The observed interaction energy of cosmic rays with atmospheric nuclei
reaches up to a PeV in the center of mass. We compute nucleon-nucleon and
nucleon-neutrino cross sections for various generic parton cross sections
appearing in string and brane world scenarios for gravity and compare them with
cosmic ray data. Scenarios with effective energy scales in the TeV range and
parton cross sections with linear or stronger growth with the center of mass
energy appear strongly constrained or ruled out. String-inspired scenarios with
infinite-volume extra dimensions may require a fundamental scale above 100 eV
for which they are probably in conflict with gravity on parsec scales.Comment: 4 revtex pages, 1 postscript figur
