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Reconstruction of source location in a network of gravitational wave interferometric detectors

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This paper deals with the reconstruction of the direction of a gravitational wave source using the detection made by a network of interferometric detectors, mainly the LIGO and Virgo detectors. We suppose that an event has been seen in coincidence using a filter applied on the three detector data streams. Using the arrival time (and its associated error) of the gravitational signal in each detector, the direction of the source in the sky is computed using a chi^2 minimization technique. For reasonably large signals (SNR>4.5 in all detectors), the mean angular error between the real location and the reconstructed one is about 1 degree. We also investigate the effect of the network geometry assuming the same angular response for all interferometric detectors. It appears that the reconstruction quality is not uniform over the sky and is degraded when the source approaches the plane defined by the three detectors. Adding at least one other detector to the LIGO-Virgo network reduces the blind regions and in the case of 6 detectors, a precision less than 1 degree on the source direction can be reached for 99% of the sky.
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arXiv:gr-qc/0609118v2 28 Sep 2006
gr-qc/XXX
Reconstruction of source location in a network of gravitational
wave interferometric detectors
Fabien Cavalier,
Matteo Barsuglia, Marie-Anne Bizouard, Violette
Brisson, Andr´e-Claude Clapson, Michel Davier, Patrice Hello,
Stephane Kreckelbergh, Nicolas Leroy, and Monica Varvella
Laboratoire de l’Acc´el´erateur Lin´eaire,
IN2P3-CNRS et Universit´e Paris-Sud 11,
Centre Scientifique d ’ Orsay, B.P. 34, 91898 Orsay Cedex (France)
(Dated: July 14, 2005)
Abstract
This paper deals with the reconstruction of the d irection of a gravitational wave source using the
detection made by a network of interferometr ic detectors, mainly the L IGO and Virgo detectors.
We suppose that an event has been seen in coincidence using a filter applied on the three detector
data streams. Using the arrival time (and its associated error) of the gravitational signal in each
detector, the direction of the source in the sky is computed using a χ
2
minimization technique.
For reasonably large signals (SNR>4.5 in all detectors), the mean angular error between the real
location and the reconstructed one is about 1
. We also investigate th e effect of the networ k
geometry assuming the same angular response for all interferometric detectors. It appears that
the reconstruction quality is not uniform over the sky and is degraded when the source ap proaches
the plane defined by the three detectors. Adding at least one other detector to th e L IGO-Virgo
network reduces the blind regions and in the case of 6 detectors, a precision less than 1
on the
source direction can be reached for 99% of the sky.
PACS numbers: 04.80.Nn, 07.05.Kf
Keywords: Gravitational Waves, LIGO, Virgo, Network Data Analysis, Reconstruction of Source Direction
Electronic addres s: cavalier@lal.in2p3.fr
1
I. INTRODUCT ION
The LIGO and Virgo gravitat io nal wave interferometric detectors are approaching their
design sensitivity [1],[2] and in the near future, coincidences between the three detectors
(LIGO-Hanford, LIGO-Livingston and Virgo) will be po ssible. In order to reconstruct the
direction of the astrophysical sources in the sky, it is well known [3] that a minimum of three
detectors is mandatory even if an ambiguity remains between two positions symmetric with
respect to the plane defined by the 3 detectors. The source direction is also provided by t he
coherent searches for bursts [4], [5], [6] or coalescing binaries [7], [8], [9] where one of the
outputs of the detection algorithm is an estimation of the source direction.
In this paper, we propose a method fo r estimating the source position using only the
arrival time of the gravitational signal in each detector. The event detection is supposed to
have been previously done by dedicated algorithms ([10-2 3] for bursts, [24-29] for coalescing
binaries) and is not within t he scope of this art icle.
The direction reconstruction is based on a χ
2
minimization as described in Section II.
This technique can be easily extended to any set of detectors. Moreover, the method can
be applied t o several types of sources (burst, coalescence of binary objects ...) as soo n as an
arrival time can be defined for the event.
Section II also deals with the simulation procedure which will be used in the following
sections to evaluate the reconstruction quality in several configurations. Sections III a nd IV
describe the performances of the L IGO-Virgo network first neglecting (III), then including
(IV) the angular response of the detectors. In section V, we consider the addition o f other
gravitational wave detectors (supposing a similar sensitivity) and investigate their impact
on the reconstruction. In real conditions, systema t ic errors on arrival time are likely to exist
and their impact on the reconstruction is tackled in Section VI.
II. MODELING THE RECONSTRUCTION OF THE SOURCE DIRECTION
A. Arrival time of GW signals in interferometers
Within a network of n interferometers, we suppose that each detector D
i
measures the
arrival time t
i
of the gravit ational wave. O f course, the definition of the arrival time depends
on the source type and is a matter o f convention, for example: peak value in the case of a
2
sup ernova signal, end of the coalescence for binary events. In the following, it is assumed
that all int erferometers use consistent conventions.
The error on the arrival time, σ
i
, depends on the estimator used and on the strength of
the signal in the detector D
i
, strength (fo r a g iven distance and a given signal type) which is
related to the antenna pattern f unctions (see [5] and references therein) at time t
i
. At that
time, the antenna pattern depends on the longitude and the latitude of the detector location,
as well a s its orientation, the angle between the interferometer arms, the sky coordinates α
(Right Ascension) and δ (Declination) of the source, and the wave polarization angle ψ.
The timing uncertainty σ
i
can be parametrized by [10]:
σ
i
=
σ
0
(SNR
i
)
ζ
(1)
where SNR
i
is the measured SNR in detector D
i
, σ
0
and ζ are constants depending on
the detection algorithm and the signal shape. For example, a burst search with a 1-ms
Gaussian correlator leads to σ
0
1.4 ms and ζ = 1. Typically, f or an SNR equal to 10, the
error on the arrival time is a few tenth of milliseco nds and weakly depends on ζ for SNR
values between 4 and 1 0.
B. χ
2
Definition
The n measured arrival times t
i
and their associated errors σ
i
are the input for the
reconstruction of the source direction in the sky, direction defined by α and δ.
In the 3-detector configuration, the angles (θ a nd φ ) of the source in the detector coor-
dinate system (see Ref.[4] and [7] for exact definitions) are given by [7]:
sin θ =
1
ab
2
(2)
cos θ = ±
1
ab
2
p
(ab
2
)
2
sin φ =
b
2
(t
1
t
2
)
cos φ =
a(t
1
t
3
) b
1
(t
1
t
2
)
with (3)
= (b
1
(t
1
t
2
) a(t
1
t
3
))
2
+ (b
2
(t
1
t
2
))
2
3
where D
1
is placed at (0,0 ,0), D
2
at (a,0,0) and D
3
at (b
1
,b
2
,0).
When performing a coherent analysis of the GW detector streams the position of the
source in the sky is part of the output parameters, corresponding to the stream combination
which maximizes the SNR. However, for a burst search, it is known that thousands of possible
positions have to be tested to obtain t he solution [5], [6] or a least-square function involving
the integration of detector streams has to be minimized [4]. This minimization also implies
the test of hundreds of initial conditions in order to reach the right minimum.
Concerning coalescing binaries, it implies the definition of a five-parameter bank of filters
including the chirp mass, the three Euler angles and the inclination angle [7] of the orbital
plane or a three-parameter bank of thousands filters for the two source angles and the chirp
time [9].
In all coherent techniques, the extraction of the source direction is a n heavy pro cess
imbedded in the detection procedure.
In this paper, we propose a simpler approa ch where α and δ are found through a least-
square minimization using separately triggered events obtained by a coincidence search. We
suppo se tha t the detection is already performed applying suitable algorithms (matched filter
for coalescing binaries, r obust filters for bursts). The χ
2
is defined by:
χ
2
=
n
X
i=1
t
i
(t
0
+
Earth
i
(α, δ))
2
σ
2
i
(4)
where t
0
is the arrival time of the gravitational wave at the center of the Earth and
Earth
i
(α, δ) is the delay between the center of the Earth and the i
th
detector which only
depends on α and δ.
The first advantage of this definition is tha t it deals with absolute times recorded by each
detector rather than time differences where o ne detector has to be singled out. Otherwise,
the best choice for t he reference detector is not obvious: the detector with t he lower error
on the arrival time, the detector which gives the larger time delays or the detector leading
the best relative errors on timing differences ? This definition leads to uncorrelated errors
on fitted measurements.
The second advantage is that the network can be extended to any number of detectors
and the addition of o t her detectors is straightfor ward. Obviously, the method requires that
the event is seen by all detectors.
4
The least-square minimization provides the estimation of t
0
, α, δ and the covariance ma-
trix of the fitted para meters. When the number of detectors is greater than 3, the χ
2
value at
the minimum can also be used as a discriminating variable, as the system is overconstrained.
C. Simulation procedure for a 3-detector network
A list of coincident events are defined by the three arrival times and their associated
errors. No detection procedure is performed in t hese simulations as stated before.
The simulation proceeds in two steps in order to study two coupled effects: antenna-
patterns and location with respect to the 3-detector plane. The first step is a simplified
approach: the antenna-pattern functions are ignored and the same error σ
i
is assumed for
arrival times. The second step is more realistic: we assume the same sensitivity for each
detector and the signal strength is adjusted to have the mean (over the three detectors) SNR
equal to 10. However, this implies that sometimes, due to the antenna-pattern, the signal is
seen in a given detector with an SNR lower than 4.5, which remains an acceptable threshold
for a real detection. The same threshold equal to 4.5 will be used later to see the effect of a
reasonable detection scheme on the a ngular r econstruction error . We evaluate the errors σ
i
for each arrival time t
i
using Equation 1 with ζ = 1.
For a given simulation, the coordinates (α
true
, δ
true
) of t he source are chosen and the true
arrival times t
true
i
on each detector are computed taken t
true
0
equal to 0 (it is obvious that
the timing origin can a lways been chosen such as t
true
0
= 0)
The measured arrival times t
measured
i
are drawn according to a Gaussian distribution
centered on t
true
i
and of width σ
i
. The simulated values t
measured
i
are then used as inputs for
the least-square minimization. As the minimizat io n is an iterative procedure, some initial
values for the parameters have to be given. t
0
is initialized by the average of t
measured
i
. For the
angles, it appears that the initial values for the angles have no influence on the minimization
convergence and a random direction is a dequate. In the case of three interferometers, it is
well known that there is a twofold ambiguity for the direction in the sky which can lead to
the same arrival times in the detectors. These two solutions are symmetric with respect to
the 3-detector plane. In order to resolve t he ambiguity, a fourth detector is needed. For the
evaluatio n of the reconstruction accuracy, only the solution closest to the source is retained.
5
FIG. 1: Reconstruction accuracy for a source at the Galactic C enter at a fixed day time for LIGO-
Virgo network.
The angular error is the angu lar distance on the sphere between the true direction and the recon-
structed one. The curves on the first two plots correspond to the best Gaussian fit.
III. SOURCE DIRECTION RECONSTRUCTION: 3-DETECTOR NETWORK
GEOMETRY
In this section, we only deal with the LIGO-Virgo network and the effect of the antenna-
pattern functions are not included and it is assumed that all detectors measure the arrival
time with the same precision. As previously said, it allows to decouple the effect of the
antenna-patterns and o f the location with respect to the 3-detector plane.
First of all, as an example, in order to eva luate the accuracy of the reconstruction, we
choose a given position in the sky (coordinates of the Galactic Center α
GC
= 266.4
, δ
GC
=
28.98
) and we perform the simulation with σ
i
= 10
4
s at a fixed time. The results are
shown on Figure 1. A resolution of about 0.7 degrees can be achieved both on α and δ. The
angular error is defined as the angular distance on the sphere between the true direction and
the reconstructed o ne (it does not depend on the coordinate system and in particular there
6
is no divergence (o nly due to t he coordinate system) when δ is equal to 90 degrees). This
variable will be used in the following steps a s the estimator of the reconstruction quality.
The mean angular error is 0.8 degrees. As shown on Table I, the estimated errors (given by
the covariance matrix) obtained by the χ
2
minimization are in perfect agreement with these
resolutions.
Angle RMS of Distribution Mean of estimated errors given
by th e covariance matrix
α 0.760
0.758
δ 0.635
0.636
TABLE I: Reconstruction accuracies on α and δ and errors given by the covariance matrix. Three
digits are given in order to show the adequacy between RMS and errors given by the covariance
matrix.
Varying σ
i
in the simula t io n, it is found that the resolution is proportional to the timing
resolution for reasonable errors (σ
i
3 ms. For σ
i
3 ms, the errors become comparable
to the sky size.) and we obtain:
α
Resolution
= 0.7
σ
i
10
4
s
(5)
δ
Resolution
= 0.6
σ
i
10
4
s
Mean Angular error = 0.8
σ
i
10
4
s
For the Galactic Center direction and with σ
i
= 0.1 ms, we evaluate the reconstruction
performances over o ne entire day, still neglecting the antenna-pattern.
It clearly a ppears o n Figure 2 that the resolution varies, as expected, during the day and
lies between 0.8
and 4.3
(see Table II fo r details).
The error maximum which appears around t = 20 h (the time origin is arbitrary) cor-
responds to directions for which the source approaches the detector plane. The angular
distance between the two possible solutions (the source and its mirror image) gives a good
estimator of this closeness (see Fig. 2). This effect is even clearer on Figure 3 where the
source is located at δ = 0
. In t his case, the source crosses the 3-detector plane twice a day
(t = 2.4 h, t = 14.4 h) and the angular distance between the real source and its mirror image
7
FIG. 2: Angular Error for a source at the Galactic Center for one day (t=0 arbitrary chosen) for
LIGO-Virgo network.
The top-left plot presents the angular error as a function of time. The bottom-left plot shows
the angular distance between the source (S) and its mirror image (M) (other direction in the sky
which gives the same time delays). The top-right plot is the distribution of angular errors. The
bottom-right gives the fraction of events with an angular error below a given threshold versus this
threshold. σ
i
is set to 0.1 ms for all detectors.
is null. This degradation of the angular reconstruction is not related to the χ
2
minimization
technique. It is a g eometrical property of the network which releases constraints when the
source belongs to the 3-detector plane. For the δ = 0
configuration, angular resolutions as
given in Table II are similar to the ones obtained for the Galactic Center.
This degradation can be easily understood with only two detectors located at (±d/2, 0, 0)
and a source in the (x,y) plane defined by its angle θ with the x axis. In this 2-detector
case, the arrival time difference t
21
is given by:
t
21
=
d
c
cos θ (6)
and thus, if t
21
is measured with an error δt, it will induce an error δθ =
t
d| sin θ|
on the
8
δ(
) Minimal Error (
) Maximal Error (
) Mean Error (
) Median Error (
)
-28.98 (GC) 0.8 4.3 1.9 0.95
0 1.3 3.1 1.8 1.5
All values 0.7 4.5 1.6 1.1
TABLE II: Angular resolution for Galactic Center, δ = 0
position and averaged on all possible
values of δ taken over the whole day. No anten na-pattern effect is included.
source direction which diverges when the source belongs to the 2-detector line (θ = 0 or π).
Of course, this simple estimation of the angular error supposes that Eq. 6 between t
21
and
θ can always be inverted, assumption which fails when t
21
becomes greater than
d
c
due to
measurement errors. We can wonder how to handle this case. In the 2-detector example,
we minimize χ
2
defined by:
χ
2
(θ) =
1
δt
2
t
measured
21
d
c
cos θ
2
(7)
which becomes minimal for θ = acos(t
measured
21
×
c
d
) if |t
measured
21
×
c
d
| 1 or θ = 0 when
the previous condition is not satisfied. We observe similar effect in the 3-detector case when
the source approaches the 3-detector plane.
Figure 4 presents the angular errors averaged on all possible value of δ. In this case, the
angular resolutions a r e given in Table II, ranging 0.7 to 4.5 degrees. Without considering
antenna-pattern effect, it means that an angular error lower than about one degree can be
reached for half of the sky a nd 90 % is below 3.5 degrees. The intersection of the 3-detector
with the celestial sphere appears as a zone with worst angular resolutions ( error ranging
from 3 degrees to 4.5 degrees) of the source direction.
IV. INCLUDING THE ANTENNA-PATTERN EFFECT IN A 3-DETECTOR
NETWORK
As in previous section, we use the LIGO-Virgo network as benchmark. Now, the strength
of the signal seen by each detector is computed ta king into account the antenna-pattern
functions. The errors on arrival times are estimated using Eq.1 with ζ = 1 and σ
0
= 1ms.
In order to remain close to the case described in Section III (all timing errors equal to
9
FIG. 3: Angular error f or a source at δ = 0
for one day (t=0 arbitrary chosen) for LIGO-Virgo
network.
The top-left plot presents the angular error as a function of time. The bottom-left plot shows
the angular distance between the source (S) and its mirror image (M) (other direction in the sky
which gives the same time delays). The top-right plot is the distribution of angular errors. The
bottom-right gives the fraction of events with an angular error below a given threshold versus this
threshold. σ
i
is set to 0.1 ms for all detectors.
0.1 ms), we impose that the mean SNR over the three detectors is equal to 10. A linear
polarization has been assumed for the incoming signal.
Figure 5 presents the angular error as a function of the day time for a source located at
δ = 0
and can be compared to Fig. 3. The two broad peaks (t = 2.4 hours and t = 14.4
hours), corresponding to the time when the source belongs to the 3-detector plane, are still
present (bar ely seen due to the change of scale) but six sharp diverging peaks show up. They
correspond to blind regions for at least one detector, as expected.
Of course, in real conditions, a minimal threshold will be applied fo r the event selection.
For this purpo se, we define a SNR thresho ld equal to 4.5 on each detector which leads
10
FIG. 4: Reconstruction resolution for the whole sky for LIGO-Virgo network.
The top plot p resents the angular error as a function of time for all possible values of δ. The
bottom-left plot is the distribution of angular errors. The bottom-right gives the fraction of events
with an angular error below a given threshold versus this threshold. The white line gives the
trajectory of the Galactic Center during the day. σ
i
is set to 0.1 ms for all detectors.
(supposing a Gaussian noise) to a false alarm rate about 10
6
Hz when requiring a triple
coincidence between Virgo, LIGO-Hanford and LIGO-Livingston. About 79 % of the events
satisfy this constraint and a median error of 2.0
is reached (See Table III and Fig . 6).
For a source located at the Galactic Center, the a ngular error shows several spikes related
to the antenna-pattern effect and the geometry of the network only increases the mean error
in the region around t = 21 h (Fig. 7). All peaks correspond to regions which are blind for
at least one interferometer. Imposing that all SNR are above 4.5 leaves 56.4 % of the events
and in particular all these blind regions are removed. The obtained angular resolutions are
given in Table III. They are slightly better than for the δ = 0
case.
Figure 8 presents the angular errors avera ged on all possible δ. As in Fig. 4, the in-
tersection between the 3-detector plane and the celestial sphere is visible. The regions of
11
FIG. 5: Angular Er ror for a s ource at δ = 0
as a function of the day time (t=0 arbitrary chosen)
for LIGO-Virgo network.
The top plot presents the angular error as a function of time. The middle plot shows the angular
distance between the source and its mirror image (the other sky direction giving the same time
delays). The S NR values in each interferometer are plotted on the bottom figure. The effect of the
antenna-pattern functions is included and the mean SNR is set to 10.
largest errors (> 15
) corresponds to the blind regions of the various detectors. Without
any conditions on the SNR in each detector, we obtain a median error of 1.7
(see Table
III). The SNR condition is satisfied for 60% of the events and leads to a median error of
1.3
with 90 % of the events below 3.7
. In the most favorable cases, we can even reach a
precision of 0.7
. A resolution better than 1
is obtained in 30 % of the cases.
V. ADDITION OF OTHER GRAVITATIONAL WAVE DETECT ORS
First of all, t he 2-kilometer long Hanford interferomet er has been included in the network.
In this case, we supposed that the SNR seen by by Ha nford 2k is half o f the SNR seen by
Hanford 4k. As expected, adding Hanford 2k does not change significantly the resolutions
12
FIG. 6: Distribution of angular errors for a source at δ = 0
for LIGO-Virgo network.
The top-left plot shows the distribution of angular error for all events. The bottom-left one presents
the fraction of events with an angular error below a given threshold versus this threshold. T he
right plots are the same imposing that the SNR seen by each detector is greater than 4.5.
obtained in previous sections.
Then, we add TAMA [30], GEO [31] and AIGO [32] assuming (for sake of simplicity) the
same sensitivities as LIGO and Virgo. We still impose a mean SNR equal to 10. Figure 9,
which has to be compared to Figure 5, shows the angular reconstruction for a source located
at δ = 0
as a function of t he day time a dding only a fourth detector to the LIGO-Virgo
network. Of course, there is no longer an ambiguity in the possible solutions and all former
blind regions are attenuated except the one around t = 8.5h for which effective low SNRs
are obtained with both Hanford and Virgo. The mean error (3
for LIGO-Virgo Table III) is
lowered to 2.0
adding GEO, to 1.7
adding TAMA and 0.7
adding AIGO. Not surprisingly,
the larger the network baseline, the better the resolution.
In order to evaluate t he performances of a strongly overconstrained network, we evaluate
the reconstruction resolution for the full sky using the 6 detectors (still using t he same
13
δ(
) Event Fraction(%) Min. Error (
) Mean Error (
) Median Error (
)
-28.98 (GC) 100 0.7 4.0 1.8
Any SNR
-28.98 (GC) 56.4 0.7 1.8 1.25
All SNR 4.5
0 100 1.2 3. 2.2
Any SNR
0 79.1 1.2 2.1 2.
All SNR 4.5
All values 100 0.7 2.7 1.7
Any SNR
All values 59.8 0.7 1.8 1.3
All SNR 4.5
TABLE III: Angular resolution for Galactic Center, δ = 0
position and averaged on all possible
values of δ taken over the whole day. The antenna-pattern effect is includ ed . The column “Event
Fraction gives the fraction of events satisfying the SNR selection criterion.
sensitivity). The results a r e shown in Figure 10 similar to Figure 8. The mean error is
about 0.4
and 99% of the sky is covered with a resolution below 0.8
. The regions with
errors between 1 and 3 degrees correspond to directions for which several interferometers
have very low SNRs (as the region ar ound t = 8.5h in the previous figure).
VI. EFFECTS OF SYSTEMATIC ERRORS ON ARRIVAL TIMES
All angular errors quoted previously suppose that arrival time measurements are only
subject to Gaussian noise. Systematic biases can also be introduced by the analysis and
their effect can be evaluated. In order t o do so, we modify Equation 8 introducing a timing
bias for only one detector:
t
Measured
i
= t
true
i
+ GaussianRandom × σ
i
+ Bias (8)
As in sections III and IV, we only consider the LIGO-Virgo network. It appears that
14
FIG. 7: Angular Error for a source at th e Galactic Center as a function of the day time (t=0
arbitrary chosen) for LI GO-Virgo network.
The top plot presents the angular error as a function of time. The middle plot shows the angular
distance between the source and its mirror image (the other sky direction giving the same time
delays). The values of the antenna-pattern functions are plotted on the bottom figure. The effect
of the anten na-pattern fu nctions is included and the mean SNR is set to 10.
Bias (ms) Bias on α(
) Bias on δ(
) Angular Bias (
)
.1 -0.24 0.65 0.68
.2 -0.48 1.26 1.33
.3 -0.72 1.79 1.90
.4 -0.97 2.55 2.69
.5 -1.2 3.17 3.35
1. -2.6 6.20 6.63
TABLE IV: Effect of a timing bias on angular reconstruction.
The bias is app lied to the arrival time at Livingstone interferometer. The RMS of the statistical
errors has been set to .1 ms. For this direction, the statistical angular error is about 0.8
.
15
FIG. 8: R econstruction resolution for the full sky for LIGO-Virgo network. The top graph gives
the angular error as a function of α an d sin(δ). For clarity, the values above 20
have been set to
20
. The bottom plots presents the angular error distribution and the fraction of events below a
given angular threshold imposing or not the SNR condition (all SNR>4.5).
the widths of the distribution for reconstructed α and δ are not modified by the bias but
the central values a r e shifted from the true ones. The differences between the reconstructed
value and the tr ue one ar e proportional to the bias and are significantly different from zero
when the bias and the statistical error have the same o rder of magnitude. Table IV shows the
effect of the bias for a given direction (we check that the effect is independant of the source
location). In this example, the bias has been applied to the Livingstone interferometer. The
width of statistical errors on arrival time was .1 ms leading to a statistical angular error
about 0.8
. For the tested configurations, we do not observe significant differences between
the three interferometers of the network.
16
FIG. 9: Reconstruction resolution for a source at δ = 0
as a function of the day time (t=0 arbitrary
chosen) adding a fourth detector to the LIGO-Virgo network.
The top figure presents the angular error as a function of time adding TAMA, th e middle one
adding GEO and the bottom one adding AIGO.
VII. CONCLUSION
We described a method for the reconstruction of the source direction using the timing
information (arrival time and associated error ) delivered by gravitational wave detectors
such as LIGO a nd Virgo. The reconstruction is performed using a least-square minimization
which allows to retrieve the angular position of the source and the arrival time at the center
of the Earth. The minimization also gives an estimation of errors and correlations on fitted
variables. For a given position, the angular error is pro portional to the timing resolution
and the systematic errors (if they exist) introduce a significant bias on reconstructed angles
when they reach the level of the statistical one.
When the antenna-patt ern effect is included and imposing a mean SNR value of 10 in
the LIGO-Virgo network, a precision of 1.7
can be reached for half of the sky. In or der to
reproduce a realistic case, we apply a threshold on the SNR in each detector (SNR> 4.5
17
FIG. 10: Reconstruction resolution for the full sky with a network of 6 detectors. The top graph
gives the angular error as a function of α and sin(δ).For clarity, the values above 1
have been set
to 1
.The bottom plots presents the angular error distribution and the fraction of events below a
given angular threshold.
leading to a false alarm rate about 10
6
Hz when performing a threefold coincidence). This
condition is satisfied for 60 % of the sky and the median angular error in this case is 1.3
. As
a resolution of 1
is obtained for 30 % of the events satisfying the SNR condition, it means
that about 20 % of the whole sky is seen with an angular error lower than 1
.
Adding o t her gravita t io na l waves detectors allows to reduce the blind regions and to lower
the mean resolution. In the best considered case (6 detectors), the resolution is about 0.4
and 99% of the sky is seen with a resolution lower than 0.8
.
All quoted resolutions (about one degree) are similar to those delivered by γ-ray satellites
when the first GRB counterpa r t s have been identified. So, we can expect it will be also
sufficient for the first identification of gravita t io nal wave sources.
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