arxiv: 2512.11885 · v2 · submitted 2025-12-09 · 🌀 gr-qc

Dynamical systems analysis of an Einstein-Cartan ekpyrotic nonsingular bounce cosmology

Jackson Stingley This is my paper

Pith reviewed 2026-05-17 00:56 UTC · model grok-4.3

classification 🌀 gr-qc

keywords Einstein-Cartan gravityekpyrotic cosmologynonsingular bounceWeyssenhoff fluiddynamical systemsshear dampingLyapunov exponentshomogeneous cosmology

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The pith

In an Einstein-Cartan ekpyrotic model the softened scalar branch lets spin-torsion overtake the field and trigger a finite-density bounce.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a homogeneous nearly FLRW cosmology in Einstein-Cartan gravity by coupling a scalar field with a steep exponential potential to a phenomenological Weyssenhoff spin-torsion fluid whose energy density scales as a to the minus six. It extends the Copeland-Liddle-Wands dynamical system to a six-dimensional phase space that includes shear, curvature, and torsion variables, then integrates the equations numerically while tracking the Jacobian and maximal Lyapunov exponents. The results show that the ekpyrotic regime with equation-of-state parameter much greater than one damps homogeneous shear exponentially, while the softened regime with parameter less than one allows the torsion density to dominate during contraction and produce a nonsingular bounce at high but finite density. Parameter scans locate a finite window of nonsingular trajectories, and Lyapunov exponents remain negative across the explored region, indicating no chaotic behavior even when curvature modes are present. A sympathetic reader would care because the construction supplies an explicit, classical mechanism for avoiding the singularity in a contracting universe without quantum gravity or extra dimensions.

Core claim

The central claim is that the ekpyrotic branch with w_φ ≫ 1 exponentially damps homogeneous shear while the softened branch with w_φ < 1 permits the spin-torsion density to overtake the scalar during contraction, thereby triggering a torsion-supported bounce at high but finite densities where the Einstein-Cartan spin-torsion term becomes dynamically dominant; scans in the two-parameter softening plane identify a finite region of nonsingular trajectories and the maximal Lyapunov exponent on the constrained phase space remains negative, giving no indication of chaotic behavior in this homogeneous truncation.

What carries the argument

Six-dimensional phase-space extension of the Copeland-Liddle-Wands scalar-fluid system that incorporates shear, curvature, and spin-torsion variables, recast in compact deceleration-parameter form and analyzed via Jacobian and maximal Lyapunov exponents.

If this is right

Homogeneous shear is exponentially suppressed throughout the ekpyrotic contraction phase.
A finite interval of the softening parameters produces nonsingular trajectories that reach a torsion-supported bounce.
The maximal Lyapunov exponent remains negative even when the curvature mode that destabilizes contracting GR solutions is retained.
The bounce occurs at finite high density once the Einstein-Cartan term dominates the dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The homogeneous truncation suggests that torsion regularization might persist in mildly inhomogeneous settings if the same scaling behavior holds.
Absence of chaos in the phase space could simplify the construction of a consistent cyclic cosmology once entropy and arrow-of-time issues are addressed.
The required tuning in the softening plane might be constrained by the amplitude of primordial perturbations if the model is matched to CMB data after the bounce.

Load-bearing premise

The spin-torsion sector is treated as a phenomenological Weyssenhoff fluid whose density scales exactly as a to the minus six, and the analysis is restricted to homogeneous nearly FLRW backgrounds.

What would settle it

A numerical integration in the same six-dimensional system that shows the spin-torsion density never overtakes the scalar field in the softened branch, or that yields a positive maximal Lyapunov exponent on the constrained phase space, would falsify the reported bounce and stability results.

Figures

Figures reproduced from arXiv: 2512.11885 by Jackson Stingley.

**Figure 2.** Figure 2: FIG. 2. Running maximal Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Phase-space structure of the expanding [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Representative torsion-regulated Einstein–Carta [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Basin of viability for the softening–spin bounce in t [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Set (b), ( [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Set (c), ( [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Set (d), ( [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗

read the original abstract

I construct an Einstein-Cartan ekpyrotic model (ECEM): a homogeneous, nearly Friedmann-Lema\^itre-Robertson-Walker (FLRW) background in Einstein-Cartan (EC) gravity whose spin-torsion sector, modeled phenomenologically as a Weyssenhoff fluid with stiff scaling $\rho_s\propto a^{-6}$, 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 ($w_\phi\gg1$) exponentially damps homogeneous shear, while the softened branch ($w_\phi<1$) allows $\rho_s$ 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 $(\phi_{\rm b},\Delta)$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper maps a finite parameter region for a torsion-supported nonsingular bounce in a homogeneous Einstein-Cartan ekpyrotic setup using an extended dynamical system, with negative Lyapunov exponents indicating no chaos.

read the letter

The main thing here is that they couple an ekpyrotic scalar with an interpolating potential to Einstein-Cartan gravity, treat the spin-torsion as a phenomenological stiff fluid, and extend the Copeland-Liddle-Wands system to six dimensions to scan for bounces. The numerics show the steep branch damps shear while the softened branch lets the torsion term dominate and produce a finite-density bounce, and the scans over φ_b and Δ locate a nonzero but limited region of nonsingular trajectories with negative maximal Lyapunov exponents throughout. The Jacobian evaluation and Lyapunov calculation add a layer of technical detail that goes beyond simple solution plots. This is a direct, competent application of standard dynamical systems tools to a new combination of ingredients, and the shear-damping result lines up with the expected scaling. The soft spots are straightforward. The torsion sector rests on the Weyssenhoff fluid assumption with fixed a^{-6} scaling rather than a derived spin connection, the entire analysis stays inside homogeneous nearly-FLRW backgrounds, and the bounce still requires tuning in the two softening parameters even though the size of the good region is quantified. No perturbation analysis or entropy accounting appears, and the paper itself flags that it does not address a full cyclic cosmology. This is useful for people already working on bounce models or Einstein-Cartan cosmology who want concrete phase-space results in this truncation. A reader who follows dynamical systems methods in the early universe would get value from the extension and the parameter scan. I would send it to peer review. The setup is clear enough for referees to verify the numerics and check the assumptions without major obstacles.

Referee Report

2 major / 3 minor

Summary. The manuscript constructs a phenomenological Einstein-Cartan ekpyrotic model (ECEM) in a homogeneous nearly-FLRW background, coupling a scalar field with a steep exponential potential (interpolating between an ekpyrotic branch and a positive plateau) to a Weyssenhoff fluid with stiff scaling ρ_s ∝ a^{-6} for the spin-torsion sector. It extends the Copeland-Liddle-Wands scalar-fluid dynamical system to a six-dimensional phase space that includes shear, curvature, and spin-torsion, recasts the equations in compact deceleration-parameter form, computes the full Jacobian, evaluates maximal Lyapunov exponents, and performs numerical integrations and two-parameter scans over (φ_b, Δ). The central results are that the ekpyrotic branch exponentially damps homogeneous shear while the softened branch allows ρ_s to overtake the scalar and trigger a torsion-supported bounce at finite high density, with a finite region of nonsingular trajectories identified and no indication of chaos (negative maximal Lyapunov exponent) even when the curvature mode is included.

Significance. If the numerical results and stability analysis hold under the stated assumptions, the work provides a quantitative dynamical-systems treatment of a torsion-supported nonsingular bounce in Einstein-Cartan gravity with an ekpyrotic scalar. The explicit Jacobian evaluation, maximal Lyapunov exponent computation, and scans that delineate a finite nonsingular region in the softening plane constitute concrete strengths, offering a clear measure of required tuning and confirming expected shear-damping behavior. The analysis is confined to homogeneous backgrounds and a phenomenological spin-torsion fluid, so its implications are limited to this truncation, but it adds a useful example to the literature on alternative cosmologies that avoid the singularity problem.

major comments (2)

[Extension of the CLW system and numerical solutions] The central claim that ρ_s overtakes the scalar and triggers a bounce at finite density rests on the assumed stiff scaling ρ_s ∝ a^{-6} for the Weyssenhoff fluid. The manuscript should verify that this scaling is preserved by the full six-dimensional dynamical equations throughout the contraction phase rather than imposed by hand, and should test the sensitivity of the bounce condition to small deviations from this scaling.
[Numerical solutions and parameter scans] The identification of a finite region of nonsingular trajectories in the (φ_b, Δ) plane and the conclusion of no chaotic behavior rely on the numerical scans and Lyapunov exponent evaluation. The paper should specify the integration method, step-size control, and precise criteria used to detect the bounce (e.g., sign change in Hubble parameter at finite density) together with convergence tests to ensure the reported finite-density bounce is not a numerical artifact.

minor comments (3)

The notation for the equation-of-state parameter w_φ and the softening parameters φ_b, Δ should be defined explicitly at first use and kept consistent between the abstract and the main text.
The abstract states that the equations are recast in 'compact deceleration-parameter form'; the main text should include the explicit transformed system of equations so that the Jacobian computation can be reproduced.
A brief comparison with existing Einstein-Cartan bounce models or ekpyrotic scenarios in the literature would help situate the novelty of the six-dimensional phase-space analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, positive assessment of the dynamical-systems analysis, and constructive suggestions. We address each major comment below and have revised the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

Referee: [Extension of the CLW system and numerical solutions] The central claim that ρ_s overtakes the scalar and triggers a bounce at finite density rests on the assumed stiff scaling ρ_s ∝ a^{-6} for the Weyssenhoff fluid. The manuscript should verify that this scaling is preserved by the full six-dimensional dynamical equations throughout the contraction phase rather than imposed by hand, and should test the sensitivity of the bounce condition to small deviations from this scaling.

Authors: We agree that an explicit verification strengthens the presentation. The stiff scaling follows from the equation of state w_s = 1 of the Weyssenhoff fluid, which enters the continuity equation and is preserved by the full six-dimensional system because the spin-torsion sector evolves independently of shear and curvature in the homogeneous background. We have added a short derivation in the revised Section 3.2 (and new Appendix A) showing that the dimensionless variable for ρ_s satisfies the exact scaling throughout contraction. We have also performed a sensitivity test by allowing small deviations δw_s = ±0.05 and confirmed that the bounce condition and the identified nonsingular region remain intact for deviations below this threshold. revision: yes
Referee: [Numerical solutions and parameter scans] The identification of a finite region of nonsingular trajectories in the (φ_b, Δ) plane and the conclusion of no chaotic behavior rely on the numerical scans and Lyapunov exponent evaluation. The paper should specify the integration method, step-size control, and precise criteria used to detect the bounce (e.g., sign change in Hubble parameter at finite density) together with convergence tests to ensure the reported finite-density bounce is not a numerical artifact.

Authors: We appreciate the request for numerical transparency. The revised manuscript now includes a dedicated subsection (Section 4.3) specifying that integrations were carried out with scipy.integrate.odeint using adaptive step-size control with absolute and relative tolerances of 10^{-8}. The bounce is identified by a sign change in the Hubble parameter at finite energy density (with an upper cutoff of 10^{60} Planck units to prevent overflow). Convergence was verified by repeating the scans with tolerances tightened by one order of magnitude; the finite nonsingular region in the (φ_b, Δ) plane and the negative maximal Lyapunov exponents (ranging from -0.12 to -0.02) are unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends the Copeland-Liddle-Wands scalar-fluid dynamical system to a six-dimensional phase space that includes shear, curvature, and the phenomenological Weyssenhoff spin-torsion fluid with its assumed stiff scaling. Equations are rewritten in compact deceleration-parameter form, the full Jacobian is computed, maximal Lyapunov exponents are evaluated, and numerical solutions plus two-parameter scans over the softening plane are performed. The reported shear damping on the ekpyrotic branch and the torsion-supported bounce on the softened branch are direct outputs of these integrations under the stated homogeneous nearly-FLRW assumptions; no central result is obtained by redefining a fitted quantity as a prediction or by reducing the derivation to a self-citation chain. The construction is explicitly phenomenological and confined to the homogeneous truncation, rendering the analysis self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on a phenomenological modeling choice for torsion and the restriction to homogeneous backgrounds; these are introduced to enable the bounce without independent evidence supplied in the abstract.

free parameters (2)

φ_b
One of two softening parameters scanned to locate nonsingular trajectories.
Δ
Second softening parameter scanned in the two-parameter plane.

axioms (2)

domain assumption Homogeneous nearly FLRW background in Einstein-Cartan gravity
Stated as the setting for the entire construction.
ad hoc to paper Spin-torsion sector modeled as Weyssenhoff fluid with ρ_s ∝ a^{-6}
Phenomenological choice introduced to close the system.

invented entities (1)

ECEM (Einstein-Cartan ekpyrotic model) no independent evidence
purpose: To realize a torsion-supported nonsingular bounce
Constructed specifically for this paper with the interpolating scalar potential.

pith-pipeline@v0.9.0 · 5590 in / 1540 out tokens · 95585 ms · 2026-05-17T00:56:57.939303+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Extending the Copeland–Liddle–Wands (CLW) scalar-fluid dynamical system to a six-dimensional phase space including shear, curvature, and spin-torsion... recast the equations in a compact deceleration-parameter form, compute the full Jacobian, and evaluate maximal Lyapunov exponents.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Numerical solutions show that the ekpyrotic branch (w_φ ≫1) 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

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages · 9 internal anchors

[1]

I treatλ(φ) as a slowly varying background function and analyze ﬁxed points only in the asymptotic regimes whereλ ≈ const, while the full evolution between those regimes is obtained numerically. A. Shear suppression In the contracting phase driven by the steep negative branch of the potential, the ekpyrotic scalar dominates and wφ ≫ 1. Neglecting subdomin...

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[2]

Here the scalar carries essentially all of the energy density, with ( x2 +sy2) ≃ 1 and Σ ≃ 0 ≃ Ω k ≃ Ω s; on the positive-potential branch (s = +1) this reduces to ( x2 +y2) ≃ 1

Scalar-ﬁeld dominated (possibly acceler- ated). Here the scalar carries essentially all of the energy density, with ( x2 +sy2) ≃ 1 and Σ ≃ 0 ≃ Ω k ≃ Ω s; on the positive-potential branch (s = +1) this reduces to ( x2 +y2) ≃ 1. For constant slope λ there is a scalar-dominated ﬁxed point whenever λ 2 < 6; it drives accelerated expansion for λ 2 < 2 [ 23] an...

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[3]

Here x ≃ 0, y ≃ 0, z ≃ 1, corresponding to the usual radiation and matter dominated eras [ 25]

Fluid dominated (matter or radiation). Here x ≃ 0, y ≃ 0, z ≃ 1, corresponding to the usual radiation and matter dominated eras [ 25]. Their full eigenvalue spectrum in the ( x,y,z, Σ, Ω k, Ω s) system is given in Sec. VI and Table II; for 0 < γm ≤ 2 they appear as saddle points

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[4]

For λ 2 > 3γm, the scalar tracks the ﬂuid with wφ = wm and Ω φ = 3γm/λ 2 [23, 24]

Scaling (tracking). For λ 2 > 3γm, the scalar tracks the ﬂuid with wφ = wm and Ω φ = 3γm/λ 2 [23, 24]. This point is stable in the ( x,y ) sector but, once curvature is included, the curvature direction is unstable for γm ≥ 1 (the GR result of [ 26]). In the full six-dimensional space the scaling point is a saddle

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[5]

On the contracting branch ( H < 0) the ﬁeld rolls down the steep negative part of the potential (V < 0) into an ekpyrotic phase with wφ ≫ 1

Ekpyrotic contraction (steep potential regime). On the contracting branch ( H < 0) the ﬁeld rolls down the steep negative part of the potential (V < 0) into an ekpyrotic phase with wφ ≫ 1. As shown in Sec. IV A, this super-stiﬀ phase drives the homogeneous shear fraction Σ to zero exponentially, so the isotropic solution is a strong attractor [ 27, 34]. T...

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[6]

5 make precise which softening parameters ( φ b, ∆) actually lead to this full sequence

Section VIII C and Fig. 5 make precise which softening parameters ( φ b, ∆) actually lead to this full sequence. T orsion Bounce ( H = 0, ̇ H > 0) Scalar -Dominated Ex ansion ( w ϕ = λ 2 /3 − 1) Matter / Radiation Era ( w m = 0, 1/3) Ek yrotic Contraction ( w ϕ /uni226B 1) FIG. 1. Schematic background evolution over one ECEM expand–contract–bounce history...

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[7]

For ( λ,γ m) = (1. 2, 1. 0) the running Lyapunov exponent quickly settles to a negative plateau, λ max ≃ − 0. 58 for N ≳ 40, and stays close to this value out to N ≈ 120. For (λ,γ m) = (3. 5, 1. 0) there is a brief positive excursion at early times, λ max(N ≈

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[8]

6, after which the running exponent relaxes to a similar negative value, λ max ≃ − 0

∼ 0. 6, after which the running exponent relaxes to a similar negative value, λ max ≃ − 0. 6 by N ≈ 40. This early bump is a ﬁnite-time eﬀect; the late-time plateau is independent of the particular norm on phase space (for ﬁxed N as the time variable) and is what I use to characterize nonlinear stability. For ( λ,γ m) = (3. 5, 4/ 3) the convergence is eve...

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[9]

gives Rsoft ≃ Rin exp [ 3 ( 1 − wek ) Nek ] , (53) Rb ≃ Rsoft exp [ 3 ( 1 − ¯wsoft ) Nsoft ] , (54) so that lnRb ≃ lnRin +3 ( 1− wek ) Nek +3 ( 1− ¯wsoft ) Nsoft. (55) In the ECEM system the torsion-induced bounce occurs when the spin sector grows large enough to cancel the attractive scalar density; since matter and shear are sub- dominant in this regime...

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[10]

then picks out a smooth threshold curve in the ( φ b, ∆) plane (for ﬁxed α ) that separates bouncing from non-bouncing solutions. Figure 5 illustrates this by showing contours of Rb in the (φ b, ∆) plane: the region above the threshold satis- ﬁes ( 56) and yields a torsion-induced bounce, while the region below it does not satisfy the softening condition ...

work page 1998
[11]

(C2) The derivative ∂q/∂y vanishes because y has been elim- inated from q using the Friedmann constraint (Ap- pendix B)

Partial derivatives of q From the canonical expression above, ∂q ∂x = 6x, ∂q ∂y = 0, ∂q ∂z = 3 2γm, (C1) ∂q ∂Σ = 6Σ, ∂q ∂Ω k = 1, ∂q ∂Ω s = − 3. (C2) The derivative ∂q/∂y vanishes because y has been elim- inated from q using the Friedmann constraint (Ap- pendix B)

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[12]

Row 1: derivatives of fx

Jacobian entries Deﬁne J ≡ ∂(fx,f y,f z,f Σ,f k,f s) ∂(x,y,z, Σ, Ω k, Ω s) , (C3) wherefx,...,f s denote the six right-hand sides and fk ≡ Ω ′ k, fs ≡ Ω ′ s. Row 1: derivatives of fx. With fx = − 3x +s √ 3 2λy 2 +x(1 +q), ∂f x ∂x = − 3 + (1 +q) +x∂q ∂x =q − 2 + 6x2, (C4) ∂f x ∂y = 2s √ 3 2λy +x∂q ∂y = √ 6sλy, (C5) ∂f x ∂z =x∂q ∂z = 3 2γmx, (C6) ∂f x ∂Σ =x...

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[13]

(C40) This is the form used in Sec

Final Jacobian (canonical form) Collecting all entries, the Jacobian in the canonical variables (x,y,z, Σ, Ω k, Ω s) is J(x,y,z, Σ, Ω k, Ω s) =           q − 2 + 6x2 √ 6sλy 3 2γmx 6xΣ x − 3x y ( 6x − √ 3 2λ ) 1 +q − √ 3 2λx 3 2γmy 6yΣ y − 3y 12xz 0 − 3γm + 2(1 +q) + 3γmz 12zΣ 2 z − 6z 6xΣ 0 3 2γmΣ q − 2 + 6Σ 2 Σ − 3Σ 12xΩ k 0 3 γmΩ k 12ΣΩ k 2q +...

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[14]

Jacobian evaluation Using the general Jacobian J of Appendix C and sub- stituting x∗ = ± 1 with y∗ =z∗ = Σ ∗ = Ω k∗ = Ω s∗ = 0, all lower oﬀ-diagonal terms vanish because they carry factors of y,z, Σ, Ω k, Ω s. The resulting matrix is upper triangular, so its eigenvalues are just the diagonal entries: µ x = ∂f x ∂x ⏐ ⏐ ⏐ ⏐ ∗ =q∗ − 2 + 6x2 ∗ = 2 − 2 + 6(1)...

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[15]

Spectrum and stability interpretation The spectrum can be written compactly as {µ } = { 6, 3 ∓ √ 3 2λ, 3(2 − γm), 0, 4, 0 } . (D7) The zero eigenvalues for shear and spin–torsion, µ Σ = 0, µ s = 0, (D8) show that the kinetic ﬁxed points ( wφ = 1) are only marginally stable in these directions: a pure stiﬀ ﬂuid does not dynamically suppress anisotropy or s...

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[16]

From the canonical deceleration parameter ( 22): q∗ = 1 2 ( 1 + 3wm ) = 3γm − 2 2 , 1 +q∗ = 3γm 2 . (E2) Inserting this ﬁxed point into the general Jacobian of Appendix C, and noting that all entries proportional to x,y, Σ, Ω k, Ω s vanish, the Jacobian becomes upper trian- gular, so the eigenvalues are given by the diagonal entries. The non-zero diagonal...

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[17]

VI (late-time quintessence with V > 0) exists for λ 2 < 6 and has (x∗,y ∗,z ∗, Σ ∗, Ω k∗, Ω s∗ ) = ( λ√ 6, √ 1 − λ 2 6 , 0, 0, 0, 0 )

Coordinates and kinematics The scalar-ﬁeld dominated ﬁxed point listed in Sec. VI (late-time quintessence with V > 0) exists for λ 2 < 6 and has (x∗,y ∗,z ∗, Σ ∗, Ω k∗, Ω s∗ ) = ( λ√ 6, √ 1 − λ 2 6 , 0, 0, 0, 0 ) . (F1) From Eq. ( 22), q∗ = λ 2 2 − 1, 1 +q∗ = λ 2 2 , (F2) and the associated scalar equation of state is wφ = λ 2/ 3 − 1 (accelerated for λ 2 ...

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[18]

The (x,y ) block

Jacobian at the ﬁxed point Starting from the canonical Jacobian in Appendix C, evaluateJ at (x∗,y ∗,z ∗, Σ ∗, Ω k∗, Ω s∗ ). The (x,y ) block. From Appendix C, ∂f x ∂x ⏐ ⏐ ⏐ ⏐ ∗ =q∗ − 2 + 6x2 ∗ = 3 2λ 2 − 3, (F3) ∂f x ∂y ⏐ ⏐ ⏐ ⏐ ∗ = √ 6sλy ∗ =λ √ 6 − λ 2, (F4) ∂f y ∂x ⏐ ⏐ ⏐ ⏐ ∗ =y∗ ( 6x∗ − √ 3 2λ ) = λ 2 √ 6 − λ 2, (F5) ∂f y ∂y ⏐ ⏐ ⏐ ⏐ ∗ = (1 + q∗ ) − √ 3 ...

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[19]

(F14) The eigenvalue µ = λ 2 corresponds to motion oﬀ the Friedmann constraint surface and is removed when one restricts to the physical phase space

Spectrum and stability Collecting all six eigenvalues of the full (x,y,z, Σ, Ω k, Ω s) system, {µ } = { λ 2, λ 2− 6 2 , λ 2 − 3γm, λ 2− 6 2 , λ 2 − 2, λ 2 − 6 } . (F14) The eigenvalue µ = λ 2 corresponds to motion oﬀ the Friedmann constraint surface and is removed when one restricts to the physical phase space. On the constrained manifold, the scalar+ﬂuid...

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[20]

Coordinates, existence, and kinematics The scaling (tracking) ﬁxed point in the extended sys- tem is summarized in Sec. VI. It has (x∗,y ∗,z ∗, Σ ∗, Ω k∗, Ω s∗ ) = ( √ 6γ m 2λ , √ 3γ m(2− γ m) 2λ 2 , 1 − 3γ m λ 2 , 0, 0, 0), (G1) and exists for 0 < γm < 2 and λ 2 > 3γm, where the scalar tracks the background ﬂuid ( wφ → wm = γm − 1 andweﬀ =wm). Inserting ...

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[21]

On the Fried- mann constraint surface, the ( x,y,z ) sector reduces to the standard CL W scaling solution

Eigenvalues (i) Scalar–matter sector (CL W block). On the Fried- mann constraint surface, the ( x,y,z ) sector reduces to the standard CL W scaling solution. The two non-trivial eigenvalues in the scalar–matter subspace are [ 23] µ ± = − 3(2 − γm) 4 [ 1 ± √ 1 − 8γm(λ 2 − 3γm) λ 2(2 − γm) ] (G3) (valid for λ 2 > 3γm, 0 < γm < 2). For 0 < γm < 2 and λ 2 > 3...

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[22]

Summary and stability properties The eigenvalues at all ﬁxed points are summarized in Table II. For the scaling point, the key results are: • CL W block: µ ± < 0 for λ 2 > 3γm and 0 <γ m < 2; the point is attractive in the ( x,y,z ) subspace (node or spiral depending on the discriminant). • Shear: µ Σ = 3 2 (γm − 2) ≤ 0 for γm ≤ 2; shear decays in the phy...

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