  We consider the coupled Einstein-Maxwell-Boltzmann system with cosmological
constant in presence of a massive scalar field. The background metric is that
of Friedman-Lema\^itre-Robertson-Walker space time in the spatially homogeneous
case where the unknown functions only depend on time and not on the space
variables $(x^i)$, $i=1,2,3$. By combining the energy estimates method with
that of characteristics we derive under suitable conditions on the chock kernel
(see Eq. 2.20), a local (in time) solution of the coupled system. Further,
under the hypotheses that the data are small in some appropriate norms and that
the cosmological constant satisfies $\Lambda > -4\pi m^2\Phi_0^2$, we derive a
unique global (in time) solution (Theorem 6.1).
