Reconstructing the sky location of gravitational-wave detected compact binary systems: Methodology for testing and comparison

The problem of reconstructing the sky position of compact binary coalescences detected via gravitational waves is a central one for future observations with the ground-based network of gravitational-wave laser interferometers, such as Advanced LIGO and Advanced Virgo. Different techniques for sky localization have been independently developed. They can be divided in two broad categories: fully coherent Bayesian techniques, which are high latency and aimed at in-depth studies of all the parameters of a source, including sky position, and “triangulation-based” techniques, which exploit the data products from the search stage of the analysis to provide an almost real-time approximation of the posterior probability density function of the sky location of a detection candidate. These techniques have previously been applied to data collected during the last science runs of gravitational-wave detectors operating in the so-called initial configuration. Here, we develop and analyze methods for assessing the self consistency of parameter estimation methods and carrying out fair comparisons between different algorithms, addressing issues of efficiency and optimality. These methods are general, and can be applied to parameter estimation problems other than sky localization. We apply these methods to two existing sky localization techniques representing the two above-mentioned categories, using a set of simulated inspiral-only signals from compact binary systems with a total mass of $\le 20M_{\odot }$ and nonspinning components. We compare the relative advantages and costs of the two techniques and show that sky location uncertainties are on average a factor $\approx 20$ smaller for fully coherent techniques than for the specific variant of the triangulation-based technique used during the last science runs, at the expense of a factor $\approx 1000$ longer processing time.

Figure 1

An output PDF of the sky position from the two codes. The contour lines label the 50% and 90% credible regions for Timing++ while the light and dark shaded regions show the 50% and 90% credible regions, respectively, for LALInference. The star indicates the source location.

Figure 2

For each CL we plot the number of injections that fall within the associated minimum credible region ${\mathrm{CR}}_{\min }$ for all the signals analyzed with LALInference, bottom (red) curve, and Timing++, top (green) curve. The error bars correspond to the binomial error; see text for more details. A self-consistent algorithm gives results that lie along the diagonal line of this plot. Results that fall above the expected line, as is the case for Timing++, highlight an algorithm that is overcautious in its estimation of ${\mathrm{CR}}_{\min }$.

Figure 3

The fraction of detected signals whose associated true (corrected) 50% ${\mathrm{CR}}_{\min }$ covers less than a given area on the sky. We can see that LALInference gives much tighter constraints than Timing++ on the location of a source.

Figure 4

The fraction of sources where the injection would have been imaged after searching less than the given area in a telescope greedy algorithm.

Figure 5

The sky area of the 50% true (corrected) minimum credible region for each of the sources as a function of the optimal network SNR of the signal. While there is some scatter, the areas from LALInference [solid (red) dots] scale as $\propto 1/{\mathrm{SNR}}^2$, as one would expect, while the areas from Timing++ [open (green) circles] are closer to $\propto 1/\mathrm{SNR}$.

Figure 6

The ratio of recovered areas of the 50% true (corrected) CRs using LALInference as the baseline. While there is some scatter, LALInference is consistently producing smaller areas than Timing++ by a factor which is roughly 10 for low SNRs and approximately scales with SNR.

Figure 7

The cumulative distribution of wall times for lalinference_mcmc to output a new independent sample across the runs performed to generate the results reported in this paper. With 10 cores used for each run, CPU times were a factor of 10 larger.