Exploring the sky localization and early warning capabilities of third generation gravitational wave detectors in three-detector network configurations

This work characterizes the sky localization and early warning performance of different networks of third generation gravitational wave detectors, consisting of different combinations of detectors with either the Einstein Telescope or Cosmic Explorer configuration in sites in North America, Europe, and Australia. Using a Fisher matrix method which includes the effect of earth rotation, we estimate the sky localization uncertainty for $1.4\text{ }\text{ }{M}_{\ensuremath{\bigodot}}--1.4\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ binary neutron star mergers at distances 40, 200, 400, 800, and 1600 Mpc, to characterize each network's performance for binary neutron star observations at a given distance. We also characterize each network's sky localization capability for an assumed astrophysical population up to redshift $\ensuremath{\le}2$. Furthermore, we also study the capabilities for the different networks to localize a binary neutron star merger prior to merger (early warning) and characterize the network performance for sky localization uncertainty between 1 and 30 square degrees. We find that, for example, for binary neutron star mergers at 200 Mpc and a network consisting of the Einstein Telescope, Cosmic Explorer, and an extra Einstein Telescope-like detector in Australia (2ET1CE), the upper limit of the size of the 90% credible region for the best localized 90% signals is $0.25\text{ }\text{ }{\mathrm{deg}}^{2}$. For the simulated astrophysical distribution this upper limit is $91.79\text{ }\text{ }{\mathrm{deg}}^{2}$. If the Einstein Telescope-like detector in Australia is replaced with a Cosmic Explorer-like detector (1ET2CE), for signals at 200 Mpc, the size of the 90% credible region for the best localized 90% signals is $0.18\text{ }\text{ }{\mathrm{deg}}^{2}$, while the corresponding value for the best localized 90% sources following the astrophysical distribution is $56.77\text{ }\text{ }{\mathrm{deg}}^{2}$. We note that the 1ET2CE network can detect 7.2% more of the simulated astrophysical population than the 2ET1CE network. In terms of early warning performance (e.g., 200 Mpc), we find that a network of 2ET1CE and 1ET2CE networks can both provide early warnings of the order of one hour prior to merger with sky localization uncertainties of 30 square degrees or less. In some cases, the 2ET1CE network is capable of estimating the sky location with an uncertainty of five square degrees or less on timescales of about one hour prior to merger. Our study concludes that the 1ET2CE network is a good compromise between binary neutron stars detection rate, sky localization, and early warning capabilities.