Geometric algorithm for efficient coincident detection of gravitational waves

Data from a network of gravitational-wave detectors can be analyzed in coincidence to increase detection confidence and reduce nonstationarity of the background. We propose and explore a geometric algorithm to combine the data from a network of detectors. The algorithm makes optimal use of the variances and covariances that exist among the different parameters of a signal in a coincident detection of events. The new algorithm essentially associates with each trigger ellipsoidal regions in parameter space defined by the covariance matrix. Triggers from different detectors are deemed to be in coincidence if their ellipsoids have a nonzero overlap. Compared to an algorithm that uses uncorrelated windows separately for each of the signal parameters, the new algorithm greatly reduces the background rate thereby increasing detection efficiency at a given false alarm rate.

Figure 1

(a) plots the contact function Eq. (3.4) for two pairs of three-dimensional ellipsoids taken from a search for binaries consisting of nonspinning compact objects characterized by parameters $(t_c,{\tau }_0,{\tau }_3)$ [see Sec. 5, in particular, Eq. (5.4)]. (b)–(d) are the projections of the ellipsoids in $(t_c,{\tau }_0)$, $(t_c,{\tau }_3)$, and $({\tau }_0,{\tau }_3)$ orthogonal planes, respectively. Solid lines refer to the case of nonoverlapping ellipsoids and dashed lines are for overlapping (i.e., coincident) triggers. Note that in the latter case the maximum of the contact function is $\le 1$, which is the test that is carried out to determine if a pair of triggers are in coincidence.

Figure 2

The ${log}_{10}$ of the ratio of the volume of the bounding box to the volume of the ellipsoid as a function of location in $({\tau }_0,{\tau }_3)$ space. The plots shown are (clockwise from top right) for the initial LIGO, advanced LIGO, Virgo, and Einstein Telescope. The low frequency cutoff is chosen to be 20 Hz.