Not yet reviewed by Pith; the record is open.

The formation and evolution of double neutron stars (DNS) have traditionally been studied using binary population synthesis. In this work, we take an alternative approach by focusing only on the second supernova (SN) forming the DNS and the subsequent orbital decay and merger due to gravitational wave radiation. Using analytic and numerical methods, we explore how different NS natal kick velocity distributions, pre-SN orbital separations, and progenitor He-star masses affect the post-SN orbital periods, eccentricities, merger times, systemic velocities, and distances traveled by the system before merging. Comparison with the set of 17 known DNSs in the Milky Way shows that DNSs have pre-SN orbital separations ranging between 1 and 44 $R_{\odot}$. Those DNSs with pre-SN separations $\sim$1 $R_{\odot}$ have merger time distributions that peak $\sim$10-100 Myr after formation, regardless of the kick velocity received by the NS. These DNSs are typically formed with systemic velocities $\sim$10$^2$ km s$^{-1}$ and may travel $\sim$1-10 kpc before merging. Depending on progenitor mass of the second-born NS, the short merger time can account for the $r$-process enrichment observed in compact stellar systems such as ultra-faint dwarf galaxies. For Milky Way-mass galaxies only DNSs with the tightest pre-SN orbits and large kick velocities ($\gtrsim$10$^2$ km s$^{-1}$) can escape. However, those DNSs that do escape may travel as far as $\sim$Mpc before merging, which as previous studies have pointed out has implications for identifying the host galaxies to short gamma ray bursts and gravitational wave events.