Limits on Gravitational-Wave Emission from Selected Pulsars Using LIGO Data
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PRL
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0031-9007
=
05
=
94(18)
=
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181103-1
2005 The American Physical Society
S. Yoshida,
24
K. D. Zaleski,
27
M. Zanolin,
12
I. Zawischa,
30
L. Zhang,
11
R. Zhu,
1
N. Zotov,
16
M. Zucker,
14
and J. Zweizig
11
(LIGO Scientific Collaboration)
*
1
Albert-Einstein-Institut, Max-Planck-Institut fu
̈
r Gravitationsphysik, D-14476 Golm, Germany
2
Albert-Einstein-Institut, Max-Planck-Institut fu
̈
r Gravitationsphysik, D-30167 Hannover, Germany
3
Australian National University, Canberra, 0200, Australia
4
California Institute of Technology, Pasadena, California 91125, USA
5
California State University Dominguez Hills, Carson, California 90747, USA
6
Caltech-CaRT, Pasadena, California 91125, USA
7
Cardiff University, Cardiff, CF2 3YB, United Kingdom
8
Carleton College, Northfield, Minnesota 55057, USA
9
Hobart and William Smith Colleges, Geneva, New York 14456, USA
10
Inter-University Centre for Astronomy and Astrophysics, Pune-411007, India
11
LIGO-California Institute of Technology, Pasadena, California 91125, USA
12
LIGO-Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
13
LIGO Hanford Observatory, Richland, Washington 99352, USA
14
LIGO Livingston Observatory, Livingston, Louisiana 70754, USA
15
Louisiana State University, Baton Rouge, Louisiana 70803, USA
16
Louisiana Tech. University, Ruston, Louisiana 71272, USA
17
Loyola University, New Orleans, Louisiana 70118, USA
18
Max Planck Institut fu
̈
r Quantenoptik, D-85748, Garching, Germany
19
Moscow State University, Moscow, 119992, Russia
20
NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
21
National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
22
Northwestern University, Evanston, Illinois 60208, USA
23
Salish Kootenai College, Pablo, Montana 59855, USA
24
Southeastern Louisiana University, Hammond, Louisiana 70402, USA
25
Stanford University, Stanford, California 94305, USA
26
Syracuse University, Syracuse, New York 13244, USA
27
The Pennsylvania State University, University Park, Pennsylvania 16802, USA
28
The University of Texas at Brownsville and Texas Southmost College, Brownsville, Texas 78520, USA
29
Trinity University, San Antonio, Texas 78212, USA
30
Universita
̈
t Hannover, D-30167 Hannover, Germany
31
Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
32
University of Birmingham, Birmingham, B15 2TT, United Kingdom
33
University of Florida, Gainesville, Florida 32611, USA
34
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
35
University of Michigan, Ann Arbor, Michigan 48109, USA
36
University of Oregon, Eugene, Oregon 97403, USA
37
University of Rochester, Rochester, New York 14627, USA
38
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA
39
Washington State University, Pullman, Washington 99164, USA
M. Kramer and A. G. Lyne
University of Manchester, Jodrell Bank Observatory, Macclesfield, Cheshire, SK11 9DL, United Kingdom
(Received 29 October 2004; published 12 May 2005)
We place direct upper limits on the amplitude of gravitational waves from 28 isolated radio pulsars by a
coherent multidetector analysis of the data collected during the second science run of the LIGO
interferometric detectors. These are the first
direct
upper limits for 26 of the 28 pulsars. We use
coordinated radio observations for the first time to build radio-guided phase templates for the expected
gravitational-wave signals. The unprecedented sensitivity of the detectors allows us to set strain upper
limits as low as a few times
10
24
. These strain limits translate into limits on the equatorial ellipticities of
the pulsars, which are smaller than
10
5
for the four closest pulsars.
DOI: 10.1103/PhysRevLett.94.181103
PACS numbers: 04.80.Nn, 07.05.Kf, 95.55.Ym, 97.60.Gb
A worldwide effort is underway to detect gravitational
waves (GWs) and thus test a fundamental prediction of
general relativity. In preparation for long-term operations,
the Laser Interferometric Gravitational Wave Observatory
PRL
94,
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PHYSICAL REVIEW LETTERS
week ending
13 MAY 2005
181103-2
(LIGO) and GEO experiments conducted their first science
run (S1) during 17 days in 2002. The detectors and the
pulsar analysis of the S1 data are described in [1,2],
respectively. LIGO’s second science run (S2) was carried
out from 14 February to 14 April 2003, with dramatically
improved sensitivity compared to S1. During S2 the GEO
detector was not operating.
A spinning neutron star is expected to emit GWs if it is
not perfectly symmetric about its rotation axis. The strain
amplitude
h
0
of the emitted signal is proportional to the
star’s deformation as measured by its ellipticity
[3].
Using data from S2, this Letter reports
direct
observational
limits on the GW emission and corresponding ellipticities
from the 28 most rapidly rotating isolated pulsars for which
radio data are complete enough to guide the phase of our
filters with sufficient precision. These are the first such
limits for 26 of the pulsars.
The limits reported here are still well above the indirect
limits inferred from observed pulsar spin-down, where
available (Fig. 1). However, fourteen of our pulsars are in
globular clusters, where local gravitational accelerations
produce Doppler effects that mask the intrinsic pulsar spin-
down, sometimes even producing apparent spin-up. For
these pulsars our observations therefore place the first
limits that are inherently independent of cluster dynamics,
albeit at levels well above what one would expect if all
globular cluster pulsars are similar to field pulsars.
Our most stringent ellipticity upper limit is
4
:
5
10
6
.
While still above the maximum expected from conven-
tional models of nuclear matter, distortions of this size
would be permitted within at least one exotic theory of
neutron star structure [4].
Detectors. —
Each of LIGO’s three detectors is a power-
recycled Michelson interferometer with Fabry-Perot
cavities in the long arms [1]. Two detectors, the 4 km
arm H1 and the 2 km arm H2 detectors, are collocated in
Hanford, WA. The 4 km arm L1 detector is situated in
Livingston Parish, LA. Improvements in noise perform-
ance between S1 and S2 were approximately an order of
magnitude over a broad frequency range. Modifications
that were made between S1 and S2 to aid in noise reduction
and improve stability include (i) increased laser power to
reduce high-frequency noise, (ii) better angular control of
the mirrors of the interferometer, and (iii) the use of lower
noise digital test mass suspension controllers in all
detectors.
During S2, the LIGO detectors’ noise performance in
the band 40 – 2000 Hz was better than any previous inter-
ferometer. The best strain sensitivity, achieved by L1, was
3
10
22
Hz
1
=
2
near 200 Hz, translating via Eq. (2.2)
of [2] to a detectable amplitude for a continuous signal of
about
3
10
24
, as shown in Fig. 1. The relative timing
stability between the interferometers was also significantly
improved. Monitored with GPS-synchronized clocks to be
better than
10
s
over S2, it allowed the coherent combi-
nation of the strain data of all three detectors to form joint
upper limits.
Analysis method. —
In [2] a search for gravitational
waves from the millisecond pulsar
J1939
2134
using
S1 data was presented. In that work, two different data
analysis methods were used, one in the time domain and
the other in the frequency domain. Here we extend the
former method [2,5] and apply it to 28 isolated pulsars.
Following [2] we model the sources as nonprecessing
triaxial neutron stars showing the same rotational phase
evolution as is present in the radio signal and perform a
complex heterodyne of the strain data from each detector at
the instantaneous frequency of the expected gravitational-
wave signal, which is twice the observed radio rotation
frequency. These data are then down-sampled to
1
=
60 Hz
and are referred to as
B
k
. Any gravitational signal in the
data would show a residual time evolution reflecting the
antenna pattern of the detector, varying over the day as the
source moved through the pattern, but with a functional
form that depended on several other source-observer pa-
rameters: the antenna responses to plus and cross polar-
izations, the amplitude of the gravitational wave
h
0
, the
angle between the line of sight to the pulsar and its spin
axis
, the polarization angle of the gravitational radiation
(all defined in [3]), and the phase
0
of the gravitational-
wave signal at some fiducial time
t
0
. Let
a
be a vector in
parameter space with components (
h
0
,
,
,
0
).
The analysis proceeds by determining the posterior
probability distribution function (PDF) of
a
given the
data
B
k
and the signal model:
p
a
jf
B
k
g /
p
a
p
f
B
k
gj
a
;
(1)
where
p
f
B
k
gj
a
is the likelihood and
p
a
the prior PDF
we assign to the model parameters. We have used a uni-
form prior for
cos
,
0
,
, and
h
0
(
h
0
>
0
), in common
with [2]. A uniform prior for
h
0
has been chosen for its
simplicity and for the easy comparison of our results to
other observations. This prior favors high values of
h
0
(which comprise the majority of the parameter space)
and therefore generates a somewhat conservative upper
limit for its value. Indeed, the reader might prefer to regard
our resulting posterior PDFs for
h
0
as marginalized like-
lihoods rather than probabilities for
h
0
— these are func-
tionally identical using our priors.
As in [2] we use a Gaussian joint likelihood for
p
f
B
k
gj
a
. In [2] the S1 noise floor was estimated over a
60 s period from a 4 Hz band about the expected signal
frequency. This gave a reliable point estimate for the noise
level but was sensitive to spectral contamination within the
band, as demonstrated in the analysis of the GEO S1 data.
In this Letter we exploit the improved stationarity of the
instruments that make it reasonable to assume the noise
floor is constant over periods of 30 min [5]. In addition, we
restrict the bandwidth to
1
=
60 Hz
, which makes it possible
to search for signals from pulsars at frequencies close to
PRL
94,
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week ending
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181103-3
strong spectral disturbances. However, the noise level now
determined is less certain as the estimate relies on fewer
data. We take account of this increased uncertainty by
explicitly marginalizing with a Jeffreys prior over the
constant but unknown noise level for each 30 min period
of data [6]. The likelihood for this analysis is then the
combined likelihood for all the 30 min stretches of data,
labeled by
j
in Eq. (2), taken as independent:
p
f
B
k
gj
a
/
Y
j
X
k
2
j
k
k
1
j
j
B
k
y
k
j
2
m
;
(2)
where
y
k
is the signal model given by Eq. (4.10) in [2] and
m
k
2
j
k
1
j
1
30
is the number of
B
k
data points
in a 30 min segment.
In principle the period over which the data are assumed
stationary need not be fixed, and can be adjusted dynami-
cally to reflect instrumental performance over the run. We
have limited our analysis to continuous 30 min stretches of
data, which included more than 88% of the S2 science data
set. Inclusion of shorter data sections would at best have
resulted in a
6%
improvement on the strain upper limits
reported here.
Validation by hardware injections. —
The software was
validated by checking its performance on fake pulsar sig-
nals injected in artificial and real detector noise, both in
software [2] and in hardware. In particular, two artificial
signals (P1, P2) were added digitally to the interferometer
length sensing and control systems (responsible for main-
taining a given interferometer on resonance), resulting in a
differential length dither in the optical cavities of the
detector. These injections were designed to give an end-
to-end validation of the search pipeline starting from as far
up the observing chain as possible.
The pulsar signals were injected for 12 h at frequencies
of 1279.123 Hz (P1) and 1288.901 Hz (P2) with frequency
derivatives of zero and
10
8
Hz s
1
, respectively, and
strain amplitudes of
2
10
21
. In the case of the 4 km
instruments, the displacement induced by this strain was up
to
4000 m
2
10
21
8
10
18
m
.
These
give
signal-to-noise ratios [as defined by Eq. (79) of [3] ] of
26 and 40 for P1 in H1 and L1, respectively, and of 38 and
34 for P2. The signals were modulated and Doppler shifted
to simulate sources at fixed positions on the sky with
0
,
cos
0
, and
0
0
. To illustrate, posterior PDFs for
the recovered P1 signal are shown in Fig. 2. The results
derived from the different detectors are in broad statistical
agreement, confirming that the relative calibrations are
consistent and that the assessments of uncertainty (ex-
pressed in the posterior widths) are reasonable. Results
for P2 were very similar to these.
The phase stability of the detectors in S2 allowed us to
implement a
joint
coherent analysis based on data from all
three participating instruments. This technique was noted
in [2], but could not be performed on the S1 data because of
timing uncertainties that existed when those observations
024
x 10
−21
0
2
4
x 10
21
h
0
p(h
0
)
−20
0
20
0
0.05
0.1
0.15
φ
0
(degrees)
p(
φ
0
)
−10
0
10
0
0.2
0.4
ψ
(degrees)
p(
ψ
)
−0.2
0
0.2
0
10
20
30
cos
ι
p(cos
ι
)
FIG. 2 (color).
Marginalized PDFs for the parameters of the
artificial pulsar P1. The vertical dotted lines show the values used
to generate the signal, the colored lines show the results from the
individual detectors (H1 green, H2 blue, L1 red), and the black
lines show the joint result from combining coherently data from
all three.
10
2
10
3
10
−28
10
−27
10
−26
10
−25
10
−24
10
−23
10
−22
10
−21
10
−20
10
−19
Frequency (Hz)
Gravitational wave amplitude h
0
H1
H2
L1
S2 joint sensitivity
LIGO design: 1 year
Joint upper limit
Spin-down upper limit
FIG. 1 (color).
Upper curves:
h
0
amplitudes detectable from a
known generic source with a 1% false alarm rate and 10% false
dismissal rate, as given by Eq. (2.2) in [2] for single detector
analyses and for a joint detector analysis. All the curves use
typical S2 sensitivities and observation times. Lower curve:
LIGO design sensitivity for 1 yr of data. Stars: upper limits
found in this Letter for 28 known pulsars. Circles: spin-down
upper limits for the pulsars with negative frequency derivative
values if
all
the measured rotational energy loss were due to
gravitational waves and assuming a moment of inertia of
10
45
gcm
2
.
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94,
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181103-4
were performed. The black lines in Fig. 2 show margin-
alizations of the joint posterior from H1, H2, and L1, i.e.,
p
a
j
H1
;
H2
;
L1
/
p
a
p
H1
j
a
p
H2
j
a
p
L1
j
a
:
(3)
For three detectors of similar sensitivities and operational
times, these coherent results would be approximately
3
p
times tighter than the individual results. The posteriors for
0
clearly highlight the relative coherence between the
instruments and verify that joint methods can be used to set
upper limits on our target pulsars.
Results. —
We selected 28 targets from the ATNF pulsar
catalogue [7]. For 18 of these, we obtained updated timing
solutions from regular timing observations made at the
Jodrell Bank Observatory using the Lovell and the Parkes
telescopes, adjusted for a reference epoch centered on the
epoch of the S2 run (starred pulsars in Table I). Details of
the techniques that were used to do this can be found in [8].
We also checked that none of these pulsars exhibited a
glitch during this period.
The list includes globular cluster pulsars (including
isolated pulsars in 47 Tuc and NGC6752), the S1 target
millisecond pulsar (
J1939
2134
), and the Crab pulsar
(
B0531
21
). Although Table I shows only approximate
pulsar frequencies and frequency derivatives, further phase
corrections were made for pulsars with measured second
derivatives of frequency. Timing solutions for the Crab
were taken from the Jodrell Bank online ephemeris [9],
and adjustments were made to its phase over the period of
S2 using the method of [10].
The analysis used 910 h of data from H1, 691 h from H2,
and 342 h from L1. There was no evidence of strong
spectral contamination in any of the bands investigated,
such as might be caused by an instrumental feature or a
potentially detectable pulsar signal. A strong gravitational
signal would generate a parameter PDF prominently
peaked off zero with respect to its width, as for the hard-
ware injections. Such a PDF would trigger a more detailed
investigation of the pulsar in question. No such triggers
occurred in the analysis of these data, and we therefore
present upper limits.
The upper limits are presented as the value of
h
0
bound-
ing 95% of the cumulative probability of the marginalized
strain PDF from
h
0
0
. The joint upper limit
h
95%
0
there-
fore satisfies
0
:
95
Z
h
95%
0
h
0
0
dh
0
Z
Z
Z
p
a
j
H1
;
H2
;
L1
dd d
0
;
(4)
consistent with [2]. The uncertainty in the noise floor
estimate is already included, as outlined above.
The remaining uncertainties in the upper limit values of
Table I stem from the calibration of the instrument and
from the accuracy of the pulsar timing models. For L1 and
H2, the amplitude calibration uncertainties are conserva-
tively estimated to be 10% and 8%, respectively. For H1,
the maximum calibration uncertainty is 18%, with typical
values at the 6% level. Phase calibration uncertainties are
negligible in comparison: less than 10
in all detectors.
Biases due to pulsar timing errors are estimated to be 3% or
less for
J0030
0451
, and 1% or less for the remaining
pulsars (see [2] for a discussion of the effect of these
uncertainties).
Discussion. —
The improved sensitivity of the LIGO in-
terferometers is clear from the strain upper limit for pulsar
J1939
2134
, which is more than a factor of 10 lower than
was achieved with the S1 data [2]. In this analysis the
lowest limit is achieved for pulsar
J1910
5959D
at the
level of
1
:
7
10
24
, largely reflecting the lower noise
floor around 200 Hz.
Table I also gives approximate limits to the ellipticities
[3] of these pulsars from the simple quadrupole model
TABLE I. The 28 pulsars targeted in the S2 run, with approxi-
mate spin parameters. The right-hand two columns show the
95% upper limit on
h
0
and ellipticity. These upper limits do
not
include the uncertainties due to calibration and to pulsar timing
accuracy, which are discussed in the text, nor uncertainties in the
pulsar’s distance,
r
.
Spin
Spin-down
h
95%
0
Pulsar
f
Hz
_
f
Hz s
1
=
10
24
=
10
5
B0021
72C
173.71
1
:
50
10
15
4.3
16
B0021
72D
186.65
1
:
19
10
16
4.1
14
B0021
72F
381.16
9
:
37
10
15
7.2
5.7
B0021
72G
247.50
2
:
58
10
15
4.1
7.5
B0021
72L
230.09
6
:
46
10
15
2.9
6.1
B0021
72M
271.99
2
:
84
10
15
3.3
5.0
B0021
72N
327.44
2
:
34
10
15
4.0
4.3
J0030
0451
205.53
4
:
20
10
16
3.8
0.48
B0531
21
29.81
3
:
74
10
10
41
2100
J0711
6830
182.12
4
:
94
10
16
2.4
1.8
J1024
0719
193.72
6
:
95
10
16
3.9
0.86
B1516
02A
180.06
1
:
34
10
15
3.6
21
J1629
6902
166.65
2
:
78
10
16
2.3
2.7
J1721
2457
285.99
4
:
80
10
16
4.0
1.8
J1730
2304
123.11
3
:
06
10
16
3.1
2.5
J1744
1134
245.43
5
:
40
10
16
5.9
0.83
J1748
2446C
118.54
8
:
52
10
15
3.1
24
B1820
30A
183.82
1
:
14
10
13
4.2
24
B1821
24
327.41
1
:
74
10
13
5.6
7.1
J1910
5959B
119.65
1
:
14
10
14
2.4
8.5
J1910
5959C
189.49
7
:
90
10
17
3.3
4.7
J1910
5959D
110.68
1
:
18
10
14
1.7
7.2
J1910
5959E
218.73
2
:
09
10
14
7.5
7.9
J1913
1011
27.85
2
:
61
10
12
51
6900
J1939
2134
641.93
4
:
33
10
14
13
2.7
B1951
32
25.30
3
:
74
10
12
48
4400
J2124
3358
202.79
8
:
45
10
16
3.1
0.45
J2322
2057
207.97
4
:
20
10
16
4.1
1.8
PRL
94,
181103 (2005)
PHYSICAL REVIEW LETTERS
week ending
13 MAY 2005
181103-5
’
0
:
237
h
0
10
24
r
1kpc
1Hz
2
f
2
10
45
gcm
2
I
zz
;
(5)
where
r
is the pulsar’s distance, which we take as the
dispersion measure distance using the model of Taylor
and Cordes [11], and
I
zz
its principal moment of inertia
about the rotation axis, which we take as
10
45
gcm
2
.
As expected, none of these upper limits improves on
those inferred from simple arguments based on the gravi-
tational luminosities achievable from the observed loss of
pulsar rotational kinetic energy. However, as discussed in
the introduction, for pulsars in globular clusters such argu-
ments are complicated by cluster dynamics, which the
direct limits presented here avoid.
The result for the Crab pulsar (
B0531
21
) is within a
factor of about 30 of the spin-down limit and over an order
of magnitude better than the previous direct upper limit of
[12]. The equatorial ellipticities of the four closest pulsars
(
J0030
0451
,
J2124
3358
,
J1024
0719
,
and
J1744
1134
) are constrained to be less than
10
5
.
Once the detectors operate at design sensitivity for a
year, the observational upper limits will improve by more
than an order of magnitude. The present analysis will also
be extended to include pulsars in binary systems, signifi-
cantly increasing the population of objects under
inspection.
The authors gratefully acknowledge the support of the
United States National Science Foundation for the con-
struction and operation of the LIGO Laboratory and the
Particle Physics and Astronomy Research Council of the
United Kingdom, the Max-Planck-Society, and the State of
Niedersachsen/Germany for support of the construction
and operation of the GEO600 detector. The authors also
gratefully acknowledge the support of the research by these
agencies and by the Australian Research Council, the
Natural Sciences and Engineering Research Council of
Canada, the Council of Scientific and Industrial Research
of India, the Department of Science and Technology of
India, the Spanish Ministerio de Ciencia y Tecnologia, the
John Simon Guggenheim Foundation, the Leverhulme
Trust, the David and Lucile Packard Foundation, the
Research
Corporation,
and
the
Alfred
P.
Sloan
Foundation. This document has been assigned LIGO
Laboratory Document No. LIGO-P040008-C-Z.
*Electronic address: http://www.ligo.org
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et al.
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PRL
94,
181103 (2005)
PHYSICAL REVIEW LETTERS
week ending
13 MAY 2005
181103-6