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In this talk I will present two models involving a very heavy dark matter candidate of mass as large as the EeV scale. In the first part I will present an original paradigm for producing such a dark-matter candidate in which the only contact between the Standard Model and dark matter is the inflationary sector. I will show that such a framework is strongly constrained by the study of inflationary observables and the presence of an early domination era. Given that such a dark-matter candidate can be produced in the early universe in one way or another, I will then expose the possibility that a heavy dark-matter particle couples to right-handed neutrinos and has some signature in the neutrino sector, interpreting the ANITA anomalous events and IceCube PeV events reported recently as coming from a decay of dark matter in the galaxy.