Proton-air interactions at ultra-high energies in muon-depleted air showers with different depths

The hardness of the energy spectrum of neutral pions produced in proton-air interactions at ultra-high energies, above 10 18 eV, is constrained by the steepness of the shower-to-shower distribution of the number of muons in muon-depleted extensive air showers. In this work, we find that this steepness, quantified by the parameter Λ μ , evolves with the depth of the shower maximum, X max , assuming a universal value for shallow showers and an enhanced dependence on the high-energy hadronic interaction model for deep showers. We show that X max probes the so-called hadronic activity of the first proton-air interaction, thus allowing direct access to the energy spectrum of neutral pions in different regions of its kinematic phase space. We verify that the unbiased measurement of Λ μ is possible for realistic mass composition expectations. Finally, we infer that the statistical precision in Λ μ required to distinguish between hadronic interaction models can be achieved in current extensive air shower detectors, given their resolution and exposure.