  Dark matter particles could be superheavy, provided their lifetime is much
longer than the age of the universe. Using the sensitivity of the Pierre Auger
Observatory to ultra-high energy neutrinos and photons, we constrain a specific
extension of the Standard Model of particle physics that meets the lifetime
requirement for a superheavy particle by coupling it to a sector of ultra-light
sterile neutrinos. Our results show that, for a typical dark coupling constant
of 0.1, the mixing angle $\theta_m$ between active and sterile neutrinos must
satisfy, roughly, $\theta_m \lesssim 1.5\times
10^{-6}(M_X/10^9~\mathrm{GeV})^{-2}$ for a mass $M_X$ of the dark-matter
particle between $10^8$ and $10^{11}~$GeV.
