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Six-Field Rational Extended Thermodynamics of Polyatomic Gases in Curved Spacetime

T0 review · 2 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A kinetically-derived six-field relativistic fluid model is lifted to curved spacetime; its bulk-viscous pressure obeys the strong energy condition, so it cannot drive cosmic acceleration alone, but combined with Λ it recreates ΛCDM-like expansion.

desk verdict A genuinely new covariant RET6 model with a correct kinetic SEC no-go theorem and a de Sitter stability result; the main blemish is a correctable typo in Eq. (V.17). read the letter →

arxiv 2607.09463 v2 pith:KLA3XVWO submitted 2026-07-10 gr-qc math-phmath.MP

Luca Gallerani , Andrea Giusti , Andrea Mentrelli , Tommaso Ruggeri This is my paper · ORCID
classification gr-qcmath-phmath.MP
keywords modelpolyatomicspacetimeconstantdynamicalpressurerelativisticrelaxation
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

Rational extended thermodynamics (RET) builds fluid equations from kinetic theory instead of guessing them. The 'six-field' model used here tracks density, temperature, the fluid velocity, plus one extra quantity, the dynamical pressure Π, which measures how far internal molecular motions are from equilibrium. In a uniform expanding universe Π is the only non-equilibrium effect that survives.

The authors lift this model from special to general relativity by the standard 'minimal coupling' rule: replace flat-space derivatives with covariant ones. Their main structural result is a no-go theorem: any stress-energy tensor that comes from a non-negative distribution function (a requirement for a physically admissible kinetic closure) satisfies the strong energy condition. In an FLRW universe this means the scale factor cannot accelerate: the RET6 gas alone cannot replace dark energy. Adding a cosmological constant to make ΛRET6, they show for a diatomic gas with constant relaxation time that the system has a de Sitter fixed point and that it is locally stable using linear stability analysis. Numerical runs starting at recombination show the dynamical pressure relaxes quickly and the expansion history becomes practically indistinguishable from ΛCDM.

The main caveats: the maximum-entropy closure is only guaranteed valid near equilibrium, so the no-go theorem applies in the admissible kinetic regime; no code or data files are shipped; and one displayed eigenvalue formula is misprinted, though the corrected Jacobian still gives the claimed stability.

Extended reading notes

Core claim

Theorem 1 states that 'any stress-energy tensor induced by a non-negative relativistic one-particle distribution function satisfies the strong energy condition' in the polyatomic RET setting; consequently in FLRW 'accelerated expansion cannot arise from such polyatomic kinetic matter alone' (Corollary 1). Second: for the diatomic equation of state and constant τ>0, the ΛRET6 system admits a de Sitter fixed point at (x=0, ξ=0, Π̄=0) and it is locally stable. If correct, acceleration requires Λ-type physics, but late-time ΛRET6 expansion approaches ΛCDM.

Load-bearing premise

That the RET6 closure operates in its admissible kinetic regime, where the MEP distribution is non-negative. The authors state the closure is local and 'need not remain positive far from equilibrium' (Sec. II); Theorem 1 requires f≥0, and the numerical runs are said to remain close to local equilibrium. If the physical trajectories used in the stability or cosmology sections left that neighborhood, the no-go theorem and the ΛRET6 field equations would no longer be justified. This is an assumption about the model's domain of validity, not the logic of the theorem.

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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 extends the six-field Rational Extended Thermodynamics (RET6) model for relativistic polyatomic gases to arbitrary curved spacetimes by minimal coupling, couples it to Einstein gravity, and analyzes its FLRW cosmology. The two central results are (i) a kinetic-theory no-go theorem (Theorem 1) stating that any stress-energy tensor induced by a non-negative polyatomic one-particle distribution satisfies the strong energy condition, with the corollary that a RET6 gas alone cannot drive accelerated FLRW expansion; and (ii) a local stability proof for a de Sitter fixed point in the combined ΛRET6 model for the diatomic equation of state with constant positive relaxation time, supplemented by numerical integrations showing rapid convergence to ΛCDM for post-recombination initial data.

Significance. If correct, the paper supplies a clean structural result: within the kinetically admissible regime, polyatomic RET closures inherit the strong energy condition, so cosmic acceleration cannot be sourced by such kinetic matter alone and requires Λ-type physics or something beyond standard kinetic matter. The work also provides a generally covariant, kinetically derived bulk-viscosity-type cosmological model whose late-time dynamics can be checked explicitly. The paper does not fit parameters to force the ΛCDM convergence; τ and Π̄0 are model inputs and the convergence is a qualitative consequence of a rapidly damped transient. The analytic parts are largely machine-checkable: Theorem 1 is an elementary positivity argument, and the ΛRET6 stability analysis is based on an explicit Jacobian and low-temperature expansions. These are strengths worth acknowledging.

major comments (2)
  1. [§V.A, Eq. (V.17)] The displayed eigenvalue formula is inconsistent with the Jacobian (V.16). From the 2×2 thermodynamic block one obtains µ2,3 = (1/(2τ))[−2H_dS τ −1 ± sqrt(4H_dS²τ² − (4/5)H_dS τ + 1)], not the printed expression containing the spurious '5√5' term. The subsequent claim that the stability condition reduces to 24H_dS τ > 0 does not follow from the printed formula. The conclusion of local stability is nevertheless correct once the formula is corrected, since the discriminant is positive and both roots are negative for all H_dS,τ>0. Please correct Eq. (V.17) and the surrounding verification so that the displayed algebra supports the stated result.
  2. [Appendix B, Eq. (B.7)] The limits lim_{ξ→0} L1(ξ)/ξ² = −9/2 and lim_{ξ→0} L2(ξ)/ξ² = 2/3 are asserted as Taylor expansions but no derivation or reference is given. These limits are essential for the Jacobian and hence for the de Sitter stability result. Please provide an explicit derivation or a precise pointer to the relevant equations in Ref. [17] so that the expansion in Eq. (B.8) can be verified independently.
minor comments (5)
  1. [§VI] The numerical results are presented without code or data files; the data availability statement offers them only 'upon reasonable request.' Given that the equations and parameters are fully stated, this is not an obstacle to reproducibility in principle, but depositing a small script or data file would strengthen the paper.
  2. [§VI, Figs. 3–5] For the boundary-motivated initial data, the paper states that the closure remains meaningful because the trajectories remain close to local equilibrium, but no quantitative bound on Π̄ or on the MEP distribution's non-negativity is given. Since Theorem 1's hypothesis is f≥0 and the MEP closure is only guaranteed to be admissible near equilibrium, please add a sentence or a diagnostic quantifying the distance from equilibrium in these runs.
  3. [§V, Eq. (V.11)] The comparison of q with the Λ+dust model at fixed H is somewhat terse. In particular, the definition of the 4πGρ/H² factor and the treatment of c in the units would benefit from an explicit statement, to avoid ambiguity in the 'Δq' formula.
  4. [§VII] There is a grammatical typo: 'The explicit model is fixed once the relaxation time, have been specified.' should read '...once the relaxation time has been specified.' Similar small typos appear in the captions (e.g., 'Figs. 3(b)' vs 'Figure 4(b)').
  5. [§V.A, Eq. (V.17)] In addition to the algebraic error in the eigenvalue formula, the notation is inconsistent: the relaxation-time parameter is written as both τ and τ̄ in different places. Please harmonize throughout.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: SEC and de Sitter-stability results are derived from stated kinetic definitions and prior published closure, not from fitted inputs.

full rationale

The derivation chain is self-contained. Theorem 1's SEC result is obtained by direct contraction of the kinetic moment (II.5) with the reverse Cauchy–Schwarz inequality; the paper explicitly states that it is not claiming this positivity as new ('The point of the present theorem is therefore not to claim such positivity as a new fact for kinetic theory in general'), so this is an application of a classical kinetic fact to the polyatomic RET hierarchy, not a renamed-input prediction. Corollary 1 follows purely from the Raychaudhuri equation plus e>0 and p+Π≥0. The ΛRET6 fixed point at (0,0,0) and its local stability are derived by linearizing the system (V.12)–(V.14) with the low-temperature expansions of ω, L1, L2 in Appendix B; no parameter is fitted to produce the ΛCDM-like numerical result, and the text explicitly frames the numerical comparison as a proof-of-concept. The heavy reliance on prior RET papers ([16], [17], [25], [31]) is proper citation of published, independently derivable closure/structural results, not circularity: those results do not assume the SEC theorem or the de Sitter stability result. The only real defects are local and non-circular: Eq. (V.17)'s eigenvalue formula does not match the Jacobian (V.16) (the correct 2×2 eigenvalues are (1/2)[-(2H_dS+1/τ) ± sqrt(4H_dS^2 - (4/5)H_dS/τ + 1/τ^2)], both negative), and the Conclusions sentence 'The explicit model is fixed once the relaxation time, have been specified' is incomplete. The admissible-regime caveat (f≥0 only near equilibrium) is an explicit domain-of-validity condition, not a circularity.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The model carries over the dynamical pressure Π, the internal-energy variable I, and the RET6 principal-subsystem structure from prior literature; it introduces no new particle, force, field, or conserved quantity. The only adjustable inputs are the relaxation time and initial conditions, which are not fitted to the target results. The axioms are mostly domain assumptions standard for RET and kinetic theory, plus the usual mathematical inequalities.

free parameters (3)
  • Relaxation time τ (dimensionless τ̄=τH0) = varied; e.g., τ̄=1; constant positive
    BGK-type collision input from Ref. [31]; not derived in the paper; controls decay time of Π and size of non-equilibrium corrections; stability theorem assumes τ>0 constant.
  • Initial dynamical pressure Π̄0 = -0.9/γ0 ≈ -2.5×10^-10 (recombination); also -9, -0.9, 0.1 in illustrations
    Initial condition for the relaxation equation; chosen by hand as representative or to explore the hyperbolicity region; no theory fixes it.
  • Effective particle mass m = m_proton
    Needed to set γ0 = mc²/(k_B T0) in the numerical runs; the authors explicitly treat the matter as an effective diatomic test gas rather than a literal cosmic medium.
assumptions (8)
  • domain assumption Polyatomic Boltzmann-Chernikov equation with scalar internal-energy variable I and density of states φ(I) is the correct kinetic basis
    Sec. II; adopted from [16,17] without independent derivation.
  • domain assumption RET6 is a principal subsystem of RET15 obtained by setting λ⟨μν⟩=0; inherited closure relations and production term are valid
    Eqs. (II.6)-(II.10); from [17,25].
  • domain assumption BGK-type collision term with constant relaxation time τ and q=0 gives Q=(uαpα)/(c²τ)(f_E-f)
    Sec. II; from [31].
  • domain assumption Minimal coupling prescription (η→g, ∂→∇, no curvature terms in constitutive relations)
    Sec. III; standard but unproven for the internal-energy variable I.
  • domain assumption Admissible distribution functions are non-negative: f≥0, φ≥0, p future-directed on mass shell
    Theorem 1 hypothesis; defines 'admissible kinetic regime' used to derive SEC.
  • domain assumption Spatially flat FLRW geometry (K=0) and separately conserved radiation with p_R=e_R/3
    Secs. III and VI; observational motivation cited, but restricts the application.
  • standard math Low-temperature asymptotics for diatomic EOS: ω≈1+5ξ/2, L1/ξ²→ -9/2, L2/ξ²→2/3
    Appendix B; used to compute the Jacobian at the de Sitter fixed point.
  • standard math Reverse Cauchy-Schwarz inequality for future-directed timelike vectors
    Theorem 1 proof.

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Pith. "Pith review of Six-Field Rational Extended Thermodynamics of Polyatomic Gases in Curved Spacetime." pith.science (2026). https://pith.science/paper/KLA3XVWO

@misc{pith2026260709463,
  author       = {Pith},
  title        = {Pith review of: Six-Field Rational Extended Thermodynamics of Polyatomic Gases in Curved Spacetime},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KLA3XVWO}},
  note         = {Machine review of arXiv:2607.09463}
}
abstract

We formulate a generally covariant six-field Rational Extended Thermodynamics model (RET$_6$) for relativistic polyatomic gases, with the dynamical pressure as the only non-equilibrium variable. The model is based on a polyatomic extension of the Boltzmann-Chernikov kinetic equation, where the one-particle distribution depends also on an internal-energy variable, and on the Maximum Entropy closure of the associated relativistic moment hierarchy. The resulting field equations, closure relations, and production term are therefore fixed by the underlying kinetic structure rather than postulated phenomenologically. We extend the RET$_6$ model from Minkowski spacetime to a general curved spacetime by the minimal coupling prescription and couple it to the Einstein equations. As a first structural result, we prove a kinetic-theory no-go theorem in this polyatomic RET setting stating that any stress-energy tensor induced by a non-negative relativistic one-particle distribution function satisfies the strong energy condition. We then specialize the theory to a homogeneous and isotropic Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) spacetime. In this setting the dynamical pressure modifies the expansion dynamics with respect to the perfect-fluid Euler case, but the no-go theorem excludes acceleration driven by the RET$_6$ gas alone. Finally, we reintroduce a cosmological constant and study the combined $\Lambda$RET$_6$ model. For the diatomic equation of state and a constant positive relaxation time, we prove the existence and local stability of a de Sitter attractor at late times. Numerical integrations show that, for representative post-recombination initial data and constant relaxation times, the expansion history rapidly approaches that of $\Lambda$CDM, with small non-equilibrium corrections controlled by the relaxation time and by the initial value of the dynamical pressure.

Figures

Figures reproduced from arXiv: 2607.09463 by the authors.

Figure 1
Figure 1. Region in the (γ, Π) ¯ -plane for which 0 < λ2 < 1. The lower curve marks the boundary p + Π = 0; the dashed line Π = 25 ¯ /27 is the upper bound and Π = 0 ¯ the lower bound in the limit γ → ∞. being cV = dε/dT and cp, respectively, the specific heat at constant volume and at constant pressure, and L1 := A′ 1 A1 , L2 := θ ′ 0,2 − θ ′ 1,2 A1 , (II.23) with L1 = L1(γ) and L2 = L2(γ). Taking into account (II.1) and (II… view at source ↗
Figure 1
Figure 1. Region in the (γ, Π) ¯ -plane for which 0 < λ2 < 1. The lower curve marks the boundary p + Π = 0; the dashed line Π = 25 ¯ /27 is the upper bound and Π = 0 ¯ the lower bound in the limit γ → ∞. C. Subcharacteristic condition The Euler system is a principal subsystem of the RET6 model. Since the entropy is convex at equilibrium, by a theorem stated in [25] the subcharacteristic condition holds, namely λEq > λEul, ∀γ … view at source ↗
Figure 2
Figure 2. Phase portrait of the de Sitter attractor for the system (V.76), projected in the [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figures from the paper (13 more)
Figure 2
Figure 2. Figure 2: Phase portrait of the de Sitter attractor for the system (V.14), projected in the [PITH_FULL_IMAGE:figures/full_fig_p015_2.png]
Figure 3
Figure 3. Figure 3: Evolution of a, γ, Π¯ with initial conditions a0 = 1, γ0 = 1, Π¯ 0 = −0.9, and three different values of τ¯ (ΩM,0 ≡ ΩM,∗ = 0.32, ΩR,0 ≡ 0, H0 = 67.4 km s−1 Mpc−1 , m = mproton). A straightforward calculation then yields 24HdSτ > 0. (V.81) Since HdS > 0 and τ > 0, this …
Figure 3
Figure 3. Figure 3: Evolution of a, γ, Π¯ with initial conditions a0 = 1, γ0 = 1, Π¯ 0 = −0.9, and three different values of τ¯ (ΩM,0 ≡ ΩM,∗ = 0.32, ΩR,0 ≡ 0, H0 = 67.4 km s−1 Mpc−1 , m = mproton). order to make the non-equilibrium effects visible on the scale of the plots. This is useful…
Figure 4
Figure 4. Figure 4: Evolution of a, γ, Π¯ with initial conditions a0 = 1, γ0 = 10−1 , Π¯ 0 = −9, and three different values of τ¯ (ΩM,0 ≡ ΩM,∗ = 0.32, ΩR,0 ≡ 0, m = mproton) [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 4
Figure 4. Figure 4: Evolution of a, γ, Π¯ with initial conditions a0 = 1, γ0 = 10−1 , Π¯ 0 = −9, and three different values of τ¯ (ΩM,0 ≡ ΩM,∗ = 0.32, ΩR,0 ≡ 0, m = mproton) [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: Evolution of a, γ, Π¯ with initial conditions a0 = 1, γ0 = 1, Π¯ 0 = 0.1, and three different values of τ¯ (ΩM,0 ≡ ΩM,∗ = 0.32, ΩR,0 ≡ 0, m = mproton). magnitude of the initial non-equilibrium pressure set in the simulation of [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 5
Figure 5. Figure 5: Evolution of a, γ, Π¯ with initial conditions a0 = 1, γ0 = 1, Π¯ 0 = 0.1, and three different values of τ¯ (ΩM,0 ≡ ΩM,∗ = 0.32, ΩR,0 ≡ 0, m = mproton). The matter, radiation and cosmological constant density parameters are normalized at this initial time from the prese…
Figure 6
Figure 6. Figure 6: Comparison of the evolution of the scale factor [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 6
Figure 6. Figure 6: Comparison of the evolution of the scale factor [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Evolution of the temperature T and the dimensionless dynamical pressure Π¯ (same parameters and initial data as in [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 7
Figure 7. Figure 7: Evolution of the temperature T and the dimensionless dynamical pressure Π¯ (same parameters and initial data as in [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Evolution of the density parameters ΩM, ΩR, ΩΛ and of the deceleration parameter q obtained with the ΛRET6 model. Initial data: a0 = 1/ (1 + z0) with z0 = 1090 (recombination era), T0 = 2.7255/a0 ≈ 2973.5 K, γ0 = mc2 /kBT0 ≈ 3.7 × 109 , Π¯ 0 = −0.9/γ0 ≈ −2.5 × 10−10 (τ…
Figure 8
Figure 8. Figure 8: Evolution of the density parameters ΩM = ΩM,0 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

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Works this paper leans on

40 extracted references · 3 linked inside Pith

  1. [17]

    Arima, M

    T. Arima, M. C. Carrisi, S. Pennisi, and T. Ruggeri, Entropy24, 43 (2022)

    2022

  2. [1]

    G. V. Vereshchagin and A. G. Aksenov,Relativistic Kinetic Theory(Cambridge University Press, 2017)

    2017

  3. [2]

    Müller,Zur Ausbreitungsgeschwindigkeit von Störungen in kontinuierlichen Medien, Ph.D

    I. Müller,Zur Ausbreitungsgeschwindigkeit von Störungen in kontinuierlichen Medien, Ph.D. thesis, Technische Hochschule Aachen (1966)

    1966

  4. [3]

    Israel, Ann

    W. Israel, Ann. Phys.100, 310 (1976)

    1976

  5. [4]

    Müller and T

    I. Müller and T. Ruggeri,Rational Extended Thermodynamics, 2nd ed., Vol. 37 (Springer, 1998)

    1998

  6. [5]

    Ruggeri and M

    T. Ruggeri and M. Sugiyama,Classical and Relativistic Rational Extended Thermodynamics of Gases(Springer, 2021)

    2021

  7. [6]

    I. Liu, I. Müller, and T. Ruggeri, Ann. Phys.169, 191 (1986)

    1986

  8. [7]

    N. A. Chernikov, Acta Phys. Polon.27, 465 (1964). 24

    1964

Show all 40 references
  1. [8]

    J. L. Synge,The Relativistic Gas(North-Holland, Amsterdam, 1957)

    1957

  2. [9]

    Cercignani and G

    C. Cercignani and G. M. Kremer,The Relativistic Boltzmann Equation: Theory and Applications(Birkhäuser, Basel, 2002)

    2002

  3. [10]

    M. N. Kogan,Rarefied Gas Dynamics(Plenum Press, New York, 1969)

    1969

  4. [11]

    Dreyer, J

    W. Dreyer, J. Phys. A20, 6505 (1987)

    1987

  5. [12]

    Müller and T

    I. Müller and T. Ruggeri,Extended Thermodynamics, Vol. 37 (Springer, 1993)

    1993

  6. [13]

    C. D. Levermore, J. Stat. Phys.83, 1021 (1996)

    1996

  7. [14]

    Boillat and T

    G. Boillat and T. Ruggeri, Continuum Mech. Thermodyn.9, 205 (1997)

    1997

  8. [15]

    Dreyer, inISIMM Symposium on Kinetic Theory and Extended Thermodynamics, edited by I

    W. Dreyer, inISIMM Symposium on Kinetic Theory and Extended Thermodynamics, edited by I. Müller and T. Ruggeri (Pitagora Editrice, Bologna, 1987) pp. 107–123

    1987

  9. [16]

    Pennisi and T

    S. Pennisi and T. Ruggeri, Ann. Phys.377, 414 (2017)

    2017

  10. [18]

    Arima and T

    T. Arima and T. Ruggeri, inAdvances in Continuum Physics: In Memoriam Wolfgang Dreyer, edited by J. Fuhrmann, D. Hömberg, W. H. Müller, and W. Weiss (Springer, 2025) pp. 765–794

    2025

  11. [19]

    monatomic

    by means of the relativistic counterpart of Grad’s classical method [20] (see [4, 15, 17]). This agreement confirms that the macroscopic closure is not postulated independently of the microscopic kinetic description. For the broader mathematical and physical background on rela...

  12. [20]

    Marle, Annales de l’Institut Henri Poincaré

    C. Marle, Annales de l’Institut Henri Poincaré. Section A, Physique Théorique10, 127 (1969)

    1969

  13. [21]

    Grad, Commun

    H. Grad, Commun. Pure Appl. Math.2, 331 (1949)

    1949

  14. [22]

    A. M. Anile,Relativistic Fluids and Magneto-Fluids: With Applications in Astrophysics and Plasma Physics, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 1989)

    1989

  15. [23]

    G. M. Kremer, AIP Conf. Proc.1501, 160 (2012)

    2012

  16. [24]

    G. M. Kremer, J. Stat. Mech.2013, P04016 (2013), Errata: J. Stat. Mech. (2013) E05001, E10001

    2013

  17. [25]

    G. M. Kremer, Phys. A: Stat. Mech. Appl.393, 76 (2014)

    2014

  18. [26]

    Boillat and T

    G. Boillat and T. Ruggeri, Arch. Ration. Mech. Anal.137, 305 (1997)

    1997

  19. [27]

    Ehlers, inGeneral Relativity and Cosmology, edited by R

    J. Ehlers, inGeneral Relativity and Cosmology, edited by R. K. Sachs (Academic Press, New York, 1971) pp. 1–70

    1971

  20. [28]

    S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time(Cambridge University Press, Cambridge, 1973)

    1973

  21. [29]

    R. M. Wald,General Relativity(University of Chicago Press, Chicago, 1984)

    1984

  22. [30]

    Andréasson, Living Rev

    H. Andréasson, Living Rev. Relativ.14, 10.12942/lrr-2011-4 (2011), arXiv:1106.1367 [gr-qc]

    2011 arXiv

  23. [31]

    Sarbach and T

    O. Sarbach and T. Zannias, AIP Conference Proceedings1548, 134 (2013), arXiv:1303.2899 [gr-qc]

    2013 arXiv

  24. [32]

    Pennisi and T

    S. Pennisi and T. Ruggeri, Journal of Physics: Conference Series1035, 012005 (2018)

    2018

  25. [33]

    G. F. R. Ellis, R. Maartens, and M. A. H. MacCallum,Relativistic Cosmology(Cambridge University Press, 2012)

    2012

  26. [34]

    In Ref. [17], p. 20, two typographical errors are present. In Eq. (72), the factorΠshould appear inside the bracket after A1, as in Eq. (II.10) of the present paper. Furthermore, in the definition ofA1 between Eqs. (72) and (73), the last two terms in the numerator are written...

  27. [35]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions(Dover, New York, 1965)

    1965

  28. [36]

    Arima, S

    T. Arima, S. Taniguchi, T. Ruggeri, and M. Sugiyama, Phys. Lett. A376, 2799 (2012)

    2012

  29. [37]

    Pennisi and T

    S. Pennisi and T. Ruggeri, J. Math. Phys.59, 043102 (2018)

    2018

  30. [38]

    J. M. S. Oliveira, M. P. Machado Ramos, and A. J. Soares, Continuum Mech. Thermodyn.34, 681 (2022)

    2022

  31. [39]

    Astrophys.641, A6 (2020), arXiv:1807.06209 [astro-ph.CO]

    Planck Collaboration, Astron. Astrophys.641, A6 (2020), arXiv:1807.06209 [astro-ph.CO]

    2020 arXiv

  32. [40]

    A. A. Coley,Dynamical systems and cosmology(Springer, 2003)

    2003

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