Eur. Phys. J. C (2025) 85: 607
https://doi.org/10.1140/epjc/s10052-025-14332-5

Regular Article - Theoretical Physics

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

Department of Mechanical Systems Engineering, Kanazawa University, Kakuma-machi, 920-1192, Kanazawa, Ishikawa, Japan

^a komatsu@se.kanazawa-u.ac.jp

Received: 5 March 2025
Accepted: 20 May 2025
Published online: 2 June 2025

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–Lemaître–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, $f_{\mathrm{\Lambda }}(t)$ and $h_{\text{B}}(t)$, which are usually used for, e.g., time-varying $\mathrm{\Lambda }(t)$ 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 (Komatsu in Phys. Rev. D 99:043523, 2019). It is found that a deviation $S_{\mathrm{\Delta }}$ of $S_H$ from the Bekenstein–Hawking entropy plays important roles in the two terms. The second law should constrain the upper limits of $f_{\mathrm{\Lambda }}(t)$ and $h_{\text{B}}(t)$ in our late Universe. The orders of the two terms are likely consistent with the order of the cosmological constant ${\mathrm{\Lambda }}_{\text{obs}}$ measured by observations. In particular, when the deviation $S_{\mathrm{\Delta }}$ is close to zero, $h_{\text{B}}(t)$ and $f_{\mathrm{\Lambda }}(t)$ should reduce to zero and a constant value (consistent with the order of ${\mathrm{\Lambda }}_{\text{obs}}$), respectively, as if a consistent and viable scenario could be obtained from thermodynamics.