Directional Hubble parameters:
 Ha=aa,Hb=bb,Hc=cc,H_a = \frac{\dot{a}}{a}, \quad H_b = \frac{\dot{b}}{b}, \quad H_c = \frac{\dot{c}}{c},
Mean expansion rate: H=13(Ha+Hb+Hc)H = \frac{1}{3}(H_a + H_b + H_c),
Shear scalar:
 2=12[(HaH)2+(HbH)2+(HcH)2],\sigma^2 = \frac{1}{2} \left[ (H_a - H)^2 + (H_b - H)^2 + (H_c - H)^2 \right],
Vorticity scalar 2=12\omega^2 = \frac{1}{2} \omega_{\mu\nu} \omega^{\mu\nu}.
The EC-corrected Friedmann-like equation becomes:
H2=8G3(s2)+223ka2,H^2 = \frac{8\pi G}{3} \left( \rho - \alpha s^2 \right) + \frac{\sigma^2 - \omega^2}{3} - \frac{k}{a^2},
where k=+1k = +1 for closed (hyperspherical) topology.
The evolution equations are:
H=4G(+p2s2)+22,\dot{H} = -4\pi G (\rho + p - 2\alpha s^2) + \sigma^2 - \omega^2, s+3Hs=0,\dot{s} + 3H s = 0,
showing that spin density decays with expansion, while its early-time dominance drives cosmic rotation.
This dynamical system, when numerically integrated, can be used to:
