Ultra-high Energy Predictions of proton-air Cross Sections from Accelerator Data

We predict $\sigma_{p{-}\rm air}^{\rm prod}$, the proton--air inelastic production cross section, at $pp$ center-of-mass energies $2\le\sqrt s \le 100000$ TeV, using high energy predictions from a saturated Froissart bound parameterization of accelerator data on forward $\bar pp$ and $pp$ scattering amplitudes, together with Glauber theory. The parameterization of the $\bar pp$ and $pp$ cross sections incorporates analyticity constraints and unitarity, allowing accurate extrapolations to ultra-high energies. Our predictions are in excellent agreement with cosmic ray extensive air shower measurements, both in magnitude and in energy dependence