Limits on gravitational-wave emission from selected pulsars using LIGO data.

Abstract

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.

Full text

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arXiv:gr-qc/0410007 v1 1 Oct 2004
Limits on gravitational wave emission from selected pulsars using LIGO data
B. Abbott,12R. Abbott,15R. Adhikari,13A. Ageev,20,27B. Allen,39R. Amin,34S. B. Anderson,12
W. G. Anderson,29M. Araya,12H. Armandula,12M. Ashley,28F. Asiri,12, aP. Aufmuth,31C. Aulbert,1
S. Babak,7R. Balasubramanian,7S. Ballmer,13B. C. Barish,12C. Barker,14D. Barker,14M. Barnes,12, b
B. Barr,35M. A. Barton,12K. Bayer,13R. Beausoleil,26, cK. Belczynski,23R. Bennett,35, dS. J. Berukoff,1, e
J. Betzwieser,13B. Bhawal,12I. A. Bilenko,20G. Billingsley,12E. Black,12K. Blackburn,12L. Blackburn,13
B. Bland,14B. Bochner,13, fL. Bogue,12R. Bork,12S. Bose,40P. R. Brady,39V. B. Braginsky,20J. E. Brau,37
D. A. Brown,39A. Bullington,26A. Bunkowski,2,31A. Buonanno,6, gR. Burgess,13D. Busby,12W. E. Butler,38
R. L. Byer,26L. Cadonati,13G. Cagnoli,35J. B. Camp,21C. A. Cantley,35L. Cardenas,12K. Carter,15
M. M. Casey,35J. Castiglione,34A. Chandler,12J. Chapsky,12, bP. Charlton,12S. Chatterji,13S. Chelkowski,2,31
Y. Chen,6V. Chickarmane,16, hD. Chin,36N. Christensen,8D. Churches,7T. Cokelaer,7C. Colacino,33
R. Coldwell,34M. Coles,15, iD. Cook,14T. Corbitt,13D. Coyne,12J. D. E. Creighton,39T. D. Creighton,12
D. R. M. Crooks,35P. Csatorday,13B. J. Cusack,3C. Cutler,1E. D’Ambrosio,12K. Danzmann,31,2E. Daw,16, j
D. DeBra,26T. Delker,34, kV. Dergachev,36R. DeSalvo,12S. Dhurandhar,11A. Di Credico,27M. D´iaz,29H. Ding,12
R. W. P. Drever,4R. J. Dupuis,35J. A. Edlund,12, bP. Ehrens,12E. J. Elliffe,35T. Etzel,12M. Evans,12
T. Evans,15S. Fairhurst,39C. Fallnich,31D. Farnham,12M. M. Fejer,26T. Findley,25M. Fine,12L. S. Finn,28
K. Y. Franzen,34A. Freise,2, lR. Frey,37P. Fritschel,13V. V. Frolov,15M. Fyffe,15K. S. Ganezer,5J. Garofoli,14
J. A. Giaime,16A. Gillespie,12, mK. Goda,13G. Gonz´ alez,16S. Goßler,31P. Grandcl´ ement,23, nA. Grant,35
C. Gray,14A. M. Gretarsson,15D. Grimmett,12H. Grote,2S. Grunewald,1M. Guenther,14E. Gustafson,26, o
R. Gustafson,36W. O. Hamilton,16M. Hammond,15J. Hanson,15C. Hardham,26J. Harms,19G. Harry,13
A. Hartunian,12J. Heefner,12Y. Hefetz,13G. Heinzel,2I. S. Heng,31M. Hennessy,26N. Hepler,28A. Heptonstall,35
M. Heurs,31M. Hewitson,2S. Hild,2N. Hindman,14P. Hoang,12J. Hough,35M. Hrynevych,12, pW. Hua,26
M. Ito,37Y. Itoh,1A. Ivanov,12O. Jennrich,35, qB. Johnson,14W. W. Johnson,16W. R. Johnston,29D. I. Jones,28
L. Jones,12D. Jungwirth,12, rV. Kalogera,23E. Katsavounidis,13K. Kawabe,14S. Kawamura,22W. Kells,12
J. Kern,15, sA. Khan,15S. Killbourn,35C. J. Killow,35C. Kim,23C. King,12P. King,12S. Klimenko,34
S. Koranda,39K. K¨ otter,31J. Kovalik,15, bD. Kozak,12B. Krishnan,1M. Landry,14J. Langdale,15B. Lantz,26
R. Lawrence,13A. Lazzarini,12M. Lei,12I. Leonor,37K. Libbrecht,12A. Libson,8P. Lindquist,12S. Liu,12
J. Logan,12, tM. Lormand,15M. Lubinski,14H. L¨ uck,31,2T. T. Lyons,12, tB. Machenschalk,1M. MacInnis,13
M. Mageswaran,12K. Mailand,12W. Majid,12, bM. Malec,2,31F. Mann,12A. Marin,13, uS. M´ arka,12E. Maros,12
J. Mason,12, vK. Mason,13O. Matherny,14L. Matone,14N. Mavalvala,13R. McCarthy,14D. E. McClelland,3
M. McHugh,18J. W. C. McNabb,28G. Mendell,14R. A. Mercer,33S. Meshkov,12E. Messaritaki,39C. Messenger,33
V. P. Mitrofanov,20G. Mitselmakher,34R. Mittleman,13O. Miyakawa,12S. Miyoki,12, wS. Mohanty,29G. Moreno,14
K. Mossavi,2G. Mueller,34S. Mukherjee,29P. Murray,35J. Myers,14S. Nagano,2T. Nash,12R. Nayak,11
G. Newton,35F. Nocera,12J. S. Noel,40P. Nutzman,23T. Olson,24B. O’Reilly,15D. J. Ottaway,13A. Ottewill,39, x
D. Ouimette,12, rH. Overmier,15B. J. Owen,28Y. Pan,6M. A. Papa,1V. Parameshwaraiah,14C. Parameswariah,15
M. Pedraza,12S. Penn,10M. Pitkin,35M. Plissi,35R. Prix,1V. Quetschke,34F. Raab,14H. Radkins,14R. Rahkola,37
M. Rakhmanov,34S. R. Rao,12K. Rawlins,13S. Ray-Majumder,39V. Re,33D. Redding,12, bM. W. Regehr,12, b
T. Regimbau,7S. Reid,35K. T. Reilly,12K. Reithmaier,12D. H. Reitze,34S. Richman,13, yR. Riesen,15K. Riles,36
B. Rivera,14A. Rizzi,15, zD. I. Robertson,35N. A. Robertson,26,35L. Robison,12S. Roddy,15J. Rollins,13
J. D. Romano,7J. Romie,12H. Rong,34, mD. Rose,12E. Rotthoff,28S. Rowan,35A. R¨ udiger,2P. Russell,12
K. Ryan,14I. Salzman,12V. Sandberg,14G. H. Sanders,12, aaV. Sannibale,12B. Sathyaprakash,7P. R. Saulson,27
R. Savage,14A. Sazonov,34R. Schilling,2K. Schlaufman,28V. Schmidt,12, bbR. Schnabel,19R. Schofield,37
B. F. Schutz,1,7P. Schwinberg,14S. M. Scott,3S. E. Seader,40A. C. Searle,3B. Sears,12S. Seel,12F. Seifert,19
A. S. Sengupta,11C. A. Shapiro,28, ccP. Shawhan,12D. H. Shoemaker,13Q. Z. Shu,34, ddA. Sibley,15X. Siemens,39
L. Sievers,12, bD. Sigg,14A. M. Sintes,1,32J. R. Smith,2M. Smith,13M. R. Smith,12P. H. Sneddon,35R. Spero,12, b
G. Stapfer,15D. Steussy,8K. A. Strain,35D. Strom,37A. Stuver,28T. Summerscales,28M. C. Sumner,12
P. J. Sutton,12J. Sylvestre,12, eeA. Takamori,12D. B. Tanner,34H. Tariq,12I. Taylor,7R. Taylor,12R. Taylor,35
K. A. Thorne,28K. S. Thorne,6M. Tibbits,28S. Tilav,12, ffM. Tinto,4, bK. V. Tokmakov,20C. Torres,29
C. Torrie,12G. Traylor,15W. Tyler,12D. Ugolini,30C. Ungarelli,33M. Vallisneri,6, ggM. van Putten,13S. Vass,12
A. Vecchio,33J. Veitch,35C. Vorvick,14S. P. Vyachanin,20L. Wallace,12H. Walther,19H. Ward,35B. Ware,12, b
K. Watts,15D. Webber,12A. Weidner,19U. Weiland,31A. Weinstein,12R. Weiss,13H. Welling,31L. Wen,12
S. Wen,16J. T. Whelan,18S. E. Whitcomb,12B. F. Whiting,34S. Wiley,5C. Wilkinson,14P. A. Willems,12
P. R. Williams,1, hhR. Williams,4B. Willke,31A. Wilson,12B. J. Winjum,28, eW. Winkler,2S. Wise,34
A. G. Wiseman,39G. Woan,35R. Wooley,15J. Worden,14W. Wu,34I. Yakushin,15H. Yamamoto,12S. Yoshida,25
LIGO−P040008−A−Z

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K. D. Zaleski,28M. Zanolin,13I. Zawischa,31, iiL. Zhang,12R. Zhu,1N. Zotov,17M. Zucker,15and J. Zweizig12
(The LIGO Scientific Collaboration, http://www.ligo.org)
M. Kramer41and A. G. Lyne41
1Albert-Einstein-Institut, Max-Planck-Institut f¨ ur Gravitationsphysik, D-14476 Golm, Germany
2Albert-Einstein-Institut, Max-Planck-Institut f¨ ur Gravitationsphysik, D-30167 Hannover, Germany
3Australian National University, Canberra, 0200, Australia
4California Institute of Technology, Pasadena, CA 91125, USA
5California State University Dominguez Hills, Carson, CA 90747, USA
6Caltech-CaRT, Pasadena, CA 91125, USA
7Cardiff University, Cardiff, CF2 3YB, United Kingdom
8Carleton College, Northfield, MN 55057, USA
9Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
10Hobart and William Smith Colleges, Geneva, NY 14456, USA
11Inter-University Centre for Astronomy and Astrophysics, Pune - 411007, India
12LIGO - California Institute of Technology, Pasadena, CA 91125, USA
13LIGO - Massachusetts Institute of Technology, Cambridge, MA 02139, USA
14LIGO Hanford Observatory, Richland, WA 99352, USA
15LIGO Livingston Observatory, Livingston, LA 70754, USA
16Louisiana State University, Baton Rouge, LA 70803, USA
17Louisiana Tech University, Ruston, LA 71272, USA
18Loyola University, New Orleans, LA 70118, USA
19Max Planck Institut f¨ ur Quantenoptik, D-85748, Garching, Germany
20Moscow State University, Moscow, 119992, Russia
21NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
22National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
23Northwestern University, Evanston, IL 60208, USA
24Salish Kootenai College, Pablo, MT 59855, USA
25Southeastern Louisiana University, Hammond, LA 70402, USA
26Stanford University, Stanford, CA 94305, USA
27Syracuse University, Syracuse, NY 13244, USA
28The Pennsylvania State University, University Park, PA 16802, USA
29The University of Texas at Brownsville and Texas Southmost College, Brownsville, TX 78520, USA
30Trinity University, San Antonio, TX 78212, USA
31Universit¨ at Hannover, D-30167 Hannover, Germany
32Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
33University of Birmingham, Birmingham, B15 2TT, United Kingdom
34University of Florida, Gainesville, FL 32611, USA
35University of Glasgow, Glasgow, G12 8QQ, United Kingdom
36University of Michigan, Ann Arbor, MI 48109, USA
37University of Oregon, Eugene, OR 97403, USA
38University of Rochester, Rochester, NY 14627, USA
39University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
40Washington State University, Pullman, WA 99164, USA
41University of Manchester, Jodrell Bank Observatory, Macclesfield, Cheshire, SK11 9DL, United Kingdom
(Dated: October 5, 2004)
We place direct upper limits on the strain of the gravitational waves from 28 isolated radio
pulsars by a coherent multi-detector 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 allow 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 10−5for the four closest pulsars.
PACS numbers: 04.80.Nn, 95.55.Ym, 97.60.Gb, 07.05.Kf
aCurrently at Stanford Linear Accelerator Center
bCurrently at Jet Propulsion Laboratory
cPermanent Address: HP Laboratories
dCurrently at Rutherford Appleton Laboratory
eCurrently at University of California, Los Angeles
fCurrently at Hofstra University
2

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A worldwide effort is underway to detect gravitational
waves (GWs) and thus test a fundamental prediction of
General Relativity. In preparation for long-term opera-
tions, the LIGO and GEO experiments conducted their
first science run (S1) during 17 days in 2002. The de-
tectors and the analyses of the S1 data are described in
Refs. [1] and [2]-[5], respectively. LIGO’s second science
run (S2) was carried out from 14 Feb - 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 h0of the emitted signal is proportional
to the star’s deformation as measured by its ellipticity ǫ
[6]. Using data from S2, this paper reports direct obser-
vational limits on the GW emission and corresponding
ellipticities from the 28 most rapidly rotating isolated
pulsars for which radio data is complete enough to guide
the phase of our filters with sufficient precision. These
are the first such limits for 26 of the pulsars. We con-
centrate on isolated pulsars with known phase evolutions
and sky positions to ensure that our targeted search re-
quires relatively few unknown parameters.
The limits reported here are still well above the indi-
rect limits inferred from observed pulsar spindown, where
available (Fig. 1). However, fourteen of our pulsars are in
globular clusters, where local gravitational accelerations
produce Doppler effects that mask the true pulsar spin-
down, sometimes even producing apparent spinup. For
gPermanent Address: GReCO, Institut d’Astrophysique de Paris
(CNRS)
hCurrently at Keck Graduate Institute
iCurrently at National Science Foundation
jCurrently at University of Sheffield
kCurrently at Ball Aerospace Corporation
lCurrently at European Gravitational Observatory
mCurrently at Intel Corp.
nCurrently at University of Tours, France
oCurrently at Lightconnect Inc.
pCurrently at W.M. Keck Observatory
qCurrently at ESA Science and Technology Center
rCurrently at Raytheon Corporation
sCurrently at New Mexico Institute of Mining and Technology /
Magdalena Ridge Observatory Interferometer
tCurrently at Mission Research Corporation
uCurrently at Harvard University
vCurrently at Lockheed-Martin Corporation
wPermanent Address: University of Tokyo, Institute for Cosmic
Ray Research
xPermanent Address: University College Dublin
yCurrently at Research Electro-Optics Inc.
zCurrently at Institute of Advanced Physics, Baton Rouge, LA
aaCurrently at Thirty Meter Telescope Project at Caltech
bbCurrently at European Commission, DG Research, Brussels, Bel-
gium
ccCurrently at University of Chicago
ddCurrently at LightBit Corporation
eePermanent Address: IBM Canada Ltd.
ffCurrently at University of Delaware
ggPermanent Address: Jet Propulsion Laboratory
hhCurrently at Shanghai Astronomical Observatory
iiCurrently at Laser Zentrum Hannover
these pulsars our observations therefore place the first
limits that are inherently independent of cluster dynam-
ics, 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 [7].
Detectors.—The LIGO observatory is composed of
three detectors. Each detector is a power-recycled
Michelson interferometer, with Fabry-Perot cavities in
the long arms. A passing GW produces a time-varying
differential strain in these arms, and the resulting dif-
ferential displacement of the cavity test mass mirrors is
sensed interferometrically. Two detectors, the 4km arm
H1 and the 2km arm H2 detectors, are collocated in Han-
ford WA. The 4km arm L1 detector is situated in Liv-
ingston Parish LA. Improvements in noise performance
between S1 and S2 were approximately an order of mag-
nitude 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 detec-
tors.
During S2, the LIGO detectors’ noise performance in
the band 40-2000Hz was better than any previous in-
terferometer. The best strain sensitivity, achieved by L1,
was ∼ 3×10−22Hz−1/2near 200Hz (Fig. 1). The relative
timing stability between the interferometers was also sig-
nificantly improved. Monitored with GPS-synchronized
clocks to be better than 10µs over S2, it allowed the
coherent combination of the strain data of all three de-
tectors to form joint upper limits.
Analysis method.—In [2] a search for gravitational
waves from the millisecond pulsar PSR J1939+2134 us-
ing 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 [8] and apply it to 28 isolated pulsars.
Following [2] we model the sources as non-precessing
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 detec-
tor at the instantaneous frequency of the expected grav-
itational wave signal, which is twice the observed radio
rotation frequency. These data are then downsampled
to 1/60Hz and will be referred to as Bk. Any gravi-
tational 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 parameters: the antenna
responses to plus and cross polarisations, the amplitude
of the gravitational wave h0, the angle between the line-
of-sight to the pulsar and its spin axis ι, the polarisation
angle of the gravitational radiation ψ (all defined in [6])
3

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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 h0
H1
H2
L1
LIGO design: 1 year
Joint upper limit
Spindown upper limit
FIG. 1: Upper curves: characteristic amplitudes detectable
from a known source with a 1% false alarm rate and 10%
false dismissal rate, as given by Eq. (2.2) in [2], using S2 sen-
sitivities and observation times. Lower curve: LIGO design
sensitivity for 1yr of data. Stars: upper limits found in this
paper for 28 known pulsars. Circles: spindown upper limits
for the pulsars with negative spindown values if all the mea-
sured loss of angular momentum were due to gravitational
waves and assuming a moment of inertia of 1045g · cm2.
and the phase φ0of the gravitational wave signal at some
fiducial time t0. Let a be a vector in parameter space
with components (h0,ι,ψ,φ0).
The analysis proceeds by determining the posterior
probability distribution function (pdf) of a given the
data Bkand the signal model:
p(a|{Bk}) ∝ p(a)p({Bk}|a),
where p({Bk}|a) is the likelihood and p(a) the prior pdf
we assign to the model parameters. We have used a uni-
form prior for cosι, φ0, ψ and h0(h0> 0), in common
with [2]. A uniform prior for h0has been chosen for its
simplicity and so that our results can readily be compared
with other observations. This prior favors high values of
h0(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 re-
gard our resulting posterior pdfs for h0as marginalised
likelihoods rather than probabilities for h0— these would
be functionally identical using our priors.
As in [2] we use a Gaussian joint likelihood for
p({Bk}|a). In [2] the S1 noise floor was estimated over
a 60s period from a 4Hz band about the expected sig-
nal 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 paper we exploit the improved
stationarity of the instruments and take the noise floor
to be constant over periods of 30min. In addition we
restrict the bandwidth to 1/60Hz, which makes it possi-
ble to search for signals from pulsars at frequencies close
(1)
02
h0
4
x 10
−21
0
2
4
x 10
21
p(h0)
−200 20
0
0.05
0.1
0.15
φ0 (degrees)
p(φ0)
−100 10
0
0.2
0.4
ψ (degrees)
p(ψ)
−0.200.2
0
10
20
30
cosι
p(cosι)
FIG. 2: Parameters of the artificial pulsar P1, recovered from
12h of strain data from the Hanford and Livingston inter-
ferometers. The results are displayed as marginal pdfs for
each of the four signal parameters. 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 the three.
to 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 marginalising with a Jeffreys prior over the
constant but unknown noise level for each 30min period
of data [9]. The likelihood for this analysis is then the
combined likelihood for all the 30min stretches of data,
labeled by j in Eq. (2), taken as independent:
p({Bk}|a) ∝
?
j
p({Bk}j|a),(2)
p({Bk}j|a) ∝


k2(j)
?
k=k1(j)
|Bk− yk|2


−m
,(3)
where yk is the signal model given by Eq. (4.10) in [2]
and m = k2(j)− k1(j)+1 = 30 is the number of Bkdata
points in a 30min segment.
In principle the period over which the data are as-
sumed stationary need not be fixed, and can be adjusted
dynamically to reflect instrumental performance over the
run. We have limited our analysis to continuous 30min
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 (Eq. (2.2) of Ref. [2]).
Validation by hardware injections.—The analysis soft-
ware was validated by checking its performance on fake
pulsar signals injected in artificial and real detector noise
both in software ([2]) and in hardware. In particular,
4

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two artificial signals (P1, P2) were injected into all three
detectors by modulating the mirror positions via the ac-
tuation control signals with the strain signal we should
expect from a hypothetical pulsar. 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 a 12h period at fre-
quencies of 1279.123Hz (P1) and 1288.901Hz (P2) with
frequency derivatives of zero and −10−8Hzs−1respec-
tively, and strain amplitudes of 2 × 10−21. The signals
were modulated and Doppler shifted to simulate sources
at fixed positions on the sky with ψ = 0, cosι = 0 and
φ0 = 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 (expressed 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 were performed. The solid lines in Fig. 2
show marginalisations of the joint posterior from H1, H2
and L1, i.e.,
p(a|H1,H2,L1) ∝ p(a)p(H1|a)p(H2|a)p(L1|a).
With three detectors of roughly similar sensitivities and
operational periods these coherent results should be ap-
proximately√3 times tighter than the individual results.
The posteriors for φ0 clearly highlight the relative co-
herence between the instruments and verify that similar
joint methods can be used to set upper limits on our
target pulsars.
Results.—From theATNF
(www.atnf.csiro.au/research/pulsar/psrcat/)
we selected 28 isolated pulsars with rotational frequen-
cies greater than 20Hz and for which good timing data
were available (Table I). For 18 of these, we obtained up-
dated timing solutions from regular timing observations
made at the Jodrell Bank Observatory using the Lovell
and the Parkes telescopes, adjusted for a reference epoch
centred on the period of the S2 run (starred pulsars in
Table I). Details of the techniques that were used to do
this can be found in [10]. We also checked that none of
these pulsars exhibited a glitch during this period.
The list includes globular cluster pulsars (including iso-
lated pulsars in 47 Tuc and NGC6752), the S1 target mil-
lisecond pulsar (PSR J1939+2134) and the Crab pulsar
(PSR B0531+21). Although Table I only shows approx-
imate pulsar frequencies and frequency derivatives, fur-
ther phase corrections were made for pulsars with mea-
sured second derivatives of frequency. Timing solutions
for the Crab were taken from the Jodrell Bank online
ephemeris [11], and adjustments were made to its phase
over the period of S2 using the method of [12].
(4)
pulsarcatalogue
pulsarspin
f (Hz)
173.71 +1.50×10−15
186.65 +1.19×10−16
381.16 −9.37×10−15
247.50 +2.58×10−15
230.09 +6.46×10−15
271.99 +2.84×10−15
327.44 +2.34×10−15
205.53 −4.20×10−16
29.81 −3.74×10−10
182.12 −4.94×10−16
193.72 −6.95×10−16
180.06 −1.34×10−15
166.65 −2.78×10−16
285.99 −4.80×10−16
123.11 −3.06×10−16
245.43 −5.40×10−16
spindown
˙f (Hzs−1) /10−24/10−5
h95%
0
ǫ
B0021−72C∗
B0021−72D∗
B0021−72F∗
B0021−72G∗
B0021−72L∗
B0021−72M∗
B0021−72N∗
J0030+0451
B0531+21∗
J0711−6830
J1024−0719∗
B1516+02A
J1629−6902
J1721−2457
J1730−2304∗
J1744−1134∗
J1748−2446C 118.54 +8.52×10−15
B1820−30A∗
183.82 −1.14×10−13
B1821−24∗
327.41 −1.74×10−13
J1910−5959B 119.65 +1.14×10−14
J1910−5959C 189.49 −7.90×10−17
J1910−5959D 110.68 −1.18×10−14
J1910−5959E218.73 +2.09×10−14
J1913+1011∗
27.85 −2.61×10−12
J1939+2134∗
641.93 −4.33×10−14
B1951+32∗
25.30 −3.74×10−12
J2124−3358∗
202.79 −8.45×10−16
J2322+2057∗
207.97 −4.20×10−16
4.3
4.1
7.2
4.1
2.9
3.3
4.0
3.8
41
2.4
3.9
3.6
2.3
4.0
3.1
5.9
3.1
4.2
5.6
2.4
3.3
1.7
7.5
51
13
48
3.1
4.1
16
14
5.7
7.5
6.1
5.0
4.3
0.48
2100
1.8
0.86
21
2.7
1.8
2.5
0.83
24
24
7.1
8.5
4.7
7.2
7.9
6900
2.7
4400
0.45
1.8
TABLE I: The 28 pulsars targeted in the S2 run, with approx-
imate spin parameters. Pulsars for which radio timing data
were taken over the S2 period are starred (*). The right-
hand two columns show the 95% upper limit on h0, based on
a coherent analysis using all the S2 data, and corresponding
ellipticity values (ǫ, see text). These upper limit values do
not include the uncertainties due to calibration and to pulsar
timing accuracy, which are discussed in the text, nor uncer-
tainties in r.
The analysis used 910 hours of data from H1, 691 hours
from H2, and 342 hours from L1. There was no evidence
of strong spectral contamination in any of the bands in-
vestigated, such as might be caused by an instrumen-
tal 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 hardware 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 simply present upper limits.
The upper limits are presented as the value of h0
bounding 95% of the cumulative probability of the
marginalised strain pdf from h0 = 0. The joint upper
5

Page 6

limit h95%
0
therefore satisfies
0.95 =
?h95%
h0=0
0
dh0
???
p(a|H1,H2,L1)dιdψ dφ0,(5)
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
conservatively 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
interferometers is clear from the strain upper limit for
PSR J1939+2134, which is more than a factor of ten
lower than was achieved with the S1 data [2]. In this anal-
ysis the lowest limit is achieved for PSR J1910−5959D at
the level of 1.7×10−24, largely reflecting the lower noise
floor around 200Hz.
Table I also gives approximate limits to the ellipticities
[6] of these pulsars from the simple quadrupole model
ǫ ≃ 0.237
h0
10−24
r
1kpc
1Hz2
f2
1045gcm2
Izz
(6)
where r is the pulsar’s distance, which we take as the
dispersion measure distance using the model of Taylor
and Cordes [13], and Izzits principal moment of inertia
about the rotation axis, which we take as 1045gcm2.
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
arguments are complicated by cluster dynamics, which
the direct limits presented here avoid.
The result for the Crab pulsar (PSR B0531+21) is
within a factor of about 30 of the spindown limit and over
an order of magnitude better than the previous direct up-
per limit of [14]. The equatorial ellipticities of the four
closest pulsars (J0030+0451, J2124+3358, J1024−0719,
and J1744−1134) are constrained to 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,
significantly increasing the population of objects under
inspection.
Acknowledgments.—The authors gratefully acknowl-
edge the support of the United States National Science
Foundation for the construction and operation of the
LIGO Laboratory and the Particle Physics and Astron-
omy 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 acknowl-
edge the support of the research by these agencies and
by the Australian Research Council, the Natural Sci-
ences 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
number LIGO-P040008-A-Z.
[1] B. Abbott et al. (The LIGO Scientific Collaboration),
Nuclear Inst. and Methods in Physics Research A 517/1-
3 154 (2004).
[2] B. Abbott et al. (The LIGO Scientific Collaboration),
Phys. Rev. D 69 082004 (2004).
[3] B. Abbott et al. (The LIGO Scientific Collaboration),
Phys. Rev. D 69 122001 (2004).
[4] B. Abbott et al. (The LIGO Scientific Collaboration),
Phys. Rev. D 69 102001 (2004).
[5] B. Abbott et al. (The LIGO Scientific Collaboration),
Phys. Rev. D 69 122004 (2004).
[6] P. Jaranowski, A. Kr´ olak and B. Schutz, Phys. Rev. D
58 063001 (1998).
[7] B.J. Owen and D.I. Jones (2004), LIGO Technical
Report T040192-00-Z,
http://www.ligo.caltech.edu/docs/T/T040192-00.pdf
[8] R. Dupuis and G. Woan, in preparation (2004).
[9] G.L. Bretthorst “Bayesian spectrum analysis and pa-
rameter estimation” Lecture Notes in Statistics, Vol 48
(Springer-Verlag, Berlin) (1988).
[10] G. Hobbs, A.G. Lyne, M. Kramer, C.E. Martin, and
C. Jordan, MNRAS, in press (2004).
[11] Jodrell Bank Crab Pulsar Timing: Monthly Ephemeris,
http://www.jb.man.ac.uk/research/pulsar/crab.html.
[12] M. Pitkin and G. Woan, Classical and Quantum Gravity
21 S843 (2004).
[13] J.H. Taylor and J.M. Cordes, ApJ 411 674 (1993).
[14] T. Suzuki, in First Edoardo Amaldi Conference on
Gravitational Wave Experiments, edited by E. Coccia,
G. Pizzella and F. Ronga (World Scientific, Singapore,
1995), p. 115.
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