Partially relativistic self-gravitating Bose-Einstein condensates with a stiff equation of state

Because of their superfluid properties, some compact astrophysical objects such as neutron stars may contain a significant part of their matter in the form of a Bose-Einstein condensate (BEC). We consider a partially relativistic model of self-gravitating BECs where the relation between the pressure and the restmass density is assumed to be quadratic (as in the case of classical BECs) but pressure effects are taken into account in the relation between the energy density and the rest-mass density. At high densities, we get a stiff equation of state in which the speed of sound equals the speed of light. We determine the maximum mass of general relativistic BEC stars described by this equation of state using the formalism of Tooper (1965). This maximum mass is slightly larger than the maximum mass obtained by Chavanis and Harko (2012) using a fully relativistic model. We also consider the possibility that dark matter is made of BECs and apply the partially relativistic model of BECs to cosmology. In this model, we show that the universe experiences a stiff matter era, followed by a dust matter era, and finally by a dark energy era due to the cosmological constant. Interestingly, the Friedmann equations can be solved analytically in that case and provide a simple generalization of the Lambda CDM model including a stiff matter era. We point out, however, the limitations of the partially relativistic model for BECs and show the need for a fully relativistic one.