Htorsions2(hij)2.\mathcal{H}_{\text{torsion}} \sim \alpha s^2 \left( \frac{\delta \Gamma}{\delta h_{ij}} \right)^2.
This formulation allows consistent evolution of spin fields along with the geometry, preserving constraints and enabling incorporation into numerical relativity codes.
Appendix D: Stability Analysis of Dynamical System
To understand the long-term behavior of the cosmological model, we analyze the autonomous system formed by the evolution equations derived from the Einstein--Cartan--Bianchi IX framework.
We define dimensionless variables:
x=2H2,y=2H2,z=s2,=3H2.x = \frac{\sigma^2}{H^2}, \quad y = \frac{\omega^2}{H^2}, \quad z = \frac{\alpha s^2}{\rho}, \quad \Omega = \frac{\rho}{3H^2}.
The dynamical system becomes:
{dxdN=f1(x,y,z),dydN=f2(x,y,z),dzdN=f3(x,y,z),\begin{cases} \frac{dx}{dN} = f_1(x, y, z), \\ \frac{dy}{dN} = f_2(x, y, z), \\ \frac{dz}{dN} = f_3(x, y, z), \end{cases}
with N=lna(t)N = \ln a(t) being the e-fold number.
Fixed points (x,y,z)(x^*, y^*, z^*) are found by solving:
fi(x,y,z)=0for i=1,2,3.f_i(x^*, y^*, z^*) = 0 \quad \text{for } i = 1, 2, 3.
