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

Exact results for the Casimir force of a three-dimensional model of relativistic Bose gas in a film geometry