Detecting gravitational waves from inspiraling binaries with a network of detectors: Coherent versus coincident strategies

We compare two strategies of multidetector detection of compact binary inspiral signals, namely, the coincidence and the coherent. For simplicity we consider here two identical detectors having the same power spectral density of noise, that of initial LIGO, located in the same place and having the same orientation. We consider the cases of independent noise as well as that of correlated noise. The coincident strategy involves separately making two candidate event lists, one for each detector, and from these choosing those pairs of events from the two lists which lie within a suitable parameter window, which then are called coincidence detections. The coherent strategy on the other hand involves combining the data phase coherently, so as to obtain a single network statistic which is then compared with a single threshold. Here we attempt to shed light on the question as to which strategy is better. We compare the performances of the two methods by plotting the receiver operating characteristics (ROC) for the two strategies. Several of the results are obtained analytically in order to gain insight. Further we perform numerical simulations in order to determine certain parameters in the analytic formulae and thus obtain the final complete results. We consider here several cases from the relatively simple to the astrophysically more relevant in order to establish our results. The bottom line is that the coherent strategy although more computationally expensive in general than the coincidence strategy, is superior to the coincidence strategy—considerably less false dismissal probability for the same false alarm probability in the viable false alarm regime.

Figure 1

Plot of the contour of the ambiguity function at $\mathrm{MM}=0.97$. The fiducial frequency which scales both the axes is $f_a=40\mathrm{Hz}$.

Figure 2

Plots of the autocorrelation functions $c(\Delta t_c)$ (dotted curve) and $c_0(\Delta t_c)$ (solid curve). For reference, the autocorrelation curve for one of the quadrature components $c_0$ is also plotted. This curve is obtained both analytically in 28 and here numerically through simulations. The curve for $c$ is obtained purely numerically via simulations.

Figure 3

The number of false alarms $N_{\mathrm{FA}}$ as a function of the threshold ${\Lambda }^{*}$ for various fixed values of $\Delta t_c$.

Figure 4

The natural log of the number of false alarms is plotted versus the threshold ${\Lambda }^{*}$. In the large threshold regime, a straight line is fitted to this curve and extended to obtain intercepts on both axes. The intercept on the vertical axis is $\sim 22.96$.

Figure 5

Distribution of the difference of the detected parameters, (a) $t_c$, (b) ${\tau }_0$, and (c) ${\tau }_3$, for an injected signal of $\mathrm{SNR}=10$.

Figure 6

Distribution of SNR for an injected signal of $\mathrm{SNR}=10$.

Figure 7

The ROC curves for single detector, uncorrelated coincident, and coherent analysis for an injected signal of $\mathrm{SNR}=10$. The solid line, dashed line, and dotted line correspond to the theoretical ROC curves for coherent, coincident, and single detectors derived in Sec. 3, with $N_{\mathrm{ind}}$ determined in Sec. 4b. We also take $\alpha =1$ irrespective of the threshold.

Figure 8

The ROC curves for coherent and coincidence detection for two correlated detectors with ${ϵ}_0$ values (a) 0.2 and (b) 0.3 for an injected signal of $\mathrm{SNR}=10$.

Figure 9

ROC curves for single (dotted line), coherent (solid line), and coincident (dashed line) detectors (uncorrelated) for a 1 yr data train sampled at 2048 Hz and searched over $1M_{\odot }–40M_{\odot }$, for an injected signal of $\mathrm{SNR}=10$.

Figure 10

ROC curves for single (dotted line,) coherent (solid line), and coincident (dashed line) detectors (uncorrelated) for a 1 yr data train. The sources are distributed uniformly in distance between 0.4 Mpc and 26 Mpc and also in orientation.