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Parity violating Friedmann Universes

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Pith's one-line read This paper claims that in a varying-Lambda Einstein-Cartan theory, homogeneous and isotropic universes can violate parity through a torsional component that carries Weyl curvature and a genuinely new gravitational degree of freedom.

desk verdict Finds a genuinely new branch in the FRW minisuperspace of a varying-Lambda Einstein-Cartan theory, but the 'new degree of freedom' headline outruns the proof. read the letter →

arxiv 1908.05184 v3 pith:UEUGSBQ6 submitted 2019-08-14 gr-qc

João Magueijo , Tom Z{l}ośnik This is my paper · ORCID
classification gr-qc
keywords Einstein-CartantheorytorsionvaryingcosmologicalconstantparityviolationFRWcosmologyWeylcurvatureHamiltonianconstraintanalysisImmirziparameter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that in theories where the cosmological constant is promoted to a variable at the price of allowing torsion, the usual assumption that homogeneous and isotropic (FRW) universes are automatically parity preserving collapses. The parity-odd part of the torsion, $P$, can be nonzero even while the metric is perfectly FRW, and it generates Weyl curvature where general relativity would have none. The Hamiltonian analysis shows that setting $P=0$ and $P\neq 0$ leads to two different branches: the parity-even branch is conformally invariant and has no new degrees of freedom beyond general relativity, while the parity-odd branch has one fewer constraint and one genuinely new degree of freedom. The paper concludes that a new torsional degree of freedom can act at the background cosmological level, with direct consequences for the expansion history and for observables such as cosmic polarization and lensing.

What carries the argument

The object that carries the argument is the parity-odd torsion component $P$, or its conformal version $c=Pa$. It is the only FRW-compatible torsional degree of freedom with no tetrad counterpart, so its canonical momentum vanishes; the constraint algebra must then be closed in one of two ways. Imposing $c=0$ produces Theory 2, with the extra constraint that represents conformal invariance, while forming the combination of constraints that commutes with the vanishing momentum produces Theory 1, with only the Hamiltonian constraint. The Chern-Simons time constructed from $b$ and $c$ and the momentum conjugate to $\Lambda^{-1}$ are what allow the reduced action to be written in a form where this branching and the degree-of-freedom count become visible.

What would settle it

Perform a full 3+1 Hamiltonian constraint analysis of the action (7) without imposing FRW symmetry and count the constraints in the parity-odd sector. If new first-class or second-class constraints appear that remove the $P$ mode or change the counting back to the general-relativity values, then the claimed new degree of freedom is an artifact of the symmetric reduction.

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Extended reading notes

Core claim

The central discovery is that the varying-$\Lambda$ Einstein-Cartan action (7), a self-dual combination of Palatini, Euler, Nieh-Yan, and Pontryagin terms with Immirzi parameter $\gamma$, admits Friedmann-Lemaitre-Robertson-Walker solutions with non-vanishing parity-odd torsion. With tetrad $e^0=dt$, $e^i=a\,dx^i$, homogeneity and isotropy permit $T^0=0$ and $T^i=-T\,e^0e^i+P\,\epsilon^i_{\ jk}e^je^k$; setting $P=0$ is an extra choice rather than a symmetry requirement. When $P\neq0$ the curvature has Weyl components proportional to $gP\,\epsilon^i_{\ jk}e^je^k$ and $(aP)^{\cdot}a^{-1}\epsilon^{ij}_{\ k}e^0e^k$, so FRW models can carry Weyl curvature. A Hamiltonian analysis of the FRW-reduced action then splits the theory into two branches: $P\neq0$ (Theory 1) has one Hamiltonian constraint and one degree of freedom in vacuum, two with matter, while $P=0$ (Theory 2) has an additional first-class constraint, conformal invariance, and no new degrees of freedom relative to general relativity. The paper concludes that the parity-odd mode is a genuinely new gravitational degree of freedom and that the two branches are effectively separate phases of the same theory.

Load-bearing premise

The load-bearing premise is that the degree-of-freedom count made from the homogeneous-isotropic reduced actions captures the full theory; if a full 3+1 analysis introduces additional spatial constraints, the parity-odd mode could be pure gauge or have different dynamics, and the central claim would fail.

Editorial extensions

If this is right

  • On the parity-odd branch, the statement that FRW spacetimes have zero Weyl curvature is not a symmetry theorem but a consequence of the field equations, and it fails in this class of torsion theories.
  • The parity-even branch has no new gravitational degree of freedom: in vacuum $\Lambda$ is pure gauge under conformal transformations, and with non-conformal matter $\Lambda$ is fixed by the matter density, so that branch is background-indistinguishable from general relativity.
  • Theory 1 carries one extra degree of freedom relative to general relativity both in vacuum and with matter, so the parity-odd mode can modify the expansion history without introducing a new matter field.
  • For $\gamma\to\infty$ the tracking solutions force the parity-odd variable to contribute an effective energy density equal to half of the dominant matter component, or to behave like spatial curvature, which conflicts with data; at small finite $\gamma$ the required fraction drops to $\sim\gamma^2/9$, making a viable cosmology possible.
  • A finite Immirzi/Pontryagin coupling generally rules out $P=0$ FRW solutions except in special cases, so turning on the Pontryagin term pushes the universe onto the parity-odd branch.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the Hamiltonian result survives a full 3+1 analysis, the parity-odd mode should also alter tensor and scalar perturbations, changing the graviton content relative to general relativity; the paper leaves this explicitly to future work.
  • A homogeneous isotropic Weyl background is a new observational handle: CMB TB polarization and vector-mode weak lensing, named in the paper as promising, would be natural places to look for this parity-odd signal.
  • Because $c^2$ enters the Hamiltonian constraint like negative spatial curvature, the parity-odd mode could masquerade as curvature or as a dark radiation component; separating it from $\Omega_k$ and $\Omega_r$ will be necessary in any observational test.
  • The two-branch structure implies a symmetry-induced phase split: turning on parity violation removes the conformal constraint and changes the number of propagating degrees of freedom, a feature worth probing in a quantum treatment if the full theory confirms the count.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies an Einstein-Cartan theory in which the cosmological constant is promoted to a variable through the introduction of torsion, with the dynamics defined by the action (7). It shows that under FRW symmetry the parity-odd torsion component P is allowed, leading to nonvanishing Weyl curvature even in homogeneous and isotropic models, Eqs. (24)-(25). The authors derive the FRW field equations (26)-(30), then restrict to |gamma| -> infinity and obtain non-self-dual vacuum solutions (47)-(49), tracking and non-tracking matter solutions with severe phenomenological problems (60)-(63), and a preliminary finite-gamma mechanism to suppress the parity-odd contribution. The Hamiltonian analysis of the FRW-reduced action, Eqs. (70)-(74), reveals two branches: the parity-even Theory 2, with two first-class constraints and no new degree of freedom, and the parity-odd Theory 1, with one first-class constraint and one additional degree of freedom; Section VII reproduces this structure as a bifurcation in the treatment of the second-class pair (c, Pi_c). The paper concludes that the parity-odd branch contains a genuinely new degree of freedom.

Significance. If the central claim is fully established, this is a substantial contribution. The paper shows explicitly that standard statements about FRW universes — zero Weyl curvature and forced parity invariance — fail once torsion is allowed, identifies the extra constraint of the parity-even branch with conformal invariance, and exhibits a concrete two-branch Hamiltonian structure in which a torsion connection component acts as a new degree of freedom. The internal derivations are explicit and largely coherent: the gamma-to-infinity equations (32)-(36), the non-SD vacuum solutions (47)-(49), the scaling solutions (60)-(63), and the constraint algebras in Sections V-VII are all checkable and appear consistent. The paper also makes falsifiable statements about the observational failures of the gamma-to-infinity tracking solutions and about the role of finite gamma in suppressing the parity-odd mode. The main caveat is that the headline 'genuinely new degree of freedom' is established only in the minisuperspace reduction, and the finite-gamma results are tied to one specific action realization; these limitations are acknowledged in the body but not in the abstract.

major comments (2)
  1. [Section V, Eqs. (70)-(73), Table I; Section VIII] The central claim that the parity-odd branch contains a genuinely new degree of freedom rests entirely on the canonical analysis of the FRW-reduced action, not on a full 3+1 Dirac-Bergmann analysis of the theory defined by action (7). The paper itself states in Step 2 of Section V that 'within this approximation (spatial homogeneity and isotropy), this exposes the fact that c is a connection degree of freedom which does not have a conjugate metric variable,' and Section VIII defers tensor and scalar perturbations to future work. A minisuperspace reduction freezes all spatial gradients and imposes isotropy before the canonical analysis, so spatial constraints, Bianchi identities, or conjugate pairs associated with vector/tensor modes in the full theory cannot be seen in Table I. Without a full canonical analysis, or at minimum a linearized analysis around the parity-odd FRW background, the abstract's statement that the parity violating branch 'contains a genuinely new degree of freedom' is not supported. The authors should either provide the missing analysis or explicitly restrict the claim to the minisuperspace model.
  2. [Section II, Eq. (7); Section IV.B, Eqs. (64)-(66)] The action (7) is explicitly described by the authors as 'a possible answer' and 'not the most general' realization of the stated requirements, with the Pontryagin prefactor admitted to be arbitrary up to dimensional analysis. Nevertheless, the finite-gamma results — including the tracking equations (64)-(65), the small-gamma solution R_c ~ gamma^2/9 in Eq. (66) and Fig. 1, and the conclusion in the Introduction that 'finite gamma is needed for a viable cosmology' — are derived from this particular realization with the Immirzi parameter tied to the Pontryagin coefficient. If the prefactor can indeed be any function of Lambda, as the text states, then the finite-gamma phenomenological rescue is not a robust prediction of the framework. The abstract and conclusions should identify clearly which claims are specific to the chosen action (7) and which would survive for other duality-invariant realizations.
minor comments (5)
  1. [Section III, after Eq. (30)] The text refers to 'equations (14)-(42)' and later to 'the Euler term in (42)'; these should be Eqs. (14)-(16) and Eq. (30), respectively, and the sentence about the last (Pontryagin) term in Eq. (42) should be reworded to refer to Eq. (30).
  2. [Section V.D, paragraph before Table I] The sentence 'For Theory 2 in the presence of radiation we have three two-class constraints, two second-class constraints' should read 'two first-class constraints and two second-class constraints'; Table I itself, with F=2 and S=2, is correct.
  3. [Section IV.A, Eqs. (24)-(25) and following text] The sentence 'From (24) and (25) we see that W_ij = 0 for these solutions, but W_01 ≠ 0' should refer to the components W_0i, since the Weyl tensor here has two tetrad indices; the notation W_01 conflicts with the earlier notation W_0i.
  4. [Section III, Eq. (15)] The notation T[a eb] in Eq. (15) is not defined; please specify the antisymmetrization convention for the tetrad index a.
  5. [Throughout] There are several typographical errors, including 'Enstein' in Section IV, 'orgin' in Section V, and 'manifestion' in the conclusions; a careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the parity-odd branch and its degree-of-freedom count are derived in-paper from the stated action and constraint algebra, not imported from prior fits or self-citations.

full rationale

The paper's central claim is that the parity-odd torsion component P produces a distinct Hamiltonian branch with one additional degree of freedom. This claim is derived in the paper itself. The parity-odd component is introduced as the generally allowed homogeneous and isotropic torsion form (Eqs. 3-4), not as a quantity defined by the desired conclusion. The field equations (26)-(30) are obtained by varying the stated action (7), and the new branch is found by solving the algebraic equation (71), which explicitly factorizes into c = 0 and V = 2Na/b. The Hamiltonian actions (73) and (74) are obtained by rewriting and eliminating variables within the paper, and the constraint counts in Table I follow from computed Poisson brackets such as (81), (131)-(135). Nothing is fitted to data, and no target quantity is used as an input. The self-citations to [1,2,15,16] supply the starting action and the Chern-Simons time motivation, but the branch analysis and degree-of-freedom counting are performed in this paper and do not reduce to those citations. The acknowledged limitation that the counting is done at the FRW-minisuperspace level, so that a full 3+1 analysis might reveal extra constraints, is a correctness or scope concern rather than circularity; the paper itself restricts the claim to homogeneous and isotropic space-times in Section VIII. No circular step is identifiable.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central claims rest on the stated action (7), the FRW symmetry reduction, the assumption that matter has no spin current, and the minisuperspace Hamiltonian analysis. No data are fitted; the only hand-chosen number is gamma (and the Pontryagin pre-factor), which is set to infinity for the main solutions.

free parameters (2)
  • Immirzi parameter gamma = Infinity for the main solutions; finite in Sections IV.C and V.E
    Controls the strength of the Pontryagin term in action (11). The core new-branch solutions take |gamma| to infinity; the proposed observational rescue uses finite gamma. It is a coupling constant chosen by hand, not fitted to data.
  • Pontryagin pre-factor coefficient = -3/(gamma*Lambda) in action (11)
    Section II states the pre-factor 'could be any function of Lambda' and the chosen form is selected for simplicity and aesthetic appeal; results may depend on this choice.
assumptions (5)
  • domain assumption The most general homogeneous and isotropic torsion is T0=0, Ti=-T e0 ei + P epsilon_ijk ej ek (Eqs. 3-4).
    Used in Section III to reduce the field equations to FRW; P is the parity-odd component whose existence is the basis of the paper.
  • domain assumption The matter action does not depend on the spin connection (delta S_M / delta omega = 0).
    Stated in Section II; it ensures no spin current sources torsion from matter and is called 'likely to be a very good approximation in cosmology'.
  • ad hoc to paper Action (7) is a valid realization of the four requirements and the duality (5) is the organizing symmetry.
    The entire derivation starts from this action; the paper explicitly notes it is not the most general action satisfying the requirements, so results may be action-dependent.
  • domain assumption The FRW minisuperspace action (70)/(73) faithfully represents the Hamiltonian constraint structure of the full theory.
    Sections V-VII count degrees of freedom from the symmetry-reduced action; no full 3+1 decomposition is given, and Section VIII scopes the result to homogeneous and isotropic spacetimes.
  • standard math Dirac's conjecture that all first-class constraints generate gauge symmetries.
    Used in Section VI to identify the extra constraint in Theory 2 with conformal invariance; the paper cites [26,27] for this.
invented entities (1)
  • Parity-odd torsion component c = Pa
    purpose: Carries the new parity-violating gravitational degree of freedom in FRW; produces Weyl curvature (Eqs. 24-25).
    This is a geometric component of the torsion tensor, previously considered by Cartan and by Baekler-Hehl-Nester, not a new particle. The paper proposes no quantitative observable signature; CMB TB and weak lensing are mentioned as promising but not computed.

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Pith. "Pith review of Parity violating Friedmann Universes." pith.science (2026). https://pith.science/paper/UEUGSBQ6

@misc{pith2026190805184,
  author       = {Pith},
  title        = {Pith review of: Parity violating Friedmann Universes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UEUGSBQ6}},
  note         = {Machine review of arXiv:1908.05184}
}
abstract

We revisit models, within the Einstein-Cartan theory, where the cosmological constant $\Lambda$ is promoted to a variable, at the cost of allowing for torsion even in the absence of spinors. We remark that some standard notions about FRW Universes collapse in these theories, most notably spatial homogeneity and isotropy may now co-exist with violations of parity invariance. The parity violating solutions have non-vanishing Weyl curvature even within FRW models. The presence of parity violating torsion opens up the space of possible such theories with relevant FRW modifications: in particular the Pontryagin term can play an important role even in the absence of spinorial matter. We present a number of parity violating solutions with and without matter. The former are the non-self dual vacuum solutions long suspected to exist. The latter lead to tracking and non-tracking solutions with a number of observational problems, unless we invoke the Pontryagin term. An examination of the Hamiltonian structure of the theory reveals that the parity even and the parity violating solutions belong to two distinct branches of the theory, with different gauge symmetries (constraints) and different numbers of degrees of freedom. The parity even branch is nothing but standard relativity with a cosmological constant which has become pure gauge under conformal invariance if matter is absent, or a slave of matter (and so not an independent degree of freedom) if non-conformally invariant matter is present. In contrast, the parity violating branch contains a genuinely new degree of freedom.

Figures

Figures reproduced from arXiv: 1908.05184 by the authors.

Figure 1
Figure 1. FIG. 1. Plot of a solution for [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Typical cosmological evolution of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

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Works this paper leans on

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