When including torsion, the Einstein--Cartan--Bianchi IX system becomes a dynamical system of coupled ODEs involving:
Hi=ai/aiH_i = \dot{a}_i/a_i for each direction,
Shear tensor \sigma_{\mu\nu},
Vorticity tensor \omega_{\mu\nu},
Spin density s(t)s(t),
Torsion source terms.
2.4 Formulation of Cosmic Angular Momentum
In the EC--Bianchi IX framework, the cosmic angular momentum arises naturally from the spin-torsion coupling in the early universe.
The total angular momentum density LL^{\mu\nu} is a sum of orbital and spin contributions:
L=xT0xT0+S0.L^{\mu\nu} = x^\mu T^{0\nu} - x^\nu T^{0\mu} + S^{0\mu\nu}.
Due to the presence of torsion, conservation of total angular momentum takes the modified form:
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