s00s_0 \neq 0 (Planckian or sub-Planckian spin contributions),
2H2\omega^2 \ll H^2 but non-zero, to test small rotational seeding.
4.2 Behavior of Rotational Modes and Curvature
The evolution of the rotational component (t)\omega(t) is governed by:+(2H+)=f(t;,s),\dot{\omega} + (2H + \Gamma) \omega = f(t; \sigma, s),
where \Gamma encodes damping effects due to torsion-spin interactions. In the presence of torsion, early-time dynamics can amplify or sustain \omega longer than in GR, especially if spin-torsion coupling is significant.
Numerical integration of the modified Friedmann--Cartan system shows:
Amplification phase: In the early universe (high s2s^2), torsion counteracts expansion damping, allowing \omega to remain significant for longer durations.
Decay phase: As s(t)0s(t) \to 0, torsion fades, and a3\omega \sim a^{-3}, restoring isotropy.
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Curvature evolution: Positive curvature (k=+1k=+1) is preserved but reduced effectively due to anisotropic stretching, providing a mechanism to reconcile apparent curvature tension.
Importantly, residual vorticity can survive until late times at small amplitudes---potentially explaining low-\ell anomalies in the CMB.
4.3 Attractor Solutions and Stability
