Acceptances for Space-Based and Ground-Based Fluorescence Detectors, and Inference of the Neutrino-Nucleon Cross-Section above 10¹9 eV

Detection of ultra-high energy neutrinos will be useful for unraveling the dynamics of the most violent sources in the cosmos and for revealing the neutrino cross-section at extreme energy. Above ~ 10^20 eV, neutrinos may well be the only cosmic primariies. Thus, it is important to know the acceptances (event rate/flux) of proposed air-shower experiments for detecting ``horizontal'' neutrino events initiated in our atmosphere, and ``Earth-skimming'' events initiated in the Earth's surface rock or ocean. We calculate these acceptances for fluorescence detectors, both space-based as with the EUSO and OWL proposals, and ground-based, as with Auger, HiRes and Telescope Array. The neutrino cross-section sigma_nuN is not measured at energies above 5.2 x 10^13 eV. Although QCD extrapolations offer motivated guesses for sigma_nuN, new physics could intervene to provide a surprise. Therefore, we present the acceptances of horizontal (HAS) and upgoing (UAS) air showers as a function of sigma_nuN over the interesting range 10^(-34) to 10^(-30) cm^2. The dependences of acceptances on neutrino energy, shower-threshold energy, shower length and column density, and cloud layers are also studied. For UAS, we present acceptances for events over land (rock), and over the ocean (water). The latter are larger by about an order of magnitude, thus favoring space-based detectors. We revisit the idea of Kusenko and Weiler to infer sigma_nuN at E_nu ~ 10^20 eV from the ratio of HAS-to-UAS events, and find favorable results. Our calculation is mostly analytic. Included in the UAS calculation are realistic energy-losses for taus, and Earth-curvature effects.