Einstein–Cartan (EC) gravity is general relativity with torsion. At high fermion densities, intrinsic spin sources torsion algebraically, and integrating out the torsion degrees of freedom leaves a repulsive{' '}{' '}correction in the Friedmann equation. During contraction this term grows steeply and can, at a finite density, force{' '}{' '}to zero, giving a nonsingular bounce instead of a singularity. The spin–torsion sector is modeled phenomenologically as a Weyssenhoff fluid, an effective description of spin-polarized fermions at high densities without resolving the full spinor microphysics.
A bounce alone is not enough: any initial anisotropy (shear) also scales as{' '}{' '}during contraction, so the background would generically be strongly anisotropic by the time the bounce occurs. The ekpyrotic mechanism addresses this. A scalar field on a steep negative potential reaches{' '} making the scalar energy density grow faster than the shear during contraction{' '} which drives the normalized shear{' '}{' '}exponentially to zero. When the potential then softens onto a positive plateau{' '}{' '}the spin–torsion term can overtake the scalar and trigger the bounce.
To analyze the full system I extend the Copeland–Liddle–Wands (CLW) framework, which casts scalar-field cosmology as a flow on a constrained surface in dimensionless phase-space variables, to six dimensions by adding shear{' '} curvature{' '} and spin–torsion{' '} The paper computes fixed points and their full Jacobian spectrum, and evaluates maximal Lyapunov exponents with the Benettin algorithm. Main results: shear is exponentially damped during ekpyrotic contraction; a two-parameter scan of the softening potential{' '}{' '}reveals a finite bounce basin; and the maximal Lyapunov exponent is negative in every case tested, with no sign of chaos in this homogeneous truncation.
);
}
// ============== Published Fig. 1 ==============
function PublishedFig1() {
return (
Fig. 1. A positively curved{' '} ECEM background integrated over multiple cycles. Closed spatial geometry allows the expansion to reverse at a classical turnaround{' '} without any torsion, while each subsequent contraction is halted by the spin–torsion bounce{' '} The top panel traces alternating between these two types of zero crossing; the bottom shows the scale factor staying bounded and oscillatory rather than running into a singularity.
);
}
// ============== Conceptual Cycle Diagram ==============
const CYCLE_PHASES = [
{
id: 'bounce',
angle: 90,
title: 'Torsion-supported bounce',
short: 'Torsion Bounce',
condTex: 'H = 0,\\;\\dfrac{dH}{dt} > 0',
detail: <>
As the universe contracts, the spin–torsion density{' '}{' '}grows until it matches the total energy density. At that point the EC correction, which enters the Friedmann equation with a minus sign as{' '}, forces{' '}{' '}to zero. Provided the equation of state has softened below{' '} the Raychaudhuri equation gives{' '}{' '}there, so{' '}{' '}crosses zero smoothly at a finite density{' '} The bounce is obtained from the modified field equations rather than imposed as a cutoff.
,
color: 'var(--bounce)',
},
{
id: 'expand',
angle: 0,
title: 'Scalar-dominated expansion',
short: 'Scalar Expansion',
condTex: 'w_\\varphi = \\dfrac{\\lambda^2}{3} - 1',
detail: <>
After the bounce the scalar sits on the positive plateau of the potential{' '}{' '}with equation of state{' '} The long-term dynamics are governed by the CLW attractor: for{' '}{' '}the fixed point drives accelerated expansion; for{' '}{' '}it still attracts but without acceleration; above that the scalar and fluid scale together in a tracking solution.
,
color: 'var(--scalar)',
},
{
id: 'matter',
angle: 270,
title: 'Matter / radiation era',
short: 'Matter / Rad.',
condTex: 'w_m = 0\\ \\text{or}\\ \\dfrac{1}{3}',
detail: <>
Between the bounce and ekpyrotic contraction sits a conventional fluid-dominated era. In the CLW variables{' '}, which measure the scalar's kinetic and potential energy fractions relative to the Hubble budget, the system idles near{' '}{' '}while matter or radiation carries the energy. This is a saddle in the full phase space: the trajectory passes through it rather than stopping there, eventually giving way to the scalar or to contraction.
,
color: 'var(--matter)',
},
{
id: 'ekpyrosis',
angle: 180,
title: 'Ekpyrotic contraction',
short: 'Ekpyrosis',
condTex: 'w_\\varphi \\gg 1',
detail: <>
The scalar rolls down a steep negative branch of the potential, reaching{' '} far stiffer than radiation. In this regime{' '}{' '}grows much faster during contraction than the shear{' '} so the normalized shear{' '}{' '}is driven exponentially toward zero. This is the key feature of ekpyrosis: whatever anisotropy the background starts with, the geometry is smoothed before the bounce.
,
color: 'var(--ekpyrotic)',
},
];
function CycleDiagram() {
const [active, setActive] = useStateI('bounce');
const phase = CYCLE_PHASES.find(p => p.id === active);
const cx = 250, cy = 220;
const R = 130;
const positions = CYCLE_PHASES.map(p => ({
...p,
x: cx + R * Math.cos(p.angle * Math.PI / 180),
y: cy - R * Math.sin(p.angle * Math.PI / 180),
}));
// Arc paths between neighboring phases, following the source order.
const arcs = positions.map((p, i) => {
const next = positions[(i + 1) % positions.length];
// Curve each connector away from the center so the loop remains readable.
const mx = (p.x + next.x) / 2, my = (p.y + next.y) / 2;
// Unit normal pointing away from the center.
const nx = (mx - cx) / Math.sqrt((mx - cx) ** 2 + (my - cy) ** 2);
const ny = (my - cy) / Math.sqrt((mx - cx) ** 2 + (my - cy) ** 2);
const cxC = mx + nx * 40;
const cyC = my + ny * 40;
return { from: p, to: next, cx: cxC, cy: cyC };
});
return (
Conceptual Guide
Four phases discussed in the ECEM model
Click any phase
{phase.id.toUpperCase()}
{phase.title}
{phase.detail}
Conceptual guide. Not a journal figure, but a map of the four dynamical regimes discussed in the paper. The published Fig. 1 above shows a numerically integrated{' '}{' '}trajectory cycling through these phases. Click any node to read what is happening physically at that stage.
