Gravitino Decay in High Scale Supersymmetry with R-parity Violation

We consider the effects of R-parity violation due to the inclusion of a bilinear $\mu^\prime L H_u$ superpotential term in high scale supersymmetric models with an EeV scale gravitino as dark matter. Although the typical phenomenological limits on this coupling (e.g. due to lepton number violation and the preservation of the baryon asymmetry) are relaxed when the supersymmetric mass spectrum is assumed to be heavy (in excess of the inflationary scale of $3 \times 10^{13}$ GeV), the requirement that the gravitino be sufficiently long-lived so as to account for the observed dark matter density, leads to a relatively strong bound on $\mu^\prime \lesssim 20$ GeV. The dominant decay channels for the longitudinal component of the gravitino are $Z \nu, W^\pm l^\mp$, and $h\nu$. To avoid an excess neutrino signal in IceCube, our limit on $\mu'$ is then strengthened to $\mu' \lesssim 50$ keV. When the bound is saturated, we find that there is a potentially detectable flux of mono-chromatic neutrinos with EeV energies.