Using a Milky Way double neutron star (DNS) merger rate of 210 Myr$^{-1}$, as derived by the Laser Interferometer Gravitational-Wave Observatory (LIGO), we demonstrate that the Laser Interferometer Space Antenna (LISA) will detect on average 240 (330) DNSs within the Milky Way for a 4-year (8-year) mission with a signal-to-noise ratio greater than 7. Even adopting a more pessimistic rate of 42 Myr$^{-1}$, as derived by the population of Galactic DNSs, we find a significant detection of 46 (65) Milky Way DNSs. These DNSs can be leveraged to constrain formation scenarios. In particular, traditional NS-discovery methods using radio telescopes are unable to detect DNSs with $P_{rm orb}$ $lesssim$1 hour (merger times $lesssim$10 Myr). If a fast-merging channel exists that forms DNSs at these short orbital periods, LISA affords, perhaps, the only opportunity to observationally characterize these systems; we show that toy models for possible formation scenarios leave unique imprints on DNS orbital eccentricities, which may be measured by LISA for values as small as $sim$10$^{-2}$.
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