Revisit of Nonlinear Landau Damping for Electrostatic Instability Driven by Blazar-induced Pair Beams

We revisit the effect of nonlinear Landau (NL) damping on the electrostatic instability of blazar-induced pair beams, using a realistic pair-beam distribution. We employ a simplified 2D model in ${\boldsymbol{k}}$-space to study the evolution of the electric-field spectrum and to calculate the relaxation time of the beam. We demonstrate that the 2D model is an adequate representation of the 3D physics. We find that nonlinear Landau damping, once it operates efficiently, transports essentially the entire wave energy to small wave numbers where wave driving is weak or absent. The relaxation time also strongly depends on the intergalactic medium temperature, T IGM, and for T IGM Lt 10 eV, and in the absence of any other damping mechanism, the relaxation time of the pair beam is longer than the inverse Compton (IC) scattering time. The weak late-time beam energy losses arise from the accumulation of wave energy at small k, that nonlinearly drains the wave energy at the resonant ${\boldsymbol{k}}$ of the pair-beam instability. Any other dissipation process operating at small k would reduce that wave-energy drain and hence lead to stronger pair-beam energy losses. As an example, collisions reduce the relaxation time by an order of magnitude, although their rate is very small. Other nonlinear processes, such as the modulation instability, could provide additional damping of the nonresonant waves and dramatically reduce the relaxation time of the pair beam. An accurate description of the spectral evolution of the electrostatic waves is crucial for calculating the relaxation time of the pair bea