Indication of population effect in UHECR sources

The Ultra-High Energy Cosmic Rays (UHECRs) are nuclei carrying the highest energy ever measured on Earth. The first particle with an energy above $10^{20}$ eV was already observed in the 1960s. But, after 60 years of observations, the sources of UHECRs remains uncertain. To tackle this unresolved question, the Pierre Auger Observatory, the largest observatory ever built, has recorded the detection of UHECRs over the past 20 years and reconstructed with an unprecedented precision their energy spectrum, mass composition and arrival directions, together with its smaller counterpart in the Northern hemisphere, the Telescope Array. Studies assuming a distribution of identical sources, which aim at reproducing the energy spectrum and the mass composition data above $10^{17.8}$ eV, have led to the paradigma of acceleration processes of UHECRs in extragalactic accelerators that depend on the magnetic rigidity of the particles. Coupled with the interaction processes in the intergalactic medium, such models result in a very hard spectral index at escape from the source. Such models have recently been improved considering the in-source interactions and the resulting escape of neutrons, which yields a softer spectral index for the proton component. In this contribution, we evaluate approaches that go beyond the hypothesis of identical sources and account for the variety of the acceleration and escape mechanisms that a population of sources may encompass. We discuss the implications of the fact that, while the range of the maximum rigidity is limited, a large diversity of spectral indices appears to be preferred by the modeling of the data.