  The dispersion relation in matter of ultrahigh-energy neutrinos above the
pole of the $W$ resonance ($E_{\nu} \gsim {\rm 10}^{7} {\rm GeV} $), is
studied. We perform our calculation using the real-time formulation of Thermal
Field Theory in which the massless limit for the $W$ boson is taken. The range
of active-to-sterile neutrino oscillation parameters for which there is
significant mixing enhancement during propagation through the interior of the
Earth, and therefore significant attenuation of neutrino beams in the Earth at
high energies, is estimated. Finally, this range is considered in view of the
cosmological and astrophysical constraints.
