Estimating parameters of coalescing compact binaries with proposed advanced detector networks

The advanced versions of the LIGO and Virgo ground-based gravitational-wave detectors are expected to operate from three sites: Hanford, Livingston, and Cascina. Recent proposals have been made to place a fourth site in Australia or India, and there is the possibility of using the Large Cryogenic Gravitational Wave Telescope in Japan to further extend the network. Using Bayesian parameter-estimation analyses of simulated gravitational-wave signals from a range of coalescing-binary locations and orientations at fixed distance or signal-to-noise ratio, we study the improvement in parameter estimation for the proposed networks. We find that a fourth detector site can break degeneracies in several parameters; in particular, the localization of the source on the sky is improved by a factor of $\sim 3–4$ for an Australian site, or $\sim 2.5–3.5$ for an Indian site, with more modest improvements in distance and binary inclination estimates. This enhanced ability to localize sources on the sky will be crucial in any search for electromagnetic counterparts to detected gravitational-wave signals.

Figure 1

Cumulative histogram of the 95% confidence interval for the area of the sky in square degrees, estimated using the Inspnest analysis of 728 signals which were detected in all network configurations, covering the range of sky locations and orientations of the binary. The extended networks show significantly better performance than the three-site network, with AHLV offering the highest fraction of signals resolved to better than $10{\mathrm{deg}}^2$. HILV and HHJLV show similar performance to each other for a large fraction of the signals, but the HHJLV network avoids the tail of poorly resolved signals located in regions of parameter space with poor sensitivity in HILV.

Figure 2

Cumulative histograms comparing the ability of the networks to resolve the individual parameters of main interest. Plots show fractions of events found within a given width of the confidence interval. The chirp mass $\mathcal{M}$ and symmetric mass ratio $\eta$ (top row) do not benefit significantly from a fourth site, except through better uniformity of SNR across the sky. The distance and inclination angle (middle row) show slight improvements in their resolvability but the effect is most marked in the 2D confidence intervals (see Fig. 6). The polarization angle $\psi$ shows interesting behavior, with the additional sites able to resolve a greater fraction of the signal to within a given interval of $\psi$, but with the median ratio of intervals being approximately unity (see Table ). This is explained by the sizable fraction of signals where the polarization angle is not well resolved. The resolution of the time of coalescence parameter $t_c$ is similar for all four-site networks, with 90% of signals resolved within 5 ms. The longer tail of less well-resolved times agrees with Figs. 1, 3 which show a hint of longer tail in sky resolution for AHLV compared to HILV.

Figure 3

A cumulative histogram over injections of the estimated accuracies in sky localization obtained via the Fisher information matrix technique. Blue, green, red, and cyan are the distributions of sky-localization uncertainties for the AHLV, HHLV, HHJLV, and HILV networks, respectively. All injections are independently normalized to an SNR of 30 for each network configuration, to avoid concerns about the trustworthiness of FIM results at low SNRs. The relative shapes of the histograms are thus more relevant than their actual values.

Figure 4

An example of the ability of a four-site network to break the degeneracy between multiple modes in $\alpha -\delta$ and $\cos \iota -d_L$ parameter space. This example shows the 95% confidence intervals from the HHLV (red, solid), AHLV (green, dashed), HILV (blue, dot-dashed), and HHJLV (black, dotted) networks. The sky position can be confined to one region with the four-site network, while the partial inclination angle degeneracy upon reflection is broken. The location of the injection in parameter space is indicated with a star.

Figure 5

Comparison of the one-dimensional PDFs for a typical source as detected by the HHLV network (red, ascending strokes) and AHLV network (blue, descending strokes). Note the bimodal posteriors in right ascension and declination for the HHLV network vs unimodal ones for the AHLV network. The latter network also allows for better estimates of the posteriors for inclination and luminosity distance, which is not properly reflected by the simple estimators of the PDF width used in Table . Dashed lines indicate the true injected values (different true values of the luminosity distance were used for the HHLV and AHLV injections so that the total network SNR is 15 in both cases).

Figure 6

Cumuative histogram comparing the ability of the networks to resolve the signal in 2D $d_L\mathrm{\text{−}}\iota$ space. The improvement in parameter estimation with the four-site networks due to elimination of the secondary mode is more striking here than in the individual $d_L$ and $\iota$ parameters (cf. Fig. 2).