Upper limits on gravitational wave emission from 78 radio pulsars
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,
7,15
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,
2,37
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. Dı
́
az,
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. Gonza
́
lez,
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. Lu
̈
ck,
2,37
B. Machenschalk,
1
M. MacInnis,
18
M. Mageswaran,
15
K. Mailand,
15
M. Malec,
37
V. Mandic,
15
S. Marano,
46
S. Ma
́
rka,
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. Mu
̈
ller-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. Ru
̈
diger,
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,
7,15
D. Sigg,
16
S. Sinha,
31
A. M. Sintes,
1,36
B. J. J. Slagmolen,
4
J. Slutsky,
19
J. R. Smith,
2
M. R. Smith,
15
K. Somiya,
1,2
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. Thu
̈
ring,
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
PHYSICAL REVIEW D
76,
042001 (2007)
1550-7998
=
2007
=
76(4)
=
042001(20)
042001-1
©
2007 The American Physical Society
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,
2,37
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 Mu
̈
hlen,
37
and J. Zweizig
15
(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
Andrews University, Berrien Springs, Michigan 49104 USA
4
Australian National University, Canberra, 0200, Australia
5
California Institute of Technology, Pasadena, California 91125, USA
6
California State University Dominguez Hills, Carson, California 90747, USA
7
Caltech-CaRT, Pasadena, California 91125, USA
8
Cardiff University, Cardiff, CF2 3YB, United Kingdom
9
Carleton College, Northfield, Minnesota 55057, USA
10
Charles Sturt University, Wagga Wagga, NSW 2678, Australia
11
Columbia University, New York, New York 10027, USA
12
Embry-Riddle Aeronautical University, Prescott, Arizona 86301 USA
13
Hobart and William Smith Colleges, Geneva, New York 14456, USA
14
Inter-University Centre for Astronomy and Astrophysics, Pune – 411007, India
15
LIGO –California Institute of Technology, Pasadena, California 91125, USA
16
LIGO Hanford Observatory, Richland, Washington 99352, USA
17
LIGO Livingston Observatory, Livingston, Louisiana 70754, USA
18
LIGO –Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
19
Louisiana State University, Baton Rouge, Louisiana 70803, USA
20
Louisiana Tech University, Ruston, Louisiana 71272, USA
21
Loyola University, New Orleans, Louisiana 70118, USA
22
Moscow State University, Moscow, 119992, Russia
23
NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
24
National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
25
Northwestern University, Evanston, Illinois 60208, USA
26
Rochester Institute of Technology, Rochester, New York 14623, USA
27
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX United Kingdom
28
San Jose State University, San Jose, California 95192, USA
29
Southeastern Louisiana University, Hammond, Louisiana 70402, USA
30
Southern University and A&M College, Baton Rouge, Louisiana 70813, USA
31
Stanford University, Stanford, California 94305, USA
32
Syracuse University, Syracuse, New York 13244, USA
33
The Pennsylvania State University, University Park, Pennsylvania 16802, USA
34
The University of Texas at Brownsville and Texas Southmost College, Brownsville, Texas 78520, USA
35
Trinity University, San Antonio, Texas 78212, USA
36
Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
37
Universita
̈
t Hannover, D-30167 Hannover, Germany
38
University of Adelaide, Adelaide, SA 5005, Australia
39
University of Birmingham, Birmingham, B15 2TT, United Kingdom
40
University of Florida, Gainesville, Florida 32611, USA
41
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
42
University of Maryland, College Park, Maryland 20742 USA
43
University of Michigan, Ann Arbor, Michigan 48109, USA
44
University of Oregon, Eugene, Oregon 97403, USA
45
University of Rochester, Rochester, New York 14627, USA
46
University of Salerno, 84084 Fisciano (Salerno), Italy
47
University of Sannio at Benevento, I-82100 Benevento, Italy
48
University of Southampton, Southampton, SO17 1BJ, United Kingdom
49
University of Strathclyde, Glasgow, G1 1XQ, United Kingdom
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
042001 (2007)
042001-2
50
University of Washington, Seattle, Washington, 98195, USA
51
University of Western Australia, Crawley, WA 6009, Australia
52
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA
53
Vassar College, Poughkeepsie, New York 12604, USA
54
Washington State University, Pullman, Washington 99164, USA
M. Kramer and A. G. Lyne
University of Manchester, Jodrell Bank Observatory, Macclesfield, SK11 9DL, United Kingdom
(Received 4 April 2007; revised manuscript received 20 June 2007; published 3 August 2007; publisher error corrected
29 February 2008)
We present upper limits on the gravitational wave emission from 78 radio pulsars based on data from
the third and fourth science runs of the LIGO and GEO 600 gravitational wave detectors. The data from
both runs have been combined coherently to maximize sensitivity. For the first time, pulsars within binary
(or multiple) systems have been included in the search by taking into account the signal modulation due to
their orbits. Our upper limits are therefore the first measured for 56 of these pulsars. For the remaining 22,
our results improve on previous upper limits by up to a factor of 10. For example, our tightest upper limit
on the gravitational 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
greater than the fiducial spin-down limit.
DOI:
10.1103/PhysRevD.76.042001
PACS numbers: 04.80.Nn, 07.05.Kf, 95.55.Ym, 97.60.Gb
I. INTRODUCTION
This paper details the results of a search for gravitational
wave signals from known radio pulsars in data from the
third and fourth LIGO and GEO 600 science runs (denoted
S3 and S4). These runs were carried out from 31 October
2003 to 9 January 2004 and from 22 February 2005 to 23
March 2005, respectively. We have applied, and extended,
the search technique of Dupuis and Woan [
1
] to generate
upper limits on the gravitational wave amplitude from a
selection of known radio pulsars, and infer upper limits on
their equatorial ellipticities. The work is a natural exten-
sion of 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 mag-
netic field (see Sec. III of Ref. [
4
] for a recent review). Such
mechanisms generally result in a triaxial neutron star
which, in the quadrupole approximation and with rotation
and angular momentum axes aligned, would produce
gravitational waves at twice the rotation frequency. These
waves would have a characteristic strain amplitude at the
Earth (assuming optimal orientation of the rotation axis) of
h
0
16
2
G
c
4
"I
zz
2
r
;
(1.1)
where
is the neutron star’s spin frequency,
I
zz
its princi-
pal moment of inertia,
"
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
. For instance, if the
star is undergoing free precession there will be gravita-
tional wave emission at (or close to) both
and
2
[
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 unless sustained by some mecha-
nism (e.g. Ref. [
7
]), and even with such a mechanism,
calculations give strain amplitudes which would probably
be too low compared to LIGO sensitivities [
7
,
8
]. For these
reasons, and for the reason discussed in Sec. 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 components, 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 gravita-
tional wave searches as their positions and frequencies are
generally well known through radio or x-ray observations.
As a result the signal search covers a much smaller pa-
rameter space than is necessary when searching for signals
from unknown sources, giving a lower significance thresh-
old. In addition, the deterministic nature of the waves
allows a building up of the signal-to-noise ratio by observ-
ing coherently for a considerable time. The main drawback
in a search for gravitational waves from the majority of
*
matthew@astro.gla.ac.uk
UPPER LIMITS ON GRAVITATIONAL WAVE EMISSION
...
PHYSICAL REVIEW D
76,
042001 (2007)
042001-3
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 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
2
I
zz
_
), then the gravitational wave amplitude at the
Earth (assuming optimal orientation of the rotation axis)
would be
h
sd
5
2
GI
zz
j
_
j
c
3
r
2
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 ap-
proaching 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 gravita-
tional waves from known pulsars. Resonant mass
gravitational wave detectors are only sensitive in a rela-
tively narrow band around their resonant frequency and so
cannot 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 aluminum quadrupole
antenna [
9
,
10
] giving a
1
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 conducted 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 gravi-
tational waves at twice the rotation frequency from PSR
J1939
2134
[
2
]. Two techniques were used in this
search: one a frequency domain, frequentist 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 upper
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 ellipticity upper
limits less than
10
5
. The Crab pulsar was also studied
in this run, giving an upper limit a factor 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 pulsars 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., nontargeted)
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 con-
tinuous wave search using the distributed computing
project
Einstein
@
home
2
has also been performed on
S3 data [
15
]. These searches use the same search algo-
rithms, are fully coherent and are ongoing using data from
more recent (and therefore more sensitive) runs. Additional
continuous wave searches using incoherent techniques are
also being performed on LIGO data [
16
,
17
].
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 sensi-
tivity of VIRGO [
18
] and Advanced LIGO [
19
] 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
:
T
0
2
f
0
T
t
0
1
2
_
0
T
t
0
2
1
6
0
T
t
0
3
...
g
;
(1.3)
where
0
is the initial (epoch) spin phase,
0
and its time
derivatives are the pulsar spin frequency and spin-down
coefficients at
t
0
, and
T
is the pulsar proper time.
1
The catalogue is continually updated and as such now con-
tains more objects.
2
http://einstein.phys.uwm.edu
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
042001 (2007)
042001-4
The expected signal in an interferometer from a triaxial
pulsar is
h
t
1
2
F
t
;
h
0
1
cos
2
cos2
t
F
t
;
h
0
cos
sin2
t
;
(1.4)
where
t
is the phase evolution in the detector time
t
,
F
and
F
are the detector antenna patterns for the plus and
cross polarizations of gravitational waves,
is the wave
polarization angle, and
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 effects 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 barycenter (SSB)
t
b
via
t
b
t
t
t
r
^
n
c
E
S
;
(1.5)
where
r
is the position of the detector with respect to the
SSB,
^
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 negligible (see Sec. 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 Sec. III B.
II. INSTRUMENTAL PERFORMANCE IN S3/S4
The S3 and S4 runs used all three LIGO interferometers
(H1 and H2 at the Hanford Observatory in Washington, and
L1 at the Livingston Observatory in Louisiana) in the U.S.
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 different duty factors
and sensitivities.
A. LIGO
For S3 the H1 and H2 interferometers maintained rela-
tively high duty factors of 69.3% and 63.4%, respectively.
The L1 interferometer was badly affected by anthropo-
genic 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 interferometers
during S4 can be seen in Fig.
1
. This shows the LIGO
detectors reach their best sensitivities at about 150 Hz,
while 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 sensitivity
to signals around 1 kHz. The first period of GEO 600
participation in S3 was between 5 and 11 November
2003, called S3 I, during which the detector operated
with a 95.1% duty factor. Afterwards, GEO 600 was taken
offline to allow further commissioning work aimed at
improving sensitivity and stability. Then from 30
December 2003 to 13 January 2004 GEO 600 rejoined
S3, called S3 II, with an improved duty factor of 98.7%
and with more than 1 order of magnitude 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 performance of GEO 600 during S3
see Ref. [
21
].
GEO 600 participated in S4 from 22 February to 24
March 2005, with a duty factor of 96.6%. It was operated 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 2 orders or magnitude around 100 Hz. For more infor-
mation about GEO 600 during S4 see Ref. [
22
].
C. Data quality
When a detector is locked on resonance and all control
loops are in their nominal running states and there are no
10
2
10
3
10
−23
10
−22
10
−21
10
−20
10
−19
10
−18
10
−17
freq
u
enc
y
(Hz)
a
m
p
lit
ud
e s
p
ectr
a
l
d
ensit
y
h/Hz
1/2
LHO 4k
LLO 4k
LHO 2k
GEO600
FIG. 1 (color online).
Median strain amplitude spectral density
curves for the LIGO and GEO 600 interferometers during the S4
run.
UPPER LIMITS ON GRAVITATIONAL WAVE EMISSION
...
PHYSICAL REVIEW D
76,
042001 (2007)
042001-5
on-site work activities that are known to compromise the
data, then the data are said to be
science mode
. All science
mode data are not of sufficient quality to be analyzed
however, and may be flagged for exclusion. Examples of
such data quality flags are ones produced for epochs of
excess seismic noise, and the flagging of data corrupted by
overflows of photodiode analogue-to-digital converters.
For this analysis we use all science mode data for which
there is no corresponding data quality flag. For S3 this
gives observation 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 using
the phase model
t
to precisely unwind the phase evo-
lution 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 rebinned 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 delay given in Eq. (
1.5
),
which transforms the signal to the SSB. Signals from
binary pulsar systems contain an extra modulation term,
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 frequency
by some small factor dependent on unknown details of the
stellar structure [
7
]. As our search technique requires pre-
cise 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
;
, from their (Bayesian) posterior probabil-
ity distribution function (pdf ) over this parameter space,
assuming Gaussian noise. The data are broken up into time
segments over which the noise can be assumed stationary
and we analytically marginalize over the unknown noise
floor, giving a Student’s t-likelihood for the parameters for
each segment (see Ref. [
1
] for the method). Combining the
segments gives an overall likelihood of
p
f
B
k
gj
a
/
Y
M
j
X
P
j
i
1
m
i
k
1
P
j
1
i
1
m
i
Re
f
B
k
g
Re
f
y
k
g
2
Im
f
B
k
g
Im
f
y
k
g
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
;
h
0
1
cos
2
e
i
2
0
i
2
F
t
k
;
h
0
cos
e
i
2
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 minute
data segments, and data that were contiguous only on
shorter time scales, and which could not be fitted into
one of these segments, were thrown out. In the analysis
presented here, we have allowed segment lengths to vary
from 5 to 30 minute, so we maximize the number of 30-
minute segments while also allowing shorter segments at
the end of locked stretches to contribute. The likelihood in
Eq. (
3.1
) assumes that the data are stationary over each of
these 30 minute (or smaller) segments. This assumption
holds well for our data. Large outliers can also be identified
and vetoed from the data, for example, those at the begin-
ning of a data segment caused by the impulsive ringing of
the low-pass filter applied after the data are heterodyned.
The prior probabilities for each of the parameters are
taken as uniform over their respective ranges. Upper limits
on
h
0
are set by marginalizing 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
jf
B
k
g
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
sensitivity of the search. This was done by forming the
joint likelihood from the three
independent
data sets:
p
B
k
j
a
Joint
p
B
k
j
a
H1
p
B
k
j
a
H2
p
B
k
j
a
L1
:
(3.4)
This is valid provided the data acquisition is coherent
between detectors, and supporting evidence for this is
presented in Sec. V. It is of course a simple matter to
extend 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.
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
042001 (2007)
042001-6
An example of the posterior pdfs for the four unknown
pulsar parameters of PSR
J0024
7204C
(each marginal-
ized over the three other parameters) is shown in Fig.
2
.
The pdfs in Fig.
2
are from the joint analysis of the three
LIGO detectors using the S3 and S4 data, all combined
coherently. The shaded area in the
h
0
posterior 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 Sec. V).
B. Binary models
Our previous known pulsar searches [
2
,
3
] have excluded
pulsars within binary systems, despite the majority 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 barycenter (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 software
TEMPO
[
23
]. The
algorithm and its testing are discussed more thoroughly in
Ref. [
24
].
There are five principal parameters describing a
Keplerian orbit: the time of periastron,
T
0
; the longitude
of periastron,
!
0
; the eccentricity,
e
; the period,
P
b
; and the
projected semimajor 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 summarized by Taylor and Weisberg [
20
] and
Lange
et al.
[
25
], and are those used in
TEMPO
. The trans-
formation 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 propagation
time across the binary orbit,
E
is the Einstein delay which
gives gravitational redshift and time dilation corrections,
and
S
is the Shapiro delay which gives the general
relativistic correction (see Ref. [
20
] for definitions of these
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 as-
sumptions about the system and/or are specialized to ac-
count for certain system features. For example, the ELL1
model is used in cases where the eccentricity is very small,
and therefore periastron is very hard to define, in which
case the time and longitude of periastron will be highly
correlated and have to be reparametrized to the Laplace-
Lagrange parameters [
25
]. When a binary pulsar’s parame-
ters are estimated from radio observations 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 gravita-
tional wave frequency cutoff (pulsar spin frequency of
25 Hz) is somewhat arbitrary, but it also loosely reflects
the split between the population of fast (millisecond/re-
cycled 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
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 parame-
ter 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 mea-
surements will better reflect the current state of the pulsar.
Importantly, there is near-continuous monitoring 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
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 marginalized posterior pdfs for the four unknown
pulsar parameters
h
0
,
0
,
cos
, and
, for PSR
J0024
7204C
using the joint data from the three LIGO detectors over S3 and
S4.
UPPER LIMITS ON GRAVITATIONAL WAVE EMISSION
...
PHYSICAL REVIEW D
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042001 (2007)
042001-7
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 pulsars
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 parameter un-
certainties on this extrapolation is discussed in Sec. IV B,
but is only important in its effect on the extrapolated 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 original list of 154)
were not timed with sufficient confidence and were ex-
cluded from the search. This included many of the newly
discovered pulsars (for example the 21 millisecond 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 uncer-
tainties for all the target pulsars, and for pulsars in a binary
system there are also uncertainties associated with all the
binary orbital parameters. Some of these uncertainties are
correlated; for example, the error on frequency could affect
the accuracy of the first frequency derivative, and the
binary time of periastron and longitude 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 combination
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
t
S3
given by Eq. (
1.3
) (ignoring,
for simplicity, the
0
and
terms) is the best-fit phase over
the time span of S3,
t
S3
, the maximum phase uncertainty is
err
max
j
t
S3
2
f
t
S3
t
S3
1
2
_
_
t
S3
t
S3
2
...
gj
;
(4.1)
where the
’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 accept-
ability. A 30
phase drift could possibly give a factor of
1
cos30
0
:
13
in loss of sensitivity for a signal.
Nine of these are in binary systems (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
!
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 uncertain-
ties 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; however 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 stability,
generically known as timing noise. The existence of timing
noise has been clear since the early days of pulsar astron-
omy 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
]asan
activity parameter
A
, referenced to that of the Crab pulsar, 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 period.
8
is not based on the
stochastic nature of the Crab pulsar’s timing noise and is
defined for a fixed time (
10
8
s
)as
8
log
1
6
j
j
10
8
s
3
:
(4.2)
This assumes that the measured value of
is dominated by
the timing noise rather than the pulsar’s intrinsic second
spin-down derivative. Although generally true, this as-
sumption is not valid for the Crab pulsar and PSR
J0537
6910
, where a nontiming noise dominated
can be mea-
sured between glitches.
3
This quantity relates to the pulsar
3
These two pulsars are among the most prolific glitchers, and
in any global fit to their parameters the value of
would most
likely be swamped by the glitch events.
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
042001 (2007)
042001-8
clock error caused by timing noise. The value of
is so
small as to be unmeasurable for most pulsars, although an
upper limit can often be defined. Arzoumanian
et al.
[
32
]
deduce, by eye, a linear relationship between
8
and
log
_
P
of
8
6
:
6
0
:
6 log
_
P;
(4.3)
where
_
P
_
=
2
is the period derivative.
As defined,
8
is a somewhat imprecise indicator of the
timing noise, not least because the time span of
10
8
seconds
chosen by Arzoumanian
et al.
was simply
the length of their data set. A preferred measure may
simply be the magnitude and sign of
P
, but we shall
continue to use the
8
parameter as our timing noise
magnitude estimate for the current analysis. A thorough
study of timing noise, comparing and contrasting the vari-
ous measures used, will be given in Ref. [
33
] (also see
Refs. [
28
,
34
]).
There is a definite correlation between the
8
parame-
ters, spin-down rate, and age. Young pulsars, like the Crab
pulsar, generally show the most timing noise. The catego-
rization of the type of timing noise (i.e., phase, frequency,
or frequency derivative) in Ref. [
31
] allowed them to
ascribe different processes for each. The majority of pul-
sars studied showed frequency-type noise, possibly a result
of random fluctuations in the star’s moment of inertia. The
actual mechanism behind the process is still unknown, with
Cordes and Greenstein [
35
] positing and then ruling out
several mechanisms inconsistent with observations.
Timing noise intrinsically linked to motions of the elec-
tromagnetic emission source or fluctuations in the magne-
tosphere, rather than the rotation of the pulsar, is important
in the search for gravitational waves as it may allow the
relative phase of the electromagnetic and gravitational
signals to drift. The implications of timing noise in this
context are discussed by Jones [
36
]. He gives three cate-
gories of timing noise, not necessarily related to the three
types of timing noise given by Cordes and Helfand [
31
],
having different effects on any search. If all parts of the
neutron star are strongly coupled on short time scales, there
should be no difference between the electromagnetic phase
and the gravitational wave phase. If the timing noise were
purely a magnetospheric fluctuation, then phase wandering
caused by timing noise would not be seen in the gravita-
tional wave emission. The third possibility, whereby the
electromagnetic emission source wanders with respect to
the mass quadrupole, could result from a weak exchange of
angular momentum between the parts of the star respon-
sible for electromagnetic and gravitational wave emission.
Jones describes the ratio of the electromagnetic and gravi-
tational timing noise phase residuals (
) by a parameter
gw
=
em
, with the three types of timing noise
described above corresponding to
1
, 0 and
I
em
=I
gw
respectively, where the
I
’s represent the moments of inertia
of the electromagnetic and gravitational wave producing
components. In principle, this factor could be included as
another search parameter. However, given the cost of in-
cluding an extra parameter in this search, and given that it
is plausible that all parts of a neutron star are tightly
coupled on the time scales of interest here, we will assume
rigid coupling between the two components, i.e. set
1
,
corresponding to the gravitational and electromagnetic
signals remaining perfectly in phase.
The Crab pulsar is regularly monitored [
27
] on time
scales that are sufficiently short to allow its timing noise
to be effectively removed using a second heterodyne pro-
cedure [
37
]. Like the Crab pulsar, PSR
J0537
6910
is
young, has a high glitch rate, and also shows high levels of
timing noise [
29
]. Unfortunately, unlike the Crab pulsar,
we have no regular ephemeris for it that covers our data set,
and timing irregularities are likely to be too great for
historical data to be of use. We therefore have excluded
PSR
J0537
6910
from the analysis. For less noisy pul-
sars we still need a method of estimating the effect of
timing noise on phase evolution that does not rely on
continuous observation. One such estimate is the
8
pa-
rameter given by Eq. (
4.2
), which can provide a measure of
the cumulative phase error. For those pulsars with a mea-
sured
we use this estimate to obtain a corresponding
value of
8
as shown in Fig.
3
.
This should provide a reasonable estimate of the timing
noise over the time span of the pulsar observation. Again
we apply our criterion that cumulative phase errors of
>
30
are unacceptable. In Fig.
3
there are four pulsars
(those with the four largest
8
values), with measured
,
for which this is the case, and therefore timing noise could
be a problem (having already noted the Crab pulsar and
−20
−18
−16
−14
−11
−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
log(dP/dt)
∆
8
FIG. 3.
The values of
8
for our selection of pulsars with
measured
.
UPPER LIMITS ON GRAVITATIONAL WAVE EMISSION
...
PHYSICAL REVIEW D
76,
042001 (2007)
042001-9