Pion condensation of quark matter in a static Einstein universe

In the framework of an extended Nambu–Jona-Lasinio model we study pion condensation in quark matter with an asymmetric isospin composition. We treat this against the gravitational field of a static Einstein universe at finite temperature and chemical potential. This particular choice of the gravitational field configuration enables us to investigate phase transitions of the system with exact consideration of the role of this field in the formation of quark and pion condensates. Also, we point out its influence on the phase portraits. We demonstrate the effect of oscillations of the thermodynamic quantities as functions of the curvature and also refer to a certain similarity between the behavior of these quantities as functions of curvature and finite temperature. Finally, the role of quantum fluctuations for spontaneous symmetry breaking in the case of a finite volume of the universe is briefly discussed.