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arxiv: 1907.02238 · v1 · pith:JEJRKPGPnew · submitted 2019-07-04 · 🌀 gr-qc

Bulk viscous cosmological model in f(R,T) theory of gravity

Partha Sarathi Debnath This is my paper

Pith reviewed 2026-05-25 09:31 UTC · model grok-4.3

classification 🌀 gr-qc
keywords f(R,T) gravitybulk viscosityEckart theoryIsrael-Stewart theoryFRW cosmologymodified gravitycosmological modelsobservational data
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The pith

Bulk viscous cosmological models in f(R,T) gravity with the form f(R,T)=R+2λT are compatible with observations in Eckart, truncated Israel-Stewart, and full Israel-Stewart theories.

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

The paper develops cosmological models for a flat expanding universe that include bulk viscous effects within the f(R,T) theory of gravity. Using the particular form f(R,T)=R+2λT, solutions are found under three different treatments of bulk viscosity. These models are then checked against observational data on the universe's expansion history. A sympathetic reader would care because the approach offers a way to incorporate dissipative effects into modified gravity while remaining consistent with measured cosmic evolution.

Core claim

In this work, exact cosmological solutions are derived for a flat FRW metric in f(R,T) gravity with f(R,T)=R+2λT in the presence of bulk viscous fluid. Solutions are obtained separately within the Eckart theory, the truncated Israel-Stewart theory, and the full Israel-Stewart theory. Detailed analysis of physical and geometrical properties, including the behavior of bulk viscous pressure, Hubble parameter, and deceleration parameter, demonstrates that these models remain consistent with current cosmological observations.

What carries the argument

The linear modification f(R,T)=R+2λT combined with bulk viscosity prescriptions from Eckart and Israel-Stewart theories applied to a flat FRW metric, which generates the scale factor and thermodynamic quantities.

If this is right

  • The scale factor and Hubble parameter evolve to produce a late-time transition from deceleration to acceleration.
  • Bulk viscous pressure stays negative and contributes to the effective negative pressure needed for observed expansion.
  • Energy density and pressure profiles decrease with time in a way that matches standard cosmological expectations.
  • The models remain viable without requiring additional exotic matter fields beyond the viscous fluid.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the models hold, viscosity in modified gravity may substitute for a separate dark energy component in driving acceleration.
  • Predictions for perturbation growth or gravitational wave propagation could be derived to test the framework further.
  • The same construction might be applied to non-flat or anisotropic metrics to check broader consistency.

Load-bearing premise

The specific linear form f(R,T)=R+2λT together with the applicability of standard bulk viscosity theories remains valid when embedded in this modified gravity framework for a flat FRW metric.

What would settle it

A measurement of the deceleration parameter or Hubble parameter that deviates from the derived time evolution in all three viscosity theories would rule out the claimed compatibility.

Figures

Figures reproduced from arXiv: 1907.02238 by Partha Sarathi Debnath.

Figure 1
Figure 1. Figure 1: shows the plot of q vs z for Eckart theory in f(R, T) theory of gravity for different values of λ for a given set of other parameters. 3.1.1. Power law model : In this particular case scale factor of the universe exhibits power law expansion a(t) = a0t D, where a0 and D are constants. For power law model, the expression of energy density and bulk viscosity stress are given by ρ = ρ0t −2 , Π = −Π0t −2 , (17… view at source ↗
Figure 2
Figure 2. Figure 2: shows the plot of a vs z for Eckart theory in f(R, T) theory of gravity for different values of λ for a given set of other parameters. __ __ __Β=0.8 _ _ _ _Β=0.6 ..........Β=0.4 _____Β=0.2 Λ=0.01 P>0 Region 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Ω D [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the plot of D vs ω for power-law evaluation for different values of β in Eckart theory of f(R, T) gravity with λ = 0.01. __ __ __ Λ=0 _ _ _ _Λ=0.1 ..........Λ= - 0.1 _____ Λ=0.2 Ω=0.33, Β=0.1, s=0, H0= 1 14 . 0 1 2 3 4 5 -1.0 -0.5 0.0 0.5 1.0 z q [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the plot of q vs z for TIS theory in f(R, T) theory of gravity for different values of λ for a given set of other parameters [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the plot of a vs z for TIS theory in f(R, T) theory of gravity for different values of λ for a given set of other parameters. __ __ __Β=0.6 _ _ _ _Β=0.5 ..........Β=0.4 _____Β=0.2 Λ=0.01 P>0 Region 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Ω D [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows the plot of D vs ω for Power-law evaluation in TIS theory for different values of β in f(R, T) gravity with λ = 0.01. __ __ __ Λ=0 _ _ _ _Λ=0.1 ..........Λ= 0.2 _____ Λ=0.5 Ω=0.33, Β=0.1, s=0, H0= 1 14 . 0 1 2 3 4 5 -1.0 -0.5 0.0 0.5 1.0 z q [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the plot of q vs z for FIS theory in f(R, T) theory of gravity for different values of λ for a given set of other parameters [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: shows the plot of a vs z for FIS theory in f(R, T) theory of gravity for different values of λ for a given set of other parameters. __ __ __Β=0.8 _ _ _ _Β=0.6 ..........Β=0.4 _____Β=0.2 Λ=0.01 P>0 Region 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Ω D [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: shows the plot of D vs ω for Power-law evaluation in FIS theory for different values of β in f(R, T) gravity with λ = 0.01. 2.2 > D > 1.9 0.0 0.5 1.0 1.5 30 35 40 45 z Μ [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: shows the plot of µ vs z for supernova data and power law evolution of the universe for D = 2.2 (solid line) and D = 1.9 (dashed line) with a0 = 1 × 10−8 [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: shows the plot of µ vs z for supernova data and exponential evolution of the Universe with H0 = 0.5 × 10−3 (solid line) and H0 = 0.15 × 10−3 (dotted line) [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
read the original abstract

In this paper, we have presented bulk viscous cosmological model of the universe in the modified gravity theory in which the Lagragian of the gravitational action contains a general function $f(R, T) $, where $R$ and $T$ denote the curvature scalar and the trace of the energy-momentum tensor respectively, in the framework of a flat Friedmann-Robertson-Walker model with isotropic fluid. We obtain cosmological solution in $f(R,T)$ theory of gravity, specially of particular choice $f(R,T)=R+2\lambda T$, where $\lambda$ is a constant, in the presence of bulk viscosity that are permitted in Eckart theory, Truncated Israel Stewart theory and Full Israel Stewart theory. The physical and geometrical properties of the models in Eckart, Truncated Israel Stewart theory and Full Israel Stewart theory are studied in detail. The analysis of the variation of bulk viscous pressure, energy density, scale factor, Hubble parameter and deceleration parameter with cosmic evolution are done in the respective theories. The models are analyzed by comparison with recent observational data. The cosmological models are compatible with observations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs bulk viscous cosmological models in f(R,T) gravity for a flat FRW metric using the specific form f(R,T)=R+2λT. It derives solutions under Eckart, truncated Israel-Stewart, and full Israel-Stewart bulk-viscosity prescriptions, examines the evolution of energy density, scale factor, Hubble parameter, deceleration parameter, and viscous pressure, and concludes that the resulting models are compatible with recent observational data.

Significance. If the transport equations are shown to be consistent with the modified field equations and the observational comparison is performed with explicit data and error analysis, the work would provide a concrete example of dissipative cosmology in f(R,T) gravity and could motivate further study of entropy production in modified gravity. The parameter-free aspects of the derivations (if any) and any machine-checked consistency checks would strengthen the contribution.

major comments (2)
  1. [Sections deriving the field equations and viscous models (likely §§3–4)] The central modeling step—direct insertion of the standard Eckart relation π=−3Hζ and the truncated/full Israel-Stewart evolution equations for π into the f(R,T) Friedmann and continuity equations—is not justified. The extra −2λT contribution to the effective energy-momentum tensor modifies both the Friedmann equations and the continuity equation; no section demonstrates that the thermodynamic identity, Gibbs relation, or entropy-production formula used to derive the Israel-Stewart equations survive this modification. This assumption is load-bearing for all three models.
  2. [Abstract and concluding section] The claim of observational compatibility is asserted in the abstract and conclusion but is not supported by any tabulated data, χ² values, best-fit parameters, or exclusion criteria. Without these, it is impossible to assess whether the models are genuinely compatible or merely adjustable via λ and the viscosity coefficients.
minor comments (2)
  1. [Abstract] Typo: 'Lagragian' should be 'Lagrangian'.
  2. [Field-equations section] Notation for the viscosity coefficients and the constant λ should be introduced with explicit definitions before use in the solutions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and will revise the paper accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Sections deriving the field equations and viscous models (likely §§3–4)] The central modeling step—direct insertion of the standard Eckart relation π=−3Hζ and the truncated/full Israel-Stewart evolution equations for π into the f(R,T) Friedmann and continuity equations—is not justified. The extra −2λT contribution to the effective energy-momentum tensor modifies both the Friedmann equations and the continuity equation; no section demonstrates that the thermodynamic identity, Gibbs relation, or entropy-production formula used to derive the Israel-Stewart equations survive this modification. This assumption is load-bearing for all three models.

    Authors: We acknowledge that the original manuscript did not explicitly demonstrate the survival of the standard thermodynamic relations under the f(R,T) modification. In the revision we will insert a new subsection (in §3) that justifies the use of the Eckart and Israel-Stewart transport equations. The argument is that the viscous pressure π enters the matter energy-momentum tensor in the usual way, while the f(R,T) term only alters the gravitational field equations; the local Gibbs relation and entropy-production formula for the fluid therefore remain unchanged to the order considered. We will support this with citations to analogous treatments in other modified-gravity viscous cosmologies. revision: yes

  2. Referee: [Abstract and concluding section] The claim of observational compatibility is asserted in the abstract and conclusion but is not supported by any tabulated data, χ² values, best-fit parameters, or exclusion criteria. Without these, it is impossible to assess whether the models are genuinely compatible or merely adjustable via λ and the viscosity coefficients.

    Authors: We agree that the original manuscript presented only qualitative statements of observational compatibility. In the revision we will add a dedicated section that performs a quantitative comparison with recent supernova and Hubble-parameter data sets, reporting best-fit values of λ and the viscosity coefficients together with the corresponding χ² statistics and 1σ uncertainties. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from modified field equations to solutions and external data comparison

full rationale

The paper selects the ansatz f(R,T)=R+2λT, inserts standard Eckart and Israel-Stewart viscous pressure expressions into the resulting Friedmann and continuity equations for flat FRW, solves for scale factor/Hubble/deceleration evolution, and then compares the resulting curves to observational data. No quoted step reduces a claimed prediction to a fitted input by construction, nor does any load-bearing premise rest solely on a self-citation chain whose cited result is itself unverified. The compatibility statement is an external check rather than an internal tautology. The validity of transplanting the transport equations into f(R,T) is an assumption whose correctness is debatable but does not constitute circularity under the enumerated patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only; free parameters and axioms cannot be exhaustively audited. The model relies on the choice of f(R,T) form and standard cosmological assumptions whose independence from the final fit is unclear.

free parameters (1)
  • λ
    Constant appearing in the chosen f(R,T)=R+2λT; its value is not derived from first principles in the abstract.
axioms (1)
  • domain assumption Flat FRW metric with isotropic fluid
    Standard background assumption invoked for the cosmological model.

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