The observed velocities of pulsars suggest the possibility that sterile neutrinos with mass of several keV are emitted from a cooling neutron star. The same sterile neutrinos could constitute all or part of cosmological dark matter. The neutrino-driven kicks can exhibit delays depending on the mass and the mixing angle, which can be compared with the pulsar data. We discuss the allowed ranges of sterile neutrino parameters, consistent with the latest cosmological and X-ray bounds, which can explain the pulsar kicks for different delay times.PACS numbers: PACS numbers: 97.60. Gb, 14.60.Pq, 97.60.Bw Observed velocities of pulsars [1] can be explained by an anisotropic emission of sterile neutrinos [2,3,4,5] or other light particles [6]. Sterile neutrinos are firmly rooted in particle physics [7], because the gauge singlet (right-handed) neutrinos are needed to account for the observed neutrino masses in what is called the seesaw Lagrangian [8]. If some of the gauge singlets turn out to be light, they can appear below the electroweak scale as sterile neutrinos. There are further hints in favor of this intriguing possibility: in addition to explaining the pulsar kicks, sterile neutrinos with the same parameters could make up the cosmological dark matter [9,10,11,12,13] and could play an important role in the formation of the first stars [14]. This form of "warm" dark matter may be in good agreements with observational inferences regarding the small-scale structure [15,16]. In contrast, the active neutrino oscillations cannot explain the pulsar kicks, unless they have very large magnetic moments [17], or the mass difference is large enough to allow the Mikheev-Smirnov-Wolfenstein [18] resonance at density 10 11 − 10 12 g/cm 3 [19,20], which is excluded by the current data on neutrino masses.The most promising way to discover sterile neutrinos is by observing X-ray photons from decays of the relic sterile neutrinos, which have lifetimes longer than the age of the universe but which can, nevertheless, produce a detectable signal [21]. The X-rays produced by the dark matter decays, and the production rate of sterile neutrinos in a supernova are both governed by two parameters: the sterile neutrino mass and the mixing angle. It is, therefore, important to understand the allowed ranges of these parameters. The range implied by the pulsar kicks can help focus the X-ray searches. In Ref.[3], the allowed range was discussed for both the resonance and the off-resonance production. In this paper we will reconsider this range of parameters, apply the present constraints from the X-ray observations and the Lymanalpha bounds, and we will also relate the range of sterile neutrino masses and mixing angles to the delays in the onset of the kick. This delay can have some observable consequences and can be determined from the studies of the pulsar populations [22].In applying the cosmological constraints, we distinguish between two different issues: the particle's existence and its ability to account for all of dark matter. Assumin...