High energy cosmic ray self-confinement close to extragalactic sources

The ultra-high energy cosmic rays observed at the Earth are most likely accelerated in extra-galactic sources. For the typical luminosities invoked for such sources, the electric current associated to the flux of cosmic rays that leave them is large. The associated plasma instabilities create magnetic fluctuations that can efficiently scatter particles. We argue that this phenomenon forces cosmic rays to be self-confined in the source proximity for energies $E<E_{\rm cut}$, where $E_{\rm cut}\approx 10^{7} L_{44}^{2/3}$ GeV for low background magnetic fields ($B_{0}\ll nG$). For larger values of $B_{0}$, cosmic rays are confined close to their sources for energies $E<E_{\rm cut}\approx 2\times 10^{8} \lambda_{10} L_{44}^{1/4} B_{-10}^{1/2}$ GeV, where $B_{-10}$ is the field in units of $0.1$ nG, $\lambda_{10}$ is its coherence lengths in units of 10 Mpc and $L_{44}$ is the source luminosity in units of $10^{44}$ erg/s.