Localization of binary neutron star mergers with a single cosmic explorer

Next-generation ground-based gravitational-wave detectors, such as Cosmic Explorer (CE), are expected to be sensitive to gravitational-wave signals with frequencies as low as 5 Hz, allowing signals to spend a significant amount of time in the detector frequency band. As a result, the effects caused by the rotation of the Earth become increasingly important for such signals. Additionally, the length of the arms of these detectors can be comparable to the wavelength of detectable gravitational waves, which introduces frequency-dependent effects that are not significant in current-generation detectors. These effects are expected to improve the ability to localize compact binary coalescences in the sky even when using only one detector. This study aims to understand how much these effects can help in localization. We present the first comprehensive Bayesian parameter estimation framework that accounts for all of these effects using bilby, a commonly used Bayesian parameter estimation tool. We focus on sky localization constraints for binary neutron star events with an optimal signal-to-noise ratio of 1000 with one detector at the projected CE sensitivity. We find that these effects help localize sources using one detector with sky areas as low as 10 square degrees. Moreover, we explore and discuss how ignoring these effects in the parameter estimation can lead to biases in the inference.

Figure 1

Histograms of 90% sky areas (top panel), 90% cosmic volumes (middle panel) and 90% fractional distance uncertainties (bottom panel) for two sets of 48 injections corresponding to two values of inclination angles ${\theta }_{\mathrm{JN}}=0.2$ shown in orange and ${\theta }_{\mathrm{JN}}=1.0$ shown in blue.

Figure 2

Distribution of 90% sky areas/volumes (denoted by colored dots at the injected positions) overlayed on the antenna pattern (where bright yellow represents directions of high average sensitivity where darker blue represents directions of low average sensitivity). Note that the color bar applies to the plotted points rather than the overlayed antenna pattern. The $+$ and $\times$ signs denote the best and worst localization, respectively. The skymap of the source marked with a $\star$ is plotted in Fig. 3.

Figure 3

Skymap at the $\star$ position in Fig. 2. This is a typical skymap at an SNR of 1000. See Appendix pp1 for sky localization posteriors for every injection considered in this study.

Figure 4

The solid black line represents the absolute value of the difference between the magnitude (top panel: $\parallel h_{\mathrm{CE}}|-|h_{\mathrm{ig}}\parallel$) and phase (bottom panel: angle of $h_{\mathrm{CE}}/h_{\mathrm{ig}}$) of two waveforms, one taking into account all the effects due to the Earth’s rotation and detector size ($h_{\mathrm{CE}}$) and the other ignoring the same effects ($h_{\mathrm{ig}}$). The dashed blue line, dash-dotted orange line, and dotted green line plot the same replacing $h_{\mathrm{CE}}$ by a waveform that takes into account only the amplitude modulation due to the rotation of the Earth in CE, the size of CE, and the phase modulation due to rotation of the Earth in CE, respectively. The left and right panels correspond to the best and worst sky localizations at ${\theta }_{\mathrm{JN}}=1.00$.

Figure 5

Posterior distributions showing the 2D sky areas for all the injections used in this paper using a Mollweide projection in an Earth-Centric coordinate system. The blue and orange contours refer to injections with ${\theta }_{\mathrm{JN}}$ values of 1.0 and 0.2, respectively.

Figure 6

2D sky localization posterior (left) and inferred luminosity distance posterior (right) for the best-localized event (marked by $+$ in Fig. 2) with ${\theta }_{\mathrm{JN}}=1.0$. The black contour/ histogram is for the inference considering all effects. The blue, orange, and green contours/ histograms ignore Earth-rotation amplitude modulation, detector-size amplitude modulation, and Earth-rotation time delay, respectively.

Figure 7

2D sky localization posterior (left) and inferred luminosity distance posterior (right) for an event on one of the dark spots with ${\theta }_{\mathrm{JN}}=1.0$. The black contour/ histogram is for the inference considering all effects. The blue, orange, and green contours/ histograms ignore Earth-rotation amplitude modulation, detector-size amplitude modulation, and Earth-rotation time delay, respectively.

Figure 8

2D sky localization posterior (left) and inferred luminosity distance posterior (right) for the worst-localized event (marked by $\times$ in Fig. 2) with ${\theta }_{\mathrm{JN}}=1.0$. The black contour/ histogram is for the inference considering all effects. The blue, orange, and green contours/ histograms ignore Earth-rotation amplitude modulation, detector-size amplitude modulation, and Earth-rotation time delay, respectively.