  We provide a simple analytical solution of the Friedmann equations for a
universe made of stiff matter, dust matter, and dark energy. A stiff matter era
is present in the cosmological model of Zel'dovich (1972) where the primordial
universe is assumed to be made of a cold gas of baryons. It also occurs in
certain cosmological models where dark matter is made of relativistic
self-gravitating Bose-Einstein condensates (BECs). When the energy density of
the stiff matter is positive, the primordial universe is singular. It starts
from a state with a vanishing scale factor and an infinite density. We consider
the possibility that the energy density of the stiff matter is negative
(anti-stiff matter). This happens, for example, when the BECs have an
attractive self-interaction. In that case, the primordial universe is
non-singular. It starts from a state in which the scale factor is finite and
the energy density is equal to zero. For the sake of generality, we consider a
cosmological constant of arbitrary sign. When the cosmological constant is
positive, the universe asymptotically reaches a de Sitter phase where the scale
factor increases exponentially rapidly. This can account for the accelerating
expansion of the universe that we observe at present. When the cosmological
constant is negative (anti-de Sitter), the evolution of the universe is cyclic.
Therefore, depending on the sign of the energy density of the stiff matter and
of the dark energy, we obtain singular and non-singular expanding or cyclic
universes.
