T=~[].T^\lambda_{\;\mu\nu} = \tilde{\Gamma}^\lambda_{\;[\mu\nu]}.
The torsion field equations, obtained by varying the Einstein--Cartan action with respect to the connection, yield:
T=8G(S+SS).T^\lambda_{\;\mu\nu} = 8\pi G \left( S^\lambda_{\;\mu\nu} + \delta^\lambda_\mu S^\sigma_{\;\nu\sigma} - \delta^\lambda_\nu S^\sigma_{\;\mu\sigma} \right).
In the presence of a spin-polarized fluid, we can express the spin density tensor as:
S=us(t),S^{\lambda\mu\nu} = \epsilon^{\lambda\mu\nu\sigma} u_\sigma s(t),
where uu^\mu is the four-velocity and s(t)s(t) is the scalar spin density. Substituting this form into the field equations leads to a purely time-dependent torsion field that modifies the cosmological dynamics.
The effective energy-momentum tensor becomes:
T(eff)=T(s2uu+12s2g),T^{\text{(eff)}}_{\mu\nu} = T_{\mu\nu} - \alpha \left( s^2 u_\mu u_\nu + \frac{1}{2} s^2 g_{\mu\nu} \right),
where \alpha is a coupling constant related to GG. This correction term enters directly into the Friedmann-like equations and modifies the pressure and energy density in the early universe.
Appendix C: ADM Decomposition with Torsion
To analyze the Hamiltonian structure of Einstein--Cartan theory and perform numerical simulations, it is useful to apply the Arnowitt--Deser--Misner (ADM) decomposition, separating spacetime into space + time foliation:
