Data-analysis strategy for detecting gravitational-wave signals from inspiraling compact binaries with a network of laser-interferometric detectors

Abstract

A data-analysis strategy based on the maximum-likelihood method (MLM) is presented for the detection of gravitational waves from inspiraling compact binaries with a network of laser-interferometric detectors having arbitrary orientations and arbitrary locations around the globe. For simplicity, we restrict ourselves to the Newtonian inspiral wave form. However, the formalism we develop here is also applicable to a wave form with post-Newtonian (PN) corrections. The Newtonian wave form depends on eight parameters: the distance r to the binary, the phase ${\ensuremath{\delta}}_{c}$ of the wave form at the time of final coalescence, the polarization-ellipse angle $\ensuremath{\psi},$ the angle of inclination $\ensuremath{\epsilon}$ of the binary orbit to the line of sight, the source-direction angles ${\ensuremath{\theta},\ensuremath{\varphi}},$ the time of final coalescence ${t}_{c}$ at the fiducial detector, and the chirp time $\ensuremath{\xi}.$ All these parameters are relevant for a chirp search with multiple detectors, unlike the case of a single detector. The primary construct on which the MLM is based is the network likelihood ratio (LR). We obtain this ratio here. For the Newtonian inspiral wave form, the LR is a function of the eight signal parameters. In the MLM-based detection strategy, the LR must be maximized over all of these parameters. Here, we show that it is possible to maximize it analytically with respect to four of the eight parameters, namely, ${r,{\ensuremath{\delta}}_{c},\ensuremath{\psi},\ensuremath{\epsilon}}.$ Maximization over the time of arrival is handled most efficiently by using the fast-Fourier-transform algorithm, as in the case of a single detector. This not only allows us to scan the parameter space continuously over these five parameters but also cuts down substantially on the computational costs. The analytical maximization over the four parameters yields the optimal statistic on which the decision must be based. The value of the statistic also depends on the nature of the noises in the detectors. Here, we model these noises to be mainly Gaussian, stationary, and uncorrelated for every pair of detectors. Instances of non-Gaussianity, as are present in detector outputs, can be accommodated in our formalism by implementing vetoing techniques similar to those applied for single detectors. Our formalism not only allows us to express the likelihood ratio for the network in a very simple and compact form, but also is at the basis of giving an elegant geometric interpretation to the detection problem. Maximization of the LR over the remaining three parameters is handled as follows. Owing to the arbitrary locations of the detectors in a network, the time of arrival of a signal at any detector will, in general, be different from those at the others and, consequently, will result in signal time delays. For a given network, these time delays are determined by the source-direction angles ${\ensuremath{\theta},\ensuremath{\varphi}}.$ Therefore, to maximize the LR over the parameters ${\ensuremath{\theta},\ensuremath{\varphi}}$ one needs to scan over the possible time delays allowed by a network. We opt for obtaining a bank of templates for the chirp time and the time delays. This means that we construct a bank of templates over $\ensuremath{\xi},$ $\ensuremath{\theta},$ and $\ensuremath{\varphi}.$ We first discuss ``idealized'' networks with all the detectors having a common noise curve for simplicity. Such an exercise nevertheless yields useful estimates about computational costs, and also tests the formalism developed here. We then consider realistic cases of networks comprising the LIGO and VIRGO detectors: These include two-detector networks, which pair up the two LIGOs or VIRGO with one of the LIGOs, and the three-detector network that includes VIRGO and both the LIGOs. For these networks we present the computational speed requirements, network sensitivities, and source-direction resolutions.