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Thermodynamics and stability of $f(T,B)$ gravity with viscous fluid by observational constraints

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Pith's one-line read This paper argues that a viscous f(T,B) modified-gravity model, fitted to Hubble data, produces accelerated expansion, late-time stability, and a valid generalized second law of thermodynamics.

desk verdict Central results are hand-selected, not data-constrained; the GSL condition is a tautology. read the letter →

arxiv 1908.11595 v1 pith:7OFRJEBO submitted 2019-08-30 gr-qc astro-ph.CO

A. Pourbagher , Alireza Amani This is my paper · ORCID
classification gr-qcastro-ph.CO PACS 98.80.-k98.80.Es95.35.+d
keywords equationofstateparameterf(TB)gravityviscousfluidgeneralizedsecondlawthermodynamicsdarkenergyacceleratedexpansionsoundspeedstabilityobservationalconstraints
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 aims to establish that a modified-teleparallel gravity theory, $f(T,B)$ gravity, combined with a bulk-viscous fluid and an interaction between matter and dark energy, can describe the late-time universe. Using a power-law form $f(T,B)=\alpha B^m+\beta T^n$ and a cubic parametrization of the Hubble parameter fitted to 38 Hubble-parameter measurements, the authors derive the dark-energy density, pressure, equation of state, and sound speed as functions of redshift. They obtain accelerated expansion with $\omega_d(0)=-1.09$, late-time stability with $c_s^2(0)=0.59$, and a generalized second law of thermodynamics that is valid in the whole universe. If these results hold, the model is a stable, thermodynamically consistent dark-energy construction in modified gravity.

What carries the argument

The central machinery is the $f(T,B)$ action $S=\int d^4x\,e(f(T,B)/\kappa^2+L_m)$ together with the power-law function $f(T,B)=\alpha B^m+\beta T^n$, where $T=-6H^2$ is the torsion scalar and $B=-6(\dot H+3H^2)$ is the boundary term. The identity $R=-T+B$ connects the torsion and curvature formulations, which lets $f(T,B)$ recover both $f(T)$ and $f(R)$ limits. The argument then inserts the fitted cubic $E(z)$ into the derived formulas for the dark-energy density and pressure, uses these to plot $\omega_d=p_d/\rho_d$ and the sound speed, and applies horizon thermodynamics (horizon entropy and the Gibbs equation) to derive the generalized-second-law condition.

What would settle it

Recompute $\omega_d(z)$ and $c_s^2(z)$ using the fitted coefficients $A_3=-0.16$, $A_2=2.39$, $A_1=-3.80$, $A_0=2.57$ while keeping $\alpha=\beta=1$, $m=1$, $n=-2$, $\xi=0.5$, and require the dark-energy density to stay positive and pressure negative. If the equation of state no longer crosses the phantom divide, or if $c_s^2(0)$ turns negative, the paper's late-time acceleration and stability claims are artifacts of the hand-picked parameters rather than consequences of the data.

Watch

Extended reading notes

Core claim

The paper's central claim is that the viscous $f(T,B)$ model with $f(T,B)=\alpha B^m+\beta T^n$ and $E(z)=A_3(1+z)^3+A_2(1+z)^2+A_1(1+z)+A_0$ produces a dark-energy component whose equation of state crosses the phantom divide, reaching $\omega_d(0)=-1.09$ today, while the adiabatic sound speed $c_s^2=\partial_z p_d/\partial_z \rho_d$ remains positive at late times, with $c_s^2(0)=0.59$. The same construction is claimed to satisfy the generalized second law of thermodynamics at the apparent horizon through the condition $\dot H^2/(2GH^4)\ge 0$, written in redshift form as $(1+z)^2 E'^2(z)/(2G E^2(z))\ge 0$. The conclusion is that the model is compatible with the accelerated expansion of the universe and is stable in the late-time regime.

Load-bearing premise

The load-bearing step is the paper's choice to plot the model with hand-selected parameter values chosen so that the dark-energy density comes out positive and the pressure negative, instead of with the coefficient values obtained from the Hubble-data fit; if those hand-picked values do not faithfully represent the data-constrained model, the reported $\omega_d$, $c_s^2$, and stability results do not follow from the observations.

Editorial extensions

If this is right

  • If the model is right, this particular $f(T,B)$ construction is a late-time dark-energy candidate that accelerates the expansion while staying classically stable.
  • The phantom-crossing value $\omega_d(0)=-1.09$ means the model can mimic the observational behavior associated with $\omega<-1$ without introducing a phantom scalar field.
  • Because the generalized second law holds, the model is not excluded by horizon thermodynamics, a common consistency check for modified gravity cosmologies.
  • The cubic parametrization with the reported fitted coefficients reproduces the 38 Hubble-parameter measurements, so the background kinematics are consistent with the data used.
  • The interaction $Q=3b^2H\rho$ changes the matter dilution law to $\rho=\rho_0 a^{-3(1-b^2+\omega)}$, so energy exchange between matter and dark energy is built into the model's evolution equations.

Reading between the lines

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

  • The generalized-second-law condition Eq (31) is nonnegative for any differentiable $E(z)$, so this thermodynamic check is automatic and does not distinguish the model from others; a more informative test would compute the separate rates $\dot S_{ih}$ and $\dot S_{oh}$.
  • Replacing the hand-selected coefficients with the fitted values $A_3=-0.16$, $A_2=2.39$, $A_1=-3.80$, $A_0=2.57$ would show whether the reported acceleration and stability are genuinely data-driven.
  • Treating the viscosity coefficient $\xi$ and the interaction strength $b$ as free parameters to be fit, rather than fixing them, would turn the qualitative redshift plots into actual parameter constraints.
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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

4 major / 5 minor

Summary. The paper studies f(T,B) gravity in a flat FRW universe with a bulk-viscous fluid, reconstructing the dark-energy density, pressure, and equation of state in terms of redshift for a power-law ansatz f(T,B)=αB^m+βT^n. The Hubble parameter is parameterized by a cubic polynomial E(z)=A3(1+z)^3+A2(1+z)^2+A1(1+z)+A0 and fitted to 38 H(z) data points. Using a set of hand-selected coefficients, the authors report an accelerating universe with ω_d(0)=-1.09, a late-time stability c_s^2(0)=0.59, and validity of the generalized second law of thermodynamics via a non-negative condition on ˙H^2. The central claims are that this f(T,B) model with viscous fluid is observationally compatible, stable, and thermodynamically consistent.

Significance. If correct, the paper would demonstrate a viable dark-energy model that combines f(T,B) gravity, bulk viscosity, and interaction with matter, supported by H(z) data. The authors do derive the f(T,B) field equations in tetrad form and provide a systematic redshift-space reconstruction, which are useful steps. However, the central results do not follow from the analysis: the observational fit is discarded in favor of parameters chosen to force the desired sign of energy density and pressure, the redshift-space pressure equation contains chain-rule coefficient errors, and the generalized second law reduces to the square of a quantity, holding identically for any E(z). These issues undermine the paper's main conclusions. The manuscript is not yet suitable for publication.

major comments (4)
  1. [Sec. IV, after Eq. (22)] The fitted coefficients from the 38 H(z) data points are reported as A3=-0.16±0.30, A2=2.39±1.71, A1=-3.80±2.89, A0=2.57, but all subsequent physical results (Figs. 2–4, ω_d(0)=-1.09, c_s^2(0)=0.59) are computed with A3=0.14, A2=0.75, A1=-1.45, A0=1.56, and α=β=1, m=1, n=-2, ξ=0.5. The authors state that these coefficients were 'selected with the motivation that the energy density of dark energy is greater than zero and the pressure of dark energy is less than zero to confirm the accelerated expansion of the universe.' This is circular: the observational fit is not used to constrain the model, and no error bars are propagated. The conclusion that the model is compatible with observational data is therefore unsupported.
  2. [Eq. (20b)] The conversion of the pressure equation (15b) to redshift space contains incorrect chain-rule coefficients. For the term ∂B¨f, using d/dt = -H(1+z)d/dz with H=H0√E gives ∂B¨f = (1/2)H0^2(1+z)^2E'∂Bf' + H0^2(1+z)E∂Bf' + H0^2(1+z)^2E∂Bf''. The published Eq. (20b) instead has coefficient 1 on (1+z)^2E'∂Bf' and coefficient 2 on (1+z)E∂Bf'. This error propagates into p_d, ω_d, and c_s^2, so the quantitative claims about the equation of state and stability are not reliable.
  3. [Eq. (31), Sec. V] The generalized-second-law condition is stated as ˙H^2/(2GH^4) = (1+z)^2E'^2(z)/(2GE^2(z)). This equality has a factor-of-4 error: substituting ˙H = -(1/2)H0^2(1+z)E' and H^2=H0^2E gives ˙H^2/H^4 = (1/4)(1+z)^2E'^2/E^2, so the right-hand side should be (1+z)^2E'^2/(8GE^2). More importantly, the expression is a square and is therefore non-negative for any function E(z). The claimed validity of the generalized second law is thus an identity that holds for every f(T,B) model, regardless of the viscous fluid, the interaction, or the fitted parameters. It cannot serve as a test or validation of the model.
  4. [Secs. III–IV, Eqs. (15b), (18), (20b)] The treatment of the viscous term is inconsistent. Eq. (15b) defines p_d as the pure f(T,B) contribution to the dark-energy pressure, and Eq. (18) defines the equation of state using ¯p_d = p_d - 3ξH. However, Eq. (20b) appends '-3ξH0√E' directly to the expression for p_d. If Eq. (20b) is meant to be p_d, then substituting it into Eq. (18) double-counts the viscosity; if Eq. (20b) is meant to be ¯p_d, the notation conflicts with Eq. (15b). The plotted equation of state is therefore ambiguous, and this ambiguity affects the reported ω_d values.
minor comments (5)
  1. [Abstract and Sec. I] The paper states that '38 supernova data' are used, but Ref. [85] is a compilation of Hubble parameter measurements, not supernova data; the wording should be corrected.
  2. [Sec. IV, around Eq. (22)] The constraint A3+A2+A1+A0=1 is used, but the fitted values are reported with uncertainties that are not propagated into any derived quantity; the absence of a goodness-of-fit statistic (e.g., χ^2) also makes the quality of the fit difficult to assess.
  3. [Fig. 1] It is unclear whether the 'our model' curve in Fig. 1 uses the fitted coefficients or the later hand-selected coefficients; this should be stated explicitly.
  4. [Sec. IV] The sentence 'From the result of fitting, we obtain the coefficients ... which Fig. 1 shows the matter' is grammatically incomplete and should be rewritten.
  5. [Throughout] There are numerous typographical errors (e.g., 'tortion', 'paramet erize', 'institing') and awkward phrasings that would need copyediting.

Circularity Check

2 steps flagged · score 8.0 of 10

The central acceleration and stability results are manufactured by hand-selecting coefficients to force rho_d>0 and p_d<0, and the GSL check reduces to a square being nonnegative.

  1. self definitional [Section IV, paragraph after Eq. (22)]
    "Here we note that the free parameters play an very important role in our results, so we try to choose them as α = β = 1, m = 1, n = −2, ξ = 0.5, A3 = 0.14, A2 = 0.75, A1 = −1.45 and A0 = 1.56. These coefficients are selected with the motivation that the energy density of dark energy is greater than zero and the pressure of dark energy is less than zero to confirm the accelerated expansion of the universe."

    The data fit reported just above gives A3 = −0.16, A2 = 2.39, A1 = −3.80, A0 = 1 − A3 − A2 − A1, but all subsequent quantities (Figs. 2-4, ωd(0) = −1.09, c_s^2(0) = 0.59) are computed with the different, hand-picked A3 = 0.14, A2 = 0.75, A1 = −1.45, A0 = 1.56. These coefficients are explicitly chosen so that ρd > 0 and pd < 0, i.e., so that the universe accelerates. The conclusion 'accelerated expansion, ωd ≈ −1.09' is therefore an input to the coefficient selection, not an output of the observational constraints, making the central viability claim circular by construction.

  2. other [Section V, Eq. (31)]
    "now by inserting (11) and (24) into aforesaid relationship, we can clearly find the condition for the validity of the generalized second law of thermodynamics in the following form ˙H 2 2GH 4 = (1 + z)2E′2(z) 2GE2(z) ≥ 0. The result shows us that the validity of the generalized second law of thermodynamics be satisfied by condition of thermodynamics equilibrium."

    The final expression is (1+z)^2 E'^2 / (2 G E^2), which is nonnegative for every real differentiable E(z) because it is a ratio of a square to positive quantities. Thus Eq. (31) is an identity, not a model-specific condition, and the claimed validity of the generalized second law holds for any E(z) in this framework. The GSL test therefore contains no information about the f(T,B) viscous-fluid model and cannot confirm or falsify it.

full rationale

The paper's headline numerical results do not follow from the observational fit. After fitting Eq. (22) to 38 H(z) points and reporting A3 = −0.16, A2 = 2.39, A1 = −3.80, the analysis switches to A3 = 0.14, A2 = 0.75, A1 = −1.45, A0 = 1.56, selected explicitly to make ρd > 0 and pd < 0, and then reads off accelerated expansion (ωd(0) = −1.09) and late-time stability (c_s^2(0) = 0.59). These outputs are therefore consequences of the parameter choice, not of the data or of the f(T,B) dynamics; the phrase 'by observational constraints' is not realized for the plotted model. The thermodynamic check in Eq. (31) is likewise vacuous, reducing the GSL to a square that is nonnegative for any E(z). These are internal, textually documented reductions of the central claims to their own inputs, so the appropriate score is 8; no external benchmark or independent derivation rescues the central claim.

Assumptions & free parameters 10 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new entity, but it rests on a large set of hand-picked constants and two ad hoc functional forms. The conclusion of stability and acceleration is enforced by the parameter selection, and the GSL result is an identity, so the ledger shows the model purchases its results with freedom rather than data.

free parameters (10)
  • alpha (amplitude of B^m term) = 1
    Hand-selected as 1 in Sec IV; not fitted to data.
  • beta (amplitude of T^n term) = 1
    Hand-selected as 1 in Sec IV; not fitted to data.
  • m (power of B) = 1
    Hand-selected as 1 in Sec IV.
  • n (power of T) = -2
    Hand-selected as -2 in Sec IV; yields a T^{-2} term.
  • xi (bulk viscosity coefficient) = 0.5
    Hand-selected in Sec IV; enters pb=-3*xi*H.
  • A3 (coefficient of (1+z)^3 in E(z)) = 0.14 (plots) / -0.16±0.30 (fit)
    Fitted to 38 H(z) points as -0.16, then replaced by 0.14 for all physical plots.
  • A2 (coefficient of (1+z)^2 in E(z)) = 0.75 (plots) / 2.39±1.71 (fit)
    Fitted as 2.39, replaced by 0.75 for physical plots.
  • A1 (coefficient of (1+z) in E(z)) = -1.45 (plots) / -3.80±2.89 (fit)
    Fitted as -3.80, replaced by -1.45 for physical plots.
  • A0 (constant term in E(z)) = 1.56 (plots) / 2.57 (fit)
    Fixes E(0)=1; different in the two sets of coefficients.
  • b (interaction strength in Q=3b^2H*rho) = not specified
    Introduced in Sec III but never assigned a value; affects Eq (17) but not used in the dark energy plots.
assumptions (7)
  • domain assumption Flat FRW metric with tetrad diag(1,a,a,a) and Weitzenbock connection yield T=-6H^2 and B=-6(Hdot+3H^2).
    Standard teleparallel setup, used in Eqs (5)-(7).
  • domain assumption Bulk viscosity pressure pb=-3*xi*H with constant xi.
    Introduced in Eq (8); assumed without microphysical derivation.
  • domain assumption Interaction term Q=3b^2H*rho between matter and dark energy.
    Adopted after Eq (16); one of many possible interaction forms.
  • ad hoc to paper Power-law form f(T,B)=alpha*B^m+beta*T^n.
    Eq (21); chosen for tractability, not derived from a more fundamental principle.
  • ad hoc to paper Polynomial E(z)=A3(1+z)^3+A2(1+z)^2+A1(1+z)+A0.
    Eq (22); inspired by Ref [84], coefficients are fit or selected as free parameters.
  • domain assumption Thermal equilibrium between apparent horizon and fluid.
    Sec V; required for the entropy derivation; limits the GSL result to equilibrium states.
  • standard math Bekenstein-Hawking entropy S=A/(4G) and Gibbs equation for the fluid within the horizon.
    Standard black hole thermodynamics, cited to Refs [96-98].

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Pith. "Pith review of Thermodynamics and stability of $f(T,B)$ gravity with viscous fluid by observational constraints." pith.science (2026). https://pith.science/paper/7OFRJEBO

@misc{pith2026190811595,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics and stability of $f(T,B)$ gravity with viscous fluid by observational constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7OFRJEBO}},
  note         = {Machine review of arXiv:1908.11595}
}
abstract

In this paper, we study the model of $f(T, B)$ gravity with viscous fluid in flat-FRW metric, in which $T$ and $B$ are torsion scalar and boundary term, respectively. We obtain the Friedmann equations in the framework of modified teleparallel gravity by tetrad components. We consider an interacting model between matter and dark energy so that universe dominates by viscous fluid. Then, we write the corresponding cosmological parameters in terms of the redshift parameter, and, we parameterize the Hubble parameter with experimental data. In what follows, we plot the corresponding cosmological parameters for dark energy components in terms of redshift, thereafter we investigate the accelerated expansion of the universe. Moreover, we discuss the stability of the model by using the sound speed parameter. Finally, we investigate the validity of the generalized second law of thermodynamics.

Figures

Figures reproduced from arXiv: 1908.11595 by the authors.

Figure 1
Figure 1. FIG. 1: The graph of the Hubble parameter in terms of redshift par [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The graph of the energy density and the pressure of dark [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The graph of the EoS of dark energy in terms of redshift pa [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The graph of the sound speed in terms of redshift paramet [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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