Several gravitational-wave detectors can observe gravitational waves from the same source. For example a network of bar detectors can observe a gravitational-wave burst from the same supernova explosion, or a network of laser interferometers can detect the inspiral of the same compact binary system. The space-borne LISA detector can be considered as a network of three detectors that can make three independent measurements of the same gravitational-wave signal. Simultaneous observations are also possible among different types of detectors. For example, a search for supernova bursts can be performed simultaneously by resonant and interferometric detectors [21].
Let us consider a network consisting of gravitational-wave detectors and let us denote by
the
data collected by the
th detector (
). We assume that noises in all detectors are additive,
so the data
is a sum of the noise
in the
th detector and eventually a gravitational-wave signal
registered by the
th detector,
The analysis is greatly simplified if the cross spectrum matrix is diagonal. This means that the
noises in various detectors are uncorrelated. This is the case when the detectors of the network are in
widely separated locations, like, for example, the two LIGO detectors. However, this assumption
is not always satisfied. An important case is the LISA detector where the noises of the three
independent responses are correlated. Nevertheless for the case of LISA one can find a set of three
combinations for which the noises are uncorrelated [108, 112]. When the cross spectrum matrix is
diagonal the network log likelihood function is just the sum of the log likelihood functions for each
detector.
Derivation of the likelihood function for an arbitrary network of detectors can be found in [49].
Applications of optimal filtering for observations of gravitational-wave signals from coalescing
binaries by networks of ground-based detectors are given in [63, 41, 62, 103], and for the case of
stellar-mass binaries observed by LISA space-borne detector in [78, 118]. The single-detector
-statistic for nearly monochromatic gravitational waves from spinning neutron stars was
generalized to the case of a network of detectors (also with time-varying noise curves) in [42] (in this
work the
-statistic was also generalized from the usual single-source case to the case of a
collection of known sources). The reduced Fisher matrix [defined in Eq. (81
)] for the case of
a network of interferometers observing spinning neutron stars has been derived and studied
in [110].
Network searches for gravitational-wave burst signals of unknown shape are often based on
maximization of the network likelihood function over each sample of the unknown polarization waveforms
and
and over sky positions of the source [52, 95]. A least-squares-fit solution for the estimation
of the sky location of the source and the polarization waveforms by a network of three detectors for the case
of a broadband burst was obtained in [56].
There is also another important method for analyzing the data from a network of detectors – the search
for coincidences of events among detectors. This analysis is particularly important when we search for
supernova bursts, the waveforms of which are not very well known. Such signals can be easily mimicked by
the non-Gaussian behavior of the detector noise. The idea is to filter the data optimally in each of the
detectors and obtain candidate events. Then one compares parameters of candidate events, like, for
example, times of arrivals of the bursts, among the detectors in the network. This method is widely used in
the search for supernovae by networks of bar detectors [24]. A new geometric coincident algorithm of
combining the data from a network of detectors was proposed in [117]. This algorithm employs the
covariances between signal’s parameters in such a way that it associates with each candidate
event an ellipsoidal region in parameter space defined by the covariance matrix. Events from
different detectors are deemed to be in coincidence if their ellipsoids have a nonzero overlap. The
coincidence and the coherent strategies of multidetector detection of gravitational-wave signals from
inspiralling compact binaries have been compared in [96, 97
, 32
]. [96] considered detectors in
pairs located in the same site and [97] pairs of detectors at geographically separated sites. The
case of three detectors (like the network of two LIGO detectors and the Virgo detector) has
been considered in detail in [32], where it was demonstrated that the hierarchical coherent
pipeline on Gaussian data has a better performance than the pipeline with just the coincident
stage.
A general framework for studying the effectiveness of networks of interferometric gravitational-wave detectors has been proposed in [125]. Using this framework it was shown that adding a fourth detector to the existing network of LIGO/VIRGO detectors can dramatically increase, by a factor of 2 to 4, the detected event rate by allowing coherent data analysis to reduce the spurious instrumental coincident background.
http://www.livingreviews.org/lrr-2012-4 |
Living Rev. Relativity 15, (2012), 4
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