Ultrarelativistic Bose-Einstein condensation in the Einstein universe and energy conditions

We study a relativistic boson gas of particles and antiparticles in the Einstein universe at high temperatures and densities. We obtain the thermodynamic potential of the boson system in curved spacetime. The result shows that ultrarelativistic Bose-Einstein condensation can occur at very high densities in the Einstein universe, and by implication in the early stages of a dynamically changing universe. We show that in a slowly changing universe with a weak self-interaction this ultrarelativistic Bose-Einstein condensate would violate the strong energy condition which enters into the cosmological singularity theorems.