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REVIEW 3 major objections 4 minor 87 references

Cosmological scenario based on the first and second laws of thermodynamics: Thermodynamic constraints on a generalized cosmological model

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that horizon thermodynamics alone fixes dark-energy driving terms to the observed Λ scale.

desk verdict Careful reformulation, but the claimed thermodynamic fix of the cosmological-constant scale is an input (C = Λ/3) rather than an output, and the 'constraints' reduce to standard energy conditions. read the letter →

arxiv 2412.19032 v3 pith:VBA5TWVD submitted 2024-12-26 gr-qc astro-ph.COhep-ph

Nobuyoshi Komatsu This is my paper · ORCID
classification gr-qcastro-ph.COhep-ph MSC 83F05 PACS 98.80.-k95.30.Tg
keywords cosmologicalconstantproblemfirstlawofthermodynamicssecondhorizonentropyBekenstein-Hawkingtime-varyingLambdacosmologybulkviscousFLRWuniverse
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 attempts to show that the first and second laws of thermodynamics, applied to the entropy of the cosmological horizon, form a consistent scenario for the cosmological constant problem. Starting from an arbitrary horizon entropy $S_H$, the first law is used to derive a generalized Friedmann equation containing two extra driving terms, $f_{\Lambda}(t)$ and $h_{\mathrm B}(t)$, corresponding to a time-varying vacuum term and a bulk-viscous term. The second law then imposes inequalities that bound these terms: $f_{\Lambda}(t) \leq H^2$ today and $-H_0^2 \lesssim h_{\mathrm B}(t) \lesssim H_0^2$, placing both at the order of the observed cosmological constant $\Lambda_{\mathrm obs}$. The author concludes that when the horizon entropy deviates only slightly from the Bekenstein--Hawking entropy, the model reduces to $\Lambda$CDM-like behavior with $f_{\Lambda}$ approaching a constant of the right order, as if thermodynamics alone could avoid the 60--120 order-of-magnitude discrepancy.

What carries the argument

The load-bearing object is the deviation $S_{\Delta} = S_H - S_{\mathrm{BH}}$ of the horizon entropy from the Bekenstein--Hawking value, and specifically the ratio $(\partial S_{\Delta}/\partial S_{\mathrm{BH}})$, which enters both driving terms: $f_{\Lambda}(t) = C - \int (\partial S_{\Delta}/\partial S_{\mathrm{BH}})\, d(H^2)$ and $h_{\mathrm B}(t) = -\dot H\,(\partial S_{\Delta}/\partial S_{\mathrm{BH}})$. The second law enters through the identity $(\partial S_H/\partial S_{\mathrm{BH}}) = \dot S_H/\dot S_{\mathrm{BH}}$, which converts $\dot S_H \geq 0$ into an inequality on $h_{\mathrm B}(t)/\dot H$; after substituting the general Friedmann equation, this inequality becomes the constraint $f_{\Lambda}(t) \leq H^2$. This chain is what turns a purely geometric entropy deviation into an order-of-magnitude bound on the extra driving terms.

What would settle it

The central bound is $f_{\Lambda}(t) \leq H^2$; since $H^2 = 8\pi G\rho/3 + f_{\Lambda}$, this is equivalent to $\rho \geq 0$. A concrete falsifier: construct any horizon entropy for which the first law gives $f_{\Lambda}(t) > H_0^2$ at late times while $\rho \geq 0$ and the second law $\dot S_H \geq 0$ holds; no such construction is shown in the paper, and finding one would overturn the claimed constraint.

Watch

Extended reading notes

Core claim

Starting from the first law $-dE_{\mathrm{bulk}} + W\,dV = T_H\,dS_H$ with an arbitrary horizon entropy $S_H$ and the Kodama--Hayward temperature, the author derives a generalized Friedmann equation whose integration constant $C$ he identifies with $\Lambda/3$. Writing $S_H = S_{\mathrm{BH}} + S_{\Delta}$, the equations are reformulated so that the two extra driving terms appear explicitly: $f_{\Lambda}(t) = C - \int (\partial S_{\Delta}/\partial S_{\mathrm{BH}})\, d(H^2)$ and $h_{\mathrm B}(t) = -\dot H\,(\partial S_{\Delta}/\partial S_{\mathrm{BH}})$. Using the second law $\dot S_H \geq 0$ together with $H > 0$, $\dot H < 0$, and $\dot H \geq -2H^2$, the paper derives $f_{\Lambda}(t) \leq H^2$ and $-2H^2 \leq \dot H \leq h_{\mathrm B}(t) \leq (3/2)(1+w)H^2$, hence $O(f_{\Lambda}) \precsim O(H_0^2)$ and $O(-H_0^2) \precsim O(h_{\mathrm B}) \precsim O(H_0^2)$ in the late universe. In the near-Bekenstein--Hawking limit $S_{\Delta} \to 0$, $h_{\mathrm B}(t)$ reduces to zero and $f_{\Lambda}(t)$ approaches a constant whose order matches $\Lambda_{\mathrm obs}$, which the author presents as a thermodynamically consistent scenario for the cosmological constant problem.

Load-bearing premise

The integration constant $C$ in the first-law Friedmann equation is assumed from the start to equal $\Lambda/3$, the observed cosmological constant, so the claimed order agreement $O(C) \approx O(\Lambda_{\mathrm obs})$ mainly restates that identification; if $C$ were left free, the second-law bound $f_{\Lambda} \leq H^2$ only requires the energy density to stay non-negative, not that the scale be the observed one.

Editorial extensions

If this is right

  • In any first-law-derived cosmology with an arbitrary horizon entropy, the second law excludes late-time models in which the vacuum-like term exceeds $H_0^2$, provided the energy density is non-negative.
  • The bulk-viscous term $h_{\mathrm B}(t)$ is bounded between $-H_0^2$ and $(3/2)(1+w)H_0^2$ in order of magnitude, so large positive or negative viscous contributions are thermodynamically forbidden in the late universe.
  • When the entropy deviation $S_{\Delta}$ is close to zero, the scenario forces $h_{\mathrm B} \to 0$ and $f_{\Lambda} \to C$ with $C$ of order $\Lambda_{\mathrm obs}$, recovering a $\Lambda$CDM-like expansion from thermodynamics.
  • The 60--120 order-of-magnitude discrepancy between the observed and quantum-field-theory vacuum energy is avoided because the thermodynamically selected scale is $H_0^2$, not the Planck scale.

Reading between the lines

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

  • My reading: the inequality $f_{\Lambda}(t) \leq H^2$ is algebraically the same as requiring non-negative energy density in the Friedmann equation, so the second law itself does not single out the observed scale; the scale enters through the initial identification $C = \Lambda/3$.
  • A testable extension would be to apply the framework to a specific nonextensive entropy (e.g., Barrow or Tsallis), compute $f_{\Lambda}$ and $h_{\mathrm B}$, and check whether the resulting background evolution satisfies supernova and cosmic-microwave-background constraints; the paper computes only the power-law example and leaves the evolution for future work.
  • The near-Bekenstein--Hawking limit implies a sharp prediction: for the scenario to reproduce $\Lambda$CDM, the horizon entropy must deviate from Bekenstein--Hawking by just enough that the integrated correction lands within $H_0^2$, which could be compared with entropy proposals from quantum gravity if their correction parameters are ever pinned down.
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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

3 major / 4 minor

Summary. The manuscript derives cosmological equations in a flat FLRW universe from the first law of thermodynamics applied to a cosmological horizon with an arbitrary entropy SH, following and reformulating earlier work by Odintsov et al. It expresses the resulting Friedmann and acceleration equations in a general form with two extra driving terms fΛ(t) and hB(t), Eqs. (39) and (40). It then imposes the second law in the form Sdot_H≥0 and derives the inequalities fΛ(t)≤H², hB(t)≥Hdot, and -2H²≤hB(t)≤(3/2)(1+w)H². The central claim is that these thermodynamic constraints imply O(fΛ(t))⪅O(H0²) and O(-H0²)⪅O(hB(t))⪅O(H0²), so that the driving terms have the order of the observed cosmological constant; in the near-Bekenstein-Hawking limit fΛ approaches a constant C with O(C)≈O(H0²). A worked example with power-law-corrected entropy is given in Appendix B.

Significance. If the central claim were valid, the paper would provide a notable result: the second law alone would fix the scale of the two driving terms to the observed dark-energy scale and offer a thermodynamic route to the cosmological constant problem. The derivation from the first law to Eqs. (39)-(40) is algebraically sound, and the inequalities (46)-(59) follow formally from the stated assumptions. The paper is also clearly organized and gives explicit formulas for a concrete entropy choice. However, the scale O(H0²) is not an output of the second law: it is inserted through the identification C=Λ/3, and the past-to-present version of the bounds contains a directional error. The constraints reduce essentially to ρ≥0, w≥-1, and the assumed Hdot≥-2H², none of which selects the observed vacuum-energy scale. The central conclusion is therefore not supported.

major comments (3)
  1. [Sec. IV, below Eq. (27); Appendix A, Eq. (A9)] Below Eq. (27) and in Eq. (A9), the integration constant C is identified from the outset with the observed cosmological constant: "C is an integral constant and should be given by Λ/3". This identification is an input, not a consequence of the second law. Equation (62) then concludes O(C)≈O[fΛ(t)]⪅O(H0²), which is just Eq. (9), O(Λ_obs/3)=O(H0²), restated. If C were treated as a free parameter, Eq. (41) shows that only the combination fΛ(t)=C-∫(∂S∆/∂SBH)d(H²) is bounded by Eq. (55), and fΛ≤H² is equivalent to ρ≥0 via Eq. (36), so it carries no information about C; a Planck-scale C could be compensated by a large entropy integral. The claimed thermodynamic fixing of the Λ scale is therefore circular.
  2. [Sec. V, Eqs. (55)-(57), (59)-(60)] Equation (56) states that, because H0≤H for past times, the "strictest constraint from the past to the present" is fΛ(t)≤H0²≤H². This is the wrong direction: H²(t)≥H0² means the upper bound fΛ(t)≤H²(t) is weaker, not stronger, at earlier times. The inference would be valid only at t=t0 or under an additional monotonicity assumption on fΛ that is never stated. Consequently the order estimate in Eq. (57) is not established as a past-to-present statement. The same problem invalidates Eq. (60): from -2H²≤hB≤(3/2)(1+w)H² and H≥H0 one cannot conclude the interval [-2H0²,(3/2)(1+w)H0²] for past times. At best, Eqs. (55), (47), and (58) are local bounds at the present time.
  3. [Sec. V, Eq. (58)] The upper bound on hB(t) in Eq. (58) uses fΛ(t)≥0, which is not a consequence of the second law for arbitrary horizon entropy; the paper notes only that such a non-negative fΛ can be obtained for the power-law-corrected entropy in Appendix B. Thus the claimed universal thermodynamic constraint on hB is conditional. Moreover, the lower bound -2H²≤hB(t) is exactly the assumed condition Hdot≥-2H², and the upper bound with fΛ≥0 follows automatically from hB=Hdot+(3/2)(1+w)H²(1-fΛ/H²), so the inequalities in Eq. (59) encode the energy conditions w≥-1 and ρ≥0 together with the assumed temperature positivity, rather than new second-law information.
minor comments (4)
  1. [Sec. V, Eq. (44)] The replacement (∂SH/∂SBH)=˙SH/˙SBH assumes SH and SBH depend on t only through H(t); since SH is introduced as arbitrary, this functional-dependence assumption should be stated explicitly before Eq. (44).
  2. [Sec. V, Eq. (60)] The notation O(hB(t)) for a quantity that can be negative should be defined; presumably it refers to the order of magnitude of |hB(t)|, but the text should say so.
  3. [Sec. V, around Eq. (43)] The second law is applied to the horizon entropy alone, with the justification that the horizon entropy dominates [87]; because the first law is also written for the horizon, the paper should clarify that the total-entropy inequality is being approximated and discuss the neglected matter contribution.
  4. [Appendix B, Eq. (B6)] The constant C1=C-C0 is "considered to be non-negative" without derivation; since C is identified with Λ/3 and is not constrained by the second law, this assumption should be justified or relaxed.

Circularity Check

4 steps flagged · score 8.0 of 10

The claimed Λ-scale output is an input: C is set to Λ/3 before the second-law analysis, so Eq. (62) restates O(Λ/3)=O(H0²); the fΛ≤H² “constraint” reduces to ρ≥0 and does not fix C.

  1. self definitional [Sec. IV below Eq. (27); Appendix A, Eq. (A9); Eqs. (9) and (62).]
    "From Eq. (A9), the Friedmann equation from the first law is written as ∫(∂SH/∂SBH)d(H²)=8πG/3 ρ+C, where C is an integral constant and should be given by Λ/3. … O(C) ≈O(C+ǫ1) ≈O[fΛ(t)] ⪅ O(H²0). Equation (62) implies that the order of C is consistent with the order of Λ_obs."

    C is fixed to the observed Λ scale before any thermodynamic analysis: Eq. (9) already states O(Λ_obs/3)≈O(H0²), and Appendix A sets C=Λ/3 by convention. Equation (62) then reports O(C)≈O(H0²) as if it were a second-law output, but it is exactly the input C=Λ/3. If C were left free, Eq. (41) would only constrain the combination fΛ=C−∫(∂S∆/∂SBH)d(H²); the indefinite entropy integral can absorb a constant, so the second law alone cannot select the observed scale.

  2. self definitional [Sec. V, Eqs. (54)–(55), using Eq. (36).]
    "1 − fΛ(t)/H² ≥ 0, or equivalently, fΛ(t) ≤ H². Equations (54) and (55) imply an upper limit of fΛ(t)."

    Substituting the model’s own Friedmann equation, H²=8πG/3 ρ+fΛ(t), the “second-law upper limit” fΛ≤H² is algebraically identical to ρ≥0. The entropy bound therefore adds no independent information about the scale of fΛ; it is a restatement of the standard positive-energy condition. Consequently, the later order claim O(fΛ)⪅O(H0²) receives its scale from the pre-imposed C=Λ/3, not from ˙SH≥0.

2 more flagged steps
  1. other [Sec. V, Eqs. (56)–(57).]
    "When 0 < H and H0 ≤ H (obtained from ˙H < 0), the strictest constraint from the past to the present is given by fΛ(t) ≤ H²0 ≤ H², and the order of fΛ(t) can be written as O(fΛ(t)) ⪅ O(H²0)."

    The premise fΛ(t)≤H²(t) together with H0≤H(t) does not imply fΛ(t)≤H0²: at earlier times H² is larger, so the allowed upper bound is H²(t), which can exceed H0². Deriving fΛ≤H0² requires assuming the desired present-scale bound or an unstated monotonicity of fΛ. Thus Eqs. (56)–(57) are not a consequence of the second law; they smuggle in the scale that the paper claims to predict.

  2. other [Sec. V, Eqs. (59)–(60).]
    "−2H² ≤ ˙H ≤ hB(t) ≤ 3/2(1+w)H² (for ˙H <0). … Applying 0 < H0 ≤ H to Eq. (59) gives the order of hB(t), written as O(−H²0) ⪅ O(hB(t)) ⪅ O(H²0)."

    From H0≤H, the lower bound hB≥−2H² is weaker (more negative) than −2H0² in the past, and the upper bound (3/2)(1+w)H² is larger than (3/2)(1+w)H0² in the past. Therefore Eq. (60) does not follow from Eq. (59). It becomes true only if one restricts to the present epoch and substitutes H≈H0, which is exactly the observed scale being fed into the result rather than derived from the second law.

full rationale

The first-law derivation of the generalized Friedmann and acceleration equations from an arbitrary horizon entropy is self-contained, and the identity hB(t)=f_dot_Λ(t)/(2H) follows from the standard continuity equation. The problem is the paper’s headline claim: that the second law fixes fΛ and hB at the observed vacuum-energy scale. That conclusion is forced by inputs. Appendix A declares the integration constant C to be Λ/3, and Eq. (9) already defines O(Λ_obs/3)=O(H0²); Eq. (62) then reports O(C)≈O(H0²) as a thermodynamic output. Independently, the central upper bound fΛ≤H² is algebraically equivalent to ρ≥0 through Eq. (36), so the second law adds no scale information. The transitions to H0² bounds in Eqs. (56)–(57) and (59)–(60) also reverse the inequality H0≤H, assuming the conclusion rather than deriving it. The self-citations to the author’s earlier works [34–36] are method precedents and are not the source of this circularity; the score is high because the main order-of-magnitude claim reduces by construction to the C=Λ/3 input.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the arbitrary horizon entropy S_H, the identification C = Λ/3, and energy conditions (ρ ≥ 0, w ≥ -1) that make the second-law constraints equivalent to standard physics. No new entities are introduced; the driving terms are functions of the entropy deviation.

free parameters (4)
  • C (integration constant) = Λ/3 (observed value)
    Identified as Λ/3 in Sec. IV below Eq. (27); the claimed order consistency O(C) ≈ O(Λ_obs) in Eq. (62) is this input.
  • S_H(H) or S_Δ(H) (arbitrary horizon entropy deviation) = unspecified (except power-law example)
    The model's driving terms depend on the arbitrary entropy; no universal form is fixed.
  • α and Ψ_α (power-law entropy parameters) = 0 < α < 4, Ψ_α > 0, otherwise free
    Introduced in Appendix B to define a specific entropy deviation; not constrained by the paper.
  • w (equation of state parameter) = w > -1, otherwise unspecified
    Assumed > -1; the hB lower bound is equivalent to w ≥ -1.
assumptions (7)
  • domain assumption Flat FLRW universe with Hubble horizon as apparent horizon
    Set at the start of Sec. II.
  • domain assumption First law: -dE_bulk + WdV = T_H dS_H with Kodama-Hayward temperature
    Eq. (17), Sec. IV.
  • domain assumption Standard continuity equation holds, implying hB = fΛdot/(2H)
    Eqs. (6)-(7), Sec. II.
  • domain assumption Second law applied to horizon entropy only: S_Hdot ≥ 0
    Eq. (43), Sec. V, justified by horizon entropy dominance.
  • domain assumption ρ ≥ 0, w > -1, Hdot < 0, Hdot ≥ -2H²
    Used in Sec. V to derive constraints; these make the second-law constraints equivalent to energy conditions.
  • ad hoc to paper C = Λ/3
    Sec. IV below Eq. (27); central to claimed order consistency.
  • domain assumption fΛ(t) ≥ 0
    Assumed for upper bound on hB in Eq. (58); noted as satisfied by power-law entropy.

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Pith. "Pith review of Cosmological scenario based on the first and second laws of thermodynamics: Thermodynamic constraints on a generalized cosmological model." pith.science (2026). https://pith.science/paper/VBA5TWVD

@misc{pith2026241219032,
  author       = {Pith},
  title        = {Pith review of: Cosmological scenario based on the first and second laws of thermodynamics: Thermodynamic constraints on a generalized cosmological model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBA5TWVD}},
  note         = {Machine review of arXiv:2412.19032}
}
abstract

The first and second laws of thermodynamics should lead to a consistent scenario for discussing the cosmological constant problem. In the present study, to establish such a thermodynamic scenario, cosmological equations in a flat Friedmann-Lema\^{i}tre-Robertson-Walker universe were derived from the first law, using an arbitrary entropy $S_{H}$ on a cosmological horizon. Then, the cosmological equations were formulated based on a general formulation that includes two extra driving terms, $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$, which are usually used for, e.g., time-varying $\Lambda (t)$ cosmology and bulk viscous cosmology, respectively. In addition, thermodynamic constraints on the two terms are examined using the second law of thermodynamics, extending a previous analysis [Phys. Rev. D 99, 043523 (2019) (arXiv:1810.11138)]. It is found that a deviation $S_{\Delta}$ of $S_{H}$ from the Bekenstein-Hawking entropy plays important roles in the two terms. The second law should constrain the upper limits of $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$ in our late Universe. The orders of the two terms are likely consistent with the order of the cosmological constant $\Lambda_{\textrm{obs}}$ measured by observations. In particular, when the deviation $S_{\Delta}$ is close to zero, $h_{\textrm{B}}(t)$ and $f_{\Lambda}(t)$ should reduce to zero and a constant value (consistent with the order of $\Lambda_{\textrm{obs}}$), respectively, as if a consistent and viable scenario could be obtained from thermodynamics.

Figures

Figures reproduced from arXiv: 2412.19032 by the authors.

Figure 1
Figure 1. FIG. 1: Thermodynamic constraints on the two driving terms [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

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