  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 $\Lambda$CDM model. We point out, however, the
limitations of the partially-relativistic model for BECs and show the need for
a fully-relativistic one.
