Best network chirplet chain: Near-optimal coherent detection of unmodeled gravitational wave chirps with a network of detectors

The searches of impulsive gravitational waves (GW) in the data of the ground-based interferometers focus essentially on two types of waveforms: short unmodeled bursts from supernova core collapses and frequency modulated signals (or chirps) from inspiralling compact binaries. There is room for other types of searches based on different models. Our objective is to fill this gap. More specifically, we are interested in GW chirps “in general,” i.e., with an arbitrary phase/frequency vs time evolution. These unmodeled GW chirps may be considered as the generic signature of orbiting or spinning sources. We expect the quasiperiodic nature of the waveform to be preserved independently of the physics which governs the source motion. Several methods have been introduced to address the detection of unmodeled chirps using the data of a single detector. Those include the best chirplet chain (BCC) algorithm introduced by the authors. In the next years, several detectors will be in operation. Improvements can be expected from the joint observation of a GW by multiple detectors and the coherent analysis of their data, namely, a larger sight horizon and the more accurate estimation of the source location and the wave polarization angles. Here, we present an extension of the BCC search to the multiple detector case. This work is based on the coherent analysis scheme proposed in the detection of inspiralling binary chirps. We revisit the derivation of the optimal statistic with a new formalism which allows the adaptation to the detection of unmodeled chirps. The method amounts to searching for salient paths in the combined time-frequency representation of two synthetic streams. The latter are time series which combine the data from each detector linearly in such a way that all the GW signatures received are added constructively. We give a proof of principle for the full-sky blind search in a simplified situation which shows that the joint estimation of the source sky location and chirp frequency is possible.

Figure 1

Coordinate transformations.—(a) $\mathcal{O}({\varphi }_e,{\theta }_e,{\psi }_e)$: $\mathrm{\text{Earth frame}}{\mathbf{x}}_E\mathrm{\text{:}}(x_E,y_E,z_E)\to \mathrm{\text{wave frame}}{\mathbf{x}}_w\mathrm{\text{:}}(x_w,y_w,z_w)$, (b) $\mathcal{O}(\alpha ,\beta ,\gamma )$: $\mathrm{\text{Earth frame}}{\mathbf{x}}_E\mathrm{\text{:}}(x_E,y_E,z_E)\to \mathrm{\text{detector frame}}{\mathbf{x}}_d\mathrm{\text{:}}(x_d,y_d,z_d)$. The latitude $l$ and longitude $L$ of the detector are related to the Euler angles by Eqs. (8, 9).

Figure 2

Degeneracy of the network response.—We show here the inverse of $\mathrm{cond}(\mathbf{\Pi })$ for various detector networks (the abbreviated detector names are listed in Table ). The brighter regions of the sky correspond to the large conditioning number $\mathrm{cond}(\mathbf{\Pi })$. The fraction of the sky where $1/\mathrm{cond}(\mathbf{\Pi })>0.1$ is (a) 27%, (b) 4.8%, (c) and (d) 0%. Since the LIGO detectors are almost aligned, this pair of detectors has the largest percentage of degeneracy.

Figure 3

Benefits of a coherent network analysis (SNR enhancement).—We display the polar maps of the following quantities for the L1-H1-V1 network (a) ${min}_{\psi ,ϵ}\rho /{\rho }_{\mathrm{best}}$ and (b) ${max}_{\psi ,ϵ}\rho /{\rho }_{\mathrm{best}}$. Here, ${\rho }_{\mathrm{best}}$ designates the best SNR of the detectors in the network. The maximum, minimum are taken over all the polarization angles $\{\psi ,ϵ\}$.

Figure 4

SNR per synthetic streams and benefits of a coherent network analysis (SNR enhancement).—We display the polar maps of the following quantities for the L1-H1-V1 network: (a) $⟨{\rho }_2/{\rho }_1{⟩}_{\psi ,ϵ}$ and (b) ${max}_{\psi ,ϵ}(\rho /{max}_{l=1,2}{\rho }_l)$ where ${\rho }_l$ denotes the SNR of synthetic stream $l=1$ or 2, as defined in Eq. (57). The maximum and average are taken over all the polarization angles $\{\psi ,ϵ\}$.

Figure 5

Coherent detection/estimation of a “random” chirp with network of three GW antennas.—In the data of three GW antennas (the two LIGO and Virgo), we inject (a) a random GW chirp emitted from a source at the position marked with a “$+$” at $\varphi =2.8\mathrm{rad}$ and $\theta =0.4\mathrm{rad}$. We perform a full-sky search using the best network chirplet chain algorithm. It produces a likelihood landscape (b) where we select the maximum. This is the detection point and it is indicated with “$\times$”. In (c) we show the combined WV distribution of the synthetic streams at the detection point. In (d), we compare the exact frequency of the chirp (solid/blue line) with the estimation (dashed/red line) obtained at the detection point.

Figure 6

Coherent detection/estimation of a random chirp with network of three GW antennas. Standard and regularized statistic.—We compare the likelihood landscape (left) and frequency estimation (right) obtained using the standard (top) and Tykhonov-regularized (bottom) versions of the network statistic. The test signal is a random GW chirp injected at the sky location marked with a “$+$”. This location has been chosen because of the associated large value of the conditioning number, namely $\mathrm{cond}(\mathbf{\Pi })\approx 192$.