Expected sensitivity of the AugerPrime Radio Detector to the masses of ultra-high-energy cosmic rays using inclined air showers

Despite enormous efforts in the last several decades, the origin of ultra-high-energy cosmic rays (UHECRs) -- their acceleration sites and acceleration mechanism(s) -- remains unidentified and is subject of active research. The progress made during that time, in particular by the Pierre Auger Observatory, established that significant advances in our understanding of the nature of UHECRs are only achieved with a better knowledge of their mass composition, i.e., through more precise measurements. To this end, the Pierre Auger Observatory is upgrading its large-aperture Surface Detector (SD) to enhance its mass sensitivity for the detection of the highest-energy cosmic rays ($E \gtrsim 4 \times 10^{19}\,$eV). As part of this effort, the AugerPrime Radio Detector (RD) will consist of over 1600 dual-polarized radio antennas mounted on top of each of the SD's water-Cherenkov detector (WCD) stations. The RD will be measuring the electromagnetic radiation in the 30$\,$MHz to 80$\,$MHz frequency band produced by highly inclined air showers with zenith angles $\gtrsim\,$65$^\circ$. Thus, the RD will allow us to determine the cosmic-ray energy by measuring the shower's electromagnetic component, which is largely independent of the cosmic-ray mass. In contrast, since most particles in highly-inclined air showers are absorbed in the atmosphere and do not reach the ground, the WCDs will mainly record muons from the muonic shower component, which is highly correlated to the cosmic-ray mass. The combination of that complementary information allows us to infer the cosmic-ray mass with high precision. With this work, I have laid the foundation to process, reconstruct, and analyze data measured by the RD. To develop a signal and reconstruction model for the radio detection of inclined air showers, I have conducted comprehensive studies of the nature of the radio emission from inclined air showers by utilizing numerical CoREAS simulations. In particular, I have investigated the origin of the radio emission within the extensive particle cascades and studied the correlation between the emission strength and ambient conditions. Furthermore, I have identified and characterized a refractive displacement of the radio-emission footprints at the ground, caused by the propagation of the electromagnetic radiation through the Earth's atmosphere. This causes the radio emission from an 85$^\circ$ air shower to be displaced by about 1.5$\,$km and thus has essential implications for the description of the radio-emission footprint and the interpretation of the reconstructed geometry for very inclined air showers with zenith angles above 80$^\circ$. With that at hand, I have developed a signal model of the 2-dimensional lateral distribution of the radio emission in the 30$\,$MHz to 80$\,$MHz frequency band. This model enables the reconstruction of the (electromagnetic) shower energy with sparse radio-antenna arrays and an intrinsic resolution of below 5\% without taking into account instrumental uncertainties. As the electromagnetic energy can be reconstructed without any dependency on the cosmic-ray mass, this model is suitable to perform precise studies of the mass(-composition) of UHECRs, for example, with RD-SD hybrid detections of the AugerPrime Observatory. In addition, I have evaluated the possibility of improving this mass sensitivity by measuring the slant depth of the shower maximum $X_\mathrm{max}$ with a newly-proposed interferometric reconstruction technique. I have worked out, that the RD does not meet the specifications for an accurate reconstruction of $X_\mathrm{max}$, and that a time synchronization between antenna stations of $\lesssim\,$1$\,$ns and a signal multiplicity of $\gtrsim\,20$ are required to achieve accurate results. With this theoretical framework, I have thoroughly studied the expected performance of the RD to detect and reconstruct inclined air showers and its potential to determine the mass(-composition) of UHECRs with RD-SD hybrid measurements. These studies utilize Monte-Carlo-generated air showers, perform end-to-end simulations of the RD instrumental response including measured noise, and a reconstruction of all relevant air shower observables with the here-developed signal model. I have found that the RD will be fully efficient to detect inclined air showers with zenith angles above 70$^\circ$ and energies above $6.3 \times 10^{18}\,$eV. For a 10-year operation period, the RD will collect over 3900 events with energies above $10^{19}\,$eV and around 570 events for energies above $4 \times 10^{19}\,$eV. An accurate reconstruction of the shower energy with the RD is already possible for air showers measured with 5 radio antennas and zenith angles above 68$^\circ$. For current assumptions on the instrumental response of the RD, I have obtained an expected energy resolution of well below 10\% for energies above $10^{19}\,$eV and find no bias in the reconstructed electromagnetic energy for air showers induced by different primary particles. This study is concluded with an assessment of possible systematic uncertainties. By combining the RD-reconstructed (electromagnetic) energy and the SD-reconstructed number of muons, I assessed the potential discrimination between different primary particle types and to measure the average mass composition of UHECRs. The separation for proton- and iron-induced air showers with zenith angles above 70$^\circ$ and electromagnetic energies above $10^{19}\,$eV is quantified with a figure of merit $\text{FOM} \approx 1.6$. The (simulated) measurements of the mean muon number with the RD and SD were found to reproduce the injected mass compositions. Hence, RD-SD hybrid measurements carry the potential to extend such measurements currently performed with the Fluorescence Detector and SD to higher energies, and thereby, to distinguish between different astrophysical scenarios that could explain the nature of UHECRs. With the reconstruction model and mass-composition analysis developed in this work, the Pierre Auger Observatory is well-prepared for the arrival of experimental data from AugerPrime of inclined air showers.