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Relativistic self-gravitating Bose-Einstein condensates and cold baryons 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 rest-mass 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 similar to the one considered by Zel'dovich (1961) in the context of baryon stars in which the baryons interact through a vector meson field. We determine the maximum mass of general relativistic BEC stars described by this equation of state by 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 phase, followed by a dust matter phase, and finally by a dark energy phase (equivalent to a cosmological constant). The same evolution is obtained in Zel'dovich (1972) model which assumes that initially, near the cosmological singularity, the universe is filled with cold baryons. Interestingly, the Friedmann equations can be solved analytically in that case and provide a simple generalization of the $Λ$CDM model. We point out, however, the limitations of the partially-relativistic model for BECs and show the need for a fully-relativistic one.