Initial Conditions from Quantum Gravity: Generate spin and shear seeds from quantum models (e.g. spin foams or bounce cosmology).
Evolution of Perturbations: Solve the perturbed Einstein--Cartan equations for scalar, vector, and tensor modes. Study mode coupling due to torsion and anisotropy.
Synthetic CMB Maps: Produce mock CMB skies including rotational and torsional effects. Compare statistical properties (e.g. alignment statistics, power asymmetry) with Planck and upcoming missions.
Boltzmann Code Extension: Modify Boltzmann solvers (e.g. CAMB, CLASS) to incorporate torsion terms and anisotropic metrics.
Such numerical work will be crucial for quantitative model selection and for placing stringent constraints on the spin--torsion coupling parameter \alpha and initial vorticity values.
Final Remark
The synthesis of torsion geometry, spin matter coupling, and cosmic rotation offers not just a resolution to current observational tensions, but a profound reinterpretation of the very fabric of spacetime. The Einstein--Cartan--Bianchi IX cosmology opens a path forward where structure, entropy, and geometry are not imposed externally, but arise naturally from the dynamical interplay of spin, curvature, and time itself.
Appendices
Appendix A: Derivation of Bianchi IX Metric Tensor Components
The Bianchi IX model represents the most general homogeneous but anisotropic cosmological model with a closed spatial topology (S3\mathbb{S}^3). It is characterized by the structure constants of the SU(2) Lie algebra, making it ideal for modeling hyperspherical geometries with anisotropic scaling.
The metric in synchronous coordinates is:
