Bose-Einstein condensation in an Einstein universe

Taking into account the possibility of particle-antiparticle pair production, the authors have investigated the onset of Bose-Einstein condensation in an ideal relativistic Bose gas confined to an Einstein universe. Through an extensive use of the Poisson summation formula, they have carried out an explicit evaluation of the summations-over-states appearing in the problem, to derive rigorously the condensate fraction, rho 0/ rho , and the specific heat at constant volume, crho , of the system as smooth functions of temperature-from T>or=Tc down to T=0K. A detailed asymptotic analysis of the growth of the condensate and of the specific-heat maximum is then carried out and, in each case, finite-size corrections to the standard bulk results are obtained in explicit terms. Special cases of the non-relativistic and extreme relativistic versions of the model are examined at some length and, wherever possible, a comparison is made with the findings of the previous authors.