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Cited by 180Cosmology and Gravitation TheoriesRelativity and Gravitational TheoryNoncommutative and Quantum Gravity Theories

Nonsingular, big-bounce cosmology from spinor-torsion coupling

Open access
Popławski, Nikodem·2012·Physical Review D
DOI 10.1103/physrevd.85.107502
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Abstract

The Einstein-Cartan-Sciama-Kibble theory of gravity removes the constraint of general relativity that the affine connection be symmetric by regarding its antisymmetric part, the torsion tensor, as a dynamical variable. The minimal coupling between the torsion tensor and Dirac spinors generates a spin-spin interaction which is significant in fermionic matter at extremely high densities. We show that such an interaction averts the unphysical big-bang singularity, replacing it with a cusp-like bounce at a finite minimum scale factor, before which the Universe was contracting. This scenario also explains why the present Universe at largest scales appears spatially flat, homogeneous and isotropic.

AI snapshot

Written from the paper’s full text.

TL;DR

In Einstein-Cartan-Sciama-Kibble gravity, minimal coupling between torsion and Dirac spinors generates a spin-spin interaction that replaces the big-bang singularity with a cusp-like nonsingular bounce at a finite minimum scale factor, preceded by contraction. The same coupling can explain why the present Universe appears spatially flat, homogeneous, and isotropic without inflation.

Key findings

  • The spinor-torsion interaction yields a nonsingular bounce: \(a \ge a_{\rm cr} > 0\). At the bounce the expansion scalar jumps from \(\theta = -3v/a_{\rm cr}\) to \(\theta = 3v/a_{\rm cr}\), preventing \(\theta \to -\infty\) and avoiding a curvature singularity.
  • For standard-model particle content \(g_b = 28\) and \(g_f = 90\), the bounce temperature is given by Eq. (16) [formula not decoded in the supplied text]. Using \(T_{\rm eq} \approx 0.75\ {\rm eV}\), \(z_{\rm eq} \approx 3200\), and \(a_0 \approx 2.9 \times 10^{27}\ {\rm m}\), the bounce scale factor is enormous and the density parameter differs from 1 by an extremely small amount.
  • The scenario solves the flatness and horizon problems without inflation: in the radiation era \(\Omega(T)-1 \sim T^{-2}\), and rapid post-bounce expansion produces \(N \sim v_{\rm ant}^3\) causally disconnected volumes. The torsion contribution weakens as \(T^{-6}\), giving a natural transition to the radiation-dominated era.
  • For \(T \ll T_{\rm cr}\), the scale factor reduces to \(a = a_r T_r/T\), and \(a \propto T^{-1} \propto t^{1/2}\), recovering radiation-dominated expansion.

Methods

  • Theoretical derivation in ECSK gravity: torsion is treated as a dynamical variable, the Cartan equations relate torsion to spin density, and the Dirac Lagrangian is minimally coupled to gravity. The spin tensor is taken as completely antisymmetric, following directly from the Dirac field.
  • A closed, homogeneous, isotropic FLRW universe (\(k=1\)) is assumed, with negligible cosmological constant in the early Universe. Friedman equations are written for ultrarelativistic matter in kinetic equilibrium: \(\epsilon(T)=\frac{\pi^2}{30}g_\star(T)T^4\), \(p=\epsilon/3\), and \(n(T)=\frac{\zeta(3)}{\pi^2}g_n(T)T^3\).
  • The analysis derives \(a(T)\), a parametric form for \(\beta(t)=T^{-1}\), and the bounce conditions. Numerical estimates assume only known standard-model particles (\(g_b=28\), \(g_f=90\)) and reference values at matter-radiation equality: \(T_{\rm eq}\approx 0.75\ {\rm eV}\), \(z_{\rm eq}\approx 3200\), \(a_0\approx 2.9\times 10^{27}\ {\rm m}\).

Limitations

  • Stated: the spin-fluid approximation of fermionic matter violates the cosmological principle, and the particle approximation for Dirac fields is not self-consistent; this paper instead uses the Dirac form of the spin tensor. The model assumes a closed \(k=1\) FLRW universe, negligible cosmological constant in the early Universe, constant \(g_\star\) and \(g_n\) over the temperature range considered, and standard-model particle content for numerical estimates.
  • Evident: the work is purely theoretical, with no observational fit or data analysis. The bounce values depend on the assumed particle content and reference epoch, and the apparent fine tuning of \(\Omega(T_{\rm cr})\) is attributed to \(|\Omega_S| \ll 1\) rather than independently constrained here.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

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Crossref · 10.1103/physrevd.85.107502

Cites

48 of 48 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

3 of 34 cited sources checked · 1 supported · 0 flagged · accuracy not yet measured

  • Premisecitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 1

    The minimal coupling between the torsion tensor and Dirac spinors generates a spin-spin interaction that is significant in fermionic matter at extremely high densities.

    “The minimal coupling between the torsion tensor and Dirac spinors generates a spin-spin interaction which is significant in fermionic matter at extremely high densities.”
    Quote in the paperSubstantial claim1 of 2 sources checked
    • cites [2] T. W. B. Kibble, J. Math. Phys. (N.Y.) 2 , 212 (1961); D. W. Sciama, in Recent Developments in General Relativity (Pergamon, Oxford, 1962), p. 415; Rev. Mod. Phys. 36 , 463 (1964); 36 , 1103(E) (1964).

      Not checked in the free run.

    • cites [3] F. W. Hehl, Phys. Lett. A 36 , 225 (1971); Gen. Relativ. Gravit. 4 , 333 (1973); 5 , 491 (1974); F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys. 48 , 393 (1976); E. A. Lord, Tensors, Relativity and Cosmology (McGraw-Hill, New Delhi, 1976); V. de Sabbata and M. Gasperini, Introduction to Gravitation (World Scientific, Singapore, 1985); V. de Sabbata and C. Sivaram, Spin and Torsion in Gravitation (World Scientific, Singapore, 1994); I. L. Shapiro, Phys. Rep. 357 , 113 (2002); R. T. Hammond, Rep. Prog. Phys. 65 , 599 (2002); N. J. Popglyph[suppress] lawski, arXiv:0 source
  • Premisecitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 1

    The ECSK theory passes all observational and experimental tests of GR.

    “The ECSK gravity therefore passes all observational and experimental tests of GR [3].”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [3] F. W. Hehl, Phys. Lett. A 36 , 225 (1971); Gen. Relativ. Gravit. 4 , 333 (1973); 5 , 491 (1974); F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys. 48 , 393 (1976); E. A. Lord, Tensors, Relativity and Cosmology (McGraw-Hill, New Delhi, 1976); V. de Sabbata and M. Gasperini, Introduction to Gravitation (World Scientific, Singapore, 1985); V. de Sabbata and C. Sivaram, Spin and Torsion in Gravitation (World Scientific, Singapore, 1994); I. L. Shapiro, Phys. Rep. 357 , 113 (2002); R. T. Hammond, Rep. Prog. Phys. 65 , 599 (2002); N. J. Popglyph[suppress] lawski, arXiv:0 source

      In “General relativity” (pp. 110–111), the paper says torsion is negligibly small in almost all physical situations and that the Einstein equations of GR follow for spinless matter. This supports the claim that ECSK generally reproduces GR’s predictions, but the copy does not establish that it passes all observational and experimental tests of GR. (LLM, conf=0.82)

      Experimental — accuracy not yet measured
  • Premisecitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 1

    Below the Cartan density, the predictions of ECSK theory do not differ from the predictions of GR.

    “Below this density, the predictions of the ECSK theory do not differ from the predictions of GR.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [3] F. W. Hehl, Phys. Lett. A 36 , 225 (1971); Gen. Relativ. Gravit. 4 , 333 (1973); 5 , 491 (1974); F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys. 48 , 393 (1976); E. A. Lord, Tensors, Relativity and Cosmology (McGraw-Hill, New Delhi, 1976); V. de Sabbata and M. Gasperini, Introduction to Gravitation (World Scientific, Singapore, 1985); V. de Sabbata and C. Sivaram, Spin and Torsion in Gravitation (World Scientific, Singapore, 1994); I. L. Shapiro, Phys. Rep. 357 , 113 (2002); R. T. Hammond, Rep. Prog. Phys. 65 , 599 (2002); N. J. Popglyph[suppress] lawski, arXiv:0 source

      The General relativity section (p. 110) says torsion can be approximated as zero when its effects are negligible, and Structure of Einstein equations (p. 111) calls the GR equations an accurate approximation under that condition. The paper does not establish that predictions are identical at every density below a specified Cartan density; it describes exact reduction to GR for spinless matter. (LLM, conf=0.86)

      Experimental — accuracy not yet measured
  • Findingcitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 6

    The big-bounce value of Δ = Ω - 1 is on the order of Ω_S ΔΩ_R^2|_0, where Ω_S is the spinor-torsion density parameter.

    “The big-bounce value of ∆ = Ω -1 is on the order of Ω S ∆ Ω 2 R ∣ ∣ 0 , where Ω S is the spinor-torsion density parameter [16], subscript R denotes radiation, and 0 denotes the present value.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [16] N. J. Popglyph[suppress] lawski, Phys. Lett. B 694 , 181 (2010); 701 , 672(E) (2011).

      Not checked in the free run.

  • Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 6

    Such extreme values result from a_r T_r >> 1.

    “Such extreme values result from a r T r ≫ 1.”
    Quote in the paperNot a substantial claim
  • May combine two claims. 21
  • Classified as probably not a substantial claim. 6
  • The quote only partly states this claim. 2

  • May combine two claims.

    Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 6

    The value (30) is enormous, whereas (31) differs from 1 by a quantity that is extremely small in magnitude.

    “The value (30) is enormous, whereas (31) differs from 1 by a quantity that is extremely small in magnitude, like in the spin-fluid cosmology with torsion [16].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [16] N. J. Popglyph[suppress] lawski, Phys. Lett. B 694 , 181 (2010); 701 , 672(E) (2011).

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Findingnone given
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 5

    For the cosmological fluid in the comoving frame, the FLRW metric gives θ = (ln √-g),0 = 3 ̇a/a.

    “For the cosmological fluid in the comoving frame, the FLRW metric gives θ = (ln √ -g ) , 0 = 3˙ a a .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The Universe is closed, homogeneous and isotropic, described by the FLRW metric with k = 1.

  • May combine two claims.

    Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 3

    The Universe is nonsingular: a ≥ a_cr.

    “The Universe is therefore nonsingular: a ≥ a cr .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 3

    The scale factor as a function of temperature is not monotonic; as T increases, a decreases until a critical temperature T_cr, then increases.

    “The function a ( T ) (15) is not monotonic. As T increases, a decreases until T reaches a critical temperature T cr given by da dT ( T cr ) = 0,”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The early Universe matter is ultrarelativistic and in kinetic equilibrium, with energy density, pressure and number density given by the standard equilibrium expressions.
    • The authors state this as an assumption: h_star and h_n can be assumed constant in the range of T considered.

  • May combine two claims.

    Findingcitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 6

    If the early Universe contained only known standard-model particles, then g_b = 28 and g_f = 90, so the temperature at the big bounce was Eq. (28).

    “If we assume that the early Universe contained only known standard-model particles, then g b = 28 and g f = 90 [18], so the temperature at the big bounce (16) was”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [18] J. Rich, Fundamentals of Cosmology (Springer-Verlag, Berlin, 2001). source
    • The authors state this as an assumption: The early Universe contained only known standard-model particles.

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

    Findingcitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 5

    For a homogeneous Universe, in the comoving frame of an irrotational cosmological fluid, the tensor (9) and the Dirac equation give W = 1/2 m ̄ψψ + 3/2 κ s^2, suggesting that torsion enhances the formation of singularities.

    “For a homogeneous Universe, in the comoving frame of reference of an irrotational cosmological fluid, the tensor (9) and the Dirac equation give W = 1 2 m ¯ ψψ + 3 2 κ s 2 , suggesting that torsion enhances the formation of singularities [6].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [6] G. D. Kerlick, Phys. Rev. D 12 , 3004 (1975); Ann. Phys. 99 , 127 (1976).

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

  • May combine two claims.

    Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 4

    The condition ̇a = 0 would be satisfied at a temperature T_st given by h_star T^4 - α h_n^2 T^6 = 0, and this temperature equals Eq. (23).

    “The condition ˙ a = 0, defining the scale factor at a stationary state a = a st , would be satisfied at a temperature T st given by (11) without the negligible term k = 1: h ⋆ T 4 -αh 2 n T 6 = 0. This temperature is equal to”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The curvature term k = 1 is negligible in the relevant Friedman equations.

  • May combine two claims.

    Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 3

    For T << T_cr, the scale factor reduces to a = a_r T_r/T, which is satisfied in the radiation-dominated era.

    “For T ≪ T cr , (15) reduces to a = a r T r T , which is satisfied in the radiation-dominated era.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 4

    For β >> β_cr, a ∝ T^-1 ∝ t^1/2, which is characteristic of the radiation-dominated era.

    “For β ≫ β cr , (19) and (22) give a ∝ T -1 ∝ t 1 / 2 , which is characteristic to the radiation-dominated era.”
    Quote in the paperSubstantial claim

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

    Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 5

    This result agrees with the contribution to W from the torsion-induced spin-spin interaction, ̃ε + 3 ̃p = 2αn^2 > 0.

    “This result agrees with the contribution to W from the torsion-induced spin-spin interaction, ˜ ϵ + 3˜ p = 2 αn 2 &gt; 0.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 5

    When the Universe reaches the radiation-dominated era, T << T_cr, v_ant(T) ∼ T and Ω(T)-1 ∼ T^-2.

    “When the Universe reaches the radiation-dominated era, T ≪ T cr , (26) reduces to v ant ( T ) ∼ T and (27) reduces to Ω( T ) -1 ∼ T -2 .”
    Quote in the paperNot a substantial claim

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

    Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 5

    As the Universe expands, the antipodal velocity decreases according to (26) and the density parameter increases according to (27).

    “As the Universe expands, the antipodal velocity decreases according to”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 5

    The Universe never reaches T = T_st and ̇a = 0.

    “so the Universe never reaches T = T st and ˙ a = 0.”
    Quote in the paperSubstantial claim

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

    Findingreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 6

    The values at the big bounce of the scale factor, antipodal velocity and density parameter were, respectively, Eqs. (29), (30), (31).

    “Accordingly, the values at the big bounce of the scale factor (17), antipodal velocity and density parameter were, respectively:”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The reference values are taken at matter-radiation equality, with T_r = T_eq ≈ 0.75 eV, a_r = a_eq = a0/(1+z_eq), a0 ≈ 2.9×10^27 m, and z_eq ≈ 3200.

  • May combine two claims.

    Premisecitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 5

    Dirac fields in ECSK gravity satisfy the strong energy condition for the Hawking-Penrose singularity theorems, and the torsion-induced spin-spin interaction enhances that condition.

    “Kerlick has shown that Dirac fields in the ECSK gravity satisfy the strong energy condition for the HawkingPenrose singularity theorems [19], and that the torsion-induced spin-spin interaction (the second term on the right of (9)) enhances that condition [6].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [19] R. M. Wald, General Relativity (Univ. of Chicago Press, Chicago, 1984). source
    • cites [6] G. D. Kerlick, Phys. Rev. D 12 , 3004 (1975); Ann. Phys. 99 , 127 (1976).

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

  • May combine two claims.

    Premisecitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 2

    This spin-fluid behavior is significant at extremely high densities, even without spin polarization, leading to gravitational repulsion and avoidance of curvature singularities by violating the energy condition of the singularity theorems.

    “This behavior is significant in spin fluids at extremely high densities, even without spin polarization, leading to gravitational repulsion and avoidance of curvature singularities by violating the energy condition of the singularity theorems [8].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [8] F. W. Hehl, P. von der Heyde, and G. D. Kerlick, Phys. Rev. D 10 , 1066 (1974).

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

  • May combine two claims.

    Premisecitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 2

    The Dirac form of the spin tensor for fermionic matter, s_ijk = s_[ijk], follows directly from the Dirac Lagrangian and is consistent with the cosmological principle.

    “In this paper, we use the Dirac form of the spin tensor for fermionic matter, s ijk = s [ ijk ] [2, 6, 12], which follows directly from the Dirac Lagrangian and is consistent with the cosmological principle [15].”
    Quote in the paperSubstantial claim0 of 4 sources checked
    • cites [2] T. W. B. Kibble, J. Math. Phys. (N.Y.) 2 , 212 (1961); D. W. Sciama, in Recent Developments in General Relativity (Pergamon, Oxford, 1962), p. 415; Rev. Mod. Phys. 36 , 463 (1964); 36 , 1103(E) (1964).

      Not checked in the free run.

    • cites [6] G. D. Kerlick, Phys. Rev. D 12 , 3004 (1975); Ann. Phys. 99 , 127 (1976).

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

    • cites [12] R. F. O'Connell, Phys. Rev. D 16 , 1247 (1977).

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

    • cites [15] P. Minkowski, Phys. Lett. B 173 , 247 (1986).

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

  • May combine two claims.

    Premisecitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 5

    Because of the Raychaudhuri equation, timelike geodesics for matter satisfying the strong energy condition converge (θ decreases), resulting in a caustic, θ → -∞, and thus in a singularity within a finite proper time.

    “Because of the Raychaudhuri equation, timelike geodesics for the matter satisfying the strong energy condition converge ( θ decreases), resulting in a caustic, θ →-∞ (if θ decreases continuously), and thus in a singularity within a finite proper time [20].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [20] A. Raychaudhuri, Phys. Rev. 98 , 1123 (1955).

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

  • May combine two claims.

    Interpretationreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 5

    The discontinuity of the expansion scalar at the cusp-like big bounce prevents θ from decreasing to -∞ and reaching a curvature singularity, and guarantees that timelike geodesics in the Universe with Dirac fields coupled to torsion continue through the bounce.

    “The discontinuity of the expansion scalar at the cusp-like big bounce therefore prevents θ from decreasing to -∞ and reaching a curvature singularity, and guarantees that timelike geodesics in the Universe with Dirac fields coupled to torsion continue through the bounce.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 6

    Such a coupling naturally explains why the present Universe at largest scales appears nearly flat, solving the flatness problem without introducing exotic matter fields necessary for cosmological inflation.

    “Accordingly, such a coupling naturally explains why the present Universe at largest scales appears nearly flat, solving the flatness problem without introducing exotic matter fields necessary for cosmological inflation.”
    • Also stated in: § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 1 · plain
      “This scenario also explains why the present Universe at largest scales appears spatially flat, homogeneous and isotropic.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The present Universe at largest scales appears spatially flat, homogeneous and isotropic.

  • May combine two claims.

    Premisereasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 1

    In ECSK theory, torsion is proportional to spin density and vanishes outside material bodies.

    “These equations are linear and algebraical: torsion is proportional to spin density and vanishes outside material bodies.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [2] T. W. B. Kibble, J. Math. Phys. (N.Y.) 2 , 212 (1961); D. W. Sciama, in Recent Developments in General Relativity (Pergamon, Oxford, 1962), p. 415; Rev. Mod. Phys. 36 , 463 (1964); 36 , 1103(E) (1964).

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 5

    The Raychaudhuri equation describes the time evolution of the expansion scalar for timelike congruences.

    “The singularity theorems, however, have another important component: the Raychaudhuri equation, which describes the time evolution of the expansion scalar for timelike congruences [20].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [20] A. Raychaudhuri, Phys. Rev. 98 , 1123 (1955).

      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).

    Premisecitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 5

    A similar discontinuity completes timelike geodesics in the gravitational field of an Einstein-Rosen bridge.

    “A similar discontinuity completes timelike geodesics in the gravitational field of an Einstein-Rosen bridge [21].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [21] N. J. Popglyph[suppress] lawski, Phys. Lett. B 687 , 110 (2010).

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 2

    Such repulsion replaces the big-bang singularity with a nonsingular big bounce, before which the Universe was contracting.

    “Trautman, Kuchowicz, and others have shown that such a repulsion replaces the big-bang singularity with a nonsingular big bounce, before which the Universe was contracting [10, 11].”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [10] W. Kopczy´ nski, Phys. Lett. A 39 , 219 (1972); 43 , 63 (1973); A. Trautman, Nature (London) 242 , 7 (1973); J. Tafel, Phys. Lett. A 45 , 341 (1973); Acta Phys. Pol. B 6 , 537 (1975); B. Kuchowicz, Acta Phys. Pol. B 6 , 555 (1975); Acta Cosmologica Z 3 , 109 (1975); Astrophys. Space Sci. 39 , 157 (1976); 40 , 167 (1976); M. Gasperini, Gen. Relativ. Gravit. 30 , 1703 (1998).

      Not checked in the free run.

    • cites [11] B. Kuchowicz, Gen. Relativ. Gravit. 9 , 511 (1978).

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

  • May combine two claims.

    Interpretationreasoning
    § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 1

    The spin-spin interaction from torsion-Dirac coupling averts the unphysical big-bang singularity, replacing it with a cusp-like bounce at a finite minimum scale factor, before which the Universe was contracting.

    “We show that such an interaction averts the unphysical big-bang singularity, replacing it with a cusp-like bounce at a finite minimum scale factor, before which the Universe was contracting.”
    • Also stated in: § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 2 · boosted
      “We show that the minimal coupling between the torsion tensor and Dirac fermions, despite enhancing the energy condition, also averts the big-bang singularity.”
    • Also stated in: § NONSINGULAR, BIG-BOUNCE COSMOLOGY FROM SPINOR-TORSION COUPLING · p. 4 · plain
      “The unphysical big-bang singularity appearing in general-relativistic cosmology is replaced in the ECSK gravity by a nonsingular (with respect to curvature) big bounce that follows a contracting phase of the Universe [11].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [11] B. Kuchowicz, Gen. Relativ. Gravit. 9 , 511 (1978).

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

    • The authors state this as an assumption: The Universe is closed, homogeneous and isotropic, described by the FLRW metric with k = 1.
    • The authors state this as an assumption: The cosmological constant is negligible in the early Universe.
    • The authors state this as an assumption: The early Universe matter is ultrarelativistic and in kinetic equilibrium, with energy density, pressure and number density given by the standard equilibrium expressions.
    • The authors state this as an assumption: h_star and h_n can be assumed constant in the range of T considered.
25 of 25

  • The authors state this as an assumption.

    The Universe is closed, homogeneous and isotropic, described by the FLRW metric with k = 1.

    “As in [16], we consider a closed, homogeneous and isotropic universe, described by the Friedman-Lemaˆ ıtreRobertson-Walker (FLRW) metric.”

  • The authors state this as an assumption.

    The cosmological constant is negligible in the early Universe.

    “(the cosmological constant is negligible in the early Universe)”

  • The authors state this as an assumption.

    The early Universe matter is ultrarelativistic and in kinetic equilibrium, with energy density, pressure and number density given by the standard equilibrium expressions.

    “As in [17], we use ϵ , p and n for ultrarelativistic matter in kinetic equilibrium: ϵ ( T ) = π 2 30 g ⋆ ( T ) T 4 , p ( T ) = ϵ ( T ) 3 and n ( T ) = ζ (3) π 2 g n ( T ) T 3 , where T is the temperature of the early Universe [18].”

  • The authors state this as an assumption.

    h_star and h_n can be assumed constant in the range of T considered.

    “where h ⋆ = π 2 30 g ⋆ ( T ) and h n = ζ (3) π 2 g n ( T ) can be assumed constant in the range of T considered.”

  • The authors state this as an assumption.

    The early Universe contained only known standard-model particles.

    “If we assume that the early Universe contained only known standard-model particles, then g b = 28 and g f = 90 [18]”

  • The authors state this as an assumption.

    The reference values are taken at matter-radiation equality, with T_r = T_eq ≈ 0.75 eV, a_r = a_eq = a0/(1+z_eq), a0 ≈ 2.9×10^27 m, and z_eq ≈ 3200.

    “As the reference values, we can take the temperature and scale factor at the matter-radiation equality, where the radiation-dominated era ends. We therefore have T r = T eq ≈ 0 . 75 eV and a r = a eq = a 0 1+ z eq , where a 0 ≈ 2 . 9 × 10 27 m is the present scale factor [16] and z eq ≈ 3200 is the redshift at the matter-radiation equality [18].”

  • The authors state this as an assumption.

    The curvature term k = 1 is negligible in the relevant Friedman equations.

    “without the negligible term k = 1”

  • The paper relies on this without support.

    The present Universe at largest scales appears spatially flat, homogeneous and isotropic.

    “This scenario also explains why the present Universe at largest scales appears spatially flat, homogeneous and isotropic.”
8 of 8

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • We showed in Popławski (2012a) that a semiclassical spin-spin interaction, generated by the minimal coupling between the torsion tensor and Dirac spinors, averts the unphysical big bang singularity and replaces it with a cusp-like big bounce at a finite minimum scale factor, before which the universe was contracting.

    “We showed in Popławski (2012a) that a semiclassical spin-spin interaction, generated by the minimal coupling between the torsion tensor and Dirac spinors, averts the unphysical big bang singularity and replaces it with a cusp-like big bounce at a finite minimum scale factor, before which the universe was contracting.”

    From UNIVERSE IN A BLACK HOLE IN EINSTEIN–CARTAN GRAVITY (2016) · cites this paper as Popławski 2012a

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Torsion term is said to scale as T−6 but is written as T6

    Minor

    The paper says the torsion contribution weakens according to T−6, but its displayed temperature-dependent term is proportional to T6. The stated inverse scaling contradicts that expression.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 4

  1. A. Trautman, 1973, Nature (London), 242, 7
  2. J. Tafel, 1975, Acta Phys. Pol. B, 6, 537
  3. B. Kuchowicz, 1975, Acta Phys. Pol. B, 6, 555
  4. B. Kuchowicz, 1975, Acta Cosmologica Z, 3, 109

Cited by

Sorted by citations

Related

Authors
Popławski, N.
Affiliations
Indiana University Bloomington (US)
Corresponding author
Popławski, Nikodem
ORCID
Popławski, Nikodem
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
2012-05-29
Volume
85
Issue
10
ISSN
  • 0556-2821 (ISSN-L)
  • 1550-7998
  • 1550-2368
DOI
10.1103/physrevd.85.107502
OpenAlex
W2036707580
Cited by
180
References
44
Citations by year
2026: 10 · 2025: 10 · 2024: 17 · 2023: 7 · 2022: 6 · 2021: 13 · 2020: 26 · 2019: 15 · 2018: 12 · 2017: 18 · 2016: 10 · 2015: 12 · 2014: 15 · 2013: 8 · 2012: 1
FWCI
6.30
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • arXiv (Cornell University) (publishedVersion)
  • IUScholarWorks (Indiana University) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
Keywords
Einstein-Cartan theory, torsion tensor, Big Bang singularity resolution, bounce cosmology, spin-spin coupling, fermionic matter, spatial flatness, homogeneous isotropic universe

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-30.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-30.

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Statistics and arithmetic are recomputed, not judged.

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