of 22
First all-sky upper limits from LIGO on the strength of periodic gravitational waves using
the Hough transform
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PHYSICAL REVIEW D
72,
102004 (2005)
1550-7998
=
2005
=
72(10)
=
102004(22)$23.00
102004-1
©
2005 The American Physical Society
D. Webber,
12
A. Weidner,
19,2
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31
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12
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13
H. Welling,
31
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1
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L. Zhang,
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R. Zhu,
1
N. Zotov,
17
M. Zucker,
15
and J. Zweizig
12
(LIGO Scientific Collaboration)
an
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
Australian National University, Canberra, 0200, Australia
4
California Institute of Technology, Pasadena, California 91125, USA
5
California State University Dominguez Hills, Carson, California 90747, USA
6
Caltech-CaRT, Pasadena, California 91125, USA
7
Cardiff University, Cardiff, CF2 3YB, United Kingdom
8
Carleton College, Northfield, Minnesota 55057, USA
9
Columbia University, New York, New York 10027, USA
10
Hobart and William Smith Colleges, Geneva, New York 14456, USA
11
Inter-University Centre for Astronomy and Astrophysics, Pune – 411007, India
12
LIGO —California Institute of Technology, Pasadena, California 91125, USA
13
LIGO —Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
14
LIGO Hanford Observatory, Richland, Washington 99352, USA
15
LIGO Livingston Observatory, Livingston, Louisiana 70754, USA
16
Louisiana State University, Baton Rouge, Louisiana 70803, USA
17
Louisiana Tech University, Ruston, Louisiana 71272, USA
18
Loyola University, New Orleans, Louisiana 70118, USA
19
Max Planck Institut fu
̈
r Quantenoptik, D-85748, Garching, Germany
20
Moscow State University, Moscow, 119992, Russia
21
NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
22
National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
23
Northwestern University, Evanston, Illinois 60208, USA
24
Salish Kootenai College, Pablo, Montana 59855, USA
25
Southeastern Louisiana University, Hammond, Louisiana 70402, USA
26
Stanford University, Stanford, California 94305, USA
27
Syracuse University, Syracuse, New York 13244, USA
28
The Pennsylvania State University, University Park, Pennsylvania 16802, USA
29
The University of Texas at Brownsville and Texas Southmost College, Brownsville, Texas 78520, USA
30
Trinity University, San Antonio, Texas 78212, USA
31
Universita
̈
t 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 Kingdom
34
University of Florida, Gainesville, Florida 32611, USA
35
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
36
University of Michigan, Ann Arbor, Michigan 48109, USA
37
University of Oregon, Eugene, Oregon 97403, USA
38
University of Rochester, Rochester, New York 14627, USA
39
University of Wisconsin –Milwaukee, Milwaukee, Wisconsin 53201, USA
40
Vassar College, Poughkeepsie, New York 12604, USA
41
Washington State University, Pullman, Washington 99164, USA
(Received 13 September 2005; published 28 November 2005)
We perform a wide parameter-space search for continuous gravitational waves over the whole sky and
over a large range of values of the frequency and the first spin-down parameter. Our search method is
based on the Hough transform, which is a semicoherent, computationally efficient, and robust pattern
recognition technique. We apply this technique to data from the second science run of the LIGO detectors
and our final results are all-sky upper limits on the strength of gravitational waves emitted by unknown
isolated spinning neutron stars on a set of narrow frequency bands in the range
200
400 Hz
. The best
upper limit on the gravitational-wave strain amplitude that we obtain in this frequency range is
4
:
43

10

23
.
DOI:
10.1103/PhysRevD.72.102004
PACS numbers: 04.80.Nn, 07.05.Kf, 95.55.Ym, 97.60.Gb
B. ABBOTT
et al.
PHYSICAL REVIEW D
72,
102004 (2005)
102004-2
I. INTRODUCTION
Continuous gravitational signals emitted by rotating
neutron stars are promising sources for interferometric
gravitational-wave detectors such as GEO 600 [1,2], the
Laser Interferometer Gravitational-Wave Observatory
(LIGO) [3,4], TAMA 300 [5] and VIRGO [6]. There are
several physical mechanisms which might cause a neutron
star to emit periodic gravitational waves. The main possi-
bilities considered in the literature are (i) nonaxisymmetric
distortions of the solid part of the star [7–10], (ii) unstable
r
modes in the fluid [7,11,12], and (iii) free precession (or
‘‘wobble’’) [13,14]. The detectability of a signal depends
on the detector sensitivity, the intrinsic emission strength,
the source distance and its orientation. If the source is not
known, the detectability also depends on the available
computational resources. For some search methods the
detectability of a signal also depends on the source model
used, but an all-sky wide-band search such as detailed here
can detect any of the sources described above.
Previous searches for gravitational waves from rotating
neutron stars have been of two kinds. The first is a search
targeting pulsars whose parameters are known through
radio observations. These searches typically use matched
filtering techniques and are not very computationally ex-
pensive. Examples of such searches are [15,16] which
targeted known radio pulsars, at twice the pulsar frequency,
using data from the first and second science runs of the
GEO 600 and LIGO detectors [17]. No signals were de-
tected and the end results were upper limits on the strength
of the gravitational waves emitted by these pulsars and
therefore on their ellipticity.
The second kind of search looks for as yet undiscovered
rotating neutron stars. An example of such a search is [18]
in which a two-day long data stretch from the Explorer bar
detector is used to perform an all-sky search in a narrow
frequency band around the resonant frequency of the de-
tector. Another example is [19] which uses data from the
LIGO detectors to perform an all-sky search in a wide
frequency band using 10 h of data. The same paper also
describes a search for a gravitational-wave signal from the
compact companion to Sco X-1 in a large
orbital
parameter
space using 6 h of data. The key issue in these wide
parameter-space searches is that a fully coherent all-sky
search over a large frequency band using a significant
amount of data is computationally limited. This is because
looking for weak continuous wave signals requires long
observation times to build up sufficient signal-to-noise
ratio; the amplitude signal-to-noise ratio increases as the
square root of the observation time. On the other hand, the
number of templates that must be considered, and therefore
the computational requirements, scale much faster than
linearly with the observation time. We therefore need
methods which are suboptimal but computationally less
expensive [20 – 24]. Such methods typically involve semi-
coherent combinations of the signal power in short
stretches of data, and the Hough transform is an example
of such a method.
The Hough transform is a pattern recognition algorithm
which was originally invented to analyze bubble chamber
pictures from CERN [25]. It was later patented by IBM
[26], and it has found many applications in the analysis of
digital images [27]. A detailed discussion of the Hough
transform as applied to the search for continuous gravita-
tional waves can be found in [23,28]. In this paper, we
apply this technique to perform an all-sky search for iso-
lated spinning neutron stars using two months of data
collected in early 2003 from the second science run of
the LIGO detectors (henceforth denoted as the S2 run). The
a
Present address: Stanford Linear Accelerator Center.
b
Present address: Jet Propulsion Laboratory.
c
Permanent address: HP Laboratories.
d
Present address: Rutherford Appleton Laboratory.
e
Present address: University of California, Los Angeles.
f
Present address: Hofstra University.
g
Permanent address: GReCO, Institut d’Astrophysique de
Paris (CNRS).
h
Present address: Charles Sturt University, Australia.
i
Present address: Keck Graduate Institute.
j
Present address: National Science Foundation.
k
Present address: University of Sheffield.
l
Present address: Ball Aerospace Corporation.
m
Present address: European Gravitational Observatory.
n
Present address: Intel Corp..
o
Present address: University of Tours, France.
p
Present address: Embry-Riddle Aeronautical University.
q
Present address: Lightconnect Inc..
r
Present address: W.M. Keck Observatory.
s
Present address: ESA Science and Technology Center.
t
Present address: Raytheon Corporation.
u
Present address: New Mexico Institute of Mining and
Technology/Magdalena Ridge Observatory Interferometer.
v
Present address: Mission Research Corporation.
w
Present address: Harvard University.
x
Permanent address: Columbia University.
y
Present address: Lockheed-Martin Corporation.
z
Permanent address: University of Tokyo, Institute for Cosmic
Ray Research.
aa
Permanent address: University College Dublin.
ab
Present address: Research Electro-Optics Inc..
ac
Present address: Institute of Advanced Physics, Baton Rouge,
LA.
ad
Present address: Thirty Meter Telescope Project at Caltech.
ae
Present address: European Commission, DG Research,
Brussels, Belgium.
af
Present address: University of Chicago.
ag
Present address: LightBit Corporation.
ah
Permanent address: IBM Canada Ltd..
ai
Present address: The University of Tokyo.
aj
Present address: University of Delaware.
ak
Permanent address: Jet Propulsion Laboratory.
al
Present address: Shanghai Astronomical Observatory.
am
Present address: Laser Zentrum Hannover.
an
Electronic address: http://www.ligo.org.
FIRST ALL-SKY UPPER LIMITS FROM LIGO ON THE
...
PHYSICAL REVIEW D
72,
102004 (2005)
102004-3
main results of this paper are all-sky upper limits on a set of
narrow frequency bands within the range
200
400 Hz
and
including one spin-down parameter.
Our best
95%
frequentist upper limit on the
gravitational-wave strain amplitude for this frequency
range is
4
:
43

10

23
. As discussed below in Sec. III,
based on the statistics of the neutron star population with
optimistic assumptions, this upper limit is about 1 order of
magnitude larger than the amplitude of the strongest ex-
pected signal. For this reason, the present search is unlikely
to discover any neutron stars, and we focus here on setting
upper limits. As we shall see later, with 1 yr of data at
design sensitivity for initial LIGO, we should gain about 1
order of magnitude in sensitivity, thus enabling us to detect
signals smaller than what is predicted by the statistical
argument mentioned above. Substantial improvements in
the detector noise have been achieved since the S2 obser-
vations. A third science run (S3) took place at the end of
2003 and a fourth science run (S4) at the beginning of
2005. In these later runs LIGO instruments collected data
of improved sensitivity, but still less sensitive than the
instruments’ design goal. Several searches for various
types of gravitational-wave signals have been completed
or are under way using data from the S2 and S3 runs
[16,19,29 – 35]. We expect that the method presented
here, applied as part of a hierarchical scheme and used
on a much larger data set, will eventually enable the direct
detection of periodic gravitational waves.
This paper is organized as follows: Sec. II describes the
second science run of the LIGO detectors; Sec. III summa-
rizes the current understanding of the astrophysical targets;
Sec. IV reviews the waveform from an isolated spinning
neutron star; Sec. V presents the general idea of our search
method, the Hough transform, and summarizes its statisti-
cal properties; Sec. VI describes its implementation and
results on short Fourier transformed data. The upper limits
are given in Sec. VII. Section VIII presents a validation of
our search method using hardware injected signals, and
finally Sec. IX concludes with a summary of our results
and suggestions for further work.
II. THE SECOND SCIENCE RUN
The LIGO detector network consists of a 4 km interfer-
ometer in Livingston, Louisiana (L1), and two interfer-
ometers in Hanford, Washington, one 4 km and the other
2 km (H1 and H2). Each detector is a power-recycled
Michelson interferometer with long Fabry-Perot cavities
in each of its orthogonal arms. These interferometers are
sensitive to quadrupolar oscillations in the space-time met-
ric due to a passing gravitational wave, measuring directly
the gravitational-wave strain amplitude.
The data analyzed in this paper were produced during
LIGO’s 59 d second science run. This run started on
February 14 and ended April 14, 2003. Although the GEO
detector was not operating at the time, all three LIGO
detectors were functioning at a significantly better sensi-
tivity than during S1, the first science run [17], and had
displacement spectral amplitudes near
10

18
m
-
Hz

1
=
2
between
200
and 400 Hz. The strain sensitivities in this
science run were within an order of magnitude of the
design sensitivity for the LIGO detectors. For a description
of the detector configurations for S2 we refer the reader to
[29] Sec. IV and [30] Sec. II.
The reconstruction of the strain signal from the error
signal of the feedback loop, used to control the differential
length of the interferometer arms, is referred to as the
calibration. Changes in the calibration were tracked by
injecting continuous, fixed-amplitude sinusoidal excita-
tions into the end test mass control systems, and monitor-
ing the amplitude of these signals at the measurement error
point. Calibration uncertainties at the three LIGO detectors
during S2 were estimated to be smaller than
11%
[36].
The data were acquired and digitized at a rate of
16 384 Hz
. The duty cycle for the interferometers, defined
as the fraction of the total run time when the interferometer
was locked (i.e., all interferometer control servos operating
in their linear regime) and in its low noise configuration,
were similar to those of the previous science run, approxi-
mately
37%
for L1,
74%
for H1 and
58%
for H2. The
longest continuous locked stretch for any interferometer
during S2 was 66 h for H1. The smaller duty cycle for L1
was due to anthropogenic diurnal low-frequency seismic
noise which prevented operations during the day on week-
days. Recently installed active feedback seismic isolation
has successfully addressed this problem.
Figure 1 shows the expected sensitivity for the Hough
search by the three LIGO detectors during S2. Those
h
0
values correspond to the amplitudes detectable from a
generic continuous gravitational-wave source, if we were
performing a targeted search, with a 1% false alarm rate
and 10% false dismissal rate, as given by Eq. (17). The
differences among the three interferometers reflect differ-
ences in the operating parameters, hardware implementa-
tion of the three instruments, and the duty cycles. Figure 1
also shows the expected sensitivity (at the same false alarm
and false dismissal rates) for initial LIGO
4km
interfer-
ometers running at design sensitivity assuming an obser-
vation time of 1 yr. These false alarm and false dismissal
values are chosen in agreement with [15,19] only for
comparison purposes. Because of the large parameter
search we perform here, it would be more meaningful to
consider a lower false alarm rate, say of
10

10
and then the
sensitivity for a targeted search would get worse by a factor
1
:
5
. The search described in this paper is not targeted and
this degrades the sensitivity even further. This will be
discussed in Sec. VII.
At the end of the S2 run, two fake artificial pulsar signals
were injected for a 12 h period into all three LIGO inter-
ferometers. These hardware injections were done by mod-
ulating the mirror positions via the actuation control
B. ABBOTT
et al.
PHYSICAL REVIEW D
72,
102004 (2005)
102004-4
signals. These injections were designed to give an end-to-
end validations of the search pipelines starting from as far
up the observing chain as possible. See Sec. VIII for
details.
III. ASTROPHYSICAL TARGETS
The target population of this search consists of isolated
rotating neutron stars that are not observed in electromag-
netic waves. Current models of stellar evolution suggest
that our Galaxy contains of order
10
9
neutron stars and that
of order
10
5
are active pulsars [37]. Up to now, only of
order
10
3
objects have been identified as neutron stars,
either by observation as pulsars, or through their x-ray
emission, of which about 90% are isolated [38– 41].
Most neutron stars will remain unobserved electromagneti-
cally for many reasons such as the nonpulsed emission
being faint or the pulses being emitted in a beam which
does not sweep across the Earth. Therefore, there are many
more neutron stars in the target population than have al-
ready been observed.
Although there is great uncertainty in the physics of the
emission mechanism and the strength of an individual
source, we can argue for a robust upper limit on the
strength of the strongest source in the galactic population
that is almost independent of individual source physics.
The details of the argument and an overview of emission
mechanisms can be found in a forthcoming paper [19].
Here we do not repeat the details but merely summarize the
result. For an upper limit we make optimistic assump-
tions — that neutron stars are born rapidly rotating and
spinning down due to gravitational waves, and that they
are distributed uniformly throughout the galactic disc —
and the plausible assumption that the overall galactic
birthrate
1
=
b
is steady. By converting these assumptions
to a distribution of neutron stars with respect to
gravitational-wave strain and frequency, we find there is
a
50%
chance that the strongest signal between frequencies
f
min
and
f
max
has an amplitude of at least
h
0

4

10

24

30 yr

b

ln
f
max
f
min

1
=
2
:
(1)
Of course, with less optimistic assumptions this value
would be smaller.
Comparing Eq. (1) to Fig. 1, a search of S2 data is not
expected to result in a discovery. However, it is still pos-
sible that the closest neutron star is closer than the typical
distance expected from a random distribution of super-
novae (for example due to recent star formation in the
Gould belt as considered in Ref. [42]). It is also possible
that a ‘‘blind’’ search of this sort may discover some
previously unknown class of compact objects not born in
supernovae. More importantly, future searches for previ-
ously undiscovered rotating neutron stars using the meth-
ods presented here will be much more sensitive. The goal
of initial LIGO is to take a year of data at design sensitivity,
which means a factor 10 decrease in the amplitude strain
noise relative to S2, and a factor
10
increase in the length of
the data set. These combine to reduce
h
0
to somewhat
below the value in Eq. (1), and thus initial LIGO at full
sensitivity will have some chance of observing a periodic
signal.
IV. THE EXPECTED WAVEFORM
In order to describe the expected signal waveform we
will use the same notation as [15]. We will briefly summa-
rize it in the next paragraphs for convenience. The form of
the gravitational wave emitted by an isolated spinning
neutron star, as seen by a gravitational-wave detector, is
h

t

F


t;

h


t

F


t;

h


t

;
(2)
where
t
is time in the detector frame,
is the polarization
angle of the wave, and
F

;

are the detector antenna
pattern functions for the two polarizations. If we assume
the emission mechanism is due to deviations of the pulsar’s
shape from perfect axial symmetry, then the gravitational
waves are emitted at a frequency which is exactly twice the
rotational rate
f
r
. Under this assumption, the waveforms
for the two polarizations
h

;

are given by
h


h
0
1

cos
2

2
cos

t

;
(3)
h


h
0
cos

sin

t

;
(4)
10
2
10
3
10
−24
10
−23
10
−22
10
−21
10
−20
10
−19
Frequency (Hz)
h
e
d
u
t
i
l
p
m
a
e
v
a
w
l
a
n
o
i
t
a
t
i
v
a
r
G
0
H1
H2
L1
Design
FIG. 1 (color online).
Characteristic amplitude detectable from
a known generic source with a 1% false alarm rate and 10% false
dismissal rate, as given by Eq. (17). All curves use typical
sensitivities of the three LIGO detectors during S2 and observa-
tion times corresponding to the up-time of the detectors during
S2. The thin line is the expected characteristic amplitude for the
same false alarm and false dismissal rates, but using the initial
LIGO design goal for the
4km
instruments and an effective
observation time of
1yr
.
FIRST ALL-SKY UPPER LIMITS FROM LIGO ON THE
...
PHYSICAL REVIEW D
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where

is the angle between the neutron star’s spin axis
and the direction of propagation of the waves, and
h
0
is the
amplitude:
h
0

16

2
G
c
4
I
zz
f
2
r
d
;
(5)
where
G
is Newton’s gravitational constant,
c
the speed of
light,
I
zz
is the principal moment with the
z
axis being its
spin axis,

:

I
xx

I
yy

=I
zz
is the equatorial ellipticity of
the star, and
d
is the distance to the star.
The phase


t

takes its simplest form in the Solar
System Barycenter (SSB) frame where it can be expanded
in a Taylor series up to second order:


t


0

2


f
0

T

T
0

1
2
_
f

T

T
0

2

:
(6)
Here
T
is time in the SSB frame and
T
0
is a fiducial start
time. The phase

0
, frequency
f
0
and spin-down parame-
ter
_
f
are defined at this fiducial start time. In this paper, we
include only one spin-down parameter in our search; as we
shall see later in Sec. VI B, our frequency resolution is too
coarse for the higher spin-down parameters to have any
significant effect on the frequency evolution of the signal
(for the spin-down ages we consider).
Modulo relativistic effects which are unimportant for
this search, the relation between the time of arrival
T
of the
wave in the SSB frame and in the detector frame
t
is
T

t

n

r
c
;
(7)
where
n
is the unit vector from the detector to the neutron
star, and
r
is the detector position in the SSB frame.
The instantaneous frequency
f

t

of the wave as ob-
served by the detector is given, to a very good approxima-
tion, by the familiar nonrelativistic Doppler formula:
f

t

^
f

t

^
f

t

v

t

n
c
;
(8)
where
^
f

t

is the instantaneous signal frequency in the SSB
frame at time
t
:
^
f

t

f
0

_
f

t

t
0


r

t

n
c

;
(9)
where
t
0
is the fiducial detector time at the start of the
observation and

r

t

r

t

r

t
0

. It is easy to see that
the

r

n
=c
term can safely be ignored so that, to an
excellent approximation
^
f

t

f
0

_
f

t

t
0

:
(10)
V. THE HOUGH TRANSFORM
In this paper, we use the Hough transform to find the
pattern produced by the Doppler shift (8) and the spin
down (10) of a gravitational-wave signal in the time-
frequency plane of our data. This pattern is independent
of the source model used and therefore of the emission
mechanisms. We only assume that the gravitational-wave
signal is emitted by an isolated spinning neutron star.
The starting point for our search is a set of data seg-
ments, each corresponding to a time interval
T
coh
. Each of
these data segments is Fourier transformed to produce a set
of
N
short time-baseline Fourier transforms (SFTs). From
this set of SFTs, calculating the periodograms (the square
modulus of the Fourier transform) and selecting frequency
bins (peaks) above a certain threshold, we obtain a time-
frequency map of our data. In the absence of a signal the
peaks in the time-frequency plane are distributed in a
random way; if signal is present, with high enough
signal-to-noise ratio, some of these peaks will be distrib-
uted along the trajectory of the received frequency of the
signal.
The Hough transform maps points of the time-frequency
plane into the space of the source parameters

f
0
;
_
f;
n

. The
result of the Hough transform is a histogram, i.e., a collec-
tion of integer numbers, each representing the detection
statistic for each point in parameter space. We shall refer to
these integers as the number count. The number counts are
computed in the following way: For each selected bin in
the SFTs, we find which points in parameter space are
consistent with it, according to Eq. (8), and the number
count in all such points is increased by unity. This is
repeated for all the selected bins in all the SFTs to obtain
the final histogram.
To illustrate this, let us assume the source parameters are
only the coordinates of the source in the sky, and this
source is emitting a signal at a frequency
f
0
. Moreover
we assume that at a given time
t
a peak at frequency
f
has
been selected in the corresponding SFT. The Hough trans-
form maps this peak into the loci of points, on the celestial
sphere, where a source emitting a signal with frequency
f
0
could be located in order in order to produce at the detector
a peak at
f
. By repeating this for all the selected peaks in
our data we will obtain the final Hough map. If the peaks in
the time-frequency plane were due only to signal, all the
corresponding loci would intersect in a region of the
Hough map identifying the source position.
An advantage of the Hough transform is that a large
region in parameter space can be analyzed in a single pass.
By dropping the amplitude information of the selected
peaks, the Hough search is expected to be computationally
efficient, but at the cost of being somewhat less sensitive
than others semicoherent methods, e.g., the stack-slide
search [21]. On the other hand, discarding this extra infor-
mation makes the Hough transform more robust against
transient spectral disturbances because no matter how large
a spectral disturbance is in a single SFT, it will contribute at
the most

1
to the number count. This is not surprising
since the optimal statistic for the detection of weak signals
in the presence of a Gaussian background with large non
Gaussian outliers is effectively cut off above some value
B. ABBOTT
et al.
PHYSICAL REVIEW D
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[43,44]. This is, in practice, what the Hough transform
does to large spectral outliers.
With the above short summary at hand, we now give the
relevant notation and equations that will be used later. For
further details and derivations of the equations below, we
refer the reader to [23].
Frequency bins are selected by setting a threshold

th
on
the normalized power

k
defined as

k

2
j
~
x
k
j
2
T
coh
S
n

f
k

;
(11)
where
~
x
k
is the discrete Fourier transform of the data, the
frequency index
k
corresponds to a physical frequency of
f
k

k=T
coh
, and
S
n

f
k

is the single sided power spectral
density of the detector noise. The
k
th frequency bin is
selected if

k

th
, and rejected otherwise. In this way,
each SFT is replaced by a collection of zeros and ones
called a peak-gram.
Let
n
be the number count at a point in parameter space,
obtained after performing the Hough transform on our
data. Let
p

n

be the probability distribution of
n
in the
absence of a signal, and
p

n
j
h

the distribution in the
presence of a signal
h

t

. It is clear that
0
n
N
, and
it can be shown that for stationary Gaussian noise,
p

n

is a
binomial distribution with mean
Nq
where
q
is the proba-
bility that any frequency bin is selected:
p

n

N
n

q
n

1

q

N

n
:
(12)
For Gaussian noise in the absence of a signal, it is easy to
show that

k
follows an exponential distribution so that
q

e


th
. In the presence of a signal, the distribution is
ideally also a binomial but with a slightly larger mean
N
where, for weak signals,

is given by


q

1


th
2


O


2


:
(13)

is the signal-to-noise ratio within a single SFT, and for
the case when there is no mismatch between the signal and
the template:


4
j
~
h

f
k
j
2
T
coh
S
n

f
k

;
(14)
with
~
h

f

being the Fourier transform of the signal
h

t

.
The approximation that the distribution in the presence of a
signal is binomial breaks down for reasonably strong sig-
nals. This is due to possible nonstationarities in the noise,
and the amplitude modulation of the signal which causes

to vary from one SFT to another.
Candidates in parameter space are selected by setting a
threshold
n
th
on the number count. The false alarm and
false dismissal rates for this threshold are defined, respec-
tively, in the usual way:


X
N
n

n
th
p

n

;

X
n
th

1
n

0
p

n
j
h

:
(15)
We choose the thresholds

n
th
;
th

based on the Neyman-
Pearson criterion of minimizing
for a given value of

.It
can be shown [23] that this criteria leads, in the case of
weak signals, large
N
, and Gaussian stationary noise, to

th

1
:
6
. This corresponds to
q

0
:
20
, i.e., we select
about
20%
of the frequency bins from each SFT. This value
of

th
turns out to be independent of the choice of

and
signal strength. Furthermore,
n
th
is also independent of the
signal strength and is given by
n
th

Nq


2
Nq

1

q

q
erfc

1

2


;
(16)
where
erfc

1
is the inverse of the complementary error
function. These values of the thresholds lead to a false
dismissal rate
which is given in [23]. The value of
of
course depends on the signal strength, and on the average,
the weakest signal which will cross the above thresholds at
a false alarm rate

and false dismissal
is given by
h
0

5
:
34
S
1
=
2
N
1
=
4

S
n
T
coh
s
;
(17)
where
S

erfc

1

2


erfc

1

2

:
(18)
Equation (17) gives the smallest signal which can be
detected by the search, and is therefore a measure of the
sensitivity of the search.
VI. THE SEARCH
A. The SFT data
The input data to our search is a collection of calibrated
SFTs with a time baseline
T
coh
of 30 min. While a larger
value of
T
coh
leads to better sensitivity, this time baseline
cannot be made arbitrarily large because of the frequency
drift caused by the Doppler effect (and also the spin down);
we would like the signal power of a putative signal to be
concentrated in less than half the frequency resolution
1
=T
coh
. It is shown in [23] that at
300 Hz
, we could ideally
choose
T
coh
up to
60 min
. On the other hand, we should
be able to find a significant number of such data stretches
during which the interferometers are in lock, the noise is
stationary, and the data are labeled satisfactory according
to certain data quality requirements. Given the duty cycles
of the interferometers during S2 and the nonstationarity of
the noise floor, it turns out that
T
coh

30 min
is a good
compromise which satisfies these constraints. By demand-
ing the data in each 30 min stretch to be continuous
(although there could be gaps in between the SFTs) the
number
N
of SFTs available for L1 data is 687, 1761 for
H1 and 1384 for H2, reducing the nominal duty cycle for
this search.
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The SFT data are calibrated in the frequency domain by
constructing a response function
R

f;t

that acts on the
error signal of the feedback loop used to control the
differential length of the interferometer arms. The response
function
R

f;t

varies in time, primarily due to changes in
the amount of light stored in the Fabry-Perot cavities of the
interferometers. During S2, changes in the response were
computed every 60 sec, and an averaging procedure was
used to estimate the response function used on each SFT.
The SFTs are windowed and high-pass filtered as described
in Sec. IV C 1 of [15]. No further data conditioning is
applied, although the data are known to contain many
spectral disturbances, including the
60 Hz
power line har-
monics and the thermally excited violin modes of test mass
suspension wires.
B. The parameter space
This section describes the portion of parameter space

f
0
;
_
f;
n

we search over, and the resolution of our grid.
Our template grid is not based on a metric calculation (as in
e.g. [20,21]), but rather on a cubic grid which covers the
parameter space as described below. Particular features of
this grid are used to increase computational efficiency as
described in Sec. VI C.
We analyze the full data set from the S2 run with a total
observation time
T
obs
5
:
1

10
6
sec
. The exact value of
T
obs
is different for the three LIGO interferometers [45].
We search for isolated neutron star signals in the frequency
range
200
400 Hz
with a frequency resolution
f

1
T
coh

5
:
556

10

4
Hz
:
(19)
The choice of the range
200
400 Hz
for the analysis is
motivated by the low noise level, and therefore our ability
to set the best upper limits for
h
0
, as seen from Fig. 1.
The resolution
_
f
in the space of first spin-down pa-
rameters is given by the smallest value of
_
f
for which the
intrinsic signal frequency does not drift by more than a
single frequency bin during the total observation time [46]:
_
f

f
T
obs

1
T
obs
T
coh
1
:
1

10

10
Hz
-
s

1
:
(20)
We choose the range of values

_
f
max
_
f
0
, where the
largest spin-down parameter
_
f
max
is about
1
:
1

10

9
Hz
-
s

1
. This yields 11 spin-down values for each
intrinsic frequency. In other words, we look for neutron
stars whose spin-down age is at least

min

^
f=
_
f
max
. This
corresponds to a minimum spin-down age of
5
:
75

10
3
yr
at
200 Hz
, and
1
:
15

10
4
yr
at
400 Hz
. These values of
_
f
max
and

min
are such that all known pulsars have a
smaller spin-down rate than
_
f
max
and, except for a few
supernova remnants, all of them have a spin-down age
significantly greater than the numbers quoted above.
With these values of

min
, it is easy to see that the second
spin-down parameter can be safely neglected; it would take
about
10 yr
for the largest second spin-down parameter to
cause a frequency drift of half a frequency bin.
As described in [23], for every given time, value of the
intrinsic frequency
f
0
and spin down
_
f
, the set of sky
locations
n
consistent with a selected frequency
f

t

cor-
responds to a constant value of
v

n
given by (8). This is a
circle in the celestial sphere. It can be shown that every
frequency bin of width
f
corresponds to an annulus on the
celestial sphere whose width is at least


min

c
v
f
^
f
;
(21)
with
v
being the magnitude of the average velocity of the
detector in the SSB frame.
The resolution
in sky positions is chosen to be
frequency dependent, being at most

1
2


min
.To
choose the template spacing only, we use a constant value
of
v=c
equal to
1
:
06

10

4
. This yields:

9
:
3

10

3
rad

300 Hz
^
f
!
:
(22)
This resolution corresponds to approximately
1
:
5

10
5
sky locations for the whole sky at
300 Hz
. For that, we
break up the sky into 23 sky patches of roughly equal area
and, by means of the stereographic projection, we map
each portion to a plane, and set a uniform grid with spacing
in this stereographic plane. The stereographic projec-
tion maps circles in the celestial sphere to circles in the
plane thereby mapping the annuli in the celestial sphere,
described earlier, to annuli in the stereographic plane. We
ensure that the dimensions of each sky-patch are suffi-
ciently small so that the distortions produced by the ste-
reographic projection are not significant. This is important
to ensure that the number of points needed to cover the full
sky is not much larger than if we were using exactly the
frequency resolution given by Eq. (22).
This adds up to a total number of templates per
1Hz
band at
200 Hz
1
:
9

10
9
while it increases up to
7
:
5

10
9
at
400 Hz
.
C. The implementation of the Hough transform
This section describes in more detail the implementation
of the search pipeline which was summarized in Sec. V.
The first step in this semicoherent Hough search is to
select frequency bins from the SFTs and construct the
peak-grams. As mentioned in Sec. V, our criteria for select-
ing frequency bins is to set a threshold of
1
:
6
on the
normalized power (11), thereby selecting about
20%
of
the frequency bins in every SFT.
The power spectral density
S
n
appearing in Eq. (11) is
estimated by means of a running median applied to the
periodogram of each individual SFT. The window size we
employ for the running median is
w

101
corresponding
B. ABBOTT
et al.
PHYSICAL REVIEW D
72,
102004 (2005)
102004-8
to
0
:
056 Hz
[47]. The running median is a robust method to
estimate the noise floor [48–50] which has the virtue of
discarding outliers which appear in a small number of bins,
thereby providing an accurate estimate of the noise floor in
the presence of spectral disturbances and possible signals.
The use of the median (instead of the mean) to estimate the
power spectral density introduces a minor technical com-
plication (see Appendix A for further details).
The next step is to choose a tiling of the sky. As
described before, we break up the sky into 23 patches, of
roughly equal area. By means of the stereographic projec-
tion, we map each portion to a two dimension plane and set
a uniform grid with a resolution
in this plane. All of our
calculations are performed on this stereographic plane, and
are finally projected back on to the celestial sphere.
In our implementation of the Hough transform, we treat
sky positions separately from frequencies and spin downs.
In particular, we do not obtain the Hough histogram over
the entire parameter space in one go, but rather for a given
sky patch, a search frequency
f
0
and a spin-down
_
f
value.
These are the so-called
Hough maps
(HMs). Repeating this
for every set of frequency and spin-down parameters and
the different sky patches we wish to search over, we obtain
a number of HMs. The collection of all these HMs repre-
sent our final histogram in parameter space.
The HMs could be produced by using a ‘‘brute force’’
approach, i.e., using all the peaks in the time-frequency
plane. But there is an alternative way of constructing them.
Let us define a
partial Hough map
(PHM) as being a
Hough histogram, in the space of sky locations, obtained
by performing the Hough transform using the peaks from a
single
SFT and for a
single
value of the intrinsic signal
frequency and no spin down. This PHM therefore consists
of only zeros and ones, i.e., the collection of the annuli
corresponding to all peaks present in a single peak-gram.
Then each HM can be obtained by summing the appropri-
ate PHMs produced from different SFTs. If we add PHMs
constructed by using the same intrinsic frequency, then the
resulting HM refers to the same intrinsic frequency and no
spin down. But note that the effect of a spin down in the
signal is the same as having a time varying intrinsic fre-
quency. This suggests a strategy to reuse PHMs computed
for different frequencies at different times in order to
compute the HM for a nonzero spin-down case.
Given the set of PHMs, the HM for a given search
frequency
f
0
and a given spin down
_
f
is obtained as
follows: using Eq. (10) calculate the trajectory
^
f

t

in the
time-frequency plane corresponding to
f
0
and
_
f
. If the mid
time stamps of the SFTs are
f
t
i
g
(
i

1...
N
), calculate
^
f

t
i

and find the frequency bin that it lies in; select the
PHM corresponding to this frequency bin. Finally, add all
the selected PHMs to obtain the Hough map. This proce-
dure is shown in Fig. 2.
This approach saves computations because it recognizes
that the same sky locations can refer to different values of
frequency and spin down, and avoids having to redeter-
mine such sky locations more than once. Another advan-
tage of proceeding in this fashion is that we can use
look up
tables
(LUTs) to construct the PHMs. The basic problem to
construct the PHMs is that of drawing the annuli on the
celestial sphere, or on the corresponding projected plane.
The algorithm we use based on LUTs has proved to be
more efficient than other methods we have studied, and this
strategy is also employed by other groups [51].
A LUT is an array containing the list of borders of all the
possible annuli, for a given value of
v
and
^
f
, clipped on the
sky-patch we use. Therefore it contains the coordinates of
the points belonging to the borders that intersect the sky-
patch, in accordance to the tiling we use, together with
information to take care of edge effects. As described in
[23], it turns out that the annuli are relatively insensitive to
the value of the search frequency and, once a LUT has been
constructed for a particular frequency, it can be reused for a
large number of neighboring frequencies thus allowing for
computational savings. The value of
v
used to construct the
LUTs corresponds to the average velocity of the detector in
the SSB frame during the 30 minutes interval of the cor-
responding SFT.
In fact, the code is further sped up by using
partial
Hough map derivatives
(PHMDs) instead of the PHMs,
in which only the borders of the annuli are indicated. A
PHMD consists of only ones, zeros, and minus ones, in
such a way that by integrating appropriately over the
different sky locations one recovers the corresponding
PHM. This integration is performed at a later stage, and
just once, after summing the appropriate PHMDs, to obtain
the final Hough map.
In the pipeline, we loop over frequency and spin-down
values, taking care to update the set of PHMDs currently
used, and checking the validity of the LUTs. As soon as the
LUTs are no longer valid, the code recomputes them again
together with the sky grid. Statistical analyses are per-
One spin−down
parameter
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
PHM
012 34
No spin−down
f
0
Tim
e
in
de
x
o
f
S
FT
Search frequency
FIG. 2.
A partial Hough map (PHM) is a histogram in the
space of sky locations obtained by performing the Hough trans-
form on a single SFT and for a given value of the instantaneous
frequency. A total Hough map is obtained by summing over the
appropriate PHMs. The PHMs to be summed over are deter-
mined by the choice of spin-down parameters which give a
trajectory in the time-frequency plane.
FIRST ALL-SKY UPPER LIMITS FROM LIGO ON THE
...
PHYSICAL REVIEW D
72,
102004 (2005)
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formed on the Hough maps in order to compress the output
size. These include finding the maximum, minimum, mean
and standard deviation of the number counts for each
individual map, the parameters of the loudest event, and
also of all the candidates above a certain threshold. We also
record the maximum number count per frequency ana-
lyzed, maximized over all spin-down values and sky loca-
tions, and a histogram of the number counts for each
1Hz
band. The schematic search pipeline is shown in Fig. 3.
As a technical implementation detail, the search is per-
formed by dividing the
200 Hz
frequency band into
smaller bands of
1Hz
and distributed using Condor [52]
on a computer cluster. Each CPU analyzes a different
1Hz
band using the same pipeline (as described in Fig. 3). The
code itself takes care to read in the proper frequency band
from the SFTs. This includes the search band plus an extra
interval to accommodate for the maximum Doppler shift,
spin down, and the block sized used by the running median.
The analysis described here was carried out on the Merlin
cluster at AEI [53]. The full-sky search for the entire S2
data from the three detectors distributed on 200 CPUs on
Merlin lasted less than half a day.
The software used in the analysis is available in the
LIGO Scientific Collaboration’s CVS archives (see [54]),
together with a suite of test programs, especially for visual-
izing the Hough LUTs. The full search pipeline has also
been validated by comparing the results with indepen-
dently written code that implements a less efficient but
conceptually simpler approach, i.e., for each point in pa-
rameter space

f
0
;
_
f;
n

, it finds the corresponding pattern
in the time-frequency plane and sums the corresponding
selected frequency bins.
D. Number counts from L1, H1 and H2
In the absence of a signal, the distribution of the Hough
number count ideally is a binomial distribution. En-
vironmental and instrumental noise sources can excite
the optically sensed cavity length, or get into the output
signal in some other way, and show up as spectral distur-
bances, such as lines. If no data conditioning is applied,
line interference can produce an excess of number counts
in the Hough maps and mask signals from a wide area in
the sky. Figure 4 shows the comparison of the theoretical
binomial distribution Eq. (12) with the distributions that
we obtain experimentally in two bands:
206
207 Hz
and
343
344 Hz
. The first band contains very little spectral
disturbances while the second band contains some violin
modes. As shown in Fig. 4, the Hough number count
follows the expected binomial distribution for the
clean
band while it diverges from the expected distribution in the
presence of strong spectral disturbances, such as the violin
modes in this case. We have verified good agreement in
several different frequency bands that were free of strong
spectral disturbances.
The sources of the disturbances present in the S2 data
are mostly understood. They consist of calibration lines,
broad
60 Hz
power line harmonics, multiples of
16 Hz
due
to the data acquisition system, and a number of mechanical
resonances, as, for example, the violin modes of the mirror
suspensions [36,55]. The
60 Hz
power lines are rather
broad, with a width of about
0
:
5Hz
, while the calibration
lines and the
16 Hz
data acquisition lines are confined to a
single frequency bin. A frequency comb is also present in
the data, having fundamental frequency at
36
:
867 Hz
for
L1,
36
:
944 Hz
for H1 and
36
:
975 Hz
for H2, some of them
accompanied with side lobes at about
0
:
7Hz
, created by
up-conversion of the pendulum modes of some core optics,
Generate sky−grid
Calculate time−frequency
path and sum PHMDs
Compute Hough map
Compute LUTs and
PHMDs
YES
Increase search
frequency
statistics
Store candidates and
Loop over
sky−patches
Read in SFTs and
generate peak−grams
Loop over
spin−downs
Are
LUTs
valid?
NO
FIG. 3.
The schematic of the analysis pipeline. The input data
are the SFTs and the search parameters. The first step is to select
frequency bins from the SFTs and generate the peak-grams.
Then, the Hough transform is computed for the different sky
patches, frequencies and spin-down values, thus producing the
different Hough maps. The search uses LUTs that are computed
for a given tiling of the sky-patch. The sky grid is frequency
dependent, but it is fixed for the frequency range in which the
LUTs are valid. Then, a collection of PHMDs is built, and for
each search frequency
f
0
and given spin down
_
f
, the trajectory in
the time-frequency plane is computed and the Hough map
obtained by summing and integrating the corresponding
PHMDs. The code loops over frequency and spin-down parame-
ters, updating the sky grid and LUTs whenever required.
Statistical analyses are performed on each map in order to reduce
the output size.
B. ABBOTT
et al.
PHYSICAL REVIEW D
72,
102004 (2005)
102004-10