  We investigate the well-known phenomenon of the beam-plasma instability in
the gravitational sector, when a fast population of particles interacts with
the massive scalar mode of an Horndeski theory of gravity, resulting into the
linear growth of the latter amplitude. Following the approach used in the
standard electromagnetic case, we start from the dielectric representation of
the gravitational plasma, as introduced in a previous analysis of the Landau
damping for the scalar Horndeski mode. Then, we set up the modified
Vlasov-Einstein equation, using at first a Dirac delta function to describe the
fast beam distribution. This way, we provide an analytical expression for the
dispersion relation and we demonstrate the existence of non-zero growth rate
for the linear evolution of the Horndeski scalar mode. A numerical
investigation is then performed with a trapezoidal beam distribution function,
which confirms the analytical results and allows to demonstrate how the growth
rate decreases as the beam spread increases.
