Nonlinear feedback of the electrostatic instability on the blazar-induced pair beam and GeV cascade

Relativistic pair beams produced in the cosmic voids by TeV gamma rays from blazars are expected to produce a detectable GeV-scale cascade that is missing in the observations. The suppression of this secondary cascade implies either the deflection of the pair beam by intergalactic magnetic fields or, alternatively, an energy loss of the beam due to the beam-plasma instability. Here, we study how the beam-plasma instability feeds back on the beam, using a realistic two-dimensional beam distribution. We find that the instability broadens the beam opening angles significantly without any significant energy loss, thus confirming a recent feedback study on a simplified one-dimensional beam distribution. However, narrowing diffusion feedback of the beam particles with Lorentz factors less than $10^6$ might become relevant even though initially it is negligible. Finally, when considering the continuous creation of TeV pairs, we find that the beam distribution and the wave spectrum reach a new quasi-steady state, in which the scattering of beam particles persists and the beam opening angle may increase by a factor of hundreds. Understanding the implications on the GeV cascade emission requires accounting for inverse-Compton cooling.