  Neutron stars close to the Galactic center are expected to swim in a dense
background of dark matter. For models in which the dark matter has efficient
interactions with neutrons, they are expected to accumulate their own local
cloud of dark matter, making them appealing targets for observations seeking
signs of dark matter annihilation. For theories with very light mediators, the
dark matter may annihilate into pairs of mediators which are sufficiently
long-lived to escape the star and decay outside it into neutrinos. We examine
the bounds on the parameter space of heavy ($\sim$TeV to $\sim$PeV) dark matter
theories with long-lived mediators decaying into neutrinos based on
observations of high energy neutrino observatories, and make projections for
the future. We find that these observations provide information that is
complementary to terrestrial searches, and probe otherwise inaccessible regimes
of dark matter parameter space.
