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

TL;DR

This work develops a homogeneous Einstein--Cartan cosmology incorporating a canonical scalar with a softened exponential potential and a Weyssenhoff spin fluid to create a nonsingular expand--contract--bounce history. It extends the Copeland--Liddle--Wands dynamical-systems framework to a six-dimensional phase space, derives the full Jacobian, and analyzes fixed points and Lyapunov stability to map phase-space structure and stability. Numerical phase-space trajectories show that an ekpyrotic contraction with $w_φ>1$ damps homogeneous shear, while a late-time softened scalar potential allows the spin-torsion term to dominate at high densities and trigger a torsion-driven bounce at sub-Planckian densities; a finite bounce basin is identified in the $(φ_b,Δ)$ plane. Across the explored parameter ranges, the maximal Lyapunov exponent on the constrained phase space is negative, indicating no chaotic behavior in this homogeneous truncation, though the authors caution that inhomogeneous modes could alter dynamics. The work remains phenomenological and homogeneous, outlining future steps to include perturbations, entropy considerations, and observational tests within a broader cyclic cosmology framework.

Abstract

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.

Figures (10)

• Figure 1: Schematic background evolution over one ECEM expand--contract--bounce history. Starting at the torsion-regulated bounce $(H=0,\dot H>0)$, the background passes through a scalar-dominated expansion phase, a matter/radiation era, and an ekpyrotic contracting phase that approaches a second high-density regime in this effective description. Each segment corresponds to a fixed point or asymptotic regime in the extended phase space. The qualitative sequence resembles earlier cyclic proposals Tolman1931PeriodicTolman1931EntropySteinhardt2002Cyclic, but here only a single expand--contract--bounce episode is constructed explicitly, with a torsion-induced bounce replacing a singular turnaround.
• Figure 2: Running maximal Lyapunov exponent $\lambda_{\max}(N)$ for the extended Einstein--Cartan system, computed with the Benettin algorithm (Sec. \ref{['sec:lyapunov']}). Panels show (a) $(\lambda,\gamma_m) = (1.2,1)$, (b) $(\lambda,\gamma_m) = (3.5,1)$, and (c) $(\lambda,\gamma_m) = (3.5,4/3)$. In all cases $\lambda_{\max}$ tends toward a negative plateau on the expanding branch, indicating no evidence of chaotic mixing in these homogeneous backgrounds for the constant-slope ($\lambda=\mathrm{const}$) slices explored here.
• Figure 3: Phase-space structure of the expanding $s=+1$ branch for $(\lambda,\gamma_m)=(1.20,1.00)$. (a) Projection onto $(x,y)$. Starting from small anisotropy, curvature, and spin--torsion seeds, trajectories inside the allowed domain converge to the scalar-field dominated expansion fixed point $(x_*,y_*)$ identified in the linear stability analysis. The torsion-regulated bounce lies in the $V<0$ branch and is not part of this constant-$\lambda$ slice. (b) Three-dimensional flow in $(x,y,z)$ for the same run, showing the decay of the fluid fraction $z$ as the orbit approaches the scalar attractor.
• Figure 4: Representative torsion-regulated Einstein--Cartan bounce and numerical diagnostics. (a) Cosmic-time evolution of the scale factor, Hubble parameter, normalized total density $\rho_{\rm tot}/\rho_{\rm b}$, and equation-of-state parameters for a contracting solution that undergoes a nonsingular bounce governed by the modified Friedmann equation \ref{['eq:EC-Friedmann']}. The universe contracts from a dust-dominated phase to a finite minimum scale factor $a_{\rm b}$ at $t=0$, where $H$ crosses zero with $\dot H>0$. The total fluid equation of state $w_{\rm tot}=p_{\rm tot}/\rho_{\rm tot}$ approaches dust far from the bounce, while the effective equation of state $w_{\rm eff}$ inferred from $H(t)$ briefly drops below $-1/3$ in a neighborhood of the bounce (where $H\neq 0$), reflecting torsion-driven acceleration rather than exotic matter in the physical sector. (b) Left: Friedmann-constraint residual along the same solution, which remains close to zero throughout the evolution. Right: component energy densities on a logarithmic scale, showing that background matter dominates far from the bounce, while the repulsive spin--torsion density $\rho_s$ becomes comparable to $\rho_{\rm tot}$ at the bounce, with scalar and shear contributions remaining subdominant.
• Figure 5: Basin of viability for the softening--spin bounce in the $(\phi_{\rm b},\Delta)$ plane for fixed $(\alpha,V_{\rm soft})$ and pre-softening initial data. The color scale shows the spin-to-scalar ratio $R_{\rm b} \equiv \rho_s/\rho_\phi$ evaluated at the bounce for those trajectories that undergo a torsion-induced reversal of contraction ($H=0,\dot H>0)$; non-bouncing trajectories are left blank. To isolate the scalar--spin competition, these basin runs use a reduced background matter density $\rho_{m0}=10^{-5}$ and a tuned spin--torsion parameter $\alpha=10^{-4}$, distinct from the high-density, dust-dominated single-bounce example discussed earlier. The solid curve marks the numerically determined boundary between bouncing and non-bouncing trajectories and lies close to the contour $R_{\rm b}\simeq 1$ suggested by \ref{['eq:bounce_inequality']}: parameter combinations on and above the curve satisfy the softening condition and realize a torsion-dominated bounce within the Einstein--Cartan regime, while points below the curve do not achieve a softening--spin bounce and typically evolve toward a singular crunch.