Conformal time in a black-hole universe with torsion

In the Einstein-Cartan-Sciama-Kibble theory of gravity, the intrinsic spin of fermionic matter generates spacetime torsion and induces gravitational repulsion at extremely high densities. This repulsion prevents the collapsing spin-fluid matter in a black hole from forming a singularity. Instead, the interior of a black hole with a stiff equation of state becomes a new universe, which contracts until a (big) bounce and then expands. We derive the equations describing the dynamics of our Universe, formed in such a scenario, in terms of the conformal time which is a convenient variable for testing signatures of the contracting phase in the Cosmic Microwave Background radiation.