Signatures of cosmic tau-neutrinos

The importance and signatures of cosmic tau--(anti)neutrinos have been studied for upward-- and downward--going $\mu^-+\mu^+$ and hadronic shower event rates relevant for present and future underground water or ice detectors, utilizing the unique and reliable ultrasmall--$x$ predictions of the dynamical (radiative) parton model. The upward--going $\mu^- +\mu^+$ event rates calculated just from cosmic $\nu_{\mu}+\bar{\nu}_{\mu}$ fluxes are sizeably enhanced by taking into account cosmic $\nu_{\tau}+ \bar{\nu}_{\tau}$ fluxes and their associated $\tau^- +\tau^+$ fluxes as well. The coupled transport equations for the upward--going $\stackrel{(-)}{\nu}_{\tau}$ flux traversing the Earth imply an enhancement of the attenuated and regenerated $\stackrel{(-)}{\nu}_{\tau}$ flux typically around $10^4-10^5$ GeV with respect to the initial cosmic flux. This enhancement turns out to be smaller than obtained so far, in particular for flatter initial cosmic fluxes behaving like $E_{\nu}^{-1}$. Downward--going $\mu^- +\mu^+$ events and in particular the background--free and unique hadronic `double bang' and `lollipop' events allow to test downward--going cosmic $\nu_{\tau} +\bar{\nu}_{\tau}$ fluxes up to about $10^9$ GeV.