Dynamical Systems Analysis of an Einstein-Cartan Ekpyrotic Nonsingular Bounce Cosmology

I construct an Einstein-Cartan ekpyrotic model (ECEM): a homogeneous, nearly Friedmann-Lemaître-Robertson-Walker (FLRW) background in Einstein-Cartan (EC) gravity whose spin-torsion sector, modeled phenomenologically as a Weyssenhoff fluid with stiff scaling $\propto a^{-6}$, is coupled to a scalar field with a steep exponential potential that interpolates between a negative ekpyrotic branch and a positive plateau. Extending the Copeland-Liddle-Wands (CLW) scalar-fluid dynamical system to a six-dimensional phase space including shear, curvature, and spin-torsion, I recast the equations in a compact deceleration-parameter form, compute the full Jacobian, and evaluate maximal Lyapunov exponents. Numerical solutions show that the ekpyrotic branch ($w_ϕ\gg1$) exponentially damps homogeneous shear, while the softened branch ($w_ϕ<1$) allows $ρ_s$ to overtake the scalar during contraction and trigger a torsion-supported bounce at high but finite densities where the EC spin-torsion term becomes dynamically dominant. Scans in a two-parameter softening plane $(ϕ_{\rm b},Δ)$ identify a finite region of nonsingular trajectories and quantify the required tuning; in the parameter ranges explored the maximal Lyapunov exponent on the constrained phase space is negative, giving no indication of chaotic behavior in this homogeneous truncation even when the usual curvature mode that destabilizes contracting General Relativity (GR) backgrounds is included. The construction is purely phenomenological and confined to homogeneous backgrounds: it does not address entropy accumulation, the cosmological arrow of time, or a complete cyclic cosmology.

💡 Research Summary

In this work the author builds a phenomenological bounce model within Einstein‑Cartan (EC) gravity that combines a canonical scalar field with a steep exponential potential and a spin‑torsion sector modeled as a Weyssenhoff fluid. The scalar potential is designed to interpolate between a negative ekpyrotic branch (V≈−V₀ e^{−λ₀φ}, λ₀>√6) and a positive plateau (V≈V_soft). During the ekpyrotic phase the scalar equation‑of‑state w_φ≫1, which rapidly damps homogeneous shear (Σ) and suppresses anisotropies, a prerequisite for a nonsingular bounce. When the field reaches a transition value φ_b, a smooth tanh‑shaped switching function S(φ) turns the potential positive; w_φ then falls below unity, causing the scalar energy density to redshift more slowly than the spin‑torsion density ρ_s∝a⁻⁶. In a contracting universe this allows ρ_s to overtake the total energy density, satisfying ρ_tot=ρ_s at H=0 while the total equation‑of‑state remains w_tot<1, guaranteeing ˙H>0 and thus a bounce.

The dynamical system is extended from the three‑dimensional Copeland‑Liddle‑Wands (CLW) variables (x, y, z) to a six‑dimensional phase space (x, y, z, Σ, Ω_k, Ω_s). Here x and y are the normalized kinetic and potential contributions of the scalar, z the normalized matter (or radiation) density, Σ the normalized shear, Ω_k the curvature density parameter, and Ω_s the spin‑torsion density parameter. The Friedmann constraint becomes x² + s y² + z + Σ² + Ω_k – Ω_s = 1, where the minus sign in front of Ω_s encodes the repulsive nature of torsion. All evolution equations are rewritten in a compact deceleration‑parameter form q = −1 − Ḣ/H², yielding a closed autonomous system.

A full 6×6 Jacobian is derived and its eigenvalue spectrum is examined at three principal fixed points: (i) a scalar‑dominated expanding solution (Ω_s≈0, Σ≈0), (ii) an ekpyrotic contracting solution (w_φ≫1, Σ≈0, Ω_s≈0) and (iii) a torsion‑dominated bounce point (Ω_s≈1, q≈−1). The eigenvalues at each point are all negative (or have negative real parts), indicating linear stability. Notably, the inclusion of curvature (Ω_k≠0) does not destabilize the flow because the torsion term suppresses the usual curvature‑driven instability present in contracting GR backgrounds.

To assess possible chaotic behavior, the maximal Lyapunov exponent λ_max is computed numerically on the constrained five‑dimensional manifold (the Friedmann constraint reduces the dimensionality by one). Across the explored parameter ranges λ_max remains negative, showing no evidence of chaos in this homogeneous truncation. This result is significant because it demonstrates that the repulsive torsion term can neutralize the BKL‑type chaotic mixmaster dynamics that would otherwise arise in a contracting universe.

A systematic scan over the “softening plane” (φ_b, Δ) reveals a finite O(1) region where nonsingular trajectories exist. The bounce basin is not a fine‑tuned line but a sizable area, implying that modest tuning of the transition scale and width suffices to align the onset of the w_φ<1 phase with the density range where torsion becomes dominant. The magnitude of the torsion coefficient α (or equivalently Ω_s at the bounce) and the plateau height V_soft must be chosen so that the bounce occurs well below the Planck density, but these are treated as phenomenological inputs rather than derived from an underlying microphysical ECSK spinor theory.

The paper emphasizes its limitations: the analysis is confined to homogeneous (Bianchi I‑type) backgrounds, neglecting inhomogeneous perturbations, entropy production, and the arrow of time—issues essential for a complete cyclic cosmology. Nevertheless, the work provides the first global dynamical‑systems treatment of an Einstein‑Cartan cosmology that includes a scalar field, a barotropic fluid, shear, curvature, and a phenomenological spin‑torsion fluid, mapping fixed points, basins of attraction, and Lyapunov stability. Future extensions should incorporate linear and nonlinear perturbations, connect the Weyssenhoff fluid to a concrete Dirac‑fermion sector, and confront observational signatures such as CMB anisotropies, primordial gravitational waves, and large‑scale structure.