Binary neutron star mergers and third generation detectors: Localization and early warning

For third generation gravitational wave detectors, such as the Einstein Telescope, gravitational wave signals from binary neutron stars can last up to a few days before the neutron stars merge. To estimate the measurement uncertainties of key signal parameters, we develop a Fisher matrix approach which accounts for effects on such long duration signals of the time-dependent detector response and the Earth's rotation. We use this approach to characterize the sky localization uncertainty for gravitational waves from binary neutron stars at 40, 200, 400, 800, and 1600 Mpc, for the Einstein Telescope and Cosmic Explorer individually and operating as a network. We find that the Einstein Telescope alone can localize the majority of detectable binary neutron stars at a distance of $\ensuremath{\le}200\text{ }\text{ }\mathrm{Mpc}$ to within $100\text{ }\text{ }{\mathrm{deg}}^{2}$ with 90% confidence. A network consisting of the Einstein Telescope and Cosmic Explorer can enhance the sky localization performance significantly---with the 90% credible region of $\mathcal{O}(1)\text{ }\text{ }{\mathrm{deg}}^{2}$ for most sources at $\ensuremath{\le}200\text{ }\text{ }\mathrm{Mpc}$ and $\ensuremath{\le}100\text{ }\text{ }{\mathrm{deg}}^{2}$ for most sources at $\ensuremath{\le}1600\text{ }\text{ }\mathrm{Mpc}$. We also investigate the prospects for third generation detectors identifying the presence of a signal prior to merger. To do this, we require a signal to have a network signal-to-noise ratio of $\ensuremath{\ge}12$ and $\ensuremath{\ge}5.5$ for at least two interferometers, and to have a 90% credible region for the sky localization that is no larger than $100\text{ }\text{ }{\mathrm{deg}}^{2}$. We find that the Einstein Telescope can send out such ``early-warning'' detection alerts 1--20 hours before merger for 100% of detectable binary neutron stars at 40 Mpc and for $\ensuremath{\sim}58%$ of sources at 200 Mpc. For sources at a distance of 400 Mpc, a network of the Einstein telescope and Cosmic Explorer can produce detection alerts up to $\ensuremath{\sim}3$ hours prior to merger for 98% of detectable binary neutron stars.