Friedmann equations from nonequilibrium thermodynamics of the Universe: A unified formulation for modified gravity
Bibliographic record
Abstract
Inspired by the Wald-Kodama entropy $S=A/(4{G}_{\text{eff}})$ where $A$ is the horizon area and ${G}_{\mathrm{eff}}$ is the effective gravitational coupling strength in modified gravity with field equation ${R}_{\ensuremath{\mu}\ensuremath{\nu}}\ensuremath{-}R{g}_{\ensuremath{\mu}\ensuremath{\nu}}/2=\phantom{\rule{0ex}{0ex}}8\ensuremath{\pi}{G}_{\mathrm{eff}}{T}_{\ensuremath{\mu}\ensuremath{\nu}}^{(\text{eff})}$, we develop a unified and compact formulation in which the Friedmann equations can be derived from thermodynamics of the Universe. The Hawking and Misner-Sharp masses are generalized by replacing Newton's constant $G$ with ${G}_{\mathrm{eff}}$, and the unified first law of equilibrium thermodynamics is supplemented by a nonequilibrium energy dissipation term $\mathcal{E}$ which arises from the revised continuity equation of the perfect-fluid effective matter content and is related to the evolution of ${G}_{\mathrm{eff}}$. By identifying the mass as the total internal energy, the unified first law for the interior and its smooth transit to the apparent horizon yield both Friedmann equations, while the nonequilibrium Clausius relation with entropy production for an isochoric process provides an alternative derivation on the horizon. We also analyze the equilibrium situation ${G}_{\mathrm{eff}}=G=\text{constant}$, provide a viability test of the generalized geometric masses, and discuss the continuity/conservation equation. Finally, the general formulation is applied to the Friedmann-Robertson-Walker cosmology of minimally coupled $f(R)$, generalized Brans-Dicke, scalar-tensor-chameleon, quadratic, $f(R,\mathcal{G})$ generalized Gauss-Bonnet and dynamical Chern-Simons gravity. In these theories we also analyze the $f(R)$-Brans-Dicke equivalence, find that the chameleon effect causes extra energy dissipation and entropy production, geometrically reconstruct the mass ${\ensuremath{\rho}}_{m}V$ for the physical matter content, and show the self-inconsistency of $f(R,\mathcal{G})$ gravity in problems involving ${G}_{\mathrm{eff}}$.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".