of 17
LIGO-P1700009
First search for nontensorial gravitational waves from known pulsars
The LIGO Scientific Collaboration and the Virgo Collaboration
(Dated: September 28, 2017)
We present results from the first directed search for nontensorial gravitational waves. While general
relativity allows for tensorial (plus and cross) modes only, a generic metric theory may, in principle,
predict waves with up to six different polarizations. This analysis is sensitive to continuous signals
of scalar, vector or tensor polarizations, and does not rely on any specific theory of gravity. After
searching data from the first observation run of the advanced LIGO detectors for signals at twice
the rotational frequency of 200 known pulsars, we find no evidence of gravitational waves of any
polarization. We report the first upper limits for scalar and vector strains, finding values comparable
in magnitude to previously-published limits for tensor strain. Our results may be translated into
constraints on specific alternative theories of gravity.
Introduction.
The first gravitational waves
(GWs) detected by the Advanced Laser Interferometer
Gravitational-wave Observatory (aLIGO) have already
been used to place some of the most stringent constraints
on deviations from the general theory of relativity (GR) in
the highly-dynamical and strong-field regimes of gravity
[
1
4
]. However, it has not been possible to unambigu-
ously confirm GR’s prediction that the associated metric
perturbations are of a tensor nature (helicity
±
2), rather
than vector (helicity
±
1), or scalar (helicity 0) [
5
]. This is
unfortunate, since the presence of nontensorial modes is
a key prediction of many extensions to GR [
6
10
]. Most
importantly, the detection of a scalar or vector compo-
nent, no matter how small, would automatically falsify
Einstein’s theory [8, 9].
In order to experimentally determine whether gravita-
tional waves are tensorial or not in a model-independent
way, one needs a local measurement of the signal’s polar-
ization content that breaks the degeneracies between the
five distinguishable modes supported by a generic metric
theory of gravity [
6
,
7
]. For transient waves like those
detected so far, this cannot be achieved with LIGO’s two
detectors, as at least five noncooriented differential-arm
antennas would be required to break
all
such degenera-
cies [
9
,
11
]. Constraints on the magnitude of non-GR
polarizations inferred from indirect measurements, like
that of the rate of orbital decay of binary pulsars, are
only meaningful in the context of specific theories (see
e.g. [12, 13], or [14, 15] for reviews).
Theory-independent polarization measurements could
instead be carried out with current detectors in the
presence of signals sufficiently long to probe the detec-
tor antenna patterns, which are themselves polarization-
sensitive [
16
19
]. Such is the case, for instance, for the
continuous, almost-monochromatic waves expected from
spinning neutron stars with an asymmetric moment of
inertia [
20
]. Known galactic pulsars are one of the main
candidates for searches for such signals in data from
ground-based detectors, and analyses targeting them have
already achieved sensitivities that are comparable to, or
even surpass, their canonical spin-down limit (i.e. the
strain that would be produced if the observed slowdown
in the pulsar’s rotation was completely due to gravita-
tional radiation) [21].
However, all previous targeted searches have been, by
design, restricted to tensorial gravitational polarizations
only
. This leaves open the possibility that, due to a
departure from GR, the neutron stars targeted in previous
searches may indeed be emitting strong continuous waves
with nontensorial content, in spite of the null results of
standard searches.
In this paper, we present results from a search for con-
tinuous gravitational waves in aLIGO data that makes no
assumptions about how the gravitational field transforms
under spatial rotations, and is thus sensitive to any of the
six polarizations allowed by a generic metric theory of
gravity. We targeted 200 known pulsars using data from
aLIGO’s first observation run (O1), and assuming GW
emission at twice the rotational frequency of the source.
Our data provide no evidence for the emission of gravi-
tational signals of tensorial or nontensorial polarization
from any of the pulsars targeted. For sources in the most
sensitive band of our detectors, we constrain the strain
of the scalar and vector modes to be below 1
.
5
×
10
26
at 95% credibility. These are the first direct upper limits
for scalar and vector strain ever published, and may be
used to constrain beyond-GR theories of gravity.
Analysis.
We search aLIGO O1 data from the Hanford
(H1) and Livingston (L1) detectors for continuous waves
of any polarization (tensor, scalar or vector) by applying
the Bayesian time-domain method of [
22
], generalized
to non-GR modes as described in [
17
] and summarized
below. Our analysis follows closely that of [
21
], and uses
the exact same interferometric data.
Calibrated detector data are heterodyned and filtered
using the timing solutions for each pulsar obtained from
electromagnetic observations. The maximum calibration
uncertainties estimated over the whole run give a limit
on the combined H1 and L1 amplitude uncertainties of
14%—this is the conservative level of uncertainty on the
strain upper limits [21, 23].
The data streams start on 2015 Sep 11 at 01:25:03
UTC for H1 and 18:29:03 UTC for L1, and finish on
2016 Jan 19 at 17:07:59 UTC at both sites. The pulsar
arXiv:1709.09203v1 [gr-qc] 26 Sep 2017
2
timing solutions used are also the same as in [
21
] and were
obtained from the 42-ft telescope and Lovell telescope at
Jodrell Bank (UK), the 26-m telescope at Hartebeesthoek
(South Africa), the Parkes radio telescope (Australia),
the Nancay Decimetric Radio Telescope (France), the
Arecibo Observatory (Puerto Rico) and the Fermi Large
Area Telescope (LAT).
As described in detail in [
17
], we construct a Bayesian
hypothesis that captures signals of any polarization con-
tent (our
any-signal
hypothesis,
H
S
) by combining the
sub-hypotheses corresponding to the signal being com-
posed of tensor, vector, scalar modes, or any combination
thereof. Each of these sub-hypotheses corresponds to a
different signal model; in particular, the least restrictive
template includes contributions from all polarizations and
can be written as:
h
(
t
) =
p
F
p
(
t
;
α,δ,ψ
)
h
p
(
t
)
,
(1)
where the sum is over the five independent polarizations:
plus (+), cross (
×
), vector-x (x), vector-y (y) and scalar
(s) [
7
]. The two scalar modes, breathing and longitudinal,
are degenerate for networks of quadrupolar antennas [
9
],
so we do not make a distinction between them.
Each term in Eq. (1) is the product of an antenna
pattern function
F
p
and an intrinsic strain function
h
p
.
We define the different polarizations in a wave-frame such
that the
z
-axis points in the direction of propagation,
x
lies in the plane of the sky along the line of nodes (here
defined to be the intersection of the equatorial plane of
the source with the plane of the sky), and
y
completes the
right-handed system, such that the polarization angle
ψ
is the angle between the
y
-axis and the projection of the
celestial North onto the plane of the sky (see e.g. [
24
]).
We can thus write the
F
p
’s as implicit functions of the
source’s right ascension
α
, declination
δ
and polarization
ψ
. (For the sources targeted here,
α
and
δ
are always
known to high accuracy, while
ψ
is usually unknown.)
The antenna patterns acquire their time dependence from
the sidereal rotation of the Earth; explicit expressions for
the
F
p
’s are given in [16–18, 25, 26].
For a continuous wave, the polarizations take the simple
form:
h
p
(
t
) =
a
p
cos(
φ
(
t
) +
φ
p
)
,
(2)
where
a
p
is a time-independent strain amplitude,
φ
(
t
) is
the intrinsic phase evolution, and
φ
p
a phase offset for each
polarization. The nature of these three quantities depends
on the specifics of the underlying theory of gravity and
the associated emission mechanism (for different emission
mechanisms within GR, see e.g. [
27
29
]). While we treat
a
p
and
φ
p
as free parameters, we take
φ
(
t
) to be the same
as in the traditional GR analysis [21]:
φ
(
t
) = 2
π
N
j
=0
(
j
)
t
f
0
(
j
+ 1)!
[
t
T
0
+
δt
(
t
)]
(
j
+1)
,
(3)
TABLE I. Existing orientation information for pulsars in our
band, obtained from observations of the pulsar wind nebulae
(see Table 3 in [30], and [31, 32] for measurement details).
ι
ψ
J0534+2200
62
.
2
±
1
.
9
35
.
2
±
1
.
5
J0537–6910
92
.
8
±
0
.
9
41
.
0
±
2
.
2
J0835–4510
63
.
6
±
0
.
6
40
.
6
±
0
.
1
J1833–1034
85
.
4
±
0
.
3
45
±
1
J1952+3252
N/A
11
.
5
±
8
.
6
where
(
j
)
t
f
0
is the
j
th
time derivative of the gravitational-
wave frequency measured at the fiducial time
T
0
;
δt
(
t
)
is the time delay from the observatory to the solar sys-
tem barycenter (including the known Rømer, Shapiro
and Einstein delays), and can also include binary system
corrections to transform to the time coordinate to a frame
approximately inertial with respect to the source;
N
is
the order of the series expansion (1 or 2 for most sources).
The gravitational-wave frequency
f
is related to the
rotational frequency of the source
f
rot
, which is in turn
known from electromagnetic observations. Although arbi-
trary theories of gravity and emission mechanisms may
predict gravitational emission at any multiple of the rota-
tional frequency, here we assume
f
= 2
f
rot
, in accordance
with the most favored emission model in GR [
20
]. This re-
striction arises from practical considerations affecting our
specific implementation. However, the constraint can be
loosely justified by the expectation that, in most viable
theories, conservation laws will still preclude emission
at the first harmonic (
f
=
f
rot
), while higher frequen-
cies will be suppressed, as in most multiple expansions.
Nevertheless, this assumption will be relaxed in a future
study.
For convenience, we define
effective strain amplitudes
for tensor, vector and scalar modes respectively by
h
t
a
2
+
+
a
2
×
,
(4)
h
v
a
2
x
+
a
2
y
,
(5)
h
s
a
s
,
(6)
in terms of the intrinsic
a
p
amplitudes of Eq. (2). These
quantities may serve as proxy for the total power in each
polarization group.
One may recover the GR hypothesis considered in pre-
vious analysis by setting:
a
+
=
h
0
(1 + cos
2
ι
)
/
2
, φ
+
=
φ
0
,
(7)
a
×
=
h
0
cos
ι , φ
×
=
φ
0
π/
2
,
(8)
a
x
=
a
y
=
a
s
= 0
,
(9)
3
where
ι
is the inclination (angle between the line of sight
and the spin axis of the source), and
h
0
,
φ
0
are free pa-
rameters. (As with
ψ
,
ι
is unknown for most pulsars.)
This corresponds to the standard triaxial-star emission
mechanism (see e.g. [
33
]). We use this parameterization
only when we wish to incorporate known orientation in-
formation as explained below; otherwise, we parametrize
the tensor polarizations directly in terms of
a
+
,
a
×
,
φ
+
and
φ
×
.
Templates of the form of Eq. (1), together with appro-
priate priors, allow us to compute Bayes factors (marginal-
ized likelihood ratios) for the presence of signals in the
data vs Gaussian noise. We do this using an extension
of the nested sampling implementation presented in [
34
]
(see [
17
] for details specific to the non-GR polarizations).
The Bayes factors corresponding to each signal model
may be combined into the odds
O
S
N
that the data contain
a continuous signal of any polarization vs Gaussian noise:
O
S
N
=
P
(
H
S
|
B
)
/P
(
H
N
|
B
)
,
(10)
i.e. the ratio of the posteriors probabilities that the data
B
contain a signal of any polarizations (
H
S
) vs just Gaussian
noise (
H
N
). We compute these odds by setting model pri-
ors such that
P
(
H
S
) =
P
(
H
N
); then, by Bayes’ theorem,
O
S
N
=
B
S
N
, with the Bayes factor
B
S
N
P
(
B
|H
S
)
/P
(
B
|H
N
)
.
(11)
Built into the astrophysical signal hypothesis,
H
S
, is
the requirement of coherence across detectors, which must
be satisfied by a real GW signal. In order to make the
analysis more robust against non-Gaussian instrumental
features in the data, we also define an
instrumental feature
hypothesis,
H
I
, that identifies non-Gaussian noise arti-
facts by their lack of coherence across detectors [
21
,
35
].
In particular, we define
H
I
to capture Gaussian noise
or
a detector-incoherent signal (i.e. a feature that mimics an
astrophysical signal in a single instrument, but is not re-
covered consistently across the network) in each detector
[
17
]. We may then compare this to
H
S
by means of the
odds
O
S
I
. For
D
detectors, this is given by:
log
O
S
I
= log
B
S
N
D
d
=1
log
(
B
S
d
N
d
+ 1
)
,
(12)
where
B
S
d
N
d
is the signal vs noise Bayes factor computed
only from data from the
d
th
detector. This choice im-
plicitly assigns prior weight to the models such that
P
(
H
S
) =
P
(
H
I
)
×
0
.
5
D
[
17
]. For an in depth analy-
sis of the behavior of the different Bayesian hypotheses
considered here, in the presence absence of simulated sig-
nals of all polarizations, we again refer the reader to the
paper methods [17].
We compute likelihoods by taking source location, fre-
quency and frequency derivatives as known quantities. In
1
.
2
0
.
8
0
.
4
0
.
0
log
10
O
S
I
10
2
10
3
f
GW
(Hz)
0
.
6
0
.
3
0
.
0
0
.
3
log
10
O
S
N
FIG. 1.
Log-odds vs GW frequency
. Log-odds comparing the
any-signal hypothesis to the instrumental (top) and Gaussian
noise (bottom) hypotheses, as a function of assumed GW
frequency,
f
= 2
f
rot
, for each pulsar. Looking at the top plot
for
log
O
S
I
, notice that the instrumental noise hypothesis is
clearly favored for all pulsars except one, for which the analysis
is inconclusive. (This is J1932+17, the same non-significant
outlier identified in [
21
].) These results were obtained without
incorporating any information on the source orientation, and
are tabulated in Table III in the supplementary material.
Expressions for both odds are given in Eq. (10) and Eq. (12).
computing Bayes factors, we employ priors uniform in
the logarithm of amplitude parameters (
h
0
or
h
p
’s), since
these are the least informative priors for scaling coeffi-
cients [
36
]; we bound these amplitudes to the 10
28
–10
24
range [
37
]. On the other hand, flat amplitude priors are
used to compute upper limits, to facilitate comparison
with published GR results in [
21
]. In all cases, flat priors
are placed over all phase offsets (
φ
0
and all the
φ
p
’s).
For those few cases in which some orientation infor-
mation exists (see Table I in supplementary material),
we analyze the data a second time using the triaxial
parametrization of tensor modes, Eqs.
(7)
and
(8)
, tak-
ing that information into account by marginalizing over
ranges of
cos
ι
and
ψ
in agreement with measurement
uncertainties.
Results.
We find no evidence of continuous-wave sig-
nals of any polarization, tensorial or otherwise, from any
of the 200 pulsars analyzed. Odds and 95%-credible up-
per limits are summarized in the supplementary material:
Table II, for pulsars with measured orientations (using the
triaxial parameterization of tensor modes), and in Table
III, for all pulsars without incorporating any orientation
information (using the unconstrained parameterization of
tensor modes). Odds values are reported with an error
of 5% at 90% confidence; errors on the upper limits due
to the use of finite samples in estimating posterior proba-
bility distributions are at most 10% at 90% confidence,
4
0
.
8
0
.
6
0
.
4
0
.
2
0
.
0
0
.
2
0
.
4
log
10
O
S
N
1
.
4
1
.
2
1
.
0
0
.
8
0
.
6
0
.
4
0
.
2
0
.
0
0
.
2
log
10
O
S
I
0
5
10
15
20
Count
FIG. 2.
Log-odds distributions
. Distributions of log-odds com-
paring the any-signal hypothesis to the instrumental (ordinate
axis, right) and Gaussian noise (abscissa axis, top) hypotheses
for all pulsars. This plot contains the same information as
Fig. 1 and displays the same non-significant outlier. These
results were obtained without incorporating any information
on the source orientation, and are tabulated in Table III in
the supplementary material. Expressions for both odds in this
plot are given in Eq. (10) and Eq. (12). We underscore that,
although this plot looks similar to Fig. 2 in [
21
], the signal
hypothesis here incorporates scalar, vector and tensor modes,
in all their combinations.
which is slightly less than the 15% error expected from
calibration uncertainties.
The main quantity of interest is
log
O
S
I
, defined in Eq.
(12), since it encodes the probability that the data contain
a signal vs just instrumental noise (Gaussian or otherwise).
This quantity, together with the log-odds for signal vs
Gaussian noise, is presented as a function of assumed GW
frequency for each pulsar in Fig. 1, and histogrammed in
Fig. 2. Importantly, the outliers in Fig. 1 lose significance
once
log
O
S
I
is taken into account; indeed, Figs. 1 and 2
reveal the usefulness of
log
O
S
I
in increasing the robustness
of the search against non-Gaussian instrumental artifacts.
Based on the intrinsic probabilistic meaning of
log
O
S
I
in terms of betting odds, it is standard to demand at least
log
O
S
I
>
1 to conclude that the signal model is favored
(see e.g. the table in Sec. 3.2 of [
38
], or Jeffrey’s original
criteria in [
39
] or [
40
]). Since none of the odds obtained
meet this criterion, we conclude that there is no evidence
for signals from any of the pulsars targeted. In most
cases,
log
O
S
I
<
0 and the noise model is clearly favored;
the single exception is J1932+17, for which
log
O
S
I
0, so that we can make no conclusive statement about
which hypothesis is preferred. (The presence of this non-
significant outlier is to be expected, as it was already
identified in [21].)
The distribution of the odds corresponding to the sub-
hypotheses making up
H
S
are summarized in the box plots
of Fig. 3. These correspond to tensor-only (t), scalar-only
(s), vector-only (v), scalar-vector (sv), scalar-tensor (st),
vector-tensor (vt), and scalar-vector-tensor (stv) mod-
els. The mean of these distributions decreases with the
number of degrees of freedom in the model, which is to
be expected from the associated Occam penalties [
17
].
The right-most panel in Fig. 3 shows the distribution
of
log
O
S
N
, which results from the combination of all the
other odds; this is the same quantity histogrammed on
the abscissa of Fig. 2.
In the absence of any discernible signals, we produce
upper limits for the magnitude of scalar, vector and tensor
polarizations, with a 95% credibility. As usual in Bayesian
analyses, upper limits are obtained by integrating poste-
rior probability distributions for the relevant parameters
up to the desired credibility (see e.g. [
17
]). Using the
effective amplitude definitions of Eqs.
(4)
(6)
, these quan-
tities are presented in Fig. 4 as a function of assumed GW
frequency, and the supplementary material. The plotted
upper limits are computed under the assumption of a sig-
nal model that includes all five independent polarizations
(
H
svt
); limits obtained assuming other signal models may
be found online in [
41
]. Previous work has demonstrated
that the presence or absence of a GR component does not
affect the non-GR upper limits (Fig. 13 in [17]).
As expected, the upper limits presented here are com-
parable in magnitude to the upper limits on the GR strain
obtained by the traditional searches [
21
]. However, con-
straints on the scalar amplitude are, on average, around
20% less stringent than those on the vector or tensor
amplitudes. This is a consequence of the fact that, for
most source locations in the sky, the LIGO detectors are
intrinsically less sensitive to continuous waves of scalar
polarization [17].
Technically, traditional all-sky searches for continu-
ous gravitational waves are also sensitive to nontensorial
modes, because they are generally designed to look for
any signal of sidereal and half-sidereal periodicities in
the data [
42
44
]. However, as can be seen by comparing
the magnitude of all-sky upper limits (e.g. Fig. 9 in [
42
])
to those in shown here in Fig. 4, the sensitivity of these
searches would be substantially poorer than that of a
targeted search like this one. This is especially true if the
search is optimized for a given signal polarization (e.g.
circular combination of plus and cross).
Conclusion.
We have presented the results of the first
direct search for nontensorial gravitational waves. This
is also the first search for gravitational waves targeted
at known pulsars that is sensitive to any of the six po-
larizations of the gravitational perturbation allowed by a
generic metric theory of gravity. From the analysis of O1
data from both aLIGO observatories, we have found no
5
t
1
.
5
1
.
0
0
.
5
0
.
0
0
.
5
log
10
O
m
N
s
v
sv
st
vt
svt
S
FIG. 3.
Sub-hypothesis odds
. Box plots for the distribution of the signal vs noise log-odds for each of the sub-hypotheses
considered, for all of the pulsars analyzed. The sub-hypotheses are: (st), vector-tensor (tv), scalar-vector-tensor tensor-only (t),
scalar-only (s), vector-only (v), scalar-vector (sv), scalar-tensor (st), vector-tensor (vt), and scalar-vector-tensor (stv); these are
all combined into the signal hypothesis (S). The quantity represented is
log
B
m
N
, which is the same as
log
O
m
N
if neither
H
m
nor
H
N
are favored
a priori
(hence the label on the ordinate axis). The horizontal red line marks the median of the distribution,
while each gray box extends from the lower to upper quartile, and the whiskers mark the full range of the distribution of
log
O
m
N
for the 200 pulsars analyzed. These results were produced without incorporating any information on the source orientation, and
are tabulated in Table III in the supplementary material.
evidence of signals from any of the 200 pulsars targeted.
In the absence of a clear signal, we have produced
the first direct upper limits for scalar and vector strains
(Fig. 4, and tables in the supplementary material). The
values of the 95%-credible upper limits are comparable
in magnitude to previously-published GR constraints,
reaching
h
1
.
5
×
10
26
for pulsars whose frequency is
in the most sensitive band of our instruments.
Our results have been obtained in a model-independent
fashion. However, our upper limits on nontensorial strain
can be translated into model-dependent constraints by
picking a specific alternative theory and emission mecha-
nism. To do so, one should use the upper limits produced
under the assumption of a signal model that incorporates
the polarizations matching those allowed by the theory
one wishes to constrain; these may not necessarily be
those in Fig. 4 (e.g. for limits on a scalar-tensor theory,
one needs upper limits from
H
st
). However, this also re-
quires non-trivial knowledge of the dynamics of spinning
neutron stars under the theory of interest.
While it is conventional to compare the sensitivity of
continuous wave searches to the canonical spin-down limit
for each pulsar, it is not possible to do so here without
committing to a specific theory of gravity. This is because
doing so would require specific knowledge of how each
polarization contributes to the effective GW stress-energy,
how matter couples to the gravitational field, how the
waves propagate (dispersion and dissipation), and what
the angular dependence of the emission pattern is. How-
ever, analogues of the canonical spin-down limit for spe-
cific theories may be obtained from the results presented
here by using the strain upper limits obtained assuming
the sub-hypotheses with polarizations corresponding to
that theory, as mentioned above.
We have demonstrated the robustness of searches for
generalized polarization states (tensor, vector, or scalar)
in gravitational waves from spinning neutron stars. Fur-
thermore, even in the absence of a gravitational-wave
detection, we were able to obtain novel constraints on the
strain amplitude of nontensorial polarizations. In the fu-
ture, once a signal is detected, similar methods will allow
us to characterize the gravitational polarization content
and, in so doing, perform novel tests of general relativ-
ity. Although this search assumed a gravitational-wave
frequency of twice the rotational frequency of the source,
this restriction will be relaxed in future analyses.
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 construction
and operation of the GEO600 detector. Additional sup-
port 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 acknowl-
edge research support from these agencies as well as by
the Council of Scientific and Industrial Research of India,
Department of Science and Technology, India, Science &
Engineering Research Board (SERB), India, Ministry of