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

May combine two claims.

In EC theory, the intrinsic spin of fermions sources spacetime torsion, which at very high densities acts like a negative stiff component and can support a nonsingular bounce in homogeneous EC models without introducing new propagating degrees of freedom beyond GR plus matter.

“In EC gravity the intrinsic spin of fermions sources spacetime torsion, which at very high densities acts like a negative stiff component [11][12][13].This purely geometric effect can, in homogeneous EC models, support a nonsingular bounce without introducing new propagating degrees of freedom beyond GR plus matter.”

• cites [11] Hehl, F.W., Heyde, P., Kerlick, G.D., Nester, J.M.: General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys. 48, 393–416 (1976). https://doi.org/10.1103/RevModPhys.48.393 sourceNot checked — no full text of the source
• cites [12] Shapiro, I.L.: Physical aspects of the space-time torsion. Phys. Rep. 357, 113–213 (2002). https://doi.org/10.1016/S0370-1573(01)00030-8 sourceSupported
• cites [13] Popławski, N.J.: Cosmology with torsion: an alternative to cosmic inflation. Phys. Lett. B 694(3), 181–185 (2010). https://doi.org/10.1016/j.physletb.2010.09.056. Erratum: Phys. Lett. B 701 (2011) 672 sourceSupported

May combine two claims.

In ECSK theory, torsion is sourced algebraically by the spin current and does not propagate; for a Dirac field it is fixed locally by the axial spin current.

“The coupling of spin to torsion is described by the Einstein-Cartan-Sciama-Kibble action, where the affine connection has an antisymmetric part and the spin current sources torsion [11,12].”

• Also stated in: § III. EINSTEIN-CARTAN FIELD EQUATIONS · plain

  “so torsion does not propagate; it is fixed locally by the axial spin current.”

• cites [11] Hehl, F.W., Heyde, P., Kerlick, G.D., Nester, J.M.: General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys. 48, 393–416 (1976). https://doi.org/10.1103/RevModPhys.48.393 sourceNot checked — no full text of the source
• cites [12] Shapiro, I.L.: Physical aspects of the space-time torsion. Phys. Rep. 357, 113–213 (2002). https://doi.org/10.1016/S0370-1573(01)00030-8 sourceSupported

May combine two claims.

On the contracting branch, the field rolls down the steep negative potential into an ekpyrotic phase with wφ≫1; this super-stiff phase drives the homogeneous shear fraction Σ to zero exponentially, so the isotropic solution is a strong attractor.

“On the contracting branch (H < 0) the field rolls down the steep negative part of the potential (V < 0) into an ekpyrotic phase with w φ ≫ 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].”

• cites [27] Erickson, J.K., Wesley, D.H., Steinhardt, P.J., Turok, N.: Kasner and mixmaster behavior in universes with equation of state $$w \ge 1$$. Phys. Rev. D 69, 063514 (2004). https://doi.org/10.1103/PhysRevD.69.063514. arXiv:hep-th/0312009 sourceSupported
• cites [34] Garfinkle, D., Lim, W.C., Pretorius, F., Steinhardt, P.J.: Evolution to a smooth universe in an ekpyrotic contracting phase with $$w > 1$$. Phys. Rev. D 78(8), 083537 (2008). https://doi.org/10.1103/PhysRevD.78.083537 sourcePartly supported

  Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured

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During ekpyrotic contraction with wφ>1, the shear energy density ratio ρσ/ρφ ∝ a^{3(1+wφ)-6} → 0 as a→0, so the anisotropic contribution becomes negligible and the ekpyrotic phase acts as a strong attractor towards isotropy.

“Thus the anisotropic contribution becomes negligible during ekpyrotic contraction, and at the homogeneous level the ekpyrotic phase acts as a strong attractor towards isotropy, suppressing the growth of shear anisotropy.”

• cites [27] Erickson, J.K., Wesley, D.H., Steinhardt, P.J., Turok, N.: Kasner and mixmaster behavior in universes with equation of state $$w \ge 1$$. Phys. Rev. D 69, 063514 (2004). https://doi.org/10.1103/PhysRevD.69.063514. arXiv:hep-th/0312009 sourceSupported
• cites [34] Garfinkle, D., Lim, W.C., Pretorius, F., Steinhardt, P.J.: Evolution to a smooth universe in an ekpyrotic contracting phase with $$w > 1$$. Phys. Rev. D 78(8), 083537 (2008). https://doi.org/10.1103/PhysRevD.78.083537 sourceSupported

May combine two claims.

Most of these papers focus on the spin-sector microphysics and on particular background or perturbative solutions; for example, Ref. [22] studies exact spinor-driven ekpyrotic bounce backgrounds and their perturbations, rather than the global phase-space structure of a scalar-fluid-spin system in a dynamical-systems picture.

“As noted above, most of these papers focus on the spin-sector microphysics and on particular background or perturbative solutions.For example, Ref. [22] studies exact spinordriven ekpyrotic bounce backgrounds and their perturbations, rather than the global phase-space structure of a scalar-fluid-spin system in a dynamical-systems pic-ture.”

• cites [22] Farnsworth, S., Lehners, J.-L., Qiu, T.: Spinor driven cosmic bounces and their cosmological perturbations. Phys. Rev. D 96(8), 083530 (2017). https://doi.org/10.1103/PhysRevD.96.083530 sourceSupported

May combine two claims.

The ratio of the fiducial bounce density to the Planck density is ρb/ρPl ∼ (E_b/M_Pl)^4 ∼ 10^-10, so the bounce lies comfortably below ρPl and in the same general regime usually assumed in semi-classical bounce scenarios such as loop quantum cosmology.

“The ratio to the Planck density is ρ b ρ Pl ∼ E b M Pl 4 ∼ 10 -10 ,(44) so the bounce lies comfortably below ρ Pl and in the same general regime usually assumed in semi-classical bounce scenarios such as loop quantum cosmology [35].”

• cites [35] Ashtekar, A., Singh, P.: Loop quantum cosmology: a status report. Class. Quantum Grav. 28(21), 213001 (2011). https://doi.org/10.1088/0264-9381/28/21/213001. arXiv:1108.0893 sourcePartly supported

  Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured

• AssumptionThe authors state this as an assumption: The torsion-induced bounce occurs at densities sufficiently below the Planck density that higher-curvature and higher-torsion corrections remain subdominant.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Billyard et al. showed that the CLW scaling solution is stable to shear in expanding GR backgrounds, while in contracting GR backgrounds a canonical scalar is generically unstable to anisotropy (the shear instability), which motivates ekpyrotic phases with wφ≫1.

“Billyard et al. showed that the CLW scaling solution is stable to shear in expanding GR backgrounds [26], while in contracting GR backgrounds a canonical scalar is generically unstable to anisotropy (the shear instability), which motivates ekpyrotic phases with w φ ≫ 1 [27].”

• cites [26] Billyard, A.P., Coley, A.A., Hoogen, R.J.: The stability of cosmological scaling solutions. Class. Quant. Grav. 15, 3063–3079 (1998). arXiv:gr-qc/9805085 sourceNot checked yet

  Not checked in the free run.

• cites [27] Erickson, J.K., Wesley, D.H., Steinhardt, P.J., Turok, N.: Kasner and mixmaster behavior in universes with equation of state $$w \ge 1$$. Phys. Rev. D 69, 063514 (2004). https://doi.org/10.1103/PhysRevD.69.063514. arXiv:hep-th/0312009 sourcePartly supported

  Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured

May combine two claims.

At the background level the bounce is a nonsingular ρ2-type bounce, similar in spirit to the effective Friedmann equation in loop quantum cosmology and to other torsion-based bouncing models.

“At the background level this is a nonsingular ρ 2 -type bounce, similar in spirit to the effective Friedmann equation in loop quantum cosmology [35] and to other torsion-based bouncing models [22,29].”

• cites [35] Ashtekar, A., Singh, P.: Loop quantum cosmology: a status report. Class. Quantum Grav. 28(21), 213001 (2011). https://doi.org/10.1088/0264-9381/28/21/213001. arXiv:1108.0893 sourceSupported
• cites [22] Farnsworth, S., Lehners, J.-L., Qiu, T.: Spinor driven cosmic bounces and their cosmological perturbations. Phys. Rev. D 96(8), 083530 (2017). https://doi.org/10.1103/PhysRevD.96.083530 sourceSupported
• cites [29] Alam, S., Sen, S., Sengupta, S.: Bouncing cosmologies in modified gravity with space time torsion. arXiv e-prints (2025) arXiv:2509.03508 [gr-qc] sourceSupported

May combine two claims.

A steep negative potential with wφ ≫ 1 makes the ekpyrotic regime a strong homogeneous attractor, with Σ ∝ e^{(q-2)N} → 0 along the contracting branch.

“By contrast, as shown in Sec.IV A, a steep negative potential with w φ ≫ 1 makes the ekpyrotic regime a strong homogeneous attractor [27,34]: Σ ∝ e (q-2)N → 0 along the contracting branch.”

• cites [27] Erickson, J.K., Wesley, D.H., Steinhardt, P.J., Turok, N.: Kasner and mixmaster behavior in universes with equation of state $$w \ge 1$$. Phys. Rev. D 69, 063514 (2004). https://doi.org/10.1103/PhysRevD.69.063514. arXiv:hep-th/0312009 sourceSupported
• cites [34] Garfinkle, D., Lim, W.C., Pretorius, F., Steinhardt, P.J.: Evolution to a smooth universe in an ekpyrotic contracting phase with $$w > 1$$. Phys. Rev. D 78(8), 083537 (2008). https://doi.org/10.1103/PhysRevD.78.083537 sourcePartly supported

  The Results (pp. 5–7) link the steep negative exponential potential and w ≫ 1 to smoothing and isotropization during contraction, but report that some regions remain anisotropic. The paper does not state the claimed decay law Σ ∝ e^{(q−2)N}; its displayed formulas are undecoded in our copy. It therefore supports the qualitative behavior, not the specific law or an unqualified attractor claim. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

May combine two claims.

In EC theory, algebraically eliminating torsion gives a quadratic spin correction that enters the effective FLRW background equations with negative sign, so the torsion sector acts as an effective fluid with ρtor=-ρs, ptor=-ρs and stiff equation of state wtor=1.

“In Einstein-Cartan theory, algebraically eliminating torsion gives a quadratic spin correction.With the conventions used here, that correction enters the effective FLRW background equations with negative sign.I denote its magnitude by ρ s > 0 and write the torsion sector as an effective fluid, ρ tor ≡ -ρ s , p tor ≡ -ρ s ,”

• Also stated in: § III. EINSTEIN-CARTAN FIELD EQUATIONS · plain

  “so it has stiff equation of state (w tor = 1) while contributing negative effective density and pressure.”

• cites [11] Hehl, F.W., Heyde, P., Kerlick, G.D., Nester, J.M.: General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys. 48, 393–416 (1976). https://doi.org/10.1103/RevModPhys.48.393 sourceNot checked — no full text of the source
• cites [12] Shapiro, I.L.: Physical aspects of the space-time torsion. Phys. Rep. 357, 113–213 (2002). https://doi.org/10.1016/S0370-1573(01)00030-8 sourcePartly supported

  Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured
• cites [13] Popławski, N.J.: Cosmology with torsion: an alternative to cosmic inflation. Phys. Lett. B 694(3), 181–185 (2010). https://doi.org/10.1016/j.physletb.2010.09.056. Erratum: Phys. Lett. B 701 (2011) 672 sourceSupported

May combine two claims.

In this construction, a single scalar field drives both late-time acceleration and ekpyrotic smoothing at the background level, while the spin-torsion sector supplies the repulsive ρs ∝ a^-6 contribution that regulates the high-density regime, instead of introducing separate ingredients for each phase.

“In this restricted sense the construction is a direct extension of familiar quintessence dynamics [31,32]: a single scalar field drives both late-time acceleration and ekpyrotic smoothing at the background level, while the spin-torsion sector supplies the repulsive ρ s ∝ a -6 contribution that regulates the high-density regime, instead of introducing separate ingredients for each phase.”

• cites [31] Chevallier, M., Polarski, D.: Accelerating universes with scaling dark matter. Int. J. Mod. Phys. D. 10(02), 213–224 (2001). https://doi.org/10.1142/S0218271801000822 sourcePartly supported

  Split into 2 atoms — 0 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 1 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured
• cites [32] Linder, E.V.: Exploring the expansion history of the universe. Phys. Rev. Lett. 90, 091301 (2003). https://doi.org/10.1103/PhysRevLett.90.091301 sourcePartly supported

  Split into 2 atoms — 0 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 1 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured

• AssumptionThe authors state this as an assumption: The spin-torsion contribution can be modeled as an a^-6 component parameterized by α.

May combine two claims.

For a conserved fermion current in an expanding or contracting FLRW background, n ∝ a^-3, so the spin-torsion contribution behaves as a stiff component with ρs ∝ n2 ∝ a^-6.

“For a conserved fermion current one has n ∝ a -3 in an expanding or contracting FLRW background, so the spin-torsion contribution behaves as a stiff component, ρ s ∝ n 2 ∝ a -6 .(14)”

• cites [11] Hehl, F.W., Heyde, P., Kerlick, G.D., Nester, J.M.: General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys. 48, 393–416 (1976). https://doi.org/10.1103/RevModPhys.48.393 sourceNot checked — no full text of the source
• cites [13] Popławski, N.J.: Cosmology with torsion: an alternative to cosmic inflation. Phys. Lett. B 694(3), 181–185 (2010). https://doi.org/10.1016/j.physletb.2010.09.056. Erratum: Phys. Lett. B 701 (2011) 672 sourceSupported
• cites [33] Weyssenhoff, J., Raabe, A.: Relativistic dynamics of spin-fluids and spin-particles. Acta Phys. Polon. 9, 7–19 (1947)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Farnsworth, Lehners, and Qiu constructed a single-field Einstein-Cartan model in which a spinor condensate drives both ekpyrotic contraction and a torsion-induced bounce, and analyzed the associated cosmological perturbations in detail.

“In particular, Farnsworth, Lehners, and Qiu constructed a singlefield Einstein-Cartan model in which a spinor condensate drives both ekpyrotic contraction and a torsion-induced bounce, and analyzed the associated cosmological perturbations in detail [22].”

• cites [22] Farnsworth, S., Lehners, J.-L., Qiu, T.: Spinor driven cosmic bounces and their cosmological perturbations. Phys. Rev. D 96(8), 083530 (2017). https://doi.org/10.1103/PhysRevD.96.083530 sourceSupported

May combine two claims.

In the ekpyrotic regime, the effective shear eigenvalue μeffΣ≃(3/2)(wφ-1) is positive for wφ>1, so along the contracting branch the shear mode decays and the ekpyrotic regime is a strong homogeneous attractor towards isotropy, unlike the marginally stable kinetic points with wφ=1.

“In the fixed-point language it is convenient to encode this as an effective shear eigenvalue µ eff Σ ≃ 3 2 w φ -1 ,(33) which is positive for w φ > 1.Along the contracting branch N → -∞, a mode with Σ ′ = µ eff Σ Σ decays as Σ(N ) ∝ e µ eff Σ N N →-∞ -----→ 0,(34) so the ekpyrotic regime is a strong homogeneous attractor towards isotropy [27], unlike the marginally stable kinetic points with w φ =”

• cites [27] Erickson, J.K., Wesley, D.H., Steinhardt, P.J., Turok, N.: Kasner and mixmaster behavior in universes with equation of state $$w \ge 1$$. Phys. Rev. D 69, 063514 (2004). https://doi.org/10.1103/PhysRevD.69.063514. arXiv:hep-th/0312009 sourceSupported

May combine two claims.

On the positive-potential branch s=+1, when λ2>3γm there is a scaling solution; for dust or radiation γm≥1, μk>0 and μΣ,μs<0, so shear and spin-torsion are damped but curvature grows, and the scaling solution is a saddle in the full six-dimensional phase space.

“For dust or radiation (γ m ≥ 1) one has µ k > 0 and µ Σ , µ s < 0. Shear and spin-torsion are damped, but curvature grows.In the full six-dimensional phase space the scaling solution is a saddle: it is stable in (x, y, Σ, Ω s ) but unstable in the curvature direction [26].”

• cites [26] Billyard, A.P., Coley, A.A., Hoogen, R.J.: The stability of cosmological scaling solutions. Class. Quant. Grav. 15, 3063–3079 (1998). arXiv:gr-qc/9805085 sourcePartly supported

  Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured

May combine two claims.

For λ2>3γm, the scalar tracks the fluid with wφ=wm and Ωφ=3γm/λ2; this scaling point is stable in the (x,y) sector but, once curvature is included, the curvature direction is unstable for γm≥1, so in the full six-dimensional space it is a saddle.

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

• cites [23] Copeland, E.J., Liddle, A.R., Wands, D.: Exponential potentials and cosmological scaling solutions. Phys. Rev. D 57(4), 4686–4690 (1998). https://doi.org/10.1103/PhysRevD.57.4686 sourcePartly supported

  Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured
• cites [24] Heard, I.P.C., Wands, D.: Cosmology with positive and negative exponential potentials. Class. Quant. Grav. 19(21), 5435–5448 (2002). https://doi.org/10.1088/0264-9381/19/21/309. arXiv:gr-qc/0206085 sourcePartly supported

  Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured
• cites [26] Billyard, A.P., Coley, A.A., Hoogen, R.J.: The stability of cosmological scaling solutions. Class. Quant. Grav. 15, 3063–3079 (1998). arXiv:gr-qc/9805085 sourceSupported

May combine two claims.

In any scenario with repeated expand-contract-bounce episodes, integrated quantities (comoving distances, accumulated lensing, etc.) can carry memory of earlier phases, so fitting ΛCDM-based templates to such a background could bias inferred parameters.

“In any scenario with repeated expandcontract-bounce episodes, integrated quantities (comoving distances, accumulated lensing, etc.) can carry memory of earlier phases, so fitting ΛCDM-based templates to such a background could bias inferred parameters [4].”

• cites [4] Abdalla, E., Abellán, G.F., Aboubrahim, A., et al.: Cosmology intertwined: a review of the cosmological tensions and anomalies. J. High Energy Astrophys. 34, 49–211 (2022). https://doi.org/10.1016/j.jheap.2022.04.002 sourcePartly supported

  Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured

May combine two claims.

The construction is complementary to microphysics-focused ECSK spinor models and to spinor-driven ekpyrotic bounce scenarios such as Ref. [22]: the emphasis is on global phase-space structure of a two-component (scalar + spin fluid) system, mapping fixed points, basins of attraction, and Lyapunov stability, while treating parameters such as the Weyssenhoff coefficient α and the softening scale as

“In this way the construction is complementary to microphysics-focused ECSK spinor models and to spinor-driven ekpyrotic bounce scenarios such as Ref. [22]: the emphasis here is on the global phase-space structure of a two-component (scalar + spin fluid) system, mapping fixed points, basins of attraction, and Lyapunov stability, while treating parameters such as the Weyssenhoff coefficient α and”

• cites [22] Farnsworth, S., Lehners, J.-L., Qiu, T.: Spinor driven cosmic bounces and their cosmological perturbations. Phys. Rev. D 96(8), 083530 (2017). https://doi.org/10.1103/PhysRevD.96.083530 sourceSupported

Probably not a substantial claim: may not pass S3 (load-bearing).

Torsion-induced bounces have been realized in Einstein-Cartan and related torsionful models, including spinor-based constructions in which a single Dirac field drives both ekpyrotic contraction and a torsion-supported bounce.

“Torsion-induced bounces have also been realized in Einstein-Cartan and related torsionful models, including spinor-based constructions in which a single Dirac field drives both ekpyrotic contraction and a torsionsupported bounce [11-13, 22, 28, 29].”

• cites [11] Hehl, F.W., Heyde, P., Kerlick, G.D., Nester, J.M.: General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys. 48, 393–416 (1976). https://doi.org/10.1103/RevModPhys.48.393 sourceNot checked — no full text of the source
• cites [12] Shapiro, I.L.: Physical aspects of the space-time torsion. Phys. Rep. 357, 113–213 (2002). https://doi.org/10.1016/S0370-1573(01)00030-8 sourcePartly supported

  Section 2.6 (p. 23) describes an Einstein–Cartan cosmological solution in which torsion prevents a singularity and produces a bounce. It does not describe a single Dirac field driving ekpyrotic contraction as well as the bounce; its example instead assumes a nonzero average spinor current. (LLM, conf=0.88)

  Experimental — accuracy not yet measured
• cites [13] Popławski, N.J.: Cosmology with torsion: an alternative to cosmic inflation. Phys. Lett. B 694(3), 181–185 (2010). https://doi.org/10.1016/j.physletb.2010.09.056. Erratum: Phys. Lett. B 701 (2011) 672 sourcePartly supported

  The Abstract describes an ECKS cosmology in which torsion prevents a singularity and the universe expands from a minimum radius. Sections 2–3 model many fermions as a spin fluid, not a single Dirac field; our copy does not support the attributed ekpyrotic contraction or single-field construction. Page numbers are unavailable in the indexed passages. (LLM, conf=0.88)

  Experimental — accuracy not yet measured
• cites [22] Farnsworth, S., Lehners, J.-L., Qiu, T.: Spinor driven cosmic bounces and their cosmological perturbations. Phys. Rev. D 96(8), 083530 (2017). https://doi.org/10.1103/PhysRevD.96.083530 sourceSupported
• cites [28] Popławski, N.: Big bounce and inflation from spin and torsion. Universe 6(7), 99 (2020). https://doi.org/10.3390/universe6070099 sourceCould not verify

  The Introduction discusses particle injection into relativistic jets, and §4.2 discusses electromagnetic cascades near black holes. Searches for Einstein–Cartan torsion, spinor cosmology, ekpyrotic contraction, and a Dirac-field bounce found no passage addressing the claim in our copy. (LLM, conf=0.76)

  Experimental — accuracy not yet measured
• cites [29] Alam, S., Sen, S., Sengupta, S.: Bouncing cosmologies in modified gravity with space time torsion. arXiv e-prints (2025) arXiv:2509.03508 [gr-qc] sourcePartly supported

  The Abstract and Section 4.2 describe a nonsingular bounce in a generalized Einstein–Cartan model with torsion, supporting the general claim that torsionful models can realize bounces. They describe scalar and torsion fields, however, not a single Dirac field driving ekpyrotic contraction and a bounce; searches for the spinor and ekpyrotic terms did not substantiate that specific construction in this paper. (LLM, conf=0.87)

  Experimental — accuracy not yet measured

May combine two claims.

In the CLW limit, the fixed points and their stability properties coincide with the original CLW analysis: the scalar-field dominated point is stable for λ^2 < 3γ_m and drives accelerated expansion when λ^2 < 2; the scaling solution exists and is stable for λ^2 > 3γ_m; and the kinetic ± points are always unstable.

“In this limit the fixed points and their stability properties coincide with the original CLW analysis.The scalar-field dominated point is stable for λ 2 < 3γ m and drives accelerated expansion when λ 2 < 2; the scaling (tracking) solution exists and is stable for λ 2 > 3γ m ; and the kinetic ± points are always unstable.”

• cites [23] Copeland, E.J., Liddle, A.R., Wands, D.: Exponential potentials and cosmological scaling solutions. Phys. Rev. D 57(4), 4686–4690 (1998). https://doi.org/10.1103/PhysRevD.57.4686 sourceSupported

May combine two claims.

The extended EC model reduces consistently to the CLW system in the flat, isotropic, torsionless case, and the numerics are benchmarked against this baseline.

“confirming that the extended EC model reduces consistently to CLW in the flat, isotropic, torsionless case and that the numerics are benchmarked against this baseline.”

• cites [23] Copeland, E.J., Liddle, A.R., Wands, D.: Exponential potentials and cosmological scaling solutions. Phys. Rev. D 57(4), 4686–4690 (1998). https://doi.org/10.1103/PhysRevD.57.4686 sourcePartly supported

  Section II (p. 1) presents the CLW autonomous system for a spatially flat FRW universe, supporting its identification as the baseline model. The paper does not examine an extended EC model or establish that it reduces to CLW, and it does not report benchmarking the citing authors’ numerics against that baseline. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In many scenarios with a stiff or ekpyrotic phase, the background dynamics can enhance high-frequency stochastic gravitational waves and tilt the tensor spectrum blue.

“Third, in many scenarios with a stiff or ekpyrotic phase, the background dynamics can enhance highfrequency stochastic gravitational waves and tilt the tensor spectrum blue [40].”

• cites [40] Giovannini, M.: Spikes in the relic graviton background from quintessential inflation. Class. Quantum Grav. 16(9), 2905–2913 (1999). https://doi.org/10.1088/0264-9381/16/9/308 sourcePartly supported

  The discussion on p. 2 says that a post-inflationary stiff phase produces relic-graviton spectra that are blue and increase with frequency; the conclusion on p. 8 says the resulting signal can be much larger than after a direct transition to radiation domination. The paper addresses quintessential inflation with a stiff phase, but our copy does not address ekpyrotic phases or establish the claim across many scenarios. (LLM, conf=0.87)

  Experimental — accuracy not yet measured

May combine two claims.

Near the scaling saddle, curvature is a growing mode with μk = 3γm - 2 > 0 for γm ≥ 1, but once the ekpyrotic field dominates it is strongly suppressed.

“Near the scaling saddle, curvature is a growing mode with µ k = 3γ m -2 > 0 for γ m ≥ 1 [26], but once the ekpyrotic field dominates it is strongly suppressed.”

• cites [26] Billyard, A.P., Coley, A.A., Hoogen, R.J.: The stability of cosmological scaling solutions. Class. Quant. Grav. 15, 3063–3079 (1998). arXiv:gr-qc/9805085 sourcePartly supported

  The Bianchi VIIh section (p. 6) says curvature-mode eigenvalues are proportional to 3γ−2 and that the scaling solution is a saddle for realistic matter (γ≥1). The Curved FRW section (p. 6) likewise reports instability to curvature perturbations. Our copy does not establish the claimed exact equality μk=3γm−2; the citation precedes the separate statement about ekpyrotic-field dominance. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

May combine two claims.

The eigenvalues at the scaling point are μ± = -[3(2 - γ_m)/4][1 ± sqrt(1 - 8γ_m(λ^2 - 3γ_m)/(λ^2(2 - γ_m)))], μΣ = (3/2)(γ_m - 2), μk = 3γ_m - 2, and μs = 3(γ_m - 2).

“The two non-trivial eigenvalues in the scalar-matter subspace are [23]”

• Also stated in: § 2. Eigenvalues · plain

  “µ ± = - 3(2 -γ m ) 4 1 ± 1 - 8γ m (λ 2 -3γ m ) λ 2 (2 -γ m )(G3)”

• Also stated in: § 2. Eigenvalues · plain

  “µ Σ = 3 2 (γ m -2), µ k = 3γ m -2, µ s = 3(γ m -2).”

• cites [23] Copeland, E.J., Liddle, A.R., Wands, D.: Exponential potentials and cosmological scaling solutions. Phys. Rev. D 57(4), 4686–4690 (1998). https://doi.org/10.1103/PhysRevD.57.4686 sourcePartly supported

  The Appendix (p. 5) identifies two eigenvalues, m±, for the scaling solution, supporting the citation’s reference to two eigenvalues in the scalar–matter phase plane. Their formula is not decoded in our copy, so the stated expression for μ± cannot be checked. The citation is attached to the sentence about those two eigenvalues, not to the additional μΣ, μk, and μs values. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

• AssumptionThe authors state this as an assumption: For the linear stability analysis, λ is treated as constant, as in an exponential potential or the local value of λ(φ) at a fixed point.

May combine two claims.

In the tuned setup, a steep negative branch of the potential produces an ekpyrotic phase that damps homogeneous shear, and for trajectories in which the spin-torsion sector already dominates the stiff component, the Weyssenhoff fluid can overtake curvature and trigger a torsion-supported bounce below the Planck scale.

“In this tuned setup I show that a steep negative branch of the potential produces an ekpyrotic phase that damps homogeneous shear, and that for trajectories in which the spin-torsion sector already dominates the stiff component, the Weyssenhoff fluid can overtake curvature and trigger a torsion-supported bounce below the Planck scale.”

May combine two claims.

The solution contracts until the total density approaches ρb at a finite scale factor ab, at which point H(t) changes sign while a(t) stays nonzero, realizing a smooth torsion-regularized bounce.

“The solution contracts until the total density approaches ρ b at a finite scale factor a b , at which point H(t) changes sign while a(t) stays nonzero, realizing a smooth torsion-regularized bounce.”

May combine two claims.

Numerical solutions show the ekpyrotic branch (wφ≫1) exponentially damps homogeneous shear, while the softened branch (wφ<1) allows ρs 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.

“Numerical solutions show that the ekpyrotic branch (w φ ≫ 1) exponentially damps homogeneous shear, while the softened branch (w φ < 1) allows ρs 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.”

May combine two claims.

On the positive-potential branch s=+1 and for λ2<6, the scalar-dominated critical point exists; on the expanding branch, λ2<3γm and λ2<2 (with λ2<6) ensure all geometric and matter perturbations decay, so it is a late-time attractor.

“On the positive-potential branch s = +1 and for λ 2 < 6 there is a scalar-dominated critical point x * , y * , z * , Σ * , Ω * k , Ω * s = λ √ 6 , 1 -λ 2 6 , 0, 0, 0, 0 ,(37) with Jacobian spectrum given in Table II.”

• Also stated in: § D. Scalar-field dominated point · plain

  “On the expanding branch, the conditions λ 2 < 3γ m and λ 2 < 2 (together with the existence condition λ 2 < 6) ensure that all geometric and matter perturbations decay, so the point is a late-time attractor.”

Probably not a substantial claim: may not pass S4 (based).

Homogeneous shear contributes an effective stiff component with energy density ρσ=3H2Σ2 and equation of state wσ=1.

“where the homogeneous shear contributes an effective stiff component with energy density ρ σ = 3H 2 Σ 2 , w σ = 1.(7)”

May combine two claims.

Contraction ends when the potential softens so that wφ<1 for a finite interval, allowing the spin-torsion density Ωs∝a^-6 to overtake the scalar density and trigger a nonsingular bounce, for softening parameters in the finite bounce basin identified in Sec.VIII C (Fig. 5).

“Contraction ends when the potential softens so that w φ < 1 for a finite interval, allowing the spin-torsion density Ω s ∝ a -6 to overtake the scalar density and trigger a nonsingular bounce, for softening parameters in the finite bounce basin identified in Sec.VIII C (Fig. 5).”

May combine two claims.

A single scalar field with a softening potential produces scalar-dominated expansion, a fluid-dominated era, ekpyrotic contraction, and a torsion-regulated bounce in one homogeneous setup.

“A single scalar field with a softening potential produces scalar-dominated expansion, a fluiddominated era, ekpyrotic contraction, and a torsionregulated bounce in one homogeneous setup.”

May combine two claims.

In the contracting ekpyrotic phase driven by the steep negative branch, wφ≫1, the deceleration parameter q≃(1+3wφ)/2≫2, and homogeneous shear Σ decays exponentially as the universe contracts toward the bounce.

“In the contracting phase driven by the steep negative branch of the potential, the ekpyrotic scalar dominates and w φ ≫ 1. Neglecting subdominant components, the deceleration parameter is approximately q ≃ 1 2 1 + 3w φ ≫ 2, (26 )”

• Also stated in: § A. Shear suppression · plain

  “Since the universe is contracting (N → -∞) and q -2 > 0, the shear Σ decays exponentially as the universe evolves toward the bounce.”

May combine two claims.

The scale factor reaches a nonzero minimum while H crosses zero smoothly with Ḣ > 0, giving a nonsingular Einstein-Cartan bounce.

“The scale factor reaches a nonzero minimum while H crosses zero smoothly with Ḣ > 0, giving a nonsingular Einstein-Cartan bounce.”

May combine two claims.

In the Fig. 4 benchmark, the Friedmann-constraint residual stays small throughout, and the densities show matter dominates far from the bounce while the spin-torsion term becomes comparable to the total density at tb.

“Panel (b) shows the Friedmann-constraint residual over the same interval together with the component densities: the residual stays small throughout, and the densities illustrate how matter dominates far from the bounce while the spin-torsion term becomes comparable to the total density at t b .”

Probably not a substantial claim: may not pass S3 (load-bearing).

The three-dimensional trajectories in Fig. 3(b) show the same flows in (x, y, z), making the decay of the fluid fraction z explicit as the orbit approaches the scalar attractor.

“The three-dimensional trajectories in Fig. 3(b) show the same flows in (x, y, z), making the decay of the fluid fraction z explicit as the orbit approaches the scalar attractor.”

May combine two claims.

The scaling solution, although attractive in the scalar-fluid subspace, is destabilized by curvature and remains a saddle in the extended Einstein-Cartan phase space.

“The scaling solution, although attractive in the scalar-fluid subspace, is destabilized by curvature and remains a saddle in the extended Einstein-Cartan phase space.”

May combine two claims.

The kinetic-dominated points (±1,0,0,0,0,0) correspond to a stiff fluid wφ=1, have q=2, and two eigenvalues vanish associated with shear and spin-torsion; a purely stiff component neither damps nor amplifies anisotropy and does not by itself solve the anisotropy problem of contracting GR backgrounds.

“There are two kinetic-dominated solutions corresponding to a stiff fluid (w φ = 1): (x, y, z, Σ, Ω k , Ω s ) = (±1, 0, 0, 0, 0, 0). (31 ) At these points q = 2.”

• Also stated in: § A. Kinetic-dominated points · plain

  “Two eigenvalues vanish, associated with shear and spin-torsion (µ Σ = 0 and µ s = 0).A purely stiff component neither damps nor amplifies anisotropy relative to the background (Σ ′ = 0) and does not by itself solve the anisotropy problem of contracting GR backgrounds; addressing that requires an ekpyrotic phase with w φ > 1.”

May combine two claims.

The contraction-to-expansion bounce is demonstrated by direct cosmic-time integration in a spatially flat benchmark run (k=0, Fig. 4), and a separate positively curved run (k=+1, Fig. 1) shows repeated turnaround and bounce crossings in the same homogeneous system.

“The contraction-to-expansion bounce is demonstrated by direct cosmic-time integration in a spatially flat benchmark run (k = 0, Fig. 4), and a separate positively curved run (k = +1, Fig. 1) shows repeated turnaround and bounce crossings in the same homogeneous system.”

May combine two claims.

Scans in a two-parameter softening plane (φb, ∆) identify a finite region of nonsingular trajectories and quantify the required tuning.

“Scans in a two-parameter softening plane (φ b , ∆) 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”

May combine two claims.

The bounce is not a fixed point; it occurs when ρs≃ρtot≡ρb and, provided wtot<1, the EC Friedmann and Raychaudhuri equations allow H to pass through zero with Ḣ>0.

“The bounce itself is not a fixed point; it occurs when the spin density reaches the EC scale at which ρ s ≃ ρ tot ≡ ρ b and, provided w tot < 1, the Einstein-Cartan Friedmann and Raychaudhuri equations allow H to pass through zero with Ḣ > 0.”

May combine two claims.

The softened potential necessarily passes through a zero at some φ=φ0 with V(φ0)=0; at that point λ(φ) diverges, but the combination sλy2 that enters the x′ equation remains finite.

“Because the softened potential (10) interpolates between a negative exponential branch and a positive plateau, it necessarily passes through a zero at some φ = φ 0 with V (φ 0 ) = 0.At that point λ(φ) diverges, but the combination s λy 2 that enters the x ′ equation remains finite, since s λy 2 = - V ,φ3H 2 is regular.”

May combine two claims.

For a pure exponential potential with constant λ, the system (23)-(25) is autonomous in (x,y,z,Σ,Ωk,Ωs) and admits the familiar CLW fixed points.

“For a pure exponential potential with constant λ the system (23)-( 25) is autonomous in (x, y, z, Σ, Ω k , Ω s ) and admits the familiar CLW fixed points.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

For φ≪φb, S→0 and V≃-V0 e^{-λ0φ} (ekpyrotic regime); for φ≫φb, S→1 and V→Vsoft>0, creating a regime with wφ<1 in which the EC spin-torsion sector can dominate and produce a bounce.

“For φ ≪ φ b one has S → 0 and V ≃ -V 0 e -λ0φ (the ekpyrotic regime); for φ ≫ φ b one has S → 1 and V → V soft > 0, creating a regime with w φ < 1 in which the EC spintorsion sector can dominate and produce a bounce.”

May combine two claims.

For standard fluids 0<γm≤2, the fluid-dominated point is not a late-time attractor on the expanding branch; at least one scalar or curvature direction has a positive eigenvalue (for dust or radiation, μk>0), so it is a saddle in the full six-dimensional phase space.

“For standard fluids 0 < γ m ≤ 2 this point is not a late-time attractor on the expanding branch: at least one of the scalar or curvature directions has a positive eigenvalue (for dust or radiation, µ k > 0), so the fluid point is a saddle in the full six-dimensional phase space.”

May combine two claims.

In the N-time formulation, the shear mode has an effective eigenvalue μeffΣ ≃ 3/2(wφ - 1) > 0, so the ekpyrotic regime is attractive as N → -∞ and homogeneous shear decays exponentially along the contracting branch.

“Its homogeneous stability is already fixed by the linear analysis: as shown in Secs.IV A and VI, the shear mode has an effective eigenvalue µ eff Σ ≃ 3 2 (w φ -1) > 0 in the N -time formulation, so the ekpyrotic regime is attractive as N → -∞ and homogeneous shear decays exponentially along the contracting branch.”

May combine two claims.

In the extended ECEM phase space, λ2<6 ensures that anisotropy and spin-torsion perturbations decay, while curvature perturbations decay only in the accelerated regime λ2<2.

“In the extended Einstein-Cartan ekpyrotic model (ECEM) phase space, λ 2 < 6 ensures that anisotropy and spin-torsion perturbations decay, while curvature perturbations decay only in the accelerated regime λ 2 < 2 (see Sec. VI).”

Classified as probably not a substantial claim.

Section VIII C and Fig. 5 make precise which softening parameters (φb, ∆) lead to a torsion-induced bounce rather than a singular crunch in the high-density contracting leg.

“Section VIII C and Fig. 5 make precise which softening parameters (φ b , ∆) lead to a torsion-induced bounce rather than a singular crunch in the high-density contracting leg.”

May combine two claims.

Purely kinetic-dominated solutions with wφ = 1 behave as stiff fluids and leave the shear eigenvalue μΣ = 0, so Σ is only marginally stable.

“Purely kinetic-dominated solutions with w φ = 1 behave as stiff fluids and leave the shear eigenvalue µ Σ = 0, so Σ is only marginally stable.”

May combine two claims.

Both shear and spin-torsion contributions behave as stiff components, ρσ ∝ a^-6 and ρs ∝ a^-6, so ρσ/ρs = const.

“Both shear and spin-torsion contributions behave as stiff components, ρ σ ∝ a -6 , ρ s ∝ a -6 , ⇒ ρ σ ρ s = const.”

May combine two claims. The quote only partly states this claim.

The scalar-field dominated fixed point exists for λ^2 < 6 and has coordinates (λ/√6, sqrt(1 - λ^2/6), 0, 0, 0, 0), with q* = λ^2/2 - 1.

“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)”

• Also stated in: § Additional information · plain

  “q * = λ 2 2 -1, 1 + q * = λ 2 2 ,(F2)”

• AssumptionThe authors state this as an assumption: For the linear stability analysis, λ is treated as constant, as in an exponential potential or the local value of λ(φ) at a fixed point.

May combine two claims.

This appears as a single event in the k = 0 benchmark and as a repeated event in the k = +1 cyclic trajectory.

“This appears as a single event in the k = 0 benchmark and as a repeated event in the k = +1 cyclic trajectory.”

May combine two claims.

Figure 6 shows that for (λ, γ_m) = (1.20, 1.00) and Set (a), the orbit converges to the scalar-field dominated node with (x*, y*) ≈ (0.49, 0.87), geometric variables decay, and w_eff ≈ -0.52, realizing a late-time accelerating attractor.

“The orbit converges to the scalar-field dominated node with (x * , y * ) = λ √ 6 , 1 -λ 2 6 ≈ (0.49, 0.87), and the geometric variables decay, z, Ω k , Σ, Ω s → 0.”

• Also stated in: § Choice of seeds. · plain

  “The effective equation of state approaches w eff ≃ λ 2 /3 -1 ≈ -0.52, corresponding to a late-time accelerating attractor on the expanding branch.”

May combine two claims.

Evaluating the 6×6 Jacobian in the extended system and restricting to (x, y, z) reproduces the CLW eigenvalues; in the CLW limit (Σ, Ω_k, Ω_s) = 0 the full 6×6 Jacobian is block-upper-triangular, so the extra geometric directions do not affect the (x, y, z) eigenvalues.

“Evaluating the 6 × 6 Jacobian in the extended system and then restricting to (x, y, z) reproduces the CLW eigenvalues; in the CLW limit (Σ, Ω k , Ω s ) = 0 the full 6 × 6 Jacobian is block-upper-triangular, so the extra geometric directions do not affect the (x, y, z) eigenvalues.”

May combine two claims.

For approximately constant equation of state w, the curvature variable obeys Ω'k = 2q Ωk = (1 + 3w) Ωk; thus Ωk = 0 is an invariant manifold of the normalized system, and a spatially flat (k = 0) trajectory cannot dynamically develop curvature or recollapse.

“For approximately constant equation of state w, the curvature variable obeys Ω ′ k = 2q Ω k = (1 + 3w) Ω k . (48 ) Thus Ω k = 0 is an invariant manifold of the normalized system: a spatially flat (k = 0) trajectory cannot dynamically develop curvature or recollapse, so the expansion-to-contraction turnaround requires the closed branch (k = +1).”

May combine two claims.

The trajectories in Fig. 3 and the bounce solution in Fig. 4 are consistent with the picture that the expanding branch approaches the scalar attractor with z → 0, while the bounce occurs in a regime where the spin sector is comparable to the total density, the shear fraction remains small throughout, and ρσ/ρs stays approximately constant.

“The trajectories in Fig. 3 and the bounce solution in Fig. 4 are consistent with this picture: the expanding branch approaches the scalar attractor with z → 0, while the bounce occurs in a regime where the spin sector is comparable to the total density, the shear fraction remains small throughout, and ρ σ /ρ s stays approximately constant.”

May combine two claims.

In the 20×20 grid scan, there are 176 bouncing trajectories, 224 crunch trajectories, and 0 undecided trajectories.

“On the 20 × 20 grid shown here (with the same run settings), the scan yields 176 bouncing trajectories, 224 crunch trajectories, and 0 undecided trajectories.”

May combine two claims.

Each fixed point or asymptotic regime has the interpretation already summarized in Secs. V and VII: a scalar-dominated phase with wφ = λ^2/3 - 1 (accelerated for λ^2 < 2), fluid-dominated saddles corresponding to the usual matter and radiation eras, and an ekpyrotic contracting regime that suppresses shear anisotropy.

“Each fixed point or asymptotic regime has the interpretation already summarized in Secs.V and VII: a scalar-dominated phase with w φ = λ 2 /3 -1 (accelerated for λ 2 < 2), fluid-dominated saddles corresponding to the usual matter and radiation eras, and an ekpyrotic contracting regime that suppresses shear anisotropy (Sec.IV A).”

Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

The Jacobian of the autonomous system in canonical variables (x, y, z, Σ, Ω_k, Ω_s) is given by the matrix collected in Appendix C.

“Collecting all entries, the Jacobian in the canonical variables (x, y, z, Σ, Ω k , Ω s ) is”

• AssumptionThe authors state this as an assumption: For the linear stability analysis, λ is treated as constant, as in an exponential potential or the local value of λ(φ) at a fixed point.

May combine two claims.

Figure 7 shows for Set (b), (λ, γ_m) = (3.50, 1.00), the flow spirals towards the matter-scaling point (x, y) ≈ (0.35, 0.35) with z ≈ 0.75, and curvature Ω_k grows because it redshifts more slowly than matter, eventually dominating and driving the system away from the flat scaling regime.

“The flow spirals towards the matter-scaling point (x, y) ≈ (0.35, 0.35) with z ≈ 0.75.(b) For a representative trajectory, curvature Ω k grows because it redshifts more slowly than matter, so Ω k eventually dominates and the system is driven away from the flat scaling regime.”

May combine two claims.

Figure 1 provides an explicit homogeneous expansion→contraction→bounce trajectory in cosmic time for a representative k = +1 run, with alternating turnarounds (H = 0, Ḣ < 0) and torsion-induced bounces (H = 0, Ḣ > 0); in this example a(t) is bounded and oscillatory rather than showing net cycle-to-cycle growth.

“This provides an explicit homogeneous expansion→contraction→bounce trajectory in cosmic time; in this example a(t) is bounded and oscillatory rather than showing net cycle-to-cycle growth.”

May combine two claims.

The scaling (tracking) fixed point exists for 0 < γ_m < 2 and λ^2 > 3γ_m, where the scalar tracks the background fluid (wφ → wm = γ_m - 1 and w_eff = wm).

“and exists for 0 < γ m < 2 and λ 2 > 3γ m , where the scalar tracks the background fluid (w φ → w m = γ m -1 and w eff = w m ).”

• AssumptionThe authors state this as an assumption: For the linear stability analysis, λ is treated as constant, as in an exponential potential or the local value of λ(φ) at a fixed point.

May combine two claims.

The torsion-regulated bounce lies in the V < 0 branch and is not part of the constant-λ slice.

“The torsion-regulated bounce lies in the V < 0 branch and is not part of this constantλ slice.”

May combine two claims.

In numerical experiments, failure to bounce typically occurs in two limiting ways: if the softening transition happens too far above the torsion scale, the spin sector never catches up and the model simply crunches; if it happens too far below, the system has already exited the ekpyrotic attractor and evolves toward a scalar-dominated singularity.

“In numerical experiments this failure typically occurs in two limiting ways: if the softening transition happens too far above the torsion scale, the spin sector never catches up and the model simply crunches; if it happens too far below, the system has already exited the ekpyrotic attractor and evolves toward a scalar-dominated singularity.”

May combine two claims.

The fluid fixed point has coordinates (0, 0, 1, 0, 0, 0), q* = (3γ_m - 2)/2, and eigenvalues {3/2(γ_m - 2), 3/2 γ_m, 3γ_m, (γ_m - 2), 3γ_m - 2, 3(γ_m - 2)}.

“The fluid fixed point in Sec.VI has (x * , y * , z * , Σ * , Ω k * , Ω s * ) = (0, 0, 1, 0, 0, 0), (E1)”

• Also stated in: § Additional information · plain

  “Collecting these, the full spectrum at the fluid point is{µ} = 3 2 (γ m -2), 3 2 γ m , 3γ m , (γ m -2), 3γ m -2, 3(γ m -2) ,”

• AssumptionThe authors state this as an assumption: For the linear stability analysis, λ is treated as constant, as in an exponential potential or the local value of λ(φ) at a fixed point.

Probably not a substantial claim: may not pass S3 (load-bearing).

The evolution equations for (x, y, z, Σ, Ω_k, Ω_s) in Appendix C are equivalent to Eqs. (23)-(25) in the main text.

“These are equivalent to Eqs. ( 23)-(25) in the main text; I retain the compact q-form because it makes the structure of the dynamics and the stability analysis more transparent.”

May combine two claims.

At the scaling fixed point, q* = 3γ_m/2 - 1, consistent with q* = (1/2)(1 + 3w_m).

“Inserting these coordinates into Eq.( 22) gives”

• Also stated in: § 3. Spectrum and stability · plain

  “1 + q * = 3γ m 2 , q * = 3γ m 2 -1,(G2)”

• Also stated in: § 3. Spectrum and stability · plain

  “consistent with q * = 1 2 (1 + 3w m ).”

• AssumptionThe authors state this as an assumption: For the linear stability analysis, λ is treated as constant, as in an exponential potential or the local value of λ(φ) at a fixed point.

May combine two claims.

The kinetic-dominated fixed points have coordinates (±1, 0, 0, 0, 0, 0), q* = 2, and eigenvalues {6, 3∓(3/2)λ, 3(2-γ_m), 0, 4, 0}.

“The kinetic-dominated fixed points listed in Sec.VI have (x * , y * , z * , Σ * , Ω k * , Ω s * ) = (±1, 0, 0, 0, 0, 0) and q * = 2 from Eq. ( 22).”

• Also stated in: § Additional information · plain

  “The spectrum can be written compactly as {µ} = 6, 3 ∓ 3 2 λ, 3(2γ m ), 0, 4, 0 .(D7)”

• AssumptionThe authors state this as an assumption: For the linear stability analysis, λ is treated as constant, as in an exponential potential or the local value of λ(φ) at a fixed point.

May combine two claims.

During ekpyrotic contraction with wφ ≃ w_ek > 1, ρφ ∝ a^{-3(1+w_ek)}, so that ρσ/ρφ(N_soft) ∼ (ρσ/ρφ)_in exp[-3(w_ek - 1)N_ek].

“During ekpyrotic contraction with w φ ≃ w ek > 1 one has ρ φ ∝ a -3(1+w ek ) , so that ρ σ ρ φ (N soft ) ∼ ρ σ ρ φ in exp -3 w ek -1 N ek ,”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The compact canonical form of the deceleration parameter is q = 3x^2 - 1 + (3/2)γ_m z + 3Σ^2 + Ω_k - 3Ω_s, coinciding with Eq. (22) used in the main text.

“gives the compact canonical form”

• Also stated in: § A. Future work and outlook · plain

  “q = 3x 2 -1 + 3 2 γ m z + 3Σ 2 + Ω k -3Ω s .(B5)”

• Also stated in: § A. Future work and outlook · plain

  “This coincides with the canonical expression ( 22) used in the main text.”

May combine two claims.

Figure 9 shows for Set (d), (λ, γ_m) = (1.20, 1.00) with larger initial curvature Ω_k0 = -5×10^-3, all trajectories shown converge to the same scalar-field node as Set (a), and curvature, shear, and spin-torsion all decay rapidly, with no curvature-driven deviation or bounce observed on the expanding branch.

“(a) Despite the higher initial |Ω k |, all trajectories shown converge to the same scalar-field node as in Set (a).(b) For the representative run, curvature, shear, and spin-torsion all decay rapidly, and no curvature-driven deviation or bounce is observed on the expanding branch.”

May combine two claims.

Figure 10 shows for Set (e), (λ, γ_m) = (1.20, 1.00) with larger initial spin-torsion Ω_s0 = 5×10^-2, the phase portrait remains qualitatively identical to Set (a), trajectories still converge to the scalar-field dominated node, and Ω_s decays rapidly, too quickly to produce a visible deviation from the scalar trajectory on this scale; no bounce phase appears in this forward-N evolution.

“(a) The phase portrait remains qualitatively identical to Set (a): trajectories still converge to the scalar-field dominated node.(b) The spin contribution Ωs is initially non-zero but decays rapidly (since ρs ∝ a -6 ), too quickly to produce a visible deviation from the scalar trajectory on this scale.No bounce phase appears in this forward-N evolution.”

May combine two claims.

In the ekpyrotic regime, the effective shear eigenvalue satisfies μ_effΣ ≈ (3/2)(wφ - 1) > 0, so on the contracting branch (N → -∞) Σ ∝ e^{μ_effΣ N} → 0.

“In that regime the effective shear eigenvalue satisfies”

• Also stated in: § Additional information · plain

  “µ eff Σ ≈ 3 2 (w φ -1) > 0,(D9)”

• Also stated in: § Additional information · plain

  “so on the contracting branch (N → -∞) one has Σ ∝ e µ eff Σ N → 0.”

May combine two claims.

Figure 8 shows for Set (c), (λ, γ_m) = (3.50, 4/3), convergence towards the radiation-scaling point (x, y) ≈ (0.47, 0.33) with z ≈ 0.67, and curvature Ω_k is unstable at the scaling saddle and eventually dominates at late times, driving the system away from the flat tracking regime; no contracting branch or bounce behavior is visible in the N > 0 evolution.

“(a) Phase portrait in (x, y) showing convergence towards the radiation-scaling point (x, y) ≈ (0.47, 0.33) with z ≈ 0.67.(b) As in the dust case, curvature Ω k is unstable at the scaling saddle and eventually dominates at late times, driving the system away from the flat tracking regime.For these parameters no contracting branch or bounce behavior is visible in the N > 0 evolution.”

May combine two claims.

Figure 5 shows that on the 20×20 grid, the scan yields 176 bouncing trajectories, 224 crunch trajectories, and 0 undecided trajectories; the numerically determined boundary lies close to the contour R_b ≃ 1, with parameter combinations on and above the curve satisfying the softening condition and realizing a torsion-dominated bounce, while points below do not achieve a softening-spin bounce and,yp

“On the 20 × 20 grid shown here (with the same run settings), the scan yields 176 bouncing trajectories, 224 crunch trajectories, and 0 undecided trajectories.The solid curve marks the numerically determined boundary between bouncing and non-bouncing trajectories and lies close to the contour R b ≃ 1 suggested by (56): parameter combinations on and above the curve satisfy the softening condition”

May combine two claims.

In the reported numerical runs, changing the seed amplitudes mainly affects visual prominence and transient duration of the geometric sectors, while the fixed-point structure and the curvature-instability trend of the scaling regime remain the same.

“In the runs reported here, changing these seed amplitudes mainly affects visual prominence and transient duration of the geometric sectors, while the fixedpoint structure and the curvature-instability trend of the scaling regime remain the same.”

Classified as probably not a substantial claim. May combine two claims.

The ingredients are standard: ekpyrotic contraction to suppress homogeneous shear, and EC torsion to regularize high densities.

“The ingredients are standard: ekpyrotic contraction to suppress homogeneous shear, and EC torsion to regularize high densities.”

May combine two claims.

In a contracting universe, if wφ<1, the scalar energy density scales more slowly than the spin-torsion density ρs∝a^-6, so the repulsive spin term can eventually overtake the scalar and enable a bounce.

“If w φ < 1, then ρ φ scales more slowly than ρ s , so in a contracting universe (a → 0) the repulsive spin term can eventually overtake the scalar and enable a bounce.”

May combine two claims.

A relativistic fermion bath in equilibrium at T ∼ E_b has n_th ∼ g* T^3, giving n_th ∼ 10^50 GeV^3 for g* ∼ 10^2; the required spin density is of the same order as the thermal expectation, consistent with a mildly overdense or degenerate fermion fluid rather than a trans-Planckian state.

“A relativistic fermion bath in equilibrium at T ∼ E b has n th ∼ g * T 3 , giving n th ∼ 10 50 GeV 3 for g * ∼ 10 2 .The required spin density is of the same order as the thermal expectation, consistent with a mildly overdense or degenerate fermion fluid rather than a trans-Planckian state.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

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.

“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.”

May combine two claims.

At the scalar-field-dominated point, the eigenvalue λ2 is orthogonal to the Friedmann constraint surface and is excluded from the physical stability spectrum; the stability classification is based only on the remaining eigenvalues tangent to the constraint surface.

“At the scalar-field-dominated point, the eigenvalue λ 2 is orthogonal to the Friedmann constraint surface and is not part of the physical stability spectrum on the constrained manifold; the stability classification is based only on the remaining eigenvalues tangent to the constraint surface.”

May combine two claims.

The conclusion of no evidence for chaotic mixing is restricted to the homogeneous sector; BKL-type chaotic behavior may still arise once spatial inhomogeneities are included.

“This conclusion is restricted to the homogeneous sector; BKLtype chaotic behavior may still arise once spatial inhomogeneities are included.”

May combine two claims.

If the ekpyrotic phase with wφ≫1 persisted to the equality point ρtot=ρs, then Ḣ<0 there and the EC spin-torsion term would delay rather than remove a curvature singularity.

“If the ekpyrotic phase with w φ ≫ 1 were to persist all the way to the equality point ρ tot = ρ s , one would have Ḣ < 0 there and the EC spin-torsion term would simply delay, rather than remove, a curvature singularity.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For the scaling point, only the five physical eigenvalues tangent to the constraint surface are listed; the sixth, orthogonal eigenvalue is omitted.

“For the scaling point, only the five physical eigenvalues tangent to the constraint surface are listed; the sixth, orthogonal eigenvalue is omitted.”

May combine two claims.

The early positive bump in the (3.5, 1.0) run 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).

“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.”

May combine two claims.

Because Ωk = 0 is an invariant manifold of the normalized system (Ω'k = 2q Ωk), the spatially flat (k = 0) benchmark cannot recollapse; expansion-to-contraction turnaround requires the closed (k = +1) branch.

“Because Ω k = 0 is an invariant manifold of the normalized system (Ω ′ k = 2q Ω k ), this flat benchmark cannot recollapse; expansion-to-contraction turnaround is shown separately in the closed (k = +1) trajectory of Fig. 1.”

May combine two claims.

In the model, the softening branch of the potential is a kinematic requirement: it is chosen so that wφ<1 near the would-be equality scale, driving wtot<1 and allowing the spin-torsion term to generate a nonsingular bounce.

“In the model studied here the softening branch of the potential is a kinematic requirement: it is chosen so that w φ < 1 near the would-be equality scale, driving w tot < 1 and allowing the spin-torsion term to generate a nonsingular bounce.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The spin-torsion contribution is modeled as an a^-6 component parameterized by α, and this phenomenological setup is not claimed to capture the full microscopic content of Einstein-Cartan theory or to describe generic initial data.

“Throughout I model the spin-torsion contribution as an a -6 component parameterized by α and I do not claim that this phenomenological setup captures the full microscopic content of Einstein-Cartan theory or describes generic initial data.”

• AssumptionThe authors state this as an assumption: The spin-torsion contribution can be modeled as an a^-6 component parameterized by α.

May combine two claims.

The present work does not introduce a new microscopic bounce mechanism; its novelty lies at the level of the homogeneous dynamical system.

“The present work does not introduce a new microscopic bounce mechanism.The novelty lies at the level of the homogeneous dynamical system:”

May combine two claims.

For k=+1, the curvature term tightens the bounce condition; wtot<1 remains necessary but is not in general sufficient.

“For k = +1 the curvature term tightens the bounce condition; w tot < 1 remains necessary but is not in general sufficient.”

May combine two claims.

In this construction, the spin-torsion contribution regulates high densities and connects scalar-dominated expansion, a fluid era, ekpyrotic contraction, and a torsion-induced bounce in a kinematically nonsingular model.

“In this construction, the spin-torsion contribution regulates high densities and connects scalar-dominated expansion, a fluid era, ekpyrotic contraction, and a torsion-induced bounce in a kinematically nonsingular model.”

• AssumptionThe authors state this as an assumption: The spin-torsion contribution can be modeled as an a^-6 component parameterized by α.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Microscopic ECSK realizations that enforce a specific algebraic relation ρs(ρm) (for example Weyssenhoff dust or radiation with ρs ∝ ρm^2) correspond to lower-dimensional invariant submanifolds of this extended phase space; some such choices may shrink or even eliminate the bounce basin shown in Fig. 5.

“Microscopic ECSK realizations that enforce a specific algebraic relation ρ s (ρ m ) (for example Weyssenhoff dust or radiation with ρ s ∝ ρ 2 m ) correspond to lower-dimensional invariant submanifolds of this extended phase space; some such choices may shrink or even eliminate the bounce basin shown in Fig. 5, and a systematic survey of these submanifolds is left for future work.”

May combine two claims.

The ECEM construction demonstrates both a directly integrated spatially flat (k = 0) torsion bounce benchmark (Fig. 4) and a directly integrated closed (k = +1) homogeneous trajectory with repeated turnaround and bounce crossings (Fig. 1).

“In this restricted sense the ECEM construction now demonstrates both: (i) a directly integrated spatially flat (k = 0) torsion bounce benchmark (Fig. 4), and (ii) a directly integrated closed (k = +1) homogeneous trajectory with repeated turnaround and bounce crossings (Fig. 1).”

May combine two claims.

At the scaling point, shear and spin are non-growing (decaying for γ_m < 2), while curvature has eigenvalue μk = 3γ_m - 2 that becomes positive for standard fluids (γ_m ≥ 1).

“Thus shear and spin are non-growing (indeed decaying for γ m < 2), while curvature has a distinct eigenvalue µ k = 3γ m -2 that becomes positive for standard fluids (γ m ≥ 1).”

• AssumptionThe authors state this as an assumption: For the linear stability analysis, λ is treated as constant, as in an exponential potential or the local value of λ(φ) at a fixed point.

Classified as probably not a substantial claim.

The potential is chosen so that the transition to the softened plateau occurs near the density at which the EC spin term would otherwise enforce a bounce.

“The potential is chosen so that the transition to the softened plateau occurs near the density at which the EC spin term would otherwise enforce a bounce.”

• AssumptionThe authors state this as an assumption: The softening scale V_soft and the spin-torsion parameter α are aligned by tuning at the background level.

May combine two claims.

Within this class of solutions, the isotropic manifold is a local attractor during ekpyrotic contraction and remains dynamically stable through the torsion-induced bounce at the homogeneous level.

“Within this class of solutions the isotropic manifold is a local attractor during ekpyrotic contraction and remains dynamically stable through the torsion-induced bounce at the homogeneous level.”

May combine two claims.

At the homogeneous level considered here, the Einstein-Cartan bounce is nonsingular and conservative: it modifies the high-density Friedmann equations via the spin-torsion term but does not introduce explicit dissipation or entropy sinks.

“At the homogeneous level considered here, the Einstein-Cartan bounce is nonsingular and conservative: it modifies the high-density Friedmann equations via the spin-torsion term but does not introduce explicit dissipation or entropy sinks.”

Classified as probably not a substantial claim.

In the present construction this is achieved by tuning (φb, ∆, V_soft) rather than deriving these parameters from a microscopic spinor model.

“In the present construction this is achieved by tuning (φ b , ∆, V soft ) rather than deriving these parameters from a microscopic spinor model.”

• AssumptionThe authors state this as an assumption: The softening scale V_soft and the spin-torsion parameter α are aligned by tuning at the background level.

Classified as probably not a substantial claim.

The ECEM background solutions shown here should be viewed as homogeneous templates (including both a k = 0 single-bounce benchmark and a k = +1 cyclic benchmark), not as a complete thermodynamic model of an eternal cyclic universe.

“The ECEM background solutions shown here should be viewed as homogeneous templates (including both a k = 0 single-bounce benchmark and a k = +1 cyclic benchmark), not as a complete thermodynamic model of an eternal cyclic universe.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

While the baseline ECEM evolution drives large-scale shear to very small values before the bounce, small residual anisotropies and curvature patterns could still imprint the CMB low multipoles or galaxy spin statistics.

“Second, while the baseline ECEM evolution drives large-scale shear to very small values before the bounce, small residual anisotropies and curvature patterns could still imprint the CMB low multipoles or galaxy spin statistics.”

May combine two claims.

If the bounce inequality holds, the spin term becomes comparable to the scalar term, |ρs| ∼ ρφ, and offsets the scalar contribution in the Friedmann constraint after the system leaves the pure ekpyrotic attractor, leading to a nonsingular bounce.

“If (56) holds, the spin term becomes comparable to the scalar term, |ρ s | ∼ ρ φ , and offsets the scalar contribution in the Friedmann constraint after the system leaves the pure ekpyrotic attractor, leading to a nonsingular bounce.”

May combine two claims.

Consequently, the bounce is only safe from shear domination if the initial stiff sector is already spin-dominated, ρσ ≪ ρs; in that case the hierarchy is preserved throughout contraction.

“Consequently, the bounce is only safe from shear domination if the initial stiff sector is already spin-dominated, ρ σ ≪ ρ s ; in that case the hierarchy is preserved throughout contraction.”

• AssumptionThe authors state this as an assumption: The initial stiff sector is spin-dominated, ρσ ≪ ρs, for the class of bouncing solutions considered.

May combine two claims.

In the ECEM system, the torsion-induced bounce occurs when the spin sector grows large enough to offset the scalar contribution in the Friedmann constraint; since matter and shear are subdominant in this regime, the bounce condition reduces to ρs ≈ ρφ, i.e. R_b ≈ 1.

“In the ECEM system the torsion-induced bounce occurs when the spin sector grows large enough to offset the scalar contribution in the Friedmann constraint; since matter and shear are subdominant in this regime, the bounce condition reduces to ρ s ≈ ρ φ , i.e.R b ≈ 1.”

May combine two claims.

The scalar-field dominated point is a late-time attractor with decaying matter, shear, torsion and curvature provided λ^2 < 3γ_m and, in particular, λ^2 < 2 for curvature stability and accelerated expansion.

“Thus the scalar-field dominated point is a late-time attractor with decaying matter, shear, torsion and curvature provided λ 2 < 3γ m and, in particular, λ 2 < 2 for curvature stability and accelerated expansion.”

• AssumptionThe authors state this as an assumption: For the linear stability analysis, λ is treated as constant, as in an exponential potential or the local value of λ(φ) at a fixed point.

May combine two claims.

In the ECEM background, the ekpyrotic phase does not generate a hierarchy between shear and spin: any hierarchy must be present in the initial data and is then preserved as the universe contracts.

“In the ECEM background, the ekpyrotic phase does not generate a hierarchy between shear and spin: any hierarchy must be present in the initial data and is then preserved as the universe contracts.”

May combine two claims.

A single scalar field with an appropriately chosen softening potential both drives expansion and implements ekpyrotic smoothing, while the spin-torsion sector supplies the repulsive ρs ∝ a^-6 component that triggers the bounce.

“A single scalar field with an appropriately chosen softening potential both drives expansion and implements ekpyrotic smoothing, while the spin-torsion sector supplies the repulsive ρ s ∝ a -6 component that triggers the bounce.”

• AssumptionThe authors state this as an assumption: The spin-torsion contribution can be modeled as an a^-6 component parameterized by α.

May combine two claims.

In the ekpyrotic regime with wφ > 1, the scalar redshifts faster than a^-6 and the shear fraction is dynamically driven to zero.

“By contrast, in the ekpyrotic regime with w φ > 1 the scalar redshifts faster than a -6 and the shear fraction is dynamically driven to zero (Sec.IV A).”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The overall bounce scale and initial spin density remain free parameters of the effective description and could in principle be constrained once a concrete perturbation framework and data analysis are specified.

“The overall bounce scale and initial spin density remain free parameters of the effective description and could in principle be constrained once a concrete perturbation framework and data analysis are specified.”

May combine two claims.

For the range of initial conditions explored here (with shear and curvature within observational bounds), the trajectories explored numerically remain regular and show no indication of chaotic mixing at the homogeneous level.

“For the range of initial conditions explored here (with shear and curvature within observational bounds), the trajectories explored numerically remain regular and show no indication of chaotic mixing at the homogeneous level, and the bounce-basin analysis of Sec.VIII C shows that these trajectories occupy a tuned but finite region of softening-parameter space rather than an isolated point.”

May combine two claims.

The zero eigenvalues for shear and spin-torsion at the kinetic fixed points show that these points are only marginally stable in those directions; a pure stiff fluid does not dynamically suppress anisotropy or spin relative to the background energy density.

“show that the kinetic fixed points (w φ = 1) are only marginally stable in these directions: a pure stiff fluid does not dynamically suppress anisotropy or spin relative to the background energy density, so the homogeneous shear instability is not cured by the stiff phase alone.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

The present work addresses the homogeneous (Bianchi-type) limit, in which controlling the shear sector is a necessary but not sufficient condition for taming the full BKL dynamics.

“The present work addresses the homogeneous (Bianchi-type) limit, in which controlling the shear sector is a necessary but not sufficient condition for taming the full BKL dynamics.”

Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

If one attempts to embed this type of background in a truly cyclic cosmology, a viable construction would require a microphysical scenario that tracks entropy flow cycle by cycle.

“If one attempts to embed this type of background in a truly cyclic cosmology, a viable construction would require a microphysical scenario that tracks entropy flow cycle by cycle.”

May combine two claims.

In the parameter choices used here, the bounce occurs when the spin sector becomes comparable to the scalar, ρs ∼ ρφ; therefore, if the initial data satisfy ρσ ≪ ρs in the stiff sector and the ekpyrotic phase lasts long enough that ρσ ≪ ρφ by the onset of softening, then shear remains subdominant to both dominant components at the bounce.

“In the parameter choices used here, the bounce occurs when the spin sector becomes comparable to the scalar, ρ s ∼ ρ φ .Therefore, if the initial data satisfy ρ σ ≪ ρ s in the stiff sector and the ekpyrotic phase lasts long enough that ρ σ ≪ ρ φ by the onset of softening, then shear remains subdominant to both dominant components at the bounce.”

• AssumptionThe authors state this as an assumption: The initial stiff sector is spin-dominated, ρσ ≪ ρs, for the class of bouncing solutions considered.

May combine two claims.

A steep negative exponential potential generates an ekpyrotic attractor that exponentially suppresses shear anisotropy, so homogeneous shear decays during the contracting phase within the extended Einstein-Cartan framework.

“Thus a steep negative exponential potential generates an ekpyrotic attractor that exponentially suppresses shear anisotropy, so homogeneous shear decays during the contracting phase within the extended Einstein-Cartan framework.”

May combine two claims.

The Einstein-Cartan bounce removes curvature singularities but does not by itself regulate the total entropy budget.

“This removes curvature singularities but does not by itself regulate the total entropy budget.”

May combine two claims.

The softening phase with wφ < 1 is the window in which ρs ∝ a^-6 overtakes ρφ, allowing H → 0 and Ḣ > 0 at a finite ab and producing the nonsingular bounce.

“The softening phase with w φ < 1 is the window in which ρ s ∝ a -6 overtakes ρ φ , allowing H → 0 and Ḣ > 0 at a finite a b and producing the nonsingular bounce.”