Extended clustering analyses to constrain the deflection angular scale and source density of the ultra-high-energy cosmic rays

The search of a clustering signal in the arrival directions of ultra-high-energy cosmic rays (UHECRs) is a standard method to assess the level of anisotropy of the data sets under investigation. Here, we first show how to quantify the sensitivity of a UHECR detector to the detection of anisotropy, and then propose a new method that pushes forward the study of the two-point auto-correlation function, enabling one to put astrophysically meaningful constraints on both the effective UHECR source density and the angular deflections that these charged particles suffer while they propagate through the galactic and intergalactic magnetic fields. We apply the method to simulated data sets obtained under various astrophysical conditions, and show how the input model parameters can be estimated through our analysis, introducing the notion of "clustering similarity" (between data sets), to which we give a precise statistical meaning. We also study how the constraining power of the method is influenced by the size of the data set under investigation, the minimum energy of the UHECRs to which it is applied, and a prior assumption about the underlying source distribution. We also show that this method is particularly adapted to data sets consisting of a few tens to a few hundreds of events, which corresponds to the current and near-future observational situation in the field of UHECRs.