Synthetic streams in a gravitational wave inspiral search with a multidetector network

A gravitational wave inspiral search with a global network of interferometers when carried in a phase coherent fashion mimics a search with two effective synthetic data streams. The streams are constructed by the linear combination of the overwhitened data from individual detectors. We demonstrate here that the two synthetic data streams pertaining to the two polarizations of the gravitational wave can be derived prior to the maximum-likelihood analysis in a most natural way using the technique of singular-value decomposition applied to the network signal-to-noise ratio vector. The singular-value technique combined with the matched filtering in network plus spectral space enables the construction of the synthetic streams. We further show that the network log likelihood ratio is then the sum of the log-likelihood ratios of these synthetic streams. In this formalism, the four extrinsic parameters of the nonspinning inspiral signal, namely, amplitude, initial phase, binary inclination, and the polarization $\{A_0,{\varphi }_a,\epsilon ,\Psi \}$, are mapped to the two amplitudes and two phases, namely, $\{{\mathit{\rho }}_L,{\mathit{\rho }}_R,{\mathrm{\Phi }}_L,{\mathrm{\Phi }}_R\}$. We show that the maximization over the new extrinsic parameters is a straightforward exercise closely linked to the single detector approach in the literature. Toward the end, we connect all the previous works related to the multidetector gravitational wave inspiral search and present in the same notation.

Figure 1

Directional SNR squares ${\mathit{\rho }}_{\mathit{L}}^{\mathit{2}}$, ${\mathit{\rho }}_{\mathit{R}}^{\mathit{2}}$, ${\mathit{\rho }}_{+}^{\mathit{2}}$, ${\mathit{\rho }}_{-}^{\mathit{2}}$, and ${{\mathit{\rho }}^{\mathit{2}}}_{\text{net}}$ for various three network configurations with $m_1$, $m_2=1.4$, $\epsilon =\pi /4$, $\mathrm{\Psi }=\pi /4$, and $r=150\mathrm{Mpc}$. with the same noise spectral densities for all detectors.

Figure 2

Directional SNR Squares ${\mathit{\rho }}_{\mathit{L}}^{\mathit{2}}$, ${\mathit{\rho }}_{\mathit{R}}^{\mathit{2}}$, ${\mathit{\rho }}_{+}^{\mathit{2}}$, ${\mathit{\rho }}_{-}^{\mathit{2}}$, and ${{\mathit{\rho }}^{\mathit{2}}}_{\text{net}}$ for various four detector network configurations with $m_1$, $m_2=1.4$, $\epsilon =\pi /4$, $\mathrm{\Psi }=\pi /4$, and $r=150\mathrm{Mpc}$. with the same noise spectral densities for all detectors.

Figure 3

Directional SNR squares ${\mathit{\rho }}_{\mathit{L}}^{\mathit{2}}$, ${\mathit{\rho }}_{\mathit{R}}^{\mathit{2}}$, ${\mathit{\rho }}_{+}^{\mathit{2}}$, ${\mathit{\rho }}_{-}^{\mathit{2}}$, and ${{\mathit{\rho }}^{\mathit{2}}}_{\text{net}}$ for network LHVKI with $m_1$, $m_2=1.4$, $\epsilon =\pi /4$, $\mathrm{\Psi }=\pi /4$, and $r=150\mathrm{Mpc}$, with the same noise spectral densities for all detectors.