The generalized second law as a thermodynamic selection criterion for dynamical dark energy

The generalized second law (GSL) of thermodynamics provides a general physical principle capable of discriminating among the wide variety of generalized horizon entropy proposals and their corresponding modified cosmological models. Working within a general modified Friedmann framework described by an arbitrary function f(H) of the Hubble parameter, and allowing the apparent horizon and the cosmic fluid to evolve out of thermal equilibrium, the authors derive model-independent constraints on the asymptotic scaling of the horizon entropy, SA ∝ Ak. They find that phantom evolution requires k ≥ (3ω − 1)/(2ω), with ω the total fluid equation of state, whereas quintessence imposes the complementary upper bound. Specifically, phantom evolution requires a lower bound on k, whereas quintessence evolution requires an upper bound. The two bounds converge to the unique value k = 2 as the phantom divide is approached, indicating that a smooth crossing of the phantom divide is thermodynamically associated with a quadratic entropy-area scaling, where the effective theory degenerates into a logarithmic gravity framework. In the exact thermal equilibrium limit, the dynamical coupling strictly enforces k ≤ 2, recovering the non-equilibrium quintessence bound.

Applying this criterion to representative generalized entropy models shows that many commonly used proposals are constrained by the generalized second law. Standard Bekenstein-Hawking and Barrow entropies do not satisfy the phantom bound, while Tsallis entropy remains compatible only within a restricted parameter range. Kaniadakis entropy is effectively excluded as a thermodynamically viable description for late-time accelerating expansion unless K = 0 exactly. Logarithmic quantum gravity entropy does not satisfy the phantom bound nor allow the critical scaling k = 2. In contrast, "multiparameter constructions, such as the Luciano-Saridakis entropy, possess sufficient freedom to realize different effective entropy scalings in different cosmological regimes and can naturally accommodate the limiting behavior required near the phantom divide." The five-parameter generalized entropy also shows conditional compatibility.

The analysis is extended to derivative-dependent cosmologies described by f(H, Ḣ), where the horizon entropy acquires an explicit dependence on higher-order cosmic kinematics. Nevertheless, "the generalized second law continues to impose the same qualitative thermodynamic restrictions. In particular, a smooth crossing of the phantom divide requires the effective geometric sector to become momentarily stationary, corresponding to Ḣ = 0 in the f(H) formulation and f˙ → 0 in the generalized f(H, Ḣ) framework. The authors conclude that the generalized second law can be viewed not only as a consistency condition but also as a useful theoretical tool for constraining generalized horizon entropies and the corresponding modified cosmological models, and that the phantom crossing is not an ad hoc feature requiring artificial fine-tuning, but rather a physically constrained transition governed by the generalized second law."

Improvements to AI Systems Based on This Paper:

1. Thermodynamic Consistency Checker for Cosmological Models

• The AI can automatically evaluate any proposed horizon entropy function S A proportional to A k or modified Friedmann model f(H) against the derived GSL bounds (e.g., phantom: k at least (3 omega - 1)/(2 omega), quintessence: k at most 2).

• It can flag models that violate these constraints as thermodynamically unviable, reducing the search space for theoretical physicists.

2. Automated Parameter-Space Constraint Solver

• For multi-parameter entropies (e.g., Luciano-Saridakis, five-parameter generalized entropy), the AI can solve for allowed parameter ranges that satisfy both phantom and quintessence bounds simultaneously, including the critical k = 2 limit near the phantom divide.

• This accelerates model selection by pruning incompatible parameter sets before numerical simulations.

3. Predictive Classifier for Phantom-Crossing Viability

• The AI can classify any given cosmological model (with or without derivatives f(H,)) as either allowing or forbidding a smooth phantom crossing, based on the condition = 0 (or to 0) at the crossing.

• This enables rapid screening of dark energy models for physical consistency without full dynamical analysis.

4. Thermal Equilibrium vs. Non-Equilibrium Regime Detector

• The AI can determine whether a model operates in exact thermal equilibrium (where k at most 2 strictly) or out of equilibrium (where phantom bound relaxes), by checking the coupling between horizon and fluid temperatures.

• This helps in designing simulations that correctly account for entropy production terms.

5. Automated Derivation of Effective Gravity Limits

• Given a specific entropy form, the AI can derive the corresponding effective logarithmic gravity framework near the phantom divide (where k to 2), generating explicit Lagrangians or field equations for further testing.

• This bridges entropy-based thermodynamics with geometric modified gravity theories.

6. Cross-Model Consistency Validator

• The AI can compare predictions from different entropy proposals (e.g., Tsallis vs. Kaniadakis vs. Barrow) against GSL constraints and observational data, producing a ranked list of thermodynamically viable models with confidence intervals.

• This supports meta-analysis of the literature and identifies robust candidates for late-time acceleration.

7. Real-Time Constraint Monitoring in Numerical Cosmology Codes

• The AI can be embedded into Boltzmann solvers or background evolution codes to monitor entropy scaling k during integration, alerting the user when a model crosses into a GSL-forbidden region (e.g., k < (3 omega-1)/(2 omega) during phantom phase).

• This prevents unphysical solutions from propagating into CMB or supernova predictions.

8. Generative Model Proposer

• Using the derived bounds as hard constraints, the AI can propose new entropy functions S(A) that automatically satisfy both phantom and quintessence limits, potentially discovering novel viable parameterizations beyond existing literature.

• This acts as an inverse-design tool for theoretical cosmology.