  Recently it has been suggested that relativistic Bose gas of some type can be
playing role in issues like dark matter, dark energy, and in some cosmological
problems. In the current article we investigate one known exactly solvable
model of three-dimensional statistical-mechanical model of relativistic Bose
gas that takes into account the existence of both particles and antiparticles.
We derive exact expressions for the behavior of the Casimir force for the
system subjected to film geometry under periodic boundary conditions. We show
that the Casimir force between the plates is attractive, monotonic as a
function of the temperature scaling variable, with a scaling function that
approaches at low temperatures a universal negative constant equal to the
corresponding one for two-component three dimensional Gaussian system. The
force decays with the distance in a power law near and below the bulk critical
temperature $T_c$ of the Bose condensate and exponentially above $T_c$. We
obtain closed form exact expression for the Casimir amplitude $\Delta_{\rm
Cas}^{\rm RBG} =-4\zeta(3)/(5\pi)$. We establish the precise correspondence of
the scaling function of the free energy of the model with the scaling functions
of two other well-known models of statistical mechanics - the spherical model
and the imperfect Bose gas model.
