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The existence of extra dimensions allows the possibility that the fundamentalscale of gravity is at the TeV. If that is the case, gravity could dominate theinteractions of ultra-high energy cosmic rays. In particular, the production ofmicroscopic black holes by cosmogenic neutrinos has been estimated in a numberof papers. We consider here gravity-mediated interactions at larger distances,where they can be calculated in the eikonal approximation. We show that for theexpected flux of cosmogenic neutrinos these elastic processes give a strongersignal than black hole production in neutrino telescopes. Taking the bounds onthe higher dimensional Planck mass M_D (D=4+n) from current air showerexperiments, for n=2 (6) elastic collisions could produce up to 118 (34) eventsper year at IceCube. On the other hand, the absence of any signal would imply abound of M_D>~5 TeV.Comment: 10 pages, 1 figure; version to appear in Phys. Rev. Let

Cosmogenic Neutrinos and Signals of TeV Gravity in Air Showers and Neutrino Telescopes