Search for gravitational wave ringdowns from perturbed black holes in LIGO S4 data

TLDR: In this paper, the authors present a search for gravitational waves from black hole ringdown in the fourth LIGO science run S4, during which the dominant mode of perturbed black holes with masses in the range of 10M to 500M⊙, the regime of intermediate-mass black holes to distances up to 300 Mpc.

Abstract: 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=m=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 10M⊙ to 500M⊙, 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 85M⊙ and 390M⊙ in the local universe, assuming a uniform distribution of sources, of 3.2×10-5  yr-1 Mpc-3=1.6×10-3  yr-1L10-1,where L10 is 1010 times the solar blue-light luminosity.

Figures

FIG. 2 (color online). The horizon distance versus mass and frequency for a black hole with spin of â 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).

FIG. 1. Venn diagram of the coincident detector times in hours.

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.

FIG. 4 (color online). Hanford effective distance versus ringdown frequency f0 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.

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.

Full text

PUBLISHED VERSION
Abbott, B.;...; Hosken, David John; Hough, J.;...; Munch, Jesper; Murray, P. G.;...; Ottaway,
David John; Ottens, R. S.;...; Veitch, Peter John; ... et al.; LIGO Scientific Collaboration
Search for gravitational wave ringdowns from perturbed black holes in LIGO S4 data
Physical Review D, 2009; 80(6):062001
©2009 American Physical Society
http://link.aps.org/doi/10.1103/PhysRevD.80.062001
http://link.aps.org/doi/10.1103/PhysRevD.62.093023
http://hdl.handle.net/2440/58907
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Search for gravitational wave ringdowns from perturbed black holes in LIGO S4 data
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PHYSICAL REVIEW D 80, 062001 (2009)
1550-7998=2009=80(6)=062001(9) 062001-1 Ó 2009 The American Physical Society

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(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
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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 10M

to 500M

, 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 85M

and
390M

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,
(20M

 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 500M

[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
ðtÞ¼<

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
ðtÞ¼A
GM
c
2
r
e
f
0
t=Q
cosð2f
0
tÞ: (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
gð
^
aÞ (3)
Q ¼ 2ð1 
^
aÞ
9=20
; (4)
where gð
^
aÞ¼1  0:63ð1 
^
aÞ
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
ð1=2Þ
FðQÞ
ð1=2Þ
gð
^
aÞ
ð1=2Þ
; (5)
where FðQÞ¼1 þ
7
24Q
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
=ðm
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 440M

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 80, 062001 (2009)
062001-4

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Search for gravitational wave ringdowns from perturbed black holes in ligo s4 data" ?

“ The author ( s ), and in the case of a Work Made For Hire, as defined in the U. S. Copyright Act, 17 U. S. C. §101, the employer named [ below ], shall have the following rights ( the “ Author Rights ” ): [... ] 3. The right to use all or part of the Article, including the APS-prepared version without revision or modification, on the author ( s ) ’ web home page or employer ’ s website and to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author ( s ) ’ and/or the employer ’ s use for educational or research purposes. ” 

Q2. How many simulated signals were added to the data?

The authors added 5701 simulated signals in the frequency range of 70 Hz–140 Hz and between 0.5M pc and 103 Mpc in distance to the data, over ten runs. 

Q3. What is the way to find a black hole?

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. 

Q4. How long after the innermost stable circular orbit of the binary?

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. 

Q5. How many hours of data was collected from the 4th LIGO science run?

This yielded a total of 567.4 hours of analyzable 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). 

Q6. What is the error in the MC?

The second source of error is due to the limited number of simulated signals in their Monte Carlo (MC) simulations to evaluate the efficiency. 

Q7. What is the sensitivity of the new LIGO?

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 2000M or higher. 

Q8. How do the authors determine the efficiency of the search?

The authors 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, "ðrÞ, defined as the fraction of simulated signals detected in the analysis, as a function of physical distance. 

Q9. How much confidence is given on the rate of the black hole?

The 90% confidence062001-7upper limit on the rate in these units is given byR90% ¼ 2:303TCL ; (15)which evaluates to 1:2 10 3 yr 1L 110 . 

Q10. What is the method for detecting the signal?

LIGO data is non-Gaussian and while the method is still appropriate it is not sufficient to discriminate between signal and background. 

Q11. How is the efficiency of the search determined?

Figure 4 shows that the efficiency is stronglydependent on the ringdown frequency f0, and thus for the purpose of setting an upper limit the authors restrict the calculation to the most sensitive frequency band, 70 Hz–140 Hz, corresponding to black hole masses in the range 85M –390M . 

Q12. How many errors do the authors assign to the waveform?

the authors 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]. 

Q13. What is the angular frequency of the real part of the ringdown phase?

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 . 

Q14. What is the noise spectral density of the data?

Filtering the data gives a signal to noise ratio (SNR)ðhÞ ¼ hs; hiffiffiffiffiffiffiffiffiffiffiffihh; hip ; (8) wherehs; hi ¼ 2 Z 1 1 ~sðfÞ~h ðfÞ ShðjfjÞ df: (9)Here, the noise spectral density ShðfÞ is the one appropriate for the data segment in question. 

Q15. How many errors can cause the SNR of a signal to be incorrectly quantified?

Errors in the calibration can cause the SNR of a signal to be incorrectly quantified, thereby introducing inaccuracies in the distance.