Measurement of the ultra high energy cosmic ray flux from data of very inclined showers at the Pierre Auger Observatory

This work describes the derivation of the energy dependent flux of ultra-high energy cosmic rays from data of very inclined air showers observed with the Pierre Auger Observatory. The Pierre Auger Observatory measures extensive air showers initiated by interactions of ultra-high energy cosmic rays with Earth's atmosphere. The properties of individual cosmic rays are not directly accessible and need to be reconstructed from the observed air showers. The observation is done with a combination of a surface and a fluorescence detector. The fluorescence detector measures the energy loss profile of the shower in the air which can be integrated to yield the total energy of the cosmic ray. The surface detector consists of a ground array of particle detectors which sample the lateral shower profile on the ground. This study focuses on the event class of very inclined air showers with zenith angles larger than 60 degrees. The lateral ground profile of these showers is muon dominated and not radially symmetric around the shower axis due to geomagnetic deflections and other effects. The dependency of this profile on the direction, energy and mass of the cosmic ray is discussed with a mixture of detailed Monte-Carlo simulations and a simplified analytical model of the air shower cascade. It is found in agreement with other studies that the normalized shape of the muon density profile is approximately universal over the range of cosmic ray energies and masses measured at the Pierre Auger Observatory, that the amplitude of the profile is almost proportional to the cosmic ray energy, and that its shower-to-shower fluctuations are sensitive to the mass composition of the cosmic rays. The reconstruction of the cosmic ray properties is based on the theoretical discussion. The angular reconstruction uses the measured signal arrival times in the surface detector. The shower direction is reconstructed by fitting a model of a spherical shower front to these arrival times. The energy reconstruction uses the measured signal strengths in the surface detector stations to derive an energy estimator proportional to the total number of muons arriving at the ground. The energy estimator is obtained by fitting a model of the lateral profile of the muon density on the ground, convoluted with the signal response of the surface detector to individual muons and a small contribution due to accompanying residual electromagnetic particles. The accuracy of the energy estimator depends strongly on the quality of the model input in this reconstruction. Therefore, a sophisticated phenomenological model of the lateral profile of the muon density on the ground is derived from a large set of simulated air showers and compared to an existing model. Existing models of the station signal response to muons and the electromagnetic particle contribution are investigated and improved with a pragmatic approach. The impact of the improved models on the reconstruction of the energy estimator is investigated with an analysis of simulated events and cross-checks based on real data. The relation between the energy estimator and the cosmic ray energy is established by a calibration against the energy measured by the fluorescence detector in a selected set of hybrid events. The calibration method uses a probability density model for the first time which includes the detector efficiencies and resolutions. The model is fitted with a maximum likelihood approach to the hybrid event distribution. This approach extracts the maximum amount of information out of the small data set. The analysis shows a large excess of muons in the data compared to air shower simulations. The calibration function itself contains information about the mass composition of cosmic rays. In addition, the calibration fit estimates the shower-to-shower fluctuations of total muon number on the ground as a function of the energy which further constrains assumptions about the cosmic ray composition. The energy dependent cosmic ray flux is obtained by correcting the distribution of the measured energies for detector efficiency and resolution effects with an unfolding method. Statistical and systematic uncertainties are propagated into the unfolded flux. The result is shown to be consistent within one standard deviation with a published analysis of the cosmic ray flux of Pierre Auger Observatory, which was derived from an independent data set.