We review severe constraints on asymmetric bosonic dark matter based on observations of old neutron stars. Under certain conditions, dark matter particles in the form of asymmetric bosonic WIMPs can be effectively trapped onto nearby neutron stars, where they can rapidly thermalize and concentrate in the core of the star. If some conditions are met, the WIMP population can collapse gravitationally and form a black hole that can eventually destroy the star. Based on the existence of old nearby neutron stars, we can exclude certain classes of dark matter candidates. 1. Introduction Compact stars such as neutron stars and white dwarfs can lead in general to two types of constraints regarding dark matter candidates. The first one has to do with annihilating dark matter that changes the thermal evolution of the star. Annihilation of Weakly Interacting Massive Particles (WIMPs) that are trapped inside compact stars can lead to the production of significant amount of heat that can change the temperature of old stars [1–4]. Such a phenomenon can be in principle contrasted to observations. The second type of constraints is related to asymmetric dark matter [5–12]. Asymmetric dark matter is an attractive alternative to thermally produced dark matter especially due to the intriguing possibility of relating its asymmetry to the baryonic one. For recent reviews see [13, 14]. Due to the asymmetry, WIMP annihilation is not significant in this case. If a certain amount of WIMPs is trapped inside the star, the WIMPs can quite rapidly thermalize and concentrate within a tiny radius in the core of the star. If the WIMP population grows significantly, WIMPs might become self-gravitating and they might collapse forming a mini black hole. Under certain conditions, the black hole might consume the rest of the star, thus leading to the ultimate destruction of the star. However, very old (older than a few billion years) nearby neutron stars have been well observed and studied. The simple presence of such verified old stars leads to the conclusion that no black hole has consumed the star and as we will argue, this can lead to very severe constraints on the properties of certain types of asymmetric dark matter. We should also mention that additional constraints on asymmetric dark matter can be imposed on different ways (e.g., from asteroseismology [15–17], from effects on the transport properties of the neutron stars [18], and/or from hybrid dark matter rich compact stars [19, 20]). One can easily figure out that fermionic WIMPs, due to the fact that they have to overcome Fermi
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