  Many competing linear instabilities are likely to occur in astrophysical
settings, and it is important to assess which one grows faster for a given
situation. An analytical model including the main beam plasma instabilities is
developed. The full 3D dielectric tensor is thus explained for a cold
relativistic electron beam passing through a cold plasma, accounting for a
guiding magnetic field, a return electronic current and moving protons.
Considering any orientations of the wave vector allows to retrieve the most
unstable mode for any parameters set. An unified description of the
Filamentation (Weibel), Two-Stream, Buneman, Bell instabilities (and more) is
thus provided, allowing for the exact determination of their hierarchy in terms
of the system parameters. For relevance to both real situations and PIC
simulations, the electron-to-proton mass ratio is treated as a parameter, and
numerical calculations are conducted with two different values, namely 1/1836
and 1/100. In the system parameters phase space, the shape of the domains
governed by each kind of instability is far from being trivial. For low density
beams, the ultra-magnetized regime tends to be governed by either the
Two-Stream or the Buneman instabilities. For beam densities equalling the
plasma one, up to four kinds of modes are likely to play a role, depending of
the beam Lorentz factor. In some regions of the system parameters phase space,
the dominant mode may vary with the electron-to-proton mass ratio. Application
is made to Solar Flares, Intergalactic Streams and Relativistic shocks physics.
