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Thermodynamics of sign-switching dark energy models

David Tamayo

Abstract

We perform a comprehensive thermodynamic analysis of three sign-switching dark energy models in a flat FLRW cosmology: graduated dark energy (gDE), sign-switching cosmological constant ($Λ_s$), and smoothed sign-switching cosmological constant ($Λ_t$). We systematically derive key cosmological thermodynamic quantities -- horizon temperature, horizon entropy, internal entropy, total entropy, and their first and second derivatives -- using the Generalised Second Law (GSL) as the fundamental evaluation criterion. We first confirm the compliance of the $Λ$CDM model with the GSL, establishing a baseline for comparison. We find that despite their unconventional negative-to-positive energy density transitions, both $Λ_s$ and $Λ_t$ remain thermodynamically consistent. In contrast, gDE exhibits significant issues: divergences in its equation-of-state lead to infinite horizon temperature and entropy derivatives; and asymptotically, the horizon temperature diverges while entropy approaches zero, causing entropy reduction and violating the GSL. We highlight a key insight: models with divergences in the product of the dark energy equation-of-state parameter and its energy density ($w_x Ω_x$) inevitably produce thermodynamic inconsistencies in standard cosmology. This thermodynamic approach provides a complementary criterion alongside observational constraints for evaluating the physical viability of cosmological models.

Thermodynamics of sign-switching dark energy models

Abstract

We perform a comprehensive thermodynamic analysis of three sign-switching dark energy models in a flat FLRW cosmology: graduated dark energy (gDE), sign-switching cosmological constant (), and smoothed sign-switching cosmological constant (). We systematically derive key cosmological thermodynamic quantities -- horizon temperature, horizon entropy, internal entropy, total entropy, and their first and second derivatives -- using the Generalised Second Law (GSL) as the fundamental evaluation criterion. We first confirm the compliance of the CDM model with the GSL, establishing a baseline for comparison. We find that despite their unconventional negative-to-positive energy density transitions, both and remain thermodynamically consistent. In contrast, gDE exhibits significant issues: divergences in its equation-of-state lead to infinite horizon temperature and entropy derivatives; and asymptotically, the horizon temperature diverges while entropy approaches zero, causing entropy reduction and violating the GSL. We highlight a key insight: models with divergences in the product of the dark energy equation-of-state parameter and its energy density () inevitably produce thermodynamic inconsistencies in standard cosmology. This thermodynamic approach provides a complementary criterion alongside observational constraints for evaluating the physical viability of cosmological models.

Paper Structure

This paper contains 16 sections, 44 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Sign-switching dark energy models
  3. Graduated dark energy
  4. Sign-switching $\Lambda$
  5. Sign-switch transition with $\tanh$
  6. Cosmological thermodynamics
  7. Apparent horizon entropy and temperature
  8. Entropy of the fluid within the horizon
  9. Total cosmological entropy
  10. $\Lambda$CDM thermodynamics
  11. Sign-switching dark energy thermodynamics
  12. Temperature, entropy, and entropy derivatives
  13. Entropy derivatives of gDE, $\Lambda_s$ and $\Lambda_t$
  14. Divergences and asymptotic values of thermodynamic magnitudes
  15. Thermodynamic evaluation
  16. ...and 1 more sections

Figures (6)

  • Figure 1: Normalized dark energy density, EoS parameter, and Hubble function for sign-switching dark energy models: gDE (blue), $\Lambda_s$ (red), $\Lambda_t$ (green); and $\Lambda$CDM (dashed black). For the gDE model, the parameters are $\gamma = -0.013$ and $\lambda = -20$, resulting in a sign-switch transition redshift of $z_* = 2.39$, which is also used for both $\Lambda_s$ and $\Lambda_t$. For the $\Lambda_t$ model, the smoothing parameter $\eta=50$ is arbitrary and intended solely to illustrate a rapid but not strictly instantaneous transition.
  • Figure 1: Set of key cosmological thermodynamic equations.
  • Figure 2: Evolution of the thermodynamic quantities in the $\Lambda$CDM model. The top left panel shows the horizon temperature, $T_h$ while in the top right panel the horizon entropy, $T_h$. The bottom left panel displays the first derivative of the total entropy, $S'_{\text{tot}}$ (black), with contributions from the horizon entropy, $S'_h$ (blue), and the internal entropy, $S'_{\text{in}}$ (red). The bottom right panel shows the second derivative of entropy, $S"_{\text{tot}}$ (black), along with $S"_h$ (blue) and $S"_{\text{in}}$ (red).
  • Figure 3: Evolution of the horizon temperature, horizon entropy and its first and second derivatives for gDE (blue), $\Lambda_s$ (red), and $\Lambda_t$ (green), with $\Lambda$CDM (dashed black) shown as a reference.
  • Figure 4: Evolution of the first and second derivatives of the internal entropy (top) and the total entropy (bottom) for gDE (blue), $\Lambda_s$ (red), and $\Lambda_t$ (green), with $\Lambda$CDM (dashed black) as a reference.
  • ...and 1 more figures