Einstein@Home search for periodic gravitational waves in early S5 LIGO data
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48
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=
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=
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=
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4
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17
(LIGO Scientific Collaboration)
*
1
Albert-Einstein-Institut, Max-Planck-Institut fu
̈
r Gravitationsphysik, D-14476 Golm, Germany
2
Albert-Einstein-Institut, Max-Planck-Institut fu
̈
r Gravitationsphysik, D-30167 Hannover, Germany
3
Andrews University, Berrien Springs, Michigan 49104, USA
4
Australian National University, Canberra, 0200, Australia
5
California Institute of Technology, Pasadena, California 91125, USA
6
Caltech-CaRT, Pasadena, California 91125, USA
7
Cardiff University, Cardiff, CF24 3AA, United Kingdom
8
Carleton College, Northfield, Minnesota 55057, USA
9
Charles Sturt University, Wagga Wagga, NSW 2678, Australia
10
Columbia University, New York, New York 10027, USA
11
Embry-Riddle Aeronautical University, Prescott, Arizona 86301, USA
12
Eo
̈
tvo
̈
s University, ELTE 1053 Budapest, Hungary
13
Hobart and William Smith Colleges, Geneva, New York 14456, USA
14
Institute of Applied Physics, Nizhny Novgorod, 603950, Russia
15
Inter-University Centre for Astronomy and Astrophysics, Pune - 411007, India
16
Leibniz Universita
̈
t Hannover, D-30167 Hannover, Germany
17
LIGO–California Institute of Technology, Pasadena, California 91125, USA
18
LIGO–Hanford Observatory, Richland, Washington 99352, USA
19
LIGO–Livingston Observatory, Livingston, Louisiana 70754, USA
20
LIGO–Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
21
Louisiana State University, Baton Rouge, Louisiana 70803, USA
22
Louisiana Tech University, Ruston, Louisiana 71272, USA
23
Loyola University, New Orleans, Louisiana 70118, USA
24
Montana State University, Bozeman, Montana 59717, USA
25
Moscow State University, Moscow, 119992, Russia
26
NASA/Goddard Space Flight Center, Greenbelt, Maryland 20771, USA
27
National Astronomical Observatory of Japan, Tokyo 181-8588, Japan
28
Northwestern University, Evanston, Illinois 60208, USA
29
Rochester Institute of Technology, Rochester, New York 14623, USA
30
Rutherford Appleton Laboratory, HSIC, Chilton, Didcot, Oxon OX11 0QX, United Kingdom
31
San Jose State University, San Jose, California 95192, USA
32
Sonoma State University, Rohnert Park, California 94928, USA
33
Southeastern Louisiana University, Hammond, Louisiana 70402, USA
34
Southern University and A&M College, Baton Rouge, Louisiana 70813, USA
35
Stanford University, Stanford, California 94305, USA
36
Syracuse University, Syracuse, New York 13244, USA
37
The Pennsylvania State University, University Park, Pennsylvania 16802, USA
38
The University of Melbourne, Parkville VIC 3010, Australia
39
The University of Mississippi, University, Mississippi 38677, USA
40
The University of Sheffield, Sheffield S10 2TN, United Kingdom
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
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042003-2
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The University of Texas at Austin, Austin, Texas 78712, USA
42
The University of Texas at Brownsville and Texas Southmost College, Brownsville, Texas 78520, USA
43
Trinity University, San Antonio, Texas 78212, USA
44
Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
45
University of Adelaide, Adelaide, SA 5005, Australia
46
University of Birmingham, Birmingham, B15 2TT, United Kingdom
47
University of Florida, Gainesville, Florida 32611, USA
48
University of Glasgow, Glasgow, G12 8QQ, United Kingdom
49
University of Maryland, College Park, Maryland 20742, USA
50
University of Massachusetts–Amherst, Amherst, Massachusetts 01003, USA
51
University of Michigan, Ann Arbor, Michigan 48109, USA
52
University of Minnesota, Minneapolis, Minnesota 55455, USA
53
University of Oregon, Eugene, Oregon 97403, USA
54
University of Rochester, Rochester, New York 14627, USA
55
University of Salerno, 84084 Fisciano (Salerno), Italy
56
University of Sannio at Benevento, I-82100 Benevento, Italy
57
University of Southampton, Southampton, SO17 1BJ, United Kingdom
58
University of Strathclyde, Glasgow, G1 1XQ, United Kingdom
59
University of Western Australia, Crawley, Washington 6009, Australia, USA
60
University of Wisconsin–Milwaukee, Milwaukee, Wisconsin 53201, USA
61
Washington State University, Pullman, Washington 99164, USA
D. P. Anderson
University of California at Berkeley, Berkeley, California 94720, USA
(Received 29 May 2009; published 11 August 2009)
This paper reports on an all-sky search for periodic gravitational waves from sources such as deformed
isolated rapidly spinning neutron stars. The analysis uses 840 hours of data from 66 days of the fifth LIGO
science run (S5). The data were searched for quasimonochromatic waves with frequencies
f
in the range
from 50 to 1500 Hz, with a linear frequency drift
_
f
(measured at the solar system barycenter) in the range
f=<
_
f<
0
:
1
f=
, for a minimum spin-down age
of 1000 years for signals below 400 Hz and
8000 years above 400 Hz. The main computational work of the search was distributed over approximately
100 000 computers volunteered by the general public. This large computing power allowed the use of a
relatively long coherent integration time of 30 hours while searching a large parameter space. This search
extends Einstein@Home’s previous search in LIGO S4 data to about 3 times better sensitivity. No
statistically significant signals were found. In the 125–225 Hz band, more than 90% of sources with
dimensionless gravitational-wave strain tensor amplitude greater than
3
10
24
would have been
detected.
DOI:
10.1103/PhysRevD.80.042003
PACS numbers: 04.80.Nn, 95.55.Ym, 97.60.Gb, 07.05.Kf
I. INTRODUCTION
Gravitational waves (GWs) are predicted by Einstein’s
general theory of relativity, but have so far eluded direct
detection. The Laser Interferometer Gravitational-wave
Observatory (LIGO) [
1
,
2
] has been built for this purpose
and is currently the most sensitive gravitational-wave de-
tector in operation.
Rapidly rotating neutron stars are expected to generate
periodic gravitational-wave signals through various
mechanisms [
3
–
9
]. Irrespective of the emission mecha-
nism, these signals are quasimonochromatic with a slowly
changing intrinsic frequency. Additionally, at a terrestrial
detector, such as LIGO, the data analysis problem is com-
plicated by the fact that the periodic GW signals are
Doppler modulated by the detector’s motion relative to
the solar system barycenter (SSB).
A previous paper [
10
] reported on the results of the
Einstein@Home search for periodic GW signals in the
data from LIGO’s fourth science run (S4). The present
work extends this search, using more sensitive data from
66 days of LIGO’s fifth science run (S5).
Because of the weakness of the GW signals buried in the
detector noise, the data analysis strategy is critical. A
powerful detection method is given by coherent matched
filtering. This means one convolves all available data with
a set of template waveforms corresponding to all possible
putative sources. The resulting detection statistic is derived
in Ref. [
11
] and is commonly referred to as the
F
-statistic.
The parameter space to be scanned for putative signals
from isolated neutron stars is four-dimensional, with two
parameters required to describe the source sky position
*
http://www.ligo.org/
Einstein@Home SEARCH FOR PERIODIC
...
PHYSICAL REVIEW D
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042003 (2009)
042003-3
using standard astronomical equatorial coordinates
(right ascension) and
(declination), and additional coor-
dinates
ð
f;
_
f
Þ
denoting the intrinsic frequency and fre-
quency drift, respectively. To achieve the maximum
possible sensitivity, the template waveforms must match
the source waveforms to within a fraction of a cycle over
the entire observation time (months or years for current
data samples). So one must choose a very closely spaced
grid of templates in this four-dimensional parameter space.
This makes the computational cost of the search very high,
and therefore limits the search sensitivity [
12
].
To maximize the possible integration time, and hence
achieve a more sensitive search, the computation was
distributed via the volunteer computing project
Einstein@Home [
13
]. This large computing power al-
lowed the use of a relatively long coherent integration
time of 30 h, despite the large parameter space searched.
Thus, this search involves coherent matched filtering in the
form of the
F
-statistic over 30-hour-long data segments
and subsequent incoherent combination of
F
-statistic re-
sults via a coincidence strategy.
The methods used here are further described in Secs.
II
,
III
, and
IV
. Estimates of the sensitivity of this search and
results are in Secs.
V
and
VI
, respectively. Previously, other
all-sky searches for periodic GW sources using LIGO S4
and S5 data, which combine power from many short co-
herent segments (30-minute intervals) of data, have been
reported by the LIGO Scientific Collaboration [
14
,
15
].
However, this Einstein@Home search explores large re-
gions of parameter space which have not been analyzed
previously with LIGO S5 data. The sensitivity of the
results here are compared with previous searches in
Sec.
VII
, and conclusions are given in Sec.
VIII
.
II. DATA SELECTION AND PREPARATION
The data analyzed in the present work were collected
between November 19, 2005 and January 24, 2006. The
total data set covering frequencies from 50 to 1500 Hz
consisted of 660 h of data from the LIGO Hanford 4-km
(H1) detector and 180 h of data from the LIGO Livingston
4-km (L1) detector.
The data preparation method is essentially identical to
that of the previous S4 analysis [
10
]. Therefore only a brief
summary of the main aspects is given here; further details
are found in [
10
] and references therein. The data set has
been divided into segments of 30 h each. However, the 30-
hour-long data segments are not contiguous, but have time
gaps. Since the number of templates required increases
rapidly with observation span, the 30 h of data for each
segment were chosen to lie within a time span of less than
40 h. In what follows, the notion of ‘‘segment’’ will always
refer to one of these time stretches, each of which contains
exactly
T
¼
30 h
of data. The total time spanned by a
given data segment
j
is denoted by
T
span
;j
and conforms
to
30 h
<T
span
;j
<
40 h
.
Given the above constraints, a total of
N
seg
¼
28
data
segments (22 from H1, 6 from L1) were obtained from the
early S5 data considered. These data segments are labeled
by
j
¼
1
;
...
;
28
. Table
I
lists the global positioning system
(GPS) start time along with the time span of each segment.
In this analysis, the maximum frequency shift of a signal
over the length of any given data segment and parameter-
space range examined is dominated by the Doppler modu-
lation due to the Earth’s orbital motion around the SSB,
while the effects of frequency change resulting from in-
trinsic spin-down of the source are smaller. The orbital
velocity of the Earth is about
v=c
10
4
; hence a signal
will always remain in a narrow frequency band smaller
than
0
:
15 Hz
around a given source frequency.
Therefore, for each detector the total frequency range
from 50 to 1500 Hz is broken up into 2900 slices, each
of 0.5 Hz bandwidth plus overlapping wings of 0.175 Hz on
either side.
The detector data contain numerous narrow-band noise
artifacts, so-called ‘‘lines,’’ which are of instrumental ori-
gin, such as harmonics of the 60 Hz mains frequency. Prior
to the analysis, line features of understood origin (at the
TABLE I. Segments of early S5 data used in this search. The
columns are the data segment index
j
, the GPS start time
t
j
and
the time spanned
T
span
;j
.
j
Detector
t
j
[s]
T
span
;j
[s]
1
H1
816 397 490
140 768
2
H1
816 778 879
134 673
3
H1
816 993 218
134 697
4
H1
817 127 915
137 962
5
H1
817 768 509
142 787
6
H1
817 945 327
143 919
7
H1
818 099 543
139 065
8
H1
818 270 501
143 089
9
H1
818 552 200
134 771
10
H1
818 721 347
138 570
11
H1
818 864 047
134 946
12
H1
819 337 064
143 091
13
H1
819 486 815
120 881
14
H1
819 607 696
116 289
15
H1
819 758 149
136 042
16
H1
820 482 173
143 904
17
H1
820 628 379
138 987
18
H1
821 214 511
126 307
19
H1
821 340 818
126 498
20
H1
821 630 884
141 913
21
H1
821 835 537
138 167
22
H1
821 973 704
142 510
23
L1
818 812 286
130 319
24
L1
819 253 562
140 214
25
L1
819 393 776
126 075
26
L1
819 547 883
138 334
27
L1
820 015 400
121 609
28
L1
821 291 797
140 758
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
80,
042003 (2009)
042003-4
time before the launch of the search) were removed
(‘‘cleaned’’) from the data by substitution of the
frequency-domain data bins with random Gaussian noise.
Table
III
in the appendix shows the frequencies of lines
excluded from the data. The harmonic-mean noise strain
amplitude spectra of the final cleaned H1 and L1 data sets
are shown in Fig.
1
.
III. DATA PROCESSING
The paper describing the previous Einstein@Home
search in S4 data [
10
] presented in detail the data process-
ing scheme. For the purpose of the present search the same
data processing infrastructure is employed. Hence, here
only a short summary thereof is given, pointing out the
minimal changes applied in setting up the present analysis.
The total computation of the search is broken up into
16 446 454 workunits. Each workunit represents a separate
computing task and is processed using the Berkeley Open
Infrastructure for Network Computing (BOINC) [
16
–
18
].
To eliminate errors and weed out results that are wrong,
each workunit is independently processed by at least two
different volunteers. Once two successful results for a
workunit are returned back to the Einstein@Home server,
they are compared by an automatic validator, which dis-
cards results that differ by more than some allowed toler-
ance. New workunits are generated and run independently
again for such cases.
In searching for periodic gravitational-wave signals,
each workunit examines a different part of parameter
space. A key design goal is that the computational effort
to conduct the entire analysis should take about 6–
7 months. An additional design goal is to minimize the
download burden on the Einstein@Home volunteers’
Internet connections and also on the Einstein@Home
data servers. This is accomplished by letting each workunit
use only a small reusable subset of the total data set, so that
Einstein@Home volunteers are able to carry out useful
computations on a one-day time scale.
Each workunit searches only one data segment over a
narrow frequency range, but covering all of the sky and the
entire range of frequency derivatives. The workunits are
labeled by three indices
ð
j;k;‘
Þ
, where
j
¼
1
;
...
;
28
de-
notes the data segment,
k
¼
1
;
...
;
2900
labels the 0.5 Hz
frequency band and
‘
¼
1
;
...
;M
ð
j;k
Þ
enumerates the
individual workunits pertinent to data segment
j
and fre-
quency band
k
.
In each segment the
F
-statistic is evaluated on a grid in
parameter space. Each parameter-space grid is constructed
such that grid points (templates) are not further apart from
their nearest neighbor by more than a certain distance. The
distance measure is defined from a metric on parameter
space, first introduced in [
19
,
20
], representing the frac-
tional loss of squared signal-to-noise ratio (
SNR
2
) due to
waveform mismatch between the putative signal and the
template. For any given workunit, the parameter-space grid
is a Cartesian product of uniformly spaced steps
df
in
frequency, uniformly spaced steps
d
_
f
in frequency deriva-
tive, and a two-dimensional sky grid, which has nonuni-
form spacings determined by the metric [
10
,
21
].
For frequencies in the range [50, 400) Hz, the maximal
allowed mismatch was chosen as
m
¼
0
:
15
(corresponding
to a maximal loss in
SNR
2
of 15%), while in the range
[400, 1500) Hz, the maximal mismatch was
m
¼
0
:
4
.It
can be shown [
10
,
21
] that these choices of maximal mis-
match enable a coherent search of near-optimal sensitivity
at fixed computational resources.
The step size in frequency
f
obtained from the metric
depends on
T
span
;j
of the
j
th data segment:
df
j
¼
2
ffiffiffiffiffiffiffi
3
m
p
=
ð
T
span
;j
Þ
. In the low-frequency range this results
in frequency spacings in the range
df
j
2
½
2
:
97
;
3
:
67
Hz
, while for high-frequency workunits
df
j
2½
4
:
85
;
6
:
0
Hz
.
The range of frequency derivatives
_
f
searched is defined
in terms of the ‘‘spin-down age’’
f=
_
f
, namely,
1000 years
for low-frequency and
8000 years
for
high-frequency workunits. As in the S4 Einstein@Home
search, these ranges were guided by the assumption that a
nearby very young neutron star would correspond to a
historical supernova, supernova remnant, known pulsar,
or pulsar wind nebula. The search also covers a small
‘‘spin-up’’ range, so the actual ranges searched are
_
f
2
½
f=;
0
:
1
f=
.In
_
f
the grid points are spaced according
FIG. 1. Strain amplitude spectral densities
ffiffiffiffiffiffiffiffiffiffiffi
S
h
ð
f
Þ
p
of the
cleaned data from the LIGO detectors H1 and L1. The curves
in the top (bottom) panel are the harmonic mean of the 22 H1
(6 L1) 30-hour segments of S5 data used this Einstein@Home
analysis.
Einstein@Home SEARCH FOR PERIODIC
...
PHYSICAL REVIEW D
80,
042003 (2009)
042003-5
to
d
_
f
j
¼
12
ffiffiffiffiffiffiffi
5
m
p
=
ð
T
2
span
;j
Þ
, resulting in resolutions
d
_
f
j
2
½
1
:
60
;
2
:
44
10
10
Hz
=
s
for low-frequency workunits,
and
d
_
f
j
2½
2
:
61
;
3
:
99
10
10
Hz
=
s
for high-frequency
workunits.
The resolution of the search grid in the sky depends on
both the start time
t
j
and duration
T
span
;j
of the segment, as
well as on the frequency
f
. The number of grid points on
the sky scales as
/
f
2
, and approximately as
/
T
2
:
4
span
;j
for
the range of
T
span
;j
30
–
40 h
used in this search. As was
done in the previous S4 analysis [
10
], to simplify the
construction of workunits and limit the number of different
input files to be sent, the sky grids are fixed over a fre-
quency range of 10 Hz, but differ for each data segment
j
.
The sky grids are computed at the higher end of each 10 Hz
band, so they are slightly ‘‘over-covering’’ the sky at lower
frequencies within the band. The search covers in total a
frequency band of 1450 Hz; thus there are 145 different sky
grids for each data segment.
The output from one workunit in the low- (high-) fre-
quency range contains the top 1000 (10 000) candidate
events with the largest values of the
F
-statistic. In order
to balance the load on the Einstein@Home servers, a low-
frequency workunit returns a factor of 10 fewer events,
because low-frequency workunits require runtimes ap-
proximately 10 times shorter than high-frequency work-
units. For each candidate event five values are reported:
frequency (hertz), right ascension angle (radians), declina-
tion angle (radians), frequency derivative (hertz per sec-
ond) and
2
F
(dimensionless). The frequency is the
frequency at the SSB at the instant of the first data point
in the corresponding data segment. Returning only the
‘‘loudest’’ candidate events effectively corresponds to a
floating threshold on the value of the
F
-statistic. This
avoids large lists of candidate events being produced in
regions of parameter space containing non-Gaussian noise,
such as instrumental artifacts that were not removed
a priori
from the input data because of unknown origin.
IV. POST-PROCESSING
After results for each workunit are returned to the
Einstein@Home servers by project volunteers, post-
processing is conducted on those servers and on dedicated
computing clusters. The post-processing has the goal of
finding candidate events that appear in many of the 28 dif-
ferent data segments with consistent parameters.
In this search, the post-processing methods are the same
as used for the Einstein@Home S4 search [
10
]. Therefore,
this section only summarizes the main steps; a more de-
tailed description can be found in [
10
].
A consistent (coincident) set of ‘‘candidate events’’ is
called a ‘‘candidate.’’ Candidate events from different data
segments are considered coincident if they cluster closely
together in the four-dimensional parameter space. By using
a grid of ‘‘coincidence cells,’’ the clustering method can
reliably detect strong signals, which would produce can-
didate events with closely matched parameters in many of
the 28 data segments. The post-processing pipeline oper-
ates in 0.5 Hz-wide frequency bands, and performs the
following steps described below.
A. The post-processing steps
A putative source with nonzero spin-down would gen-
erate candidate events with different apparent frequency
values in each data segment. To account for these effects,
the frequencies of the candidate events are shifted back to
the same frequency value at fiducial time
t
fiducial
via
f
ð
t
fiducial
Þ¼
f
ð
t
j
Þþð
t
fiducial
t
j
Þ
_
f
, where
_
f
and
f
ð
t
j
Þ
are
the spin-down rate and frequency of a candidate event
reported by the search code in the result file, respectively,
and
t
j
is the time stamp of the first datum in the
j
th data
segment. The fiducial time is chosen to be the GPS
start time of the earliest (
j
¼
1
) data segment,
t
fiducial
¼
t
1
¼
816 397 490 s
.
A grid of cells is then constructed in the four-
dimensional parameter space to find coincidences among
the 28 different data segments. The coincidence search
algorithm uses rectangular cells in the coordinates
ð
f;
_
f;
cos
;
Þ
. The dimensions of the cells are adapted
to the parameter-space search grid (see below). Each can-
didate event is assigned to a particular cell. In cases where
two or more candidate events from the same data segment
j
fall into the same cell, only the candidate event having the
largest value of
2
F
is retained in the cell. Then the number
of candidate events per cell coming from distinct data
segments is counted, to identify cells with more coinci-
dences than would be expected by random chance.
To ensure that candidate events located on opposite sides
of a cell border are not missed, the entire cell coincidence
grid is shifted by half a cell width in all possible
2
4
¼
16
combinations of the four parameter-space dimensions.
Hence, 16 different coincidence-cell grids are used in the
analysis.
B. Construction of coincidence windows
The coincidence cells are constructed to be as small as
possible to reduce the probability of false alarms. However,
since each of the 28 different data segments uses a different
parameter-space grid, the coincidence cells must be chosen
to be large enough that the candidate events from a source
(which would appear at slightly different points in parame-
ter space in each of the 28 data segments) would still lie in
the same coincidence cell.
In the frequency direction, the size
f
for the coinci-
dence cell is given by the largest search grid spacing in
f
(for the smallest value of
T
span
;j
) plus the largest possible
offset in spin-down:
f
¼
max
j
ð
df
j
þ
td
_
f
j
Þ
, where the
maximization over
j
selects the data segment with the
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
80,
042003 (2009)
042003-6
smallest
T
span
;j
(which is
j
¼
6
) and
t
¼j
max
j
t
j
min
j
t
j
j¼
t
22
t
1
¼
5 576 214 s
is the total time span
between the latest and earliest data segments. For safety,
e.g. against noise fluctuations that could shift a candidate
peak,
f
has been increased by a further 30%, so that the
width of the coincidence cell in
f
below 400 Hz is
f
¼
1
:
78 mHz
and
f
¼
2
:
9 mHz
above 400 Hz.
In the frequency-derivative direction, the size of the
coincidence cell is given by the largest
d
_
f
j
spacing in the
parameter-space grid, which is also determined by the
smallest value of
T
span
;j
. For safety this is also increased
by 30%, so that
_
f
¼
3
:
18
10
10
Hz s
1
below 400 Hz
and
_
f
¼
5
:
19
10
10
Hz s
1
above 400 Hz.
In sky position, the size of the coincidence cells is
guided by the behavior of the parameter-space metric. As
described in [
10
], the density of grid points in the sky is
approximately proportional to
j
cos
ð
Þ
sin
ð
Þj/j
sin
ð
2
Þj
,
and it follows from [
10
] that
cos
ð
Þ
d
¼j
sin
ð
Þj
d
¼
const
. Because of the singularity when
!
0
, a useful
model for the coincidence-window size varying with dec-
lination is given by
ð
Þ¼
ð
0
Þ
=
cos
ð
Þ
;
ð
Þ¼
ð
0
Þ
if
j
j
<
c
;
ð
0
Þ
=
j
sin
ðj
j
ð
0
ÞÞj
if
j
j
c
:
(1)
To ensure continuity at
¼
c
, the transition point
c
is
defined by the condition
ð
0
Þ
=
j
sin
ðj
c
j
ð
0
ÞÞj¼
ð
0
Þ
. The tuning parameter
is chosen based on visual
inspection to be
¼
1
:
5
in this search. The values of
ð
0
Þ
and
ð
0
Þ
are directly determined from the sky
grids (see [
10
] for details). Figure
2
shows these parame-
ters for all-sky grids as a function of frequency. As stated
above, the sky grids are constant for 10 Hz-wide steps in
frequency, and so these parameters vary with the same step
size.
C. Output of the post-processing
The output of the post-processing is a list of the candi-
dates with the greatest number of coincidences. The pos-
sible number of coincidences ranges from a minimum of 0
to a maximum of 28 (the number of data segments ana-
lyzed). The meaning of
C
coincidences is that there are
C
candidate events from different data segments within a
given coincidence cell. In each frequency band of
coincidence-window width
f
, the coincidence cell con-
taining the largest number of candidate events is found.
The pipeline outputs the average frequency of the coinci-
dence cell, the average sky position and spin-down of the
candidate events, the number of candidate events in the
coincidence cell, and the ‘‘significance’’ of the candidate.
The significance of a candidate, first introduced in [
22
] and
explained in [
10
], is defined by
S
¼
X
C
q
¼
1
ð
F
q
ln
ð
1
þ
F
q
ÞÞ
;
(2)
where
F
q
is the
F
-statistic value of the
q
th candidate
event in the same coincidence cell, which harbors a total
of
C
candidate events.
D. False alarm probability and detection threshold
The central goal of this search is to make a
confident
detection
, not to set upper limits with the broadest possible
coverage band. This is reflected in the choice of detection
threshold based on the expected false alarm rates. In this
search the background level of false alarm candidates is
expected at 10 coincidences (out of 28 possible). As a
pragmatic choice, the threshold of confident detection is
set at 20 coincidences, which is highly improbable to arise
from random noise only. These settings will be elucidated
in the following.
To calculate the false alarm probabilities, consider the
case where
E
seg
ð
k
Þ
candidate events per data segment
obtained from pure Gaussian noise are distributed uni-
formly about
N
cell
ð
k
Þ
independent coincidence cells in a
given 0.5 Hz band
k
. Assuming the candidate events are
independent, the probability
p
F
ð
k
;
C
max
Þ
per coincidence
cell of finding
C
max
or more candidate events from different
data segments has been derived in [
10
] and is given by the
binomial distribution
p
F
ð
k
;
C
max
Þ¼
X
N
seg
n
¼
C
max
N
seg
n
½
ð
k
Þ
n
½
1
ð
k
Þ
N
seg
n
;
(3)
FIG. 2. The parameters
ð
0
Þ
and
ð
0
Þ
of the sky
coincidence-window model as a function of the 10 Hz frequency
band. The vertical dashed line at 400 Hz indicates the separation
between the low- and high-frequency ranges.
Einstein@Home SEARCH FOR PERIODIC
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PHYSICAL REVIEW D
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042003 (2009)
042003-7
where
ð
k
Þ
denotes the probability of populating any given
coincidence cell with one or more candidate events in a
given data segment, obtained as
ð
k
Þ¼
1
1
1
N
cell
ð
k
Þ
E
seg
ð
k
Þ
:
(4)
Finally, the probability
P
F
ð
k
;
C
max
Þ
that there are
C
max
or
more coincidences in
one or more
of the
N
cell
cells per
0.5 Hz band
k
is
P
F
ð
k
;
C
max
Þ¼
1
½
1
p
F
ð
k
;
C
max
Þ
N
cell
:
(5)
Figure
3
shows the dependence of
P
F
ð
k
;
C
max
Þ
on the
frequency bands for different values of
C
max
. One finds that
the average false alarm probability of obtaining 10 or more
coincidences is approximately
10
3
. This means, in our
analysis of 2900 half-Hz frequency bands, only a few
candidates are expected to have 10 or more coincidences.
Thus this will be the anticipated background level of
coincidences, because from pure random noise one would
not expect candidates of
more than
10 coincidences in this
analysis. In contrast, the false alarm probability of reaching
the detection threshold of 20 or more coincidences per
0.5 Hz averaged over all frequency bands is about
10
21
.
Therefore, this choice of detection threshold makes it
extremely improbable to be exceeded in the case of random
noise.
During parts of the LIGO S5 run ten simulated periodic
GW signals were injected at the hardware level by modu-
lating the interferometer mirror positions via signals sent to
voice actuation coils surrounding magnets glued near the
mirror edges. The hardware injections were scheduled with
an overall duty cycle of about 50% during S5 to minimize
potential interference for other GW searches. Thus, in only
12 (of the 28) data segments chosen for this search were
these hardware injections active more than 90% of the
time. Therefore, the hardware injections are not expected
to meet the detection condition defined above, simply
because they were inactive during a large fraction of the
data used in this analysis. For future science runs improved
understanding will allow the hardware injections to be
activated permanently.
V. ESTIMATED SENSITIVITY
The methods used here would be expected to yield very
high confidence if a strong signal were present. To estimate
the sensitivity of this detection scheme, Monte Carlo meth-
ods are used to simulate a population of sources. The goal
is to find the strain amplitude
h
0
at which 10%, 50%, or
90% of sources uniformly populated over the sky and in
their ‘‘nuisance parameters’’ would be confidently de-
tected. In this analysis, ‘‘detectable’’ means ‘‘produces
coincident events in 20 or more distinct data segments.’’
As discussed above, the false alarm probability for obtain-
ing such a candidate in a given 0.5 Hz band is of order
10
21
. This is therefore an estimate of the signal strength
required for high-confidence detection. For this purpose,
the pipeline developed in [
10
] is run here, using the input
data of the present analysis. A large number of distinct
simulated sources (trials) are tested for detection. A ‘‘trial’’
denotes a single simulated source which is probed for
FIG. 3. False alarm probabilities
P
F
ð
k
;
C
max
Þ
as a function of
frequency band (labeled by
k
) for different values of
C
max
2
f
10
;
14
;
17
;
20
;
25
g
. The dashed horizontal lines represent the
corresponding average across all frequencies. The vertical
dashed line at 400 Hz indicates the separation between the
low- and high-frequency ranges.
FIG. 4. Estimated sensitivity of the Einstein@Home search for
isolated periodic GW sources in the early S5 LIGO data. The set
of three curves shows the source strain amplitudes
h
0
at which
10% (bottom), 50% (middle) and 90% (top) of simulated sources
would be confidently detected (i.e., would produce at least
20 coincidences out of 28 possible) in this Einstein@Home
search.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
80,
042003 (2009)
042003-8
detection. For a detailed description of the methodology,
the reader is referred to [
10
].
Figure
4
shows the resulting search sensitivity curves as
functions of frequency. Each data point on the plot denotes
the results of 1000 independent trials. These show the
values of
h
0
as defined in [
11
] such that 10%, 50%, and
90% of simulated sources are confidently detected in the
post-processing pipeline.
The dominant sources of error in these sensitivity curves
are uncertainties in calibration of the LIGO detector re-
sponse functions (cf. [
10
,
15
]). The uncertainties range
typically from about 8% to 15%, depending on frequency.
The behavior of the curves shown in Fig.
4
essentially
reflects the instrument noise given in Fig.
1
. One may fit the
curves obtained in Fig.
4
to the shape of the harmonic-
mean averaged strain noise power spectral density
S
h
ð
f
Þ
.
Then the three sensitivity curves in Fig.
4
are described by
h
D
0
ð
f
Þ
R
D
ffiffiffiffiffiffiffiffiffiffiffiffi
S
h
ð
f
Þ
30 h
s
;
(6)
where the prefactors
R
D
for different detection probabil-
ities levels
D
¼
90%
, 50% and 10% are well fit below
400 Hz by
R
90%
¼
29
:
4
,
R
50%
¼
18
:
5
, and
R
10%
¼
11
:
6
,
and above 400 Hz by
R
90%
¼
30
:
3
,
R
50%
¼
19
:
0
, and
R
10%
¼
11
:
8
.
VI. RESULTS
A. Vetoing instrumental-noise lines
At the time the instrument data were prepared and
cleaned, narrow-band instrumental line features of known
origin were removed, as previously described in Sec.
II
.
However, the data also contained stationary instrumental
line features that were not understood, or were poorly
understood, and thus were not removed
a priori
. After
the search had been conducted, at the time the post-
processing started, the origin of more stationary noise lines
became known. Therefore, these lines, whose origin was
tracked down after the search, are excluded (cleaned
a posteriori
) from the results. A list of the polluted fre-
quency bands which have been cleaned
a posteriori
is
shown in Table
IV
in the appendix.
However, noise features still not understood instrumen-
tally at this point were not removed from the results. As a
consequence, the output from the post-processing pipeline
contains instrumental artifacts that in some respects mimic
periodic GW signals. But these artifacts tend to cluster in
certain regions of parameter space, and in many cases they
can be automatically identified and vetoed as done in
previous searches [
10
,
24
]. The method used here is derived
in [
23
] and a detailed description of its application is found
in [
10
].
For a coherent observation time baseline of 30 h the
parameter-space regions where instrumental lines tend to
appear are determined by global-correlation hypersurfaces
[
23
] of the
F
-statistic. On physical grounds, in these
parameter-space regions there is little or no frequency
Doppler modulation from the Earth’s motion, which can
lead to a relatively stationary detected frequency. Thus, the
locations of instrumental-noise candidate events are de-
scribed by
_
f
þ
f
v
j
c
^
n
<;
(7)
where
c
denotes the speed of light,
^
n
is a unit vector
pointing to the source’s sky location in the SSB frame
and relates to the equatorial coordinates
and
by
^
n
¼
ð
cos
cos
;
cos
sin
;
sin
Þ
, and
v
j
is the orbital velocity
of the Earth at the midpoint of the
j
th data segment (
j
v
j
j
10
4
c
). The parameter
accounts for a certain tolerance
needed due to the parameter-space gridding and can be
understood as
¼
f=N
c
T
, where
f
denotes width in
frequency (corresponding to the coincidence-cell width in
the post-processing) up to which candidate events can be
resolved during the characteristic length of time
T
, and
N
c
represents the size of the vetoed or rejected region,
measured in coincidence cells. In this analysis
T
¼
5 718 724 s
(
66 days
) is the total time interval spanned
by the input data.
Because false alarms are expected at the level of 10 co-
incidences, candidates that satisfy Eq. (
7
) for more than
10 data segments are eliminated (vetoed). The fraction of
parameter space excluded by this veto is determined by
Monte Carlo simulations to be about 13%. From Eq. (
7
)it
follows that for fixed frequency the resulting fraction of
sky excluded by the veto (uniformly averaged over spin-
down) is greatest at lowest frequencies and decreases
approximately as
f
1
for higher frequencies.
Appendix A of Ref. [
10
] presents an example calculation,
illustrating the parameter-space volume excluded by this
vetoing method.
B. Post-processing results
Figures
5
and
6
summarize all post-processing results
from the entire search frequency range of 50 to 1500 Hz,
for each frequency coincidence cell maximized over the
entire sky and full spin-down range.
In Fig.
5(a)
all candidates that have 7 or more coinci-
dences are shown in a sky projection. The color scale is
used to indicate the number of coincidences. The most
prominent feature still apparent forms an annulus of high
coincidences in the sky, including the ecliptic poles, a
distinctive fingerprint of the instrumental-noise lines
[
23
]. To obtain the results shown in Fig.
5(b)
, the set of
candidates is cleaned
a posteriori
by removing strong
instrumental-noise lines, whose origin became understood
after the search was begun, and excluding the hardware
injections. Finally, in Fig.
5(c)
the parameter-space veto is
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PHYSICAL REVIEW D
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042003 (2009)
042003-9
applied and coincidence cells which contain candidate
events from a single detector only are excluded, too.
In Fig.
6(a)
the coincidences and significance of all
candidates that have 7 or more coincidences are shown
as a function of frequency. From this set of candidates the
hardware injections are excluded, strong instrumental-
noise lines of known origin are removed, the parameter-
space veto is applied and finally single-detector candidates
are excluded to obtain Fig.
6(b)
.
As can be seen from Figs.
5(c)
and
6(b)
there are no
candidates that exceed the predefined detection threshold
FIG. 5 (color). Sky maps of post-processing results.
Candidates having more than 7 coincidences are shown in
Hammer-Aitoff projections of the sky. The color bar indicates
the number of coincidences of a particular candidate (cell). The
top plot (a) shows the coincidence analysis results. In (b),
a posteriori
strong lines of known instrumental origin and
hardware injections are removed. The bottom plot (c) is
obtained by additionally applying the parameter-space veto
and excluding single-detector candidates. Note that in every
sky map the regions of lower coincidences near the equatorial
plane (colored dark blue) are due to the sky-grid construction (cf.
Fig. 3 in [
10
]).
FIG. 6. The top plot (a) shows the post-processing candidates
having more than 7 coincidences as function of frequency. The
light-gray shaded rectangular regions highlight the frequency
bands of the hardware injections. The dark-gray data points show
the candidates resulting from the hardware-injected GW signals.
In (b), the final results are shown after exclusion of instrumental
lines of known origin and hardware injections, application of
parameter-space veto and exclusion of single-detector candi-
dates.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
80,
042003 (2009)
042003-10