Search for gravitational wave ringdowns from perturbed black holes in LIGO S4 data
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M. E. Zucker,
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H. zur Mu
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and J. Zweizig
17
(The LIGO Scientic 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
80,
062001 (2009)
062001-2
41
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, WA 6009, Australia
60
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA
61
Washington State University, Pullman, Washington 99164, USA
(Received 22 June 2009; published 9 September 2009)
According to general relativity a perturbed black hole will settle to a stationary configuration by the
emission of gravitational radiation. Such a perturbation will occur, for example, in the coalescence of a
black hole binary, following their inspiral and subsequent merger. At late times the waveform is a
superposition of quasinormal modes, which we refer to as the ringdown. The dominant mode is expected
to be the fundamental mode,
l
¼
m
¼
2
. Since this is a well-known waveform, matched filtering can be
implemented to search for this signal using LIGO data. We present a search for gravitational waves from
black hole ringdowns in the fourth LIGO science run S4, during which LIGO was sensitive to the
dominant mode of perturbed black holes with masses in the range of
10
M
to
500
M
, the regime of
intermediate-mass black holes, to distances up to 300 Mpc. We present a search for gravitational waves
from black hole ringdowns using data from S4. No gravitational wave candidates were found; we place a
90%-confidence upper limit on the rate of ringdowns from black holes with mass between
85
M
and
390
M
in the local universe, assuming a uniform distribution of sources, of
3
:
2
10
5
yr
1
Mpc
3
¼
1
:
6
10
3
yr
1
L
1
10
;
where
L
10
is
10
10
times the solar blue-light luminosity.
DOI:
10.1103/PhysRevD.80.062001
PACS numbers: 95.85.Sz, 04.80.Nn, 07.05.Kf, 97.60.Jd
I. INTRODUCTION
The existence of intermediate mass black holes, IMBHs,
(
20
M
M
10
6
M
) has been under debate for several
decades. While general relativity does not preclude
IMBHs, there had been no observational evidence for their
existence until recently. Electromagnetic observations
have indicated that ultraluminous X-ray sources, that is,
sources radiating above the Eddington luminosity for a
stellar mass black hole, may be powered by IMBHs.
Strong evidence in support of this argument has recently
been reported with the discovery of a source whose lumi-
nosity implies the presence of black hole with mass of at
least
500
M
[
1
]. Further information pertaining to IMBHs
may be found in recent comprehensive review articles
[
2
,
3
].
Predictions have been made for the rate of detection of
ringdowns from IMBHs in Advanced LIGO. Reference [
4
]
predicts a rate of 10 events per year from IMBH-IMBH
binary coalescences. When scaled to the sensitivity of the
data set under consideration in this investigation, this
prediction becomes
10
4
yr
1
[
5
]. Ringdowns following
coalescences of stellar-mass BHs with IMBHs could also
be detectable with Advanced LIGO, with possible rates of
tens of events per year [
6
].
Detection of gravitational radiation from IMBHs how-
ever, would provide unambiguous evidence of their exis-
tence. In order for such an object to reveal itself through
gravitational radiation it must come to be in a perturbed
state, for example, as the remnant of the coalescence of two
IMBHs. Current ground-based gravitational wave detec-
tors, such as the Laser Interferometer Gravitational-Wave
Observatory (LIGO), operate in an optimal frequency
range for the detection of the ringdown phase of the binary
coalescence of IMBH binaries. In this paper we describe a
search for ringdown waveforms in data from the fourth
LIGO science run, S4.
SEARCH FOR GRAVITATIONAL WAVE RINGDOWNS FROM
...
PHYSICAL REVIEW D
80,
062001 (2009)
062001-3
II. THE RINGDOWN WAVEFORM
A series of studies within a linearized approximation to
Einstein’s equations and also full-blown numerical simu-
lations have shown that the gravitational wave signal from
a perturbed black hole consists of roughly three stages [
7
]:
(i) A prompt response at early times, which depends
strongly on the initial conditions, and is the counterpart
to light-cone propagation; (ii) An exponentially decaying
‘‘ringdown’’ phase at intermediate times, where quasinor-
mal modes, QNMs, dominate the signal, which depends
entirely on the final black hole’s parameters; (iii) A late-
time tail, usually a power-law falloff of the field. The
ringdown phase, which is the focus of this work, starts
roughly when the perturbing source reaches the peak of the
potential barrier around the black hole, and consists of a
superposition of quasinormal modes. For instance, during
the merger of two black holes, the start of the ringdown is
roughly associated with the formation of a common appar-
ent horizon, which also corresponds to the peak of the
gravitational-wave amplitude. For black holes in the
LIGO band this is on the order of tens of milliseconds
after the innermost stable circular orbit of the binary. Each
quasinormal mode has a characteristic complex angular
frequency
!
lm
; the real part is the angular frequency and
the imaginary part is the inverse of the damping time
.
Numerical simulations (for example [
8
]) have demon-
strated that the dominant mode is the fundamental mode,
l
¼
m
¼
2
, and that far from the source the waveform can
be approximated by
h
0
ð
t
Þ¼<
A
GM
c
2
r
e
i
!
22
t
;
(1)
where
A
is the dimensionless amplitude of the
l
¼
m
¼
2
mode,
r
is the distance to the source,
M
is the black hole
mass,
c
is the speed of light, and
G
is the gravitational
constant. This is usually expressed in terms of the oscil-
lation frequency
f
0
¼<ð
!
22
Þ
=
2
and the quality factor
Q
¼
f
0
=
=ð
!
22
Þ
,
h
0
ð
t
Þ¼
A
GM
c
2
r
e
f
0
t=Q
cos
ð
2
f
0
t
Þ
:
(2)
Under the assumption that the waveform is completely
known, we can implement the method of matched filtering
[
9
], in which the data is correlated with a bank of signal
templates parametrized by the ringdown frequency and
quality factor. An analytic fit by Echeverria [
10
]to
Leaver’s numerical calculations [
11
] relates the waveform
parameters to the black hole’s physical parameters, mass
M
, and dimensionless spin factor, defined in terms of the
spin angular momentum
J
, for the fundamental mode
^
a
¼
Jc=GM
2
:
f
0
¼
1
2
c
3
GM
g
ð
^
a
Þ
(3)
Q
¼
2
ð
1
^
a
Þ
9
=
20
;
(4)
where
g
ð
^
a
Þ¼
1
0
:
63
ð
1
^
a
Þ
3
=
10
. Thus, if we detect the
l
¼
m
¼
2
mode, these formulas will provide the mass and
spin of the black hole [
12
].
The amplitude is given by [
5
]
A
¼
ffiffiffiffiffi
5
2
q
Q
ð
1
=
2
Þ
F
ð
Q
Þ
ð
1
=
2
Þ
g
ð
^
a
Þ
ð
1
=
2
Þ
;
(5)
where
F
ð
Q
Þ¼
1
þ
7
24
Q
2
. In addition to a frequency and
quality factor dependence, the amplitude of the waveform
also depends on the fraction of the final black hole’s mass
radiated as gravitational waves,
. This quantity scales
with the square of the symmetric mass ratio
, where
¼
m
1
m
2
=
ð
m
1
þ
m
2
Þ
2
, and thus is largest for an equal mass
binary [
14
,
15
]. Numerical simulations of the merger of
equal mass binaries have shown that approximately 1%
of the final black hole’s mass is emitted in gravitational
waves [
8
]. In this search we do not attempt to evaluate
;
we use the output of the filter to calculate the effective
distance to a source emitting 1% of its mass as gravita-
tional waves. The effective distance is the distance to an
optimally located and oriented source.
III. DATA SET
This search uses data from the 4th LIGO science run
(S4), which took place between February 22nd and March
24th, 2005. This yielded a total of 567.4 hours of analyz-
able data from the 4 km interferometer in Hanford, WA
(H1), 571.3 hours from the 2 km interferometer in Hanford,
WA (H2), and 514.7 hours from the 4 km interferometer in
Livingston, LA (L1). In this analysis we require that data
be available from at least two detectors at any given time.
This results in approximately 364 hours of triple coinci-
dence and 210 hours of double coincidence, as shown in
Fig.
1
. During S4, the LIGO detectors operated signifi-
cantly below their design sensitivity; this was attained in
the subsequent science run, S5 [
16
].
The LIGO detectors are sensitive to gravitational waves
in the frequency band of
50 Hz
to 2 kHz. This corre-
sponds to a mass range of 11 to
440
M
for a black hole
with
^
a
¼
0
:
9
oscillating in its fundamental mode. Using a
FIG. 1. Venn diagram of the coincident detector times in
hours.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
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062001 (2009)
062001-4
typical instantiation of the S4 noise power spectrum we can
estimate the horizon distance,
D
H
, the distance out to
which a specified source with optimal location and orien-
tation produces an SNR of 8 in the detector. We consider a
spinning black hole with
^
a
¼
0
:
9
and
¼
1%
. This quan-
tity is shown as a function of mass, for each of the three
LIGO detectors and the LIGO design sensitivity in Fig.
2
.
During each science run there are times when disturban-
ces couple into the data stream, introducing noise tran-
sients that can trigger a matched filter with high SNR. A
careful study of correlations between LIGO data and aux-
iliary channels allowed us to identify data segments with
an excessive noise transient rate, or with known artifacts.
We refer to these as data quality flags. These are then
categorized according to the severity of the disturbance
[
17
]; triggers occurring during category 1 times are not
analyzed as the excess of noise is likely to contaminate the
estimation of the power spectral density. Data occurring
during category 2 or 3 times are less problematic, and thus
to avoid excessive segmentation of the data, these are
vetoed during the analysis. Any gravitational wave candi-
dates occurring during category 4 times are cautiously
examined.
IV. PIPELINE
When the signal is known, the optimal method of ex-
tracting the signal from Gaussian noise is matched filter-
ing. LIGO data is non-Gaussian and while the method is
still appropriate it is not sufficient to discriminate between
signal and background. We employ an analysis pipeline
which was designed for this purpose, and closely resem-
bles that of the inspiral searches described in [
18
,
19
]. Here
we summarize the main steps. The first stage of the pipe-
line involves reading in and conditioning the data
s
ð
t
Þ
from
each of the three LIGO detectors. We read in uncalibrated
data, the differential arm length signal, at a sample rate of
16 384 s
1
and down-sample to
8192 s
1
. This is con-
verted to strain by applying the detector response function.
The one-sided power spectral density
S
h
ð
f
Þ
is calculated
for each 2176 s long segment of the calibrated data. The
data is then broken further into sets of 16 overlapping
blocks, 256 s in length, and filtered using a bank of ring-
down templates. The templates are positioned in
f
0
and
Q
according to the metric [
20
]
ds
2
¼
1
8
3
þ
16
Q
4
Q
2
ð
1
þ
4
Q
2
Þ
2
dQ
2
2
3
þ
4
Q
2
f
0
Q
ð
1
þ
4
Q
2
Þ
dQdf
0
þ
3
þ
8
Q
2
f
2
0
df
2
0
;
(6)
such that the maximum mismatch,
ds
2
, between any point
within the template bank and the nearest template is 3%.
We search within the most sensitive portion of the LIGO
frequency band, 50 Hz to 2 kHz and in quality factor
between 2 and 20. With these parameters we obtain a
bank of 583 templates with five different values of
Q
.
The same template bank is used throughout the run.
Numerical simulations [
21
–
24
] have demonstrated that
the maximum spin attained by the final black hole in a
binary black hole merger is less than 0.96, corresponding to
a quality factor of 8.5. In this search, we chose to cover a
larger parameter space, and explore spin values between 0
(
Q
¼
2
) and 0.994 (
Q
¼
20
). Each filter has the form
h
ð
t
Þ¼
cos
ð
2
f
0
t
Þ
e
f
0
t=Q
;
0
t
t
max
(7)
with a length of ten e-folding times,
t
max
¼
10
, where
¼
Q=f
0
. Filtering the data gives a signal to noise ratio
(SNR)
ð
h
Þ¼
h
s;h
i
ffiffiffiffiffiffiffiffiffiffiffiffi
h
h;h
i
p
;
(8)
where
h
s;h
i¼
2
Z
1
1
~
s
ð
f
Þ
~
h
ð
f
Þ
S
h
ðj
f
jÞ
df:
(9)
Here, the noise spectral density
S
h
ð
f
Þ
is the one appropriate
for the data segment in question. For each filter, only
triggers exceeding a predefined threshold,
¼
5
:
5
are
retained. These are then clustered using a peak finding
algorithm over a minimum of 1 s, with the loudest triggers
for each filter written out to a file. Approximately
10
6
triggers were written out for each detector for the entire
S4 run.
In order to claim a detection of a gravitational wave
ringdown we require coincidence between at least two
detectors in the time of arrival of the signal and in the
FIG. 2 (color online). The horizon distance versus mass and
frequency for a black hole with spin of
^
a
¼
0
:
9
and
¼
0
:
01
.
From top to bottom, the curves show the horizon distance for the
Initial LIGO reference design (black), the Hanford 4 km detector
H1 (red), the Livingston 4 km detector L1 (green), and the
Hanford 2 km detector H2 (blue).
SEARCH FOR GRAVITATIONAL WAVE RINGDOWNS FROM
...
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waveform parameters. The time requirement is enforced
first, only allowing triggers that appear within 4 ms of each
other for colocated Hanford detectors and 14 ms for the
Livingston-Hanford pairs (the two observatories are sepa-
rated by 10 ms of light travel time) through to the next
stage. The second coincidence test is based on the metric
used to lay out the template bank, Eq. (
6
), with the size of
the coincidence window around each template varying
depending on its position within the parameter space. At
this stage we also veto triggers occurring during times
when category 2 and 3 data quality flags were on, and
implement an amplitude consistency test between triggers
appearing in both Hanford detectors, such that the pair of
triggers is retained only if the ratio of the H1 effective
distance to the H2 effective distance is less than 2. This
results in list of coincident triggers found in two or three
detectors, hereafter referred to as doubles and triples,
respectively. The final step is to cluster this coincidences
over a time window of 10 s, retaining only the coincidences
with the highest value of a detection statistic
DS
, a mea-
sure of the relative significance of the coincidences. For
triple coincidences (H1H2L1) the detection statistic was
DS
¼
2
H1
þ
2
H2
þ
2
L1
:
(10)
The high level of false alarms associated with two detector
coincidences (H1L1, H1H2, or H2L1), shown in Fig.
3
,
warranted a different detection statistic,
DS
¼
min
f
ifo1
þ
ifo2
;a
ifo1
þ
b;a
ifo2
þ
b
g
;
(11)
where suitable values of
a
and
b
were found to be 2 and 2.2,
respectively, from an evaluation of the false alarm rate.
(The evaluation of false alarm rates is discussed in Sec.
V
.)
Note that in this ranking we do not take the square root of
Eq. (
10
) so as to emphasize the triple coincidences over the
doubles. The ten coincidences with the highest value of the
detection statistic in each clustered coincidence category
(triples or doubles) are followed up upon as detection
candidates.
V. TUNING THE SEARCH
As with previous matched filtering searches in LIGO we
implement a ‘‘blind analysis’’ of the data, that is, the
constraints are decided upon prior to looking at the full
data set. We chose values of the constraints that maximize
the number of ringdown signals recovered while minimiz-
ing the false alarm rate. The constraints under considera-
tion include the SNR threshold, the size of the coincidence
windows, the detection statistic, and the amplitude consis-
tency test.
The signal is modeled by adding simulated signals in
software to the data stream and running the pipeline in the
same manner as one would with the real data. Figure
4
displays a plot of Hanford effective distance (the distance
to an optimally located and oriented source) as a function
of frequency for simulated signals that were found in all
three detectors and in all combinations of two detectors
(H1H2, H1L1, H2L1). The plot shows that the search is
most sensitive to ringdown signals occurring in the 70 Hz–
140 Hz band, where detector noise is lowest.
The background, or false alarm rate, is estimated by
shifting two data streams in time with respect to one-
another and running the pipeline described in Sec.
IV
.
The L1 data stream is slid 50 times in multiples of
10 s
and H2 is slid 50 times in multiples of
5s
. As the time
FIG. 3 (color online). Plot of background events (black dot)
and simulated signal (red x’s) found in double coincidence. The
green lines denote the lines of constant detection statistic for two
detector coincidences.
FIG. 4 (color online). Hanford effective distance versus ring-
down frequency
f
0
for simulated signals found in coincidence.
Simulated signals found in triple coincidence are marked as
green x’s, simulated signals found in double coincidence are
shown as red dots and those signals found in double coincidence
because the third detector was vetoed are circled in black.
B. P. ABBOTT
et al.
PHYSICAL REVIEW D
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steps are much larger than the longest possible delay
between detectors in receiving a real gravitational wave
signal, any coincidences found cannot be due to gravita-
tional waves, and are therefore considered false coinci-
dences. We use these as estimates of the background due to
false in-time coincidence.
As a final sanity check before the search is unblinded we
look at approximately one tenth of the data to ensure that
the false alarm rate is consistent with our background
studies. In order to avoid any potential bias in the tuning
procedure affecting the upper limits, these data are ex-
cluded from the upper limit calculation.
VI. RESULTS
Once the tuning is complete and all thresholds and cuts
are finalized the pipeline described above is run on the full
data set. Unblinding the search with the cuts and thresholds
described above revealed no triple coincident events. A
number of double coincidences were found to satisfy the
constraints; however, these events are consistent with back-
ground as shown in Fig.
5
. The loudest candidate events
were subjected to further investigation, and in each case
there was insufficient evidence that the coincidence could
not be attributed to noise in the detector.
We calculate an upper limit on the rate of ringdowns for
a given population of black holes using simulated signals
to evaluate the efficiency of the search,
"
ð
r
Þ
, defined as the
fraction of simulated signals detected in the analysis, as a
function of physical distance. The simulated signals were
uniform in orientation and direction. Given the expected
high false alarm rate in double coincidence from back-
ground studies, we set an upper limit from times when all
three detectors were recording data, a total of
T
¼
0
:
0375 years
. Figure
4
shows that the efficiency is strongly
dependent on the ringdown frequency
f
0
, and thus for the
purpose of setting an upper limit we restrict the calculation
to the most sensitive frequency band, 70 Hz–140 Hz, cor-
responding to black hole masses in the range
85
M
–
390
M
. We added 5701 simulated signals in the
frequency range of 70 Hz–140 Hz and between 0.5M pc
and
10
3
Mpc
in distance to the data, over ten runs. Of
these, 3010 were recovered in triple coincidence. Figure
6
displays the efficiency as a function of distance for the
frequency band of interest. (Note that the efficiency is not
equal to 1 at nearby distances because we apply vetoes and
treat these as a loss of efficiency rather than a loss in live-
time.) From the efficiency we can calculate the effective
volume we are sensitive to,
V
eff
V
eff
¼
4
Z
"
ð
r
Þ
r
2
dr:
(12)
For this band
V
eff
¼
2
:
6
10
6
Mpc
3
. This corresponds to
a typical distance to a source of
85 Mpc
. The 90%
confidence upper limit on the rate is given by [
25
]
R
90%
¼
2
:
303
TV
eff
;
(13)
which, for the 70 Hz–140 Hz band is
2
:
4
10
5
yr
1
Mpc
3
. Even though our knowledge of the
population of intermediate mass black holes is poor, as
discussed in Sec.
I
, we do know that the formation of stars
in general scales with the blue-light luminosity emitted by
galaxies, and as it is expected that the rate of binary
coalescence (including ringdown) follows the rate of star
formation, we can work under the assumption that the rate
of ringdowns also scales with blue light luminosity [
26
].
We introduce the cumulative blue luminosity,
C
L
, observ-
able within the range of the search,
C
L
¼
L
V
eff
:
(14)
C
L
has units of
L
10
, where
L
10
¼
10
10
L
B;
and
L
B;
is the
B
-band solar luminosity (1 MWEG is equivalent to 1.7
L
10
), and
L
¼ð
1
:
98
0
:
16
Þ
10
2
L
10
Mpc
3
is the av-
erage blue luminosity density [
27
]. The 90% confidence
FIG. 5 (color online). Cumulative histogram of coincident
triggers (red circles) and an estimate of the background (black
x’s): H1L1 doubles in triple time.
FIG. 6. The efficiency of detecting simulated signal injections
in triple coincidence in the 70 Hz–140 Hz band, fit to a sigmoid.
SEARCH FOR GRAVITATIONAL WAVE RINGDOWNS FROM
...
PHYSICAL REVIEW D
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upper limit on the rate in these units is given by
R
90%
¼
2
:
303
TC
L
;
(15)
which evaluates to
1
:
2
10
3
yr
1
L
1
10
.
Because of our lack of knowledge about the population
of black holes we assign no error to the astrophysical
source population. Similarly, we assign no error to the
waveform: comparison with numerical relativity results
has shown that exponentially-damped-sinusoid templates
perform well at detecting the signal and characterizing the
black hole parameters [
7
]. We limit ourselves to evaluating
systematic errors associated with the experimental appara-
tus and analysis method. The only error that we associate
with the former is calibration of the data, and with the latter
is with the limited number of Monte Carlo simulations
used to evaluate the efficiency. Errors in the calibration
can cause the SNR of a signal to be incorrectly quantified,
thereby introducing inaccuracies in the distance. As the
efficiency is a function of distance and frequency, care has
to be taken to adjust the efficiency curve appropriately. The
fractional uncertainty in amplitude was found from cali-
bration studies [
28
] to be 5%. This results in an error of
3
:
5
10
5
Mpc
3
in
V
eff
. The second source of error is due
to the limited number of simulated signals in our
Monte Carlo (MC) simulations to evaluate the efficiency.
Assuming binomial errors we calculate the variance of the
efficiency
2
MC
, which corresponds to an error in
V
eff
of
2
:
3
10
5
Mpc
3
. These errors are summed in quadrature,
and multiplied by 1.64 to give a 90% confidence interval of
6
:
9
10
5
Mpc
3
. To obtain an upper limit we apply a
downward excursion to the effective volume and obtain
R
90%
¼
3
:
2
10
5
yr
1
Mpc
3
¼
1
:
6
10
3
yr
1
L
1
10
.
VII. CONCLUSION
We have performed the first search for gravitational
waves signals from black hole ringdowns in LIGO data,
and demonstrated with simulated signals that this pipeline
is an effective method of detecting such signals in coinci-
dence between all three LIGO detectors. The search en-
compassed black holes in the mass range of
10
M
–
500
M
,
the regime of intermediate mass black holes, with spins
ranging from 0 to 0.994. No gravitational-wave events were
found, and an upper limit of
R
90%
¼
1
:
6
10
3
yr
1
L
1
10
was placed on the rate of ringdowns from black holes
formed from binary mergers, in the mass range of
85
M
–
390
M
.
A search for black hole ringdowns in data from the fifth
LIGO science run is currently underway. This 2 yr long
science run was the first at LIGO design sensitivity. With
the increase in sensitivity of the LIGO detectors between
the two runs, the ringdown horizon distance of S5 is greater
than that of S4, allowing access to a greater number of
potential sources from which a detection could be made. A
further increase in sensitivity will come with Advanced
LIGO, allowing us to detect compact binary coalescence to
cosmological distances, and the improved sensitivity at
lower frequency will make us sensitive to black holes
with masses up to
2000
M
or higher.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the support of the
United States National Science Foundation for the con-
struction and operation of the LIGO Laboratory and the
Science and Technology Facilities Council of the United
Kingdom, the Max-Planck-Society, and the State of
Niedersachsen/Germany for support of the construction
and operation of the GEO600 detector. The authors also
gratefully acknowledge the support of the research by these
agencies and by the Australian Research Council, the
Council of Scientific and Industrial Research of India,
the Istituto Nazionale di Fisica Nucleare of Italy, the
Spanish Ministerio de Educacio
́
n y Ciencia, the
Conselleria d’Economia, Hisenda i Innovacio
́
of the
Govern de les Illes Balears, the Scottish Funding
Council, the Scottish Universities Physics Alliance, The
National Aeronautics and Space Administration, the
Carnegie Trust, the Leverhulme Trust, the David and
Lucile Packard Foundation, the Research Corporation,
and the Alfred P. Sloan Foundation. This paper was as-
signed LIGO document number LIGO-P080093.
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