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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 =

cδt

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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