arXiv:gr-qc/0605028v2 31 May 2006
Coherent searches for periodic gravitational waves from un
known isolated sources and
Scorpius X-1: results from the second LIGO science run.
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L. Zhang,
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and J. Zweizig
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(The LIGO Scientific Collaboration, http://www.ligo.org)
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
Australian National University, Canberra, 0200, Australi
a
4
California Institute of Technology, Pasadena, CA 91125, US
A
5
California State University Dominguez Hills, Carson, CA 90
747, USA
6
Caltech-CaRT, Pasadena, CA 91125, USA
7
Cardiff University, Cardiff, CF2 3YB, United Kingdom
8
Carleton College, Northfield, MN 55057, USA
9
Columbia University, New York, NY 10027, USA
10
Hobart and William Smith Colleges, Geneva, NY 14456, USA
11
Inter-University Centre for Astronomy and Astrophysics, P
une - 411007, India
12
LIGO - California Institute of Technology, Pasadena, CA 911
25, USA
13
LIGO - Massachusetts Institute of Technology, Cambridge, M
A 02139, USA
14
LIGO Hanford Observatory, Richland, WA 99352, USA
15
LIGO Livingston Observatory, Livingston, LA 70754, USA
16
Louisiana State University, Baton Rouge, LA 70803, USA
17
Louisiana Tech University, Ruston, LA 71272, USA
18
Loyola University, New Orleans, LA 70118, USA
19
Max Planck Institut f ̈ur Quantenoptik, D-85748, Garching,
Germany
20
Moscow State University, Moscow, 119992, Russia
21
NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
22
National Astronomical Observatory of Japan, Tokyo 181-858
8, Japan
23
Northwestern University, Evanston, IL 60208, USA
24
Salish Kootenai College, Pablo, MT 59855, USA
25
Southeastern Louisiana University, Hammond, LA 70402, USA
26
Stanford University, Stanford, CA 94305, USA
27
Syracuse University, Syracuse, NY 13244, USA
28
The Pennsylvania State University, University Park, PA 168
02, USA
29
The University of Texas at Brownsville and Texas Southmost C
ollege, Brownsville, TX 78520, USA
30
Trinity University, San Antonio, TX 78212, USA
31
Universit ̈at Hannover, D-30167 Hannover, Germany
32
Universitat de les Illes Balears, E-07122 Palma de Mallorca
, Spain
33
University of Birmingham, Birmingham, B15 2TT, United King
dom
34
University of Florida, Gainesville, FL 32611, USA
35
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
36
University of Maryland, College Park, MA 20742, USA
37
University of Michigan, Ann Arbor, MI 48109, USA
38
University of Oregon, Eugene, OR 97403, USA
39
University of Rochester, Rochester, NY 14627, USA
40
University of Wisconsin-Milwaukee, Milwaukee, WI 53201, U
SA
41
Vassar College, Poughkeepsie, NY 12604
42
Washington State University, Pullman, WA 99164, USA
(Dated: Revision: 1.420 Date: 2006/05/31 17:36:27 )
We carry out two searches for periodic gravitational waves u
sing 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 prev
ious analyses which requires considerable
algorithm development and computational power. The first se
arch is targeted at isolated, previously
unknown neutron stars, covers the entire sky in the frequenc
y 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. Du
e to the high computational cost of
these searches we limit the analyses to the most sensitive 10
hours and 6 hours of data respectively.
2
Both searches look for coincidences between the Livingston
and Hanford 4-km interferometers.
Given the limited sensitivity and duration of the analyzed d
ata 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 volume
s of the parameter space. For isolated
neutron stars our 95% confidence upper limits on the gravitat
ional 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 prese
nted in this paper are the
first broad-band wide parameter space upper limits on period
ic gravitational waves from coherent
search techniques. The methods developed here lay the found
ations for upcoming hierarchical
searches of more sensitive data which may detect astrophysi
cal signals.
PACS numbers: 04.80.Nn, 95.55.Ym, 97.60.Gb, 07.05.Kf
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
generate continuous gravitational waves (GW) through
a
Currently at Stanford Linear Accelerator Center
b
Currently at SunGard Trading and Risk Systems
c
Currently at Jet Propulsion Laboratory
d
Permanent Address: HP Laboratories
e
Currently at Rutherford Appleton Laboratory
f
Currently at University of California, Los Angeles
g
Currently at Hofstra University
h
Currently at Charles Sturt University, Australia
i
Currently at Keck Graduate Institute
j
Currently at National Science Foundation
k
Permanent Address: Jet Propulsion Laboratory
l
Currently at University of Sheffield
m
Currently at Ball Aerospace Corporation
n
Currently at European Gravitational Observatory
o
Currently at Intel Corp.
p
Currently at University of Tours, France
q
Currently at Embry-Riddle Aeronautical University
r
Currently at Lightconnect Inc.
s
Currently at W.M. Keck Observatory
t
Currently at ESA Science and Technology Center
u
Currently at Raytheon Corporation
v
Currently at New Mexico Institute of Mining and Technology /
Magdalena Ridge Observatory Interferometer
w
Currently at Univa Corporation
x
Currently at Mission Research Corporation
y
Currently at Harvard University
z
Currently at Lockheed-Martin Corporation
aa
Permanent Address: Science and Technology Corporation
bb
Permanent Address: University of Tokyo, Institute for Cosm
ic
Ray Research
cc
Permanent Address: University College Dublin
dd
Currently at Universit ́a di Trento and INFN, Trento, Italy
ee
Currently at Research Electro-Optics Inc.
ff
Currently at Institute of Advanced Physics, Baton Rouge, LA
gg
Currently at Thirty Meter Telescope Project at Caltech
hh
Currently at European Commission, DG Research, Brussels, B
el-
gium
ii
Currently at University of Chicago
jj
Currently at LightBit Corporation
kk
Permanent Address: IBM Canada Ltd.
ll
Currently at The University of Tokyo
mm
Currently at University of Delaware
nn
Currently at Universit ́a di Pisa, Pisa, Italy
oo
Currently at Continental AG, Hannover, Germany
pp
Currently at Japan Corporation, Tokyo
qq
Currently at Laser Zentrum Hannover
a variety of mechanisms, including nonaxisymmetric dis-
tortions of the star [1, 2, 3, 4, 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 quasi-periodic wave whose frequency changes slowly
during the observation time due to energy loss through
gravitational wave emission, and possibly other mecha-
nisms. At an Earth-based detector the signal exhibits
amplitude and phase modulations due to the motion of
the Earth with respect to the source. The intrinsic grav-
itational wave amplitude is likely to be several orders
of magnitude smaller than the typical root-mean-square
value of the detector noise, hence detection 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 sensitiv-
ity is achieved through hierarchical search schemes [10,
11, 12]. Such schemes alternate incoherent and coherent
search stages in order to first efficiently identify statisti-
cally significant candidates and then follow them up with
more sensitive, albeit computationally intensive, meth-
ods. Hierarchical search schemes have been investigated
only theoretically, under the simplified assumption of
Gaussian and stationary instrumental noise; the compu-
tational costs have been estimated only on the basis of
counts of floating point operations necessary to evalu-
ate the relevant detection statistic and have not taken
into account additional costs coming
e.g.
from data in-
put/output; computational savings obtainable through
efficient dedicated numerical implementations have also
been neglected. Furthermore, general theoretical investi
-
gations 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 col-
lected by LIGO during the second science run with the
Livingston and Hanford 4-km interferometers. The sec-
ond LIGO science run took place over the period 14 Feb.
2003 to 14 Apr. 2003. As we will show, this analysis
requires careful tuning of a variety of search parameters
and implementation choices, such as the tilings of the pa-
rameter space, the selection of the data, and the choice of
3
the coincidence windows, that are difficult to determine
on purely theoretical grounds. This paper complements
the study presented in [13] where we reported results ob-
tained by applying an incoherent analysis 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.
The search described in this paper has been the
test-bench for the core science analysis that the Ein-
stein@home [17] project is carrying out now. The de-
velopment 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 param-
eters and in the frequency bands 464 Hz – 484 Hz and
604 Hz – 624 Hz. It is the first time that a coherent anal-
ysis is carried out over such a wide frequency band, using
data in coincidence and (in one case) for a rotating neu-
tron star in a binary system; the only other example of
a somewhat similar analysis is an all-sky search over two
days of data from the Explorer resonant detector over a
0.76 Hz band around 922 Hz [18, 19, 20].
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 Section 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 Section II we
describe the instrument configuration during the second
science run and the details of the data taking. In Sec-
tion III we review the current astrophysical understand-
ing of neutron stars as gravitational wave sources, in-
cluding 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 ex-
plored in this paper. In Section IV we review the signal
model and discuss the search area considered here. In
Section V we describe the analysis pipeline. In Section VI
we present and discuss the results of the analyses. In Sec-
tion 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 ob-
servatory 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 detec-
tor GEO600; results from the analysis of those data were
presented in [15, 21, 22, 23], while the instrument sta-
tus was detailed in [24]. Significant improvements in the
strain sensitivity of the LIGO interferometers (an order
of magnitude over a broad band) culminated in the sec-
ond science run (S2), which took place from February 14
to April 14, 2003. Details of the S2 run, including de-
tector improvements between S1 and S2 can be found in
[16], Section IV of [25], and Section II of [13] and [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
(H2), while a single 4-km counterpart (L1) exists in Liv-
ingston Parish, LA. Differential motions are sensed inter-
ferometrically, and the resultant sensitivity is broadban
d
(40 Hz – 7 kHz), with spectral disturbances such as 60 Hz
power line harmonics evident in the noise spectrum (see
Fig. 1). Optical resonance, or “lock”, in a given detector
is maintained by servo loops; lock may be interrupted by,
for example, seismic transients or poorly conditioned ser-
vos. S2 duty cycles, accounting for periods in which lock
was broken and/or detectors were known to be function-
ing 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 detailed 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—
involves 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 i
n
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 pre-
4
100
200
300
400
500
600
700
10
−23
10
−22
10
−21
10
−20
10
−19
Frequency (Hz)
Strain spectral amplitude/
√
Hz
H2
H1
L1
FIG. 1: Typical one-sided amplitude spectral densities of d
e-
tector noise during the second science run, for the three LIG
O
instruments. The solid black line is the design sensitivity
for
the two 4-km instruments L1 and H1.
sented in Sec. V H.
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 strength
s.
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 ob-
jects: (1) nonaxisymmetric distortions of the solid part
of the star [1, 2, 3, 4, 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
ǫ
=
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 mo-
ment of inertia. A nonaxisymmetric neutron star rotat-
ing with frequency
ν
emits periodic gravitational waves
with amplitude
h
0
=
4
π
2
G
c
4
I
zz
f
2
d
ǫ,
(2)
where
G
is Newton’s gravitational constant,
c
is the speed
of light,
I
zz
is the principal moment of inertia of the ob-
ject,
f
(equal to 2
ν
) 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 op-
timally oriented 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]
ǫ
max
≈
5
×
10
−
7
(
σ
10
−
2
)
,
(3)
where
σ
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. Eq. (3) uses a fiducial
breaking strain of 10
−
2
since that is roughly the upper
limit for the best terrestrial alloys. However,
σ
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
feature 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
ǫ
max
≈
4
×
10
−
4
(
σ
10
−
2
)
.
(4)
This much higher value of
ǫ
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,
ǫ
max
≈
9
×
10
−
6
(
σ
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 bina-
ries have a natural way of reaching and maintaining the
maximum deformation, 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 temperature variations throughout the crust.
Through the temperature dependence of electron cap-
ture, these variations can lead to “hills” in hotter areas
which extend down to the dense inner crust, and with
a reasonable temperature variation the ellipticity might
5
reach the maximum elastic value [1]. The accreted ma-
terial can also be held up in mountains on the surface by
the magnetic field itself: The matter is a good conduc-
tor, 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 configura-
tion, accretion rate, and temperature, the ellipticity fro
m
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 immediatel
y
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
ǫ
≈
{
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 timescale.
An alternative way of generating asymmetry is the
r
-
modes, fluid oscillations dominated by the Coriolis restor-
ing force. These modes may be unstable to growth
through gravitational radiation reaction (the CFS insta-
bility) under astrophysically realistic conditions. Rath
er
than go into the many details of the physics and astro-
physics, we refer the reader to a recent review [29] of the
literature and summarize here only what is directly rel-
evant 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 be-
cause the emission is very short-lived, low amplitude, or
both. Accreting neutron stars (or quark stars) are a bet-
ter prospect for a detection of
r
-mode gravitational radi-
ation 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
(
θ
w
0
.
1
) (
1 kpc
d
)
(
ν
500 Hz
)
2
(7)
where
θ
w
is the wobble amplitude in radians. Such wob-
ble 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 mech-
anisms, the most likely sources of detectable gravitationa
l
waves are isolated neutron stars (through deformations)
and accreting neutron stars in binaries (through defor-
mations or
r
-modes).
B. Isolated neutron stars
The target population of this search is isolated rotat-
ing compact stars that have not been observed electro-
magnetically. Current models of stellar evolution sug-
gest 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 nu-
merous 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 energet-
ics 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 ar-
gument is a modification of an observation due to Bland-
ford (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 very
high spin rate and then spin down principally due to
gravitational wave emission. For simplicity we shall also
assume that all neutron stars follow the same spin-down
law ̇
ν
(
ν
) or equivalently
̇
f
(
f
), although this turns out to
be unnecessary to the conclusion. It is helpful to express
the spin-down law in terms of the spin-down timescale
τ
gw
(
f
)
≡
f
|
4
̇
f
(
f
)
|
.
(8)
For a neutron star with constant ellipticity,
τ
gw
(
f
) is the
time for the gravitational wave frequency to drift down
to
f
from some initial, much higher spin frequency—
but the argument does not place any requirements on
the ellipticity or the emission mechanism. A source’s
gravitational wave amplitude
h
0
is then related to
τ
gw
(
f
)
by
h
0
(
f
) =
d
−
1
√
5
GI
zz
8
c
3
τ
gw
(
f
)
.
(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.
6
We now consider the distribution of neutron stars
in space and frequency. Let
N
(
f
)∆
f
be the num-
ber of Galactic neutron stars in the frequency range
[
f
−
∆
f/
2
,f
+ ∆
f/
2]. We assume that the birthrate has
been roughly constant over about the last 10
9
years, so
that this distribution has settled into a statistical stead
y
state:
dN
(
f
)
/dt
= 0. Then
N
(
f
)
̇
f
is just the neutron
star birthrate 1
/τ
b
, where
τ
b
may be as short as 30 years.
For simplicity, we model the spatial distribution of neu-
tron stars in our Galaxy as that of a uniform cylindrical
disk, with radius
R
G
≈
10 kpc and height
H
≈
600 pc.
Then the density
n
(
f
) of neutron stars near the Earth,
in the frequency range [
f
−
∆
f/
2
,f
+ ∆
f/
2], is just
n
(
f
)∆
f
= (
πR
2
G
H
)
−
1
N
(
f
)∆
f
.
Let
ˆ
N
(
f,r
) be that portion of
N
(
f
) due to neutron
stars whose distance from Earth is less than
r
. For
H/
2
.
r
.
R
G
, we have
d
ˆ
N
(
f,r
)
dr
= 2
πrHn
(
f
)
(10)
= 2
N
(
f
)
r
R
2
G
(11)
(and it drops off rapidly for
r
&
R
G
). Changing variables
from
r
to
h
0
using Eqs. (8) and (9), we have
d
ˆ
N
(
f,h
0
)
dh
0
=
3
2
5
GI
zz
c
3
τ
b
R
2
G
f
−
1
h
−
3
0
.
(12)
Note that the dependence on the poorly known
τ
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
τ
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
∫
∞
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 fifty percent
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
τ
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
g cm
2
,
R
G
= 10 kpc, and
τ
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
τ
gw
(
f
) was unnecessary, since
τ
gw
(
f
)
disappeared from Eq. (12) and the subsequent equations
that led to
h
max
0
. Had we divided neutron stars into
different classes labelled by
i
and assigned each a spin-
down law
τ
i
gw
(
f
) and birthrate 1
/τ
i
b
, each would have
contributed its own term to
d
ˆ
N/dh
0
which would have
been independent of
τ
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
.
r
.
R
G
. We cannot evade the upper limit by assuming
that the neutron stars have extremely long spin-down
times (so that
r < H/
2) or extremely short ones (so that
the brightest is outside our Galaxy,
r > R
G
). If the
brightest sources are at
r < H/
2 (as happens if these
sources have long spin-down times,
τ
gw
&
τ
b
(2
R
G
/H
)
2
),
then our estimate of
h
max
0
only decreases, because at short
distances the spatial distribution of neutron stars be-
comes 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
τ
gw
(
f
) (in the LIGO range) is much shorter
than
τ
b
, then the probability that such an object ex-
ists inside our Galaxy is
≪
1. For example, a neutron
star with
τ
gw
(
f
) = 3 yr located at
r
= 10 kpc would
have
h
0
= 4
.
14
×
10
−
24
, but the probability of currently
having a neutron star with this (or shorter)
τ
gw
is only
τ
gw
/τ
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 accre-
tion, and then spin down again. This effectively increases
the neutron 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 Gaus-
sian stationary noise with a false alarm rate of 1% and a
false dismissal rate of 10% is [15]
h
h
0
(
f
)
i
= 11
.
4
√
S
h
(
f
)
T
obs
,
(17)
where
T
obs
is the integration time and the angled brackets
indicate an average source. In all-sky searches for pulsars
7
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
h
max
0
of Eq. (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 mismatch between the signal and matched filter
causes the detection statistic (see Sec. V A) 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
deformation), our search is restricted to pulsars with el-
lipticity
ǫ
less than
ǫ
sd
=
(
5
c
5
max[
̇
f
]
32
π
4
GI
zz
f
5
)
1
/
2
.
(19)
This limit, derived from combining the quadrupole for-
mula for GW luminosity
dE
dt
=
1
10
G
c
5
(2
πf
)
6
I
2
zz
ǫ
2
(20)
(the first factor is 1/10 instead of 1/5 due to time aver-
aging of the signal) with the kinetic energy of rotation
E
=
1
2
π
2
f
2
I
zz
,
(21)
(assuming
f
= 2
ν
) takes the numerical value
ǫ
sd
= 9
.
6
×
10
−
6
(
10
45
g cm
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 val-
ues of the ellipticity, using an average value for noise in
the detectors during the S2 run. The curves show the
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 1
0
hours, of about 21, which is extremely close to the value of 20
which is used in this analysis as threshold for registering c
andi-
date events. Thus combining Eqs. (2) and (17) determines the
smallest amplitude that our search pipeline could detect (c
orre-
sponding to a signal just at the threshold), provided approp
riate
follow-up studies of the registered events ensued.
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: Effective average range (defined in the text) of our
search as a function of frequency for three ellipticities: 1
0
−
6
(maximum for a normal neutron star), 10
−
5
(maximum for a
more optimistic object), and
ǫ
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.
average distance, in the sense of the definition (17), at
which a source may be detected.
The dark gray region shows that a GW pulsar with
ǫ
= 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
ǫ
= 10
−
5
could be detected if templates with suffi-
ciently 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
ǫ
= 10
−
5
spins
down too fast to be detected with the no-spin-down tem-
plates used. The thick line indicates the distance limit
for the (frequency-dependent) maximum value of epsilon
that could be detected with the templates used in this
search. At certain frequencies below 300 Hz, a GW pul-
sar could be seen somewhat farther away than 30 pc, but
only if it has
ǫ >
10
−
5
. Although
ǫ
sd
and the corre-
sponding curve were derived assuming a quadrupolar de-
formation as the emission mechanism, the results would
be similar for other mechanisms. Equation (21) includes
an implicit factor
f
2
/
(2
ν
)
2
, which results in
ǫ
sd
and the
corresponding range (for a fixed GW frequency
f
) being
multiplied by
f/
(2
ν
), 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 closest known
pulsar in the LIGO frequency band, PSR J0437
−
4715,
is about 140 pc [32]. The distance to the closest known
neutron star, RX J185635
−
3754, is about 120 pc [34].
Therefore this search would be sensitive only to nearby
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
8
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 [35]). 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 pre-
sented here will be much more sensitive. The goal of
initial LIGO is to take a year of data at design sensitiv-
ity. 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 sen-
sitivity 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 angu-
lar momentum, a different but equally robust argument
(i.e., practically independent of the details of the emis-
sion 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 ac-
creted material is tidally stripped from a low-mass com-
panion 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, 36, 37], if one as-
sumes that spin-down from GW emission is in equilib-
rium with accretion torque, then the GW amplitude
h
0
is directly related to the X-ray luminosity:
h
0
≈
5
×
10
−
27
(
300 Hz
ν
)
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 con-
nection 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
.
ν
.
620 Hz [38]. Since most neutron stars will have accreted
enough matter to spin them up to near their theoreti-
cal maximum spin frequencies, estimated at
∼
1400 Hz,
the observed spin distribution is hard to explain without
some competing mechanism, such as gravitational radia-
tion, to halt the spin-up. Since the gravitational torque
scales as
ν
5
, gravitational radiation is also a natural ex-
planation 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 neu-
tron star brightest in X-rays is also the brightest in grav-
itational 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 inter-
ferometers. The gravitational wave strains from other
accreting 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 rota-
tion frequency of the source (details are provided in Sec-
tion IV B 2). The immediate implication for a coherent
search for gravitational waves from such a neutron star
is that a very large number of discrete templates are re-
quired to cover the relevant parameter space, which in
turn dramatically increases the computational costs [39].
The optimal sensitivity that can be achieved with a co-
herent search is therefore set primarily by the length of
the data set that one can afford to process (with fixed
computational resources) and the spectral density of the
detector noise. As we discuss in Section IV B 2, the max-
imum span of the observation time set by the computa-
tional 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 de-
scribing is determined by each stage of the pipeline,
which we describe in detail in Section V B. Assuming
that the noise in the instrument can be described as a
Gaussian and stationary 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 es-
timate 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 implemented for the analysis. We
contrast this with the hypothetical case in which the Sco
X-1 parameters are known perfectly making it a single
9
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 optimall
y
selected 6 hr dataset (chosen specifically for our search ban
d).
The gray curve (second from the top) shows the sensitivity
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
for L1 (black) and H1 (gray);
S
h
(
f
)
is the typical noise spectral density that characterizes th
e 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.
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 sce-
narios is primarily due to the large parameter space we
have to search. This has two consequences, which con-
tribute to degrading the sensitivity of the analysis: (i) we
are computationally limited by the vast number of tem-
plates that we must search and therefore must reduce the
observation to a subsection of the S2 data, and (ii) sam-
pling 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 (see Sec-
tion IV B 2) is a factor of
≈
5000 less sensitive than the
characteristic amplitude given in Eq. (24). In the hy-
pothetical 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 “hierarchi-
cal pipeline” that will allow us to achieve quasi-optimal
sensitivity with fixed computational resources.
IV. SIGNAL MODEL
A. The signal at the detector
We consider a rotating neutron star with equatorial co-
ordinates
α
(right ascension) and
δ
(declination). Gravi-
tational waves propagate in the direction
ˆ
k
and the star
spins around an axis whose direction, assumed to be con-
stant, is identified by the unit vector ˆ
s
.
The strain
h
(
t
) recorded at the interferometer output
at detector time
t
is:
h
(
t
) =
h
0
[
1
2
(
1 + cos
2
ι
)
F
+
(
t
;
α,δ,ψ
) cos Φ(
t
)
+ cos
ιF
×
(
t
;
α,δ,ψ
) sin Φ(
t
)
]
,
(25)
where
ψ
is the polarization angle, defined as tan
ψ
=
[(ˆ
s
ˆ
k
) (ˆ
z
ˆ
k
)
−
(ˆ
s
ˆ
k
)]
/
ˆ
k
(ˆ
s
×
ˆ
z
), ˆ
z
is the direction to
the north celestial pole, and cos
ι
=
ˆ
k
ˆ
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) antenna beam patterns
F
+
,
×
(
t
;
α,δ,ψ
). The
term Φ(
t
) in Eq. (25) represents the phase of the received
gravitational signal.
The analysis challenge to detect weak quasi-periodic
continuous gravitational waves stems from the Doppler
shift of the gravitational phase Φ(
t
) due to the relative
motion between the detector and the source. It is con-
venient to introduce the following times:
t
, the time
measured at the detector;
T
, the solar-system-barycenter
(SSB) coordinate time; and
t
p
, the proper time in the rest
frame of the pulsar
2
.
The timing model that links the detector time
t
to the
coordinate time
T
at the SSB is:
T
=
t
+
~r
ˆ
n
c
+ ∆
E
⊙
−
∆
S
⊙
,
(26)
where
~r
is a (time-dependent) vector from the SSB to
the detector at the time of the observations, ˆ
n
is a unit
vector towards the pulsar (it identifies the source posi-
tion in the sky) and ∆
E
⊙
and ∆
S
⊙
are the solar system
Einstein and Shapiro time delays, respectively [40]. For
an isolated neutron star
t
p
and
T
are equivalent up to
an additive constant. If the source is in a binary sys-
tem, as it is the case for Sco X-1, significant additional
accelerations are involved, and a further transformation
is required to relate the proper time
t
p
to the detector
time
t
. Following [40], we have:
T
−
T
0
=
t
p
+ ∆
R
+ ∆
E
+ ∆
S
(27)
2
Notice that our notation for the three different times is diffe
r-
ent from the established conventions adopted in the radio pu
lsar
community,
e.g.
[40].
10
where ∆
R
, the Roemer time delay, is analogous to the
solar system term (
~r
ˆ
n
)
/c
; ∆
E
and ∆
S
are the orbital
Einstein and Shapiro time delay, analogous to ∆
E
⊙
and
∆
S
⊙
; and
T
0
is an arbitrary (constant) reference epoch.
For the case of Sco X-1, we consider a circular orbit
for the analysis (cf Section IV B 2 for more details) and
therefore set ∆
E
= 0. Furthermore, the binary is non-
relativistic and from the source parameters we estimate
∆
S
<
3
s which is negligible. For a circular orbit, the
Roemer time delay is simply given by
∆
R
=
a
p
c
sin(
u
+
ω
)
(28)
where
a
p
is the radius of the neutron star orbit projected
on the line of sight,
ω
the argument of the periapsis and
u
the so-called eccentric anomaly; for the case of a circular
orbit
u
= 2
π
(
t
p
−
t
p,
0
)
/P
, where
P
is the period of the bi-
nary and
t
p,
0
is a constant reference time, conventionally
referred to as the “time of periapse passage”.
In this paper we consider gravitational waves whose
intrinsic
frequency drift is negligible over the integration
time of the searches (details are provided in the next
section), both for the blind analysis of unknown isolated
neutron stars and Sco X-1. The phase model is simplest
in this case and given by:
Φ(
t
p
) = 2
πf
0
t
p
+ Φ
0
,
(29)
where Φ
0
is an overall constant phase term and
f
0
is the
frequency of the gravitational wave at the reference time.
B. Parameter space of the search
The analysis approach presented in this paper is used
for two different searches with different search parame-
ters. Both searches require exploring a three dimensional
parameter space, made up of two “position parameters”
(whose nature is different for the two searches) and the
unknown frequency of the signal. For the all-sky blind
analysis aimed at unknown isolated neutron stars one
needs to Doppler correct the phase of the signal for any
given point in the sky, based on the angular resolution of
the instrument over the observation time, and so a search
is performed on the sky coordinates
α
and
δ
. For the Sco
X-1 analysis, the sky location of the system is known,
however the system is in a binary orbit with poorly mea-
sured orbital elements; thus, one needs to search over a
range of orbital parameter values. The frequency search
parameter is for both searches the
f
0
defined by Eq. (29),
where the reference time has been chosen to be the time-
stamp of the first sample of the data set. The frequency
band over which the two analyses are carried out is also
different, and the choice is determined by astrophysical
and practical reasons. As explained in Sec. V C, the data
set in H1 does not coincide in time with the L1 data set
for either of the analyses. Consequently a signal with a
non-zero frequency derivative would appear at a differ-
ent frequency template in each data set. However, for
the maximum spin-down rates considered in this search,
and given the time lag between the two data sets, the
maximum difference between the search frequencies hap-
pens for the isolated objects search and amounts to 0
.
5
mHz. We will see that the frequency coincidence win-
dow is much larger than this and that when we discuss
spectral features in the noise of the data and locate them
based on template-triggers at a frequency
f
0
, the spec-
tral resolution is never finer than 0
.
5 mHz. So for the
practical purposes of the present discussion we can ne-
glect this difference and will often refer to
f
0
generically
as the signal’s frequency.
1. Isolated neutron stars
The analysis for isolated neutron stars covers the entire
sky and we have restricted the search to the frequency
range 160–728
.
8 Hz. The low frequency end of the band
was chosen because the depth of our search degrades sig-
nificantly below 160 Hz, see Fig. 2. The choice of the
high frequency limit at 728.8 is primarily determined by
the computational burden of the analysis, which scales
as the square of the maximum frequency that is searched
for.
In order to keep the computational costs at a reason-
able level (
<
1 month on
<
∼
800 CPUs), no explicit
search over spin-down parameters was carried out. The
length of the data set that is analyzed is approximately
10 hours, thus no loss of sensitivity is incurred for sources
with spin-down rates smaller than 4
×
10
−
10
Hz s
−
1
; see
Eq. (18). This is a fairly high spin-down rate compared
to those measured in isolated radio pulsars; however it
does constrain the sensitivity for sources above 300 Hz,
as can be seen from Fig. 2.
2. Sco X-1
Sco X-1 is a neutron star in a 18
.
9 h orbit around a low
mass (
∼
0
.
42
M
⊙
) companion at a distance
r
= 2
.
8
±
0
.
3
kpc from Earth. In this section we review our present
knowledge of the source parameters that are relevant for
gravitational wave observations. Table I contains a sum-
mary of the parameters and the associated uncertainties
that define the search area. In what follows we will as-
sume the observation time to be 6 hours. This is approx-
imately what was adopted for the analysis presented in
this paper. We will justify this choice at the end of the
section.
The most accurate determination of the Sco X-1 sky
position comes from Very Long Baseline Array (VLBA)
observations [41, 42] and is reported in Table I. The over-
all error on the source location is
∼
0
.
5 arcsec, which is
significantly smaller than the
∼
100 arcsec sky resolution
associated with a 6 hour GW search. Hence we assume
the position of Sco X-1 (
i.e.
the barycenter of the binary
11
right ascension
α
16h 19m 55
.
0850s
declination
δ
−
15
o
38
′
24
.
9
′′
proper motion (east-west direction)
x
−
0
.
00688
±
0
.
00007 arcsec yr
−
1
proper motion (north-south direction)
y
0
.
01202
±
0
.
00016 arcsec yr
−
1
distance
d
2
.
8
±
0
.
3 kpc
orbital period
P
68023
.
84
±
0
.
08 sec
time of periapse passage
̄
T
731163327
±
299 sec
projected semi-major axis
a
p
1
.
44
±
0
.
18 sec
eccentricity
e
<
3
×
10
−
3
QPOs frequency separation
237
±
5 Hz
≤
∆
ν
QPO
≤
307
±
5 Hz
TABLE I: The parameters of the low-mass X-ray binary Scorpiu
s X-1. The quoted measurement errors are all 1-
σ
. We refer
the reader to the text for details and references.
system) to be exactly known and we “point” (in software)
at that region of the sky.
Three parameters describe the circular orbit of a star
in a binary system: the orbital period (
P
), the projection
of the semi-major axis of the orbit
a
p
(which for
e
= 0
corresponds to the projected radius of the orbit) and the
location of the star on the orbit at some given reference
time. For eccentric orbits this is usually parameterized
by the time of periapse passage (or time of periastron).
In the case of a circular orbit we define the
orbital phase
reference time
̄
T
as the time at which the star crosses the
ascending node as measured by an observer at the SSB.
This is equivalent to setting the argument of periapse
(the angle between the ascending node and the direction
of periapsis) to zero.
In the case of Sco X-1,
P
is by far the most accurately
determined parameter [43], and over a 6 hour search it
can be considered known because the loss of signal-to-
noise ratio (SNR) introduced by matching two templates
with any value of
P
in the range of Table I is negligible.
P
becomes a search parameter, requiring multiple filters,
only for coherent integration times
>
∼
10
6
s. The major
orbital parameters with the largest uncertainties are the
projected semi-major axis of the orbit along the line of
sight,
a
p
, and the orbital phase reference time. The large
uncertainty on
a
p
is primarily due to the poor determi-
nation of the orbital velocity (40
±
5 km
s
−
1
[44]). The
uncertainty on the orbital phase reference time is due to
the difficulty in locating the Sco X-1 low-mass compan-
ion on the orbit. The search therefore requires a discrete
grid of filters in the (
a
p
,
̄
T
) space.
We assume that Sco X-1 is in a circular orbit, which
is what one expects for a semi-detached binary system
and which is consistent with the best fits of the orbital
parameters [45]. However, orbital fits for models with
e
6
= 0 were clearly dominated by the noise introduced by
the geometry of the Roche lobe [45]. Over an integra-
tion time of
∼
6 h, the eccentricity needs to be smaller
than
∼
10
−
4
in order for the detection statistic
F
to be
affected less than 1%; for
e
≈
10
−
3
, losses of the order
of 10% are expected and are consistent with the results
presented later in the paper. Unfortunately current ob-
servations are not able to constrain
e
to such levels of
accuracy: in this paper we adopt the strategy of analyz-
ing the data under the assumption
e
= 0, and we quantify
(for a smaller set of the parameter space) the efficiency of
the pipeline in searching for gravitational waves emitted
by a binary with non zero eccentricity; in other words, we
quote upper limits for different values of the eccentricity
that are obtained with non-optimal search templates.
The last parameter we need to search for is the fre-
quency of the gravitational radiation
f
. The rotation
frequency
ν
of Sco X-1 is inferred from the difference
of the frequency of the kHz quasi periodic oscillations
(QPOs). Unfortunately this frequency difference is not
constant, and over a 4 day observation [46] has shown a
very pronounced drift between 237
±
5 Hz to 307
±
5 Hz,
where the errors should be interpreted as the 1
σ
val-
ues [47]. This drifting of QPO frequency separation was
found to be positively correlated to the inferred mass ac-
cretion rate. It is also important to stress that there is
a still unresolved controversy as to whether the adopted
model that links
ν
to the difference of the frequency of
the kHz QPOs is indeed the correct one, and if it is valid
for all the observed LMXBs. Moreover, the gravitational
wave frequency
f
is related to
ν
in a different way, de-
pending on the model that is considered:
f
= 2
ν
if one
considers nonaxisymmetric distortions and
f
= (4
/
3)
ν
if one considers unstable
r
-modes. It is therefore clear
that a search for gravitational waves from Sco X-1 should
assume that the frequency is essentially unknown and
the whole LIGO sensitivity band (say from
≈
100 Hz to
≈
1 kHz) should be considered. Because of the heavy
computational burden, such a search requires a differ-
ent approach (this search is currently in progress). For
the analysis presented in this paper, we have decided to
confine the search to GWs emitted by nonaxisymmet-
ric distortions (
f
= 2
ν
)
and
to constrain the frequency
band to the two 20 Hz wide bands (464–484 Hz and 604–
624 Hz) that bound the range of the drift of
ν
, according
to currently accepted models for the kHz QPOs. The to-
tal computational time for the analysis can be split into
12