The Role of Structured Magnetic Fields on Constraining Properties of Transient Sources of Ultra-high-energy Cosmic Rays

We study how the properties of transient sources of ultra-high-energy cosmic rays (UHECRs) can be accessed by exploiting UHECR experiments, taking into account the propagation of UHECRs in magnetic structures which the sources are embedded in, i.e., clusters of galaxies and filamentary structures. Adopting simplified analytical models, we demonstrate that the structured extragalactic magnetic fields (EGMFs) play crucial roles in unveiling the properties of the transient sources. These EGMFs unavoidably cause significant delay in the arrival time of UHECRs as well as the Galactic magnetic field, even if the strength of magnetic fields in voids is zero. Then, we show that, given good knowledge on the structured EGMFs, UHECR observations with high statistics above $10^{20}$ eV allow us to constrain the generation rate of transient UHECR sources and their energy input per burst, which can be compared with the rates and energy release of known astrophysical phenomena. We also demonstrate that identifying the energy dependence of the apparent number density of UHECR sources at the highest energies is crucial as a clue to such transient sources. Future UHECR experiments with extremely large exposure are required to reveal the nature of transient UHECR sources.