Pith. sign in

REVIEW 3 major objections 4 minor 2 cited by

Realization of quintom dark energy after DESI DR2 in Nieh-Yan modified teleparallel gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A Nieh-Yan coupling in teleparallel gravity erases the dark-energy perturbation at the w=-1 crossing, letting a single fluid or scalar realize quintom behavior without ghost or gradient instabilities, and predicting gravitational-wave biref

desk verdict Clever new mechanism to kill quintom instabilities by constraining away the dark-energy perturbation, but the paper contradicts itself on whether the key constraint is independent. read the letter →

arxiv 2602.00506 v3 pith:OGF7KRLU submitted 2026-01-31 gr-qc astro-ph.COhep-phhep-th

Yuxuan Kang , Mingzhe Li , Changzhi Yi This is my paper · ORCID
classification gr-qcastro-ph.COhep-phhep-th
keywords quintomdarkenergyNieh-Yanmodifiedteleparallelgravityw=-1crossingperturbationsghostinstabilitygradientgravitationalwavebirefringenceDESIDR2
topics Dark Energy
open problems Dark Energy
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

The paper addresses a known obstruction: single-component dark energy models that cross the cosmological-constant boundary w=-1 suffer unavoidable ghost or gradient instabilities in their perturbations. It proposes a way around this in Nieh-Yan modified teleparallel gravity, where dark energy couples to a topological torsion term. The coupling does not change the background Friedmann equations, but it generates a constraint that forces the gauge-invariant dark energy perturbation to vanish, deleting it from the set of dynamical fields. If correct, the quintom-B behavior preferred by recent baryon acoustic oscillation data can be realized by ordinary single-fluid or single-k-essence models, with the added prediction that left- and right-handed gravitational waves travel at different speeds.

What carries the argument

The central object is the Nieh-Yan density, the topological torsion invariant TλμνT~λμν = (1/2) εμνρσ Tλμν Tλρσ, coupled to dark energy with strength c. In the teleparallel formulation, the tetrad contains a pseudoscalar perturbation λ that does not appear in the metric. The coupling's quadratic action is linear in λ; since λ is non-dynamical, its equation of motion becomes the constraint H δφ_1 + ψ φ'_1 = 0, i.e. ζ_1 = 0. This constraint is the load-bearing device: it removes the dark-energy perturbation from the dynamical system, eliminating the ghost and gradient instabilities, while the Friedmann background is untouched because the Nieh-Yan density vanishes on the homogeneous, isotropic

What would settle it

Compute the exact quadratic Hamiltonian in a fixed physical gauge after integrating out λ; if the kinetic matrix for the remaining dark-energy and metric perturbations is positive definite, the dark-energy mode is genuinely absent, whereas a zero eigenvalue tied to local Lorentz gauge symmetry would mean the instability is merely hidden. A numerical lattice simulation of the full action crossing w=-1 would provide the same decisive test.

Watch

Extended reading notes

Core claim

The central claim is that the Nieh-Yan density TλμνT~λμν vanishes on the FRW background, so a shift-symmetric coupling c/2 φ TλμνT~λμν does not feed back into background dynamics, but at first order the tetrad pseudoscalar perturbation λ enters the quadratic action only through S_NY^(2) = -∫ d⁴x a² [4c ∂i∂i λ (H δφ_1 + ψ φ'_1)]. Varying with respect to λ gives the constraint H δφ_1 + ψ φ'_1 = 0, which is exactly the vanishing of the gauge-invariant curvature perturbation ζ_1 for dark energy. After imposing this constraint, the quadratic scalar action reduces to that of the matter sector alone; the dark-energy perturbation is no longer dynamical, so the negative sound-speed and wrong-sign kin

Load-bearing premise

The result rests on treating the tetrad pseudoscalar perturbation λ as an independent auxiliary field, so that its equation of motion H δφ_1 + φ'_1 Ψ = 0 is a genuine constraint that eliminates the dark-energy perturbation; the paper itself notes that the antisymmetric field equation behind this constraint is not independent of the symmetric equation, so if λ is a gauge artifact of local Lorentz symmetry, the removal of ζ1 may be a gauge choice rather than a physical eliminat

Editorial extensions

If this is right

  • In NYTG, a single perfect fluid whose equation of state follows the CPL form w0+wa(1-a) can cross w=-1 at z≈0.44 with no perturbative instability; only background equations need to be solved.
  • A single k-essence scalar with L = X + c1√X - V(φ) can also cross w=-1 smoothly in the constructed example, so the single-scalar no-go theorem is circumvented.
  • Dark energy perturbations vanish identically at linear order, so the dark-energy sector acts as a background-only component and matter perturbations obey a standard single-field action with a modified mass term.
  • Gravitational waves of opposite circular polarizations acquire different phase velocities, v_p ≈ 1 ± cφ'_1/k, a parity-violating signal that can be searched for in existing and future gravitational-wave data.
  • The bound from current binary black hole gravitational-wave events translates to |cφ'_1/a| < 3.25×10⁻⁴² GeV, giving a concrete target for future tests of the mechanism.

Reading between the lines

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

  • An implication the paper leaves implicit is that dark energy becomes effectively non-clustering: without a dark-energy perturbation, observables such as the integrated Sachs-Wolfe effect and CMB lensing would resemble a matter-only universe with a modified background expansion; this could be tested against cosmological data.
  • The paper itself notes that the antisymmetric field equation (21) is not independent of (20), suggesting the constraint may be tied to the theory's local Lorentz redundancy. If λ is a pure-gauge mode, the removal of ζ1 might be a gauge choice rather than a physical elimination, and a fully gauge-invariant Hamiltonian analysis would be needed to confirm the instability is truly gone.
  • The same coupling could be applied beyond dark energy, for example to suppress isocurvature perturbations in multifield inflation or to freeze density perturbations of an extra component; these applications are not explored in the paper but follow directly from the constraint mechanism.
  • The paper's stability demonstration is limited to linear perturbation theory. At nonlinear order, auxiliary fields can develop kinetic terms once integrated out, so a numerical lattice simulation or full Hamiltonian analysis of the NYTG action across the crossing would test whether the healthy behavior persists beyond quadratic order.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a way to evade the quintom no-go theorem by coupling single-fluid or single-k-essence dark energy to the Nieh-Yan density in teleparallel gravity (NYTG). Because the Nieh-Yan density vanishes on the FRW background, the background dynamics are unchanged; at the perturbative level the tetrad pseudoscalar perturbation lambda enters linearly and, when integrated out, yields a constraint H*delta_phi1 + psi*phi1' = 0, i.e. zeta1 = 0. The authors then reduce the quadratic scalar actions to a single matter-sector variable, Eqs. (41) and (55), concluding that the dark-energy perturbation is removed from the dynamical spectrum and the instabilities of Section II are avoided. Two toy models are presented as quintom-B examples, and a parity-violating gravitational-wave birefringence prediction is derived. The paper is clearly written and the strategy is interesting, but the central mechanism relies on treating the above constraint as genuinely independent, which the manuscript itself does not reconcile with its earlier statement that Eq. (21) is not independent of Eq. (20).

Significance. If the central claim holds, the result is significant: it would allow a single perfect fluid or a single k-essence-like scalar to cross w = -1 while remaining perturbatively stable, without introducing additional dark-sector degrees of freedom, and it would make gravitational-wave parity violation a generic observational signature. The paper benefits from explicit reduced quadratic actions, use of the already-published NYTG framework, and a concrete falsifiable prediction. The limitation is that the main claim is exactly where the manuscript's own text creates ambiguity, so my assessment of significance is conditional on a rigorous demonstration of the constraint structure.

major comments (3)
  1. [Section III, Eq. (21); Section IV B2, Eq. (37)] The load-bearing step is treating H*delta_phi1 + psi*phi1' = 0 as an independent constraint that eliminates zeta1. But Section III states that Eq. (21), the antisymmetric field equation from which this perturbed constraint descends, 'is not independent of Eq. (20)'. If Eq. (37) is merely a combination of the other perturbed Einstein/fluid equations, it is not a new constraint; initial data for zeta1 remain free and the negative-c_s^2/ghost instabilities of Section II still propagate. The manuscript never reconciles this contradiction. Please provide an explicit count of independent scalar constraints and dynamical degrees of freedom at linear order, for example by solving the complete linearized system without imposing Eq. (37), or by a Dirac-Bergmann analysis. The reduced action (41), obtained after imposing (37), cannot by itself prove the reduction.
  2. [Section IV B2, Eq. (36); Section III] The derivation integrates out lambda because it appears linearly in S_NY. However, lambda is a tetrad perturbation that does not appear in the metric and is invariant under the listed diffeomorphisms, Eq. (29). Whether lambda is a physical auxiliary field or a pure-gauge remnant of the local Lorentz redundancy in the Weitzenbock gauge is crucial: if it is pure gauge, varying the quadratic action with respect to it may not generate a physical constraint. The paper should identify the residual local Lorentz transformation associated with lambda and show explicitly that the constraint H*delta_phi1 + psi*phi1' = 0 survives as an independent equation after that redundancy is fixed.
  3. [Sections IV and V, stability claims] All stability conclusions are based on the quadratic action at linear perturbation order. The abstract and conclusion claim that the dark-energy perturbation is 'removed from the menu of dynamical degrees of freedom', but no nonlinear Hamiltonian analysis is provided. Please either supply a nonlinear constraint/Hamiltonian analysis or explicitly state that the claim is limited to linear order. As it stands, the reader cannot tell whether the elimination of zeta1 is an exact property of the theory or an artifact of the quadratic truncation.
minor comments (4)
  1. [Section V B 1, after Eq. (50)] The sentence 'These equations are formally identical to those in Eqs. (50)' is a self-reference; it should refer to the corresponding scalar perturbation equations in Section IV, e.g. Eqs. (32), or to a different numbered set.
  2. [Throughout] Typographical issues: 'straitforwardly' in Section II, 'the the gauge-invariant' after Eq. (37), and 'Nieh-Ya n' in the title. Please proofread.
  3. [Section IV C and V C, Figs. 3 and 4] The CPL parameters in the fluid toy model are taken as the DESI best fit, while the scalar toy model parameters are simply chosen; please state in the captions or text that both are illustrative and not fits to data.
  4. [Eq. (42)] After eliminating zeta1, the matter-sector mass term still contains c_s1, which diverges at the crossing. A brief comment explaining why this does not reintroduce a problem would help, since a reader might otherwise worry about a divergent coefficient in the reduced action.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; the central ζ1=0 result follows from varying the quadratic action and is not a fitted or self-referential input. Only minor non-load-bearing self-citations are present.

full rationale

The paper's central claim—that the Nieh-Yan coupling removes the dark-energy perturbation—is derived by a direct first-order calculation: the tetrad and matter perturbations are substituted into the action, the quadratic action (36) is obtained, and variation with respect to λ gives the constraint Hδφ1 + ψφ'1 = 0 (Eq. 37), which is equivalent to ζ1 = 0. Substituting this constraint into the quadratic action yields the reduced actions (41) and (55). No fitted parameter, observational input, or previously published result is used to obtain this constraint; DESI best-fit w0, wa enter only in the illustrative background toy models and are not used to predict the perturbation removal. The scalar-field toy model is also explicitly described as 'only an illustrative example.' Self-citations to [58,59] define the NYTG model and provide the local-Lorentz gauge discussion, but the relevant perturbation equations and quadratic actions are re-derived in this paper rather than imported as conclusions. A separate technical concern is the paper's statement that Eq. (21) is 'not independent of Eq. (20)' while later referring to its perturbed form as an 'additional constraint'; this is a consistency question about constraint counting, not a circular use of inputs or fits. Hence the circularity score is low, reflecting only minor self-citations that do not carry the derivation.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central mechanism requires only the Nieh-Yan coupling constant c and the perturbative constraint structure. The toy-model parameters (CPL w0,wa; scalar c1,Λφ,mφ,λφ and initial conditions) are illustrative inputs, not derived results. No new particles, forces, or dimensions are postulated.

free parameters (5)
  • c (Nieh-Yan coupling constant)
    Free coupling governing the strength of the dark-energy/Nieh-Yan interaction; unconstrained in the paper except by the GW bound |cφ'_1/a| < 3.25e-42 GeV.
  • w0, wa (CPL parameters) = w0=-0.667, wa=-1.09
    Taken from the DESI+CMB+Union3 best fit for the fluid toy model; not derived in the paper.
  • f0 (present dark energy density) = 8.63e-61
    Sets the overall dark energy scale in the fluid toy model; follows from the assumed Ωm0≈0.31 and h≈0.69.
  • Ωm0, h = 0.31, 0.69
    Assumed background cosmological parameters used to set the toy model evolutions.
  • Scalar toy model parameters (c1, Λφ, mφ, λφ, φ1ini, φdot1ini) = c1=-1.14e-60, Λφ=5.29e-121, mφ=1.12e-60, λφ=7.69e-121, φ1ini=0.6, φdot1ini=-8.06e-61
    Hand-picked to produce a quintom-B crossing at z≈0.44; labeled by the authors as illustrative.
assumptions (5)
  • domain assumption The diagonal FRW tetrad with Weitzenböck gauge is valid, and the Nieh-Yan density vanishes on this background.
    Used in Sec. IV.A with tetrad (24); if the background had nonzero Nieh-Yan density, the background evolution would be modified.
  • domain assumption The no-go theorem for single-fluid and single-k-essence quintom is correctly captured by the quadratic actions (12) and (16).
    Sec. II relies on this to motivate the cure; adopted from Refs. [6,18-23].
  • domain assumption Linear perturbation theory around FRW is sufficient to establish physical stability.
    Secs. IV.B and V.B use truncated quadratic actions; no nonlinear or full Hamiltonian analysis is provided.
  • ad hoc to paper Eq. (21) can be used as an additional constraint even though the text states it is not independent of Eq. (20).
    Sec. III after Eq. (21) and Sec. IV.B.1 Eq. (31); the constraint-counting is not fully reconciled.
  • standard math Local Lorentz invariance permits the Weitzenböck gauge and identifies λ as a genuine non-propagating auxiliary field.
    Sec. III and Refs. [58,59]; this is the load-bearing structure for the ζ1=0 constraint.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Realization of quintom dark energy after DESI DR2 in Nieh-Yan modified teleparallel gravity." pith.science (2026). https://pith.science/paper/OGF7KRLU

@misc{pith2026260200506,
  author       = {Pith},
  title        = {Pith review of: Realization of quintom dark energy after DESI DR2 in Nieh-Yan modified teleparallel gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGF7KRLU}},
  note         = {Machine review of arXiv:2602.00506}
}
abstract

Recent observations from the DESI Collaboration indicate a preference for quintom dark energy, i.e., its equation of state evolves across the cosmological constant boundary $w=-1$. It is well known that models with single perfect fluid or single scalar field minimally coupled to Einstein gravity develop perturbative instabilities around the crossing, thereby cannot realize the quintom scenario. In this paper, we propose a method to circumvent the instability problem of these models by considering the coupling of dark energy to the Nieh-Yan density within the framework of teleparallel gravity. We show that with this coupling the background evolution is not affected, but the dark energy perturbation is removed from the menu of dynamical degrees of freedom, thus avoiding the inherent difficulties in the old models. Furthermore, the Nieh-Yan coupling causes parity violation in gravitational waves, and this can be considered as a clear prediction of this mechanism.

Figures

Figures reproduced from arXiv: 2602.00506 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Correlated parity violation in gravity and electromagnetism from five-dimensional spacetime

    gr-qc 2026-08 conditional novelty 7.0 of 10

    A five-dimensional teleparallel Kaluza-Klein action produces a single parameter-free prediction: the electromagnetic helicity dispersion shift is exactly six times the gravitational one.

  2. Dark Energy in the DESI Era: A Brief Review of Evidence, Beyond-$\Lambda$CDM Interpretations, and Tensions

    astro-ph.CO 2026-06 unverdicted novelty 3.0 of 10

    Review of DESI evidence for dynamical dark energy, its dependence on parametrization and datasets, and alternative beyond-LambdaCDM interpretations that may address cosmological tensions.

Reference graph

Works this paper leans on

85 extracted references · 50 linked inside Pith · cited by 2 Pith papers

  1. [1]

    The exist ence of ghost mode makes the Hamiltonian unbounded from below, leading to severe problems in q uantization

    This means the dark energy perturbation ζ1 is a ghost mode because it has a kinetic term with wrong sign. The exist ence of ghost mode makes the Hamiltonian unbounded from below, leading to severe problems in q uantization. Consequently, dark energy exhibiting quintom behavior cannot be d escribed successfully by a single perfect fluid minimally coupled to...

  2. [2]

    The scalar perturbations of tetrad is parameterized as follows: [ 64, 65]: e0 0 =a (1 +A), e 0 i =a∂ iχ, e i 0 =a∂ iγ, e i j =a [(1 −ψ)δij +∂i∂jκ +ǫijk∂kλ]

    Linear perturbation equations for scalar perturbation We now turn to the analysis of cosmological perturbations in our mod el. The scalar perturbations of tetrad is parameterized as follows: [ 64, 65]: e0 0 =a (1 +A), e 0 i =a∂ iχ, e i 0 =a∂ iγ, e i j =a [(1 −ψ)δij +∂i∂jκ +ǫijk∂kλ]. (28) The resulting metric perturbations have the same form as those in ( ...

  3. [3]

    P. A. R. Ade, et al., Planck 2015 results. XIII. Cosmologi cal parameters, Astron. Astrophys. 594 (2016) A13. arXiv:1502.01589

    arXiv 2015

  4. [4]

    For our pur poses, it is convenient to choose the unitary gauge, δφ2 =κ = 0, such that ζ2 =ψ

    Quadratic action for scalar perturbations To demonstrate more explicitly that the dark energy perturbation s are free from instabilities, we will analyze the quadratic action for scalar perturbations in this model. For our pur poses, it is convenient to choose the unitary gauge, δφ2 =κ = 0, such that ζ2 =ψ. Before deriving the quadratic action, we briefly ...

  5. [5]

    A. G. Adame, et al., DESI 2024 VI: cosmological constrain ts from the measurements of baryon acoustic oscillations, J CAP 02 (2025) 021. arXiv:2404.03002

    arXiv 2024

  6. [6]

    (39) It is clear that the variables δα1, δα2, δβ1 and δβ2 completely decouple from the remaining fields

    + 2∂i∂iBψ ′ + 3ψ′2 + 6HAψ′ +aδn1δφ′ 1 − 3an1ψδφ′ 1 +an1∂iv1∂iδφ1 +a2A (f2,nδn2 +f1,nδn1) − 1 2a2n2f2,n (∂iB +∂iv2)2 − 1 2a2n1f1,n (∂iB +∂iv1)2 +a2A2 (f1 +f2) + 1 2a2f1,nnδn2 1 + 1 2a2f2,nnδn2 2 +an1δα1δβ ′ 1 +an2δα2δβ ′ 2 ] . (39) It is clear that the variables δα1, δα2, δβ1 and δβ2 completely decouple from the remaining fields. Moreover, the action (

  7. [7]

    Y. Cai, X. Ren, T. Qiu, M. Li, X. Zhang, The Quintom theory o f dark energy after DESI DR2 (5 2025). arXiv:2505.24732

    arXiv 2025

  8. [8]

    Linear perturbation equations for scalar perturbation In this subsection, we investigate cosmological perturbations in th e model ( 46) to determine whether the instabilities in dark energy perturbation can be avoid. The tetrad perturbatio ns take the same form as those in ( 28), and the gauge invariant variables considered in the following analysis are: ...

Show all 85 references
  1. [9]

    Lodha, et al., Extended Dark Energy analysis using DES I DR2 BAO measurements (3 2025)

    K. Lodha, et al., Extended Dark Energy analysis using DES I DR2 BAO measurements (3 2025). arXiv:2503.14743

    2025 arXiv

  2. [10]

    The variable λI still behaves like a viscosity, generating a difference between the perturbations Φ and Ψ

    These equations are formally identical to those in Eqs.( 50), with the only difference being that the dark energy sector is now associated only with the scalar field φ1. The variable λI still behaves like a viscosity, generating a difference between the perturbations Φ and Ψ. Now...

  3. [11]

    The constraint given in Eq.(

  4. [12]

    Ratra, P

    B. Ratra, P. J. E. Peebles, Cosmological Consequences o f a Rolling Homogeneous Scalar Field, Phys. Rev. D 37 (1988) 3406

    1988

  5. [13]

    Wetterich, Cosmology and the Fate of Dilatation Symm etry, Nucl

    C. Wetterich, Cosmology and the Fate of Dilatation Symm etry, Nucl. Phys. B 302 (1988) 668–696. arXiv:1711.03844

    1988 arXiv

  6. [14]

    For simplicity we choose the unitary gauge, δφ2 = κ = 0, so that ζ2 = ψ

    Quadratic action for scalar perturbation We now proceed to analyze the quadratic action for scalar perturb ations in order to demonstrate more explicitly that dark energy perturbation in this model is free from instabilities . For simplicity we choose the unitary gauge, δφ2 = ...

  7. [15]

    + 2∂i∂iBψ ′ + 6HAψ′ + 3ψ′2 +A ( a2f2,nδn2 +ρ,Xφ′ 1δφ′ 1 ) − 1 2a2n2f2,n (∂iB +∂iv2)2 − 1 2ρ,Xδφ′ 2 1 + 1 2P,X∂iδφ1∂iδφ1 +P,Xφ′ 1∂iB∂iδφ1 +P,Xφφ′ 1δφ1(φ′ 1A −δφ′

  8. [16]

    In particular, the gener al solution for δβ2 is determined by its initial value since action (

    + 3P,Xφ′ 1δφ′ 1ψ − 1 2ρ,Xφ′ 1 2A2 +a2A2 (2XP ,X −P +f2) −a2P,φ1(A − 3ψ)δφ1 − 1 2a2P,φ1φ1δφ2 1 + 1 2a2f2,nnδn2 2 +an2δα2δβ ′ 2 ] , (53) The variables δα2 and δβ2 completely decouple from the other fields. In particular, the gener al solution for δβ2 is determined by its initial ...

  9. [17]

    Armendariz-Picon, V

    C. Armendariz-Picon, V. F. Mukhanov, P. J. Steinhardt, A Dynamical solution to the problem of a small cosmological constant and late time cosmic acceleration, Phys. Rev. Lett . 85 (2000) 4438–4441. arXiv:astro-ph/0004134

    2000 arXiv

  10. [18]

    Hence the vector sector contains no dynamical degrees of freedom, as in GR

    does not contain any vector perturbation [ 58]. Hence the vector sector contains no dynamical degrees of freedom, as in GR. We next consider the tensor sector , where tensor perturbations arise only from the tetrad ei j: ei j =a ( δij + 1 2hT ij ) . (62) 15 The tensor perturba...

  11. [19]

    Hu, Crossing the phantom divide: Dark energy interna l degrees of freedom, Phys

    W. Hu, Crossing the phantom divide: Dark energy interna l degrees of freedom, Phys. Rev. D 71 (2005) 047301. arXiv:astro-ph/0410680

    2005 arXiv

  12. [20]

    are listed below: 6H(HΦ + Ψ′) − 2∂i∂iΨ = −a2ρ,φ1δφI 1 +ρ,Xφ′ 1(φ′ 1Φ −δφI′ 1 ) −a2δρI 2, 2Ψ′ + 2HΦ = P,Xφ′ 1δφI 1 −a2(1 +w2)ρ2vI 2, 2Ψ′′ + 4HΨ′ + 2HΦ′ + 2(H2 + 2H′)Φ = a2P,φ1δφI 1 +P,Xφ′ 1(δφI′ 1 −φ′ 1Φ) +a2δP I 2 , Φ − Ψ = 2cφ′ 1λI, (50) where δρI 2 and δP I 2 are associated ...

  13. [21]

    A. G. Riess, et al., Observational evidence from superno vae for an accelerating universe and a cosmological constan t, Astron. J. 116 (1998) 1009–1038. arXiv:astro-ph/9805201. 16

    1998 arXiv

  14. [22]

    Perlmutter, et al., Measurements of Ω and Λ from 42 High Redshift Supernovae, Astrophys

    S. Perlmutter, et al., Measurements of Ω and Λ from 42 High Redshift Supernovae, Astrophys. J. 517 (1999) 565–586. arXiv:astro-ph/9812133

    1999 arXiv

  15. [23]

    As a result, varying the action S(2) N Y with respect to λ leads to the unique constraint: Hδφ1 +ψφ′ 1 = 0

    is identical to that of the model ( 8), the variable λ, which is purely a tetrad perturbation and does not appear in the metric, contribut es only to the quadratic action S(2) N Y . As a result, varying the action S(2) N Y with respect to λ leads to the unique constraint: Hδφ1...

  16. [24]

    Aghanim, et al., Planck 2018 results

    N. Aghanim, et al., Planck 2018 results. VI. Cosmologica l parameters, Astron. Astrophys. 641 (2020) A6, [Erratum: Astron.Astrophys. 652, C4 (2021)]. arXiv:1807.06209

    2018 arXiv

  17. [25]

    Feng, X.-L

    B. Feng, X.-L. Wang, X.-M. Zhang, Dark energy constraint s from the cosmic age and supernova, Phys. Lett. B 607 (2005) 35–41. arXiv:astro-ph/0404224

    2005 arXiv

  18. [26]

    are δnI ′ 1 + 3Hδn1 +n1∂i∂ivI − 3n1Ψ′ − 4cH∂i∂iλI a = 0, δα I ′ 1 = 0, δφI ′ +af1,nΦ +af1,nnδnI 1 = 0, δφ I 1 −af1,nvI 1 = 0, δβ I ′ 1 = 0. (33) 9 As mentioned, the equations in ( 26) are jointly equivalent to Eq.( 22), and it can be easily shown that the corresponding perturb...

  19. [27]

    Abdul Karim, et al., DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constrai nts (3 2025)

    M. Abdul Karim, et al., DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constrai nts (3 2025). arXiv:2503.14738

    2025 arXiv

  20. [28]

    M.-z. Li, B. Feng, X.-m. Zhang, A Single scalar field mode l of dark energy with equation of state crossing -1, JCAP 12 (2005) 002. arXiv:hep-ph/0503268

    2005 arXiv

  21. [29]

    Gu, et al., Dynamical Dark Energy in light of the DESI D R2 Baryonic Acoustic Oscillations Measurements (4 2025)

    G. Gu, et al., Dynamical Dark Energy in light of the DESI D R2 Baryonic Acoustic Oscillations Measurements (4 2025). arXiv:2504.06118

    2025

  22. [30]

    B. Popovic, et al., The Dark Energy Survey Supernova Pro gram: A Reanalysis Of Cosmology Results And Evidence For Evolving Dark Energy With An Updated Type Ia Supernova Calib ration (11 2025). arXiv:2511.07517

    2025 arXiv

  23. [31]

    M. Li, T. Qiu, Y. Cai, X. Zhang, On dark energy models of si ngle scalar field, JCAP 04 (2012) 003. arXiv:1112.4255. 17

    2012 arXiv

  24. [32]

    Langlois, M

    D. Langlois, M. Mancarella, K. Noui, F. Vernizzi, Effect ive Description of Higher-Order Scalar-Tensor Theories, J CAP 05 (2017) 033. arXiv:1703.03797

    2017 arXiv

  25. [33]

    R. R. Caldwell, R. Dave, P. J. Steinhardt, Cosmological imprint of an energy component with general equation of stat e, Phys. Rev. Lett. 80 (1998) 1582–1585. arXiv:astro-ph/9708069

    1998 arXiv

  26. [34]

    R. R. Caldwell, A Phantom menace?, Phys. Lett. B 545 (200 2) 23–29. arXiv:astro-ph/9908168

    arXiv

  27. [35]

    Chiba, T

    T. Chiba, T. Okabe, M. Yamaguchi, Kinetically driven qu intessence, Phys. Rev. D 62 (2000) 023511. arXiv:astro-ph/9912463

    2000 arXiv

  28. [36]

    Deffayet, G

    C. Deffayet, G. Esposito-Farese, A. Vikman, Covariant G alileon, Phys. Rev. D 79 (2009) 084003. arXiv:0901.1314

    2009 arXiv

  29. [37]

    Vikman, Can dark energy evolve to the phantom?, Phys

    A. Vikman, Can dark energy evolve to the phantom?, Phys. Rev. D 71 (2005) 023515. arXiv:astro-ph/0407107

    2005 arXiv

  30. [38]

    Bamba, C.-Q

    K. Bamba, C.-Q. Geng, C.-C. Lee, L.-W. Luo, Equation of s tate for dark energy in f (T ) gravity, JCAP 01 (2011) 021. arXiv:1011.0508

    2011 arXiv

  31. [39]

    DESI+CMB+Union3

    implies that δβ1 and δβ2 are time-independent, so their general solutions are determined b y their initial 10 values. Consequently, these variables can be be safely neglected in the following analysis. The variables A, B, δni, vi and λ are non-dynamical and give rise to the fo...

  32. [40]

    Zhao, J.-Q

    G.-B. Zhao, J.-Q. Xia, M. Li, B. Feng, X. Zhang, Perturba tions of the quintom models of dark energy and the effects on observations, Phys. Rev. D 72 (2005) 123515. arXiv:astro-ph/0507482

    2005 arXiv

  33. [41]

    Xia, Y.-F

    J.-Q. Xia, Y.-F. Cai, T.-T. Qiu, G.-B. Zhao, X. Zhang, Co nstraints on the Sound Speed of Dynamical Dark Energy, Int. J. Mod. Phys. D 17 (2008) 1229–1243. arXiv:astro-ph/0703202

    2008 arXiv

  34. [42]

    Y.-F. Cai, E. N. Saridakis, M. R. Setare, J.-Q. Xia, Quin tom Cosmology: Theoretical implications and observations , Phys. Rept. 493 (2010) 1–60. arXiv:0909.2776

    2010 arXiv

  35. [43]

    Qiu, Theoretical Aspects of Quintom Models, Mod

    T. Qiu, Theoretical Aspects of Quintom Models, Mod. Phy s. Lett. A 25 (2010) 909–921. arXiv:1002.3971

    2010 arXiv

  36. [44]

    Guo, Y.-S

    Z.-K. Guo, Y.-S. Piao, X.-M. Zhang, Y.-Z. Zhang, Cosmol ogical evolution of a quintom model of dark energy, Phys. Let t. B 608 (2005) 177–182. arXiv:astro-ph/0410654

    2005 arXiv

  37. [45]

    Wei, R.-G

    H. Wei, R.-G. Cai, D.-F. Zeng, Hessence: A New view of qui ntom dark energy, Class. Quant. Grav. 22 (2005) 3189–3202. arXiv:hep-th/0501160

    2005 arXiv

  38. [46]

    M. Z. Mughal, I. Ahmad, A multi-field tachyon-quintom mo del of dark energy and fate of the universe, Eur. Phys. J. Plus 136 (5) (2021) 581. arXiv:2012.13429

    2021 arXiv

  39. [47]

    J. A. V´ azquez, D. Tamayo, G. Garcia-Arroyo, I. G´ omez-Vargas, I. Quiros, A. A. Sen, Coupled multiscalar field dark e nergy, Phys. Rev. D 109 (2) (2024) 023511. arXiv:2305.11396

    2024 arXiv

  40. [48]

    Zhang, T

    X.-F. Zhang, T. Qiu, Avoiding the big-rip jeopardy in a q uintom dark energy model with higher derivatives, Phys. Let t. B 642 (2006) 187–191. arXiv:astro-ph/0603824

    2006 arXiv

  41. [49]

    Cai, M.-z

    Y.-f. Cai, M.-z. Li, J.-X. Lu, Y.-S. Piao, T.-t. Qiu, X.- m. Zhang, A String-Inspired Quintom Model Of Dark Energy, Ph ys. Lett. B 651 (2007) 1–7. arXiv:hep-th/0701016

    2007 arXiv

  42. [50]

    Therefore, no instabilities arise in dark energy perturbation at the crossing

    demonstrate that both metric perturbations Ψ and Φ remain finite when EoS parameter of dark energy crosses −1, i.e., P,X = 0. Therefore, no instabilities arise in dark energy perturbation at the crossing

  43. [51]

    Furthermore, the equations in (

    indicates that δφI 1 can be express in terms of the metric perturbation Ψ, implying that δφI 1 does not represent an independent propagating degree of freed om at linear order, when it couples to the tetrad field through the Nieh-Yan density. Furthermore, the equations in (

  44. [52]

    Langlois, M

    D. Langlois, M. Mancarella, K. Noui, F. Vernizzi, Mimet ic gravity as DHOST theories, JCAP 02 (2019) 036. arXiv:1802.03394

    2019 arXiv

  45. [53]

    Therefore, these variables ca n be safely neglected in the subsequent analysis

    forces it to be time-independent. Therefore, these variables ca n be safely neglected in the subsequent analysis. In addition, it can be directly found out that A, B, δn2, v2 and λ are non-dynamical fields. Variation of the quadratic action ( 53) with respect to these variables...

  46. [54]

    G. W. Horndeski, Second-order scalar-tensor field equa tions in a four-dimensional space, Int. J. Theor. Phys. 10 (1 974) 363–384

  47. [55]

    Nicolis, R

    A. Nicolis, R. Rattazzi, E. Trincherini, The Galileon a s a local modification of gravity, Phys. Rev. D 79 (2009) 06403 6. arXiv:0811.2197

    2009 arXiv

  48. [56]

    Deffayet, X

    C. Deffayet, X. Gao, D. A. Steer, G. Zahariade, From k-ess ence to generalised Galileons, Phys. Rev. D 84 (2011) 064039 . arXiv:1103.3260

    2011 arXiv

  49. [57]

    H. T. Nieh, M. L. Yan, An Identity in Riemann-cartan Geom etry, J. Math. Phys. 23 (1982) 373

    1982

  50. [58]

    Y.-F. Cai, S. Capozziello, M. De Laurentis, E. N. Sarida kis, f(T) teleparallel gravity and cosmology, Rept. Prog. P hys. 79 (10) (2016) 106901. arXiv:1511.07586

    2016 arXiv

  51. [59]

    Arora, P

    S. Arora, P. K. Sahoo, Crossing Phantom Divide in f(Q) f (Q) Gravity, Annalen Phys. 534 (8) (2022) 2200233. arXiv:2206.05110

    2022 arXiv

  52. [60]

    Basilakos, A

    S. Basilakos, A. Paliathanasis, E. N. Saridakis, Equiv alence of f(Q) cosmology with quintom-like scenario: The ph antom field as effective realization of the non-trivial connection , Phys. Lett. B 868 (2025) 139658. arXiv:2503.19864

    2025 arXiv

  53. [61]

    Paliathanasis, Testing non-coincident f(Q)-gravi ty with DESI DR2 BAO and GRBs, Phys

    A. Paliathanasis, Testing non-coincident f(Q)-gravi ty with DESI DR2 BAO and GRBs, Phys. Dark Univ. 49 (2025) 10199 3. arXiv:2504.11132

    2025

  54. [62]

    Li, G.-H

    T.-N. Li, G.-H. Du, Y.-H. Li, P.-J. Wu, S.-J. Jin, J.-F. Z hang, X. Zhang, Probing the sign-changeable interaction be tween dark energy and dark matter with DESI baryon acoustic oscill ations and DES supernovae data, Sci. China Phys. Mech. Astron. 69 (1) (2026) 210413. arXiv:2...

    2026

  55. [63]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, V. K. Oikonomou, Phantom cros sing and oscillating dark energy with F(R) gravity, Phys. Re v. D 112 (10) (2025) 104035. arXiv:2506.21010

    2025

  56. [64]

    Wang, R.-G

    J.-Q. Wang, R.-G. Cai, Z.-K. Guo, S.-J. Wang, Resolving the Planck-DESI tension by non-minimally coupled quintess ence (8 2025). arXiv:2508.01759

    2025 arXiv

  57. [65]

    From ( 66), one can find that gravitational waves with different helicities will ha ve different phase velocities: vA p = ωA/k ≈ 1 + pAcφ′ 1/k ≡ 1 + pAaMP V/(2k)

    is h′′ A + 2Hh′ A +ω2 AhA = 0, (66) where ω2 A = k2 + 2 pAcφ′ 1k. From ( 66), one can find that gravitational waves with different helicities will ha ve different phase velocities: vA p = ωA/k ≈ 1 + pAcφ′ 1/k ≡ 1 + pAaMP V/(2k). This phenomenon is known as velocity birefringence ...

  58. [66]

    Tsujikawa, Crossing the phantom divide in scalar-te nsor and vector-tensor theories (8 2025)

    S. Tsujikawa, Crossing the phantom divide in scalar-te nsor and vector-tensor theories (8 2025). arXiv:2508.17231

    2025

  59. [67]

    R. Chen, J. M. Cline, V. Muralidharan, B. Salewicz, Quin tessential dark energy crossing the phantom divide (8 2025) . arXiv:2508.19101

    2025

  60. [68]

    W. J. Wolf, P. G. Ferreira, C. Garc ´ ıa-Garc ´ ıa, Cosmological constraints on Galileon dark energy with broken shift symmetry (9 2025). arXiv:2509.17586

    2025 arXiv

  61. [69]

    S´ anchez L´ opez, A

    S. S´ anchez L´ opez, A. Karam, D. K. Hazra, Non-Minimall y Coupled Quintessence in Light of DESI (10 2025). arXiv:2510.14941

    2025

  62. [70]

    M. W. Toomey, E. Hughes, M. M. Ivanov, J. M. Sullivan, Kin etic Mixing and the Phantom Illusion: Axion-Dilaton Quintessence in Light of DESI DR2 (11 2025). arXiv:2511.23463

    2025

  63. [71]

    Paliathanasis, T

    A. Paliathanasis, T. Mengoni, G. Leon, O. Luongo, Const raints on Chiral-Quintom dark energy after DESI DR2 and impact on unifying dark energy with inflation (11 2025). arXiv:2512.00558

    2025

  64. [72]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, V. K. Oikonomou, Apparent Pha ntom Crossing in Gauss-Bonnet Gravity (12 2025). arXiv:2512.06279

    2025

  65. [73]

    T.-N. Li, W. Giar` e, G.-H. Du, Y.-H. Li, E. Di Valentino, J.-F. Zhang, X. Zhang, Strong Evidence for Dark Sector Interactions (1 2026). arXiv:2601.07361

    2026 arXiv

  66. [74]

    Tsujikawa, Realizing the phantom-divide crossing w ith vector and scalar fields (1 2026)

    S. Tsujikawa, Realizing the phantom-divide crossing w ith vector and scalar fields (1 2026). arXiv:2601.21274

    2026 arXiv

  67. [75]

    Aldrovandi, J

    R. Aldrovandi, J. G. Pereira, Teleparallel Gravity: An Introduction, Springer, 2013

    2013

  68. [76]

    Bahamonde, K

    S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rive ra, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. Levi Said , J. Mifsud, E. Di Valentino, Teleparallel gravity: from theo ry to cosmology, Rept. Prog. Phys. 86 (2) (2023) 026901. arXiv:2106.13793

    2023 arXiv

  69. [77]

    M. Li, H. Rao, D. Zhao, A simple parity violating gravity model without ghost instability, JCAP 11 (2020) 023. arXiv:2007.08038

    2020 arXiv

  70. [78]

    M. Li, H. Rao, Y. Tong, Revisiting a parity violating gra vity model without ghost instability: Local Lorentz covari ance, Phys. Rev. D 104 (8) (2021) 084077. arXiv:2104.05917. 18

    2021 arXiv

  71. [79]

    Jackiw, V

    R. Jackiw, V. P. Nair, S. Y. Pi, A. P. Polychronakos, Perf ect fluid theory and its extensions, J. Phys. A 37 (2004) R327–R432. arXiv:hep-ph/0407101

    2004 arXiv

  72. [80]

    De Felice, J.-M

    A. De Felice, J.-M. Gerard, T. Suyama, Cosmological per turbations of a perfect fluid and noncommutative variables, Phys. Rev. D 81 (2010) 063527. arXiv:0908.3439

    2010 arXiv

  73. [81]

    B. F. Schutz, Perfect Fluids in General Relativity: Vel ocity Potentials and a Variational Principle, Phys. Rev. D 2 (1970) 2762–2773

    1970

  74. [82]

    B. F. Schutz, R. Sorkin, Variational aspects of relativ istic field theories, with application to perfect fluids, Ann als Phys. 107 (1977) 1–43

    1977

  75. [83]

    M. Li, Z. Li, H. Rao, Ghost instability in the teleparall el gravity model with parity violations, Phys. Lett. B 834 (2 022) 137395. arXiv:2201.02357

    arXiv

  76. [84]

    Izumi, Y

    K. Izumi, Y. C. Ong, Cosmological Perturbation in f(T) G ravity Revisited, JCAP 06 (2013) 029. arXiv:1212.5774

    2013 arXiv

  77. [85]

    Q. Wu, T. Zhu, R. Niu, W. Zhao, A. Wang, Constraints on the Nieh-Yan modified teleparallel gravity with gravitational waves, Phys. Rev. D 105 (2) (2022) 024035. arXiv:2110.13870

    2022 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.