Joint measurement of the ultra-high-energy neutrino spectrum and cross section

Soon, a new generation of neutrino telescopes, presently under planning, will target the discovery of ultra-high-energy (UHE) neutrinos of cosmic origin, with energies higher than 100 PeV, that promise unique insight into astrophysics and particle physics. Yet, predictions of the UHE neutrino flux and interaction cross section -- whose measurement is co-dependent -- are laden with significant uncertainty that, if unaddressed, could misrepresent the capabilities to measure one or the other. To address this, we advocate for the joint measurement of the UHE neutrino spectrum and neutrino-nucleon cross section, including of their energy dependence, without assuming prior knowledge of either. We illustrate our methods by adopting empirical parametrizations of the neutrino spectrum, in forecasts geared to the planned radio array of the IceCube-Gen2 neutrino telescope. We warn against using simple parametrizations -- a simple power law or one augmented with an exponential cut-off -- that might fail to capture features of the spectrum that are commonplace in the predictions. We argue instead for the use of flexible parametrizations -- a piecewise power law or an interpolating polynomial -- that ensure accuracy. We report loose design targets for the detector energy and angular resolution that are compatible with those under present consideration.