Optimal detection of burst events in gravitational wave interferometric observatories

We consider the problem of detecting a burst signal of unknown shape in the data from gravitational wave interferometric detectors. We introduce a statistic which generalizes the excess power statistic proposed first by Flanagan and Hughes, and then extended by Anderson et al. to the multiple detector case. The statistic that we propose is shown to be optimal for an arbitrary noise spectral characteristic, under the two hypotheses that the noise is Gaussian, albeit colored, and that the prior for the signal is uniform. The statistic derivation is based on the assumption that a signal affects only $N_{\Vert }$ samples in the data stream, but that no other information is a priori available, and that the value of the signal at each sample can be arbitrary. This is the main difference from previous works, where different assumptions were made, such as a signal distribution uniform with respect to the metric induced by the (inverse) noise correlation matrix. The two choices are equivalent if the noise is white, and in that limit the two statistics do indeed coincide. In the general case, we believe that the statistic we propose may be more appropriate, because it does not reflect the characteristics of the noise affecting the detector on the supposed distribution of the gravitational wave signal. Moreover, we show that the proposed statistic can be easily implemented in its exact form, combining standard time-series analysis tools which can be efficiently implemented. We generalize this version of an excess power statistic to the multiple detector case, considering first a noise uncorrelated among the different instruments, and then including the effect of correlated noise. We discuss exact and approximate forms of the statistic; the choice depends on the characteristics of the noise and on the assumed length of the burst event. As an example, we show the sensitivity of the network of interferometers to a $\delta$-function burst.