We study the capture of dark matter particles by neutron stars in close binary systems. By performing a direct numerical simulation, we find that there is a sizable amplification of the rate of dark matter capture by each of the companions. In case o
f the binary pulsar PSR J1906+0746 with the orbital period of 4 hours the amplification factor is 3.5. This amplification can be attributed to the energy loss by dark matter particles resulting from their gravitational scattering off moving companions.
We argue that current neutron star observations exclude asymmetric bosonic non-interacting dark matter in the range from 2 keV to 16 GeV, including the 5-15 GeV range favored by DAMA and CoGeNT. If bosonic WIMPs are composite of fermions, the same li
mits apply provided the compositeness scale is higher than ~10^12 GeV (for WIMP mass ~1 GeV). In case of repulsive self-interactions, we exclude large range of WIMP masses and interaction cross sections which complements the constraints imposed by observations of the Bullet Cluster.
We argue that observations of old neutron stars can impose constraints on dark matter candidates even with very small elastic or inelastic cross section, and self-annihilation cross section. We find that old neutron stars close to the galactic center
or in globular clusters can maintain a surface temperature that could in principle be detected. Due to their compactness, neutron stars can acrete WIMPs efficiently even if the WIMP-to-nucleon cross section obeys the current limits from direct dark matter searches, and therefore they could constrain a wide range of dark matter candidates.
