of 21
arXiv:gr-qc/0702039v5 12 Oct 2007
LIGO-P060011-05-Z
Upper limits on gravitational wave emission from 78 radio pu
lsars
B. Abbott,
15
R. Abbott,
15
R. Adhikari,
15
J. Agresti,
15
P. Ajith,
2
B. Allen,
2, 52
R. Amin,
19
S. B. Anderson,
15
W. G. Anderson,
52
M. Arain,
40
M. Araya,
15
H. Armandula,
15
M. Ashley,
4
S. Aston,
39
P. Aufmuth,
37
C. Aulbert,
1
S. Babak,
1
S. Ballmer,
15
H. Bantilan,
9
B. C. Barish,
15
C. Barker,
16
D. Barker,
16
B. Barr,
41
P. Barriga,
51
M. A. Barton,
41
K. Bayer,
18
K. Belczynski,
25
J. Betzwieser,
18
P. T. Beyersdorf,
28
B. Bhawal,
15
I. A. Bilenko,
22
G. Billingsley,
15
R. Biswas,
52
E. Black,
15
K. Blackburn,
15
L. Blackburn,
18
D. Blair,
51
B. Bland,
16
J. Bogenstahl,
41
L. Bogue,
17
R. Bork,
15
V. Boschi,
15
S. Bose,
54
P. R. Brady,
52
V. B. Braginsky,
22
J. E. Brau,
44
M. Brinkmann,
2
A. Brooks,
38
D. A. Brown,
15,7
A. Bullington,
31
A. Bunkowski,
2
A. Buonanno,
42
O. Burmeister,
2
D. Busby,
15
W. E. Butler,
45
R. L. Byer,
31
L. Cadonati,
18
G. Cagnoli,
41
J. B. Camp,
23
J. Cannizzo,
23
K. Cannon,
52
C. A. Cantley,
41
J. Cao,
18
L. Cardenas,
15
K. Carter,
17
M. M. Casey,
41
G. Castaldi,
47
C. Cepeda,
15
E. Chalkey,
41
P. Charlton,
10
S. Chatterji,
15
S. Chelkowski,
2
Y. Chen,
1
F. Chiadini,
46
D. Chin,
43
E. Chin,
51
J. Chow,
4
N. Christensen,
9
J. Clark,
41
P. Cochrane,
2
T. Cokelaer,
8
C. N. Colacino,
39
R. Coldwell,
40
R. Conte,
46
D. Cook,
16
T. Corbitt,
18
D. Coward,
51
D. Coyne,
15
J. D. E. Creighton,
52
T. D. Creighton,
15
R. P. Croce,
47
D. R. M. Crooks,
41
A. M. Cruise,
39
A. Cumming,
41
J. Dalrymple,
32
E. D’Ambrosio,
15
K. Danzmann,
37, 2
G. Davies,
8
D. DeBra,
31
J. Degallaix,
51
M. Degree,
31
T. Demma,
47
V. Dergachev,
43
S. Desai,
33
R. DeSalvo,
15
S. Dhurandhar,
14
M. D ́ıaz,
34
J. Dickson,
4
A. Di Credico,
32
G. Diederichs,
37
A. Dietz,
8
E. E. Doomes,
30
R. W. P. Drever,
5
J.-C. Dumas,
51
R. J. Dupuis,
15
J. G. Dwyer,
11
P. Ehrens,
15
E. Espinoza,
15
T. Etzel,
15
M. Evans,
15
T. Evans,
17
S. Fairhurst,
8, 15
Y. Fan,
51
D. Fazi,
15
M. M. Fejer,
31
L. S. Finn,
33
V. Fiumara,
46
N. Fotopoulos,
52
A. Franzen,
37
K. Y. Franzen,
40
A. Freise,
39
R. Frey,
44
T. Fricke,
45
P. Fritschel,
18
V. V. Frolov,
17
M. Fyffe,
17
V. Galdi,
47
K. S. Ganezer,
6
J. Garofoli,
16
I. Gholami,
1
J. A. Giaime,
17, 19
S. Giampanis,
45
K. D. Giardina,
17
K. Goda,
18
E. Goetz,
43
L. Goggin,
15
G. Gonz ́alez,
19
S. Gossler,
4
A. Grant,
41
S. Gras,
51
C. Gray,
16
M. Gray,
4
J. Greenhalgh,
27
A. M. Gretarsson,
12
R. Grosso,
34
H. Grote,
2
S. Grunewald,
1
M. Guenther,
16
R. Gustafson,
43
B. Hage,
37
D. Hammer,
52
C. Hanna,
19
J. Hanson,
17
J. Harms,
2
G. Harry,
18
E. Harstad,
44
T. Hayler,
27
J. Heefner,
15
I. S. Heng,
41
A. Heptonstall,
41
M. Heurs,
2
M. Hewitson,
2
S. Hild,
37
E. Hirose,
32
D. Hoak,
17
D. Hosken,
38
J. Hough,
41
E. Howell,
51
D. Hoyland,
39
S. H. Huttner,
41
D. Ingram,
16
E. Innerhofer,
18
M. Ito,
44
Y. Itoh,
52
A. Ivanov,
15
D. Jackrel,
31
B. Johnson,
16
W. W. Johnson,
19
D. I. Jones,
48
G. Jones,
8
R. Jones,
41
L. Ju,
51
P. Kalmus,
11
V. Kalogera,
25
D. Kasprzyk,
39
E. Katsavounidis,
18
K. Kawabe,
16
S. Kawamura,
24
F. Kawazoe,
24
W. Kells,
15
D. G. Keppel,
15
F. Ya. Khalili,
22
C. Kim,
25
P. King,
15
J. S. Kissel,
19
S. Klimenko,
40
K. Kokeyama,
24
V. Kondrashov,
15
R. K. Kopparapu,
19
D. Kozak,
15
B. Krishnan,
1
P. Kwee,
37
P. K. Lam,
4
M. Landry,
16
B. Lantz,
31
A. Lazzarini,
15
B. Lee,
51
M. Lei,
15
J. Leiner,
54
V. Leonhardt,
24
I. Leonor,
44
K. Libbrecht,
15
P. Lindquist,
15
N. A. Lockerbie,
49
M. Longo,
46
M. Lormand,
17
M. Lubinski,
16
H. L ̈uck,
37, 2
B. Machenschalk,
1
M. MacInnis,
18
M. Mageswaran,
15
K. Mailand,
15
M. Malec,
37
V. Mandic,
15
S. Marano,
46
S. M ́arka,
11
J. Markowitz,
18
E. Maros,
15
I. Martin,
41
J. N. Marx,
15
K. Mason,
18
L. Matone,
11
V. Matta,
46
N. Mavalvala,
18
R. McCarthy,
16
D. E. McClelland,
4
S. C. McGuire,
30
M. McHugh,
21
K. McKenzie,
4
J. W. C. McNabb,
33
S. McWilliams,
23
T. Meier,
37
A. Melissinos,
45
G. Mendell,
16
R. A. Mercer,
40
S. Meshkov,
15
E. Messaritaki,
15
C. J. Messenger,
41
D. Meyers,
15
E. Mikhailov,
18
S. Mitra,
14
V. P. Mitrofanov,
22
G. Mitselmakher,
40
R. Mittleman,
18
O. Miyakawa,
15
S. Mohanty,
34
G. Moreno,
16
K. Mossavi,
2
C. MowLowry,
4
A. Moylan,
4
D. Mudge,
38
G. Mueller,
40
S. Mukherjee,
34
H. M ̈uller-Ebhardt,
2
J. Munch,
38
P. Murray,
41
E. Myers,
16
J. Myers,
16
T. Nash,
15
G. Newton,
41
A. Nishizawa,
24
F. Nocera,
15
K. Numata,
23
B. O’Reilly,
17
R. O’Shaughnessy,
25
D. J. Ottaway,
18
H. Overmier,
17
B. J. Owen,
33
Y. Pan,
42
M. A. Papa,
1, 52
V. Parameshwaraiah,
16
C. Parameswariah,
17
P. Patel,
15
M. Pedraza,
15
S. Penn,
13
V. Pierro,
47
I. M. Pinto,
47
M. Pitkin,
41
H. Pletsch,
2
M. V. Plissi,
41
F. Postiglione,
46
R. Prix,
1
V. Quetschke,
40
F. Raab,
16
D. Rabeling,
4
H. Radkins,
16
R. Rahkola,
44
N. Rainer,
2
M. Rakhmanov,
33
K. Rawlins,
18
S. Ray-Majumder,
52
V. Re,
39
T. Regimbau,
8
H. Rehbein,
2
S. Reid,
41
D. H. Reitze,
40
L. Ribichini,
2
R. Riesen,
17
K. Riles,
43
B. Rivera,
16
N. A. Robertson,
15, 41
C. Robinson,
8
E. L. Robinson,
39
S. Roddy,
17
A. Rodriguez,
19
A. M. Rogan,
54
J. Rollins,
11
J. D. Romano,
8
J. Romie,
17
R. Route,
31
S. Rowan,
41
A. R ̈udiger,
2
L. Ruet,
18
P. Russell,
15
K. Ryan,
16
S. Sakata,
24
M. Samidi,
15
L. Sancho de la Jordana,
36
V. Sandberg,
16
G. H. Sanders,
15
V. Sannibale,
15
S. Saraf,
26
P. Sarin,
18
B. S. Sathyaprakash,
8
S. Sato,
24
P. R. Saulson,
32
R. Savage,
16
P. Savov,
7
A. Sazonov,
40
S. Schediwy,
51
R. Schilling,
2
R. Schnabel,
2
R. Schofield,
44
B. F. Schutz,
1, 8
P. Schwinberg,
16
S. M. Scott,
4
A. C. Searle,
4
B. Sears,
15
F. Seifert,
2
D. Sellers,
17
A. S. Sengupta,
8
P. Shawhan,
42
D. H. Shoemaker,
18
A. Sibley,
17
J. A. Sidles,
50
X. Siemens,
15, 7
D. Sigg,
16
S. Sinha,
31
A. M. Sintes,
36, 1
B. J. J. Slagmolen,
4
J. Slutsky,
19
J. R. Smith,
2
M. R. Smith,
15
K. Somiya,
2, 1
K. A. Strain,
41
D. M. Strom,
44
A. Stuver,
33
T. Z. Summerscales,
3
K.-X. Sun,
31
M. Sung,
19
P. J. Sutton,
15
H. Takahashi,
1
D. B. Tanner,
40
M. Tarallo,
15
R. Taylor,
15
R. Taylor,
41
J. Thacker,
17
K. A. Thorne,
33
K. S. Thorne,
7
A. Th ̈uring,
37
K. V. Tokmakov,
41
C. Torres,
34
C. Torrie,
41
G. Traylor,
17
M. Trias,
36
W. Tyler,
15
D. Ugolini,
35
C. Ungarelli,
39
K. Urbanek,
31
H. Vahlbruch,
37
M. Vallisneri,
7
C. Van Den Broeck,
8
M. van Putten,
18
M. Varvella,
15
S. Vass,
15
A. Vecchio,
39
J. Veitch,
41
P. Veitch,
38
A. Villar,
15
C. Vorvick,
16
S. P. Vyachanin,
22
S. J. Waldman,
15
L. Wallace,
15
H. Ward,
41
R. Ward,
15
K. Watts,
17
D. Webber,
15
A. Weidner,
2
M. Weinert,
2
A. Weinstein,
15
R. Weiss,
18
S. Wen,
19
K. Wette,
4
J. T. Whelan,
1
D. M. Whitbeck,
33
S. E. Whitcomb,
15
B. F. Whiting,
40
S. Wiley,
6
C. Wilkinson,
16
P. A. Willems,
15
L. Williams,
40
B. Willke,
37, 2
I. Wilmut,
27
W. Winkler,
2
C. C. Wipf,
18
S. Wise,
40
A. G. Wiseman,
52
G. Woan,
41
D. Woods,
52
R. Wooley,
17
J. Worden,
16
W. Wu,
40
I. Yakushin,
17
H. Yamamoto,
15
Z. Yan,
51
S. Yoshida,
29
N. Yunes,
33
M. Zanolin,
18
J. Zhang,
43
L. Zhang,
15
C. Zhao,
51
N. Zotov,
20
M. Zucker,
18
H. zur M ̈uhlen,
37
and J. Zweizig
15
(The LIGO Scientific Collaboration, http://www.ligo.org)
M. Kramer
55
and A. G. Lyne
55
1
Albert-Einstein-Institut, Max-Planck-Institut f ̈ur Gra
vitationsphysik, D-14476 Golm, Germany
2
Albert-Einstein-Institut, Max-Planck-Institut f ̈ur Gra
vitationsphysik, D-30167 Hannover, Germany
3
Andrews University, Berrien Springs, MI 49104 USA
4
Australian National University, Canberra, 0200, Australi
a
5
California Institute of Technology, Pasadena, CA 91125, US
A
6
California State University Dominguez Hills, Carson, CA 90
747, USA
7
Caltech-CaRT, Pasadena, CA 91125, USA
8
Cardiff University, Cardiff, CF2 3YB, United Kingdom
9
Carleton College, Northfield, MN 55057, USA
10
Charles Sturt University, Wagga Wagga, NSW 2678, Australia
11
Columbia University, New York, NY 10027, USA
12
Embry-Riddle Aeronautical University, Prescott, AZ 86301
USA
13
Hobart and William Smith Colleges, Geneva, NY 14456, USA
14
Inter-University Centre for Astronomy and Astrophysics, P
une - 411007, India
15
LIGO - California Institute of Technology, Pasadena, CA 911
25, USA
16
LIGO Hanford Observatory, Richland, WA 99352, USA
17
LIGO Livingston Observatory, Livingston, LA 70754, USA
18
LIGO - Massachusetts Institute of Technology, Cambridge, M
A 02139, USA
19
Louisiana State University, Baton Rouge, LA 70803, USA
20
Louisiana Tech University, Ruston, LA 71272, USA
21
Loyola University, New Orleans, LA 70118, USA
22
Moscow State University, Moscow, 119992, Russia
23
NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
24
National Astronomical Observatory of Japan, Tokyo 181-858
8, Japan
25
Northwestern University, Evanston, IL 60208, USA
26
Rochester Institute of Technology, Rochester, NY 14623, US
A
27
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX1
1 0QX United Kingdom
28
San Jose State University, San Jose, CA 95192, USA
29
Southeastern Louisiana University, Hammond, LA 70402, USA
30
Southern University and A&M College, Baton Rouge, LA 70813,
USA
31
Stanford University, Stanford, CA 94305, USA
32
Syracuse University, Syracuse, NY 13244, USA
33
The Pennsylvania State University, University Park, PA 168
02, USA
34
The University of Texas at Brownsville and Texas Southmost C
ollege, Brownsville, TX 78520, USA
35
Trinity University, San Antonio, TX 78212, USA
36
Universitat de les Illes Balears, E-07122 Palma de Mallorca
, Spain
37
Universit ̈at Hannover, D-30167 Hannover, Germany
38
University of Adelaide, Adelaide, SA 5005, Australia
39
University of Birmingham, Birmingham, B15 2TT, United King
dom
40
University of Florida, Gainesville, FL 32611, USA
41
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
42
University of Maryland, College Park, MD 20742 USA
43
University of Michigan, Ann Arbor, MI 48109, USA
44
University of Oregon, Eugene, OR 97403, USA
45
University of Rochester, Rochester, NY 14627, USA
46
University of Salerno, 84084 Fisciano (Salerno), Italy
47
University of Sannio at Benevento, I-82100 Benevento, Ital
y
48
University of Southampton, Southampton, SO17 1BJ, United K
ingdom
49
University of Strathclyde, Glasgow, G1 1XQ, United Kingdom
50
University of Washington, Seattle, WA, 98195
2
51
University of Western Australia, Crawley, WA 6009, Austral
ia
52
University of Wisconsin-Milwaukee, Milwaukee, WI 53201, U
SA
53
Vassar College, Poughkeepsie, NY 12604
54
Washington State University, Pullman, WA 99164, USA
55
University of Manchester, Jodrell Bank Observatory, Maccl
esfield, SK11 9DL, United Kingdom
(Dated: February 5, 2008)
We present upper limits on the gravitational wave emission f
rom 78 radio pulsars based on data
from the third and fourth science runs of the LIGO and GEO 600 g
ravitational wave detectors. The
data from both runs have been combined coherently to maximis
e sensitivity. For the first time
pulsars within binary (or multiple) systems have been inclu
ded in the search by taking into account
the signal modulation due to their orbits. Our upper limits a
re therefore the first measured for
56 of these pulsars. For the remaining 22, our results improv
e on previous upper limits by up to
a factor of 10. For example, our tightest upper limit on the gr
avitational strain is 2
.
6
×
10
25
for
PSR J1603
7202, and the equatorial ellipticity of PSR J2124
3358 is less than 10
6
. Furthermore,
our strain upper limit for the Crab pulsar is only 2.2 times gr
eater than the fiducial spin-down limit.
PACS numbers: 04.80.Nn, 95.55.Ym, 97.60.Gb, 07.05.Kf
I. INTRODUCTION
This paper details the results of a search for gravita-
tional wave signals from known radio pulsars in data from
the third and fourth LIGO and GEO 600 science runs (de-
noted S3 and S4). These runs were carried out from 31
st
October 2003 to 9
th
January 2004 and 22
nd
February to
23
rd
March, 2005 respectively. We have applied, and ex-
tended, the search technique of Dupuis and Woan [1] to
generate upper limits on the gravitational wave ampli-
tude from a selection of known radio pulsars, and infer
upper limits on their equatorial ellipticities. The work is
a natural extension our previous work given in Refs. [2, 3].
A. Motivation
To emit gravitational waves a pulsar must have some
mass (or mass-current) asymmetry around its rotation
axis. This can be achieved through several mechanisms
such as elastic deformations of the solid crust or core
or distortion of the entire star by an extremely strong
misaligned magnetic field (see Sec. III of Ref. [4] for a
recent review). Such mechanisms generally result in a
triaxial neutron star which, in the quadrupole approxi-
mation and with rotation and angular momentum axes
aligned, would produce gravitational waves at twice the
rotation frequency. These waves would have a charac-
teristic strain amplitude at the Earth (assuming optimal
orientation of the rotation axis) of
h
0
=
16
π
2
G
c
4
εI
zz
ν
2
r
,
(1.1)
where
ν
is the neutron star’s spin frequency,
I
zz
its prin-
cipal moment of inertia,
ε
= (
I
xx
I
yy
)
/I
zz
its equatorial
ellipticity, and
r
its distance from Earth [5].
A rotating neutron star may also emit gravitational
waves at frequencies other than 2
ν
. For instance, if the
star is undergoing free precession there will be gravita-
tional wave emission at (or close to) both
ν
and 2
ν
[6].
In general such a precession would modulate the time-
of-arrival of the radio pulses. No strong evidence of such
a modulation is seen in any of the pulsars within our
search band, although it might go unnoticed by radio
astronomers, either because the modulation is small (as
would be the case if the precession is occurring about
an axis close to the pulsar beam axis) or because the
period of the modulation is very long. However, this
misalignment and precession will be quickly damped un-
less sustained by some mechanism (e.g. Ref. [7]), and
even with such a mechanism calculations give strain am-
plitudes which would probably be too low compared to
LIGO sensitivities [7, 8]. For these reasons, and for the
reason discussed in
§
III, we restrict our search to twice
the rotation frequency. Of course, it cannot be ruled out
that there are in fact other gravitational wave compo-
nents, perhaps caused either by a stronger than expected
precession excitation mechanism or by an event in the
pulsar’s recent past that has set it into a precessional
motion which has not yet decayed away. A search for
gravitational waves from the Crab pulsar at frequencies
other than twice the rotation frequency is currently under
way and will be presented elsewhere.
Known pulsars provide an enticing target for gravi-
tational wave searches as their positions and frequencies
are generally well-known through radio or X-ray observa-
tions. As a result the signal search covers a much smaller
parameter space than is necessary when searching for sig-
nals from unknown sources, giving a lower significance
threshold. In addition, the deterministic nature of the
waves allows a building up of the signal-to-noise ratio by
observing coherently for a considerable time. The main
drawback in a search for gravitational waves from the
majority of known pulsars is that the level of emission
is likely to be lower than can be detected with current
detector sensitivities.
Using existing radio measurements, and some reason-
able assumptions, it is possible to set an upper limit on
the gravitational wave amplitude from a pulsar based
purely on energy conservation arguments. If one assumes
3
that the pulsar is an isolated rigid body and that the
observed spin-down of the pulsar is due to the loss of
rotational kinetic energy as gravitational radiation (i.e
.,
d
E
rot
/
d
t
= 4
π
2
I
zz
ν
̇
ν
) then the gravitational wave ampli-
tude at the Earth (assuming optimal orientation of the
rotation axis) would be
h
sd
=

5
2
GI
zz
|
̇
ν
|
c
3
r
2
ν

1
/
2
.
(1.2)
Of course these assumptions may not hold, but it would
be surprising if neutron stars radiated significantly more
gravitational energy than this. With these uncertainties
in mind, searches such as the one described in this paper
place
direct
upper limits on gravitational wave emission
from rotating neutron stars, and these limits are already
approaching the regime of astrophysical interest.
B. Previous results
Before the advent of large-scale interferometric detec-
tors, there was only a limited ability to search for gravi-
tational waves from known pulsars. Resonant mass grav-
itational wave detectors are only sensitive in a relatively
narrow band around their resonant frequency and so can-
not be used to target objects radiating outside that band.
A specific attempt to search for gravitational waves from
the Crab pulsar at a frequency of
60 Hz was, however,
made with a specially designed aluminium quadrupole
antenna [9, 10] giving a 1
σ
upper limit of
h
0
2
×
10
22
.
A search for gravitational waves from what was then
the fastest millisecond pulsar, PSR J1939+2134, was con-
ducted by Hough
et al
[11] using a split bar detector,
producing an upper limit of
h
0
<
10
20
.
The first pulsar search using interferometer data was
carried out with the prototype 40 m interferometer at
Caltech by Hereld [12]. The search was again for gravita-
tional waves from PSR J1939+2134, and produced upper
limits of
h
0
<
3
.
1
×
10
17
and
h
0
<
1
.
5
×
10
17
for the first
and second harmonics of the pulsar’s rotation frequency.
A much larger sample of pulsars is accessible to broad-
band interferometers. As of the beginning of 2005 the
Australia Telescope National Facility (ATNF) online pul-
sar catalogue [13] listed
1
154 millisecond and young pul-
sars, all with rotation frequencies
>
25 Hz (gravitational
wave frequency
>
50 Hz) that fall within the design
band of the LIGO and GEO 600 interferometers, and
the search for their gravitational waves has developed
rapidly since the start of data-taking runs in 2002. Data
from the first science run (S1) were used to perform
a search for gravitational waves at twice the rotation
frequency from PSR J1939+2134 [2]. Two techniques
1
The catalogue is continually updated and as such now contain
s
more objects.
were used in this search: one a frequency domain, fre-
quentist search, and the other a time domain, Bayesian
search which gave a 95% credible amplitude upper limit
of 1
.
4
×
10
22
, and an ellipticity upper limit of 2
.
9
×
10
4
assuming
I
zz
= 10
38
kg m
2
.
Analysis of data from the LIGO S2 science run set up-
per limits on the gravitational wave amplitude from 28
radio pulsars [3]. To do this, new radio timing data were
obtained to ensure the pulsars’ rotational phases could be
predicted with the necessary accuracy and to check that
none of the pulsars had glitched. These data gave strain
upper limits as low as a few times 10
24
, and several el-
lipticity upper limits less than 10
5
. The Crab pulsar
was also studied in this run, giving an upper limit a fac-
tor of
30 greater than the spin-down limit considered
above. Prior to this article these were the most sensitive
studies made. Preliminary results for the same 28 pul-
sars using S3 data were given in Dupuis (2004) [14], and
these are expanded below.
In addition to the above, data from the LIGO S2
run have been used to perform an all-sky (i.e., non-
targeted) search for continuous wave signals from isolated
sources, and a search for a signal from the neutron star
within the binary system Sco-X1 [4]. An all-sky continu-
ous wave search using the distributed computing project
Einstein@home
2
has also been performed on S3 data [17].
These searches use the same search algorithms, are fully
coherent and are ongoing using data from more recent
(and therefore more sensitive) runs. Additional contin-
uous wave searches using incoherent techniques are also
being performed on LIGO data [18, 19].
Unfortunately the pulsar population is such that most
have spin frequencies that fall below the sensitivity band
of current detectors. In the future, the low-frequency sen-
sitivity of VIRGO [15] and Advanced LIGO [16] should
allow studies of a significantly larger sample of pulsars.
C. The signal
Following convention, we model the observed phase
evolution of a pulsar using a Taylor expansion about a
fixed epoch time
t
0
:
φ
(
T
) =
φ
0
+ 2
π
n
ν
0
(
T
t
0
) +
1
2
̇
ν
0
(
T
t
0
)
2
+
1
6
̈
ν
0
(
T
t
0
)
3
+
...
o
,
(1.3)
where
φ
0
is the initial (epoch) spin phase,
ν
0
and its time
derivatives are the pulsar spin frequency and spin-down
coefficients at
t
0
, and
T
is the pulsar proper time.
The expected signal in an interferometer from a triaxial
2
http://einstein.phys.uwm.edu
4
pulsar is
h
(
t
) =
1
2
F
+
(
t
;
ψ
)
h
0
(1 + cos
2
ι
) cos 2
φ
(
t
) +
F
×
(
t
;
ψ
)
h
0
cos
ι
sin 2
φ
(
t
)
,
(1.4)
where
φ
(
t
) is the phase evolution in the detector time
t
,
F
+
and
F
×
are the detector antenna patterns for the
plus and cross polarisations of gravitational waves,
ψ
is
the wave polarisation angle, and
ι
is the angle between
the rotation axis of the pulsar and the line-of-sight. A
gravitational wave impinging on the interferometer will
be modulated by Doppler, time delay and relativistic ef-
fects caused by the motions of the Earth and other bodies
in the solar system. Therefore we need to transform the
‘arrival time’ of a wave-crest at the detector,
t
, to its
arrival time at the solar system barycentre (SSB)
t
b
via
t
b
=
t
+
δt
=
t
+
r

ˆ
n
c
+ ∆
E
+ ∆
S
,
(1.5)
where
r
is the position of the detector with respect to the
SSB,
ˆ
n
is the unit vector pointing to the pulsar, ∆
E
is
the special relativistic Einstein delay, and ∆
S
is the
general relativistic Shapiro delay [20]. Although pulsars
can be assumed to have a large velocity with respect to
the SSB, it is conventional to ignore this Doppler term
and set
t
b
=
T
, as its proper motion is generally negli-
gible (see
§
VI A for cases where this assumption is not
the case). For pulsars in binary systems, there will be
additional time delays due to the binary orbit, discussed
in
§
III B.
II. INSTRUMENTAL PERFORMANCE IN S3/S4
The S3 and S4 runs used all three LIGO interferome-
ters (H1 and H2 at the Hanford Observatory in Washing-
ton, and L1 at the Livingston Observatory in Louisiana)
in the USA and the GEO 600 interferometer in Hannover,
Germany. GEO 600 did not run for all of S3, but had two
main data taking periods between which improvements
were made to its sensitivity. All these detectors had dif-
ferent duty factors and sensitivities.
A. LIGO
For S3 the H1 and H2 interferometers maintained rel-
atively high duty factors of 69
.
3% and 63
.
4% respec-
tively. The L1 interferometer was badly affected by an-
thropogenic seismic noise sources during the day and thus
had a duty factor of only 21
.
8%.
Between S3 and S4 the L1 interferometer was upgraded
with better seismic isolation. This greatly reduced the
amount of time the interferometer was thrown out of its
operational state by anthropogenic noise, and allowed it
to operate successfully during the day, with a duty factor
of 74
.
5% and a longest lock stretch of 18.7 h. The H1 and
H2 interferometers also both improved their duty factors
to 80
.
5% and 81
.
4%, with longest lock stretches of almost
a day. The typical strain sensitivities of all the interfer-
10
2
10
3
10
−23
10
−22
10
−21
10
−20
10
−19
10
−18
10
−17
10
−16
frequency (Hz)
amplitude spectral density h/Hz
1/2
LHO 4k
LLO 4k
LHO 2k
GEO600
FIG. 1: Median strain amplitude spectral density curves for
the LIGO and GEO 600 interferometers during the S4 run.
ometers during S4 can be seen in Fig. 1. This shows the
LIGO detectors reach their best sensitivities at about
150 Hz, whilst GEO 600 achieves its best sensitivity at
its tuned frequency of 1 kHz.
B. GEO 600
During S3 GEO 600 was operated as a dual-recycled
Michelson interferometer tuned to have greater sensi-
tivity to signals around 1 kHz. The first period of
GEO 600 participation in S3 was between 5
th
to 11
th
November 2003, called S3 I, during which the detec-
tor operated with a 95.1% duty factor. Afterwards,
GEO 600 was taken offline to allow further commission-
ing work aimed at improving sensitivity and stability.
Then from 30
th
December 2003 to 13
th
January 2004
GEO 600 rejoined S3, called S3 II, with an improved duty
factor of 98.7% and with more than one order of magni-
tude improvement in peak sensitivity. During S3 there
were five locks of longer than 24 hours and one lock longer
than 95 hours. For more information about the perfor-
mance of GEO 600 during S3 see Ref. [21].
GEO 600 participated in S4 from 22
nd
February to 24
th
March 2005, with a duty factor of 96.6%. It was oper-
ated in essentially the same optical configuration as in S3.
With respect to S3, the sensitivity was improved more
than an order of magnitude over a wide frequency range,
and close to two orders or magnitude around 100 Hz. For
more information about GEO 600 during S4 see Ref. [22].
5
C. Data quality
When a detector is locked on resonance and all con-
trol loops are in their nominal running states and there
are no on-site work activities that are known to compro-
mise the data, then the data are said to be
science mode
.
All science mode data are not of sufficient quality to be
analysed however, and may be flagged for exclusion. Ex-
amples of such data quality flags are ones produced for
epochs of excess seismic noise, and the flagging of data
corrupted by overflows of photodiode ADCs. For this
analysis we use all science mode data for which there is
no corresponding data quality flag. For S3 this gives ob-
servation times of 45.5 days for H1; 42.1 days for H2; and
13.4 days for L1. For S4 this gives observation times of
19.4 days for H1; 22.5 days for H2; and 17.1 days for L1.
III. THE SEARCH METHOD
Our search method involves heterodyning the data us-
ing the phase model
φ
(
t
) to precisely unwind the phase
evolution of the expected signal, and has been discussed
in detail in Ref. [1]. After heterodyning, the data are
low-pass filtered, using a ninth order Butterworth filter
with a knee frequency of 0.5 Hz, and re-binned from the
raw data sample rate of 16 384 Hz to 1/60 Hz i.e., one
sample per minute. The motion of the detector within
the solar system modulates the signal and this is taken
into account within the heterodyne by using a time de-
lay given in Eq. (1.5), which transforms the signal to the
SSB. Signals from binary pulsar systems have an extra
modulation term in the signal, as discussed briefly below,
and these we targeted for the first time in S3/S4.
The search technique used here is currently only able to
target emission at twice the pulsar’s rotation frequency.
Emission near the rotation frequency for a precessing
star is likely to be offset from the observed pulsation fre-
quency by some small factor dependent on unknown de-
tails of the stellar structure [7]. As our search technique
requires precise knowledge of the phase evolution of the
pulsar such an additional parameter cannot currently be
taken into account. For the emission at twice the rotation
frequency there is no extra parameter dependence on the
frequency and this is what our search was designed for.
We infer the pulsar signal parameters, denoted
a
=
(
h
0
0
,
cos
ι,ψ
), from their (Bayesian) posterior proba-
bility distribution (pdf) over this parameter space, as-
suming Gaussian noise. The data are broken up into time
segments over which the noise can be assumed stationary
and we analytically marginalise over the unknown noise
floor, giving a Student-t likelihood for the parameters for
each segment (see Ref. [1] for the method). Combining
the segments gives an overall likelihood of
p
(
{
B
k
}|
a
)
M
Y
j
P
j
i
=1
m
i
X
k
=1+
P
j
1
i
=1
m
i
(
ℜ{
B
k
}−ℜ{
y
k
}
)
2
+ (
ℑ{
B
k
}−ℑ{
y
k
}
)
2
m
j
,
(3.1)
where each
B
k
is a heterodyned sample with a sample
rate of one per minute,
M
is the number of segments
into which the whole data set has been cut,
m
j
is the
number of data points in the
j
th segment, and
y
k
, given
by
y
k
=
1
4
F
+
(
t
k
;
ψ
)
h
0
(1+cos
2
ι
)
e
i
2
φ
0
i
2
F
×
(
t
k
;
ψ
)
h
0
cos
ιe
i
2
φ
0
,
(3.2)
is the gravitational wave signal model evaluated at
t
k
,
the time corresponding to the
k
’th heterodyned sample.
In Ref. [3] the value of
m
j
was fixed at 30 to give 30 min
data segments, and data that was contiguous only on
shorter timescales, and which could not be fitted into
one of these segments, was thrown out. In the analy-
sis presented here we have allowed segment lengths to
vary from 5 to 30 min, so we maximise the number of 30-
minute segments whilst also allowing shorter segments at
the end of locked stretches to contribute. The likelihood
in Eq. (3.1) assumes that the data is stationary over each
of these 30 minute (or smaller) segments. This assump-
tion holds well for our data. Large outliers can also be
identified and vetoed from the data, for example those
at the beginning of a data segment caused by the impul-
sive ringing of the low-pass filter applied after the data
is heterodyned.
The prior probabilities for each of the parameters are
taken as uniform over their respective ranges. Upper lim-
its on
h
0
are set by marginalising the posterior over the
nuisance parameters and then calculating the
h
95%
0
value
that bounds the cumulative probability for the desired
credible limit of 95%:
0
.
95 =
Z
h
95%
0
0
p
(
h
0
|{
B
k
}
)d
h
0
.
(3.3)
A. Combining data
In the search of Ref. [3] the combined data from the
three LIGO interferometers were used to improve the sen-
sitivity of the search. This was done by forming the joint
6
likelihood from the three
independent
data sets:
p
(
B
k
|
a
)
Joint
=
p
(
B
k
|
a
)
H1
.p
(
B
k
|
a
)
H2
.p
(
B
k
|
a
)
L1
.
(3.4)
This is valid provided the data acquisition is coherent
between detectors and supporting evidence for this is
presented in
§
V. It is of course a simple matter to ex-
tend Eq. (3.4) to include additional likelihood terms from
other detectors, such as GEO 600.
In this analysis we also combine data sets from two
different science runs. This is appropriate because S3
and S4 had comparable sensitivities over a large portion
of the spectrum. Provided the data sets maintain phase
coherence between runs, this combination can simply be
achieved by concatenating the data sets from the two
runs together for each detector.
An example of the posterior pdfs for the four un-
known pulsar parameters of PSR J0024
7204C (each
marginalised over the three other parameters) is shown
in Fig. 2. The pdfs in Fig. 2 are from the joint analysis
0
2
4
6
x 10
−24
0
2
4
6
8
10
x 10
23
h
0
prob. density
0
2
4
6
0.1
0.2
0.3
0.4
0.5
0.6
φ
0
−1
0
1
0.2
0.4
0.6
0.8
1
cos
ι
−0.5
0
0.5
0.45
0.5
0.55
0.6
0.65
0.7
ψ
h
0
95%
FIG. 2: The marginalised posterior pdfs for the four unknown
pulsar parameters
h
0
,
φ
0
, cos
ι
and
ψ
, for PSR J0024
7204C
using the joint data from the three LIGO detectors over S3
and S4.
of the three LIGO detectors using the S3 and S4 data,
all combined coherently. The shaded area in the
h
0
pos-
terior shows the area containing 95% of the probability
as given by Eq. (3.3). In this example the posterior on
h
0
is peaked at
h
0
= 0, though any distribution that is
credibly close to zero is consistent with
h
0
= 0. Indeed
an upper limit can formally be set even when the bulk of
the probability is well away from zero (see the discussion
of hardware injections in
§
V).
B. Binary models
Our previous known pulsar searches [2, 3] have ex-
cluded pulsars within binary systems, despite the ma-
jority of pulsars within our detector band being in such
systems. To address this, we have included an additional
time delay to transform from the binary system barycen-
tre (BSB) to pulsar proper time, which is a stationary
reference frame with respect to the pulsar. The code for
this is based on the widely used radio pulsar timing soft-
ware TEMPO [23]. The algorithm and its testing are
discussed more thoroughly in Ref. [24].
There are five principal parameters describing a Kep-
lerian orbit: the time of periastron,
T
0
, the longitude of
periastron,
ω
0
, the eccentricity,
e
, the period,
P
b
, and the
projected semi-major axis,
x
=
a
sin
i
. These describe the
majority of orbits very well, although to fully describe the
orbit of some pulsars requires additional relativistic pa-
rameters. The basic transformation and binary models
below are summarised by Taylor and Weisberg [20] and
Lange
et al.
[25], and are those used in TEMPO. The
transformation from SSB time
t
b
to pulsar proper time
T
follows the form of Eq. (1.5) and is
t
b
=
T
+ ∆
R
+ ∆
E
+ ∆
S
,
(3.5)
where ∆
R
is the Roemer time delay giving the propaga-
tion time across the binary orbit, ∆
E
is the Einstein delay
and gives gravitational redshift and time dilation correc-
tions, and ∆
S
is the Shapiro delay and gives the general
relativistic correction (see Ref. [20] for definitions of th
ese
delays).
The majority of binary pulsars can be described by
three orbital models: the Blandford-Teukolsky (BT)
model, the low eccentricity (ELL1) model, and the
Damour-Deruelle (DD) model (see Refs. [20, 23, 25] for
further details of these models). These different models
make different assumptions about the system and/or are
specialised to account for certain system features. For ex-
ample, the ELL1 model is used in cases where the eccen-
tricity is very small, and therefore periastron is very hard
to define, in which case the time and longitude of perias-
tron will be highly correlated and have to be reparame-
terised to the Laplace-Lagrange parameters [25]. When a
binary pulsar’s parameters are estimated from radio ob-
servations using TEMPO the different models are used
accordingly. These models can be used within our search
to calculate all the associated time delays and therefore
correct the signal to the pulsar proper time, provided we
have accurate model parameters for the pulsar.
IV. PULSAR SELECTION
The noise floor of the LIGO detectors increases rapidly
below about 50 Hz, so pulsar targets were primarily se-
lected on their frequency. The choice of a 50 Hz grav-
itational wave frequency cut-off (pulsar spin frequency
of 25 Hz) is somewhat arbitrary, but it also loosely re-
flects the split between the population of fast (millisec-
ond/recycled and young) pulsars and slow pulsars.
All 154 pulsars with spin frequencies
>
25 Hz were
taken from the ATNF online pulsar catalogue [13] (de-
scribed in Ref. [26]). The accuracy of these parameters
7
varies for each pulsar and is dependent on the time span,
density of observations and the noise level of the timing
observations. Clearly it is important to ensure that pa-
rameter uncertainties do not lead to unacceptable phase
errors in the heterodyne. Pulsars are not perfect clocks,
so the epoch of the parameters is also important as more
recent measurements will better reflect the current state
of the pulsar. Importantly, there is near-continuous mon-
itoring of the Crab pulsar at Jodrell Bank Observatory,
and as such its parameters are continuously updated [27].
Precise knowledge of the phase evolution of each target
pulsar is vital for our analysis, and possible effects that
may lead to a departure from the simple second-order
Taylor expansion are discussed below.
A. Pulsar timing
Using TEMPO, we obtained the parameters of 75 pul-
sars from the regular observation programs carried out
at Jodrell Bank Observatory and the Parkes Telescope
(see Ref. [28] for details of the techniques used for this).
For 37 of these the timings spanned the period of S3.
These same model parameters were used to extrapolate
the pulsar phases to the period of S4. The effect of pa-
rameter uncertainties on this extrapolation is discussed
in
§
IV B, but is only important in its effect on the extrap-
olated phase. For those pulsars observed during S3 the
interpolation is taken to be free from significant error.
The parameters for 16 additional pulsars (for which
new timings were not available) were taken directly from
the ATNF catalogue, selected using criteria described
in the following section. The parameters of the X-ray
pulsar PSR J0537
6910 were taken from Ref. [29] and
those for the Crab pulsar from the Jodrell Bank monthly
ephemeris [27]. The remaining 61 pulsars (from the orig-
inal list of 154) were not timed with sufficient confidence
and were excluded from the search. This included many
of the newly discovered pulsars (for example the 21 mil-
lisecond pulsars in the Terzan 5 globular cluster [30]) for
which accurate timing solutions have yet to be published.
We therefore had a catalogue of 93 timed pulsars for our
gravitational wave search.
B. Error propagation in source parameters
The impact of parameter uncertainties on the search
was assessed for both the S3 and S4 runs. At some level
there are positional, frequency and frequency derivative
uncertainties for all the target pulsars, and for pulsars
in binary system there are also uncertainties associated
with all the binary orbital parameters. Some of these
uncertainties are correlated, for example the error on fre-
quency could affect the accuracy of the first frequency
derivative, and the binary time of periastron and longi-
tude of periastron are also highly correlated.
We took a ‘worst-case scenario’ approach by adding
and subtracting the quoted uncertainties from the best-
fit values of all the parameters to determine the com-
bination which gave a maximum phase deviation, when
propagated over the period of the run (either S3 or S4),
from the best fit phase value calculated over the same
time period. For example if we assume
φ
(
t
S3
) given by
Eq. (1.3) (ignoring for simplicity the
φ
0
and ̈
ν
terms)
is the best fit phase over the time span of S3,
t
S3
, the
maximum phase uncertainty is
φ
err
= max
h
φ
(t
S3
)
±
2
π
n
(
ν
±
σ
ν
)(t
S3
±
σ
t
S3
+
1
2
( ̇
ν
±
σ
̇
ν
)(
t
S3
±
σ
t
S3
)
2
+
...
o
i
,
(4.1)
where the
σ
s are the uncertainties on the individual pa-
rameters. Correlations between the parameters mean
that this represents an upper limit to the maximum phase
uncertainty, sometimes greatly overestimating its true
value.
There are 12 pulsars with overall phase uncertainty
>
30
in S3, which we take as the threshold of ac-
ceptability. A 30
phase drift could possibly give a
factor of
1
cos 30
= 0
.
13 in loss of sensitiv-
ity for a signal. Nine of these are in binary sys-
tems (PSRs J0024
7204H, J0407+1607, J0437
4715,
J1420
5625, J1518+0205B, J1709+2313, J1732
5049,
J1740
5340 and J1918
0642) and in five of these
T
0
and
ω
0
contribute most to the phase uncertainty. For the
three isolated pulsars (PSRs J0030+0451, J0537
6910,
and J1721
2457) the phase error is dominated by uncer-
tainties in frequency and/or position.
Applying the same criterion to the time-span of S4
we find that PSR J1730
2304 rises above the limit. For
this pulsar its parameter uncertainties do not affect it
for the S3 analysis as it was timed over this period, how-
ever when extrapolating over the time of the S4 run the
uncertainties become non-negligible.
In total there are 13 pulsars rejected over the combined
run. This highly conservative parameter check reduces
our 93 candidate pulsars to 80.
C. Timing noise
Pulsars are generally very stable rotators, but there
are phenomena which can cause deviations in this stabil-
ity, generically known as timing noise. The existence of
timing noise has been clear since the early days of pul-
sar astronomy and appears as a random walk in phase,
frequency or frequency derivative of the pulsar about the
regular spin-down model given in Eq. (1.3) [31]. The
strength of this effect was quantified in Ref. [31] as an
activity parameter
A
, referenced to that of the Crab pul-
sar, and in Ref. [32] as a
stability parameter
8
.
A
is
based on the logarithm of the ratio of the rms residual
phase of the pulsar, after removal of the timing model, to
that of the Crab pulsar over an approximately three-year
8