Improving the efficiency of the detection of gravitational wave signals from inspiraling compact binaries: Chebyshev interpolation

Inspiraling compact-object binary systems are promising gravitational wave sources for ground and space-based detectors. The time-dependent signature of these sources is a well-characterized function of a relatively small number of parameters; thus, the favored analysis technique makes use of matched filtering and maximum likelihood methods. As the parameters that characterize the source model vary, so do the templates against which the detector data are compared in the matched filter. For small variations in the parameters, the filter responses are closely correlated. Current analysis methodology samples a bank of filters whose parameter values are chosen so that the correlation between successive samples from successive filters in the bank is 97%. Correspondingly, the additional information available with each successive template evaluation is, in a real sense, only 3% of that already provided by the nearby templates. The reason for such a dense coverage of parameter space is to minimize the chance that a real signal, near the detection threshold, will be missed by the parameter space sampling. Here we investigate the use of Chebyshev interpolation for reducing the number of templates that must be evaluated to obtain the same analysis sensitivity. Additionally, rather than focus on the “loss” of signal-to-noise associated with the finite number of filters in the template bank, we evaluate the receiver operating characteristic (ROC) as a measure of the effectiveness of an analysis technique. The ROC relates the false alarm probability to the false dismissal probability of an analysis, which are the quantities that bear most directly on the effectiveness of an analysis scheme. As a demonstration, we compare the present “dense sampling” analysis methodology with the “interpolation” methodology using Chebyshev polynomials, restricted to one dimension of the multidimensional analysis problem by plotting the ROC curves. We find that the interpolated search can be arranged to have the same false alarm and false dismissal probabilities as the dense sampling strategy using 25% fewer templates. Generalized to the two-dimensional space used in the computationally limited current analyses, this suggests a factor of 2 increase in computational efficiency; generalized to the full seven-dimensional parameter space that characterizes the signal associated with an eccentric binary system of spinning neutron stars or black holes, it suggests an order of magnitude increase in computational efficiency. Since the computational cost of the analysis is driven almost exclusively by the matched filter evaluations, a reduction in the number of template evaluations translates directly into an increase in computational efficiency; additionally, since the computational cost of the analysis is large, the increased efficiency translates also into an increase in the size of the parameter space that can be analyzed and, thus, the science that can be accomplished with the data.

Figure 1

This figure demonstrates the interpolated search. The ambiguity function is sampled and reconstructed over the chosen parameter space of ${\tau }_0=13–17\sec$ (only a part of the parameter space has been shown in the figure) with the help of the Chebyshev interpolating polynomial. The approximate location of the peak of the interpolating function is first located and the zero of the derivative is obtained by applying successive approximations around the peak. Finally a template is placed at this estimated signal location. Note that, by placing a template at the maximum of the interpolating polynomial, the match has improved over the one obtained by simply evaluating the maximum of the interpolating polynomial.

Figure 2

Ambiguity function $\mathcal{H}$ plotted as a function of $\Delta {\tau }_0$. Horizontal lines are drawn for various matches; the dashed horizontal line is for a 97% match, which corresponds to an intertemplate separation of $\sim 30\mathrm{ms}$.

Figure 3

Normalized test signals (without noise) were injected densely at regular intervals along the ${\tau }_0$ parameter space. For each injected signal ${\tau }_0$, we plot the maximum of the interpolating polynomial (dashed curve) and the match obtained by placing a template at the maximum of the interpolating polynomial (solid curve) according to the interpolated search strategy. This figure illustrates that the match is a (nearly) periodic function of ${\tau }_0$ with the period equal to the template separation. Moreover, with just $37+1$ interpolated search templates the minimal match is $0.97$. To maintain the same minimal match $133$ templates are needed for the usual dense search.

Figure 4

The variation of the false and true positive fractions, $\alpha$ and $ϵ$ with threshold $\eta$ for the dense and interpolated search methods, each making use of 40 template evaluations. The top panel shows the false positive fraction. Note how the false positive falls much sooner for the interpolated search than for the dense search. The bottom panel shows $ϵ$ when a signal of amplitude signal-to-noise 8 is present in the range ${\tau }_0\in [13\mathrm{s},17\mathrm{s}]$. Note how the $ϵ$ is always greater for the interpolated search than for the dense search. For the same computational cost (determined by the number of template evaluations), the interpolated search will always perform better than the dense search.

Figure 5

ROC curves for dense searches (solid curves) and interpolating searches (dashed curves). For a given number of templates, the solid curves are “lower”—less false dismissal probability for the same false alarm—than the dotted curves in the regime of low false alarm showing that the interpolated search performs better than the dense search for low false alarm. The bottom panel shows an analogous plot for high minimal match (fine bank) $\sim 0.98$. Here the performance of the dense search with 160 and 140 templates is comparable to that of the interpolated search with 120 and 100 templates, respectively.