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

## TL;DR

This paper derives exact expressions for the Casimir force in a three-dimensional relativistic Bose gas model, revealing its attractive nature and temperature-dependent decay, with implications for cosmological phenomena.

## Contribution

It provides the first exact analytical results for the Casimir force in a relativistic Bose gas model considering particles and antiparticles, linking it to known statistical models.

## Key findings

- Casimir force is attractive and monotonic with temperature.
- Force decays as a power law below and exponentially above the critical temperature.
- Exact Casimir amplitude is derived as -4ζ(3)/(5π).

## Abstract

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.