Marginal Role of the Electrostatic Instability in the GeV-scale Cascade Flux from 1ES 0229+200

Abstract Relativistic pair beams produced in the intergalactic medium by TeV gamma rays from blazars are expected to generate a detectable GeV-scale electromagnetic cascade, yet this cascade is absent in the observed spectra of hard-spectrum TeV emitting blazars. This suppression is often attributed to weak intergalactic magnetic fields (IGMF) deflecting electron-positron pairs out of the line of sight. Alternatively, it has been proposed that beam-plasma instabilities could drain the energy of the beam before they produce the secondary cascades. Recent studies suggest that the modification of beam distribution due to these instabilities is primarily driven by particle scattering, rather than energy loss. In this paper, we quantitatively assess, for the blazar 1ES 0229+200, the arrival time of secondary gamma rays at Earth from the beam scattering by the electrostatic instability. We first computed the production rates of electron-positron pairs at various distances using the Monte Carlo simulation CRPropa. We then simulated the feedback of the plasma instability on the beam, incorporating production rates and inverse Compton cooling, to determine the steady-state distribution function. Our findings reveal that the time delay of the GeV secondary cascade arrival due to instability broadening is on the order of a few months. This delay is insufficient to account for the missing cascade emission in blazar spectra, suggesting that plasma instabilities do not significantly affect IGMF constraints.