Rotating cosmologies, such as those described by Bianchi models (especially types VII0_0 and IX) or Gdel-like universes, permit such anisotropies while remaining consistent with general relativity. However, their implications for observables such as CMB anisotropies and structure formation remain underexplored in the context of modern data.
1.4 Motivations for Torsion and the Einstein--Cartan Theory
To explore cosmological rotation and anisotropy beyond the limits of standard GR, one must consider extensions of Einstein's theory that naturally accommodate spin and angular momentum as sources of spacetime geometry. One such extension is the Einstein--Cartan (EC) theory, which generalizes GR by allowing for non-symmetric affine connections, thereby introducing torsion into spacetime.
Unlike curvature, which is sourced by the energy-momentum tensor, torsion is sourced by the spin density of matter. While negligible in low-density regimes, torsion can become significant under the extreme conditions of the early universe, where spin densities were non-trivial. Importantly, EC theory allows for non-zero vorticity and anisotropy without violating fundamental conservation laws.
From a theoretical standpoint, the inclusion of torsion resolves several conceptual issues:
It naturally couples gravity to intrinsic angular momentum (spin) in a gauge-invariant manner.
It avoids the singularity theorems of GR by introducing a repulsive spin-spin interaction at extremely high densities.
It provides a mechanism for seeding primordial cosmic rotation via torsion-spin coupling, potentially explaining the observed anisotropies and angular momenta of cosmic structures.
Moreover, the EC framework aligns well with approaches to quantum gravity and spin foam models, and it offers a bridge between macroscopic cosmic behavior and microscopic spin phenomena---a connection missing in standard CDM.
Summary of Section Goals
This introduction establishes the need to revisit the assumptions of cosmic isotropy and general relativity, in light of mounting observational tensions and theoretical gaps. In the following sections, we present a detailed theoretical model of a hyperspherical, rotating universe governed by Einstein--Cartan dynamics and explore its implications for resolving cosmological anomalies.
2. Theoretical Framework
