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Cited by 205Cosmology and Gravitation TheoriesRelativity and Gravitational TheoryNoncommutative and Quantum Gravity Theories

Cosmology with torsion: An alternative to cosmic inflation

Open access
Popławski, Nikodem J.·2010·Physics Letters B
DOI 10.1016/j.physletb.2010.09.056
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Abstract

We propose a simple scenario which explains why our Universe appears spatially flat, homogeneous and isotropic. We use the Einstein–Cartan–Kibble–Sciama (ECKS) theory of gravity which naturally extends general relativity to include the spin of matter. The torsion of spacetime generates gravitational repulsion in the early Universe filled with quarks and leptons, preventing the cosmological singularity: the Universe expands from a state of minimum but finite radius. We show that the dynamics of the closed Universe immediately after this state naturally solves the flatness and horizon problems in cosmology because of an extremely small and negative torsion density parameter, ΩS≈−10−69. Thus the ECKS gravity provides a compelling alternative to speculative mechanisms of standard cosmic inflation. This scenario also suggests that the contraction of our Universe preceding the bounce at the minimum radius may correspond to the dynamics of matter inside a collapsing black hole existing in another universe, which could explain the origin of the Big Bang.

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Written from the paper’s full text.

TL;DR

The paper proposes that Einstein-Cartan-Kibble-Sciama (ECKS) gravity, with spacetime torsion generated by fermion spin, prevents the cosmological singularity and naturally solves the flatness and horizon problems without cosmic inflation. It argues that a closed universe bounces at a finite minimum radius and may originate inside a collapsing black hole.

Key findings

  • Torsion from a spin fluid contributes a negative effective energy density and gravitational repulsion; the torsion density parameter is estimated as Ω_S = -8.6 × 10^-70 (abstract gives Ω_S ≈ -10^-69).
  • The expansion starts at H = 0 with â_m = 3.1 × 10^-33, corresponding to a minimum radius a_m = 9 × 10^-6 m; Ω(â) is infinite at the bounce and reaches a local minimum at √2 â_m of 1 + 8.9 × 10^-64, taking t = 5.3 × 10^-46 s and expanding by only a factor of √2.
  • For the horizon problem, the antipodal velocity reaches v_a = 1.1 × 10^32 c at √2 â_m; the antipodal distance is 32 orders of magnitude greater than the Hubble radius, giving N ≈ 10^96 causally disconnected volumes.
  • The black-hole-universe scenario uses a nonsingular bounce; at minimum radius the radiation energy density is 1.1 × 10^116 J m^-3, greater than the Planck energy density by a few orders of magnitude, and the scenario may explain the arrow of time.

Methods

  • Theoretical cosmology in ECKS gravity: torsion is dynamical, Cartan field equations relate torsion algebraically to the spin tensor, and fermions are averaged as a Weyssenhoff spin fluid.
  • A closed, homogeneous, isotropic FLRW universe (k = 1) is used; Friedman equations include torsion, with an ultrarelativistic equation of state p = ρ/3 and effective radiation plus torsion densities.
  • Density parameters are estimated using seven-year WMAP data: Ω = 1.002, H_0^-1 = 4.4 × 10^17 s, Ω_R = 8.8 × 10^-5, and relic background neutrinos with n = 5.6 × 10^7 m^-3 for each of 6 types; the analysis then solves Ω(â) and the flatness/horizon problems.

Limitations

  • The paper states that many gravitational Lagrangians with torsion imply torsion is algebraic, does not propagate, and vanishes in vacuum, limiting its detection; whether torsion couples to spin and whether other torsion kinds are physical should be tested experimentally.
  • The paper states that the contraction before the bounce does not explain what caused the contraction, just as Big Bang cosmology cannot explain what happened before the Big Bang; it proposes pair production and spin alignment as possible asymmetry mechanisms.
  • Evident from the text: no direct observational confirmation of torsion or of the black-hole-universe scenario is presented; proposed checks are indirect, such as preferred-direction corrections from rotating black holes or experimental limits on torsion couplings.

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

Integrity signals

Registry

Crossref · 10.1016/j.physletb.2010.09.056

Cites

52 of 52 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 39 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    The minimum scale factor (radius of a closed universe) is a_m = 9 × 10^-6 m.

    “corresponding to the minimum but finite scale factor (radius of a closed universe) a m = 9 × 10 -6 m.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The torsion density parameter Ω_S is extremely small and negative, approximately -10^-69 (specifically -8.6 × 10^-70).

    “Ω S ≈ -10 -69 .”
    • Also stated in: § 4. Density parameters · plain
      “Eqs. ( 13) and ( 15) then give Ω S = -8.6 × 10 -70 .(23)”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The torsion of spacetime prevents the cosmological singularity.

    “The torsion of spacetime generates gravitational repulsion in the early Universe filled with quarks and leptons, preventing the cosmological singularity: the Universe expands from a state of minimum but finite radius.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    The Universe expands from a state of minimum but finite radius.

    “The torsion of spacetime generates gravitational repulsion in the early Universe filled with quarks and leptons, preventing the cosmological singularity: the Universe expands from a state of minimum but finite radius.”
    Quote in the paperSubstantial claim
  • Findingcitation

    The Einstein-Cartan equations for a spin fluid with random spin orientation are equivalent to the Einstein equations for a perfect fluid with effective energy density -κs^2/4 and effective pressure p - κs^2/4.

    “Thus the Einstein-Cartan equations for such a spin fluid are equivalent to the Einstein equations for a perfect fluid with the effective energy density -κs 2 /4 and the effective pressure pκs 2 /4 [18,[25][26][27].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [18,[25][26][27]

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

    • The authors state this as an assumption: The spin orientation of particles in the spin fluid is random.
  • May combine two claims. 15
  • Classified as probably not a substantial claim. 3

  • May combine two claims.

    Findingreasoning

    At t given by (27), d_a is 32 orders of magnitude greater than d_H and N ≈ 10^96.

    “At t given by ( 27), d a is 32 orders of magnitude greater than d H and N ≈ 10 96 .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    At the minimum radius, the radiation energy density is ρ_R = 1.1 × 10^116 J m^-3, which is greater than the Planck energy density by a few orders of magnitude.

    “First, when the Universe has the minimum radius (24), the radiation energy density is R = 1.1 × 10 116 J m -3 , which is greater than the Planck energy density by a few orders of magnitude.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationreasoning

    This scenario could explain the origin of the Big Bang.

    “This scenario also suggests that the contraction of our Universe preceding the bounce at the minimum radius may correspond to the dynamics of matter inside a collapsing black hole existing in another universe, which could explain the origin of the Big Bang.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    Our own Universe may be the interior of a black hole existing in another universe, and the time asymmetry of motion at the boundary of this black hole may cause the perceived arrow of cosmic time.

    “Accordingly, our own Universe may be the interior of a black hole existing in another universe, and the time asymmetry of motion at the boundary of this black hole may cause the perceived arrow of cosmic time.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    The relic background neutrinos are the most abundant fermions in the Universe, with n = 5.6 × 10^7 m^-3 for each type (out of 6).

    “To estimate Ω S , we use the relic background neutrinos which are the most abundant fermions in the Universe, with n = 5.6 × 10 7 m -3 for each type (out of 6) [28].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [28]

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

  • May combine two claims.

    Interpretationreasoning

    The ECKS cosmology is a compelling alternative not only to Big-Bang cosmology (it avoids the initial singularity), but also to the standard inflationary scenario because it does not require false vacuum, scalar fields or other speculative mechanisms.

    “The ECKS cosmology is thus a compelling alternative not only to Big-Bang cosmology (it avoids the initial singularity), but also to the standard inflationary scenario because it does not require false vacuum, scalar fields or other speculative mechanisms.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation

    A universe born from a rotating black hole should inherit its preferred direction, related to the axis of rotation, introducing small corrections to the FLRW metric.

    “A universe born from a rotating black hole should inherit its preferred direction, related to the axis of rotation.Such a preferred direction should introduce small corrections to the FLRW metric, containing the Kerr radius a = M/(mc), where M is the angular momentum of a rotating black hole and m is its mass [22,43].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [22]

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

    • cites [43]

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

  • May combine two claims.

    Interpretationcitation

    The gravitational collapse of a star composed of quarks and leptons to a black hole does not create a singularity, allowing matter inside the event horizon to undergo a bounce at the nonsingular state of maximum density and then reexpand.

    “Thus the gravitational collapse of a star composed of quarks and leptons to a black hole does not create a singularity [21], allowing matter inside the event horizon to undergo a bounce at the (nonsingular) state of maximum density and then reexpand.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [21]

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

  • May combine two claims.

    Premisecitation

    The ECKS gravity does not contain free parameters (G and c can be always set to 1 by changing the units), thus it is advantageous over inflationary models that do introduce new parameters.

    “Moreover, the ECKS gravity does not contain free parameters (G and c can be always set to 1 by changing the units), thus it is advantageous over inflationary models that do introduce new parameters [2].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [2]

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

  • May combine two claims.

    Premisecitation

    WMAP data give H0^-1 = 4.4 × 10^17 s and Ω_R = 8.8 × 10^-5.

    “The WMAP data give also H -1 0 = 4.4 × 10 17 s and Ω R = 8.8 × 10 -5 .”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [29]

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

  • May combine two claims.

    Premisenone given

    The ECKS theory is a viable theory, which differs significantly from general relativity only at densities of matter much larger than the nuclear density.

    “It is a viable theory, which differs significantly from general relativity only at densities of matter much larger than the nuclear density.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The new universe is a separate spacetime branch with its own timeline; it can last infinitely long and grow infinitely large if dark energy is present.

    “The new universe is thus a separate spacetime branch with its own timeline; it can last infinitely long and grow infinitely large if dark energy is present.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation

    The particle production does not change the total (matter plus gravitational field) energy of the resulting FLRW universe, which is zero if we neglect the cosmological constant.

    “The particle production does not change the total (matter plus gravitational field) energy of the resulting FLRW universe, which is zero if we neglect the cosmological constant [37].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [37]

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

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).

    Interpretationnone given

    The contraction of our Universe preceding the bounce at the minimum radius may correspond to the dynamics of matter inside a collapsing black hole existing in another universe.

    “This scenario also suggests that the contraction of our Universe preceding the bounce at the minimum radius may correspond to the dynamics of matter inside a collapsing black hole existing in another universe, which could explain the origin of the Big Bang.”
    Quote in the paperNot a substantial claim

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

    Premisecitation

    All astrophysical black holes may be Einstein-Rosen bridges (wormholes), each with a new universe inside that formed simultaneously with the black hole.

    “We recently suggested that all astrophysical black holes may be Einstein-Rosen bridges (wormholes), each with a new universe inside that formed simultaneously with the black hole [39].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [39]

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

  • May combine two claims.

    Interpretationreasoning

    Torsion provides a viable theoretical explanation of how black-hole interiors can generate new universes, avoiding unphysical singularities and replacing speculative mechanisms of inflation.

    “Torsion thus provides a viable theoretical explanation of how black-hole interiors can generate new universes, avoiding unphysical singularities and replacing speculative mechanisms of inflation.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    GRS 1915+105, which is the heaviest and fastest spinning known stellar black hole in the Milky Way Galaxy, has a < 26 km.

    “GRS 1915+105, which is the heaviest and fastest spinning, known stellar black hole in the Milky Way Galaxy, has a < 26 km [44].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [44]

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

17 of 17

  • The paper relies on this without support.

    Big Bang cosmology does not explain the origin of the initial extremely hot and dense state of the Universe and what existed before this state.

    “Moreover, Big Bang cosmology does not explain the origin of the initial, extremely hot and dense state of the Universe and what existed before this state.”

  • The paper relies on this without support.

    The early Universe was filled with quarks and leptons.

    “The torsion of spacetime generates gravitational repulsion in the early Universe filled with quarks and leptons, preventing the cosmological singularity: the Universe expands from a state of minimum but finite radius.”

  • The authors state this as an assumption.

    The spin orientation of particles in the spin fluid is random.

    “If the spin orientation of particles is random then the macroscopic spacetime average of the spin and of the spin gradients vanish.”

  • The authors state this as an assumption.

    The closed Universe was causally connected at some instant t < 0.

    “If the closed Universe was causally connected at some instant t < 0, then it remains causally connected during its contraction until t = 0 and also during the subsequent expansion until v a reaches c.”
4 of 4

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?

  • This scenario could also explain the arrow of time and the black hole information paradox.

    “This scenario could also explain the arrow of time and the black hole information paradox (Popławski 2010c(Popławski , 2011a(Popławski , 2014)).”

    From UNIVERSE IN A BLACK HOLE IN EINSTEIN–CARTAN GRAVITY (2016) · cites this paper as Popławski 2010c

    The Discussion (section 7; page unavailable) says the proposed scenario may explain the arrow of time through the one-way motion of matter across a black-hole horizon. Our copy does not address the black hole information paradox, so it supports only the first part of the attributed claim. (LLM, conf=0.88)

    Experimental — accuracy not yet measured

  • When a trapped null surface forms in a black hole, the region within this surface is causally connected and mathematically equivalent to a new, closed universe.

    “When a trapped null surface forms in a black hole (Hochberg & Visser 1998), the region within this surface is causally connected and mathematically equivalent to a new, closed universe (Berger et al. 1998;Garfinkle 2004;Popławski 2010cPopławski , 2011a).”

    From UNIVERSE IN A BLACK HOLE IN EINSTEIN–CARTAN GRAVITY (2016) · cites this paper as Popławski 2010c

    The Discussion (section 7; page unavailable) says that, after an event horizon forms, contraction inside it looks like that of a closed universe and that this universe is initially causally connected. It does not establish mathematical equivalence, and the Horizon problem section (section 6; page unavailable) says causal connection does not persist throughout expansion. (LLM, conf=0.84)

    Experimental — accuracy not yet measured

  • Relation (15) between a and T does not depend on the presence of spin.

    “Relation (15) between a and T does not depend on the presence of spin (Popławski 2010c(Popławski , 2011a)).”

    From UNIVERSE IN A BLACK HOLE IN EINSTEIN–CARTAN GRAVITY (2016) · cites this paper as Popławski 2010c

    Section 3 (page not available) says its energy-density scaling with scale factor a has the same form in the absence of spin, supporting the spin-independence part. Our copy does not state a relation between a and temperature T, so it does not directly establish the cited relation. (LLM, conf=0.78)

    Experimental — accuracy not yet measured

  • Substituting the torsion-spin relation into the ECSK action generates a four-fermion self-interaction whose energy-density correction scales as δρtor ∼ n2, where n is fermion number density.

    “Substituting this relation back into the action generates a four-fermion self-interaction for the Dirac field, and the corresponding correction to the energy density scales as δρ tor ∼ S 2 ∼ ψγ µ γ 5 ψ ψγ µ γ 5 ψ ∼ n 2 , (13 ) where n is the fermion number density [11,13,33].”

    From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [13]

    Section 3, equations (13) and (15), states that the squared spin density of an unpolarized fermion fluid is proportional to n² and that the spin-density contribution to effective energy density is proportional to that squared density. Section 1 gives an algebraic torsion–spin relation, but the paper does not show that substituting it into the action generates a four-fermion Dirac self-interaction or the stated axial-current expression. (LLM, conf=0.88)

    Experimental — accuracy not yet measured

  • In a microscopic Einstein-Cartan treatment, the spin-torsion contribution for a fermion fluid can be written schematically as ρs ∼ ξ n^2 / M_Pl^2, where n is the fermion number density and ξ = O(1) depends on spin representation and conventions.

    “In a microscopic Einstein-Cartan treatment, the spintorsion contribution for a fermion fluid can be written schematically as ρ s ∼ ξ n 2 M 2 Pl , (45 ) where n is the fermion number density and ξ = O(1) depends on spin representation and conventions [11][12][13]33].”

    From Dynamical systems analysis of an Einstein–Cartan ekpyrotic nonsingular bounce cosmology (2026) · cites this paper as [13]

    Section 3, Eq. (13) gives the squared spin density of an unpolarized fermion fluid as proportional to n², and Eq. (15) gives its contribution to effective energy density as −κs²/4. Together these support the schematic n²/Planck-scale² dependence, with a negative contribution. The paper does not state that the coefficient varies with spin representation or conventions; it gives a specific result for the fluid considered. (LLM, conf=0.84)

    Experimental — accuracy not yet measured

Cites

Flagged first, then by citations

Research works printed but not linked: 3

  1. Weyssenhoff, 1947, Acta Phys. Pol., 9, 7
  2. Capozziello, 2001, Ann. Phys. (Leipzig), 10, 713, 10.1002/andp.20015130803 · DOI
  3. Zel'dovich, 1970, J. Exp. Theor. Phys. Lett., 12, 307

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Related

Authors
Popławski, N. J.
Affiliations
Indiana University Bloomington (US)
Corresponding author
Popławski, Nikodem J.
ORCID
Popławski, Nikodem J.
Venue
Physics Letters B
Publisher
Elsevier BV
Published
2010-11
Volume
694
Issue
3
Pages
181-185
ISSN
0370-2693 (ISSN-L)
DOI
10.1016/j.physletb.2010.09.056
OpenAlex
W2057776977
Cited by
205
References
49
Citations by year
2026: 20 · 2025: 18 · 2024: 16 · 2023: 18 · 2022: 16 · 2021: 15 · 2020: 12 · 2019: 17 · 2018: 14 · 2017: 9 · 2016: 5 · 2015: 11 · 2014: 11 · 2013: 8 · 2012: 6
FWCI
3.93
Type
journal article
Language
en
Open access
Licence
Locations
  • Physics Letters B (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion)
  • Digital Commons - New Heaven (University of New Haven) (submittedVersion, cc-by)
  • 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
torsion, flatness problem, horizon problem, bounce cosmology, closed universe, Big Bang origin, black hole interior, early universe expansion, cosmological singularity

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