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Collisionless shocks are ubiquitous in astrophysics and in the lab. Recentnumerical simulations and experiments have shown how they can arise from theencounter of two collisionless plasma shells. When the shells interpenetrate,the overlapping region turns unstable, triggering the shock formation. As afirst step towards a microscopic understanding of the process, we analyze herein detail the initial instability phase. On the one hand, 2D relativistic PICsimulations are performed where two symmetric initially cold pair plasmascollide. On the other hand, the instabilities at work are analyzed, as well asthe field at saturation and the seed field which gets amplified. For mildlyrelativistic motions and onward, Weibel modes govern the linear phase. Wederive an expression for the duration of the linear phase in good agreementwith the simulations. This saturation time constitutes indeed a lower-bound forthe shock formation time.

Collisionless shock formation, spontaneous electromagnetic fluctuations and streaming instabilities