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We study the longitudinal stability of beam-plasma systems in the presence ofa density inhomogeneity in the background plasma. Previous works have focusedon the non-relativistic regime where hydrodynamical models are used to evolvepre-existing Langmuir waves within inhomogeneous background plasmas. Here, forthe first time we study the problem with kinetic equations in afully-relativistic way. We do not assume the existence of Langmuir waves, andwe focus on the rate and the mechanism by which waves are excited in suchsystems from an initial perturbation. We derive the structure of the unstablemodes and compute an analytical approximation for their growth rates. Ourcomputation is limited to dilute and cold beams, and shows an excellentagreement with particle-in-cell simulations performed using the SHARP code. Weshow that, due to such an inhomogeneity, the virulent beam-plasma instabilitiesin the intergalactic medium are not suppressed but their counterparts in thesolar wind can be suppressed as evidenced by propagating type-III solar radiobursts.

The growth of the longitudinal beam-plasma instability in the presence of an inhomogeneous background