
[2]
, including the removal of a
short-duration noise artifact from the Livingston data
(Ref.
[31]
and discussion in Ref.
[1]
) along with other
independently measured noise sources (see, e.g., Refs.
[32
35]
), calibration
[36,37]
, marginalization over calibration
uncertainties, and whitening procedures
[38,39]
. We use the
publicly available LALInference software package through-
out
[40,41]
.
Model selection.
Using GW data from GW170817, we
perform Bayesian model selection. We compare a model
that includes linear tides, spin components aligned with the
orbital angular momentum, and PN phasing effects up to
3.5 PN phase terms (
H
!
pg
) to an extension of this model
that also includes
p
-
g
effects (
H
pg
). Since we have nested
models (
H
!
pg
is obtained from
H
pg
as
A
i
0
), we use the
Savage-Dickey density ratio (see, e.g., Refs.
[42
44]
)to
estimate the Bayes factor (
B
pg
!
pg
¼
p
ð
D
j
H
pg
Þ
=p
ð
D
j
H
!
pg
Þ
,
where
D
refers to the observed data). Because we use a
uniform-in-log
10
A
0
prior,
H
pg
does not formally include
A
i
¼
0
. Nonetheless, our lower limit on
A
i
is sufficiently
small that
H
!
pg
is effectively nested in
H
pg
. Specifically, we
sample from the model
s posterior distribution
[40,41]
and
calculate
lim
A
i
0

p
ð
A
i
j
D;
H
pg
Þ
p
ð
A
i
j
H
pg
Þ

¼
lim
A
i
0

1
p
ð
D
j
H
pg
Þ
Z
d
θ
df
i
dn
i
p
ð
D
j
θ
;A
i
;f
i
;n
i
;
H
pg
Þ
p
ð
θ
j
H
pg
Þ
p
ð
f
i
;n
i
j
A
i
;
H
pg
Þ

¼
1
p
ð
D
j
H
pg
Þ
Z
d
θ
p
ð
D
j
θ
;
H
!
pg
Þ
p
ð
θ
j
H
!
pg
Þ

p
ð
θ
j
H
pg
Þ
p
ð
θ
j
H
!
pg
Þ

Z
df
i
dn
i
p
ð
f
i
;n
i
j
A
i
;
H
pg
Þ
¼
p
ð
D
j
H
!
pg
Þ
p
ð
D
j
H
pg
Þ
p
ð
θ
j
H
pg
Þ
p
ð
θ
j
H
!
pg
Þ
p
ð
θ
j
D;
H
!
pg
Þ
;
ð
3
Þ
where
θ
refers to all parameters besides the
p
-
g
phenom-
enological parameters; we note that
R
dfdnp
ð
f
i
;n
i
j
A
i
;
H
pg
Þ¼
1
A
i
, and
h
x
i
p
denotes the average of
x
with respect to the measure defined by
p
. Assuming that
p
ð
θ
j
H
pg
Þ¼
p
ð
θ
j
H
!
pg
Þ
, we determine ln
B
pg
!
pg
from the
ratio, as
A
i
0
, of the marginal distribution of
A
i
a priori
to the distribution
a posteriori
:
ln
B
pg
!
pg
¼
lim
A
i
0
½
ln
p
ð
A
i
j
H
pg
Þ
ln
p
ð
A
i
j
D;
H
pg
Þ
:
ð
4
Þ
This allows us to directly measure ln
B
pg
!
pg
by extracting
p
ð
A
j
D;
H
pg
Þ
from Monte Carlo analyses with a known
prior
p
ð
A
j
H
!
pg
Þ
. We confirmed that this estimate agrees
with estimates from both nested sampling
[45]
and thermo-
dynamic integration
[46]
.
Figure
1
shows ln
B
pg
!
pg
as a function of
f
low
, the
minimum GW frequency considered. At a given
f
low
,
we show the distribution of ln
B
pg
!
pg
due to the sampling
uncertainty from the finite length of our MCMC chains.
The solid and dashed curves correspond to the high-spin
(
j
χ
i
j
0
.
89
) and low-spin (
j
χ
i
j
0
.
05
) priors discussed in
Refs.
[1
3]
.
For certain combinations of
f
low
and
j
χ
i
j
, we find
ln
B
pg
!
pg
>
0
, suggesting
H
pg
is more likely than
H
!
pg
.In
order to assess how likely such values are, we calculate
ln
B
pg
!
pg
foralargenumberofsimulated,high-spinsignalswith
A
i
¼
0
and distinct realizations of detector noise from times
near GW170817. We find that simulated signals without
p
-
g
effects can readily produce ln
B
pg
!
pg
at least as large as the ones
we measured from GW170817. For example, for the 30 Hz
high-spin data we obtain ln
B
pg
!
pg
¼
0
.
03
þ
0
.
70
0
.
58
(maximum
a posteriori
and 90% credible region; bottom panel of
Fig.
1
), whereas approximately half of our simulated signals
yield ln
B
pg
!
pg
at least this large, i.e., a false alarm probability
(FAP)
50%
. We focus on the 30 Hz, high-spin data because
FIG. 1. Distributions of ln
B
pg
!
pg
due to sampling uncertainty
when analyzing GW170817 data with different values of
f
low
.
The solid red curves assume high-spin priors (
j
χ
i
j
0
.
89
) and
the dashed blue curves assume low-spin priors (
j
χ
i
j
0
.
05
).
PHYSICAL REVIEW LETTERS
122,
061104 (2019)
061104-3
it corresponds to the largest bandwidth investigated and the
largest signal-to-noise ratio. The high-spin prior is the most
inclusive prior considered, and therefore allows the most
model freedom when fitting
p
-
g
effects.
In our model of the instability, the phase shift
ΔΨ
accumulates primarily at frequencies just above the satu-
ration frequency
f
f
i
. Therefore, if it is present, its impact
should become more apparent as we decrease the minimum
GW frequency considered from
f
low
f
i
to
f
low
f
i
.We
do see some indication of this behavior in Fig.
1
. However,
we note that if our phenomenological model breaks down at
f<f
i
due to poor modeling of the presaturation behavior
(e.g., if our step-function turn-on at
f
i
is not a good
approximation to the instability
s induced phase shift),
we might expect ln
B
pg
!
pg
to decrease as we lower
f
low
below
f
i
. If the fidelity of our model is sufficiently poor, we could
be insensitive to
p
-
g
effects even at frequencies above
f
low
.
Parameter inference.
We now investigate the con-
straints obtained from GW170817. Figure
2
shows the joint
posterior distributions for both
H
pg
and
H
!
pg
. We find that
H
pg
and
H
!
pg
yield similar posterior distributions for all
non-
p
-
g
parameters, including both extrinsic and intrinsic
parameters. The constraints on the chirp mass (
M
), effective
spin
χ
eff
¼ð
m
1
χ
1
þ
m
2
χ
2
Þ
=
ð
m
1
þ
m
2
Þ
, and
̃
Λ
are slightly
weaker in
H
pg
than
H
!
pg
. This is because
H
pg
provides extra
freedom to the signal
s duration in the time domain.
Regarding the
p
-
g
parameters, we find a noticeable peak
near
A
0
10
7
with a flat tail to small
A
0
. We find
A
0
3
.
3
×
10
7
assuming a uniform-in-log
10
A
0
prior and
A
0
6
.
8
×
10
7
assumingauniform-in-
A
0
prior,bothat90%con-
fidence. The upper limit with a uniform-in-
A
0
prior is larger
only because we weight larger values of
A
0
more
a priori
than with a uniform-in-log
10
A
0
prior. We also find a peak at
f
0
70
Hz. The peaks persist when we analyze the data
from each interferometer separately, with reasonably con-
sistent locations and shapes (Fig.
2
). However, we find that
the simulated signals with
A
i
¼
0
can produce similar peaks,
suggesting they may be due to noise alone. Similar to
Ref.
[27]
, we find that
n
i
is not strongly constrained and the
gradient terms in the Taylor expansions are not measurable.
Theoretical arguments suggest an upper bound of
A
0
10
6
[27]
. Therefore, our
A
0
constraint only rules out the
most extreme values of the
p
-
g
parameters.
Discussion.
While GW170817 is consistent with mod-
els that neglect
p
-
g
effects, it is also consistent with a broad
range of
p
-
g
parameters. The constraints from GW170817
imply that there are
200
excited modes at
f
¼
100
Hz,
assuming all modes grow as rapidly as possible and saturate
at their breaking amplitudes (
λ
¼
β
¼
1
in Eq. (7) of
Ref.
[27]
) and that the frequency at which modes become
unstable is well approximated by
f
0
. For comparison,
theoretical arguments suggest an upper bound of
10
3
FIG. 2. Posterior distributions for
H
!
pg
(red) and
H
pg
with Hanford, Livingston, and Virgo data (thick black, gray shading), Hanford
data only (dark blue), and Livingston data only (light blue) using GW data above 30 Hz,
j
χ
i
j
0
.
89
, and a uniform-in-log
10
A
0
prior.
Left: a subset of parameters shared by
H
!
pg
and
H
pg
. Right: a subset of parameters belonging only to
H
pg
. We only show one-
dimensional posteriors for the single instrument data, although the multidimensional posteriors are similarly consistent with the full
H
pg
data. Contours in the two-dimensional distributions represent 10%, 50%, and 90% confidence regions.
PHYSICAL REVIEW LETTERS
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061104 (2019)
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modes saturating by wave breaking
[27]
. More modes may
be excited if they grow more slowly or saturate below their
wave breaking energy.
We can also use the measured constraints to place
upper limits on the amount of energy dissipated by the
p
-
g
instability. As Fig.
3
shows,
p
-
g
effects dissipate
2
.
7
×
10
51
erg throughout the entire inspiral at 90% con-
fidence. In comparison, GWs carry away
10
53
erg. This
implies time-domain phase shifts
j
Δ
φ
j
7
.
6
rad (0.6
orbits) at 100 Hz and
j
Δ
φ
j
32
rad (2.6 orbits) at
1000 Hz after accounting for the joint uncertainty in
component masses, spins, linear tides, and
p
-
g
effects.
A
g
modewith natural frequency
f
g
is predicted to become
unstable at a frequency
f
crit
45
Hz
ð
f
g
=
10
4
λ
f
dyn
Þ
1
=
2
,
where
f
dyn
is the dynamical frequency of the NS and
λ
is
a slowly varying function typically between 0.1
1
[25,27]
.
Since the modes grow quickly, the frequency at which the
instability saturates is likely close to the frequency at which
the modes become unstable (
f
0
f
crit
). If we assume that the
observed peak near
f
0
70
Hz is not due to noise alone, then
the maximum
a posteriori
estimate for
f
0
along with
approximate values for the masses (
1
.
4
M
) and radii
(11 km) of the components
[3]
imply
f
g
0
.
5
Hz.
With several more signals comparable to GW170817, it
should be possible to improve the amplitude constraint to
A
0
10
7
. Obtaining even tighter constraints will likely
require many more detections, especially since most
events will have smaller SNR. Future measurements will
also benefit from a better understanding of how the
instability saturates. To date, there have only been detailed
theoretical studies of the instability
s threshold and growth
rate
[23
26]
, not its saturation. As a result, we cannot be
certain of the fidelity of our phenomenological model.
While this Letter was in review, related work was posted
[47]
with the conclusion that the
H
!
pg
model is strongly
favored over the
H
pg
model by a factor of at least
10
4
.In
Ref.
[48]
, some of the authors of this work investigate the
origin of the discrepancy by analyzing publicly available
posterior samples from Ref.
[47]
. Contrary to the claims in
Ref.
[47]
, they find that samples from Ref.
[47]
yield
B
pg
!
pg
1
and therefore conclude that their posterior data, like what is
presented here, do not disfavorthe
H
pg
model.Reference
[48]
suggests that the error stems from using too few temperatures
when implementing thermodynamic integration.
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 Science and Engineering Research
Council Canada, the Canadian Institute for Advanced
Research, the Brazilian Ministry of Science, Technology,
Innovations, and Communications, the International Center
FIG. 3. Upper limits on the cumulative energy dissipated by the
p
-
g
instability as a function of frequency. We note the relatively
strong constraints at lower frequencies where
p
-
g
effects are more
pronounced.
PHYSICAL REVIEW LETTERS
122,
061104 (2019)
061104-5
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.
N. Weinberg was supported in part by NASA Grant
No. NNX14AB40G.
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PHYSICAL REVIEW LETTERS
122,
061104 (2019)
061104-6
B. P. Abbott,
1
R. Abbott,
1
T. D. Abbott,
2
F. Acernese,
3,4
K. Ackley,
5
C. Adams,
6
T. Adams,
7
P. Addesso,
8
R. X. Adhikari,
1
V. B. Adya,
9,10
C. Affeldt,
9,10
B. Agarwal,
11
M. Agathos,
12
K. Agatsuma,
13
N. Aggarwal,
14
O. D. Aguiar,
15
L. Aiello,
16,17
A. Ain,
18
P. Ajith,
19
B. Allen,
9,20,10
G. Allen,
11
A. Allocca,
21,22
M. A. Aloy,
23
P. A. Altin,
24
A. Amato,
25
A. Ananyeva,
1
S. B. Anderson,
1
W. G. Anderson,
20
S. V. Angelova,
26
S. Antier,
27
S. Appert,
1
K. Arai,
1
M. C. Araya,
1
J. S. Areeda,
28
M. Ar`
ene,
29
N. Arnaud,
27,30
K. G. Arun,
31
S. Ascenzi,
32,33
G. Ashton,
5
M. Ast,
34
S. M. Aston,
6
P. Astone,
35
D. V. Atallah,
36
F. Aubin,
7
P. Aufmuth,
10
C. Aulbert,
9
K. AultONeal,
37
C. Austin,
2
A. Avila-Alvarez,
28
S. Babak,
38,29
P. Bacon,
29
F. Badaracco,
16,17
M. K. M. Bader,
13
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
S. Barnum,
14
F. Barone,
3,4
B. Barr,
45
L. Barsotti,
14
M. Barsuglia,
29
D. Barta,
48
J. Bartlett,
46
I. Bartos,
49
R. Bassiri,
50
A. Basti,
21,22
J. C. Batch,
46
M. Bawaj,
51,42
J. C. Bayley,
45
M. Bazzan,
52,53
B. B ́
ecsy,
54
C. Beer,
9
M. Bejger,
55
I. Belahcene,
27
A. S. Bell,
45
D. Beniwal,
56
M. Bensch,
9,10
B. K. Berger,
1
G. Bergmann,
9,10
S. Bernuzzi,
57,58
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59
C. P. L. Berry,
60
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61
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13
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6
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62
I. A. Bilenko,
63
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40
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1
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49
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6
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26
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59
S. Biscans,
1,14
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5
A. Bisht,
9,10
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30,22
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27
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1
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47
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6
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64
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46
S. Bloemen,
65
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9
N. Bode,
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66
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67
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66
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38
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68
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50
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13
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36
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1
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30
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64
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64
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22
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20
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70
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71
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27
,
P. Brockill,
20
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1
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56
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1
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2
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14
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74
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7
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50
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77
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1
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29
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30
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84
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85
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22
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13,20
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86
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27
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30
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22
C. B. Cepeda,
1
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66
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88
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90
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91
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29
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20
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92
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40
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93
X. Chen,
64
Y. Chen,
47
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49
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49
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61
A. Chiummo,
30
T. Chmiel,
85
H. S. Cho,
94
M. Cho,
76
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24
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95,66
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64
A. J. K. Chua,
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S. Chua,
71
K. W. Chung,
96
S. Chung,
64
G. Ciani,
52,53,49
A. A. Ciobanu,
56
R. Ciolfi,
97,98
F. Cipriano,
66
C. E. Cirelli,
50
A. Cirone,
81,61
F. Clara,
46
J. A. Clark,
77
P. Clearwater,
99
F. Cleva,
66
C. Cocchieri,
86
E. Coccia,
16,17
P.-F. Cohadon,
71
D. Cohen,
27
A. Colla,
100,35
C. G. Collette,
101
C. Collins,
60
L. R. Cominsky,
102
M. Constancio Jr.,
15
L. Conti,
53
S. J. Cooper,
60
P. Corban,
6
T. R. Corbitt,
2
I. Cordero-Carrión,
103
K. R. Corley,
104
N. Cornish,
105
A. Corsi,
84
S. Cortese,
30
C. A. Costa,
15
R. Cotesta,
38
M. W. Coughlin,
1
S. B. Coughlin,
36,91
J.-P. Coulon,
66
S. T. Countryman,
104
P. Couvares,
1
P. B. Covas,
106
E. E. Cowan,
77
D. M. Coward,
64
M. J. Cowart,
6
D. C. Coyne,
1
R. Coyne,
107
J. D. E. Creighton,
20
T. D. Creighton,
108
J. Cripe,
2
S. G. Crowder,
109
T. J. Cullen,
2
A. Cumming,
45
L. Cunningham,
45
E. Cuoco,
30
T. Dal Canton,
80
G. Dálya,
54
S. L. Danilishin,
10,9
S. D
Antonio,
33
K. Danzmann,
9,10
A. Dasgupta,
110
C. F. Da Silva Costa,
49
V. Dattilo,
30
I. Dave,
62
M. Davier,
27
D. Davis,
43
E. J. Daw,
111
B. Day,
77
D. DeBra,
50
M. Deenadayalan,
18
J. Degallaix,
25
M. De Laurentis,
79,4
S. Del ́
eglise,
71
W. Del Pozzo,
21,22
N. Demos,
14
T. Denker,
9,10
T. Dent,
9
R. De Pietri,
57,58
J. Derby,
28
V. Dergachev,
9
R. De Rosa,
79,4
C. De Rossi,
25,30
R. DeSalvo,
112
O. de Varona,
9,10
S. Dhurandhar,
18
M. C. Díaz,
108
L. Di Fiore,
4
M. Di Giovanni,
113,98
T. Di Girolamo,
79,4
A. Di Lieto,
21,22
B. Ding,
101
S. Di Pace,
100,35
I. Di Palma,
100,35
F. Di Renzo,
21,22
A. Dmitriev,
60
Z. Doctor,
93
V. Dolique,
25
F. Donovan,
14
K. L. Dooley,
36,86
S. Doravari,
9,10
I. Dorrington,
36
M. Dovale Álvarez,
60
T. P. Downes,
20
M. Drago,
9,16,17
C. Dreissigacker,
9,10
J. C. Driggers,
46
Z. Du,
83
P. Dupej,
45
S. E. Dwyer,
46
P. J. Easter,
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T. B. Edo,
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M. C. Edwards,
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A. Effler,
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H.-B. Eggenstein,
9,10
P. Ehrens,
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J. Eichholz,
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M. Eisenmann,
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R. A. Eisenstein,
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R. C. Essick,
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H. Estelles,
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D. Estevez,
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Z. B. Etienne,
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T. Etzel,
1
M. Evans,
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T. M. Evans,
6
V. Fafone,
32,33,16
H. Fair,
43
S. Fairhurst,
36
X. Fan,
83
S. Farinon,
61
B. Farr,
70
W. M. Farr,
60
E. J. Fauchon-Jones,
36
M. Favata,
114
M. Fays,
36
C. Fee,
85
H. Fehrmann,
9
J. Feicht,
1
M. M. Fejer,
50
F. Feng,
29
A. Fernandez-Galiana,
14
I. Ferrante,
21,22
E. C. Ferreira,
15
F. Ferrini,
30
F. Fidecaro,
21,22
I. Fiori,
30
D. Fiorucci,
29
M. Fishbach,
93
R. P. Fisher,
43
J. M. Fishner,
14
M. Fitz-Axen,
44
R. Flaminio,
7,115
M. Fletcher,
45
H. Fong,
116
J. A. Font,
23,117
P. W. F. Forsyth,
24
S. S. Forsyth,
77
J.-D. Fournier,
66
S. Frasca,
100,35
F. Frasconi,
22
Z. Frei,
54
A. Freise,
60
R. Frey,
70
V. Frey,
27
P. Fritschel,
14
V. V. Frolov,
6
P. Fulda,
49
M. Fyffe,
6
H. A. Gabbard,
45
B. U. Gadre,
18
S. M. Gaebel,
60
J. R. Gair,
118
L. Gammaitoni,
41
M. R. Ganija,
56
S. G. Gaonkar,
18
A. Garcia,
28
C. García-Quirós,
106
F. Garufi,
79,4
B. Gateley,
46
S. Gaudio,
37
G. Gaur,
119
V. Gayathri,
120
G. Gemme,
61
E. Genin,
30
A. Gennai,
22
D. George,
11
J. George,
62
L. Gergely,
121
V. Germain,
7
PHYSICAL REVIEW LETTERS
122,
061104 (2019)
061104-7