eff=s2,peff=ps2,\rho_{\text{eff}} = \rho - \alpha s^2, \quad p_{\text{eff}} = p - \alpha s^2,
where \alpha is a coupling constant derived from the gravitational constant and geometrical factors.
This correction becomes negligible at late times but may dominate in the early universe, offering a natural mechanism to seed global vorticity and resolve singularity issues.
2.3 Bianchi IX Metric and Hyperspherical Geometry
To model a rotating, anisotropic universe, we utilize the Bianchi type IX cosmological model. Bianchi IX is the most general homogeneous but anisotropic solution of Einstein's equations with closed topology and is often used as a generalization of the FLRW metric.
Its line element in orthonormal frame can be written as:
ds2=dt2+a(t)2(1)2+b(t)2(2)2+c(t)2(3)2,ds^2 = -dt^2 + a(t)^2 \, (\omega^1)^2 + b(t)^2 \, (\omega^2)^2 + c(t)^2 \, (\omega^3)^2,
where i\omega^i are invariant 1-forms on SU(2), satisfying:
di=12 jkijk.d\omega^i = \frac{1}{2} \epsilon^i_{\ jk} \omega^j \wedge \omega^k.
In a more explicit form, for hyperspherical coordinates, the 1-forms can be expressed in terms of Euler angles (,,)(\theta, \phi, \psi), leading to a geometry that resembles a rotating 3-sphere, or a hypersphere with embedded rotational degrees of freedom.
The scale factors a(t),b(t),c(t)a(t), b(t), c(t) evolve independently, allowing the inclusion of anisotropic shear and vorticity, crucial for modeling cosmic rotation and CMB anomalies.
