Big bounce and closed universe from spin and torsion
  We analyze the dynamics of a homogeneous and isotropic universe in the
Einstein$--$Cartan theory of gravity. The coupling between the spin and torsion
prevents gravitational singularities and replaces the Big Bang with a
nonsingular big bounce, at which the universe transitions from contraction to
expansion. We show that a closed universe exists only when the product of the
scale factor and temperature is higher than a particular threshold, contrary to
a flat universe and an open universe, which are not restricted. During
inflation, this product must increase to another threshold, so that the
universe can reach dark-energy acceleration.
