Big bounce and closed universe from spin and torsion

Understand

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.