Searches for periodic gravitational waves from unknown isolated sources and Scorpius X-1:
Results from the second LIGO science run
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PHYSICAL REVIEW D
76,
082001 (2007)
1550-7998
=
2007
=
76(8)
=
082001(35)
082001-1
©
2007 The American Physical Society
N. E. Strand,
34
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45
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34
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3
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(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, AZ 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
Max Planck Institut fu
̈
r Quantenoptik, D-85748, Garching, Germany
23
Moscow State University, Moscow, 119992, Russia
24
NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
25
National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
26
Northwestern University, Evanston, Illinois 60208, USA
27
Rochester Institute of Technology, Rochester, NY 14623, USA
28
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX United Kingdom
29
San Jose State University, San Jose, California 95192, USA
30
Southeastern Louisiana University, Hammond, Louisiana 70402, USA
31
Southern University, Baton Rouge, LA 70813, USA and A&M College, Baton Rouge, LA 70813, USA
32
Stanford University, Stanford, California 94305, USA
33
Syracuse University, Syracuse, New York 13244, USA
34
The Pennsylvania State University, University Park, Pennsylvania 16802, USA
35
The University of Texas at Brownsville, Brownsville, Texas 78520, USA
and Texas Southmost College, Brownsville, Texas 78520, USA
36
Trinity University, San Antonio, Texas 78212, USA
37
Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
38
Universita
̈
t Hannover, D-30167 Hannover, Germany
39
University of Adelaide, Adelaide, SA 5005, Australia
40
University of Birmingham, Birmingham, B15 2TT, United Kingdom
41
University of Florida, Gainesville, Florida 32611, USA
42
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
43
University of Maryland, College Park, Massachusetts 20742, USA
B. ABBOTT
et al.
PHYSICAL REVIEW D
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082001 (2007)
082001-2
44
University of Michigan, Ann Arbor, Michigan 48109, USA
45
University of Oregon, Eugene, Oregon 97403, USA
46
University of Rochester, Rochester, New York 14627, USA
47
University of Salerno, 84084 Fisciano (Salerno), Italy
48
University of Sannio at Benevento, I-82100 Benevento, Italy
49
University of Sheffield, Sheffield, S3 7RH, United Kingdom
50
University of Southampton, Southampton, SO17 1BJ, United Kingdom
51
University of Strathclyde, Glasgow, G1 1XQ, United Kingdom
52
University of Washington, Seattle, Washington 98195, USA
53
University of Western Australia, Crawley, WA 6009, Australia
54
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA
55
Vassar College, Poughkeepsie, New York 12604, USA
56
Washington State University, Pullman, Washington 99164, USA
(Received 12 June 2006; revised manuscript received 2 April 2007; published 24 October 2007)
We carry out two searches for periodic gravitational waves using the most sensitive few hours of data
from the second LIGO science run. Both searches exploit fully coherent matched filtering and cover wide
areas of parameter space, an innovation over previous analyses which requires considerable algorithm
development and computational power. The first search is targeted at isolated, previously unknown
neutron stars, covers the entire sky in the frequency band 160 –728.8 Hz, and assumes a frequency
derivative of less than
4
10
10
Hz
=
s
. The second search targets the accreting neutron star in the low-
mass x-ray binary Scorpius X-1 and covers the frequency bands 464 – 484 Hz and 604 – 624 Hz as well as
the two relevant binary orbit parameters. Because of the high computational cost of these searches we
limit the analyses to the most sensitive 10 hours and 6 hours of data, respectively. Given the limited
sensitivity and duration of the analyzed data set, we do not attempt deep follow-up studies. Rather we
concentrate on demonstrating the data analysis method on a real data set and present our results as upper
limits over large volumes of the parameter space. In order to achieve this, we look for coincidences in
parameter space between the Livingston and Hanford 4-km interferometers. For isolated neutron stars our
95% confidence level upper limits on the gravitational wave strain amplitude range from
6
:
6
10
23
to
1
10
21
across the frequency band; for Scorpius X-1 they range from
1
:
7
10
22
to
1
:
3
10
21
across
the two 20-Hz frequency bands. The upper limits presented in this paper are the first broadband wide
parameter space upper limits on periodic gravitational waves from coherent search techniques. The
methods developed here lay the foundations for upcoming hierarchical searches of more sensitive data
which may detect astrophysical signals.
DOI:
10.1103/PhysRevD.76.082001
PACS numbers: 04.80.Nn, 07.05.Kf, 95.55.Ym, 97.60.Gb
I. INTRODUCTION
Rapidly rotating neutron stars are the most likely
sources of persistent gravitational radiation in the fre-
quency band
100 Hz
–
1 kHz
. These objects may gener-
ate continuous gravitational waves (GW) through a variety
of mechanisms, including nonaxisymmetric distortions of
the star [
1
–
5
], velocity perturbations in the star’s fluid
[
1
,
6
,
7
], and free precession [
8
,
9
]. Regardless of the specific
mechanism, the emitted signal is a quasiperiodic wave
whose frequency changes slowly during the observation
time due to energy loss through gravitational wave emis-
sion, and possibly other mechanisms. At an Earth-based
detector the signal exhibits amplitude and phase modula-
tions due to the motion of the Earth with respect to the
source. The intrinsic gravitational wave amplitude is likely
to be several orders of magnitude smaller than the typical
root-mean-square value of the detector noise, hence detec-
tion can only be achieved by means of long integration
times, of the order of weeks to months.
Deep, wide parameter space searches for continuous
gravitational wave signals are computationally bound. At
fixed computational resources the optimal sensitivity is
achieved through hierarchical search schemes [
10
–
12
].
Such schemes alternate incoherent and coherent search
stages in order to first efficiently identify statistically sig-
nificant candidates and then follow them up with more
sensitive, albeit computationally intensive, methods.
Hierarchical search schemes have been investigated only
theoretically, under the simplified assumption of Gaussian
and stationary instrumental noise; the computational costs
have been estimated only on the basis of rough counts of
floating point operations necessary to evaluate some de-
tection statistic, usually not the optimal, and have not taken
into account additional costs coming e.g. from data input/
output; computational savings obtainable through efficient
dedicated numerical implementations have also been ne-
glected. Furthermore, general theoretical investigations
*
Electronic address: http://www.ligo.org
SEARCHES FOR PERIODIC GRAVITATIONAL
...
PHYSICAL REVIEW D
76,
082001 (2007)
082001-3
have not relied on the optimizations that can be introduced
on the basis of the specific area in parameter space at which
a search is aimed.
In this paper we demonstrate and characterize the co-
herent stage of a hierarchical pipeline by carrying out two
large parameter space coherent searches on data collected
by LIGO during the second science run with the Livingston
and Hanford 4-km interferometers. As we will show, this
analysis requires careful tuning of a variety of search
parameters and implementation choices, such as the tilings
of the parameter space, and the selection of the data that are
difficult to determine on purely theoretical grounds. This
paper complements the study presented in [
13
] where we
reported results obtained by applying an incoherent analy-
sis method [
14
] to data taken during the same science run.
Furthermore, here we place upper limits on regions of the
parameter space that have never been explored before. We
do this by combining the output of the coherent searches
via a coincidence scheme.
The coherent search described in this paper has been the
test-bench for the core science analysis that the
Einstein@home [
15
] project is carrying out now. The
development of analysis techniques such as the one de-
scribed here, together with the computing power of
Einstein@home in the context of a hierarchical search
scheme, will allow the deepest searches for continuous
gravitational waves.
In this paper the same basic pipeline is applied to and
tuned for two different searches: (i) for signals from iso-
lated sources over the whole sky and the frequency band
160 Hz –728.8 Hz, and (ii) for a signal from the low-mass
x-ray binary Scorpius X-1 (Sco X-1) over orbital parame-
ters and in the frequency bands 464 – 484 Hz and 604 –
624 Hz. It is the first time that a coherent analysis is carried
out over such a wide frequency band and coincidence
techniques are used among the registered candidates
from different detectors; the only other example of a some-
what similar analysis is an all-sky search over two days of
data from the Explorer resonant detector and that was over
a 0.76 Hz band around 922 Hz [
16
–
18
]. This is absolutely
the first wide parameter space search for a rotating neutron
star in a binary system.
The main scope of the paper is to illustrate an analysis
method by applying it to two different wide parameter
spaces. In fact, based on the typical noise performance of
the detectors during the run, which is shown in Fig.
1
, and
the amount of data that we were able to process in
1 month
with our computational resources (totalling about
800 CPUs over several Beowulf clusters) we do not expect
to detect gravitational waves. For isolated neutron stars we
estimate (see Sec. III for details) that statistically the
strongest signal that we expect from an isolated source is
&
4
10
24
which is a factor
*
20
smaller than the
dimmest signal that we would have been able to observe
with the present search. For Scorpius X-1, the signal is
expected to have a strength of at most
3
10
26
and our
search is a factor
5000
less sensitive. The results of the
analyses confirm these expectations and we report upper
limits for both searches.
The paper is organized as follows. In Sec. II we describe
the instrument configuration during the second science run
and the details of the data taking. In Sec. III we review the
current astrophysical understanding of neutron stars as
gravitational wave sources, including a somewhat novel
statistical argument that the strength of the strongest such
signal that we can expect to receive does not exceed
h
max
0
4
10
24
. We also detail and motivate the choice
of parameter spaces explored in this paper. In Sec. IV we
review the signal model and discuss the search area con-
sidered here. In Sec. V we describe the analysis pipeline. In
Sec. VI we present and discuss the results of the analyses.
In Sec. VII we recapitulate the most relevant results in the
wider context and provide pointers for future work.
II. INSTRUMENTS AND THE SECOND SCIENCE
RUN
Three detectors at two independent sites comprise the
Laser Interferometer Gravitational Wave Observatory, or
LIGO. Detector commissioning has progressed since the
fall of 1999, interleaved with periods in which the observ-
atory ran nearly continuously for weeks or months, the so-
called ‘‘science runs.’’ The first science run (S1) was made
in concert with the gravitational wave detector GEO600;
results from the analysis of those data were presented in
[
19
–
22
], while the instrument status was detailed in [
23
].
Significant improvements in the strain sensitivity of the
LIGO interferometers (an order of magnitude over a broad-
band) culminated in the second science run (S2), which
100
200
300
400 500 600700
10
−23
10
−22
10
−21
10
−20
10
−19
Frequency (Hz)
Strain spectral amplitude/
√
Hz
H2
H1
L1
FIG. 1 (color online).
Typical one-sided amplitude spectral
densities of detector noise during the second science run, for
the three LIGO instruments. The solid black line is the design
sensitivity for the two 4-km instruments L1 and H1.
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
082001 (2007)
082001-4
took place from February 14 to April 14, 2003. Details of
the S2 run, including detector improvements between S1
and S2 can be found in [
24
], Sec. IV of [
25
], and Sec. II of
[
13
,
26
].
Each LIGO detector is a recycled Michelson interfer-
ometer with Fabry-Perot arms, whose lengths are defined
by suspended mirrors that double as test masses. Two
detectors reside in the same vacuum in Hanford, WA,
one (denoted H1) with 4-km armlength and one with 2-
km armlength (H2), while a single 4-km counterpart (L1)
exists in Livingston Parish, LA. Differential motions are
sensed interferometrically, and the resultant sensitivity is
broadband (40 Hz –7 kHz), with spectral disturbances such
as 60 Hz powerline harmonics evident in the noise spec-
trum (see Fig.
1
). Optical resonance, or ‘‘lock,’’ in a given
detector is maintained by servo loops; lock may be inter-
rupted by, for example, seismic transients or poorly con-
ditioned servos. S2 duty cycles, accounting for periods in
which lock was broken and/or detectors were known to be
functioning not at the required level, were 74% for H1,
58% for H2, and 37% for L1. The two analyses described
in this paper used a small subset of the data from the two
most sensitive instruments during S2, L1, and H1; the
choice of the segments considered for the analysis is de-
tailed in Sec. V C.
The strain signal at the interferometer output is recon-
structed from the error signal of the feedback loop which is
used to control the differential length of the arms of the
instrument. Such a process — known as calibration — in-
volves the injection of continuous, constant amplitude
sinusoidal excitations into the end test mass control sys-
tems, which are then monitored at the measurement error
point. The calibration process introduces uncertainties in
the amplitude of the recorded signal that were estimated to
be
&
11%
during S2 [
27
]. In addition, during the run
artificial pulsar-like signals were injected into the data
stream by physically moving the mirrors of the Fabry-
Perot cavity. Such ‘‘hardware injections’’ were used to
validate the data analysis pipeline and details are presented
in Appendix C.
III. ASTROPHYSICAL SOURCES
We review the physical mechanisms of periodic gravi-
tational wave emission and the target populations of the
two searches described in this paper. We also compare the
sensitivity of these searches to likely source strengths.
A. Emission mechanisms
In the LIGO frequency band there are three predicted
mechanisms for producing periodic gravitational waves,
all of which involve neutron stars or similar compact
objects: (1) nonaxisymmetric distortions of the solid part
of the star [
1
–
5
], (2) unstable
r
-modes in the fluid part of
the star [
1
,
6
,
7
], and (3) free precession of the whole star
[
8
,
9
].
We begin with nonaxisymmetric distortions. These
could not exist in a perfect fluid star, but in realistic neutron
stars such distortions could be supported either by elastic
stresses or by magnetic fields. The deformation is often
expressed in terms of the ellipticity
I
xx
I
yy
I
zz
;
(1)
which is (up to a numerical factor of order unity) the
m
2
quadrupole moment divided by the principal moment of
inertia. A nonaxisymmetric neutron star rotating with fre-
quency
emits periodic gravitational waves with ampli-
tude
h
0
4
2
G
c
4
I
zz
f
2
d
;
(2)
where
G
is Newton’s gravitational constant,
c
is the speed
of light,
I
zz
is the principal moment of inertia of the object,
f
(equal to
2
) is the gravitational wave frequency, and
d
is
the distance to the object. Equation (
2
) gives the strain
amplitude of a gravitational wave from an optimally ori-
ented source [see Eq. (
25
) below].
The ellipticity of neutron stars is highly uncertain. The
maximum ellipticity that can be supported by a neutron
star’s crust is estimated to be [
2
]
max
5
10
7
10
2
;
(3)
where
is the breaking strain of the solid crust. The
numerical coefficient in Eq. (
3
) is small mainly because
the shear modulus of the inner crust (which constitutes
most of the crust’s mass) is small, in the sense that it is
about
10
3
times the pressure. Equation (
3
) uses a fiducial
breaking strain of
10
2
since that is roughly the upper limit
for the best terrestrial alloys. However,
could be as high
as
10
1
for a perfect crystal with no defects [
28
], or several
orders of magnitude smaller for an amorphous solid or a
crystal with many defects.
Some exotic alternatives to standard neutron stars fea-
ture solid cores, which could support considerably larger
ellipticities [
5
]. The most speculative and highest-
ellipticity model is that of a solid strange-quark star, for
which
max
4
10
4
10
2
:
(4)
This much higher value of
max
is mostly due to the higher
shear modulus, which for some strange star models can be
almost as large as the pressure. Another (still speculative
but more robust) model is the hybrid star, which consists of
a normal neutron star outside a solid core of mixed quark
and baryon matter, which may extend from the center to
nearly the bottom of the crust. For hybrid stars,
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...
PHYSICAL REVIEW D
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082001 (2007)
082001-5
max
9
10
6
10
2
;
(5)
although this is highly dependent on the poorly known
range of densities occupied by the quark-baryon mixture.
Stars with charged meson condensates could also have
solid cores with overall ellipticities similar to those of
hybrid stars.
Regardless of the maximum ellipticity supportable by
shear stresses, there is the separate problem of how to reach
the maximum. The crust of a young neutron star probably
cracks as the neutron star spins down, but it is unclear how
long it takes for gravity to smooth out the neutron star’s
shape. Accreting neutron stars in binaries have a natural
way of reaching and maintaining the maximum deforma-
tion, since the accretion flow, guided by the neutron star’s
magnetic field, naturally produces ‘‘hot spots’’ on the
surface, which can imprint themselves as lateral tempera-
ture variations throughout the crust. Through the tempera-
ture dependence of electron capture, these variations can
lead to ‘‘hills’’ in hotter areas which extend down to the
dense inner crust, and with a reasonable temperature varia-
tion the ellipticity might reach the maximum elastic value
[
1
]. The accreted material can also be held up in mountains
on the surface by the magnetic field itself: The matter is a
good conductor, and thus it crosses field lines relatively
slowly and can pile up in mountains larger than those
supportable by elasticity alone [
4
]. Depending on the field
configuration, accretion rate, and temperature, the elliptic-
ity from this mechanism could be up to
10
5
even for
ordinary neutron stars.
Strong internal magnetic fields are another possible
cause of ellipticity [
3
]. Differential rotation immediately
after the core collapse in which a neutron star is formed can
lead to an internal magnetic field with a large toroidal part.
Dissipation tends to drive the symmetry axis of a toroidal
field toward the star’s equator, which is the orientation that
maximizes the ellipticity. The resulting ellipticity is
8
>
<
>
:
1
:
6
10
6
B
10
15
G
B<
10
15
G
;
1
:
6
10
6
B
10
15
G
2
B>
10
15
G
;
(6)
where
B
is the root-mean-square value of the toroidal part
of the field averaged over the interior of the star. Note that
this mechanism requires that the external field be much
smaller than the internal field, since such strong external
fields will spin a star out of the LIGO frequency band on a
very short time scale.
An alternative way of generating asymmetry is the
r
-modes, fluid oscillations dominated by the Coriolis
restoring force. These modes may be unstable to
growth through gravitational radiation reaction [the
Chandrasekhar-Friedman-Schutz (CFS) instability] under
astrophysically realistic conditions. Rather than go into the
many details of the physics and astrophysics, we refer the
reader to a recent review [
29
] of the literature and summa-
rize here only what is directly relevant to our search: The
r
-modes have been proposed as a source of gravitational
waves from newborn neutron stars [
6
] and from rapidly
accreting neutron stars [
1
,
7
]. The CFS instability of the
r
-modes in newborn neutron stars is probably not a good
candidate for detection because the emission is very short-
lived, low amplitude, or both. Accreting neutron stars (or
quark stars) are a better prospect for a detection of
r
-mode
gravitational radiation because the emission may be long-
lived with a duty cycle near unity [
30
,
31
].
Finally we consider free precession, i.e. the wobble of a
neutron star whose symmetry axis does not coincide with
its rotation axis. A large-amplitude wobble would produce
[
8
]
h
0
10
27
w
0
:
1
1 kpc
d
500 Hz
2
;
(7)
where
w
is the wobble amplitude in radians. Such wobble
may be longer lived than previously thought [
9
], but the
amplitude is still small enough that such radiation is a
target for second generation interferometers such as
Advanced LIGO.
In light of our current understanding of emission mecha-
nisms, the most likely sources of detectable gravitational
waves are isolated neutron stars (through deformations)
and accreting neutron stars in binaries (through deforma-
tions or
r
-modes).
B. Isolated neutron stars
The target population of this search is isolated rotating
compact stars that have not been observed electromagneti-
cally. Current models of stellar evolution suggest that our
Galaxy contains of order
10
9
neutron stars, while only of
order
10
5
are active pulsars. Up to now only about 1500
have been observed [
32
]; there are numerous reasons for
this, including selection effects and the fact that many have
faint emission. Therefore the target population is a large
fraction of the neutron stars in the Galaxy.
1. Maximum expected signal amplitude at the earth
Despite this large target population and the variety of
GW emission mechanisms that have been considered, one
can make a robust argument, based on energetics and
statistics, that the amplitude of the strongest gravitational
wave pulsar that one could reasonably hope to detect on
Earth is bounded by
h
0
&
4
10
24
. The argument is a
modification of an observation due to Blandford (which
was unpublished, but credited to him in Thorne’s review in
[
33
]).
The argument begins by assuming, very optimistically,
that all neutron stars in the Galaxy are born at a very high
spin rate and then spin down principally due to gravita-
tional wave emission. For simplicity we shall also assume
that all neutron stars follow the same spin-down law
_
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
082001 (2007)
082001-6
or equivalently
_
f
f
, although this turns out to be unnec-
essary to the conclusion. It is helpful to express the spin-
down law in terms of the spin-down time scale
gw
f
f
j
4
_
f
f
j
:
(8)
For a neutron star with constant ellipticity,
gw
f
is the
time for the gravitational wave frequency to drift down to
f
from some initial, much higher spin frequency. This time
scale is independent of ellipticity and emission mecha-
nism, so long as the emission is quadrupolar. (It is similar
to the characteristic age
f=
j
2
_
f
j
used in pulsar astronomy,
except that the 2 is replaced by 4 as appropriate for
quadrupole rather than dipole radiation.) A source’s gravi-
tational wave amplitude
h
0
is then related to
gw
f
by
h
0
f
d
1
5
GI
zz
8
c
3
gw
f
s
:
(9)
Here we are assuming that the star is not accreting, so that
the angular momentum loss to GWs causes the star to slow
down. The case of accreting neutron stars is dealt with
separately, below.
We now consider the distribution of neutron stars in
space and frequency. Let
N
f
f
be the number of
Galactic neutron stars in the frequency range
f
f=
2
;f
f=
2
. We assume that the birthrate has been
roughly constant over a long enough time scale that this
distribution has settled into a statistical steady state:
dN
f
=dt
0
above the minimum frequency
f
min
of our
search. (This is not true for millisecond pulsars; see below.)
Then
N
f
_
f
is just the neutron star birthrate
1
=
b
, where
b
may be as short as 30 years. For simplicity, we model the
spatial distribution of neutron stars in our Galaxy as that of
a uniform cylindrical disk, with radius
R
G
10 kpc
and
height
H
600 pc
. Then the spatial density
n
f
of neu-
tron stars near the Earth, in the frequency range
f
f=
2
;f
f=
2
, is just
n
f
f
R
2
G
H
1
N
f
f
.
Let
^
N
f;d
be that portion of
N
f
due to neutron stars
whose distance from Earth is less than
d
.For
H=
2
&
d
&
R
G
, we have
d
^
N
f;d
d
d
2
dHn
f
(10)
2
N
f
d
R
2
G
(11)
(and it drops off rapidly for
d
*
R
G
). Changing variables
from
d
to
h
0
using Eqs. (
8
) and (
9
), we have
d
^
N
f;h
0
dh
0
5
GI
zz
c
3
b
R
2
G
f
1
h
3
0
:
(12)
Note that the dependence on the poorly known
gw
f
has
dropped out of this equation. This was the essence of
Blandford’s observation.
Now consider a search for GW pulsars in the frequency
range
f
min
;f
max
. Integrating the distribution in Eq. (
12
)
over this band, we obtain the distribution of sources as a
function of
h
0
:
dN
band
dh
0
5
GI
zz
c
3
b
R
2
G
h
3
0
ln
f
max
f
min
:
(13)
The amplitude
h
max
0
of the strongest source is implicitly
given by
Z
1
h
max
0
dN
band
dh
0
dh
0
1
2
:
(14)
That is, even given our optimistic assumptions about the
neutron star population, there is only a 50% chance of
seeing a source as strong as
h
max
0
. The integral in
Eq. (
14
) is trivial; it yields
h
max
0
5
GI
zz
c
3
b
R
2
G
ln
f
max
f
min
1
=
2
:
(15)
Inserting
ln
f
max
=f
min
1
=
2
1
(appropriate for a typical
broadband search, as conducted here), and adopting as
fiducial values
I
zz
10
45
gcm
2
,
R
G
10 kpc
, and
b
30 yr
, we arrive at
h
max
0
4
10
24
:
(16)
This is what we aimed to show.
We now address the robustness of some assumptions in
the argument. First, the assumption of a universal spin-
down function
gw
f
was unnecessary, since
gw
f
dis-
appeared from Eq. (
12
) and the subsequent equations that
led to
h
max
0
. Had we divided neutron stars into different
classes labeled by
i
and assigned each a spin-down law
i
gw
f
and birthrate
1
=
i
b
, each would have contributed its
own term to
d
^
N=dh
0
which would have been independent
of
i
gw
and the result for
h
max
0
would have been the same.
Second, in using Eq. (
10
), we have in effect assumed that
the strongest source is in the distance range
H=
2
&
d
&
R
G
. We cannot evade the upper limit by assuming that the
neutron stars have extremely long spin-down times (so that
d<H=
2
) or extremely short ones (so that the brightest is
outside our Galaxy,
d>R
G
). If the brightest sources are at
d<H=
2
(as happens if these sources have long spin-down
times,
gw
*
b
2
R
G
=H
2
), then our estimate of
h
max
0
only
decreases, because at short distances the spatial distribu-
tion of neutron stars becomes approximately spherically
symmetric instead of planar and the right-hand sides of
Eqs. (
10
) and (
12
) are multiplied by a factor
2
r=H <
1
.On
the other hand, if
gw
f
(in the LIGO range) is much
shorter than
b
, then the probability that such an object
exists inside our Galaxy is
1
. For example, a neutron
star with
gw
f
3yr
located at
r
10 kpc
would have
h
0
4
:
14
10
24
, but the probability of currently having
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...
PHYSICAL REVIEW D
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082001 (2007)
082001-7
a neutron star with this (or shorter)
gw
is only
gw
=
b
&
1
=
10
.
Third, we have implicitly assumed that each neutron star
spins down only once. In fact, it is clear that some stars in
binaries are ‘‘recycled’’ to higher spins by accretion, and
then spin down again. This effectively increases the neu-
tron star birth rate (since for our purposes the recycled stars
are born twice), but since the fraction of stars recycled is
very small the increase in the effective birth rate is also
small.
2. Expected sensitivity of the S2 search
Typical noise levels of LIGO during the S2 run were
approximately
S
h
f
1
=
2
3
10
22
Hz
1
=
2
, where
S
h
is the strain noise power spectral density, as shown in
Fig.
1
. Even for a
known
GW pulsar with an average sky
position, inclination angle, polarization, and frequency, the
amplitude of the signal that we could detect in Gaussian
stationary noise with a false alarm rate of 1% and a false
dismissal rate of 10% is [
19
]
h
h
0
f
i
11
:
4
S
h
f
T
obs
s
;
(17)
where
T
obs
is the integration time and the angled brackets
indicate an average source. In all-sky searches for pulsars
with
unknown
parameters, the amplitude
h
0
must be sev-
eral times greater than this to rise convincingly above the
background. Therefore, in
T
obs
10 hours
of S2 data,
signals with amplitude
h
0
below about
10
22
would not
be detectable. This is a factor
25
greater than the value
of
h
max
0
shown in (
16
), so our S2 analysis is unlikely to be
sensitive enough to reveal previously unknown pulsars.
The sensitivity of our search is further restricted by the
template bank, which does not include the effects of signal
spin-down for reasons of computational cost. Phase mis-
match between the signal and matched filter causes the
detection statistic (see Sec. VA) to decrease rapidly for
GW frequency derivatives
_
f
that exceed
max
_
f
1
2
T
2
obs
4
10
10
T
obs
10 h
2
Hz s
1
:
(18)
Assuming that all of the spin-down of a neutron star is due
to gravitational waves (from a mass quadrupole deforma-
tion), our search is restricted to pulsars with ellipticity
less than
sd
5
c
5
max
_
f
32
4
GI
zz
f
5
1
=
2
:
(19)
This limit, derived from combining the quadrupole formula
for GW luminosity
dE
dt
1
10
G
c
5
2
f
6
I
2
zz
2
(20)
with the kinetic energy of rotation
E
1
2
2
f
2
I
zz
;
(21)
(assuming
f
2
) takes the numerical value
sd
9
:
6
10
6
10
45
gcm
2
I
zz
1
=
2
300 Hz
f
5
=
2
(22)
for our maximum
_
f
.
The curves in Fig.
2
are obtained by combining Eqs. (
2
)
and (
17
)
1
and solving for the distance
d
for different values
of the ellipticity, using an average value for the noise in the
detectors during the S2 run. The curves show the average
distance, in the sense of the definition (
17
), at which a
source may be detected.
The black region shows that a GW pulsar with
10
6
could be detected by this search only if it were very close,
less than
5 parsecs
away. The light gray region shows the
distance at which a GW pulsar with
10
5
could be
detected if templates with sufficiently large spin-down
values were searched. However,
this
search can detect
such pulsars only below 300 Hz, because above 300 Hz a
GW pulsar with
10
5
spins down too fast to be
200
300
400
500
600
700
0
10
20
30
40
50
60
70
Frequency (Hz)
Distance (Parsecs)
ε
= 10
−6
ε
= 10
−5
ε
=
ε
sd
FIG. 2 (color online).
Effective average range (defined in the
text) of our search as a function of frequency for three elliptic-
ities:
10
6
(maximum for a normal neutron star),
10
5
(maxi-
mum for a more optimistic object), and
sd
, the spin-down limit
defined in the text. Note that for sources above 300 Hz the reach
of the search is limited by the maximum spin-down value of a
signal that may be detected without loss of sensitivity.
1
Note that the value of
h
0
derived from Eq. (
17
) yields a value
of the detection statistic
2
F
for an average source as seen with a
detector at S2 sensitivity and over an observation time of
10 hours, of about 21, which is extremely close to the value of
20 which is used in this analysis as the threshold for registering
candidate events. Thus combining Eqs. (
2
) and (
17
) determines
the smallest amplitude that our search pipeline could detect
(corresponding to a signal just at the threshold), provided ap-
propriate follow-up studies of the registered events ensued.
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
082001 (2007)
082001-8
detected with the no-spin-down templates used. The thick
line indicates the distance limit for the (frequency-
dependent) maximum value of epsilon that could be de-
tected with the templates used in this search. At certain
frequencies below 300 Hz, a GW pulsar could be seen
somewhat farther away than 30 pc, but only if it has
>
10
5
. Although
sd
and the corresponding curve were
derived assuming a quadrupolar deformation as the emis-
sion mechanism, the results would be similar for other
mechanisms. Equation (
21
) includes an implicit factor
f
2
=
2
2
, which results in
sd
and the corresponding range
(for a fixed GW frequency
f
) being multiplied by
f=
2
,
which is
1
=
2
for free precession and about
2
=
3
for
r
-modes. Even for a source with optimum inclination angle
and polarization, the range increases only by a factor
2
.
The distance to the nearest known pulsar in the LIGO
frequency band, PSR
J0437
4715
, is about 140 pc. The
other nearest neutron stars are at comparable distances
[
32
,
34
] including RX
J1856
:
5
3754
, which may be the
nearest of all and was found to have a pulsation period out
of the LIGO band after this article was submitted [
35
].
Therefore our search would be sensitive only to previously
unknown objects.
While we have argued that a detection would be very
unlikely, it should be recalled that Eq. (
16
) was based on a
statistical argument. It is always possible that there is a
GW-bright neutron star that is much closer to us than
would be expected from a random distribution of super-
novae (for example due to recent star formation in the
Gould belt as considered in [
36
]). It is also possible that
a ‘‘blind’’ search of the sort performed here could discover
some previously unknown class of compact objects not
born in supernovae.
More importantly, future searches for previously undis-
covered rotating neutron stars using the methods presented
here will be much more sensitive. The goal of initial LIGO
is to take a year of data at design sensitivity. With respect to
S2, this is a factor 10 improvement in the amplitude strain
noise at most frequencies. The greater length of the data set
will also increase the sensitivity to pulsars by a factor of a
few (the precise value depends on the combination of
coherent and incoherent analysis methods used). The net
result is that initial LIGO will have
h
0
reduced from the S2
value by a factor of 30 or more to a value comparable to
h
max
0
4
10
24
of Eq. (
16
).
C. Accreting neutron stars
1. Maximum expected signal amplitude at earth
The robust upper limit in Eq. (
16
) refers only to non-
accreting neutron stars, since energy conservation plays a
crucial role. If accretion replenishes the star’s angular
momentum, a different but equally robust argument (i.e.,
practically independent of the details of the emission
mechanism) can be made regarding the maximum strain
h
max
0
at the Earth. In this case
h
max
0
is set by the x-ray
luminosity of the brightest x-ray source.
The basic idea is that if the energy (or angular momen-
tum) lost to GWs is replenished by accretion, then the
strongest GW emitters are those accreting at the highest
rate, near the Eddington limit. Such systems exist: the low-
mass x-ray binaries (LMXBs), so-called since the accreted
material is tidally stripped from a low-mass companion
star. The accreted gas hitting the surface of the neutron star
is heated to
10
8
K
and emits x-rays. As noted several times
over the years [
1
,
37
,
38
], if one assumes that spin-down
from GW emission is in equilibrium with accretion torque,
then the GW amplitude
h
0
is directly related to the x-ray
luminosity:
h
0
5
10
27
300 Hz
1
=
2
F
x
10
8
erg cm
2
s
1
1
=
2
;
(23)
where
F
x
is the x-ray flux. In the 1970s when this connec-
tion was first proposed, there was no observational support
for the idea that the LMXBs are strong GW emitters. But
the spin frequencies of many LMXBs are now known, and
most are observed to cluster in a fairly narrow range of spin
frequencies
270 Hz
&
&
620 Hz
[
39
]. Since most neu-
tron stars will have accreted enough matter to spin them up
to near their theoretical maximum spin frequencies, esti-
mated at
1400 Hz
, the observed spin distribution is hard
to explain without some competing mechanism, such as
gravitational radiation, to halt the spin-up. Since the gravi-
tational torque scales as
5
, gravitational radiation is also a
natural explanation for why the spin frequencies occupy a
rather narrow window: a factor 32 difference in accretion
rate leads to only a factor 2 difference in equilibrium spin
rate [
1
].
If the above argument holds, then the accreting neutron
star brightest in x-rays is also the brightest in gravitational
waves. Sco X-1, which was the first extrasolar x-ray source
discovered, is the strongest persistent x-ray source in the
sky. Assuming equilibrium between GWs and accretion,
the gravitational wave strain of Sco X-1 at the Earth is
h
0
3
10
26
540 Hz
f
1
=
2
;
(24)
which should be detectable by second generation interfer-
ometers. The gravitational wave strains from other accret-
ing neutron stars are expected to be lower.
2. Expected sensitivity of S2 search for Sco X-1
The orbital parameters of Sco X-1 are poorly con-
strained by present (mainly optical) observations and large
uncertainties affect the determination of the rotation fre-
quency of the source (details are provided in Sec. IV B 2).
The immediate implication for a coherent search for gravi-
tational waves from such a neutron star is that a very large
number of discrete templates are required to cover the
SEARCHES FOR PERIODIC GRAVITATIONAL
...
PHYSICAL REVIEW D
76,
082001 (2007)
082001-9
relevant parameter space, which in turn dramatically in-
creases the computational costs [
40
]. The optimal sensi-
tivity that can be achieved with a coherent search is
therefore set primarily by the length of the data set that
one can afford to process (with fixed computational resour-
ces) and the spectral density of the detector noise. As we
discuss in Sec. IV B 2, the maximum span of the observa-
tion time set by the computational burden of the Sco X-1
pipeline (approximately one week on
100
CPUs) limits
the observation span to 6 hours.
The overall sensitivity of the search that we are describ-
ing is determined by each stage of the pipeline, which we
describe in detail in Sec. V B. Assuming that the noise in
the instrument can be described as a Gaussian and sta-
tionary process (an assumption which, however, breaks
down in some frequency regions and/or for portions of
the observation time), we can statistically model the effects
of each step of the analysis and estimate the sensitivity of
the search. The results of such modelling through the use of
Monte Carlo simulations are shown in Fig.
3
where we give
the expected upper limit sensitivity of the search. We
contrast this with the hypothetical case in which the
Sco X-1 parameters are known perfectly making it a single
filter target for the whole duration of the S2 run. The
dramatic difference (of at least an order of magnitude)
between the estimated sensitivity curves of these two
scenarios is primarily due to the large parameter space
we have to search. This has two consequences, which
contribute to degrading the sensitivity of the analysis:
(i) we are computationally limited by the vast number of
templates that we must search and therefore must reduce
the observation to a subsection of the S2 data, and
(ii) sampling a large number of independent locations
increases the probability that noise alone will produce a
high value of the detection statistic.
We note that the S2 Sco X-1 analysis is a factor of
5000
less sensitive than the characteristic amplitude given
in Eq. (
24
). In the hypothetical case in which Sco X-1 is a
single filter target and we are able to analyze the entirety of
S2 data, then we are still a factor
100
away. However, as
mentioned in the introduction, the search reported in this
paper will be one of the stages of a more sensitive ‘‘hier-
archical pipeline’’ that will allow us to achieve quasiopti-
mal sensitivity with fixed computational resources.
IV. SIGNAL MODEL
A. The signal at the detector
We consider a rotating neutron star with equatorial
coordinates
(right ascension) and
(declination).
Gravitational waves propagate in the direction
^
k
and the
star spins around an axis whose direction, assumed to be
constant, is identified by the unit vector
^
s
.
The strain
h
t
recorded at the interferometer output at
detector time
t
is
h
t
h
0
1
2
1
cos
2
F
t
;
;;
cos
t
cos
F
t
;
;;
sin
t
;
(25)
where
is the polarization angle, defined as
tan
^
k
^
s
^
z
=
^
s
^
k
^
z
^
k
^
s
^
z
,
^
z
is the direction to the
north celestial pole, and
cos
^
k
^
s
. Gravitational wave
laser interferometers are all-sky monitors with a response
that depends on the source location in the sky and the wave
polarization: this is encoded in the (time-dependent) an-
tenna beam patterns
F
;
t
;
;;
. The term
t
in
Eq. (
25
) represents the phase of the received gravitational
signal.
The analysis challenge to detect weak quasiperiodic
continuous gravitational waves stems from the Doppler
shift of the gravitational phase
t
due to the relative
motion between the detector and the source. It is conve-
nient to introduce the following times:
t
, the time measured
at the detector;
T
, the solar-system-barycenter (SSB) co-
ordinate time; and
t
p
, the proper time in the rest frame of
the pulsar.
2
200
300
400
500
600
700
800
900
1000
10
−25
10
−24
10
−23
10
−22
10
−21
10
−20
Frequency (Hz)
Strain (dimensionless)
FIG. 3.
Here we show the expected upper limit sensitivity of
the S2 Sco X-1 search. The upper black curve represents the
expected sensitivity of the S2 analysis based on an optimally
selected 6-hour data set (chosen specifically for our search
band). The gray curve (second from the top) shows the sensi-
tivity in the hypothetical case in which
all
of the Sco X-1 system
parameters are known exactly making Sco X-1 a single filter
target and the entire S2 data set is analyzed. Both curves are
based on a 95% confidence upper limit. The remaining curves
represent
S
h
f
=T
obs
p
for L1 (black) and H1 (gray);
S
h
f
is the
typical noise spectral density that characterizes the L1 and H1
data, and
T
obs
is the actual observation time (taking into account
the duty cycle, which is different for L1 and H1) for each
instrument.
2
Notice that our notation for the three different times is
different from the established conventions adopted in the radio
pulsar community, e.g. [
41
].
B. ABBOTT
et al.
PHYSICAL REVIEW D
76,
082001 (2007)
082001-10