Pith. sign in

REVIEW 2 major objections 4 minor 58 references

Dynamical analysis of bulk viscous cosmological model in $F(T)$ gravity

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read This paper claims that bulk viscous fluids in teleparallel $F(T)=(1+\gamma)T+\alpha(-T)^n$ gravity admit accelerating power-law and exponential solutions with de Sitter fixed points stable under explicit parameter conditions.

desk verdict Table 1's unconditional 'unstable' for s=1 is contradicted by the paper's own Eq. (28); the abstract's observational support claim is empty, but the model extension is legitimate. read the letter →

arxiv 2507.15754 v1 pith:J2PTDTX2 submitted 2025-07-21 gr-qc

Partha Sarathi Debnath , Chanchal Mondal , B. C. Paul This is my paper · ORCID
classification gr-qc MSC 83F0583D0535D40
keywords bulkviscosityF(T)gravityteleparallelEckarttheoryIsrael-StewartdeSitterfixedpointsdynamicalsystemsanalysislate-timecosmicacceleration
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 sets out to show that the late-time accelerating expansion of the universe can be produced by bulk viscosity within a modified torsion-based gravity, with no separate dark-energy fluid required. In flat Friedmann-Robertson-Walker spacetime with the teleparallel action $F(T) = (1+\gamma)T + \alpha(-T)^n$, where $T = -6H^2$ is the torsion scalar, it solves the viscous Friedmann equations for both the Eckart theory and the truncated Israel-Stewart theory of dissipative fluids. It claims accelerating power-law solutions $a(t) \sim t^D$ and exponential solutions $a(t) \sim e^{H_0 t}$, and it tabulates explicit stability conditions for the de Sitter fixed points as functions of the model parameters $\gamma$, $\alpha$, $n$, $\beta$, $s$, and $\omega$. A sympathetic reader would care because, if correct, the analysis makes this model family a viable phenomenological account of cosmic acceleration that connects modified gravity with dissipative fluid dynamics and reduces to ordinary viscous cosmology in the limits $n \to 1/2$, $\alpha \to 0$, $\gamma \to 0$.

What carries the argument

The load-bearing object is the bulk-viscous transport equation $\Pi + \tau\dot{\Pi} = -3\zeta H$ joined to the $F(T)$ Friedmann equations (13)-(14), closed by the ansatz $\zeta = \beta\rho^s$, $\tau = \beta\rho^{s-1}$ with constant $\beta > 0$ and $s > 0$. Because the torsion scalar is $T = -6H^2$, the torsion terms in the field equations become powers of $H$, so the system reduces to autonomous ordinary differential equations in the Hubble parameter: first-order, $\dot{H} = f(H)$, in Eckart theory ($\tau = 0$), and a two-dimensional system in $(H, y = \dot{H})$ in the truncated Israel-Stewart theory. Fixed points $H = H_0$ represent de Sitter phases, and stability is read from $f'(H_0) < 0$ or from the Jacobian matrix of the linearized system. Entropy production is evaluated through $\dot{S} = -3H\Pi/(n_1 T_1)$ with the barotropic temperature $T_1 = T_0 H^{\omega/(1+\omega)}$, which produces the positivity conditions mapped in Figure 1.

What would settle it

Two checks would settle the claim. First, a kinetic-theory or laboratory determination of the bulk viscosity of the relevant cosmic fluid as a function of $\rho$: if $\zeta(\rho)$ is not a pure power law, the fixed-point condition and Tables 1-2 no longer describe the system. Second, an observational fit: search the parameter space of Tables 1 and 2 for a point that reproduces the measured Hubble constant and deceleration parameter while keeping $\rho > 0$, $\Pi < 0$, and $|\Pi| \ll \rho$; if no such point exists, the statement that the models are supported by observations fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that bulk viscosity alone suffices to drive both power-law and exponential accelerated phases in $F(T) = (1+\gamma)T + \alpha(-T)^n$ gravity. Closing the field equations with the transport equation $\Pi + \tau\dot{\Pi} = -3\zeta H$ and the power-law ansatz $\zeta = \beta\rho^s$, $\tau = \beta\rho^{s-1}$, it derives explicit power-law solutions: in Eckart theory the exponent $D$ is given in closed form for the cases ($\alpha = 0$ or $n = 1/2$ with $s = 1/2$), ($n = 1$ with $s = 1/2$), and ($\gamma = -1$ with $s = (2n-1)/2n$), and acceleration $D > 1$ requires lower bounds on the viscosity coefficient $\beta$; the truncated Israel-Stewart theory yields analogous solutions with $D$ determined by a quadratic formula. For exponential expansion $a \sim e^{H_0 t}$, the de Sitter fixed points obey $3\beta H_0(A_\gamma H_0^2 - B_n H_0^{2n})^{s-1} = 1 + \omega$, with $A_\gamma = 3(1+\gamma)$ and $B_n = 3\alpha(2n-1)6^{n-1}$, and their stability is decided by $f'(H_0) < 0$ in Eckart theory and by the Jacobian trace and determinant in the Israel-Stewart case; the resulting stable windows are listed in Tables 1 and 2. The paper also derives entropy-evolution formulas whose positivity selects allowed parameter regions, and it states that the models are supported by observational results.

Load-bearing premise

The entire analysis rests on the ansatz that the viscosity coefficient and the relaxation time scale as exact power laws of the energy density, $\zeta = \beta\rho^s$ and $\tau = \beta\rho^{s-1}$ with constant $\beta > 0$ and $s > 0$; the paper offers no microphysical derivation, and if the real cosmic fluid's bulk viscosity does not follow that scaling, the fixed-point, stability, and entropy conclusions in Tables 1 and 2 stop applying.

Editorial extensions

If this is right

  • If the central claim holds, bulk viscosity alone can sustain an accelerated phase in this torsion-gravity family, with explicit windows on $\beta$, $s$, $\gamma$, $\alpha$, $n$, and $\omega$ where $D > 1$ or a stable de Sitter point exists.
  • The de Sitter fixed points are stable attractors under the tabulated conditions, so the exponential solutions survive small perturbations in the Hubble parameter rather than being fine-tuned.
  • The parameter regions with positive entropy production are thermodynamically viable, so requiring both stability and entropy growth can constrain the viscosity parameters in principle.
  • In the limits $n \to 1/2$, $\alpha \to 0$, $\gamma \to 0$, the field equations reduce to general-relativistic viscous cosmology, so the results connect continuously to standard cosmology.
  • Because higher $\beta$ favors accelerated power-law and exponential solutions in both Eckart and Israel-Stewart treatments, a measurement of the bulk-viscosity coefficient would directly test whether this mechanism can drive the observed acceleration.

Reading between the lines

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

  • Beyond the paper: the statement that the models are 'supported by observations' is not tied to any specific dataset in the text; a joint fit of $H_0$ and the deceleration parameter against the stable windows in Tables 1-2 would turn that assertion into a quantitative test.
  • Beyond the paper: intersecting the stability conditions with the positive-entropy windows would shrink the allowed parameter space further, since the paper presents these two sets of constraints separately.
  • Beyond the paper: the same fixed-point machinery could be applied to other torsion-gravity forms, such as $F(T)$ with a $T^2$ term, or to non-flat FRW geometries, where the de Sitter fixed-point structure would in general change.
  • Beyond the paper: if microphysics ever fixes the viscosity exponent $s$ to a specific value (simple kinetic models suggest special values such as $s = 1/2$), many stable windows in Tables 1-2 that require $s < 1/2$ would be excluded, giving a concrete way the model could be falsified.
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

2 major / 4 minor

Summary. The paper analyzes bulk-viscous FRW cosmologies in F(T) = (1+γ)T + α(−T)^n gravity. Using Eckart and truncated Israel-Stewart transport equations with ζ = βρ^s and τ = βρ^{s−1}, it derives power-law and exponential solutions, entropy-evolution formulas, and stability classifications for de Sitter fixed points. The abstract states that the resulting cosmological models are supported by observational results.

Significance. The algebraic framework is mostly self-consistent: the modified Friedmann equations (9)–(15) follow from the stated action, several fixed-point expressions and sign conditions in Tables 1 and 2 check out, and the paper carefully separates Eckart and truncated Israel-Stewart cases. If the stability classification were correct, the paper would provide a useful phenomenological map of viable viscous de Sitter models in torsional gravity. However, one row of the central Eckart stability table is demonstrably false as stated, and the observational-support claim has no quantitative basis; the paper's main claims therefore require revision.

major comments (2)
  1. [Abstract and §4] Table 1, row (v), classifies the s = 1 de Sitter fixed point H0 = (1+ω)/(3β) as unconditionally unstable for β > 0, ω > −1. This is false. For s = 1, Eq. (28) gives f(H) = 3ρ(3βH − (1+ω))/(2D) with ρ = AγH^2 − BnH^{2n} and D = Aγ − nBnH^{2n−2}; at the fixed point f'(H0) = 9βρ0/(2D0). The sign is determined by D0, not by β and ω alone. A concrete allowed counterexample is γ = 0, n = 2, α = 0.1, β = 0.1, ω = −0.8104, which yields H0 = 0.632, ρ0 = 0.336 > 0, D0 = −1.32, and f'(H0) = −0.1145 < 0, i.e., a stable de Sitter point. This point lies in the viable region of §2, case (iii), and the bracket in Eq. (30) is positive. Therefore row (v) is wrong, and the abstract's claim that the stability conditions in Tables 1–2 guarantee stable de Sitter solutions cannot stand. The full f'(H0) must be re-derived and all α ≠ 0 rows re-examined.
  2. [Abstract and §4] The abstract claims that "the cosmological models are supported by observations results," but the manuscript contains no observational dataset, no likelihood, no χ² statistic, and no error bars. The only evidence presented is algebraic derivation and parameter-space plots. This sentence should be removed or replaced by a quantitative comparison to data; as written, the claim is unsupported.
minor comments (4)
  1. [Eq. (17) and Tables 1–2] Equation (17) states s > 0, but Tables 1 and 2 list s = 0 cases. The paper should either remove the s = 0 entries or explicitly state that the ansatz is extended to s = 0 for those entries.
  2. [§3.1, Eq. (24) and following text] The text near Eq. (24) refers to "all the three coefficients (S1, S2, S3)," but Eq. (24) defines only S1 and S2. Please correct the reference or define S3.
  3. [§3.2 and Conclusion] The discussion in §3.2 says the right panel of Fig. 1 indicates that lower values of H0 and lower values of ω are suitable for TIS exponential evolution, while the Conclusion states that higher values of H0 and higher values of ω are suitable. These statements should be harmonized.
  4. [§3.2, Eq. (31)] Equation (31) is introduced as the truncated Israel-Stewart case with "ε = 0," but the parameter ε is never defined. Please clarify what truncation is being used and how it follows from Eq. (16).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the fixed points, stability conditions, and entropy constraints are solved from the stated field and transport equations under an explicit ansatz, not fitted or redefined.

full rationale

The derivation chain is self-contained. Starting from the F(T) action and FRW metric, the paper derives the field equations (11)-(14), defines the bulk viscous pressure through Eq. (15) as the difference between the effective pressure and the linear equation-of-state term, imposes the Eckart or truncated Israel-Stewart transport equation (16), and closes the system with the explicit power-law ansatz (17), zeta = beta rho^s and tau = beta rho^{s-1}. The later results are algebraic consequences of these equations: power-law exponents D solve Eq. (22) in special cases; exponential fixed points H0 are obtained from Eq. (27) in Eckart theory and Eq. (36) in TIS theory by setting H-dot = 0, i.e. 3 beta H0 rho0^{s-1} = 1 + omega; stability signs are computed from f'(H0) or the Jacobian of the autonomous system; and the entropy expressions (24), (30), (33), and (38) follow from the definitions (18)-(20). No parameter is fitted to a target observable, and no fixed point or stability condition is imported from the authors' prior work. The only self-reference, Ref. [15], is contextual and not load-bearing: it notes that a previous paper used F(T) = (1+gamma)T + alpha T^2, whereas the present work studies F(T) = (1+gamma)T + alpha(-T)^n. The abstract's statement that the models are supported by observations is not supported by any observational fitting procedure in the text, but that is an unsupported assertion rather than a circular derivation. A reader counterexample to Table 1 row (v) would indicate an algebraic or classification error, not a reduction of the claim to its own input; the stability analysis remains a straightforward linearization of the stated dynamical equation.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central results rest on six free model parameters and several domain assumptions. The viscosity ansatz ζ=βρ^s, τ=βρ^{s-1} is adopted ad hoc, as is the linear EoS p=ωρ. These are not derived from microphysics and are not constrained by data in the paper. No new particles or forces are introduced.

free parameters (6)
  • γ
    Coupling parameter in F(T)=(1+γ)T+α(−T)^n; restricted by viability cases, not fitted to data.
  • α
    Coefficient of the torsion power-law term; a free dimensional constant.
  • n
    Exponent in (−T)^n; scanned and constrained by cases, not fitted.
  • β
    Bulk viscosity coefficient; lower bounds for acceleration are derived, not fitted.
  • s
    Bulk viscosity exponent in ζ=βρ^s; special values 0, 1/2, 1, 2 are used.
  • ω
    Equation-of-state parameter for the linear EoS p=ωρ; restricted to [-1,1].
assumptions (5)
  • domain assumption Flat, homogeneous FRW geometry and linear equation of state p=ωρ for the cosmic fluid.
    Used throughout Sections 2 and 3; all solutions and stability results depend on these choices.
  • ad hoc to paper Bulk viscosity parameterization ζ=βρ^s and relaxation time τ=βρ^{s-1} with constant β>0, s>0.
    Introduced after Eq. (16) in Section 3; the derived power-law exponents, fixed points, and stability conditions depend directly on s.
  • ad hoc to paper Barotropic temperature T1=T0 H^{ω/(1+ω)} for entropy production.
    Used to derive entropy expressions in Eqs. (24), (30), (33), and (38); no justification is given beyond 'we consider'.
  • domain assumption Eckart and truncated Israel-Stewart transport equations Π+τ ˙Π = -3ζH, with τ=0 for Eckart.
    These are standard phenomenological choices in dissipative cosmology; the central solutions follow from them.
  • standard math Dynamical systems linearization: fixed points are stable if f'(H0)<0 in Eckart theory, and via Jacobian trace and determinant in TIS theory.
    Standard stability criterion used in Tables 1 and 2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dynamical analysis of bulk viscous cosmological model in $F(T)$ gravity." pith.science (2026). https://pith.science/paper/J2PTDTX2

@misc{pith2026250715754,
  author       = {Pith},
  title        = {Pith review of: Dynamical analysis of bulk viscous cosmological model in $F(T)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J2PTDTX2}},
  note         = {Machine review of arXiv:2507.15754}
}
abstract

Bulk viscous cosmological models is presented in the teleparallel ($F(T)$, where $T$ denotes torsion) gravity. In the teleparallel gravity, the Lagrangian of the gravitational action contains a general function $F(T)= T+ f(T)=(1+ \gamma) T+\alpha (-T)^n$, where $\gamma$, $n$ and $\alpha$ are dimensional constants. Cosmological solutions in Eckart theory and Truncated Israel Stewart theory are talked about in $F(T)=(1+ \gamma) T+\alpha (-T)^n$ gravity form which is one of the most generalized gravity form in the torsional gravity. The substantial and geometrical prospective of the cosmological models in Eckart theory and Truncated Israel Stewart theory in $F(T)$ gravity are deliberated for flat Friedmann-Robertson-Walker space time. Dynamical analysis of the fixed points of exponential expansion in $F(T)$ with bulk viscous cosmological models are studied here. The characteristics of the various cosmological parameters such as bulk viscous pressure, energy density, scale factor, Hubble parameter and entropy evolution are studied in Power law and Exponential models. Stability analysis of the exponential models are also argued with cosmic growth in the relevant theories by using directional plots. The cosmological models are supported by observations results.

Figures

Figures reproduced from arXiv: 2507.15754 by the authors.

Figure 1
Figure 1. In the region plots shadow regions are unsuitable for [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. The left panel shows stability analysis in Eckart the [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. The left panel shows stability analysis in TIS theory [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

58 extracted references · 50 canonical work pages

  1. [1]

    A. G. Riess, et al., Observational Evidence from Superno vae for an Accelerating Uni- verse and a Cosmological Constant, Astron. J. 116 (1998), 1009–1038

    1998

  2. [2]

    Padmanabhan, Cosmological constant—the weight of th e vacuum, Phys

    T. Padmanabhan, Cosmological constant—the weight of th e vacuum, Phys. Rept. 380 (2003), 235–320

    work page 2003

  3. [3]

    E. J. Copeland, M. Sami, S. Tsijikikawa, DYNAMICS OF DARK ENERGY, Int. J. Mod. Phys. D 15 (2006), 1753–1935

    2006

  4. [4]

    Nojiri and S

    S. Nojiri and S. D. Odintsov, Modified gravity with negati ve and positive powers of curvature: Unification of inflation and cosmic acceleration , Phys. Rev. D 68 (2003), 123512

    2003

  5. [5]

    A. A. Starobinsky, A new type of isotropic cosmological m odels without singularity, Phys. Lett. B 91 (1980), 99–102

    1980

  6. [6]

    Mukherjee, B

    S. Mukherjee, B. C. Paul, N. K. Dadhich, S. D. Maharaj and A . Beesham, Emergent universe with exotic matter, Classical and Quantum Gravity 23 (2006) 6927

    work page 2006

  7. [7]

    Nojiri, & S

    S. Nojiri, & S. D. Odintsov, INTRODUCTION TO MODIFIED GRA VITY AND GRA VITATIONAL ALTERNATIVE FOR DARK ENERGY, Int. J. Geom. Meth. Mod. Phys. 4 (2007) 115–145

    work page 2007

  8. [8]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, V. K. Oikonomou, Modified gravi ty theories on a nutshell: Inflation, bounce and late-time evolution, Phys. Rept. 692 (2017) 1–104. August 12, 2025 15:20 WSPC/INSTRUCTION FILE vis-T˙2˙Stability Dynamical analysis of bulk viscous cosmological model in F (T ) gravity 15

    work page 2017

Show all 58 references
  1. [9]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, Unified cosmic history in modifi ed gravity: From theory to Lorentz non-invariant models, Phys. Rept. 505 (2011) 59–144

    2011

  2. [10]

    Nojiri, & S

    S. Nojiri, & S. D. Odintsov, Modified f(R) gravity consis tent with realistic cosmology: From a matter dominated epoch to a dark energy universe, Phys. Rev. D 74 (2006) 086005

    2006

  3. [11]

    Harko, F

    T. Harko, F. S. N. Lobo, S. Nojiri, and S. D. Odintsov, f (R, T) gravity, Phys. Rev. D 84 (2011) 024020

    2011

  4. [12]

    R. K. Tiwari, A. Beesham, R. Singh, L. K. Tiwari, Time var ying G and Λ cosmology in f (R, T) gravity theory, Astrophys Space Sci 362 (2017) 143

    2017

  5. [13]

    B. Li, ,J. D. Barrow, and D. F. Mota, Cosmology of modified Gauss-Bonnet gravity, Phys. Rev. D 76 (2007) 044027

    2007

  6. [14]

    D. C. Maurya, Accelerating scenarios of viscous fluid un iverse in modified f(T) grav- ity, Int. J. Geom. Meth. Mod. Phys. 19 (2022) 2250144

    2022

  7. [15]

    P. S. Debnath, C. Mondal, B. C. Paul, Bulk Viscous Cosmol ogical model in F (T ) gravity, Int. J. Geom. Meth. Mod. Phys. 21 (2024) 2440036

    2024

  8. [16]

    Krssak, R

    M. Krssak, R. J. van den Hoogen, J. G. Pereira, C. G. Boehm er, A. A. Coley, Telepar- allel Theories of gravity: Illuminating a Fully Invariant A pproach, Class. Quant. Grav. 36 (2019) 186001

    2019

  9. [17]

    Capozziello, M

    Yi-Fu Cai, S. Capozziello, M. D. Laurentis and E. N. Sari dakis, f (T ) teleparallel gravity and cosmology, Rep. Prog. Phys. 79 (2016) 106901

    2016

  10. [18]

    C. Xu, E. N.Saridakis and G.Leon, Phase-Space analysis of Teleparallel Dark Energy, JCAP. 07 (2012) 005

    2012

  11. [19]

    M. A. Skugoreva, E. N. Saridakis and A. V. Toporensky, Dy namical features of scalar-torsion theories, Phys. Rev. D 91(2015) 044023

    2015

  12. [20]

    Cai, S.-H

    Y.-F. Cai, S.-H. Chen, J. B. Dent, S. Dutta, , & E. N. Sarid akis, Matter bounce cosmology with the f (T ) gravity, Classical and Quantum Gravity 28 (2011) 215011

    2011

  13. [21]

    Bamba, C.-Q

    K. Bamba, C.-Q. Geng, C.-C. Lee, & L.-W. Luo, Equation of state for dark energy in f (T ) gravity, JCAP. 01 (2011) 021

    2011

  14. [22]

    Ferraro, F

    R. Ferraro, F. Fiorini, Modified teleparallel gravity: inflation without inflaton, Phys. Rev. D 75 (2007) 084031

    2007

  15. [23]

    Paliathanasis, J

    A. Paliathanasis, J. D. Barrow and P.G.L. Leach, Cosmol ogical Solutions of f(T) Gravity, Phys. Rev. D 94 (2016) 023525

    2016

  16. [24]

    Basilakos, S

    S. Basilakos, S. Capozziello, M. De Laurentis, A. Palia thanasis and M. Tsamparlis, Noether symmetries and analytical solutions in f(T)-cosmo logy: A complete study Phys. Rev. D 88 (2013) 103526

    2013

  17. [25]

    Bamba, Geng C.-Q

    K. Bamba, Geng C.-Q. Thermodynamics of cosmological ho rizons in f(T ) gravity, JCAP. 11 (2011) 008

    2011

  18. [26]

    Capozziello, V

    S. Capozziello, V. F. Cardone, H. Farajollahi, A. Ravan pak, Cosmography in f(T)- gravity, Phys. Rev. D 84 (2011) 043527

    2011

  19. [27]

    Capozziello, P

    S. Capozziello, P. A. Gonzalez, E. N. Saridakis and Y. Va squez, Exact charged black- hole solutions in D-dimensional f (T ) gravity: torsion vs curvature analysis, JHEP 2013, 039 (2013)

    2013

  20. [28]

    Nesseris, S

    S. Nesseris, S. Basilakos, E. N. Saridakis and L. Perivo laropoulos, Variable f (T ) models are practically indistinguishable from ΛCDM, Phys. Rev. D 88 (2013) 103010

    2013

  21. [29]

    J. B. Dent, S. Dutta and E. N. Sridakis, f (T ) gravity mimicking dynamical dark energy. Background and perturbation analysis, Phys. Rev. D 83 (2011) 023508

    2011

  22. [30]

    Shih-Hung Chen, J. B. Dent, S. Dutta and E. N. Sridakis, C osmological perturbations in f (T ) gravity, Phys. Rev. D 83(2011) 023508

    2011

  23. [31]

    Wu and H

    P. Wu and H. W. Yu, The dynamical behavior of f(T) theory, Phys. Lett. B 692(2010) 176. August 12, 2025 15:20 WSPC/INSTRUCTION FILE vis-T˙2˙Stability 16 Partha Sarathi Debnath, Chanchal Mondal, B. C. Paul

    2010

  24. [32]

    Bamba, S

    K. Bamba, S. D. Odintsov and E. N. Saridakis, Inationary cosmology in unimodular F(T) gravity, [arXiv:1605.02461]

    arXiv

  25. [33]

    G. L. Nashed, FR W in quadratic form of f(T) gravitationa l theories, Gen. Rel. Grav. 47, 75 (2015)

    2015

  26. [34]

    C. W. Misner, The isotropy of the universe, Astrophys. J. 151 (1968), 431-457

    1968

  27. [35]

    Zimdahl, D

    W. Zimdahl, D. J. Schwarz, A. B. Balakin, and D. Pavon, Co smic antifriction and accelerated expansion, Phys. Rev. D 64 (2001) 063501

    2001

  28. [36]

    Pavon, Nonequilibrium fluctuations in cosmic vacuum decay, Phys

    D. Pavon, Nonequilibrium fluctuations in cosmic vacuum decay, Phys. Rev. D. 43 (1991), 375

    1991

  29. [37]

    J. A. S. Lima, A. S. M. Germano and L. R. W. Abramo, FR W-typ e cosmologies with adiabatic matter creation, Phys. Rev. D 53 (1996) 4287

    1996

  30. [38]

    Brevik, E.Elizalde, S

    I. Brevik, E.Elizalde, S. D. Odintsov, A. V. Timoshkin, Inflationary universe in terms of a van der Waals viscous fluid, Int. J. Geom. Meth. Mod. Phys. 14 (2017) 1750185

    2017

  31. [39]

    Brevik, V

    I. Brevik, V. V. Obukhov, A. V. Timoshkin, Inflation in te rms of a viscous van der Waals coupled fluid, Int. J. Geom. Meth. Mod. Phys. 15 (2018) 1850150

    2018

  32. [40]

    Brevik, O

    I. Brevik, O. Gron, J. de Haro, S. D. Odintsov and E. N. Sar idakis, Viscous cosmology for early- and late-time universe, Int. J. Mod. Phys. D 26 (2017) 1730024

    2017

  33. [41]

    Khurshudyan, Cosmology with an interac ting van der Waals fluid, Int

    E.Elizalde, M. Khurshudyan, Cosmology with an interac ting van der Waals fluid, Int. J. Mod. Phys. D 27 (2018) 1850037

    2018

  34. [42]

    J. D. Barrow, R. A. Matzner, The homogeneity and isotrop y of the Universe, Mon. Not. Roy. astron. Soc. 181 (1977) 719–727

    1977

  35. [43]

    Zimdhal, Dark matter and dissipation, Phys

    D Pavon, W. Zimdhal, Dark matter and dissipation, Phys. Lett. A 179 (1993) 261– 265

    1993

  36. [44]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, Inhomogeneous equation of st ate of the universe: Phantom era, future singularity, and crossing the phantom barrier, Phys. Rev. D 72 (2005) 023003

    2005

  37. [45]

    Eckart, The Thermodynamics of Irreversible Process es

    C. Eckart, The Thermodynamics of Irreversible Process es. II. Fluid Mixtures, Phys. Rev. D 58 (1940) 269

    1940

  38. [46]

    W. A. Hiscock, Generic instabilities in first-order dis sipative relativistic fluid theories. II. Havas-Swenson-type theories, Phys. Rev. D 33 (1986) 1527

    1986

  39. [47]

    Israel, J

    W. Israel, J. M. Stewart, Transient relativistic therm odynamics and kinetic theory, Ann. Phys 118 (1979) 341–372

    1979

  40. [48]

    A. I. Arbab, A. Beesham, Causal Dissipative Cosmology W ith Variable G and Λ , Gen. Rel. Grav. 32 (2000) 615–620

    2000

  41. [49]

    Pradhan, V

    A. Pradhan, V. Rai, R. S. Singh, BULK VISCOUS COSMOLOGIC AL MODELS IN GENERAL RELATIVITY, Fizika B 16 (2007) 99–112

    2007

  42. [50]

    Colistete Jr., J

    R. Colistete Jr., J. C. Fabris, J. Tossa, W. Zimdahl, Bul k viscous cosmology, Phys. Rev. D 76 (2007) 103516

    2007

  43. [51]

    Ghosh, A

    R. Ghosh, A. Pasqua & S. Chattopadhyay, Generalized sec ond law of thermodynamics in the emergent universe for some viable models of f(T) gravi ty, The European Physics Journal Plus 128 (2013) 12

    2013

  44. [52]

    Brevik, E.Elizalde, S

    I. Brevik, E.Elizalde, S. Nojiri, S. D. Odintsov, Visco us little rip cosmology, Phys. Rev. D 84 (2011), 103508

    2011

  45. [53]

    Zimdahl, Bulk viscous cosmology, Phys.Rev

    W. Zimdahl, Bulk viscous cosmology, Phys.Rev. D 53 (1996) 5483-5493

    1996

  46. [54]

    Brevik and O

    I. Brevik and O. Gorbunova, Viscous dark cosmology with account of quantum effects, Eur. Phys. J. C. 56 (2008) 425–428

    2008

  47. [55]

    Ren and Ming-Guang Hu, Communications i n Theoretical Physics Friedmann Cosmology with a Generalized Equation of State an d Bulk Viscosity, Com- mun

    Xin-He Meng, J. Ren and Ming-Guang Hu, Communications i n Theoretical Physics Friedmann Cosmology with a Generalized Equation of State an d Bulk Viscosity, Com- mun. Theor. Phys. , 47 (2007) 379–384

    2007

  48. [56]

    D. Jou, J. C. Vazquez, G.Lebon, Extended Irreversible Thermodynamics (Spinger, August 12, 2025 15:20 WSPC/INSTRUCTION FILE vis-T˙2˙Stability Dynamical analysis of bulk viscous cosmological model in F (T ) gravity 17 New York, 2010)

    2025

  49. [57]

    S. H. Strogatz, Nonlinear Dynamics and Chaos (Westview Press, 2001)

    2001

  50. [58]

    D. W. Jordan and P. Smith, Nonlinear Ordinary Differenti al Equations (Oxford Univ. Press, 2007). August 12, 2025 15:20 WSPC/INSTRUCTION FILE vis-T˙2˙Stability 18 Partha Sarathi Debnath, Chanchal Mondal, B. C. Paul Α/EquΑl1, Γ/EquΑl/Minus 0.1, n/EquΑl0.3, T0 /EquΑl 1, n0 /EquΑl...

    2007

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.